Termodinamica e Introducao a Termoestatistica - Herbert B Callen

No text found THERMODYNAMICS AND AN INTRODUCTION TO THERMOSTATISTICS No text found THERMODYNAMICS AND AN INTRODUCTION TO THERMOSTATISTICS SECOND EDITION HERBERT B CALLEN University of Pennsylvania JOHN WILEY SONS New York Chichester Brisbane Toronto Singapore Copynght 1985 by John Wiley Sons Inc All nght reserved Published 1multaneouly in Canada Reproduction or translat10n of any part of this work beyond that permitted by Sections 107 and 108 of the 1976 Umtcd States Copynght Act without the perm1sMon of the copyright owner 1s unlawful Requests for perm1s10n or further information should be addressed to the Perm1ss1on Department John Wiley Sons I ibrary of Congress Cataloging in Publication Data Callen Herbert B Thermodynamics and an Introduction to Thermostatlstics Rev ed of Thermodynamics 1960 B1hhography p 485 Includes mdex 1 Thermodynamics 2 Stalltical Mechanics Callen Herbert B Thermodynanucs II Title III Title Thermostatistic QC31 l C25 1985 536 7 856387 Printed in the Republic of Singapore IO 9 8 To Sara and to Jill Jed Zachary and Jessica THERMODYNAMICS AND AN INTRODUCTION TO THERMOSTATISTICS SECOND EDITION HERBERT B CALLEN PREFACE Twentyfive years after writing the first edition of Thermodynamics I am gratified that the book is now the thermodynamic reference most fre quently cited in physics research literature and that the postulational formulation which it introduced is now widely accepted Nevertheless several considerations prompt this new edition and extension First thermodynamics advanced dramatically in the 60s and 70s pri marily in the area of critical phenomena Although those advances are largely beyond the scope of this book I have attempted to at least describe the nature of the problem and to introduce the critical exponents and scaling functions that characterize the nonanalytic behavior of ther modynamic functions at a secondorder phase transition This account is descriptive and simple It replaces the relatively complicated theory of secondorder transitions that in the view of many students was the most difficult section of the first edition Second I have attempted to improve the pedagogical attributes of the book for use in courses from the junior undergraduate to the first year graduate level for physicists engineering scientists and chemists This purpose has been aided by a large number of helpful suggestions from students and instructors Many explanations are simplified and numerous examples are solved explicitly The number of problems has been ex panded and partial or complete answers are given for many Third an introduction to the principles of statistical mechanics has been added Here the spirit of the first edition has been maintained the emphasis is on the underlying simplicity of principles and on the central train of logic rather than on a multiplicity of applications For this purpose and to make the text accessible to advanced undergraduates I have avoided explicit noncommutivity problems in quantum mechanics All that is required is familiarity with the fact that quantum mechanics predicts discrete energy levels in finite systems However the formulation is designed so that the more advanced student will properly interpret the theory in the noncommutative case viu Preface Fourth I have long been puzzled by certain conceptual problems lying at the foundations of thermodynamics and this has led me to an interpre tation of the meaning of thermodynamics In the final chapteran interpretive postlude to the main body of the textI develop the thesis that thermostatistics has its roots in the symmetries of the fundamental laws of physics rather than in the quantitative content of those laws The discussion is qualitative and descriptive seeking to establish an intuitive framework and to encourage the student to see science as a coherent structure in which thermodynamics has a natural and fundamental role Although both statistical mechanics and thermodynamics are included in this new edition I have attempted neither to separate them completely nor to meld them into the undifferentiated form now popular under the rubric of thermal physics I believe that each of these extreme options is misdirected To divorce thermodynamics completely from its statistical II1chanical base is to rob thermodynamics of its fundamental physical origins Without an insight into statistical mechanics a scientist remains rooted in the macroscopic empiricism of the nineteenth century cut off from contemporary developments and from an integrated view of science Conversely the amalgamation of thermodynamics and statistical me chanics into an undifferentiated thermal physics tends to eclipse ther modynamics The fundamentality and profundity of statistical mechanics are treacherously seductive thermal physics courses almost perforce give short shrift to macroscopic operational principles Furthermore the amalgamation of thermodynamics and statistical mechanics runs counter to the principle of theoretical economy the principle that predictions should be drawn from the most general and least detailed assumptions possible Models endemic to statistical mechanics should be eschewed whenever the general methods of macroscopic thermodynamics are suffi cient Such a habit of mind is hardly encouraged by an organization of the subjects in which thermodynamics is little more than a subordinate clause The balancing of the two distinct components of the thermal sciences is carried out in this book by introducing the subject at the macroscopic level by formulating thermodynamics so that its macroscopic postulates are precisely and clearly the theorems of statistical mechanics and by frequent explanatory allusions to the interrelationships of the two compo nents Nevertheless at the option of the instructor the chapters on statistical mechanics can be interleaved with those on thermodynamics in a sequence to be described But even in that integrated option the basic macroscopic structure of thermodynamics is established before statistical reasoning is introduced Such a separation and sequencing of the subjects The Amcncan Phy5ical Society Committee on Applications of Phys1c5 reported Buletm of the APS Vol 22 IO 1233 1971 that a 5urvey of mdustnal research leaders designated thermody namics above all other subJect5 as requiring increased emphasis m the undergraduate curriculum That emphaM ubsequently ha deaeased Preface ix preserves and emphasizes the hierarchical structure of science organizing physics into coherent units with clear and easily remembered interrela tionships Similarly classical mechanics is best understood as a self contained postulatory structure only later to be validated as a limiting case of quantum mechanics Two primary curricular options are listed in the menu following In one option the chapters are followed in sequence Column A alone or followed by all or part of column B In the integrated option the menu is followed from top to bottom Chapter 15 is a short and elementary statistical interpretation of entropy it can be inserted immediately after Chapter 1 Chapter 4 or Chapter 7 The chapters listed below the first dotted line are freely flexible with respect to sequence or to inclusion or omission To balance the concrete and particular against more esoteric sections instructors may choose to insert parts of Chapter 13 Properties of Materials at various stages or to insert the Postlude Chapter 21 Symmetry and Conceptual Foundations at any point in the course The minimal course for junior year undergraduates would involve the first seven chapters with Chapter 15 and 16 optionally included as time permits Philadelphia Pennsylvania Herbert B Callen Preface to the Fourth Printing In the issuance of this fourth printing of the second edition the publisher has graciously given me the opportunity to correct various misprints and minor errors I am painfully aware that no error numerical or textual is truly minor to the student reader Accordingly I am deeply grateful both to the numerous read ers who have called errors to my attention and to the charitable forbearance of the publisher in permitting their correction in this printing November 1987 Herbert Callen X Preface 1 Postulates 15 2 Conditions of Equilibrium 3 Formal Relations and Sample Sys terns 4 Reversible Processes Engines 15 Statistical Mechanics in Entropy Representation 5 Legendre Transformations 6 Extremum Principles in Legendre Representation 7 Maxwell Relations 15 16 Canonical Formalism 17 Generalized Canonical Formula tion 8 Stability 9 FirstOrder Phase Transitions 10 Critical Phenomena 18 Quantum Fluids 11 Nernst 19 Fluctuations 12 Summary of Principles 20 Variational Properties and Mean 13 Properties of Materials Field Theory 14 Irreversible Thermodynamics 21 Postlude Symmetry and the Conceptual Foundations of Thermodynamics CONTENTS PART I GENERAL PRINCIPLES OF CLASSICAL THERMODYNAMICS 1 Introduction The Nature of Thermodynamics and the Basis of Thermostatistics 2 1 THE PROBLEM AND THE POSTULATES 5 11 The Temporal Nature of Macroscopic Measurements 5 12 The Spatial Nature of Macroscopic Measurements 6 13 The Composition of Thermodynamic Systems 9 14 The Internal Energy 11 15 Thermodynamic Equilibrium 13 16 Walls and Constraints 15 17 Measurability of the Energy 16 18 Quantitative Definition of HeatUnits 18 19 The Basic Problem of Thermodynamics 25 110 The Entropy Maximum Postulates 27 2 THE CONDITIONS OF EQUILIBRIUM 35 21 Intensive Parameters 35 22 Equations of State 37 23 Entropic Intensive Parameters 40 24 Thermal Equilibrium Temperature 43 25 Agreement with Intuitive Concept of Temperature 45 26 Temperature Units 46 27 Mechanical Equilibrium 49 28 Equilibrium with Respect to Matter Flow 54 29 Chemical Equilibrium 56 xi xii Contents 3 SOME FORMAL REIA TIONSHIPS AND SAMPLE SYSTEMS 31 The Euler Equation 32 The Gibbs Duhem Relation 33 Summary of Formal Structure 34 The Simple Ideal Gas and Multicomponent Simple Ideal Gases 35 The Ideal van der Waals Fluid 36 Electromagnetic Radiation 3 7 The Rubber Band 38 Unconstrainable Variables Magnetic Systems 39 Molar Heat Capacity and Other Derivatives 4 REVERSIBLE PROCESSES AND THE MAXIMUM WORK THEOREM 41 Possible and Impossible Processes 42 QuasiStatic and Reversible Processes 43 Relaxation Times and Irreversibility 44 Heat Flow Coupled Systems and Reversal of Processes 45 The Maximum Work Theorem 46 Coefficients of Engine Refrigerator and Heat Pump Performance 4 7 The Carnot Cycle 48 Measurability of the Temperature and of the Entropy 49 Other Criteria of Engine Performance Power Output and Endo reversible Engines 410 Other Cyclic Processes 5 ALTERNATIVE FORMULATIONS AND LEGENDRE TRANSFORMATIONS 51 The Energy Minimum Principle 52 Legendre Transformations 53 Thermodynamic Potentials 54 Generalized Massieu Functions 6 THE EXTREMUM PRINCIPLE IN THE 59 59 60 63 66 74 78 80 81 84 91 91 95 99 101 103 113 118 123 125 128 131 131 137 146 151 LEGENDRE TRANSFORMED REPRESENTATIONS 153 61 The Minimum Principles for the Potentials 153 62 The Helmholtz Potential 157 63 The Enthalpy The JouleThomson or Throttling Process 160 64 The Gibbs Potential Chemical Reactions 167 65 Other Potentials 172 66 Compilations of Empirical Data The Enthalpy of Formation 173 67 The Maximum Principles for the Massieu Functions 179 7 MAXWELL RELATIONS 7 1 The Maxwell Relations 7 2 A Thermodynamic Mnemonic Diagram 73 A Procedure for the Reduction of Derivatives in SingleComponent Systems 74 Some Simple Applications 7 5 Generalizations Magnetic Systems 8 STABILITY OF THERMODYNAMIC SYSTEMS 81 Intrinsic Stability of Thermodynamic Systems 82 Stability Conditions for Thermodynamics Potentials 83 Physical Consequences of Stability 84 Le Chateliers Principle The Qualitative Effect of Fluctuations 85 The Le ChatelierBraun Principle 9 FIRSTORDER PHASE TRANSITIONS Contents xiii 181 181 183 186 190 199 203 203 207 209 210 212 215 91 FirstOrder Phase Transitions in SingleComponent Systems 215 92 The Discontinuity in the EntropyLatent Heat 222 93 The Slope of Coexistence Curves the Clapeyron Equation 228 94 Unstable Isotherms and FirstOrder Phase Transitions 233 95 General Attributes of FirstOrder Phase Transitions 243 96 FirstOrder Phase Transitions in Multicomponent SystemsGibbs Phase Rule 245 97 Phase Diagrams for Binary Systems 248 10 CRITICAL PHENOMENA 255 101 Thermodynamics in the Neighborhood of the Critical Point 255 102 Divergence and Stability 261 103 Order Parameters and Critical Exponents 263 104 Classical Theory in the Critical Region Landau Theory 265 105 Roots of the Critical Point Problem 270 106 Scaling and Universality 11 THE NERNST POSTULATE 111 Nernsts Postulate and the Principle of Thomsen and Bertholot 272 277 277 112 Heat Capacities and Other Derivatives at Low Temperatures 280 113 The Unattainability of Zero Temperature 281 12 SUMMARY OF PRINCIPLES FOR GENERAL SYSTEMS 121 General Systems 122 The Postulates 283 283 283 xw Contents 123 The Intensive Parameters 124 Legendre Transforms 125 Maxwell Relations 126 Stability and Phase Transitions 127 Critical Phenomena 128 Properties at Zero Temperature 13 PROPERTIES OF MATERIALS 131 The General Ideal Gas 132 Chemical Reactions in Ideal Gases 133 Small Deviations from Ideality The Virial Expansion 134 The Law of Corresponding States for Gases 135 Dilme Solutions Osmotic Pressure and Vapor Pressure 136 Solid Systems 14 IRREVERSIBLE THERMODYNAMICS 141 General Remarks 142 Affinities and Fluxes 143 PurelyResistive and Linear Systems 144 The Theoretical Basis of the Onsager Reciprocity 145 Thermoelectric Effects 146 The Conductivities 147 The Seebeck Effect and the Thermoelectric Power 148 The Peltier Effect 149 The Thomsen Effect PART II STATISTICAL MECHANICS 15 STATISTICAL MECHANICS IN THE ENTROPY REPRESENTATION 284 285 285 286 287 287 289 289 292 297 299 302 305 307 307 308 312 314 316 319 320 323 324 THE MICROCANONICAL FORMALISM 329 151 Physical Significance of the Entropy for Closed Systems 329 152 The Einstein Model of a Crystalline Solid 333 153 The TwoState System 337 154 A Polymer Model The Rubber Band Revisited 339 155 Counting Techniques and their Circumvention High Dimensionality 343 16 THE CANONICAL FORMALISM STATISTICAL MECHANICS IN HELMHOLTZ REPRESENTATION 349 161 The Probability Distribution 349 162 Additive Energies and Factorizability of the Partition Sum 353 163 Internal Modes in a Gas 164 Probabilities in Factorizable Systems 165 Statistical Mechanics of Small Systems Ensembles 166 Density of States and DensityofOrbital States 167 The Debye Model of Nonmetallic Crystals 168 Electromagnetic Radiation 169 The Classical Density of States 1610 The Classical Ideal Gas 1611 High Temperature PropertiesThe Equipartition Theorem 17 ENTROPY AND DISORDER GENERALIZED CANONICAL FORMULATIONS 171 Entropy as a Measure of Disorder 172 Distributions of Maximal Disorder 173 The Grand Canonical Formalism 18 QUANTUM FLUIDS 181 Quantum Particles A Fermion PreGas Model 182 The Ideal Fermi Fluid 183 The Classical Limit and the Quantum Criteria 184 The Strong Quantum Regime Electrons in a Metal 185 The Ideal Bose Fluid 186 NonConserved Ideal Bose Fluids Electromagnetic Radiation Revisited 18 7 Bose Condensation 19 FLUCTUATIONS 191 The Probability Distribution of Fluctuations 192 Moments and The Energy Fluctuations 193 General Moments and Correlation Moments 20 VARIATIONAL PROPERTIES PERTURBATION EXPANSIONS AND MEAN FIELD THEORY 201 The Bogoliubov Variational Theorem 202 Mean Field Theory 203 Mean Field Theory in Generalized Representation the Binary Alloy PART III FOUNDATIONS 21 POSTLUDE SYMMETRY AND THE CONCEPTUAL Contents xv 355 358 360 362 364 368 370 372 375 379 379 382 385 393 393 399 402 405 410 412 413 423 423 424 426 433 433 440 449 FOUNDATIONS OF THERMOSTATISTICS 455 211 Statistics 455 212 Symmetry 213 Noethers Theorem 214 Energy Momentum and Angular Momentum the Generalized First Law of Thermodynamics 215 Broken Symmetry and Goldstones Theorem 216 Other Broken Symmetry Coordinates Electric and Magnetic Moments 217 Mole Numbers and Gauge Symmetry 218 Time Reversal the Equal Probability of M1crostates and the Entropy Principle 219 Symmetry and Completeness APPENDIX A SOME RELATIONS INVOLVING PARTIAL DERIVATIVES AI Partial Denvatives A2 Taylors Expansion A3 Differentials A4 Composite Functions A5 Implicit Functions APPENDIX B MAGNETIC SYSTEMS GENERAL REFERENCES INDEX 458 460 461 462 465 466 467 469 473 473 474 475 475 476 479 485 487 PART GENERAL PRINCIPLES OF CLASSICAL THERMODYNAMICS 2 General Prmcpes of Classical Thermodynamics INTRODUCTION The Nature of Thermodynamics and the Basis of ThermoStatistics Whether we are physicists chemists biologists or engineers our primary interface with nature is through the properties of macroscopic matter Those properties are subject to universal regularities and to stringent limitations Subtle relationships exist among apparently unconnected properties The existence of such an underlying order has far reaching implications Physicists and chemists familiar with that order need not confront each new material as a virgin puzzle Engineers are able to anticipate limita tions to device designs predicated on creatively imagined but yet undis covered materials with the requisite properties And the specific form of the underlying order provides incisive clues to the structure of fundamen tal physical theory Certain primal concepts of thermodynamics are intuitively familiar A metallic block released from rest near the rim of a smoothly polished metallic bowl oscillates within the bowl approximately conserving the sum of potential and kinetic energies But the block eventually comes to rest at the bottom of the bowl Although the mechanical energy appears to have vanished an observable effect is wrought upon the material of the bowl and block they are very slightly but perceptibly warmer Even before studying thermodynamics we are qualitatively aware that the mechanical energy has merely been converted to another form that the fundamental principle of energy conservation is preserved and that the physiological sensation of warmth is associated with the thermodynamic concept of temperature Vague and undefined as these observations may be they nevertheless reveal a notable dissimilarity between thermodynamics and the other branches of classical science Two prototypes of the classical scientific paradigm are mechanics and electromagnetic theory The former ad dresses itself to the dynamics of particles acted upon by forces the latter to the dynamics of the fields that mediate those forces In each of these cases a new law is formulatedfor mechanics it is Newtons Law or Lagrange or Hamiltons more sophisticated variants for electromag netism it is the Maxwell equations In either case it remains only to explicate the consequences of the law Thermodynamics is quite different It neither claims a unique domain of systems over which it asserts primacy nor does it introduce a new fundamental law analogous to Newtons or Maxwells equations In contrast to the specificity of mechanics and electromagnetism the hall mark of thermodynamics is generality Generality first in the sense that thermodynamics applies to all types of systems in macroscopic aggrega Introduuion 3 tion and second in the sense that thermodynamics does not predict specific numerical values for observable quantities Instead thermody namics sets limits inequalities on permissible physical processes and it establishes relationships among apparently unrelated properties The contrast between thermodynamics and its counterpart sciences raises fundamental questions which we shall address directly only in the final chapter There we shall see that whereas thermodynamics is not based on a new and particular law of nature it instead reflects a commonality or universal feature of all laws In brief thermodynamics is the study of the restrictions on the possible properties of matter that follow from the symmetry properties of the fundamental laws of physics The connection between the symmetry of fundamental laws and the macroscopic properties of matter is not trivially evident and we do not attempt to derive the latter from the former Instead we follow the postulatory formulation of thermodynamics developed in the first edition of this text returning to an interpretive discussion of symmetry origins in Chapter 21 But even the preliminary assertion of this basis of thermody namics may help to prepare the reader for the somewhat uncommon form of thermodynamic theory Thermodynamics inherits its universality it nonmetric nature and its emphasis on relationships from its symmetry parentage No text found in the image 1 THE PROBLEM AND THE POSTULATES 11 THE TEMPORAL NATURE OF MACROSCOPIC MEASUREMENTS Perhaps the most striking feature of macroscopic matter is the incredi ble simplicity with which it can be characterized We go to a pharmacy and request one liter of ethyl alcohol and that meager specificatin is pragmatically sufficient Yet from the atomistic point of view we have specified remarkably little A complete mathematical characterization of the system would entail the specification of coordinates and momenta for each molecule in the sample plus sundry additional variables descriptive of the internal state of each moleculealtogether at least 1023 numbers to describe the liter of alcohol A computer printing one coordinate each microsecond would require 10 billion yearsthe age of the universeto list the atomic coordinates Somehow among the 1023 atomic coordinates or linear combinations of them all but a few are macroscopically irrele vant The pertinent few emerge as macroscopic coordinates or thermody namic coordinates Like all sciences thermodynamics is a description of the results to be obtained in particular types of measurements The character of the contemplated measurements dictates the appropriate descriptive variables these variables in turn ordain the scope and structure of thermodynamic theory The key to the simplicity of macroscopic description and the criterion for the choice of thermodynamic coordinates lies in two attributes of macroscopic measurement Macroscopic measurements are extremely slow on the atomic scale of time and they are extremely coarse on the atomic scale of distance While a macroscopic measurement is being made the atoms of a system go through extremely rapid and complex motions To measure the length of a bar of metal we might choose to calibrate it in terms of the wavelength of yellow light devising some arrangement whereby reflection 6 The Problem and the Postulates from the end of the bar produces interference fringes These fringes are then to be photographed and counted The duration of the measurement is determined by the shutter speed of the cameratypically on the order of one hundredth of a second But the characteristic period of vibration of the atoms at the end of the bar is on the order of 10 15 seconds A macroscopic observation cannot respond to those myriads of atomic coordinates which vary in time with typical atomic periods Only those few particular combinations of atomic coordinates that are essentially time independent are macroscopically observable The word essentially is an important qualification In fact we are able to observe macroscopic processes that are almost but not quite time inde pendent With modest difficulty we might observe processes with time scales on the order of 10 7 s or less Such observable processes are still enormously slow relative to the atomic scale of 10 15 s It is rational then to first consider the limiting case and to erect a theory of timeindepen dent phenomena Such a theory is thermodynamics By definition suggested by the nature of macroscopic observations ther modynamics describes only static states of macroscopic systems Of all the 10 23 atomic coordmates or combinations thereof only a few are time independent Quantities subject to conservation principles are the most obvious candidates as timeindependent thermodynamic coordinates the energy each component of the total momentum and each component of the total angular momentum of the system But there are other timeindependent thermodynamic coordinates which we shall enumerate after exploring the spatial nature of macroscopic measurement 12 THE SPATIAL NATURE OF MACROSCOPIC MEASUREMENTS Macroscopic measurements are not only extremely slow on the atomic scale of time but they are correspondingly coarse on the atomic scale of distance We probe our system always with blunt instruments Thus an optical observation has a resolving power defined by the wavelength of light which is on the order of 1000 interatomic distances The smallest resolvable volume contains approximately 109 atoms Macroscopic ob servations sense only coarse spatial averages of atomic coordinates The two types of averaging implicit in macroscopic observations to gether effect the enormous reduction in the number of pertinent variables from the initial 1023 atomic coordinates to the remarkably small number of thermodynamic coordinates The manner of reduction can be il lustrated schematically by considering a simple model system as shown in Fig 11 The model system consists not of 1023 atoms but of only 9 These atoms are spaced along a onedimensional line are constrained to The Spatial Nature of Macroscopic Measurements 7 FIGURE 11 Three normal modes of oscillation in a nineatom model system The wave lengths of the three modes are four eight and sixteen interatomic distances The dotted curves are a transverse representation of the longitudinal displacements move only along that line and interact by linear forces as if connected by springs The motions of the individual atoms are strongly coupled so the atoms tend to move in organized patterns called normal modes Three such normal modes of motion are indicated schematically in Fig 11 The arrows indicate the displacements of the atoms at a particular moment the atoms oscillate back and forth and half a cycle later all the arrows would be reversed Rather than describe the atomic state of the system by specifying the position of each atom it is more convenient and mathematically equiv alent to specify the instantaneous amplitude of each normal mode These amplitudes are called normal coordinates and the number of normal coordinates is exactly equal to the number of atomic coordinates In a macroscopic system composed of only nine atoms there is no precise distinction between macroscopic and atomic observations For the purpose of illustration however we think of a macroscopic observation as a kind of blurred observation with low resolving power the spatial coarseness of macroscopic measurements is qualitatively analo gous to visual observation of the system through spectacles that are somewhat out of focus Under such observation the fine structure of the first two modes in Fig 11 is unresolvable and these modes are rendered unobservable and macroscopically irrelevant The third mode however corresponds to a relatively homogeneous net expansion or contraction of the whole system Unlike the first two modes it is easily observable through blurring spectacles The amplitude of this mode describes the length or volume in three dimensions of the system The length or 8 The Problem and the Postulates volume remains as a thermodynamic vanable undestroyed by the spatial averaging because of its spatially homogeneous long wavelength structure The time averaging associated with macroscopic measurements aug ments these considerations Each of the normal modes of the system has a characteristic trequency the frequency being smaller for modes of longer wavelength The frequency of the third normal mode in Fig 11 is the lowest of those shown and if we were to consider systems with very large numbers of atoms the frequency of the longest wavelength mode would approach zero for reasons to be explored more fully in Chapter 21 Thus all the short wavelength modes are lost in the time averaging but the long wavelength mode corresponding to the volume is so slow that it survives the time averaging as well as the spatial averaging This simple example illustrates a very general result Of the enormous number of atomic coordinates a very few with unique symmetry proper ties survive the statistical averaging associated with a transition to a macroscopic description Certain of these surviving coordinates are me chanical in nature they are volume parameters descriptive of the shape components of elastic strain and the like Other surviving coordinates are electrical in nature they are electric dipole moments magnetic dipole moments various multipole moments and the like The study of mechanics including elasticity is the study of one set of surviving coordinates The subject of electricity including electrostatics magnetostatics and ferromag netism is the study of another set of surviving coordinates Thermodynamics in contrast is concerned with the macroscopic conse quences of the myriads of atomic coordinates that by virtue of the coarseness of macroscopic observations do not appear explicitly in a macroscopic description of a system Among the many consequences of the hidden atomic modes of motion the most evident is the ability of these modes to act as a repository for energy Energy transferred via a mechanical mode ie one associated with a mechanical macroscopic coordinate is called me chanical work Energy transferred via an electrical mode is called electri cal work Mechanical work is typified by the term P dV P is pressure Vis volume and electrical work is typified by the term Eedg Ee is electric field g is electric dipole moment These energy terms and various other mechanical and electrical work terms are treated fully in the standard mechanics and electricity references But it is equally possible to trans er energy via the hidden atomic modes of motion as well as via those that happen to be macroscopically observable An energy transfer via he hidden atomic modes is called heat Of course this descriptive characterization of heat is not a sufficient basis for the formal development of thermody namics and we shall soon formulate an appropriate operational defini tion With this contextual perspective we proceed to certain definitions and conventions needed for the theoretical development The Composttwn of Thermodinam1L Sstems 9 13 THE COMPOSITION OF THERMODYNAMIC SYSTEMS Thermodynamics is a subject of great generality applicable to systems of elaborate structure with all manner of complex mechamcal electrical and thermal properties We wish to focus our chief attention on the thermal properties Therefore it is convenient to idealize and simplify the mechanical and electrical properties of the systems that we shall study initially Similarly in mechanics we consider uncharged and unpolarized systems whereas in electricity we consider systems with no elastic com pressibility or other mechanical attributes The generality of either subject is not essentially reduced by this idealization and after the separate content of each subject has been studied it is a simple matter to combine the theories to treat systems of simultaneously complicated electrical and mechanical properties Similarly in our study of thermodynamics we idealize our systems so that their mechanical and electrical properties are almost trivially simple When the essential content of thermodynamics has thus been developed it again is a simple matter to extend the analysis to systems with relatively complex mechanical and electrical structure The essential point to be stressed is that the restrictions on the types of systems considered in the following several chapters are not basic limita tions on the generality of thermodynamic theory but are adopted merely for simplicity of exposition We temporarily restrict our attention to simple systems defined as systems that are macroscopically homogeneous isotropic and uncharged that are large enough so that surface effects can be neglected and that are not acted on by electric magnetic or gravitational fields For such a simple system there are no macroscopic electric coordinates whatsoever The system is uncharged and has neither electric nor magnetic dipole quadrupole or higherorder moments All elastic shear compo nents and other such mechanical parameters are zero The volume V does remain as a relevant mechanical parameter Furthermore a simple system has a definite chemical composition which must be described by an appropriate set of parameters One reasonable set of composition parame ters is the numbers of molecules in each of the chemically pure compo nents of which the system is a mixture Alternatively to obtain numbers of more convenient size we adopt the mole numbers defined as the actual number of each type of molecule divided by Avogadros number NA 602217 X 10 23 This definition of the mole number refers explicitly to the number of molecules and it therefore lies outside the boundary of purely maro scopic physics An equivalent definition which avoids the reference to molecules simply designates 12 grams as the molar mass of the isotope 12C The molar masses of other isotopes are then defined to stand in the same ratio as the conventional atomic masses a partial list of which is given in Table LL JO The Problem and the Postulates TABLE 11 Atomic Masses g of Some Naturally Occurring Elements Mixtures of Isotopest H 10080 F 189984 Li 6941 Na 229898 C 12011 Al 269815 N 140067 s 3206 0 159994 Cl 35453 a As adopted by the International Uruon of Pure and Applied Chemistry 1969 If a system is a mixture of r chemical components the r ratios NkL 1 N k 1 2 r are called the mole fractions The sum of all r mole fractions is unity The quantity V f 1 N is called the molar volume The macroscopic parameters V N1 N2 N have a common property that will prove to be quite significant Suppose that we are given two identical systems and that we now regard these two systems taken together as a single system The value of the volume for the composite system is then just twice the value of the volume for a single subsystem Similarly each of the mole numbers of the composite system is twice that for a single subsystem Parameters that have values in a composite system equal to the sum of the values in each of the subsystems are called extenswe parameters Extensive parameters play a key role throughout thermody namic theory PROBLEMS 131 One tenth of a kilogram of NaCl and 015 kg of sugar C 12H 220 11 are dissolved in 050 kg of pure water The volume of the resultant thermodynamic system is 055 X 10 3 m3 What are the mole numbers of the three components of the system What are the mole fractions What 1s the molar volume of the system It is sufficient to carry the calculations only to two significant figures Answer Mole fraction of NaCl 0057 molar volume 18 x 10 6 m3mole 132 Naturally occurring boron has an atomic mass of 10811 g It is a mixture of the isotopes 10B with an atomic mass of 100129 g and 11 B with an atomic mass of 110093 g What is the mole fraction of 10B in the mixture 133 Twenty cubic centimeters each of ethyl alcohol C2H 50H density 079 gcm 3 methyl alcohol CH 30H density 081 gcm 3 and water H 70 The Internal Energy 11 density 1 gcm 3 are mixed together What are the mole numbers and mole fractions of the three components of the system Answer mole fractions 017 026 057 134 A 001 kg sample is composed of 50 molecular percent H2 30 molecular percent HD hydrogen deuteride and 20 molecular percent D2 What additional mass of D2 must be added if the mole fraction of D2 in the final mixture 1s to be 03 135 A solution of sugar C12H 220u in water is 20 sugar by weight What is the mole fraction of sugar in the solution 136 An aqueous solution of an unidentified solute has a total mass of 01029 kg The mole fraction of the solute is 01 The solution is diluted with 0036 kg of water after which the mole fraction of the solute is 007 What would be a reasonable guess as to the chemical identity of the solute 137 One tenth of a kg of an aqueous solution of HCI is poured into 02 kg of an aqueous solution of NaOH The mole fraction of the HCl solution was 01 whereas that of the NaOH solution was 025 What are the mole fractions of each of the components in the solution after the chemical reaction has come to completion Answer XH20 NH20N 084 14 THE INTERNAL ENERGY The development of the principle of conservation of energy has been one of the most significant achievements in the evolution of physics The present form of the principle was not discovered in one magnificent stroke of insight but was slowly and laboriously developed over two and a half centuries The first recognition of a conservation principle by Leibniz in 1693 referred orly to the sum of the kinetic energy t mv2 and the potential energy mgh of a simple mechanical mass point in the terrestrial gravitational field As additional types of systems were considered the established form of the conservation principle repeatedly failed but in each case it was found possible to revive it by the addition of a new mathematical terma new kind of energy Thus consideration of charged systems necessitated the addition of the Coulomb interaction energy Q1Qifr and eventually of the energy of the electromagnetic field In 1905 Einstein extended the principle to the relativistic region adding such terms as the relativistic restmass energy In the 1930s Enrico Fermi postulated the exidPnP nf n I 2 The Problem and the Postulates purpose of retaining the energy conservation principle in nuclear reac tions The principle of energy conservation is now seen as a reflection of the presumed fact that the fundamental laws of physics are the same today as they were eons ago or as they will be in the remote future the laws of physics are unaltered by a shift in the scale of time t t constant Of this basis for energy conservation we shall have more to say in Chapter 21 Now we simply note that the energy conservation principle is one of the most fundamental general and significant principles of physical theory Viewing a macroscopic system as an agglomerate of an enormous number of electrons and nuclei interacting with complex but definite forces to which the energy conservation principle applies we conclude that macroscopic systems have definite and precise energies subject to a definite conservation principle That is we now accept the existence of a welldefined energy of a thermodynamic system as a macroscopic mani festation of a conservation law highly developed tested to an extreme precision and apparently of complete generality at the atomic level The foregoing justification of the existence of a thermodynamic energy function is quite different from the historical thermodynamic method Because thermodynamics was developed largely before the atomic hy pothesis was accepted the existence of a conservative macroscopic energy function had to be demonstrated by purely macroscopic means A signifi cant step in that direction was taken by Count Rumford in 1798 as he observed certain thermal effects associated with the boring of brass cannons Sir Humphry Davy Sadi Carnot Robert Mayer and finally between 1840 and 1850 James Joule carried Rumfords initial efforts to their logical fruition The history of the concept of heat as a form of energy transfer is unsurpassed as a case study in the tortuous development of scientific theory as an illustration of the almost insuperable inertia presented by accepted physical doctrine and as a superb tale of human ingenuity applied to a subtle and abstract problem The interested reader is referred to The Early Development of the Concepts of Temperature and Heat by D Roller Harvard University Press 1950 or to any standard work on the history of physics Although we shall not have recourse explicitly to the experiments of Rumford and Joule in order to justify our postulate of the existence of an energy function we make reference to them in Section 17 in our discus sion of the measurability of the thermodynamic energy Only differences of energy rather than absolute values of the energy have physical significance either at the atomic level or in macroscopic systems It is conventional therefore to adopt some particular state of a system as a fiducial state the energy of which is arbitrarily taken as zero The energy of a system in any other state relative to the energy of the system in the fiducial state is then called the thermodynamic internal energy of the system in that state and is denoted by the symbol U Like Thermodynanuc Equilibrium 13 the volume and the mole numbers the internal energy is an extensive parameter 15 THERMODYNAMIC EQUILIBRIUM Macroscopic systems often exhibit some memory of their recent history A stirred cup of tea continues to swirl within the cup Coldworked steel maintains an enhanced hardness imparted by its mechanical treat ment But memory eventually fades Turbulences damp out internal strains yield to plastic flow concentration inhomogeneities diffuse to uniformity Systems tend to subside to very simple states independent of their specific history In some cases the evolution toward simplicity is rapid in other cases it can proceed with glacial slowness But in all systems there is a tendency to evolve toward states in which the properties are determined by intrinsic factors and not by previously applied external influences Such simple terminal states are by definition time independent They are called equi librium states Thermodynamics seeks to describe these simple static equilibrium states to which systems eventually evolve To convert this statement to a formal and precise postulate we first recognize that an appropriate criterion of simplicity is the possibility of description in terms of a small number of variables It therefore seems plausible to adopt the following postulate suggested by experimental observation and formal simplicity and to be verified ultimately by the success of the derived theory Postulate I There exist particular states called equilibrium states of simple systems that macroscopically are characterized completely by the internal energy U the volume V and the mole numbers N1 N2 N of the chemical components As we expand the generality of the systems to be considered eventually permitting more complicated mechanical and electrical properties the number of parameters required to characterize an equilibrium state in creases to include for example the electric dipole moment and certain elastic strain parameters These new variables play roles in the formalism which are completely analogous to the role of the volume V for a simple system A persistent problem of the experimentalist is to determine somehow whether a given system actually is in an equilibrium state to which thermodynamic analysis can be applied He or she can of course observe whether the system is static and quiescent But quiescence is not sufficient As the state is assumed to be characterized completely by the extensive 14 The Problem and the Postulates parameters U V N1 N2 Nr it follows that the properties of the system must be independent of the past history This is hardly an operational prescription for the recognition of an equilibrium state but in certain cases this independence of the past history is obviously not satisfied and these cases give some insight into the significance of equi librium Thus two pieces of chemically identical commercial steel may have very different properties imparted by coldworking heat treatment quenching and annealing in the manufacturing process Such systems are clearly not in equilibrium Similarly the physical characteristics of glass depend upon the cooling rate and other details of its manufacture hence glass is not in equilibrium If a system that is not in equilibrium is analyzed on the basis of a thermodynamic formalism predicated on the supposition of equilibrium inconsistencies appear in the formalism and predicted results are at variance with experimental observations This failure of the theory is used by the experimentalist as an a posteriori criterion for the detection of nonequilibrium states In those cases in which an unexpected inconsistency arises in the thermodynamic formalism a more incisive quantum statistical theory usually provides valid reasons for the failure of the system to attain equilibrium The occasional theoretical discrepancies that arise are there fore of great heuristic value in that they call attention to some unsus pected complication in the molecular mechanisms of the system Such circumstances led to the discovery of ortho and parahydrogen 1 and to the understanding of the molecular mechanism of conversion between the two forms From the atomic point qf view the macroscopic equilibrium state is associated with incessant and rapid transitions among all the atomic states consistent with the given boundary conditions If the transition mecha nism among the atomic states is sufficiently effective the system passes rapidly through all representative atomic states in the course of a macro scopic observation such a system is in equilibrium However under certain unique conditions the mechanism of atomic transition may be ineffective and the system may be trapped in a small subset of atypical atomic states Or even if the system is not completely trapped the rate of transition may be so slow that a macroscopic measurement does not yield a proper average over all possible atomic states In these cases the system is not in equilibrium It is readily apparent that such situations are most likely to occur in solid rather than in fluid systems for the comparatively high atomic mobility in fluid systems and the random nature of the 1 If the two nuclei in a H 2 molecule have parallel angular momentum the molecule is called orthoH 2 if antiparallel paraH 2 The ratio of orthoH 2 to paraH 2 in a gaseous H 2 system should have a definite value in equilibnum but this ratio may not be obtained under certain conditions The resultant failure of H 2 to satisfy certain thermodynamic equations motivated the investigations of the ortho and paraforms of H 2 Walls and Constraints 15 interatomic collisions militate strongly against any restrictions of the atomic transition probabilities In actuality few systems are in absolute and true equilibrium In absolute equilibrium all radioactive materials would have decayed com pletely and nuclear reactions would have transmuted all nuclei to the most stable of isotopes Such processes which would take cosmic times to complete generally can be ignored A system that has completed the relevant processes of spontaneous evolution and that can be described by a reasonably small number of parameters can be considered to be in metastable equilibrium Such a limited equilibrium is sufficient for the application of thermodynamics In practice the criterion for equilibrium is circular Operationally a system is in an equilibrium state if its properties are consistently described by thermodynamic theory It is important to reflect upon the fact that the circular character of thermodynamics is not fundamentally different from that of mechanics A particle of known mass in a known gravitational field might be expected to move in a specific trajectory if it does not do so we do not reject the theory of mechanics but we simply conclude that some additional force acts on the particle Thus the existence of an electrical charge on the particle and the associated relevance of an electrical force cannot be known a priori It is inferred only by circular reasoning in that dynamical predictions are incorrect unless the electric contribution to the force is included Our model of a mechanical system including the assignment of its mass moment of inertia charge dipole moment etc is correct if it yields successful predictions 16 WALLS AND CONSTRAINTS A description of a thermodynamic system requires the specification of the walls that separate it from the surroundings and that provide its boundary conditions It is by means of manipulations of the walls that the extensive parameters of the system are altered and processes are initiated The processes arising by manipulations of the walls generally are associated with a redistribution of some quantity among various systems or among various portions of a single system A formal classification of thermodynamic walls accordingly can be based on the property of the walls in permitting or preventing such redistributions As a particular illustration consider two systems separated by an internal piston within a closed rigid cylinder If the position of the piston is rigidly fixed the wall prevents the redistribution of volume between the two systems but if the piston is left free such a redistribution is permitted The cylinder and the rigidly fixed piston may be said to constitute a wall restrictive with respect to the volume whereas the cylinder and the movable piston 16 The Problem and the Postulates may be said to constitute a wall nonrestrictive with respect to the volume In general a wall that constrains an extensive parameter of a system to have a definite and particular value is said to be restrictive with respect to that parameter whereas a wall that permits the parameter to change freely is said to be nonrestrictive with respect to that parameter A wall that is impermeable to a particular chemical component is restrictive with respect to the corresponding mole number whereas a permeable membrane is nonrestrictive with respect to the mole number Semipermeable membranes are restrictive with respect to certain mole numbers and nonrestrictive with respect to others A wall with holes in it is nonrestrictive with respect to all mole numbers The existence of walls that are restrictive with respect to the energy is associated with the larger problem of measurability of the energy to which we now turn our attention 17 MEASURABILITY OF THE ENERGY On the basis of atomic considerations we have been led to accept the existence of a macroscopic conservative energy function In order that this energy function may be meaningful in a practical sense however we must convince ourselves that it is macroscopically controllable and measurable We shall now show that practical methods of measurement of the energy do exist and in doing so we shall also be led to a quantitative operational definition of heat An essential prerequisite for the measurability of the energy is the existence of walls that do not permit the transfer of energy in the form of heat We briefly examine a simple experimental situation that suggests that such walls do indeed exist Consider a system of ice and water enclosed in a container We find that the ice can be caused to melt rapidly by stirring the system vigor ously By stirring the system we are clearly transferring energy to it mechanically so that we infer that the melting of the ice is associated with an input of energy to the system If we now observe the system on a summer day we find that the ice spontaneously melts despite the fact that no work is done on the system It therefore seems plausible that energy is being transferred to the system in the form of heat We further observe that the rate of melting of the ice is progressively decreased by changing the wall surrounding the system from thin metal sheet to thick glass and thence to a Dewar wall consisting of two silvered glass sheets separated by an evacuated interspace This observation strongly suggests that the metal glass and Dewar walls are progressively less permeable to the flow of heat The ingenuity of experimentalists has produced walls that are able to reduce the melting rate of the ice to a negligible value and such walls are correspondingly excellent approximations to the limiting idealization of a wall that is truly impermeable to the flow of heat Measurab1hty of the Energy 17 It is conventional to refer to a wall that is impermeable to the flow of heat as adiabatic whereas a wall that permits the flow of heat is termed diathermal If a wall allows the flux of neither work nor heat it is restrictive with respect to the energy A system enclosed by a wall that is restrictive with respect to the energy volume and all the mole numbers is said to be closed 2 The existence of these several types of walls resolves the first of our concerns with the thermodynamic energy That is these walls demonstrate that the energy is macroscopically controllable It can be trapped by restrictive walls and manipulated by diathermal walls If the energy of a system is measured today and if the system is enclosed by a wall restrictive with respect to the energy we can be certain of the energy of the system tomorrow Without such a wall the concept of a macroscopic thermodynamic energy would be purely academic We can now proceed to our second concern that of measurability of the energy More accurately we are concerned with the measurability of energy differences which alone have physical significance Again we invoke the existence of adiabatic walls and we note that for a simple system enclosed by an impermeable adiabatic wall the only type of permissible energy transfer is in the form of work The theory of me chanics provides us with quantitative formulas for its measurement If the work is done by compression displacing a piston in a cylinder the work is the product of force times displacement or if the work is done by stirring it is the product of the torque times the angular rotation of the stirrer shaft In either case the work is well defined and measurable by the theory of mechanics We conclude that we are able to measure the energy difference of two states provided that one state can be reached from the other by some mechanical process while the system is enclosed by an adiabatic impermeable wall The entire matter of controllability and measurability of the energy can be succinctly stated as follows There exist walls called adiabatic with the property that the work done in taking an adiabatically enclosed system between two given states is determined entirely by the states independent of all external conditions The work done is the difference in the internal energy of the two states As a specific example suppose we are given an equilibrium system composed of ice and water enclosed in a rigid adiabatic impermeable wall Through a small hole in this wall we pass a thin shaft carrying a propellor blade at the inner end and a crank handle at the outer end By turning the crank handle we can do work on the system The work done is equal to the angular rotation of the shaft multiplied by the viscous torque After turning the shaft for a definite time the system is allowed to come to a new equilibrium state in which some definite amount of the ice is observed 2 Tlus definit10n of closure differs from a usage common in chemistry in which closure 1mphes only a wall restrictive with respect to the transfer of matter J 8 The Problem and the Postulates to have been melted The difference in energy of the final and initial states is equal to the work that we have done in turning the crank We now inquire about the possibility of starting with some arbitrary given state of a system of enclosing the system in an adiabatic imperme able wall and of then being able to contrive some mechanical process that will take the system to another arbitrarily specified state To determine the existence of such processes we must have recourse to experimental observation and it is here that the great classical experiments of Joule are relevant His work can be interpreted as demonstrating that for a system enclosed by an adiabatic impermeable wall any two equilibrium states with the same set of mole numbers N1 N2 N can be joined by some possible mechanical process Joule discovered that if two states say A and B are specified it may not be possible to find a mechanical process consistent with an adiabatic impermeable wall to take the system from A to B but that it is always possible to find either a process to take the system from A to B or a process to take the system from B to A That is for any states A and B with equal mole numbers either the adiabatic mechanical process A B or B A exists For our purposes either of these processes is satisfactory Experiment thus shows that the methods of mechanics permit us to measure the energy difference of any two states with equal mole numbers Joules observation that only one of the processes A B or B A may exist is of profound significance This asymmetry of two given states is associated with the concept of irreversibility with which we shall subsequently be much concerned The only remaining limitation to the measurability of the energy difference of any two states is the requirement that the states must have equal mole numbers This restriction is easily eliminated by the following observation Consider two simple subsystems separated by an imperme able wall and assume that the energy of each subsystem is known relative to appropriate fiducial states of course If the impermeable wall is removed the subsystems will intermix but the total energy of the com posite system will remain constant Therefore the energy of the final mixed system is known to be the sum of the energies of the original subsystems This technique enables us to relate the energies of states with different mole numbers In summary we have seen that by employing adiabatic walls and by measuring only mechanical work the energy of any thermodynamic system relative to an appropriate reference state can be measured 18 QUANTITATIVE DEFINITION OF HEATUNITS The fact that the energy difference of any two equilibrium states is measurable provides us directly with a quantitative definition of the heat The heat flux to a system in any process at constant mole numbers is Quantitative Definition of Heat Units 19 simply the difference in internal energy between the final and initial states diminished by the work done in that process Consider some specified process that takes a system from the initial state A to the final state B We wish to know the amount of energy transferred to the system in the form of work and the amount transferred in the form of heat in that particular process The work is easily measured by the method of mechanics Furthermore the total energy difference u is measurable by the procedures discussed in Section 17 Sub trcting the work from the total energy difference gives us the heat flux in the specified process It should be noted that the amount of work associated with different processes may be different even though each of the processes initiates in the same state A and each terminates in the same state B Similarly the heat flux may be different for each of the processes But the sum of the work and heat fluxes is just the total energy difference U8 and is the same for each of the processes In referring to the total energy flux we therefore need specify only the initial and terminal states but in referring to heat or work fluxes we must specify in detail the process considered Restricting our attention to thermodynamic simple systems the quasi static work is associated with a change in volume and is given quantita tively by dWM PdV 11 where P is the pressure In recalling this equation from mechanics we stress that the equation applies only to quasistatic processes A precise definition of quasistatic processes will be given in Section 42 but now we merely indicate the essential qualitative idea of such processes Let us suppose that we are discussing as a particular system a gas enclosed in a cylinder fitted with a moving piston If the piston is pushed in very rapidly the gas immediately behind the piston acquires kinetic energy and is set into turbulent motion and the pressure is not well defined In such a case the work done on the system is not quasistatic and is not given by equation 11 If however the piston is pushed in at a vanishingly slow rate quasistatically the system is at every moment in a quiescent equilibrium state and equation 11 then applies The infinite slowness of the process is roughly the essential feature of a quasistatic process A second noteworthy feature of equation 11 is the sign convention The work is taken to be positive if it increases the energy of the system If the volume of the system is decreased work is done on the system increasing its energy hence the negative sign in equation 11 With the quantitative expression dW M P dV for the quasistatic work we can now give a quantitative expression for the heat flux In an infinitesimal quasistatic process at constant mole numbers the quasistatic heat dQ is defined by the equation dQ dU dW M at constant mole numbers 12 20 The Problem and the Postulates or dQ dU P dV at constant mole numbers 13 It will be noted that we use the terms heat and heat flux interchange ably Heat like work is only a form of energy transfer Once energy is transferred to a system either as heat or as work it is indistinguishable from energy that might have been transferred differently Thus although dQ and dW M add together to give dU the energy U of a state cannot be considered as the sum of work and heat components To avoid this implication we put a stroke through the symbol d infinitesimals such as dW M and dQ are called imperfect differentials The integrals of dW M and dQ for a particular process are the work and heat fluxes in that process the sum is the energy difference U which alone is independent of the process The concepts of heat work and energy may possibly be clarified in terms of a simple analogy A certain farmer owns a pond fed by one stream and drained by another The pond also receives water from an occasional rainfall and loses it by evaporation which we shall consider as negative rain In this analogy the pond is our system the water within it is the internal energy water transferred by the streams is work and water transferred as rain is heat The first thing to be noted is that no examination of the pond at any time can indicate how much of the water within it came by way of the stream and how much came by way of rain The term rain refers only to a method of water transfer Let us suppose that the owner of the pond wishes to measure the amount of water in the pond He can purchase flow meters to be inserted in the streams and with these flow meters he can measure the amount of stream water entering and leaving the pond But he cannot purchase a rain meter However he can throw a tarpaulin over the pond enclosing the pond in a wall impermeable to rain an adiabatic wall The pond owner consequently puts a vertical pole into the pond covers the pond with his tarpaulin and inserts his flow meters into the streams By damming one stream and then the other he varies the level in the pond at will and by consulting his flow meters he is able to calibrate the pond level as read on his vertical stick with total water content U Thus by carrying out processes on the system enclosed by an adiabatic wall he is able to measure the total water content of any state of his pond Our obliging pond owner now removes his tarpaulin to permit rain as well as stream water to enter and leave the pond He is then asked to evaluate the amount of rain entering his pond during a particular day He proceeds simply he reads the difference in water content from his vertical stick and from this he deducts the total flux of stream water as registered by his flow meters The difference is a quantitative measure of the rain The strict analogy of each of these procedures with its thermodynamic counterpart is evident Quan11ta11ve Defimtion of Heat Units 21 Since work and heat refer to particular modes of energy transfer each is measured in energy units In the cgs system the unit of energy and hence of work and heat is the erg In the mks system the unit of energy is the joule or 10 7 ergs A practical unit of energy is the calorie 3 or 41858 J Historically the calorie was introduced for the measurement of heat flux before the relationship of heat and work was clear and the prejudice toward the use of the calorie for heat and of the joule for work still persists Nevertheless the calorie and the joule are simply alternative units of energy either of which is acceptable whether the energy flux is work heat or some combination of both Other common units of energy are the British thermal unit Btu the literatmosphere the footpound and the watthour Conversion factors among energy units are given inside the back cover of this book Example 1 A particular gas is enclosed in a cylinder with a moveable piston It is observed that if the walls are adiabatic a quasistatic increase in volume results in a decrease in pressure according to the equation P 3V 5 constant for Q 0 a Find the quasistatic work done on the system and the net heat transfer to the system in each of the three processes ADB ACB and the direct linear process AB as shown in the figure 8 X 103 In the process ADB the gas is heated at constant pressure P 105 Pa until its volume increases from its initial value of 10 3 m3 to its final value of 8 x to 3 m3 The gas is then cooled at constant volume until its pressure decreases to 105 32 Pa The other processes ACB and AB can be similarly interpreted according to the figure 1 Nutritionists refer to a kilocalorie as a Calone presumably to spare calorie counters the trauma of large numbers To compound the confusion the initial capital C is often dropped so that a kJlocalorie becomes a calorie 22 The Problem and the Postulates b A small paddle is installed inside the system and is driven by an external motor by means of a magnetic couplmg through the cylinder wall The motor exerts a torque driving the paddle at an angular velocity w and the pressure of the gas at constant volume is observed to mcrease at a rate given by dP 2 w X torque dt 3 V Show that the energy difference of any two states of equal volumes can be determined by this process In particular evaluate Uc VA and VD U8 Explain why this process can proceed only in one direction vertically upward rather than downward in the P V plot c Show that any two states any two points in the PV plane can be connected by a combination of the processes in a and b In particular evaluate U D VA d Calculate the work WAD m the process A D Calculate the heat transfer QAD Repeat for D B and for C A Are these results consistent with those of a The reader should attempt to solve this problem before reading the following solution Solution a Given the equation of the adiabat for which Q 0 and AU W we find i p V 5 3 V 23 V 23 2 A A B A 3 225 100 1125 J Now consider process ADB WADB f PdV 10 5 x8 x 10 3 10 3 700J But QADB 1125 700 5875 J Note that we are able to calculate QADB but not QAD and QD8 separately for we do not yet know VD VA Similarly we find WAc 8 219 J and QAcB 906 J Also WA8 3609 J and QA8 2484 J b As the motor exerts a torque and turns through an angle dB it delivers an Quantitative Defimtwn of Heat Units 2 3 energy 4 dU torque X dO to the system But dO w dt so that 2 1 dP 3 V torque wdl or 2 1 du 3 V 3 dU VdP 2 This process is carried out at constant V and furthermore dU 0 and conse quently dP 0 The condition dU 0 follows from dU torque x dO for the sign of the rotation dO is the same as the sign of the torque that induces that rotation In particular 3 3 1 VA Uc 2 VPA Pc 2 x 10 3 x 105 32 x 105 1453 J and 3 3 1 U0 Us 2 VP 0 P8 2 X 8 X 10 3 X 105 32 X 105 1162 5 J c To connect any two points in the plane we draw an adiabat through one and an isochor V constant through the other These two curves intersect thereby connecting the two states Thus we have found using the adiabatic process that U8 VA 1125 J and using the irreversible stirrer process that U0 U8 11625 J Therefore Uv VA 1050 J Equivalently if we assign the value zero to UA then UA 0 Us 1125 J Uc 1453 J Uv 1050 J and similarly every state can be assigned a value of U d Now having U0 UA and WAD we can calculate QAD Uo UA Uo QAD 1050 700 QAD QAD J750 J Also or 625 0 QDB To check we note that QAD Q08 5875 J which is equal to QADB as found in a 4 Note that the energy output of the motor is delivered to the system as energy that cannot be classified either as work or as heatit is a nonquas1sta11c transfer of energy 24 The Problem and the Postulates PROBLEMS 181 For the system considered in Example 1 calculate the energy of the state with P 5 X 10 4 Pa and V 8 X 10 3 m3 182 Calculate the heat transferred to the system considered in Example I in the process in which it is taken in a straight line on the PV diagram from the state A to the state referred to in the preceding problem 183 For a particular gaseous system it has been determined that the energy is given by U 25PV constant The system is initially in the state P 02 MPa megaPascals V 001 m3 designated as point A in the figure The system is taken through the cycle of three processes A B B C and C A shown in the figure Calculate Q and W for each of the three processes Calculate Q and W for a process from A to B along the parabola P 105 109 X V 022 05 04 t l 03 a 02 01 00 C n A 001 002 Vm 3 003 Answer WBC 7 X 103 J QBC 95 X 103 J 184 For the system of Problem 183 find the equation of the adiabats in the P V plane ie find the form of the curves P P V such that dQ 0 along the curves Answer V 1P 5 constant The Basic Problem OJ I flermvuuu l85 The energy of a particular system of one mole is given by U AP 2V where A is a positive constant of dimensions P 1 Find the equation of the adiabats in the PV plane 186 For a particular system it is found that if the volume is kept constant at the value V0 and the pressure is changed from P0 to an arbitrary pressure P the heat transfer to the system is Q AP P0 A 0 In addition it is known that the adiabats of the system are of the form pvr constant y a positive constant Find the energy U P V for an arbitrary point in the P V plane expressing UP V in terms of P0 V0 A U0 UP 0 V0 and y aswell as P and V U U0 APr P0 PVy 11 r 1 Answer where r VV 0 187 Two moles of a particular singlecomponent system are found to have a dependence of internal energy U on pressure and volume given by U APV 2 for N 2 Note that doubling the system doubles the volume energy and mole number but leaves the pressure unaltered Write the complete dependence of U on P V and N for arbitrary mole number 19 THE BASIC PROBLEM OF THERMODYNAMICS The preliminaries thus completed we are prepared to formulate first the seminal problem of thermodynamics and then its solution Surveying those preliminaries retrospectively it is remarkable how far reaching and how potent have been the consequences of the mere choice of thermodynamic coordinates Identifying the criteria for those coordi nates revealed the role of measurement The distinction between the macroscopic coordinates and the incoherent atomic coordinates suggested the distinction between work and heat The completeness of the descrip tion by the thermodynamic coordinates defined equilibrium states The thermodynamic coordinates will now provide the framework for the solution of the central problem of thermodynamics There is in fact one central problem that defines the core of thermody namic theory All the ults of thermodynamics propagate from its SOiution 26 The Problem and the Postulates The single allencompassing problem of thermodynamics is the determina tion of the equilibrium state that eventually results after the removal of internal constraints in a closed composite system Let us suppose that two simple systems are contained within a closed cylinder separated from each other by an internal piston Assume that the cylinder walls and the piston are rigid impermeable to matter and adiabatic and that the position of the piston is firmly fixed Each of the systems is closed If we now free the piston it will in general seek some new position Similarly if the adiabatic coating is stripped from the fixed piston so that heat can flow between the two systems there will be a redistribution of energy between the two systems Again if holes are punched in the piston there will be a redistribution of matter and also of energy between the two systems The removal of a constraint in each case results in the onset of some spontaneous process and when the systems finally settle into new equilibrium states they do so with new values of the parameters ul vl Np and U2 V2 Nfl The basic prob lem of thermodynamics is the calculation of the equilibrium values of these parameters FIGURE 12 Piston Cylinder Before formulating the postulate that provides the means of solution of the problem we rephrase the problem in a slightly more general form without reference to such special devices as cylinders and pistons Given two or more simple systems they may be considered as constituting a single composite system The composite system is termed closed if it is surrounded by a wall that is restrictive with respect to the total energy the total volume and the total mole numbers of each component of the composite system The individual simple systems within a closed com posite system need not themselves be closed Thus in the particular example referred to the composite system is closed even if the internal piston is free to move or has holes in it Constraints that prevent the flow of energy volume or matter among the simple systems constituting the composite system are known as internal constraints If a closed composite system is in equilibrium with respect to internal constraints and if some of these constraints are then removed certain previously disallowed processes become permissible These processes bring the system to a new equilibrium state Prediction of the new equilibrium state is the central problem of thermodynamics The Entropy Maximum Postulates 27 t10 THE ENTROPY MAXIMUM POSTULATES The induction from experimental observation of the central principle that provides the solution of the basic problem is subtle indeed The historical method culminating in the analysis of Caratheodory is a tour de force of delicate and formal logic The statistical mechanical approach pioneered by Josiah Willard Gibbs required a masterful stroke of induc tive inspiration The symmetrybased foundations to be developed in Chapter 21 will provide retrospective understanding and interpretation but they are not yet formulated as a deductive basis We therefore merely formulate the solution to the basic problem of thermodynamics in a set of postulates depending upon a posteriori rather than a priori justification These postulates are in fact the most natural guess that we might make providing the simplest conceivable formal solution to the basic problem On this basis alone the problem might have been solved the tentative postulation of the simplest formal solution of a problem is a conventional and frequently successful mode of procedure in theoretical physics What then is the simplest criterion that reasonably can be imagined for the determination of the final equilibrium state From our experience with many physical theories we might expect that the most economical form for the equilibrium criterion would be in terms of an extremum principle That is we might anticipate the values of the extensive parameters in the final equilibrium state to be simply those that maximize 5 some function And straining our optimism to the limit we might hope that this hypothetical function would have several particularly simple mathematical properties designed to guarantee simplicity of the derived theory We develop this proposed solution in a series of postulates Postulate II There exists a function called the entropy S of the extensive parameters of any composite system defined for all equilibrium states and having the foil owing property The values assumed by the extensive parame ters in the absence of an internal constraint are those that maximize the entropy over the manifold of constrained equilibrium states It must be stressed that we postulate the existence of the entropy only for equilibrium states and that our postulate makes no reference whatsoever to nonequilibrium states In the absence of a constraint the system is free to select any one of a number of states each of which might also be realized in the presence of a suitable constraint The entropy of each of these constrained equilibrium states is definite and the entropy is largest in some particular state of the set In the absence of the constraint this state of maximum entropy is seltcted by the system 50r minimize the function this being purely a matter of convention in the choice of the sign of the function having no consequence whatever in the logical structure of the theory 28 The Problem and the Postulates In the case of two systems separated by a diathermal wall we might wish to predict the manner in which the total energy U distributes between the two systems We then consider the composite system with the internal diathermal wall replaced by an adiabatic wall and with particular values of u1 and U2 consistent of course with the restriction that u1 U2 U For each such constrained equilibrium state there is an entropy of the composite system and for some particular values of u1 and U2 this entropy is maximum These then are the values of u1 and U2 that obtain in the presence of the diathermal wall or in the absence of the adiabatic constraint All problems in thermodynamics are derivative from the basic problem formulated in Section 19 The basic problem can be completely solved with the aid of the extremum principle if the entropy of the system is known as a function of the extensive parameters The relation that gives the entropy as a function of the extensive parameters is known as a fundamental relation It therefore follows that if the fundamental relation of a particular system is known all conceivable thermodynamic information about the system is ascertainable from it The importance of the foregoing statement cannot be overemphasized The information contained in a fundamental relation is allinclusiveit is equivalent to all conceivable numerical data to all charts and to all imaginable types of descriptions of thermodynamic properties If the fundamental relation of a system is known every thermodynamic attri bute is completely and precisely determined Postulate III The entropy of a composite system is additive over the constituent subsystems The entropy is continuous and differentiable and is a monotonically increasing function of the energy Several mathematical consequences follow immediately The additivity property states that the entropy S of the composite system is merely the sum of the entropies s 0 of the constituent subsystems 14 a The entropy of each subsystem is a function of the extensive parameters of that subsystem alone 15 The additivity property applied to spatially separate subsystems re quires the following property The entropy of a simple system is a homoge neous firstorder function of the extensive parameters That is if all the extensive parameters of a system are multiplied by a constant A the The Entropy Maximum Postulates 29 entropy is multiplied by this same constant Or omitting the superscript a The monotonic property postulated implies that the partial derivative asauvN 1 N is a positive quantity iVN 1 N Q 17 As the theory develops in subsequent sections we shall see that the reciprocal of this partial derivative is taken as the definition of the temperature Thus the temperature is postulated to be nonnegative 6 The continuity differentiability and monotonic property imply that the entropy function can be inverted with respect to the energy and that the energy is a singlevalued continuous and differentiable function of S V N1 N The function 18 can be solved uniquely for V in the form V USVNi NJ 19 Equations 18 and 19 are alternative forms of the fundamental relation and each contains all thermodynamic information about the system We note that the extensivity of the entropy permits us to scale the properties of a system of N moles from the properties of a system of 1 mole The fundamental equation is subject to the identity S V V N1 N2 N NS UN V N N1N NJN 110 in which we have taken the scale factor A of equation 16 to be equal to lN lEk Nk For a singlecomponent simple system in particular SU V N NSUN VN l 111 But V N is the energy per mole which we denote by u u UN 112 6 The posibility of negative values of this derivative ie of negative temperatures has been discussed by N F Ramsey Phys Rev 103 20 1956 Such states are not equilibrium states m real systems and they do not invalidate equation 1 7 They can be produced only m certain very unique systems specifically in isolated spin systems and they spontaneously decay away Nevertheless the study of these states is of stahshcal mechanical interest elucidating the stahstical mechanical concept of temperature 30 The Problem a11d the Postulates Also V N is the volume per mole which we denote by v V VN 113 Thus SUN VN 1 Suv 1 is the entropy of a system of a single mole to be denoted by s u v su v Su v I 114 Equation 111 now becomes SU V N Nsu v 115 Postulate IV The entropy of any system vanishes in the state for which oUoSvN 1 N 0 that is at the zero of temperature We shall see later that the vanishing of the derivative au oSv N N is equivalent to the vanishing of the temperature as indicated Hene thJ fourth postulate is that zero temperature implies zero entropy It should be noted that an immediate implication of postulate IV is that S like V and N but unlike V has a uniquely defined zero This postulate is an extens10n due to Planck of the socalled Nernst postulate or third law of thermodynamics Historically it was the latest of the postulates to be developed being inconsistent with classical statistical mechanics and requiring the prior establishment of quantum statistics in order that it could be properly appreciated The bulk of thermodynamics does not require this postulate and I make no further reference to it until Chapter 10 Nevertheless I have chosen to present the postulate at this point to close the postulatory basis The foregoing postulates are the logical bases of our development of thermodynamics In the light of these postulates then it may be wise to reiterate briefly the method of solution of the standard type of thermody namic problem as formulated in Section 19 We are given a composite system and we assume the fundamental equation of each of the con stituent systems to be known in principle These fundamental equations determine the individual entropies of the subsystems when these systems are in equilibrium If the total composite system is in a constrained equilibrium state with particular values of the extensive parameters of each constituent system the total entropy is obtained by addition of the individual entropies This total entropy is known as a function of the various extensive parameters of the subsystems By straightforward differ entiation e compute the extrema of the total entropy function and then on the basis of the sign of the second derivative we classify these extrema as minima maxima or as horizontal inflections In an appropriate physi The Entropy Maximum Pmtulates 31 cal terminology we first find the equilibrium states and we then classify them on the basis of stability It should be noted that in the adoption of this conventional terminology we augment our previous definition of equilibrium that which was previously termed equiltbrium is now termed stable equilibrium whereas unstable equilibrium states are newly defined in terms of extrema other than maxima It is perhaps appropriate at this point to acknowledge that although all applications of thermodynamics are equivalent in principle to the proce dure outlined there are several alternative procedures that frequently prove more convenient These alternate procedures are developed in subsequent chapters Thus we shall see that under appropriate conditions the energy US V Ni may be minimized rather than the entropy S U V Ni maximized That these two procedures determine the same final state is analogous to the fact that a circe may be characterized either as the closed curve of minimum perimeter for a given area or as the closed curve of maximum area for a given perimeter In later chapters we shall encounter several new functions the minimization of which is logically equivalent to the minimization of the energy or to the maximiza tion of the entropy The inversion of the fundamental equation and the alternative state ment of the basic extremum principle in terms of a minimum of the energy rather than a maximum of the entropy suggests another view point from which the extremum postulate perhaps may appear plausible In the theories of electricity and mechanics ignoring thermal effects the energy is a function of various mechanical parameters and the condition of equilibrium is that the energy shall be a minimum Thus a cone is stable lying on its side rather than standing on its point because the first position is of lower energy If thermal effects are to be included the energy ceases to be a function simply of the mechanical parameters According to the inverted fundamental equation however the energy is a function of the mechanical parameters and of one additional parameter the entropy By the introduction of this additional parameter the form of the energy minimum principle is extended to the domain of thermal effects as well as to pure mechanical phenomena In this manner we obtain a sort of correspondence principle between thermodynamics and mechanics ensuring that the thermodynamic equilibrium principle reduces to the me chanical equilibrium principle when thermal effects can be neglected We shall see that the mathematical condition that a maximum of S U V Ni implies a minimum of U S V N1 is that the deriva tive iJS iJUv N be positive The motivation for the introduction of this statement in postulate III may be understood in terms of our desire to ensure that the entropymaximum principle will go over into an energy minimum principle on inversion of the fundamental equation In Parts II and III the concept of the entropy will be more deeply explored both in terms of its symmetry roots and in terms of its statistical 32 The Problem and the Postulates mechanical interpretation Pursuing those inquires now would take us too far afield In the classical spirit of thermodynamics we temporarily def er such interpretations while exploring the farreaching consequences of our simple postulates PROBLEMS 1101 The following ten equations are purported to be fundamental equations of various thermodynamic systems However five are inconsistent with one or more of postulates II III and IV and consequently are not physically acceptable In each case qualitatively sketch the fundamental relationship between S and U with N and V constant Find the five equations that are not physically permissible and indicate the postulates violated by each The quantities v0 and R are positive constants and in all cases in which fractional exponents appear only the real positive root is to be taken a S R2 13NVU13 Vo b S2r3Nr3 c S t NU Rt r d S V 3NU e S fIN2VU21f5 Vo2 S NRlnUVN 2ROv0 g S f 1NUJ 1l 2exp V22N 2vl h S R 11NU 112exp NR0v 0 i U vt f expSNR j U N 1 R expSNR 1102 For each of the five physically acceptable fundamental equations in problem 1101 find U as a function of S V and N Problems 33 1103 The fundamental equation of system A is S R2 l3NVU13 vof and similarly for system B The two systems are separated by a rigid imperme able adiabatic wall System A has a volume of 9 X 10 6 m3 and a mole number of 3 moles System B has a volume of 4 X 10 6 m3 and a mole number of 2 moles The total energy of the composite system is 80 J Plot the entropy as a function of UAUA U8 If the internal wall is now made diathermal and the system is allowed to come to equilibrium what are the internal energies of each of the individual systems As in Problem 1101 the quantities v0 and R are positive constants No text found in the image 2 THE CONDITIONS OF EQUILIBRIUM 21 INTENSIVE PARAMETERS By virtue of our interest in processes and in the associated changes of the extensive parameters we anticipate that we shall be concerned with the differential form of the fundamental equation Writing the fundamen tal equation in the form U US V N1 N2 N 21 we compute the first differential au au au dU dS dV L d as vN 1 N av sN N ji aN1 sv N 22 The various partial derivatives appearing in the foregoing equation recur so frequently that it is convenient to introduce special symbols for them They are called intensive parameters and the following notation is conven tional aaUS T the temperature VN 1 N 23 P the pressure SN 1 N 24 au the electrochemical potential of aN µ1 thejth component J SV Nk 25 35 36 The Conditwns of Equ1libr1um With this notation equation 22 becomes 26 The formal definition of the temperature soon will be shown to agree with our intuitive qualitative concept based on the physiological sensa tions of hot and cold We certainly would be reluctant to adopt a definition of the temperature that would contradict such strongly en trenched although qualitative notions For the moment however we merely introduce the concept of temperature by the formal definition 23 Similarly we shall soon corroborate that the pressure defined by equation 24 agrees in every respect with the pressure defined in mecha nics With respect to the several electrochemical potentials we have no prior definitions or concepts and we are free to adopt the definition equation 25 forthwith For brevity the electrochemical potential is often referred to simply as the chemical potential and we shall use these two terms interchangea bly1 The term P dV in equation 26 is identified as the quasistatic work dWM as given by equation II In the special case of constant mole numbers equation 26 can then be written as TdS dUdWM 27 Recalling the definition of the quasistatic heat or comparing equation 27 with equation 12 we now recognize T dS as the quasistatic heat flux dQ TdS 28 A quasistatic flux of heat into a system is associated with an increase of entropy of that system The remaining terms in equation 26 represent an increase of internal energy associated with the addition of matter to a system This type of energy flux although intuitively meaningful is not frequently discussed outside thermodynamics and does not have a familiar distinctive name We shall call E 1µ 1 d the quasistatic chemical work 29 1 However it should be noted that occasionally and particularly in the theory of solids the chemical potential is defined as the electrochemical potential p mirr be molar electrostatic energy Equations of State 3 7 Therefore dU dQ dW M dWC 210 Each of the terms TdS PdV µ 1d in equation 26 has the dimen sions of energy The matter of units will be considered in Section 26 We can observe here however that having not yet specified the units nor even the dimensions of entropy the units and dimensions of temperature remain similarly undetermined The units of µ are the same as those of energy as the mole numbers are dimensionless The units of pressure are familiar and conversion factors are listed inside the back cover of this book 22 EQUATIONS OF STATE The temperature pressure and electrochemical potentials are partial derivatives of functions of S V N 1 Nr and consequently are also functions of S V N 1 Nr We thus have a set of functional relation ships 211 P PSVN 1 NJ 212 213 Such relationships expressing intensive parameters in terms or the inde pendent extensive parameters are called equations of state Knowledge of a single equation of stale does not constitute complete knowledge of the thermodynamic properties of a system We shall see subsequently that knowledge of all the equations of state of a system is equivalent to knowledge of the fundamental equation and consequently is thermodynamically complete The fact that the fundamental equation must be homogeneous first order has direct implications for the functional form of the equations of state It follows immediately that the equations of stale are homogeneous zero order That is multiplication of each of the independent extensive parameters by a scalar A leaves the function unchanged 214 38 The Cond1tons of Eqwhhrium It therefore follows that the temperature of a portion of a system is equal to the temperature of the whole system This is certainly in agree ment with the intuitive concept of temperature The pressure and the electrochemical potentials also have the property 214 and together with the temperature are said to be intensive To summarize the foregoing considerations it is convenient to adopt a condensed notation We denote the extensive parameters V N1 Nr by the symbols X1 X2 X so that the fundamental relation takes the form U US Xi X 2 XJ The intensive parameters are denoted by aaus r rs x1 x2 x X1X 2 whence I dU TdS L PjdXJ 1l 215 216 jl2 t 217 218 It should be noted that a negative sign appears in equation 24 but does not appear in equation 217 The formalism of thermodynamics is uniform if the negative pressure P is considered as an intensive parameter analogous to T and µ 1 µ 2 Correspondingly one of the general in tensive parameters of equation 217 is P For singlecomponent simple systems the energy differential is fre quently written in terms of molar quantities Analogous to equations 111 through 115 the fundamental equation per mole is u usv 219 where s SN V VN 220 and 1 us v N US V N 221 Problems 39 Taking an infinitesimal variation of equation 219 au au du ds as dv av 222 However au au au T as v as VN as VN 223 and similarly 224 Thus du Tds Pdv 225 PROBLEMS 221 Find the three equations of state for a system with the fundamental equation U vofl R2 NV Corroborate that the equations of state are homogeneous zero order ie that T P and µ are intensive parameters 222 For the system of problem 221 findµ as a function of T V and N 223 Show by a diagram drawn to arbitrary scale the dependence of pressure on volume for fixed temperature for the system of problem 221 Draw two such isotherms corresponding to two values of the temperature and indicate which isotherm corresponds to the higher temperature 224 Find the three equations of state for a system with the fundamental equation and show that for this systemµ u 225 Express µ as a function of T and P for the system of problem 224 226 Find the three equations of state for a system with the fundamental equation 40 The Condawn of Eqwhhrium 227 A particular system obeys the relation u Av 2expsR N moles of this substance initially at temperature Tii and pressure P0 are expanded isentropically s constant until the pressure is halved What is the final temperature Answer 1j 063 T0 228 Show that in analogy with equation 225 for a system with r components r 1 du Tds Pdv µ 1 µdx 1 11 where the x1 are the mole fractions N 229 Show that if a singlecomponent system is such that PV is constant in an adiabatic process k is a positive constant the energy is where is an arbitrary function Hint PV must be a function of S so that au 8Vs gS v where gS is an unspecified function 23 ENTROPIC INTENSIVE PARAMETERS If instead of considering the fundamental equation in the form U US X1 with U as dependent we had considered S as depen dent we could have carried out all the foregoing formalism in an inverted but equivalent fashion Adopting the notation X0 for U we write S S X0 X 1 X 226 We take an infinitesimal variation to obtain 1 as dS E ax dX kO I 227 Entropc lntenme Parameter 41 The quantities as I a xk are denoted by Fk 228 By carefully noting which variables are kept constant in the vanou partial derivatives and by using the calculus of partial derivatives as reviewed in Appendix A the reader can demonstrate that 1 Fo T k 123 229 These equations also follow from solving equation 218 for dS and comparing with equation 227 Despite the close relationship between the F and the P there is a very important difference in principle Namely the P are obtained by dif ferentiating a function of S X1 and are considered as functions of these variables whereas the Fk are obtained by differentiating a function of U X1 and are considered as functions of these latter variables That is in one case the entropy is a member of the set of independent parameters and in the second case the energy is such a member In performing formal manipulations in thermodynamics it is extremely important to make a definite commitment to one or the other of these choices and to adhere rigorously to that choice A great deal of confusion results from a vacillation between these two alternatives within a single problem If the entropy is considered dependent and the energy independent as in S S U Xk we shall refer to the analysis as being in the entropy representation If the energy is dependent and the entropy is independent as in U U S X we shall refer to the analysis as being in the energy representation The formal development of thermodynamics can be carried out in either the energy or entropy representations alone but for the solution of a particular problem either one or the other representation may prove to be by far the more convenient Accordingly we shall develop the two representations in parallel although a discussion presented in one repre sentation generally requires only a brief outline in the alternate represen tation The relation S S X0 X1 is said to be the entropic fundamen tal relation the set of variables X0 X1 is called the entropic extensive parameters and the set of variables F1 is called the entropic intensive parameters Similarly the relation U US X 1 X1 is said to be the energetic fundamental relation the set of 42 The Condllwns of Equ1br1um variables S X1 is called the energetic extensive parameters and the set of variables T P 1 is called the energetic intensive parameters PROBLEMS 231 Find the three equations of state in the entropy representation for a system with the fundamental equation u d20 s5l RJ2 v12 Answer 1 2 vllfJ 25 vt5 T 5 R32 u3f5 J 1 U2S 1 12 25 T 5 R312 V 232 Show by a diagram drawn to arbitrary scale the dependence of tempera ture on volume for fixed pressure for the system of problem 231 Draw two such isobars corresponding to two values of the pressure and indicate which isobar corresponds to the higher pressure 233 Find the three equations of state in the entropy representation for a system with the fundamental equation u sie v v 234 Consider the fundamental equation S AUnvmN where A is a positive constant Evaluate the permissible values of the three constants n m and r if the fundamental equation is to satisfy the thermody namic postulates and if in addition we wish to have P increase with U V at constant N This latter condition is an intuitive substitute for stability require ments to be studied in Chapter 8 For definiteness the zero of energy is to be taken as the energy of the zerotemperature state 235 Find the three equations of state for a system with the fundamental relation Thermal Equ1lhrum Temperature 43 0 Show that the equations of state in entropy representation are homogeneous zeroorder functions b Show that the temperature is intrinsically positive c Find the mechanical equation of state P PT v d Find the form of the adiabats in the Pv plane An adiabat is a locus of constant entropy or an isentrope 24 THERMAL EQUILIBRIUM TEMPERATURE We are now in a position to illustrate several interesting implications of the extremum principle which has been postulated for the entropy Consider a closed composite system consisting of two simple systems separated by a wall that is rigid and impermeable to matter but that does allow the flow of heat The volumes and mole numbers of each of the simple systems are fixed but the energies u1 and u2 are free to change subject to the conservation restriction uoi U2 constant 230 imposed by the closure of the composite system as a whole Assuming that the system has come to equilibrium we seek the values of U1 and U2 According to the fundmental postulate the values of u 0 and U2 are such as to maxinuze the entropy Therefore by the usual mathematical condition for an extremum it follows that in the equilibrium state a virtual infinitesimal transfer of energy from system I to system 2 will produce no change in the entropy of the whole system That is dS O 231 The additivity of the entropy for the two subsystems gives the relation s so uo vl o s2 u2 vi 12 232 As u1 and U2 are changed by the virtual energy transfer the entropy change is as 1 dS au0 vu NI as2 dU 11 au2 v21 233 NI J 44 The Cond1twns of Eqwbrum or employing the definition of the temperature dS 1 dU 0 1 dU2 rll r2 234 By the conservation condition equation 230 we have 235 whence dS 1 1 du 0 rl r2 236 The condition of equilibrium equation 231 demands that dS vanish for arbitrary values of dU 1 whence 237 This is the condition of equilibrium If the fundamental equations of each of the subsystems were known then 1Tll would be a known function of u1 and of vl and Np which however are merely constants Similarly IT 2 would be a known function of u2 and the equation 1r 1 1r 2 would be one equation in u1 and u2 The conserva tion condition Ul U2 constant provides a second equation and these two equations completely determine in principle the values of uo and of U2 To proceed further and actually to obtain the values of uo and U2 would require knowledge of the explicit forms of the fundamen tal equations of the systems In thermodynamic theory however we accept the existence of the fundamental equations but we do not assume explicit forms for them and we therefore do not obtain explicit answers In practical applications of thermodynamics the fundamental equations may be known either by empirical observations in terms of measure ments to be described later or on the basis of statistical mechanical calculations based on simple models In this way applied thermodynamics is able to lead to explicit numerical answers Equation 237 could also be written as r 0 T2 We write it in the form 1r 1 1r 2 to stress the fact that the analysis is couched in the entropy representation By writing ITo we indicate a function of u 0 v 0 whereas ro would imply a function of so vl The physical significance of equation 237 however remains the equality of the temperatures of the two subsystems A second phase of the problem is the investigation of the stability of the predicted final state In the solution given we have not exploited fully the Agreement with Intwtwe Concept of Temperature 45 basic postulate that the entropy 1s a maximum in equilibrium rather we merely have investigated the consequences of the fact that it is an extremum The condition that it be a maximum requires in addition to the condition dS 0 that 238 The consequences of this condition lead to considerations of stability to which we shall give explicit attention in Chapter 8 25 AGREEMENT WITH INTUITIVE CONCEPT OF TEMPERATURE In the foregoing example we have seen that if two systems are separated by a diathermal wall heat will flow until each of the system attains the same temperature This prediction is in agreement with our intuitive notion of temperature and it is the first of several observations that corroborate the plausibility of the formal definition of the temperature Inquiring into the example in slightly more detail we suppoie that the two subsystems initially are separated by an adiabatic wall and that the temperatures of the two subsystems are almost but not quite equal In particular we assume that 239 The system is considered initially to be in equilibrium with respect to the internal adiabatic constraint If the internal adiabatic constraint now is removed the system is no longer in equilibrium heat flows across the wall and the entropy of the composite system increases Finally the system comes to a new equilibrium state determined by the condition that the final values of ro and r2 are equal and with the maximum possible value of the entropy that is consistent with the remaining constraints Compare the initial and the final states If AS denotes the entropy difference between the final and initial states AS O 240 But as in equation 236 AS 1 Auo rI r2 241 where r1 and r2 are the initial values of the temperatures By the 46 The Cond1tons of Equ1ibnum condition that Tl r2 it follows that u 1 o 242 This means that the spontaneous process that occurred was one in which heat flowed from subsystem I to subsystem 2 We conclude therefore that heat tends to flow from a system with a high value of T to a system with a low value of T This is again in agreement with the intuitive notion of temperature It should be noted that these conclusions do not depend on the assumption that r1 is approximately equal to r2 this assumption was made merely for the purpose of obtaining mathematical simplicity in equation 241 which otherwise would require a formulation in terms of integrals If we now take stock of our intuitive notion of temperature based on the physiological sensations of hot and cold we realize that it is based upon two essential properties First we expect temperature to be an intensive parameter having the same value in a part of a system as it has in the entire system Second we expect that heat should tend to flow from regions of high temperature toward regions of low temperature These properties imply that thermal equilibrium is associated with equality and homogeneity of the temperature Our formal definition of the temperature possesses each of these properties 26 TEMPERATURE UNITS The physical dimensions of temperature are those of energy divided by those of entropy But we have not yet committed ourselves on the dimensions of entropy in fact its dimensions can be selected quite arbitrarily If the entropy is multiplied by any positive dimensional constant we obtain a new function of different dimensions but with exactly the same extremum propertiesand therefore equally acceptable as the entropy We summarily resolve the arbitrariness simply by adopting the convention that the entropy is dimensionless from the more incisive viewpoint of statistical mechanics this is a physically reasonable choice Consequently the dimensions of temperature are identical to those of energy However just as torque and work have the same dimensions but are different types of quantities and are measured in different units the meterNewton and the joule respectively so the temperature and the energy should be carefully distinguished The dimensions of both energy and temperature are mass length2time 2 The units of energy are joules ergs calories and the like The units of temperature remain to be discussed In our later discussion of thermodynamic Carnot engines in Chapter 4 we shall find that the optimum performance of an engine in contact Temperature Umts 47 with two thermodynamic systems is completely determined by the ratio of the temperatures of those two systems That is the principles of thermody namics provide an experimental procedure that unambiguously determines the ratio of the temperatures of any two given systems The fact that the ratio of temperatures is measurable has immediate consequences First the zero of temperature is uniquely determined and cannot be arbitrarily assigned or shifted Second we are free to assign the value of unity or some other value to one arbitrary chosen state All other temperatures are thereby determined Equivalently the single arbitrary aspect of the temperature scale is the size of the temperature unit determined by assigning a specific tempera ture to some particular state of a standard system The assignment of different temperature values to standard states leads to different thermodynamic temperature scales but all thermodynamic temperature scales coincide at T 0 Furthermore according to equation 17 no system can have a temperature lower than zero Needless to say this essential positivity of the temperature is in full agreement with all measurements of thermodynamic temperatures The Kelvin scale of temperature which is the official Systeme Interna tional SI system is defined by assigning the number 27316 to the temperature of a mixture of pure ice water and water vapor in mutual equilibrium a state which we show in our later discussion of triple points determines a unique temperature The corresponding unit of temperature is called a kelvin designated by the notation K The ratio of the kelvin and the joule two units with the same dimen sions is 13806 X 10 23 jouleskelvin This ratio is known as Boltzmanns constant and is generally designated as k 8 Thus k 8 T is an energy The Rankine scale is obtained by assigning the temperature X 27316 491688R to the icewaterwater vapor system just referred to The unit denoted by 0 R is called the degree Rankine Rankine tempera tures are merely times the corresponding Kelvin temperature Closely related to the absolute Kelvin scale of temperature is the International Kelvin scale which is a practical scale defined in terms of the properties of particular systems in various temperature ranges and contrived to coincide as closely as possible with the absolute Kelvin scale The practical advantage of the International Kelvin scale is that it provides reproducible laboratory standards for temperature measurement throughout the temperature range However from the thermodynamic point of view it is not a true temperature scale and to the extent that it deviates from the absolute Kelvin scale it will not yield temperature ratios that are consistent with those demanded by the thermodynamic for malism The values of the temperature of everyday experiences are large num bers on both the Kelvin and the Rankine scales Room temperatures are in the region of 300 K or 540R For common usage therefore two 48 The Conditions of Equilibrium derivative scales are in common use The Celsius scale is defined as TC T K 27315 243 where T 0 C denotes the Celsius temperature for which the unit is called the degree Celsius denoted by 0 C The zero of this scale is displaced relative to the true zero of temperature so the Celsius tempera ture scale is not a thermodynamic temperature scale at all Negative temper atures appear the zero is incorrect and ratios of temperatures are not in agreement with thermodynamic principles Only temperature differences are correctly given On the Celsius scale the temperature of the triple point ice water and water vapor in mutual equilibrium is 001 C The Celsius tempera ture of an equilibrium mixture of ice and water maintained at a pressure of 1 atm is even closer to 0C with the difference appearing only in the third decimal place Also the Celsius temperature of boiling water at 1 atm pressure is very nearly 100C These near equalities reveal the historical origin2 of the Celsius scale before it was recognized that the zero of temperature is unique it was thought that two points rather than one could be arbitrarily assigned and these were taken by Anders Celsius in 1742 as the 0C and 100C just described The Fahrenheit scale is a similar practical scale It is now defined by TF T 0 R 45967 T 0 C 32 244 The Fahrenheit temperature of ice and water at 1 atm pressure is roughly 32F the temperature of boiling water at 1 atm pressure is about 212F and room temperatures are in the vicinity of 70F More suggestive of the presumptive origins of this scale are the facts that ice salt and water coexist in equilibrium at 1 atm pressure at a temperature in the vicinity of 0F and that the body ie rectal temperature of a cow is roughly 100F Although we have defined the temperature formally in terms of a partial derivative of the fundamental relation we briefly note the conventional method of introduction of the temperature concept as developed by Kelvin and Caratheodory The heat flux dQ is first defined very much as we have introduced it in connection with the energy conservation princi ple From the consideration of certain cyclic processes it is then inferred that there exists an integrating factor 1T such that the product of this integrating factor with the imperfect differential dQ is a perfect differen tial dS dS dQ 245 2A very short but fascinating review of the history of temperature scales is J by E R Jones Jr The Physics Teacher 18 S94 1980 Mechanical Equi1brium 49 The temperature and the entropy thereby are introduced by analysis of the existence of integrating factors in particular types of differential equations called Pfaffian forms PROBLEMS 261 The temperature of a system composed of ice water and water vapor in mutual equilibrium has a temperature of exactly 27316 K by definition The temperature of a system of ice and water at 1 atm of pressure is then measured as 27315 K with the third and later decimal places uncertain The temperature of a system of water and water vapor ie boiling water at 1 atm is measured as 37315 K 001 K Compute the temperature of waterwater vapor at 1 atm with its probable error on the Celsius absolute Fahrenheit and Fahrenheit scales 262 The gas constant R is defined as the product of Avogadros number NA 60225 X 1023mole and Boltzmanns constant R NAk 8 Correspond ingly R 8314 Jmole K Since the size of the Celsius degree is the same as the size of Kelvin degree it has the value 8314 Jmole 0 C Express R in units of JmoleF 263 Two particular systems have the following equations of state and 1 3 NO R TI 2 u1 1 5 N2 R T2 2 u2 where R is the gas constant Problem 262 The mole number of the first system is NCI 2 and that of the second is N2 3 The two systems are separated by a diathermal wall and the total energy in the composite system is 25 X 103 J What is the internal energy of each system in equilibrium Answer u1 7143 J 264 Two systems with the equations of state given in Problem 263 are separated by a diathermal wall The respective mole numbers are N1l 2 and N2 3 The initial temperatures are T1 250 Kand T2 350 K What are the values of u1 and u2 after equilibrium has been established What is the equilibrium temperature 27 MECHANICAL EQUILIBRIUM A second application of extremum principle for the entropy yields an even simpler result and u1erefore is useful in making the procedure 50 The Conditwns of Equ1bbnum clear We consider a closed composite system consisting of two simple systems separated by a movable diathermal wall that is impervious to the flow of matter The values of the mole numbers are fixed and constant but the values of u1 and u2 can change subject only to the closure condition ul u2 constant 246 and the values of v1 and v2 can change subject only to the closure condition v1 v2 constant 247 The extremum principle requires that no change in entropy result from infinitesimal virtual processes consisting of transfer of heat across the wall or of displacement of the wall Then where dS as1 au1 vo N I k dS 0 dU1 asl av1 u11 NP as2 ducii as2 au2 v2 N2 av2 u2 Nj2 By the closure conditions and dV2 dv 1 whence 248 dV2 249 250 251 dS dU1 dV1 0 l l p O p 2 rl r2 rl r2 252 As this expression must vanish for arbitrary and independent values of du1 and dv1 we must have 1 1 0 ro rc2 253 and pl p2 0 yl y2 Mechanical Equilibrium 51 254 Although these two equations are the equilibrium conditions in the proper form appropriate to the entropy representation we note that they imply the physical conditions of equality of both temperature and pressure yl y2 pl p2 255 256 The equality of the temperatures is just our previous result for equi librium with a diathermal wall The equality of the pressures is the new feature introduced by the fact that the wall is movable Of course the equality of the pressures is precisely the result that we would expect on the basis of mechanics and this result corroborates the identification of the function P as the mechanical pressure Again we stress that this result is a formal solution of the given problem In the entropy representation 1r 1 is a function of u1 vm and N1 an entropic equation of state so that equation 253 is formally a relationship among uI vI u2 and v2 with Nll and N2 each held fixed Similarly plr 1 is a function of uI v1 and N1 so that equation 254 is a second relationship among Ull v1 u2 and V2 The two conservation equations 246 and 247 complete the four equa tions required to determine the four soughtfor variables Again thermo dynamics provides the methodology which becomes explicit when applied to a concrete system with a definite fundamental relation or with known equations of state The case of a moveable adiabatic rather than diathermal wall presents a unique problem with subtleties that are best discussed after the for malism is developed more fully we shall return to that case in Problem 273 and in Problem 512 Example 1 Three cylinders of identical crosssectional areas are fitted with pistons and each contains a gaseous system not necessarily of the same composition The pistons are connected to a rigid bar hinged on a fixed fulcrum as indicated in Fig 21 The moment arms or the distances from the fulcrum are in the ratio of I 2 3 The cylinders rest on a heat conductive table of negligible mass the table makes no contribution to the physics of the problem except to ensure that the three cylinders are in diathermal contact The entire system is isolated and no pressure acts on the external surfaces of the pistons Find the ratio of pressures and of temperatures in the three cylinders 52 The Conditions of Equilibrium FIGURE 21 Three volumecoupled systems Example 271 Solution The closure condition for the total energy is iu1 8U2 iu3 0 and the coupling of the pistons imposes the conditions that iv2 2 iv1 and 8V3 38Vl Then the extremal property of the entropy is is 1 iu1 1 iu2 1 iu3 pu iv1 TI T2 Tl TI pC2 p3 iv 2iv 3 0 Tc2 TJ Eliminating u3 v2 and v3 is 11 iuc1 11 iui TI T3 T2 T3 23 ivuo pI p2 p3 TI T2 T3 The remaining three variations iu1 iu2 and iv1 are arbitrary and uncon strained so that the coefficient of each must vanish separately From the coeffi dent of 8u1 we find T1 T3 and from the coefficient of iu2 we find T2 T3 Hence all three systems come to a common final temperature From the coefficient of 8VI and using the equality of the temperatures we find pI 2p2 3p3 This is the expected result embodying the familiar mechanical principle of th lever Explicit knowledge of the equations of state would le us to convert this into a solution for the volumes of the three systems Problems 53 PROBLEMS 271 Three cylinders are fitted with four pistons as shown in Fig 22 The crosssectional areas of the cylinders are in the ratio A 1 A 2 A 3 1 2 3 Pairs of pistons are coupled so that their displacements linear motions are equal The walls of the cylinders are diathermal and are connected by a heat conducting bar crosshatched in the figure The entire system is isolated so that for instance there is no pressure exerted on the outer surfaces of the pistons Find the ratios of pressures in the tiree cylinders 272 Two particular systems have the following equations of state and 1 3 N1 R y 2 ul 1 5 N2 R y2 2 u2 pI NCI R y vo pc2i N2 R ycii vcii The mole number of the first system is N1 05 and that of the second is N2 075 The two systems are contained in a closed cylinder separated by a fixed adiabatic and impermeable piston The initial temperatures are yCI 200 K and Y 2 300 K and the total volume is 20 liters The setscrew which prevents the motion of the piston is then removed and simultaneously the adiabatic insulation of the piston is stripped off so that the piston becomes moveable diathermal and impermeable What is the energy volume pressure and temperature of each subsystem when equilibrium is established It is sufficient to take R 83 J mole K and to assume the external pressure to be zero Answer ljl 1700 J 273 The hypothetical problem of equilibrium in a closed composite system with an internal moveable adiabatic wall is a unique indeterminate problem Physi cally release of the piston would lead it to perpetual oscillation in the absence of viscous damping With visco imping the piston would eventually come to rest at such a position that the pressures on either side would be equal but the 54 The Conditions of Eqwhbnum temperatures in each subsystem would then depend on the relative viscosity in each subsystem The solution of this problem depends on dynamical considera tions Show that the application of the entropy maximum formalism is corre spondingly indeterminate with respect to the temperatures but determinate with respect to the pressures Hint First show that with du1 pldv1 and similarly for subsystem 2 energy conservation gives p1 p2 Then show that the entropy maximum condition vanishes identically giving no solution for r1 or T2 28 EQUILIBRIUM WITH RESPECT TO MA TIER FLOW Consideration of the flow of matter provides insight into the nature of the chemical potential We consider the equilibrium state of two simple systems connected by a rigid and diathermal wall permeable to one type of material N 1 and impermeable to aU others N 2 N3 N We seek the equilibrium values of u1 and u2 and of N11 and N The virtual change in entropy in the appropriate virtual process is 1 1 1 2 dS du 1 L dNI du 2 L dN2 257 rl rl i r2 r2 i and the closure conditions demand dU2 du 1 258 and dN2 dN 1 I l 259 whence dS 11 dUl µ11 µ2 dNI r1 r2 Tl T2 I 260 As dS must vanish for arbitrary values of both dU1 and dNp we find as the conditions of equilibrium 1 1 261 rI r2 and µI µ TI r2 hence also µ1 µf 262 Problems 55 Thus just as the temperature can be looked upon as a sort of potential for heat flux and the pressure can be looked upon as a sort of potential for volume changes so the chemical potential can be looked upon as a sort of potential for matter flux A difference in chemical potential provides a generalized force for matter flow The direction of the matter flow can be analyzed by the same method used in Section 25 to analyze the direction of the heat flow If we assume that the temperatures T1 and T2 are equal equation 260 becomes 2 1 dS Pi Pi dN1 T i 263 If µ1 1 is greater than µ dNp will be negative since dS must be positive Thus matter tends to flow from regions of high chemical poten tial to regions of low chemical potential In later chapters we shall see that the chemical potential provides the generalized force not only for the flow of matter from point to point but also for its changes of phase and for chemical reactions The chemical potential thus plays a dominant role in theoretical chemistry The units of chemical potential are joules per mole or any desired energy unit per mole PROBLEMS 281 The fundamental equation of a particular type of twocomponent system is uv 2v N1 N2 S NA NRln N 512 N1R1nN N2R1nN N NI N2 where A is an unspecified constant A closed rigid cylinder of total volume 10 liters is divided into two chambers of equal volume by a diathermal rigid membrane permeable to the first component but impermeable to the second In one chamber is placed a sample of the system with original parameters Np 05 Njl 075 v1 5 liters and T1 300 K In the second chamber is placed a sample with original parameters N 1 NP 05 v2 5 liters and T2 250 K After equilibrium is established what are the values of Np NF T po and p2 Answer T 2727 K 282 A twocomponent gaseous system has a fundamental equation of the form s Au113v113N113 BilN2 N N1 N2 56 The Cond111ons of Eqwhbnum where A and B are positive constants A closed cylinder of total volume 2V0 is separated into two equal subvolumes by a rigid diathermal partition permeable only to the first component One mole of the first component at a temperature T is introduced in the lefthand subvolume and a mixture of mole of each component at a temperature T is introduced into the righthand subvolume Find the equilibrium temperature T and the mole numbers in each subvolume when the system has come to equilibrium assuming that T 27 400 Kand that 37 B2 100A3V0 Neglect the heat capacity of the walls of the container Answer N1t 09 29 CHEMICAL EQUILIBRIUM Systems that can undergo chemical reactions bear a strong formal similarity to the diffusional systems considered in the preceding section Again they are governed by equilibrium conditions expressed in terms of the chemical potential µwhence derives its name chemical potential In a chemical reaction the mole numbers of the system change some increasing at the expense of a decrease in others The relationships among the changing mole numbers are governed by chemical reaction equations such as 264 or 265 The meaning of the first of these equations is that the changes in the mole numbers of hydrogen oxygen and water stand in the ratio of 2 1 2 More generally one writes a chemical reaction equation for a system with r components in the form 266 The v1 are the stoichiometric coefficients 2 1 2 for the reaction of hydrogen and oxygen to form water and the A 1 are the symbols for the chemical components A 1 H 2 A 2 0 2 and A 3 H 20 for the preceding reaction If the reaction is viewed in the reverse sense for instance as the dissociation of water to hydrogen plus oxygen the opposite signs would be assigned to each of the v1 this is a matter of arbitrary choice and only the relative signs of the v1 are significant Chermca Equtfbrtum 5 7 The fundamental equation of the system is S SU V N1 N2 Nr 267 In the course of the chemical reaction both the total energy U and the total volume V remain fixed the system being considered to be enclosed in an adiabatic and rigid reaction vessel This is not the most common boundary condition for chemical reactions which are more often carried out in open vessels free to interchange energy and volume with the ambient atmosphere we shall return to these open boundary conditions in Section 64 The change in entropy in a virtual chemical process is then dS r µ dN jl T 268 However the changes in the mole numbers are proportional to the stoicliometric coefficients v1 Let the factor of proportionality be denoted by dN so that dN r dS T L µ1v1 269 Jl Then the extremum principle dictates that in equilibrium r E µ1 o 1 l 270 If the equations of state of the mixture are known the equilibrium condition 2 70 permits explicit solution for the final mole numbers It is of interest to examine this solution in principle in a slightly richer case If hydrogen oxygen and carbon dioxide are introduced into a vessel the following chemical reactions may occur H 2 0 2 H 20 CO2 H 2 CO H 20 CO 0 2 CO2 In equilibrium we then have µ lµ µ H 2 2 0 2 H 20 271 272 58 The Condtwns of Equlbrmm These constitute two independent equations for the first equation is simply the sum of the two following equations just as the first chemical reaction is the net result of the two succeeding reactions The amounts of hydrogen oxygen and carbon introduced into the system in whatever chemical combinations specify three additional comtraints There are thus five constraints and there are precisely five mole numbers to be found the quantities of H 2 0 2 H 20 CO2 and CO The problem is thereby solved in prmciple As we observed earlier chemical reactions more typically occur in open vessels with only the final pressure and temperature determined The number of variables is then increased by two the energy and the volume but the specification of T and P provides two additional constraints Again the problem is determinate We shall return to a more thorough discussion of chemical reactions in Section 64 For now it is sufficient to stress that the chemical potential plays a role in matter transfer or chemical reactions fully analogous to the role of temperature in heat transfer or pressure in volume transfer PROBLEMS 291 The hydrogenation of propane C 3H 8 to form methane CH 4 proceeds by the reaction C3H 8 2H 2 3CH 4 Find the relationship among the chemical potentials and show that both the problem and the solution are formally identical to Example 1 on mechanical equilibrium 3 SOME FORMAL RELATIONSHIPS AND SAMPLE SYSTEMS 31 THE EULER EQUATION Having seen how the fundamental postulates lead to a solution of the equilihnum problem we now pause to examine in somewhat greater detail the mathematical properties of fundamental equations The homogeneous firstorder property of the fundamental relation permits that equation to be written in a particularly convenient form called the Euler form From the definition of the homogeneous firstorder property we have for any A UIS AX1 AX IUS Xi X 31 Differentiating with respect to A au AX aAS au AX aIX aAS aI aAXJ aI US Xi X 32 or au AXk au AXk X aAS S 11 aAX J US X X 33 This equation is true for any X and in particular for X 1 in which case 60 Some Formal Relatwnships and Sample Systems it takes the form u 34 35 For a simple system in particular we have 36 The relation 35 or 36 is the particularization to thermodynamics of the Euler theorem on homogeneous firstorder forms The foregoing develop ment merely reproduces the standard mathematical derivation We refer to equation 35 or 36 as the Euler relation In the entropy representation the Euler relation takes the form 37 or 38 PROBLEMS 311 Write each of the five physically acceptable fundamental equations of Problem 1 I 01 in the Euler form 32 THE GIBBSDUHEM RELATION In Chapter 2 we arrived at equilibrium criteria involving the tempera ture pressure and chemical potentials Each of the intensive parameters entered the theory in a similar way and the formalism is in fact symmetric in the several intensive parameters Despite this symmetry however the reader is apt to feel an intuitive response to tlte concepts of temperature and pressure which is lacking at least to some degree in the case of the chemical potential It is of interest then to note that the intensive parameters are not all independent There is a relation among The GbbsDuhem Relatwn 61 the intensive parameters and for a singlecomponent system µ is a function of T and P The existence of a relationship among the various intensive parameters is a consequence of the homogeneous firstorder property of the funda mental relation For a singlecomponent system this property permits the fundamental relation to be written in the form u us v as in equation 219 each of the three intensive parameters is then also a function of s and v Elimination of s and v from among the three equations of state yields a relation among T P and µ The argument can easily be extended to the more general case and it again consists of a straightforward counting of variables Suppose we have a fundamental equation in t 1 extensive variables 39 yielding in turn t 1 equations of state 310 If we choose the parameter A of equation 214 as A 1X we then have 311 Thus each of the t 1 intensive parameters is a function of just t variables Elimination of these t variables among the t 1 equations yields the desired relation among the intensive parameters To find the explicit functional relationship that exists among the set of intensive parameters would require knowledge of the explicit fundamental equation of the system That is the analytic form of the relationship varies from system to system Given the fundamental relation the procedure is evident and follows the sequence of steps indicated by equations 39 through 311 A differential form of the relation among the intensive parameters can be obtained directly from the Euler relation and is known as the Gibbs Duhem relation Taking the infinitesimal variation of equation 35 we find I I du T dS s dT L pl dXJ L X dPJ 312 1l 1l But in accordance with equation 26 we certainly know that I dU T dS L dXJ 313 1l 62 Some Formal Relat1onsh1ps and Sample Sstems whence by subtraction we find the Gibbs Duhem relation t s dT x1 d o 314 1I For a singlecomponent simple system in particular we have S dT V dP Ndµ 0 315 or dµ sdT vdP 316 The variation in chemical potential is not independent of the variations in temperature and pressure but the variation of any one can be computed in terms of the variations of the other two The GibbsDuhem relation presents the relationship among the inten sive parameters in d1ff erential form Integration of this equation yields the relation in explicit form and this is a procedure alternative to that presented in equations 39 through 311 In order to integrate the GibbsDuhem relation one must know the equations of state that enable one to write the X1 s in terms of the P1 s or vice versa The number of intensive parameters capable of independent variation is called the number of thermodynamic degrees of freedom of a given system A simple system of r components has r I thermodynamic degrees of freedom In the entropy representation the GibbsDuhem relation again states that the sum of products of the extensive parameters and the differentials of the corresponding intensive parameters vanishes 317 or 318 PROBLEMS 321 Find the relation among T P and µ for the system with the fundamental equation U v58 R 3 NV 2 Summary of Formal Strullure 63 33 SUMMARY OF FORMAL STRUCTURE Let us now summarize the structure of the thermodynamic formalism in the energy representation For the sake of clarity and in order to be explicit we consider a singlecomponent simple system The fundamental equation U US V N 319 contains all thermodynamic information about a system With the defini tions T au I as and so forth the fundamental equation implies three equations of state T T S V N T s v P PSVN Psv µ µS V N µsv 320 321 322 If all three equations of state are known they may be substituted into the Euler relation thereby recovering the fundamental equation Thus the totality of all three equations of state is equivalent to the fundamental equation and contains all thermodynamic information about a system Any single equation of state contains less thermodynamic information than the fundamental equation If two equations of state are known the Gibbs Duhem relation can be integrated to obtain the third The equation of state so obtained will contain an undetermined integration constant Thus two equations of state are sufficient to determine the fundamental equation except for an undetermined constant A logically equivalent but more direct and generally more convenient method of obtaining the fundamental equation when two equations of state are given is by direct integration of the molar relation du Tds Pdv 323 Clearly knowledge of T Ts v and P Ps v yields a differential equation in the three variables u s and v and integration gives u usv 324 which is a fundamental equation Again of course we have an unde termined constant of integration It is always possible to express the internal energy as a function of parameters other than S V and N Thus we could eliminate S from U US V N and T TS V N to obtain an equation of the form U U T V N However I stress that such an equation is not a funda mental relation and does not contain all possible thermodynamic informa 64 Some Formal Relatwnshps and Sample System u a FIGURE 31 I I I I I I I I I I I I I I I I I I I I I I I s b T u as tion about the system In fact recalling the definition of T as au as we see that U UT V N actually is a partial differential equation Even if this equation were integrable it would yield a fundamental equation with undetermined functions Thus knowledge of the relation U US V N allows one to compute the relation U U T V N but knowledge of U UT V N does not permit one inversely to compute U US V N Associated with every equation there is both a truth value and an informational content Each of the equations U US V N and U U T V N may be true but only the former has the optimum informational content These statements are graphically evident if we focus for instance on the dependence of U on Sat constant V and N Let that dependence be as shown in the solid curve in Fig 3la This curve uniquely determines the dependence of U on T shown in Fig 3lb for each point on the U S curve there is a definite u and a definite slope T au I as determining a point on the U T curve Suppose however that we are given the UT curve an equation of state and we seek to recover the fundamental US curve Each of the dotted curves in Fig 3la is equally compatible with the given U T curve for all have the same slope T at a given U The curves differ by an arbitrary displacement corre sponding to the arbitrary constant of integration in the solution of the differential equation U UaUaS Thus Fig 3la implies Fig 3lb but the reverse is not true Equivalently stated only U US is a fundamental relation The formal structure is illustrated by consideration of several specific and explicit systems in the following Sections of this book Example A particular system obeys the equations U tPV and AU32 T2 VN112 where A is a positive constant Find the fundamental equation Solution Problem Writing the two equations in the form of equations of state in the entropy representation which is suggested by the appearance of U V and N as independent parameters I A112u3f4vlf2 T P ZA lf2ulf4v 12 T Then the differential form of the molar fundamental equation the analogue of equation 323 is so that and I p dsdudv T T A 12 u 3f4vlf2 du 2ulf4v 12 dv 4A 112d ulf4vlf2 s 4A lf2ulf4vlf2 so S 4A 112u114v112Ntf4 Nso The reader should compare this method with the alternative technique of first integrating the GibbsDuhem relation to obtain µu v and then inserting the three equations of state into the Euler equation Particular note should be taken of the manner in which ds is integrated to obtain s The equation for ds in terms of du and dv is a partial differential equationit certainly cannot be integrated term by term nor by any of the familiar methods for ordinary differential equations in one independent variable We have integrated the equation by inspection simply recognizing that u 314vi12 du 2u 1i 4v 112 dv is the differential of u1i 4v1i 2 PROBLEMS 331 A particular system obeys the two equations of state 3As 2 T the thermal equation of state V 66 Some Formal Relationships and Sample SystenLf and As 3 P V 2 where A is constant the mechanical equation of state a Find µ as a function of s and v and then find the fundamental equation b Find the fundamental equation of this system by direct integration of the molar form of the equation 332 It is found that a particular system obeys the relations UPV and P BT 2 where B is constant Find the fundamental equation of this system 333 A system obeys the equations NU p NV 2AVU and U 112v112 T 2C AUN N 2AUe Find the fundamental equation Hint To integrate let where D n and mare constants to be determined 334 A system obeys the two equations u iPv and u1 2 BTv 113 Find the fundamental equation of this system 34 THE SIMPLE IDEAL GAS AND MULTI COMPONENT SIMPLE IDEAL GASES A simple ideal gas is characterized by the two equations PV NRT 325 and U cNRT 326 where c is a constant and R is the universal gas constant R NAk 8 83144 Jmole K Gases composed of noninteracting monatornic atoms such as He Ar Ne are observed to satisfy equations 325 and 326 at temperatures such that k 8 T is small compared to electronic excitation energies ie T 10 4 K and at low or moderate pressures All such monatomic ideal gases have a value of c 1 The Simple Ideal Gas and Multicomponent Simple Ideal Gases 67 Under somewhat more restrictive conditions of temperature and pres sure other real gases may conform to the simple ideal gas equations 325 and 326 but with other values of the constant c For diatomic molecules such as 0 2 or NO there tends to be a considerable region of temperature for which c c and another region of higher temperature for which c c 1 with the boundary between these regions generally occurring at tempera tures on the order of 103 K Equations 325 and 326 permit us to determine the fundamental equation The explicit appearance of the energy U in one equation of state equation 326 suggests the entropy representation Rewriting the equa tions in the correspondingly appropriate form 327 and 328 From these two entropic equations of state we find the third equation of state 1 function of u v 329 by integration of the GibbsDuhem relation 330 Finally the three equations of state will be substituted into the Euler equation S uv1N 331 Proceeding in this way the GibbsDuhem relation 330 becomes d 1 u X du v X dv cR d R d 332 and integrating cRln Rln µ µ U V T T o u0 v0 333 Here u0 and v0 are the parameters of a fixed reference state and µT 0 arises as an undetermined constant of integration Then from the Euler 68 Some Formal Rela1tonsh1ps and Sample Systems relation 331 U c V N cll S Nso NRn Uo Vo No 334 where So C R t 335 Equation 334 is the desired fundamental equation if the integration constant s0 were known equation 334 would contain all possible thermo dynamic information about a simple ideal gas This procedure is neither the sole method nor even the preferred method Alternatively and more directly we could integrate the molar equation 336 which in the present case becomes ds c du dv 337 giving on integration s s0 cR In J R In J 338 This equation is equivalent to equation 334 It should perhaps be noted that equation 337 is integrable term by term despite our injunction in Example 3 that such an approach generally is not possible The segregation of the independent variables u and v in separate terms in equation 337 is a fortunate but unusual simplification which permits term by term integration in this special case A mixture of two or more simple ideal gasesa multicomponent simple ideal gas is characterized by a fundamental equation which is most simply written in parametric form with the temperature T playing the role of the parametric variable 339 The Simple Ideal Gas and Multicomponent Simple Ideal Gases 69 Elimination of T between these equations gives a single equation of the standard form S S U V N1 Ni Comparison of the individual terms of equations 339 with the expres sion for the entropy of a singlecomponent ideal gas leads to the following interpretation often referred to as Gibbss Theorem The entropy of a mixture of ideal gases is the sum of the entropies that each gas would have if it alone were to occupy the volume V at temperature T The theorem is in fact true for all ideal gases Chapter 13 It is also of interest to note that the first of equations 339 can be written in the form 340 and the last term is known as the entropy of mixing It represents the difference in entropies between that of a mixture of gases and that of a collection of separate gases each at the same temperature and the same density as the original mixture N V and hence at the same pressure as the original mixture see Problem 3415 The close similarity and the important distinction between Gibbss theorem and the interpre tation of the entropy of mixing of ideal gases should be noted carefully by the reader An application of the entropy of mixing to the problem of isotope separation will be given in Section 44 Example 4 Gibbss theorem is demonstrated very neatly by a simple thought experiment A cylinder Fig 32 of total volume 2V0 is divided into four chambers designated as a 3 y by a fixed wall in the center and by two sliding walls The two sliding walls are coupled together so that their distance apart is always one half the length of the cylinder V0 VY and Vp Vii Initially the two sliding walls are coincident with the left end and the central fixed partition respectively so that Va Vv 0 The chamber 3 of volume V0 is filled with a mixture of N0 moles of a simple ideal gas A and N0 moles of a simple ideal gas B Chamber S is initially evacuated The entire system is maintained at temperature T The lefthand sliding wall is permeable to component A but not to component B The fixed partition is permeable to component B but not to component A The righthand sliding wall is impermeable to either component The coupled sliding walls are then pushed quasistatically to the right until Vp Vii 0 and Va V V0 Chamber a then contains pure A and chamber y contains pur B The initial mixture of volume V0 thereby is separated into two pure components each of volume V0 According to Gibbss theorem the final entropy should be equal to the initial entropy and we shall now see directly that this is in fact true 70 Some Formal Rela11onsh1ps and Sample Systems I AB I 3 I I Coupling bar A a Vacuum ll Coupling bar FIGURE 32 Separation of a mixture of ideal gases demonstrating Gibbss theorem We first note that the second of equations 339 stating that the energy is a function of only T and the mole number ensures that the final energy is equal to the initial energy of the system Thus Tl1S is equal to the work done in moving the coupled walls The condition of equilibrium with respect to transfer of component A across the lefthand wall is µAa µAP It is left to Problem 3414 to show that the conditions µAa µAP and µ 8p µ 8Y imply that That is the total force on the coupled moveable walls P 0 Pµ Py vanishes Thus no work is done in moving the walls and consequently no entropy change accompanies the process The entropy of the original mixture of A and B in a common volume V0 is precisely equal to the entropy of pure A and pure B each in a separate volume VoThis is Gibbss theorem Finally we note that the simple ideal gas considered in this section is a special case of the general ideal gas which encompasses a very wide class Problems 71 of real gases at low or moderate pressures The general ideal gas is again characterized by the mechanical equation of state PV NRT equation 325 and by an energy that again is a function of the temperature onlybut not simply a linear function The general ideal gas will be discussed in detail in Chapter 13 and statistical mechanical derivations of the fundamental equations will emerge in Chapter 16 PROBLEMS Note that Problems 341 342 343 and 348 refer to quasistatic processes such processes are to be interpreted not as real processes but merely as loci of equilibrium states Thus we can apply thermodynamics to such quasistatic processes the work done in a quasistatic change of volume from V1 to V2 is W f PdV and the heat transfer is Q JTdS The relationship of real processes to these idealized quasistatic processes will be discussed in Chapter 4 341 A constant volume ideal gas thermometer is constructed as shown schematically in Fig 33 The bulb containing the gas is constructed of a material with a negligibly small coefficient of thermal expansion The point A is a reference point marked on the stem of the bulb The bulb is connected by a flexible tube to a reservoir of liquid mercury open to the atmosphere The mercury reservoir is raised or lowered until the mercury miniscus coincides with the reference point A The height h of the mercury column is then read a Show that the pressure of the gas is the sum of the external atmospheric pressure plus the height h of the mercury column multiplied by the weight per unit volume of mercury as measured at the temperature of interest b Using the equation of state of the ideal gas explain how the temperature of the gas is then evaluated A T l Hg FIGURE33 Constantvolume ideal gas thermometer 72 Some Formal Relatwmhtps and Sample Sytems c Describe a constant pressure ideal gas thermometer in which a changing volume is directly measured at constant pressure 342 Show that the relation between the volume and the pressure of a mon atomic ideal gas undergoing a quasistatic adiabatic compression dQ T dS 0 S constant is Pv 513 P0vf 3e lso 3R e 2sJR constant Sketch a family of such adiabats in a graph of P versus V F md the corresponding relation for a simple ideal gas 343 Two moles of a monatom1c ideal gas are at a temperature of 0C and a volume of 45 liters The gas is expanded adiabatically dQ 0 and quasistati cally until its temperature falls to 50C What are its imtial and final pressures and its final volume Answer P 01 MPa 61 X 10 3 m3 344 By carrying out the integral f P dV compute the work done by the gas m Problem 343 Also compute the initial and final energies and corroborate that the difference in these energies is the work done 345 In a particular engine a gas is compressed in the initial stroke of the piston Measurements of the instantaneous temperature carried out during the compres sion reveal that the temperature increases accordmg to where T0 and V0 are the initial temperature and volume and 1 is a constant The gas is compressed to the volume V1 where V1 V0 Assume the gas to be monatomic ideal and assume the process to be quas1stat1c a Calculate the work W done on the gas b Calculate the change m energy AU of the gas c Calculate the heat transfer Q to the gas through the cylinder walls by using the results of a and b d Calculate the heat transfer directly by integratmg dQ T dS e From the result of c or d for what value of 1 is Q O Show that for this value of 1 the locus traversed coincides with an adiabat as calculated in Problem 342 346 Find the three equations of state of the simple ideal gas equation 334 Show that these equations of state satisfy the Euler relation 347 Find the fur equations of state of a twocomponent mixture of simple ideal gases equations 339 Show that these equations of state satisfy the Euler relation 11oherm 73 348 If a monatomic ideal gas is permitted to expand into an evacuated region thereby increasing its volume from V to AV and if the walls are rigid and adiabatic what is the ratio of the initial and final pressures What is the ratio of the inttial and final temperatures What is the difference of the inttial and final entropies 349 A tank has a volume of 01 m3 and 1s filled with He gas at a pressure of 5 X 106 Pa A second tank has a volume of 015 m3 and is filled with He gas at a pressure of 6 X 106 Pa A valve connecting the two tanks is opened Assuming He to be a monatomic ideal gas and the walls of the tanks to be adiabatic and ngd find the final pressure of the system Hmt Note that the internal energy is constant Answer P1 56 X 106 Pa 3410 a If the temperatures within the two tanks of Problem 349 before opening the valve had been T 300 K and 350 K respectively what would the final temperature be b If the first tank had contained He at an initial temperature of 300 K and the second had contained a diatomic ideal gas with c 52 and an initial tempera ture of 350 K what would the final temperature be Answer a 330 K b 7t 337 K 3411 Show that the pressure of a multicomponent simple ideal gas can be written as the sum of partial pressures where RT V Theie partial pressures are purely formal quantities not subject to experimental observation From the mechanistic viewpoint of kinetic theory the partial pressure P is the contribution to the total pressure that result from bombardment of the wall by molecules of species a distinction that can be made only when the molecules are nonmteracting as in an ideal gas 3412 Show that µ1 the electrochemical potential of the 1th component in a multicomponent simple ideal gas satisfies Nv 0 µ1 RTln V function of T and find the explicit form of the function of T Show that µ can be expressed m terms of the partial pressure Problem 3411 and the temperature 3413 An impermeable diathermal and rigid partition divides a container into two subvolumes each of volume V The subvolumes contam respectively one 74 Some Formal Relattonsh1ps and Sample Systems mole of Hz and three moles of Ne The system is maintained at constant temperature T The partition is suddenly made permeable to H2 but not to Ne and equilibrium is allowed to reestablish Find the mole numbers and the pressures 3414 Use the results of Problems 3411 and 3412 to estabfoh the results P0 Pr and Pp 2P0 in the demonstration of Gibbss theorem at the end of this section 3415 An impermeable diathermal and rigid partition divides a container into two subvolumes of volumes n V0 and m V0 The subvolumes contain respectively n moles of Hz and m moles of Ne each to be considered as a simple ideal gas The system is maintained at constant temperature T The partition 1s suddenly ruptured and equilibrium is allowed to reestablish Find the initial pressure in each subvolume and the final pressure Find the change in entropy of the system How is this result related to the entropy of mixing the last term in equation 340 35 THE IDEAL VAN DER WAALS FLUID Real gases seldom satisfy the ideal gas equation of state except in the limit of low density An improvement on the mechanical equation of state 328 was suggested by J D van der Waals in 1873 p RT V b V2 341 Here a and b are two empirical constants characteristic of the particular gas In strictly quantitative terms the success of the equation has been modest and for detailed practical applications it has been supplanted by more complicated empirical equations with five or more empirical con stants Nevertheless the van der Waals equation is remarkably successful in representing the qualitative features of real fluids including the gasliquid phase transition The heuristic reasoning that underlies the van der Waals equation is intuitively plausible and informative although that reasoning lies outside the domain of thermodynamics The ideal gas equation P RT v is known to follow from a model of point molecules moving independently and colliding with the walls to exert the pressure P Two simple correc tions to this picture are plausible The first correction recognizes that the molecules are not point particles but that each has a nonzero volume bNA Accordingly the volume Vin the ideal gas equation is replaced by V Nb the total volume diminished by the volume Nb occupied by the molecules themselves The second correction arises from the existence of forces between the molecules A molecule in the interior of the vessel is acted upon by The Ideal mn der Waals Fluid 7 5 intermolecular forces in all directions which thereby tend to cancel But a molecule approaching the wall of the contamer experiences a net back ward attraction lo the remaining molecules and this force in turn reduces the effective pressure that the molecule exerts on colliding with the container wall This diminution of the pressure should be proportional to the number of interacting pairs of molecules or upon the square of the number of molecules per unit volume lv 2 hence the second term in the van der Waals equation Statistical mechanics provides a more quantitative and formal deriva tion of the van der Waals equation but it also reveals that there are an infinite series of higher order corrections beyond those given in equation 341 The truncation of the higher order terms lo give the simple van der Waals equation results in an equation with appropriate qualitative fea tures and with reasonable but not optimum quantitative accuracy The van der Waals equation must be supplemented with a thermal equation of state in order lo define the system fully It is instructive not simply to appeal lo experiment but rather to inquire as lo the simplest possible and reasonable thermal equation of stale that can be paired with the van der Waals equation of state Unfortunately we are not free simply lo adopt the thermal equation of state of an ideal gas for thermodynamic formalism imposes a consistency condition between the two equations of state We shall be forced to alter the ideal gas equation slightly We write the van der Waals equation as P R a 1 T v b v2 T 342 and the sought for additional equation of state should be of the form 1 T fu v 343 These two equations would permit us to integrate the molar equation 1 p ds du dv T T 344 to obtain the fundamental equation However if ds is to be a perfect differential it is required that the mixed secondorder partial derivatives should be equal a2s a2s av au au av 345 76 Some Formal Relationships and Sample Systems or 346 whence 347 This condition can be written as 348 That is the function 1T must depend on the two variables 1v and ua in such a way that the two derivatives are equal One possible way of accomplishing this is to have 1T depend only on the sum 1v ua We first recall that for a simple ideal gas 1T cRu this suggests that the simplest possible change consistent with the van der Waals equation is 1 cR T u av 349 For purposes of illustration throughout this text we shall refer to the hypothetical system characterized by the van der Waals equation of state 341 and by equation 349 as the ideal van der Waals fluid We should note that equation 341 although referred to as the van der Waals equation of state is not in the appropriate form of an equation of state However from equations 349 and 342 we obtain p R acR 350 T v b uv2 av The two preceding equations are the proper equations of state in the entropy representation expressing 1T and P T as functions of u and v With the two equations of state we are now able to obtain the fundamental relation It is left to the reader to show that S NRlnv bu avr Ns 0 351 where s0 is a constant As in the case of the ideal gas the fundamental Problems 77 TABLE31 Van der Waals Constants and Molar Heat Capacities of Common Gasesu Gas aPam 6 bJ06m3 C He 000346 237 15 Ne 00215 171 15 H2 00248 266 25 A 0132 302 15 N2 0136 385 25 02 0138 326 25 co 0151 399 25 CO2 0401 427 35 N20 0384 442 35 H20 0544 305 31 CI2 0659 563 28 S0 2 0680 564 35 0 Adapted from Paul S Epstem Textbook of Thermodynamics Wiley New York 1937 equation does not satisfy the Nernst theorem and it cannot be valid at very low temperatures We shall see later in Chapter 9 that the ideal van der Waals fluid is unstable in certain regions of temperature and pressure and that it spontaneously separates into two phases liquid and gas The funda mental equation 351 is a very rich one for the illustration of thermody namic principles The van der Waals constants for various real gases are given in Table 31 The constants a and b are obtained by empirical curve fitting to the van der Waals isotherms in the vicinity of 273 K they represent more distant isotherms less satisfactorily The values of c are based on the molar heat capacities at room temperatures PROBLEMS 351 Are each of the listed pairs of equations of state compatible recall equation 346 If so find the fundamental equation of the system a u aPv and Pv 2 bT b u aPv 2 and Pv 2 bT c p c buv and T u v a buv a buv 352 Find the relationship between the volume and the temperature of an ideal van der Waals fluid in a quasistatic adiabatic expansion ie in an isentropic expansion with dQ T dS 0 or S constant 78 Some Formal Relatwnsh1ps and Sample Systems 353 Repeat Problem 343 for CO2 rather than for a monatomic ideal gas Assume CO2 can be represented by an ideal van der Waals fluid with constants as given in Table 31 At what approximate pressure would the term a v2 in the van der Waals equation of state make a 10 correction to the pressure at room temperature Answer vi 0091 m3 354 Repeat parts a b and c of problem 345 assuming that 1J and that the gas is an ideal van der Waals fluid Show that your results for U and for W and hence for Q reduce to the results of Problem 345 for 1J as the van der Waals constants a and b go to zero and c f Recall that lnl x x for small x 355 Consider a van der Waals gas contained in the apparatus described in Problem 341 ie in the constant volume gas thermometer a Assuming it to be known in advance that the gas obeys a van der Waals equation of state show that knowledge of two reference temperatures enables one to evaluate the van der Waals constants a and b b Knowing the constants a and b show that the apparatus can then be used as a thermometer to measure any other temperature c Show that knowledge of three reference temperatures enables one to determine whether a gas satisfies the van der Waals equation of state and if it does enables one to measure any other temperature 356 One mole of a monatomic ideal gas and one mole of Cl 2 are contained in a rigid cylinder and are separated by a moveable internal piston If the gases are at a temperature of 300 K the piston is observed to be precisely in the center of the cylinder Find the pressure of each gas Treat Cl 2 as a van der Waals gas see Table 31 Answer P 35 X 107 Pa 36 ELECTROMAGNETIC RADIATION If the walls of any empty vessel are maintained at a temperature Tit is found that the vessel is in fact the repository of electromagnetic energy The quantum theorist might consider the vessel as containing photons the engineer might view the vessel as a resonant cavity support ing electromagnetic modes whereas the classical thermodynamicist might eschew any such mechanistic models From any viewpoint the empir ical equations of state of such an electromagnetic cavity are the StefanBoltzmann Law U bVT 4 352 Problems 79 and 353 where b is a particular constant b 756 X 10 16 Jm 3 K4 which will be evaluated from basic principles in Section 168 It will be noted that these empirical equations of state are functions of U and V but not of N This observation calls our attention to the fact that in the empty cavity there exist no conserved particles to be counted by a parameter N The electromagnetic radiation within the cavity is governed by a fundamental equation of the form S S U V in which there are only two rather than three independent extensive parameters For electromagnetic radiation the two known equations of state con stitute a complete set which need only be substituted in the truncated Euler relation 1 SUV p T T 354 to provide a fundamental relation For this purpose we rewrite equations 352 and 353 in the appropriate form of entropic equations of state bl4 r 14 355 and p bl4 ur4 T 3 V 356 so that the fundamental relation becomes on substitution into 354 357 PROBLEMS 361 The universe is considered by cosmologists to be an expanding electromag netic cavity containing radiation that now is at a temperature of 27 K What will be the temperature of the radiation when the volume of the universe is twice its present value Assume the expansion to be isentropic this being a nonobvious prediction of cosmological model calculations 362 Assuming the electromagnetic radiation filling the universe to be in equi librium at T 2 7 K what is the pressure associated with this radiation Express the answer both in pascals and in atmospheres 80 Some Formal Relatwnsh1ps and Sample S1tem1 363 The density of matter primarily hydrogen atoms in intergalactic space is such that its contribution to the pressure is of the order of 10 23 Pa a What is the approximate density of matter in atomsm 3 m intergalactic space b What is the ratio of the kinetic energy of matter to the energy of radiation in intergalactic space Recall Problems 361 and 362 c What is the ratio of the total matter energy ie the sum of the kinetic energy plus the relativistic energy mc2 to the energy of radiation in intergalactic space 37 THE RUBBER BAND A somewhat different utility of the thermodynamic formalism is il lustrated by consideration of the physical properties of a rubber band thermodynamics constrains and guides the construction of simple phe nomenological models for physical systems Let us suppose that we are interested in building a descriptive model for the properties of a rubber band The rubber band consists of a bundle of longchain polymer molecules The quantities of macroscopic interest are the length L the tension fr the temperature T and the energy U of the rubber band The length plays a role analogous to the volume and the tension plays a role analogous to the negative pressure fr P An analogue of the mole number might be associated with the number of monomer units in the rubber band but that number is not generally variable and it can be taken here as constant and suppressed in the analysis A qualitative representation of experimental observations can be sum marized in two properties First at constant length the tension increases with the temperaturea rather startling property which is in striking contrast to the behavior of a stretched metallic wire Second the energy is observed to be essentially independent of the length at least for lengths shorter than the elastic limit of the rubber band a length corresponding to the unkinking or straightening of the polymer chains The simplest representation of the latter observation would be the equation 358 where c is a constant and L0 also constant is the unstretched length of the rubber band The linearity of the length with tension between the unstretched length L0 and the elastic limit length L 1 is represented by LL fr bT L Lo L 0 L L 1 359 1 where b is a constant The insertion of the factor T in this equation rather than T 2 or some othr function of T is dictated by the thermody Unrnmtramahe Vurwbles Mug11et1c Sistem 81 namic condition of consistency of the two equations of state That is as in equation 346 a L a 360 which dictates the linear factor T in equation 359 Then 1 Y dU L L 0 dSTdUTdLcL 0 UbL L dL l 0 361 and the fundamental equation correspondingly is 362 Although this fundamental equation has been constructed on the basis only of the most qualitative of information it does represent empirical properties reasonably and most important consistently The model il lustrates the manner in which thermodynamics guides the scientist in elementary model building A somewhat more sophisticated model of polymer elasticity will be derived by statistical mechanical methods m Chapter 15 PROBLEMS 371 For the rubber band model calculate the fractional change m L L0 that results from an increase lT m temperature at constant tension Expres the result m terms of the length and the temperature 372 A rubber band is stretched by an amount dl at contant T Calcultte the heat transfer dQ to the rubber band Also calculate the work done How are these related and why 373 If the energy of the unstretched rubber band were found to increase qua dratically with T so that equation 358 were to be replaced by UcL 0P would equation 359 require alteration Again find the fundamental equation of the rubber band 38 UNCONSTRAINABIE VARIABLES MAGNETIC SYSTEMS In the precedmg sections we have seen examples of several specific systems emphasizing the great d1verity of types of system to whteh thermodynamics applies and illustrating the constramts on analytic mod 82 Some Formal Rela11onsh1ps and Sample Systems eling of simple systems In this section give an example of a magnetic system Here we have an additional purpose for although the general structure of thermodynamics is represented by the examples already given particular idiosyncrasies are associated with certain thermodynamic parmeters Magnetic systems are particularly prone to such individual peculiarities and they well illustrate the special considerations that occa sionally are required In order to ensure magnetic homogeneity we focus attention on el lipsoidal samples in homogeneous external fields with one symmetry axis of the sample parallel to the external field For simplicity we assume no magnetocrystalline anisotropy or if such exists that the easy axis lies parallel to the external field Furthermore we initially consider only paramagnetic or diamagnetic systemsthat is systems in which the magnetization vanishes in the absence of an externally imposed magnetic field In our eventual consideration of phase transitions we shall include the transition to the ferromagnetic phase in which the system develops a spontaneous magnetization As shown in Appendix B the extensive parameter that characterizes the magnetic state is the magnetic dipole moment I of the system The fundamental equation of the system is of the form U US V I N In the more general case of an ellipsoidal sample that is not coaxial with the external field the single parameter I would be replaced by the three cartesian coordinates of the magnetic moment US V Ix Iy 2 N The thermodynamic structure of the problem is most conveniently illustrated in the oneparameter case The intensive parameter conjugate to the magnetic moment I is B the external magnetic field that would exist in the absence of the system B iJU e i SVN 363 The unit of Be is the tesla T and the units of I are Joulesffesla Jff It is necessary to note a subtlety of definition implicit in these identifi cations of extensive and intensive parameters see Appendix B The energy U is here construed as the energy of the material system alone in addition the vacuum occupied by the system must be assigned an energy JIABV where µ0 the permeability of free space has the value µ 0 417 X 10 7 teslametersampere Thus the total energy within the spatial region occupied by a system is U µABV Whether the vacuum term in the energy is associated with the system or is treated separately as we do is a matter of arbitrary choice but considerable confusion can arise if different conventions are not carefully distinguished To repeat the energy U is the change in energy within a particular region in the field when the material system is introduced it excludes the energy µiBV of the region prior to the introduction of the system Unconstrainable Variables Magnetic Systems 83 The Euler relation for a magnetic system is now U TS PV Bel µN 364 and the GibbsDuhem relation is S dT V dP I dBe Ndµ 0 365 An idiosyncrasy of magnetic systems becomes evident if we attempt to consider problems analogous to those of Sections 27 and 28namely the condition of equilibrium of two subsystems following the removal of a constraint We soon discover that we do not have the capability of constraining the magnetic moment in practice the magnetic moment is always unconstrained We can specify and control the magnetic field applied to a sample Uust as we can control the pressure and we thereby can bring about a desired value of the magnetic moment We can even hold that value of the magnetic moment constant by monitoring its value and by continually adjusting the magnetic fieldagain just as we might keep the volume of a system constant by a feedback mechanism that continually adjusts the external pressure But that is very different from simply enclosing the system in a restrictive wall There exist no walls restrictive with respect to magnetic moment Despite the fact that the magnetic moment is an unconstrainable variable the overall structure of thermodynamic theory still applies The fundamental equation the equations of state the GibbsDuhem and the Euler relations maintain their mutual relationships The nonavailability of walls restrictive to magnetic moment can be viewed as a mere experi mental quirk that does not significantly influence the applicability of thermodynamic theory Finally to anchor the discussion of magnetic systems in an explicit example the fundamental equation of a simple paramagnetic model system is S 2 U NRToexp NR Nill 366 where T0 and 0 are positive constants This model does not describe any particular known systemit is devised to provide a simple tractable model on which examples and problems can be based and to illustrate characteristic thermomagnetic interactions We shall leave it to the prob lems to explore some of these properties With the magnetic case always in mind as a prototype for generaliza tions we return to explicit consideration of simple systems 84 Some Formal Relatonshzpf and Sample Systems PROBLEMS 381 Calculate the three equations of state of the paramagnetic model of equation 366 That is calculate TS I N BeS I N and µS I N Note that the fundamental equation of this problem is independent of V and that more generally there would be four equations of state Show that the three equations of state satisfy the Euler relation 382 Repeat Problem 381 for a system with the fundamental equation V lx 2 Neexp2S1NR where x and E dre positive constants 39 MOLAR HEAT CAPACITY AND OTHER DERIVATIVES The first derivatives of the fundamental equation have been seen to have important physical significance The various second derivatives are descriptive of material properties and these second derivatives often are the quantities of most direct physical interest Accordingly we exhibit a few particularly useful second derivatives and illustrate their utility In Chapter 7 we shall return to study the formal structure of such second derivatives demonstrating that only a small number are independent and that all others can be related to these few by a systematic reduction scheme For simple nonmagnetic systems the basic set of derivatives to which a wide set of others can be related are just three The coefficient of thermal expansion is defined by 367 The coefficient of thermal expansion is the fractional increase in the volume per unit increase in the temperature of a system maintained at constant pressure and constant mole numbers The isothermal compressibility is defined by 368 The isothermal compressibility is the fractional decrease in volume per unit increase in pressure at constant temperature The molar heat capacity at constant pressure is defined by 369 Molar Ieat CapaCI and 1her DerrlUtwes 85 The molar heat capacity at constant pressure 1s the quasistatic heat flux per mole required to produce unit increase in the temperature of a system maintained at constant pressure For systems of constant mole number all other second derivatives can be expressed in terms of these three and these three are therefore normally tabulated as functions of temperature and pressure for a wide variety of materials The origin of the relationships among second derivatives can be under stood in principle at this point although we postpone a full exploration to Chapter 7 Perhaps the simplest such relationship 1s the identity 370 which follows directly from the elementary theorem of calculus to the effect that the two mixed second partial derivatives of U with respect to V and S are equal aau aau av as as av 371 The two quantities appearing in equation 370 have direct physical interpretations and each can be measured The quantity aT av s v is the temperature change associated with adiabatic expansion of the volume the quantity aPaSvv when written as TdPdQvN is the product of the temperature and the change in pressure associated with an intro duction of heat dQ into a system at constant volume The prediction of equality of these apparently unrelated quantities is a nontrivial result in effect the first triumph of the theory Needless to say the prediction is corroborated by experiment The analogue of equation 370 in the entropy representation is 372 and we recognize that this is precisely the identity that we invoked in equation 346 in our quest for a thermal equation of state to be paired with the van der Waals equation In Chapter 7 we show in considerable detail that these equalities are prototypes of a general class of analogous relationships ref erred to as Maxwell relations Although the Maxwell relations have the simple form of equality of two derivatives they in turn are degenerate cases of a more general theorem that asserts that there must exist a relation among any four derivatives These general relations will permit any second derivative at constant N to be expressed in terms of the basic set cP a and KT 86 Some Formal Relationshps and Sample Systems To illustrate such anticipated relatiunships we first introduce two ad ditional second derivatives of practical interest the adiabatic com pressibility Ks and the molar heat capacity at constant volume cv The adiabatic compressibility is defined by K 1 av v aP s v aP s 373 This quantity characterizes the fractional decrease in volume associated with an isentropic increase in pressure ie for a system that is adiabati cally insulated The molar heat capacity at constant volume defined by cv r L L L 374 measures the quasistatic heat flux per mole required to produce unit increase in the temperature of a system maintained at constant volume In Chapter 7 we show that and TVa 2 r Ks N Cp 375 376 Again our purpose here is not to focus on the detailed relationships 375 and 376 but to introduce definitions of cP a and K 7 to can attention to the fact that cP a and K 7 are nonna1ly tabulated as functions of T and P and to stress that all other derivatives such as cv and Ks can be related to cP a and Kr A systematic approach to an such equalities and a mnemonic device for recalling them as needed is presented in Chap ter 7 Problem 396 is particularly recommended to the student Example For a particular material cp a and r are tabulated as functions of T and P Find the molar volume v as a function of T and P Solution We consider the TP plane The quantities cp a and r are known at all points in the plane and we seek to evaluate vT P at an arbitrary point in the plane Then dv r dP P dT vKrdP vadT or dv KrdP adT V Problems 87 Jf T0 P0 is a chosen reference point in the plane and if T P is a point of interest we can integrate along the path shown or any other convenient path for the path that we have chosen the term in dT vanishes for the horizontal section of the path and the term in dP vanishes for the vertical section of the path so that f dv 1T P v aT P0 dT J TT P dP To Po or v T P In 1 aTP 0 dT f rTPdP Vo T0 P0 The value of the molar volume at the reference point v0 must be specified we are then able to relate all other volumes to this volume PROBLEMS 391 a Show that for the multicomponent simple ideal gas a lT r lP 88 and Some Formal Relationships and Sample Systems c 1 Ks c 1 p cp c lR b What is the value of c for a monatomic ideal gas c Using the values found in part a corroborate equations 375 and 376 392 Corroborate equation 370 for a multicomponent simple ideal gas showing that both the right and lefthand members of the equation equal T cV where c is defined in Problem 391 393 Compute the coefficient of expansion a and the isothermal compressibility Kr in terms of P and v for a system with the van der Waals equation of state equation 341 394 Compute Cp cv Ks and Kr for the system in Problem 110la With these values corroborate the validity of equations 375 and 376 395 From equations 375 and 376 show that cpcv KrKs 396 A simple fundamental equation that exhibits some of the qualitative properties of typical crystaline solids is u Aebvv 0s4f3e tJR where A b and v0 are positive constants a Show that the system satisfies the Nernst theorem b Show that cv is proportional to T 3 at low temperature This is commonly observed and was explained by P Debye by a statistical mechanical analysis which will be developed in Chapter 16 c Show that cv 3k 8 at high temperatures This is the equipartition value which is observed and which will be demonstrated by statistical mechanical analysis in Chapter 16 d Show that for zero pressure the coefficient of thermal expansion vanishes in this modela result that is incorrect Hint Calculate the value of v at P O 397 The density of mercury at various temperatures is given here in gramsen 13 6202 10C 135955 0C 135708 10C 135462 20C 135217 30C 134973 40C 134729 50C 133522 100C 133283 110C 131148 200c 128806 300C 128572 310C Calculate a at 0C at 45C at 105C and at 305C Should the stem of a mercuryinglass thermometer be marked off in equal divisions for equal temperature intervals if the coefficient of tJ 11al expansion of glass is assumed to be strictly constant Problems 89 398 For a particular material c p a and Kr can be represented empirically by power series in the vicinity of T0 P0 as follows Cp C AcT Bc72 Dcp Ecp 2 Erp a aO AaT Bar2 Dap EaP2 FaTP KT Ko AT Br2 Dp Ep 2 Frp where r T T0 p P P0 Find the molar volume explicitly as a function of T and P in the vicinity of To Po 399 Calculate the molar entropy sT P0 for fixed pressure P0 and for tempera ture Tin the vicinity of T0 Assume that cP a and KT are given in the vicinity of T0 P0 as in the preceding problem and assume that sT0 P0 is known 3910 By analogy with equations 370 and 371 show that for a paramagnetic system aBe aT as 1VN aJ SVN or inverting T as T a1 aBe IV N aT SVN Interpret the physical meaning of this relationship 3911 By analogy with equations 370 and 371 show that for a paramagnetic system aBe ap av SlN ai SVN 3912 The magnetic analogues of the molar heat capacities cp and cv are c8 and c Calculate c8 T Be N and c1T Be N for the paramagnetic model of equation 366 Note that no distinction need be made between cv and cP for this model because of the absence of a dependence on volume in the fundamental relation 366 Generally all four heat capacities exist and are distinct 3913 The isothermal molar magnetic susceptibility is defined by Po a1 X N aBe T Show that the susceptibility of the paramagnetic model of equation 366 varies inversely with the temperature and evaluate Xi defined as the value of x for T lK 3914 Calculate the adiabatic molar susceptibility Po a1 Xs N aBe s as a function of T and Be for e paramagnetic model of equation 366 90 Some Formal Relationships and Sample Systems 3915 Calculate the isothermal and adiadc molar susceptibilities defined in Problems 3913 and 3914 for the system with fundamental equation µ 12 U 2 Nx Neexp2SINR How are each of these related to the constant x appearing in the fundamental relation 3916 Show that for the system of Problem 382 aT aT as 0 aBe s a1 s a1 r aBe 7 and L e L B B O That is there is no coupling between the thermal and magnetic properties What is the atypical feature of the equation of state of this system that leads to these results 3917 Calculate the heat transfer to a particular system if 1 mole is taken from T0 P0 to 2T0 2P0 along a straight line in the TP plane For this system it is known that aT P a0 where a0 is a constant cpT P ci a constant KrT P Ki a constant Hint Use the relation asaPr avaTp analogous to equations 370 through 372 and to be derived systematically in Chapter 7 to establish that dQ Tds cpdT TvadP 4 REVERSIBLE PROCESSES AND THE MAXIMUM WORK THEOREM 41 POSSIBLE AND IMPOSSIBLE PROCESSES An engineer may confront the problem of designing a device to accom plish some specified taskperhaps to lift an elevator to the upper floors of a tall building Accordingly the engineer contrives a linkage or engine that conditionally permits transfer of energy from a furnace to the elevator if heat flows from the furnace then by virtue of the interconnec tion of various pistons levers and cams the elevator is required to rise But nature ie the laws of physics exercises the crucial decisionwill the proposition be accepted or wi11 the device sit dormant and inactive with no heat leaving the furnace and no rise in height of the elevator The outcome is conditioned by two criteria First the engine must obey the Jaws of mechanics including of course the conservation of energy Second the process must maximal1y increase the entropy Patent registration offices are replete with failed inventions of impecca ble conditional logic if A occurs then B must occuringenious devices that conform to al1 the laws of mechanics but that nevertheless sit stubbornly inert in mute refusal to decrease the entropy Others operate but with unintended results increasing the entropy more effectively than envisaged by the inventor If however the net changes to be effected correspond to a maximal permissible increase in the total entropy with no change in total energy then no fundamental law precludes the existence of an appropriate process It may require considerable ingenuity to devise the appropriate engine but such an engine can be assumed to be permissible in principle Example 1 A particular system is constrained to constant mole number and volume so that no work can be done on or by the system Furthermore the heat capacity of the 91 92 Reversible Processes and the Maximum Work Theorem system is C a constant The fundamental equation of the system for constant volume is S S0 C In U U0 so U CT Two such systems with equal heat capacities have initial temperatures T10 and Tio with T10 Tio An engine is to be designed to lift an elevator ie to deliver work to a purely mechanical system drawing energy from the two thermodynamic systems What is the maximum work that can be so delivered Solution The two thermal systems will be left at some common temperature 7i The change in energy of the two thermal systems accordingly will be JU 2C7i CT 10 Tzo and the work delivered to the mechanical system the elevator will be W JU or W C TIO T20 27t The change in total entropy will occur entirely in the two thermal systems for which 7t 1i 7t JS Cln Cln 2Cln TIO T20 JT10T20 To max1rmze W we clearly wish to nummize 7t cf the second equation preceding and by the third equation this dictates that we minimize JS The minimum possible JS is zero corresponding to a reversible process Hence the optimum engine will be one for which 1i JT10Tzo and W c TIO T20 2JT 10T20 As a postscript we note that the assumption that the two thermal systems are left at a common temperature is not necessary W can be minimized with respect to Tlf and T21 separately with the same result The simplifying assumption of a common temperature follows from a selfconsistent argument for if the final temperature were different we could obtain additional work by the method described Example 2 An interesting variant of Example 1 is one in which three bodies each of the type described in Example 1 with U CT have initial temperatures of 300 K 350 K and 400 K respectively It is desired to raise one body to as high a temperature as possible independent of the final temperatures of the other two and without changing the state of any external system What is the maximum achievable temperature of the single body Solution Designate the three initial temperatures measured in units of 100 K as T1 T2 and T3 T 1 3 T2 35 and T3 4 Similarly designate the high temperature Prnllbleand lmpossbe Proese 93 achieved by one of the bodies in the same urnts as Th It is evident that the two remainmg bodies will be left at the same temperature T for 1f they were to be left at different temperatures we could extract work as in Example 1 and insert it as heat to further raise the temperature of the hot body Then energy conserva tion requires Th 2Tc T1 T2 T3 105 The total entropy change is TTh tJS C ln T T T I 2 3 and the requirement that this be positive implies 42 Eliminating Tc by the energy conservation condition 5 25 h h 42 A plot of the lefthand side of this equation is shown The plot is restricted to values of Th between O and 105 the latter bound following from the energy conservation condition and the requirement that T be positive The plot indi 421 40 30 Range of possible values of Th t E 20 E I I N 8 10 4095 2 3 4 5 6 7 8 9 10 Th 94 Reversible Processes and the Maximum Work Theorem cates that the maximum value of Th for w1uch the ordinate is greater than 42 is T 4095 or T 4095 K and furthermore that this value satisfies the equality and therefore corresponds to a reversible process Another solution to this problem will be developed in Problem 467 PROBLEMS 411 One mole of a monatomic ideal gas and one mole of an ideal van der Waals fluid Section 35 with c 32 are contained separately in vessels of fixed volumes v1 and v2 The temperature of the ideal gas is T1 and that of the van der Waals fluid is 7 It is desired to bring the ideal gas to temperature T2 maintaining the total energy constant What is the final temperature of the van der Waals fluid What restrictions apply among the parameters T 1 T2 a b v1 v2 if it is to be possible to design an engine to accomplish this temperature inversion assuming as always that no external system is to be altered in the process 412 A rubber band Section 37 is initially at temperature TB and length LB One mole of a monatomic ideal gas is initially at temperature Tc and volume VG The ideal gas maintained at constant volume V0 is to be heated to a final temperature T0 The energy required is to be supplied entirely by the rubber band Need the length of the rubber band be changed and if so by what amount Answer If L 8 L 0 2 2 2b1 L L 1 JR Tc5 Tc 1 t co 1L 0 ln ZRLo TB 3Rb L 1L 0nTcTc 413 Suppose the two systems in Example 1 were to have heat capacities of the form CT DT with n 0 a Show that for such systems U U0 DT 1n 1 and S S0 DTn What is the fundamental equation of such a system b If the initial temperature of the two systems were T10 and T20 what would be the maximum delivered work leaving the two systems at a common temperature Answer bforn2 D 1 1 1 2 2 l W 3 T10 T20 fi T10 T20 Quasistatic and Reversible Processes 95 42 QUASISTATIC AND REVERSIBLE PROCESSES The central principle of entropy maximization spawns various theorems of more specific content when specialized to particular classes of processes We shall turn our attention to such theorems after a preliminary refine ment of the descriptions of states and of processes To describe and characterize thermodynamic states and then to de scribe possible processes it is useful to define a thermodynamic configura tion space The thermodynamic configuration space of a simple system is an abstract space spanned by coordinate axes that correspond to the entropy S and to the extensive parameters U V N1 Nr of the system The fundamental equation of the system S S U V N1 N defines a surface in the thermodynamic configuration space as indicated schemati cally in Fig 41 It should be noted that the surface of Fig 41 conforms to the requirements that asau X lT be positive and that U be a single valued function of S x By definition each point in the configuration space represents an equilibrium state Representation of a nonequilibrium state would require a space of immensely greater dimension The fundamental equation of a composite system can be represented by a surface in a thermodynamic configuration space with coordinate axes t s S SUX 1 u FlGURE4 l The hypersurface S S U in the thermodynamic configuration space of a simple system 96 Reversible Processes and the Maximum Work Theorem u FIGURE42 t s x1 J The hypersurface S Sull x1 U m the thermodynamic con figuration space of a composite system corresponding to the extensive parameters of all of the subsystems For a composite system of two simple subsystems the coordinate axes can be associateJ with the total entropy S and the extensive parameters of the two subsystems A more convenient choice is the total entropy S the extensive parameters of the first subsystem u1 VCl Np Np and the extensive parameters of the composite system U V N1 N2 An appropriate section of the thermodynamic configuration space of a com posite system is sketched in Fig 42 Consider an arbitrary curve drawn on the hypersurface of Fig 43 from an initial state to a tenninal state Such a curve is known as a quasistatic locus or a quasistatic process A quasistatic process is thus defined in terms of a dense succession of equilibriwn states It is to be stressed that a quasistatic process therefore is an idealized concept quite distinct from a real physical process for a real process always involves nonequilibrium intermediate states having no representation in the thermodynamic con figuration space Furthermore a quasistatic process in contrast to a real process does not involve considerations of rates velocities or time The quasistatic process simply is an ordered succession of equilibrium states whereas a real process is a temporal succession of equilibrium and nonequilibrium states Although no real process is identical to a quasistatic process it is possible to contrive real processes that have a close relationship to quasistatic processes In particular it is possible to f i a system through a succession of states that coincides at any desired 11 1ber of points with Quasistatic locus or Quasistatic process FIGURE43 t s Quasistatic and Reversible Processes 97 x1 1 Representation of a quasistatic process in the thermodynamic configuration space a given quasistatic locus Thus consider a system originally in the state A of Fig 43 and consider the quasistatic locus passing through the points A B C H We remove a constraint which permits the system to proceed from A to B but not to points further along the locus The system disappears from the point A and subsequently appears at B having passed en route through nonrepresentable nonequilibrium states If the constraint is further relaxed making the state C accessible the system disappears from B and subsequently reappears at C Repetition of the operation leads the system to states D E H By such a succession of real processes we construct a process that is an approximation to the abstract quasistatic process shown in the figure By spacing the points A B C arbitrarily closely along the quasistatic locus we approximate the quasistatic locus arbitrarily closely The identification of P dV as the mechanical work and of T dS as the heat transfer is valid only for quasistatic processes Consider a closed system that is to be led along the sequence of states A B C H approximating a quasistatic locus The system is induced to go from A to B by the removal of some internal constraint The closed system proceeds to B if and only if the state B has maximum entropy among all newly accessible states In particular the state B must have higher entropy than the state A Accordingly the physical process joining states A and B in a closed system has unique directionality It proceeds rom the state A of lowe1 tropy to the state B of higher entropy but not inversely Such processes are irreversible 98 Reversible Processes and the Maximum Work Theorem A quasistatic locus can be approx ed by a real process in a closed system only if the entropy is monotonically nondecreasing along the quasi static locus The limiting case of a quasistatic process in which the increase in the entropy becomes vanishingly small is called a reversible process Fig 44 For such a process the final entropy is equal to the initial entropy and the process can be traversed in either direction t s FIGURE44 The plane SS 0 x1 J A reversible process along a quasistatic isentropic locus PROBLEMS 421 Does every reversible process coincide with a quasistatic locus Does every quasistatic locus coincide with a reversible process For any real process starting in a state A and terminating in a state H does there exist some quasistatic locus with the same two terminal states A and H Does there exist some reversible process with the same two terminal states 422 Consider a monatomic ideal gas in a cylinder fitted with a piston The walls of the cylinder and the piston are adiabatic The system is initially in equilibrium but the external pressure is slowly decreased The energy change of the gas in the resultant expansion dV is dU P dV Show from equation 334 that dS 0 so that the quasistatic adiabatic expansion is isentropic and reversible Relaxatwn Times and lrreuersibihty 99 423 A monatomic ideal gas is permitted to expand by a free expansion from V to V dV recall Problem 348 Show that dS NR dV V In a series of such infinitesimal free expansions leading from v to J show that s NRln Whether this atypical and infamous continuous free expansion process should be considered as quasistatic is a delicate point On the positive side is the observation that the terminal states of the infinitesimal expansions can be spaced as closely as one wishes along the locus On the negative side is the realization that the system necessarily passes through nonequilibrium states during each expansion the irreversibility of the microexpansions is essential and irreducible The fact that dS 0 whereas dQ 0 is inconsistent with the presumptive applicability of the relation dQ T dS to all quasistatic processes We define by somewhat circular logic the continuous free expansion process as being essentially irreversible and nonquasistatic 424 In the temperature range of interest a system obeys the equations T Av 2s P 2Av lnss0 where A is a positive constant The system undergoes a free expansion from v0 to v1 with v1 v0 Find the final temperature in terms of the initial temperature T0 v0 and v1 Find the increase in molar entropy 43 RELAXATION TIMES AND IRREVERSIBILITY Consider a system that is to be led along the quasistatic locus of Fig 43 The constraints are to be removed step by step the system being permitted at each step to come to a new equilibrium state lying on the locus After each slight relaxation of a constraint we must wait until the system fully achieves equilibrium then we proceed with the next slight relaxation of the constraint and we wait again and so forth Although this is the theoretically prescribed procedure the practical realization of the process seldom follows this prescription In practice the constraints usu ally are relaxed continuously at some sufficiently slow rate The rate at which constraints can be relaxed as a system approximates a quasistatic locus is characterized by the relaxation time 7 of the system For a given system with a given relaxation time T processes that occur in times short compared to T are not quasistatic whereas processes that occur in times long compared to T can be approximately quasistatic The physical considerations that determine the relaxation time can be illustrated by the adiabatic expansion of a gas recall Problem 422 If JOO Reversible Processes and the Maximum Work Theorem the piston is permitted to move outward only extremely slowly the process is quasistatic and reversible If however the external pressure is de creased rapidly the resulting rapid motion of the piston is accompanied by turbulence and inhomogeneous flow within the cylinder and by an entropy increase that drives these processes The process is then neither quasistatic nor reversible To estimate the relaxation time we first recog nize that a slight outward motion of the piston reduces the density of the gas immediately adjacent to the piston If the expansion is to be reversible this local rarefaction in the gas must be homogenized by hydrodynamic flow processes before the piston again moves appreciably The rarefaction itself propagates through the gas with the velocity of sound reflects from the walls of the cylinder and gradually dissipates The mechanism of dissipation involves both diffusive reflection from the walls and viscous damping within the gas The simplest case would perhaps be that in which the cylinder walls are so rough that a single reflection would effectively dissipate the rarefaction pulseadmittedly not the common situation but sufficient for our purely illustrative purposes Then the relaxation time would be on the order of the time required for the rarefaction to I propagate across the system or T v c where the cube root of the volume is taken as a measure of the length of the system and c is the velocity of sound in the gas If the adiabatic expansion of the gas in the cylinder is performed in times much longer than this relaxation time the expansion occurs reversibly and isentropically If the expansion is performed in times comparable to or shorter than the relaxation time there is an irreversible increase in entropy within the system and the expansion though adiabatic is not isentropic PROBLEMS 431 A cylinder of length L and crosssectional area A is divided into two equalvolume chambers by a piston held at the midpoint of the cylinder by a setscrew One chamber of the cylinder contains N moles of a monatomic ideal gas at temperature T0 This same chamber contains a spring connected to the piston and to the endwall of the cylinder the unstretched length of the spring is L2 so that it exerts no force on the piston when the piston is at its initial midpoint position The force constant of the spring is Kspnnp The othr chamber of the cylinder is evacuated The setscrew is suddenly removed Find the volume and temperature of the gas when equilibrium is achieved Assume the walls and the piston to be adiabatic and the heat capacities of the spring piston and walls to be negligible Discuss the nature of the processes that lead to the final equilibrium state If there were gas in each chamber of the cylinder the probleI stated would be indeterminate Why Heat Flow Coupled Srstems and Reersa of Processes 101 44 HEAT FLOW COUPLED SYSTEMS AND REVERSAL OF PROCESSES Perhaps the most characteristic of all thermodynamic processes is the quasistatic transfer of heat between two systems and it is instructive to examine this process with some care In the simplest case we consider the trans er of heat dQ from one system at temperature T to another at the same temperature Such a process is reversible the increase in entropy of the recipient subsystem dQT being exactly counterbalanced by the decrease in entropy dQT of the donor subsystem In contrast suppose that the two subsystems have different initial temperatures TIO and T20 with TIO T20 Further let the heat capacities at constant volume be C1T and CiT Then if a quantity of heat dQ 1 is quasistatically inserted into system I at constant volume the entropy mcrease is 41 and similarly for subsystem 2 If such infinitesimal transfers of heat from the hotter to the colder body continue until the two temperatures become equal then energy conservation requires which determines The resultant change in entropy is S lr CT dT lr C2T2 dT T 1 T 2 T10 I T20 2 42 43 In the particular case in which C1 and C2 are independent of T the energy conservation condition gives 44 and the entropy increase is s c1n c21n RJ 45 t i left to Problem 443 emonstrate that this expression for S is intrmsically positive 102 Reversible Processes and the Maximum Worf1eorem Several aspects of the heat transfer process deserve reflection First we note that the process though quasistatic is irreversible it is represented in thermodynamic configuration space by a quasistatic locus of monotonically increasing S Second the process can be associated with the spontaneous flow of heat from a hot to a cold system providing a that the intermediate wall through which the heat flow occurs is thin enough that its mass and hence its contribution to the thermodynamic properties of the system is negligi ble and b that the rate of heat flow is sufficiently slow ie the thermal resistivity of the wall is sufficiently high that the temperature remains spatially homogeneous within each subsystem Third we note that the entropy of one of the subsystems is decreased whereas that of the other subsystem is increased It is possible to decrease the entropy of any particular system providing that this decrease is linked to an even greater entropy increase in some other system In this sense an irreversible process within a given system can be reversedwith the hidden cost paid elsewhere PROBLEMS 441 Each of two bodies has a heat capacity given in the temperature range of interest by C A BT where A 8 JK and B 2 X 10 2 JK 2 If the two bodies are initially at temperatures T10 400 K and T20 200 K and if they are brought into thermal contact what is the final temperature and what is the change in entropy 442 Consider again the system of Problem 441 Let a third body be available with heat capacity C3 BT and with an initial temperature of T30 Bodies 1 and 2 are separated and body 3 is put into thermal contact with body 2 What must the initial temperature 70 be in order thereby to restore body 2 to its initial state By how much is the entropy of body 2 decreased in this second process 443 Prove that the entropy change in a heat flow process as given in equation 45 is intrinsically positive 444 Show that if two bodies have equal heat capacities each of which is constant independent of temperature the equilibrium temperature achieved by direct thermal contact is the arithmetic average of the initial temperatures 445 Over a limited temperature range the heat capacity at constant volume of a particular type of system is inversely proportional to the temperature a What is the temperature dependence of the energy at constant volume for this type of system The Maximum Work Theorem I 03 b If two such systems at initial temperatures T10 and T20 are put into thermal contact what is the equilibrium temperature of the pair 446 A series of N I large vats of water have temperatures T0 T1 T2 TN with Tn T 1 A small body with heat capacity C and with a constant volume independent of temperature is initially in thermal equilibrium with the vat of temperature T0 The body is removed from this vat and immersed in the vat of temperature T1 The process is repeated until after N steps the body is in equilibrium with the vat of temperature TN The sequence is then reversed until the body is once again in the initial vat at temperature T0 Assuming the ratio of temperatures of successive vats to be a constant or and neglecting the small change in temperature of any vat calculate the change in total entropy as a the body is successively taken up the sequence from T0 to TN and b the body is brought back down the sequence from TN to T0 What is the total change in entropy in the sum of the two sequences above Calculate the leading nontrivial limit of these results as N oo keeping T0 and TN constant Note that for large N Nx 1IN 1 lnx lnx 22N 45 THE MAXIMUM WORK THEOREM The propensity of physical systems to increase their entropy can be channeled to deliver useful work All such applications are governed by the maximum work theorem Consider a system that is to be taken from a specified initial state to a specified final state Also available are two auxiliary systems into one of which work can be transferred and into the other of which heat can be transferred Then the maximum work theorem states that for all processes leading from the specified initial state to the specified final state of the primary system the delivery of work is maximum and the delivery of heat is minimum for a reversible process Furthermore the delivery of work and of heat is identical for every reversible process The repository system into which work is delivered is called a reversi ble work source Reversible work sources are defined as systems enclosed by adiabatic impermeable walls and characterized by relaxation times suffi ciently short that all processes within them are essentially quasistatic From the thermodynamic point of view the conservative nonfrictional sys tems considered in the theory of mechanics are reversible work sources 104 Reversible Processes and the Maxmum Work Theorem Reversible heat source FIGURE45 System State A State B AU UAUB Reversible work source Maximum work process The delivered work W Rw is maximum and the delivered heat QRHS is minimum if the entire process is reversible S 10 1 0 The repository system into which heat is delivered is called a reversible heat source 1 Reversible heat sources are defined as systems enclosed by rigid impermeable walls and characterized by relaxation times sufficiently short that all processes of interest within them are essentially quasistatic If the temperature of the reversible heat source is T the transfer of heat dQ to the reversible heat source increases its entropy according to the quasi static relationship dQ T dS The external interactions of a reversible heat source accordingly are fully described by its heat capacity C T the definition of the reversible heat source implies that this heat capacity is at constant volume but we shall not so indicate by an explicit subscript The energy change of the reversible heat source is dU dQ CT dT and the entropy change is dS CTT dT The various transfers envisaged in the maximum work theorem are indicated schematically in Fig 45 The proof of the maximum work theorem is almost immediate Con sider two processes Each leads to the same energy change tlU and the same entropy change tlS within the primary subsystem for these are determined by the specified initial and final states The two processes ditf er only in the apportionment of the energy ditf erence AU between the reversible work source and the reversible heat source tlU W Rws QRHsBut the process that delivers the maximum possible work to the reversible work source correspondingly delivers the least possible heat to the reversible heat source and therefore leads to the least possible entropy increase of the reversible heat source and thence of the entire system 1The use of the term source might be construed as biasmg the terminology m favor of extractwn of heat as contrasted with 1yectwn such a bias is not intended Tle Maximum Work Theorem I 05 The absolute minimum of S 10a1 for all possible processes is attained by any reversible process for all of which S 10a1 0 To recapitulate energy conservation requires U WRws QR11s 0 Wah U fixed to maximize WRws is to minimize QRHS This is achieved by minimizing si since SRHS increases monotonically with positive heat input QRHs The minimum SJ therefore is achieved by minimum S 10a1 or by S 101a1 0 The foregoing descriptive proof can be cast into more formal lan guage and this is particularly revealing in the case in which the initial and final states of the subsystem are so close that all differences can be expressed as differentials Then energy conservation requires dU dQRHS dWRWS 0 46 whereas the entropy maximum principle requires dS 101 dS dQ RHS 0 T RHS 47 It follows that 48 The quantities on the righthand side are all specified In particular dS and dU are the entropy and energy differences of the primary subsystem in the specified final and initial states The maximum work transfer dWRws corresponds to the equality sign in equation 48 and therefore in equation 47 dS 101 0 It is useful to calculate the maximum delivered work which from equation 48 and from the identity dU dQ dW becomes dWRws maximum TRHS y dQ dU 1 TRHsTdQ dW 49 That is in an infinitesimal process the maximum work that can be dewered to the reversible work source 1s the sum of a the work dW directly extracted from the subsystem b a fraction 1 TRHsT of the heat dQ directly extracted from the subsystem The fraction 1 T RHsT of the extracted heat that can be converted to work in an infinitesimal process is called the thermodynamic engine I 06 Reversible Processes and the Maximum Work Theorem efficiency and we shall return to a discussion of that quantity in Section 45 However it generally is preferable to solve maximum work problems in terms of an overall accounting of energy and entropy changes rather than to integrate over the thermodynamic engine efficiency Returning to the total noninfinitesimal process the energy conserva tion condition becomes ubsystem QRHS WRWS 0 410 whereas the reversibility condition is Slota ssubsystem f dQRHsTRHS 0 411 In order to evaluate the latter integral it is necessary to know the heat capacity CRHsT dQRHsdTRHs of the reversible heat source Given CRHsT the integral can be evaluated and one can then also infer the net heat transfer QRHS Equation 410 in turn evaluates WRws Equations 410 and 411 evaluated as described provide the solution of all problems based on the maximum work theorem The problem is further simplified if the reversible heat source is a thermal reservoir A thermal reservoir is defined as a reversible heat source that is so large that any heat trans er of interest does not alter the tempera ture of the thermal reservoir Equivalently a thermal reservoir is a reversi ble heat source characterized by a fixed and definite temperature For such a system equation 411 reduces simply to S101a1 Ssubsys1em es 0 res 412 and Qres QRHs can be eliminated between equations 410 and 412 giving W RWS TesSsubsys1em ubsys1em 413 Finally it should be recognized that the specified final state of the subsystem may have a larger energy than the initial state In that case the theorem remains formally true but the delivered work may be negative This work which must be supplied to the subsystem will then be least the delivered work remains algebraically maximum for a reversible process Example 1 One mole of an ideal van der Waals fluid is to be taken by an unspecified process from the state T0 v0 to the state v1 A second system is constrained to have a The Maximum Work Theorem 107 fixed volume and its initial temperature is T20 its heat capacity is linear in the temperature D constant What is the maximum work that can be delivered to a reversible work source Solution The solution parallels those of the problems in Section 41 despite the slightly different formulations The second system is a reversible heat source for it the dependence of energy on temperature is U2 T f C2 T dT tDT 2 constant and the dependence of entropy on temperature is S2T J C1T TdT DT constant For the primary fluid system the dependence of energy and entropy on T and v is given in equations 349 and 351 from which we find 6U cRT T I I O V V f 0 6S 1 R In cR In The second system the reversible heat source changes temperature from T20 to some as yet unknown temperature T21 so that 6U2 tD Tz T2i and 6S 2 D T21 T 20 The value of T21 is determined by the reversibility condit10n 6S 1 6S 2 Rln cRln DT 21 T 20 0 or T21 T20 RD 11n cRD 1ln io The conservation of energy then determines the work W3 delivered to the reversible work source whence W3 D T2 T2t cR T0 where we recall that is given whereas T21 has been found 108 Reversible Processes and the Maximum Work Theorem An equivalent problem but with a somewhat simpler system a mon atomic ideal gas and a thermal reservoir is formulated in Problem 451 In each of these problems we do not commit ourselves to any specific process by which the result might be realized but such a specific process is developed in Problem 452 which with 451 is strongly recommended to the reader Example 2 Isotope Separation In the separation of U 235 and U 238 to produce enriched fuels for atomic power plants the naturally occurring uranium is reacted with fluorine to form uramum hexafluonde UF6 The uranium hexafluoride is a gas at room temperature and atmospheric pressure The naturally occurring mole fraction of U 235 is 00072 or 072 It is desired to process 10 moles of natural UF6 to produce 1 mole of 2 enriched matenal leaving 9 moles of partially depleted material The UF6 gas can be represented approximately as a polyatomic multicomponent simple ideal gas with c 7 2 equation 340 Assuming the separation process to be earned out at a temperature of 300 K and a pressure of 1 atm and assuming the ambient atmosphere at 300 K to act as a thermal reservoir what is the minimum amount of work required to carry out the enrichment process Where does this work energy ultimately reside Solution The problem is an example of the maximum work theorem in which the minimum work required corresponds to the maximum work delivered The initial state of the system is 10 moles of natural UF6 at T 300 K and P 1 atm The final state of the system is I mole of ennched gas and 9 moles of depleted gas at the same temperature and pressure The cold reservoir 1s also at the same tempera ture We find the changes of entropy and of energy of the system From the fundamental equation 340 we find the equations of state to be the familiar forms U72NRT PV NRT These enable us to write the entropy as a function of T and P S ti Nso NRln NR In NR 1t xln x Tlus last termthe entropy of mixing as defined followmg equation 340is the significant term in the iolope separation process We first calculate the mole fraction of U 235 F6 m the 9 moles of depleted material this 1s found to be 0578 Accordingly the change in entropy is tS R002 ln 002 098 In 098 9R000578 ln000578 0994ln 0994 lOR I00072 In 00072 09928 In 09928 00081R 0067 JK The gas e1ects heat Problems J 09 There is no change in the energy of the gas and all the energy supplied as work is transferred to the ambient atmosphere as heat That work or heat is WRws QesTflS 300X0067 20J If there existed a semipermeable membrane permeable to U 235F6 but not to U 238F6 the separation could be accomplished simply Unfortunate1y no such membrane exists The methods employed in practice are all dynamic nonquasi static processes that exploit the sma11 mass difference of the two isotopesin ultracentrifuges in mass spectrometers or in gaseous diffusion PROBLEMS 451 One mole of a monatomic ideal gas is contained in a cylinder of volume 10 3 m3 at a temperature of 400 K The gas is to be brought to a final state of volume 2 X 10 3 m3 and temperature 400 K A thermal reservoir of temperature 300 K is available as is a reversible work source What is the maximum work that can be delivered to the reversible work source Answer WRws 300 Rln2 452 Consider the following process for the system of Problem 451 The ideal gas is first expanded adiabatically and isentropically until its temperature falls to 300 K the gas does work on the reversible work source in this expansion The gas is then expanded while in thermal contact with the thermal reservoir And finally the gas is compressed adiabatically until its volume and temperature reach the specified values 2 X 10 3 m3 and 400 K a Draw the three steps of this process on a T V diagram giving the equation of each curve and labelling the numerical coordinates of the vertices b To what volume must the gas be expanded in the second step so that the third adiabatic compression leads to the desired final state c Calculate the work and heat transfers in each step of the process and show that the overall results are identical to those obtained by the general approach of Example l 453 Describe how the gas of the preceding two problems could be brought to the desired final state by a free expansion What are the work and heat transfers in this case Are these results consistent with the maximum work theorem 454 The gaseous system of Problem 451 is to be restored to its initial state Both states have temperature 400 K and the energies of the two states are equal U 600 R Need any work be supplied and if so what is the minimum supplied work Note that the thermal reservoir of temperature 300 K remains accessible 110 Reversible Processes and the Maximum Work Theorem 455 If the thermal reservoir of Problem 451 were to be replaced by a reversible heat source having a heat capacity of the form and an initial temperature of T RHso 300 K again calculate the maximum delivered work Before doing the calculation would you expect the delivered work to be greater equal to or smaller than that calculated in Prob 451 Why 456 A system can be taken from state A to state B where SB SA either a directly along the adiabat S constant or b along the isochore AC and the isobar CB The difference in the work done by the system is the area enclosed between the two paths in a PV diagram Does this contravene the statement that the work delivered to a reversible work source is the same for every reversible process Explain 457 Consider the maximum work theorem in the case in which the specified final state of the subsystem has lower energy than the initial state Then the essential logic of the theorem can be summarized as follows Extraction of heat from the subsystem decreases its entropy Consequently a portion of the extracted heat must be sacrificed to a reversible heat source to effect a net increase in entropy otherwise the process will not proceed The remainder of the extracted heat is available as work Similarly summarize the essential logic of the theorem in the case in which the final state of the subsystem has larger energy and larger entropy than the initial state 458 If SB SA and VB VA does this imply that the delivered work is negative Prove your assertion assuming the reversible heat source to be a thermal reservoir Does postulate III which states that S is a monotonically increasing function of V disbar the conditions assumed here Explain 459 Two identical bodies each have constant and equal heat capacities C 1 C2 C a constant In addition a reversible work source is available The initial temperatures of the two bodies are TIO and T20 What is the maximum work that can be delivered to the reversible work source leaving the two bodies in thermal equilibrium What is the corresponding equilibrium temperature Is this the minimum attainable equilibrium temperature and if so why What is the maximum attainable equilibrium temperature For C 8 JK TIO 100C and T20 0C calculate the maximum delivered work and the possible range of final equilibrium temperature Answer 7tnun 46oc 7tmax 500c wmax C 2 622J Problems 111 4510 Two identical bodies each have heat capacities at constant volume of CT aT The initial temperatures are TIO and T20 with T20 T10 The two bodies are to be brought to thermal equilibrium with each other maintaining both volumes constant while delivering as much work as possible to a reversible work source What is the final equilibrium temperature and what is the maXImum work delivered to the reversible work source Evaluate your answer for Tio TIO and for Tio 2T 10 Answer W a ln9 8 if T20 2T 10 4511 Two bodies have heat capacities at constant volume of C1 aT C2 2bT The initial temperatures are T10 and T20 with T20 T10 The two bodies are to be brought to thermal equilibrium mamtaining both volumes constant while de livering as much work as possible to a reversible work source What is the final equilibrium temperature and what is the maximum work delivered to the reversible work source 4512 One mole of an ideal van der Waals fluid is contained in a cylinder fitted with a piston The initial temperature of the gas is T and the initial volume is v A reversible heat source with a constant heat capacity C and with an initial temperature T0 is available The gas is to be compressed to a volume of v1 and brought into thermal equilibrium with the reversible heat source What is the maximum work that can be delivered to the reversible work source and what is the final temperature Answer R cR 1cRC 1j vb TTo f 4513 A system has a temperatureindependent heat capacity C The system is initially at temperature T and a heat reservoir is available at temperature T with T T Find the maximum work recoverable as the system is cooled to the temperature of the reservoir 4514 If the temperature of the atmosphere is 5C on a winter day and if 1 kg of water at 90C is available how much work can be obtained as the water is cooled to the ambient temperature Assume that the volume of the water is constant and assume that the molar heat capacity at constant volume is 75 Jmole K and is independent of temperature Answer 45 X 103J J J 2 Reversible Processes and the Maximum Work Theorem 4515 A rigid cylinder contains an internal adiabatic piston separating it into two chambers of volumes Vo and V20 The first chamber contains one mole of a monatomic ideal gas at temperature T10 The second chamber contains one mole of a simple diatomic ideal gas c 52 at temperature T20 ln addition a thermal reservoir at temperature is available What is the maximum work that can be delivered to a reversible work source and what are the corresponding volumes and temperatures of the two subsystems 4516 Each of three identical bodies has a temperatureindependent heat capac ity C The three bodies have initial temperatures T3 T2 T1 What is the maximum amount of work that can be extracted leaving the three bodies at a common final temperature 4517 Each of two bodies has a heat capacity given by C A 2BT where A 8 JK and B 2 x 10 2 JK 2 If the bodies are initially at temperatures of 200 K and 400 K and if a reversible work source is available what is the minimum final common temperature to which the two bodies can be brought If no work can be extracted from the reversible work source what is the maximum final common temperature to which the two bodies can be brought What is the maximum amount of work that can be transferred to the reversible work source Answer Tmm 293K 4518 A particular system has the equations of state T Asv 112 and P T 2 4Av 1li where A is a constant One mole of this system is initially at a temperature T1 and volume V1 It is desired to cool the system to a temperature T2 while compressing it to volume Vi Ti T1 Vi V1 A second system is available It is initially at a temperature T2 Its volume is kept constant throughout and its heat capacity is Cv BT 1i2 B constant Whal is the minimum amount of work that must be supplied by an external agent to accomplish this goal 4519 A particular type of system obeys the equations u T and P avT b where a and b are constants Two such systems each of 1 mole are initially at temperatures T1 and T2 with Ti T1 and each has a volume v0 The systems are to be brought to a common temperature 7j with each at the same final volume v1 The process is to be such as to deliver maximum work to a revenible work source Coefficents of Engine Refrigerator and Heat Pump Performance 113 a What is the final temperature b How much work can be delivered Express the result in terms of Ti T2 v0 v1 and the constants a and b 4520 Suppose that we have a system in some initial state we may think of a tank of hot compressed gas as an example and we wish to use it as a source of work Practical considerations require that the system be left finally at atmo spheric temperature and pressure in equilibrium with the ambient atmosphere Show first that the system does work on the atmosphere and that the work actually available for useful purposes is therefore less than that calculated by a straightforward application of the maximum work theorem In engineering parlance this net available work is called the availability a Show that the availability is given by Availability U0 PatrFo TatmSo 1t Palm T 1mS where the subscript f denotes the final state in which the pressure is Paim and the temperature is Tam b If the original system were to undergo an internal chemical reaction during the process considered would that invalidate this formula for the availability 4521 An antarctic meteorological station suddenly loses all of its fuel It has N moles of an inert ideal van der Waals fluid at a high temperature Th and a high pressure Ph The constant temperature of the environment 1s T0 and the atmospheric pressure is P0 If operation of the station requires a continuous power J what is the longest conceivable time t max that the station can operate Calculate tmax in terms of Th T0 Ph P0 9 N and the van der Waals constants a b and c Note that this is a problem in availability as defined and discussed in Problem 4520 In giving the solution it is not required that the molar volume vh be solved explicitly in terms of Th and Ph it is sufficient simply to designate it as vhTh Ph and similarly for v0 T0 P0 4522 A geothermal power source is available to drive an oxygen production plant The geothermal source is simply a well containing 103 m3 of water initially at 100C nearby there is a huge infinite lake at 5C The oxygen is to be separated from air the separation being carried out at 1 atm of pressure and at 20C Assume air to be oxygen and nitrogen in moie fractions and assume that it can be treated as a mixture of ideal gases How many moles of 0 2 can be produced in principle ie assuming perfect thermodynamic efficiency before exhausting the power source 46 COEFFICIENTS OF ENGINE REFRIGERATOR AND HEAT PUMP PERFORMANCE As we saw in equations 46 and 47 m an infinitesimal reversible process involving a hot subsystem a cold reversible heat source and a reversible work source dQh dWh dQC dWRWS 0 414 114 Reversible Processes and The Maximum Wnk Theorem and ds aQC o h T C 415 where we now indicate the hot system by the subscript h and t cold reversible heat source by the subscript c In such a process the delivered work dWRws is algebraically maximum This fact leads to criteria for the operation of various types of useful devices The most immediately evident system of interest is a thermodynamic engine Here the hot subsystem may be a furnace or a steam boiler whereas the cold reversible heat source may be the ambient atmospher or for a large power plant a river or lake The measure of performance is the fraction of the heat aQh withdrawn 2 from the hot system that is converted to work aWRws Taking aWh 0 in equation 414 it is simply additive to the delivered work in equation 49 we find the thermodynamic engine efficiency Ee 416 The relationship of the various energy exchanges is indicated in Fig 46a For a subsystem of given temperature Th the thermodynamic engine efficiency increases as T decreases That is the lower the temperature of the cold system to which heat is delivered the higher the engine efficiency The maximum possible efficiency Ee 1 occurs if the tempera ture of the cold heat source is equal to zero If a reservoir at zero temperature were available as a heat repository heat could be freely and fully converted into work and the world energy shortage would not exist3 A refrigerator is simply a thermodynamic engine operated in reverse Fig 47b The purpose of the device is to extract heat from the cold system and with the input of the minimum amount of work to eject that heat into the comparatively hot ambient atmosphere Equations 414 and 2The problem of signs may be confusing Throughout this book the symbols Wand Q or dW and dQ indicate work and heat inputs Heat withdrawn from a system is Q or dQ Thus if S J are withdrawn from the hot subsystem we would write that the heat withdrawn is Qh 5 J whereas Qh the heat input would be 5 J For clarity in this chapter we use the parentheses to serve as a reminder that Qh is to be considered as a positive quantity in the particular example being discussed 3The energy shortage is in any case a misnomer Energy is conserved The shortage is one of entropy sinks of systems of low entropy Given such systems we could bargam with nature offering to allow the entropy of such a system to increase as by allowing a hydrocarbon to oxidize or heat to flow to a low temperature sink or a gas to expand if useful tasks were simultaneously done There is only a negentropy shortage Coefficients of Engine Refrigerator and Heat Pump Performance 115 FIGURE46 Cooling System Tc Refrigerator Tc Ambient Atmosphere Tc Energy Source Furnace Boiler Th a Ambient Atmosphere T b Building Interior Th c Machinery Power Plant Rev Work Source Power Plant Rev Work Source Engine refrigerator and heat pump In this diagram dWdW Rws 415 remain true but the coefficient of refrigerator performance represents the appropriate criterion for this device the ratio of the heat removed from the refrigerator the cold system to the work that must be purchased from the power company That is dQJ Er dWRws 417 If the temperatures Th and are equal the coefficient of refrigerator performance becomes infinite no work is then required to transfer heat from one system to the other The coefficient of performance becomes progressively smaller as the temperature decreases relative to Th And if 116 Reversible Procefses and The Maximum Work Theorem the temperature T approaches zero the coefficient of performance also approaches zero assuming T fixed It therefore requires huge amounts of work to extract even trivially small quantities of heat from a system near T 0 We now turn our attention to the heat pump In this case we are interested in heating a warm system extracting some heat from a cold system and extracting some work from a reversible work source In a practical case the warm system may be the interior of a home in winter the cold system is the outdoors and the reversible work source is again the power company In effect we heat the home by removing the door of a refngerator and pushing it up to an open window The inside of the refrigerator is exposed to the outdoors and the refrigerator attempts with negligible success further to cool the outdoors The heat extracted from this huge reservoir together with the energy purchased from the power company is ejected directly into the room from the cooling coils in the back of the refrigerator The coefficient of heat pump performance EP is the ratio of the heat delivered to the hot system to the work extracted from the reversible work source 418 PROBLEMS 461 A temperature of 0001 K is accessible in low temperature laboratories with moderate effort If the price of energy purchased from the electric utility company is 15 kW h what would be the minimum cost of extraction of one watthour of heat from a system at 0001 K The warm reservoir is the ambient atmosphere at 300 K Answer 45 462 A home is to be maintained at 70F and the external temperature is 50F One method of heating the home is to purchase work from the power company and to convert it directly into heat This is the method used in common electric room heaters Alternatively the purchased work can be used to operate a heat pump What is the ratio of the costs if the heat pump attains the ideal thermody namic coefficient of performance 463 A household refrigerator is maintained at a temperature of 35F Every time the door is opened warm material is placed inside introducing an average of 50 kcal but making only a small change in the temperature of the refrigerator Problems 117 The door 1s opened 15 times a day and the refrigerator operates at 15 of the ideal coefficient of performance The cost of work is 15 kW h What is the monthly bill for operating this refrigerator 464 Heat is extracted from a bath of liquid helium at a temperature of 42 K The hightemperature reservoir is a bath of liquid nitrogen at a temperature of 77 3 K How many Joules of heat are introduced into the nitrogen bath for each Joule extracted from the helium bath 465 Assume that a particular body has the equation of state U NCT with NC 10 JK and assume that this equation of state is valid throughout the temperature range from 05 K to room temperature How much work must be expended to cool this body from room temperature 300 K to 05 K using the ambient atmosphere as the hot reservoir Answer 162 kJ 466 One mole of a monatomic ideal gas is allowed to expand isothermally from an initial volume of 10 liters to a final volume of 15 liters the temperature being maintained at 400 K The work delivered is used to drive a thermodynamic refrigerator operating between reservoirs of temperatures 200 and 300 K What is the maximum amount of heat withdrawn from the lowtemperature reservoir 467 Give a constructive solution of Example 2 of Section 41 Your solution may be based on the following procedure for achieving maximum temperature of the hot body A thermodynamic engine is operated between the two cooler bodies extracting work until the two cooler bodies reach a common temperature This work is then used as the input to a heat pump extracting heat from the cooler pair and heating the hot body Show that this procedure leads to the same result as was obtained in the example 468 Assume that 1 mole of an ideal van der Waals fluid is expanded isother mally at temperature Th from an initial volume V to a final volume A thermal reservoir at temperature T is available Apply equation 4 9 to a differen tial process and integrate to calculate the work delivered to a reversible work source Corroborate by overall energy and entropy conservation Hint Remember to add the direct work transfer P dV to obtain the total work delivered to the reversible work source as in equation 49 469 Two moles of a monatomic ideal gas are to be taken from an initial state P V to a final state P1 B 2P VB where Bis a constant A reversible work source and a thermal reservoir of temperature Tc are available Find the maximum work that can be delivered to the reversible work source Given values of B P and Teo for what values of V is the maximum delivered work positive 4610 Assume the process in Problem 469 to occur along the locus P BV2 where B PV2 Apply the thermodynamic engine efficiency to a differential 118 Reversible Processes and The Maximum Work Theorem process and integrate to corroborate the result obtained in Problem 469 Recall the hint given in Problem 468 4611 Assume the process in Problem 469 to occur along a straightline locus in the TV plane Integrate along this locus and again corroborate the results of Problems 469 and 4610 47 THE CARNOT CYCLE Throughout this chapter we have given little attention to specific processes purposefully stressing that the delivery of maximum work is a general attribute of all reversible processes It is useful nevertheless to consider briefly one particular type of processthe Carnot cycle both because it elucidates certain general features and because this process has played a critically important role in the historical development of thermo dynamic theory A system is to be taken from a particular initial state to a given final state while exchanging heat and work with reversible heat and work sources To describe a particular process it is not sufficient merely to describe the path of the system in its thermodynamic configuration space The critical features of the process concern the manner in which the extracted heat and work are conveyed to the reversible heat and work sources For that purpose auxiliary systems may be employed The aux iliary systems are the tool or devices used to accomplish the task at hand or in a common terminology they constitute the physical engines by which the process is effected Any thermodynamic systema gas in a cylinder and piston a magnetic substance in a controllable magnetic field or certain chemical systemscan be employed as the auxiliary system It is only required that the auxiliary system be restored at the end of the process to its initial state the auxiliary system must not enter into the overall energy or entropy accounting It is this cyclic nature of the process within the auxiliary system that is reflected in the name of the Carnot cycle For clarity we temporarily assume that the primary system and the reversible heat source are each thermal reservoirs the primary system being a hot reservoir and the reversible heat source being a cold reservoir this restriction merely permits us to consider finite heat and work transfers rather than infinitesimal transfers The Carnot cycle is accomplished in four steps and the changes of the temperature and the entropy of the auxiliary system are plotted for each of these steps in Fig 47 1 The auxiliary system originally at the same temperature as the primary system the hot reservoir is placed in contact with that reservoir and with the reversible work source The auxiliary system is then caused to undergo an isothermal process by changing some convenient extensive The Carnot Cycle 119 T1 A B A t l T p C Tc D C s SB v s FIGURE47 The TS and PV diagrams for the auxiliary system in the Carnot cycle parameter if the auxiliary system is a gas it may be caused to expand isothermally if it is a magnetic system its magnetic moment may be decreased isothermally and so forth In this process a flux of heat occurs from the hot reservoir to the auxiliary system and a transfer of work f P dV or its magnetic or other analogue occurs from the auxiliary system to the reversible work source This is the isothermal step A B in Fig 47 2 The auxiliary system now in contact only with the reversible work source is adiabatically expanded or adiabatically demagnetized etc until its temperature falls to that of the cold reservoir A further transfer of work occurs from the auxiliary system to the reversible work source The quasistatic adiabatic process occurs at constant entropy of the auxiliary system as in B C of Fig 47 3 The auxiliary system is isothermally compressed while in contact with the cold reservoir and the reversible work source This compression is continued until the entropy of the auxiliary system attains its initial value During this process there is a transfer of work from the reversible work source to the auxiliary system and a transfer of heat from the auxiliary system to the cold reservoir This is the step C D in Fig 47 4 The auxiliary system is adiabatically compressed and receives work from the reversible work source The compression brings the auxiliary system to its initial state and completes the cycle Again the entropy of the auxiliary system is constant from D to A in Fig 47 The heat withdrawn from the primary system the hot reservoir in process 1 is Th llS and the heat transferred to the cold reservoir in process 3 is TllS The difference Th T lS is the net work transferred to the reversible work source in the complete cycle On the TS diagram of Fig 4 7 the heat Th llS withdrawn from the primary system is represented by the area bounded by the four points labeled ABS 8 S4 the heat ejected to the cold reservoir is represented by the area CDSAS8 and the net work delivered is represented by the area ABCD The coefficient of perfor mance is the ratio of the area ABCD to the area ABS 8SA or T TT 120 Reversible Processes and The Maximum Work Theorem The Carnot cycle can be represented on any of a number of other diagrams such as a PV diagram or a T V diagram The representation on a P V diagram is indicated in Fig 4 7 The precise form of the curve BC representing the dependence of P on V in an adiabatic isentropic process would follow from the equation of state P PS VN of the auxiliary system If the hot and cold systems are merely reversible heat sources rather than reservoirs the Carnot cycle must be carried out in infinitesimal steps The heat withdrawn from the primary hot system in process 1 is then Th dS rather than Th tS and similarly for the other steps There is clearly no difference in the essential results although Th and T are continually changing variables and the net evaluation of the process requires an integration over the differential steps It should be noted that real engines never attain ideal thermodynamic efficiency Because of mechanical friction and because they cannot be operated so slowly as to be truly quasistatic they seldom attain more than 30 or 40 thermodynamic efficiency Nevertheless the upper limit on the efficiency set by basic thermodynamic principles is an important factor in engineering design There are other factors as well to which we shall return in Section 49 Example N moles of a monatomic ideal gas are to be employed as the auxiliary system in a Carnot cycle The ideal gas is initially in contact with the hot reservoir and in the first stage of the cycle it is expanded from volume VA to volume VB4 Calculate the work and heat transfers in each of the four steps of the cycle in terms of Th Tc VA VB and N Directly corroborate that the efficiency of the cycle is the Carnot efficiency Solution The data are given in terms of T and V we therefore express the entropy and energy as functions of T V and N T 312VN S Ns 0 NR In 3 2 T0 1 V0N and U fNRT Then in the isothermal expansion at temperature Th tSAB SB SA NRln and tUAB 0 4Note that m this example quantities such as V S V Q refer to the auxiliary system rather than to the primary system the hot reservoir The Carnot Cycle 121 whence and WAB NRThln In the second step of the cycle the gas is expanded adiabatically until the temperature falls to T0 the volume meanwhile increasing to Ve From the equation for S we see that Ti V constant and and In the third step the gas is isothermally compressed to a volume Vv This volume must be such that it lies on the same adiabat as VA see Fig 47 so that Then as in step 1 and Finally in the adiabatic compression QDA Q and From these results we obtain and which is the expected Carnot efficiency 122 Reversible Processes and The Maximum Work Theorem PROBLEMS 471 Repeat the calculation of Example 5 assuming the working substance of the auxiliary system to be I mole of an ideal van der Waals fluid rather than of a monatomic ideal gas recall Section 35 472 Calculate the work and heat transfers in each stage of the Carnot cycle for the auxiliary system being an empty cylinder containing only electromagnetic radiation The first step of the cycle is again specified to be an expansion from VA to VB All results are to be expressed in terms of VA VB Th and Tc Show that the ratio of the total work transfer to the firststage heat transfer agrees with the Carnot efficiency 473 A primary subsystem in the initial state A is to be brought reversibly to a specified final state B A reversible work source and a thermal reservoir at temperature Tr are available but no auxiliary system is to be employed Is it possible to devise such a process Prove your answer Discuss Problem 452 in this context 474 The fundamental equation of a particular fluid is UN V AS R 3 where A 2 X 102 K 3mJ 3 Two moles of this fluid are used as the auxiliary system in a Carnot cycle operating between two thermal reservoirs at temperature 100C and 0C In the first isothermal expansion 106 J is extracted from the hightemperature reservoir Find the heat transfer and the work transfer for each of the four processes in the Carnot cycle Calculate the efficiency of the cycle directly from the work and heat transfers just computed Does this efficiency agree with the theoretical Carnot efficiency Hint Carnot cycle problems generally are best discussed in terms of a T S diagram for the auxiliary system 47S One mole of the simple paramagnetic model system of equation 366 is to be used as the auxiliary system of a Carnot cycle operating between reservoirs of temperature Th and Tc The auxiliary system initially has a magnetic moment I and is at temperature Th By decreasing the external field while the system is in contact with the high temperature reservoir a quantity of heat Q1 is absorbed from the reservoir the system meanwhile does work W1 on the reversible work source ie on the external system that creates the magnetic field and thereby induces the magnetic moment Describe each step in the Carnot cycle and calculate the work and heat transfer in each step expressmg each in terms of Th T0 Q1 and the parameters T0 and 0 appearmg in the fundamental equation 476 Repeat Problem 474 using the rubber band model of Section 3 7 as the auxiliary system Measurahlrtv of the Temperature and of the Entropy 123 48 MEASURABILITY OF THE TEMPERATURE AND OF THE ENTROPY The Carnot cycle not only illustrates the general principle of reversible processes as maximum work processes but it provides us with an oper ational method for measurements of temperature We recall that the entropy was introduced merely as an abstract function the maxima of which determine the equilibrium states The temperature was then defined in terms of a partial derivative of this function It is clear that such a definition does not provide a direct recipe for an operational measurement of the temperature and that it is necessary therefore for such a procedure to be formulated explicitly In our discussion of the efficiency of thermodynamic engines we have seen that the efficiency of an engine working by reversible processes between two systems of temperatures Th and is 419 The thermodynamic engine efficiency is defined in terms of fluxes of heat and work and is consequently operationally measurable Thus a Carnot cycle provides us with an operat10nal method of measuring the ratio of two temperatures Unfortunately real processes are never truly quasistatic so that real engines never quite exhibit the theoretical engine efficiency Therefore the ratio of two given temperatures must actually be determined in terms of the limiting maximum efficiency of all real engines but this is a difficulty of practice rather than of principle The statement that the ratio of temperatures is a measurable quantity is tantamount to the statement that the scale of temperature is determined within an arbitrary multiplicative constant The temperature of some arbitrarily chosen standard system may be assigned at will and the temperatures of all other systems are then uniquely determined with values directly proportional to the chosen temperature of the fiducial system The choice of a standard system and the arbitrary assignment of some definite temperature to it has been discussed in Section 26 We recall that the assignment of the number 27316 to a system of ice water and vapor in mutual equilibrium leads to the absolute Kelvin scale of temperature A Carnot cycle operating between this system and another system de termines the ratio of the second temperature to 27316 K and conse quently determines the second temperature on the absolute Kelvin scale Having demonstrated that the temperature is operationally measurable we are able almost trivially to corroborate that the entropy too is measur able The ability to measure the entropy underlies the utility of the entire 124 Reversible Processes and The Maximum Work Theorem thermodynamic formalism It is also of particular interest because of the somewhat abstract nature of the entropy concept The method of measurement to be described yields only entropy differences or relative entropiesthese differences are then converted to absolute entropies by Postulate IVthe N ernst postulate Section 110 Consider a reversible process in a composite system of which the system of interest is a subsystem The subsystem is taken from some reference state T0 P0 to the state of interest T 1 P1 by some path in the TP plane The change in entropy is 420 fTiP1 as aP dTdP CIoPo aP T aT s 421 Equation 421 follows from the elementary identity A22 of Appendix A Equation 422 is less obvious though the general methods to be developed in Chapter 7 will reduce such transformations to a straightforward proce dure an elementary but relatively cumbersome procedure is suggested in Problem 481 Now each of the factors in the integrand is directly measurable the factor a P aT s requires only a measurement of pressure and tempera ture changes for a system enclosed by an adiabatic wall Thus the entropy difference of the two arbitrary states T0 P0 and T1 P1 is obtainable by numerical integration of measurable data PROBLEMS 481 To corroborate equation 422 show that First consider the righthand side and write generally that dT u 55 ds udv so that Other Cnterw of Engine Performance 125 Similarly show that r uuvu u establishing the required iden tity 49 OTHER CRITERIA OF ENGINE PERFORMANCE POWER OUTPUT AND ENDOREVERSIBLE ENGINES As we have remarked earlier maximum efficiency is not necessarily the primary concern in design of a real engine Power output simplicity low initial cost and various other considerations are also of importance and of course these are generally in conflict An informative perspective on the criteria of real engine performance is afforded by the endoreversible engine problem 5 Let us suppose once again that two thermal reservoirs exist at tempera tures Th and Tc and that we wish to remove heat from the high temperature reservoir delivering work to a reversible work source We now know that the maximum possible efficiency is obtained by any reversible engine However considerations of the operation of such an engine immediately reveals that its power output work delivered per unit time is atrocious Consider the very first stage of the process in which heat is transferred to the system from the hot reservoir If the working fluid of the engine is at the same temperature as the reservoir no heat will flow whereas if it is at a lower temperature the heat flow process and hence the entire cycle becomes irreversible In the Carnot engine the temperature difference is made infinitely small resulting in an in finitely slow process and an infinitely small power output To obtain a nonzero power output the extraction of heat from the high temperature reservoir and the insertion of heat into the low temperature reservoir must each be done irreversibly An endoreversible engine is defined as one in which the two processes of heat transfer from and to the heat reservoirs are the only irreversible processes in the cycle To analyze such an engine we assume as usual a high temperature thermal reservoir at temperature Th a low temperature thermal reservoir at temperature Tc and a reversible work source We assume the isothermal strokes of the engine cycle to be at Tw w designating warm and T t designating tepid with Th Tw T T Thus heat flows from the high temperature reservoir to the working fluid across a temperature difference of Th Tw as indicated schematically in Fig 48 Similarly in the heat rejection stroke of the cycle the heat flows across the temperature difference T Tc 5F L Curzon and B Ahlborn Amer J Phys 43 22 1975 See also M H Rubin Phys Rev A19 1272 and 1279 1979 and references therein for a sophisticated analysis and for further generalization of the theorem 126 Reversible Processes and The Maximum Work Theorem t T T T 0 s FIGURE48 The endoreversible engine cycle Let us now suppose that the rate of heat flow from the high temperature reservoir to the system is proportional to the temperature difference Th Tw If th is the time required to transfer an amount Qh of energy then 423 where oh is the conductance the product of the thermal conductivity times the area divided by the thickness of the wall between the hot reservoir and the working fluid A similar law holds for the rate of heat flow to the cold reservoir Therefore the time required for the two isothermal strokes of the engine is 424 We assume the time required for the two adiabatic strokes of the engine to be negligible relative to th t c as these times are limited by relatively rapid relaxation times within the working fluid itself Furthermore the relaxation times within the working fluid can be shortened by appropriate design of the piston and cylinder dimensions internal baffles and the like Now Qh Qc and the delivered work W are related by the Carnot efficiency of an engine working between the temperatures Tw and T so that equation 424 becomes t w 1 1 Tw 1 1 T Jh Th Tw Tw T Jc T Tc Tw T 425 Problems 127 The power output of the engine is W t and this quantity is to be maximized with respect to the two as yet undetermined temperatures T and T The optimum intermediate temperatures are then found to be where c T cTJ12 ohTh112 ocTJl2 o2 o12 and the optimum power delivered by the engine is 426 427 428 Let Eerp denote the efficiency of such an endoreversible engine maxi mized for power for which we find 12 Eerp 1 TJTh 429 Remarkably the engine efficiency is not dependent on the conductances Jh and oc Large power plants are evidently operated close to the criterion for maximum power output as Curzon and Ahlborn demonstrate by data on three power plants as shown in Table 41 TABLE4l Efficiencies of Power Plants as Compared with the Carnot Efficiency and with the Efficiency of an Endoreversible Engine Maximized for Power Output terp 6 T Th E E Power Plant OC OC Carnot Eerp observed West Thurrock UK coal fired steam plant 25 565 064 040 0 36 CANDU Canada PHW nuclear reactor 25 300 048 028 030 Larderello Italy geothermal steam plant 80 250 032 0175 016 From Curzon and Ahlborn PROBLEMS 491 Show that the efficiency of an endoreversible engine maximized for power output is always less than Ecarno Plot the former efficiency as a function of the Carnot efficiency 128 Reversible Processes and The Maximum Work Theorem 492 Suppose the conductance ah aJ to be such that I kW is transferred to the system as heat flux if its temperature is 50 K below that of the high temperature reservoir Assuming Th 800 K and T 300 K calculate the maximum power obtainable from an endoreversible engine and find the tempera tures Tw and T for which such an engine should be designed 493 Consider an endoreversible engine for which the high temperature reservoir is boiling water 100C and the cold reservoir is at room temperature taken as 20C Assuming the engine is operated at maximum power what is the ratio of the amount of heat withdrawn from the high temperature reservoir per kilowatt hour of delivered work to that withdrawn by a Carnot engine How much heat is withdrawn by each engine per kilowatt hour of delivered work Answer Ratio 19 494 Assume that one cycle of the engine of Problem 493 takes 20 s and that the conductance ah ac 100 W K How much work is delivered per cycle Assuming the control volume ie the auxiliary system is a gas driven through a Carnot cycle plot a T S diagram of the gas during the cycle Indicate numerical values for each vertex of the diagram note that one value of the entropy can be assigned arbitrarily 410 OTHER CYCLIC PROCESSES In addition to Carnot and endoreversible engines various other engines are of interest as they conform more or less closely to the actual operation of commonplace practical engines The Otto cycle or more precisely the airstandard Otto cycle is a rough approximation to the operation of a gasoline engine The cycle is shown in Fig 49 in a VS diagram The working fluid a mixture of air and gasoline vapor in the gasoline engine is first compressed adiabatically t Sc LJ s SB B A VB Vi FIGURE49 v The Otto cycle Other Cyclc Processe5 129 A B It is then heated at constant volume B C this step crudely describes the combustion of the gasoline in the gasoline engine In the third step of the cycle the working fluid is expanded adiabatically in the power stroke C D Finally the working fluid is cooled isochorically to its initial state A In a real gasoline engine the working fluid chemically reacts burns during the process B C so that its mole number changesan effect not represented in the Otto cycle Furthermore the initial adiabatic compression is not quasistatic and therefore is certainly not isentropic Nevertheless the idealized airstandard Otto cycle does provide a rough perspective for the analysis of gasoline engines In contrast to the Carnot cycle the absorption of heat in step B C of the idealized Otto cycle does not occur at constant temperature Therefore the ideal engine efficiency is different for each infinitesimal step and the overall efficiency of the cycle must be computed by integration of the Carnot efficiency over the changing temperature It follows that the efficiency of the Otto cycle depends upon the particular properties of the working fluid It is left to the reader to corroborate that for an ideal gas with temperature independent heat capacities the Otto cycle efficiency is 430 The ratio VA V8 is called the compression ratio of the engine The Brayton or Joule cycle consists of two isentropic and two isobaric steps It is shown on a PS diagram in Fig 410 In a working engine air and fuel is compressed adiabatically A B heated by fuel combus tion at constant pressure B C expanded C D and rejected to the atmosphere The process D A occurs outside the engine and a fresh charge of air is taken in to repeat the cycle If the working gas is an ideal gas with temperature independent heat capacities the efficiency of a D C 1 D A B FJGURE410 p The Brayton or Joule cycle 130 Reversible Processes and The Maximum Work Theorem Brayton cycle is ppAB cpcp c Ee 431 The airstandard diesel cycle consists of two isentropic processes alter nating with isochoric and isobaric steps The cycle is represented in Fig 411 After compression of the air and fuel mixture A B the fuel combustion occurs at constant pressure B C The gas is adiabatically expanded C D and then cooled at constant volume D A C t t p T B A SA Sc sp FIGURE 411 The airstandard diesel cycle PROBLEMS 4101 Assuming that the working gas is a monatomic ideal gas plot a T S diagram for the Otto cycle 4102 Assuming that the working gas is a simple 1dedl gas with temperature independent heat capacities show that the engine efficiency of the Otto cycle is given by equation 430 4103 Assuming that the working gas is a simple ideal gas with temperature independent heat capacities show that the engine efficiency of the Brayton cycle is given by equation 431 4104 Assuming that the working gas is a monatofllic ideal gas plot a T S diagram of the Brayton cycle 4105 Assuming that the working gas is a monatofllic ideal gas plot a T S diagram of the airstandard diesel cycle 5 ALTERNATIVE FORMULATIONS AND LEGENDRE TRANSFORMATIONS 51 THE ENERGY MINIMUM PRINCIPLE In the preceding chapters we have inf erred some of the most evident and immediate consequences of the principle of maximum entropy Fur ther consequences will lead to a wide range of other useful and fundamen tal results But to facilitate those developments it proves to be useful now to reconsider the formal aspects of the theory and to note that the same content can be reformulated in several equivalent mathematical forms Each of these alternative formulations is particularly convenient in par ticular types of problems and the art of thermodynamic calculations lies largely in the selection of the particular theoretical formulation that most incisively fits the given problem In the appropriate formulation ther modynamic problems tend to be remarkably simple the converse is that they tend to be remarkably complicated in an inappropriate formalism Multiple equivalent formulations also appear in mechanicsNewto nian Lagrangian and Hamiltonian formalisms are tautologically equiv alent Again certain problems are much more tractable in a Lagrangian formalism than in a Newtonian formalism or vice versa But the dif ference in convenience of different formalisms is enormously greater in thermodynamics It is for this reason that the general theory of transforma tion among equivalent representations is here incorporated as a fundamental aspect of thermostatistical theory In fact we have already considered two equivalent representations the energy representation and the entropy representation But the basic ex tremum principle has been formulated only in the entropy representation If these two representations are to play parallel roles in the theory we must find an extremum principle in the energy representation analogous to the entropy maximum principle There is indeed such an extremum principle the principle of maximum entropy is equivalent to and can be 131 132 Alternatwe Formuatons and Legendre Transformatwns The plane U U0 u FIGURE 5 I xl J The equilibrium state A as a point of maximum S for constant U replaced by a principle of minimum energy Whereas the entropy maxi mum principle characterizes the equilibrium state as having maximum entropy for given total energy the energy minimum principle char acterizes the equilibrium state as having minimum energy for given total entropy Figure 51 shows a section of the thermodynamic configuration space for a composite system as discussed in Section 41 The axes labeled S and U correspond to the total entropy and energy of the composite system and the axis labeled X1 corresponds to a particular extensive parameter of the first subsystem Other axes not shown explicitly in the figure are u1 X1 and other pairs xp Xk The total energy of the composite system is a constant determined by the closure condition The geometrical representation of this closure condition is the requirement that the state of the system lie on the plane U U0 in Fig 51 The fundamental equation of the system is repre sented by the surface shown and the representative point of the system therefore must be on the curve of intersection of the plane and the surface If the parameter XIJ is unconstrained the equilibrium state is the particular state that maximizes the entropy along the permitted curve the state labeled A in Fig 51 The alternative representation of the equilibrium state A as a state of minimum energy for given entropy is illustrated in Fig 52 Through the The plane SS 0 u FIURE 5 2 The Energy Mmmum Prmczple 133 s The equilibrium state A as a point of minimum U for constant S equilibrium point A is passed the plane S S0 which determines a curve of intersection with the fundamental surface This curve consists of a family of states of constant entropy and the equilibrium state A is the state that minimizes the energy along this curve The equivalence of the entropy maximum and the energy minimum principles clearly depends upon the fact that the geometrical form of the fundamental surface is generally as shown in Fig 51 and 52 As dis cussed in Section 41 the form of the surface shown in the figures is determined by the postulates that as au o and that u is a singleval ued continuous function of S these analytic postulates accordingly are the underlying conditions for the equivalence of the two principles To recapitulate we have made plausible though we have not yet proved that the following two principles are equivalent Entropy Maximum Principle The equilibrium value of any unconstrained internal parameter is such as to maximize the entropy for the given value of the total internal energy Energy Minimum Principle The equihbrium value of any unconstrained internal parameter is such as to minimize the energy for the given value of the total entropy 134 Alternatwe Formulatwns and Legendre Transformatwm The proof of the equivalence of the two extremum criteria can be formulated either as a physical argument or as a mathematical exercise We turn first to the physical argument to demonstrate that if the energy were not minimum the entropy could not be maximum in equilibrium and inversely Assume then that the system is in equilibrium but that the energy does not have its smallest possible value consistent with the given entropy We could then withdraw energy from the system in the form of work maintaining the entropy constant and we could thereafter return this energy to the system in the form of heat The entropy of the system would increase dQ T dS and the system would be restored to its original energy but with an increased entropy This is inconsistent with the principle that the initial equilibrium state is the state of maximum entropy Hence we are forced to conclude that the original equilibrium state must have had minimum energy consistent with the prescribed entropy The inverse argument that minimum energy implies maximum entropy is similarly constructed see Problem 511 In a more formal demonstration we assume the entropy maximum principle L 0 and a2s 0 ax2 u 51 where for clarity we have written X for X1 and where it is implicit that all other Xs are held constant throughout Also for clarity we tempo rarily denote the first derivative au a X s by P Then by equation A22 of Appendix A T as o ax u 52 We conclude that U has an extremum To classify that extremum as a maximum a minimum or a point of inflection we must study the sign of the second derivativea 2uax 2s aPaXs But considering Pas a function of U and X we have L L j L u xp v 53 at P 0 54 The Energy Minimum Pnnc1ple 135 i ul 55 ax as au x u a2s a2s ax2 as axau as ax as 2 s6 au au a2s T ax2 0 as at0 ax so that U is a minimum The inverse argument is identical in form 57 As already indicated the fact that precisely the same situation is described by the two extremal criteria is analogous to the isoperimetric problem in geometry Thus a circle may be characterized either as the two dimensional figure of maximum area for given perimeter or alternatively as the two dimensional figure of minimum perimeter for given area The two alternative extremal criteria that characterize a circle are completely equivalent and each applies to every circle Yet they suggest two different ways of generating a circle Suppose we are given a square and we wish to distort it continuously to generate a circle We may keep its area constant and allow its bounding curve to contract as if it were a rubber band We thereby generate a circle as the figure of minimum perimeter for the given area Alternatively we might keep the perimeter of the given square constant and allow the area to increase thereby obtain ing a different circle as the figure of maximum area for the given perimeter However after each of these circles is obtained each satisfies both extremal conditions for its final values of area and perimeter The physical situation pertaining to a thermodynamic system is very closely analogous to the geometrical situation described Again any equilibrium state can be characterized either as a state of maximum entropy for given energy or as a state of minimum energy for given entropy But these two criteria nevertheless suggest two different ways of attaining equilibrium As a specific illustration of these two approaches to equilibrium consider a piston originally fixed at some point in a closed cylinder We are interested in bringing the system to equilibrium without the constraint on the position of the piston We can simply remove the constraint and allow the equilibrium to establish itself spontaneously the entropy increases and the energy is maintained constant by the closure condition This is the process suggested by the entropy maximum princi ple Alternatively we can permit the piston to move very slowly reversi 136 Alternatve Formulatwns and Legendre Transformations bly doing work on an external agent until it has moved to the position that equalizes the pressure on the two sides During this process energy is withdrawn from the system but its entropy remains constant the process is reversible and no heat flows This is the process suggested by the energy minimum principle The vital fact we wish to stress however is that independent of whether the equilibrium is brought about by either of these two processes or by any other process the final equilibrium state in each case satisfies both extremal conditions Finally we illustrate the energy minimum principle by using it in place of the entropy maximum principle to solve the problem of thermal equilibrium as treated in Section 24 We consider a closed composite system with an internal wall that is rigid impermeable and diathermal Heat is free to flow between the two subsystems and we wish to find the equilibrium state The fundamental equation in the energy representation is All volume and mole number parameters are constant and known The variables that must be computed are s1 and S2 Now despite the fact that the system is actually closed and that the total energy is fixed the equilibrium state can be characterized as the state that would minimize the energy if energy changes were permitted The virtual change in total energy associated with virtual heat fluxes in the two systems is dU r1 ds1 r2 ds2 59 The energy minimum condition states that dU 0 subject to the condi tion of fixed total entropy Sl Sl Constant 510 whence dU r 1 r2ds1 0 511 and we conclude that r1 r2 512 The energy minimum principle thus provides us with the same condi tion of thermal equilibrium as we previously found by using the entropy maximum principle Equation 512 is one equation in s1 and s2 The second equation is most conveniently taken as equation 58 in which the total energy U is Legendre Transformatwm 137 known and which consequently involves only the two unknown quantities s1 and S2 Equations 58 and 512 in principle permit a fully explicit solution of the problem In a precisely analogous fashion the equilibrium condition for a closed composite system with an internal moveable adiabatic wall is found to be equality of the pressure This conclusion is straightforward in the energy representation but as was observed in the last paragraph of Section 27 it is relatively delicate in the entropy representation PROBLEMS 511 Formulate a proof that the energy minimum principle implies the entropy maximum principlethe inverse argument referred to after equation 57 That is show that if the entropy were not maximum at constant energy then the energy could not be minimum at constant entropy Hint First show that the permissible mcrease in entropy in the system can be exploited to extract heat from a reversible heat source initially at the same temperature as the system and to deposit it in a reversible work source The reversible heat source is thereby cooled Continue the argument 512 An adiabatic impermeable and fixed piston separates a cylinder into two chambers of volumes V04 and 3V04 Each chamber contains 1 mole of a monatomic ideal gas The temperatures are T and the subscripts s and I referring to the small and large chambers respectively a The piston is made thermally conductive and moveable and the system relaxes to a new equilibrium state maximizing its entropy while conserving its total energy Find this new equilibrium state b Consider a small virtual change in the energy of the system maintaining the entropy at the value attained in part a To accomplish this physically we can reimpose the adiabatic constraint and quasistatically displace the piston by imposition of an external force Show that the external source of this force must do work on the system in order to displace the piston in either direction Hence the state attamed in part a is a state of minimum energy at constant entropy c Reconsider the initial state and specify how equilibrium can be established by decreasing the energy at constant entropy Find this equilibrium state d Describe an operation that demonstrates that the equilibrium state attained in c is a state of maximum entropy at constant energy 52 LEGENDRE TRANSFORMATIONS In both the energy and entropy representations the extensive parame ters play the roles of mathematically independent variables whereas the intensive parameters arise as derived concepts This situation is in direct 138 Alternative Formulations and Legendre Transformations contrast to the practical situation dictated by convenience in the labora tory The experimenter frequently finds that the intensive parameters are the more easily measured and controlled and therefore is likely to think of the intensive parameters as operationally independent variables and of the extensive parameters as operationally derived quantities The extreme instance of this situation is provided by the conjugate variables entropy and temperature No practical instruments exist for the measurement and control of entropy whereas thermometers and thermostats for the mea surement and control of the temperature are common laboratory equipment The question therefore arises as to the possibility of recasting the mathematical formalism in such a way that intensive parameters will replace extensive parameters as mathematically independent variables We shall see that such a reformulation is in fact possible and that it leads to various other thermodynamic representations It is perhaps superfluous at this point to stress again that thermody namics is logically complete and selfcontained within either the entropy or the energy representations and that the introduction of the transformed representations is purely a matter of convenience This is admittedly a convenience without which thermodynamics would be almost unusably awkward but in principle it is still only a luxury rather than a logical necessity The purely formal aspects of the problem are as follows We are given an equation the fundamental relation of the form Y YX 0 X Xi 513 and it is desired to find a method whereby the derivatives 514 can be considered as independent variables without sacrificing any of the informational content of the given fundamental relation513This formal problem has its counterpart in geometry and in several other fields of physics The solution of the problem employing the mathematical tech nique of Legendre transformations is most intuitive when given its geometrical interpretation and it is this geometrical interpretation that we shall develop in this Section For simplicity we first consider the mathematical case in which the fundamental relation is a function of only a single independent vari able X Y YX 515 Geometrically the fundamental relation is represented by a curve in a Legendre Transformatons 139 y X FIGURE 53 space Fig 53 with cartesian coordinates X and Y and the derivative 516 is the slope of this curve Now if we desire to consider P as an independent variable in place of X our first impulse might be simply to eliminate X between equations 515 and 516 thereby obtaining Y as a function of P Y YP 517 A moments reflection indicates however that we would sacrifice some of the mathematical content of the given fundamental relation 515 for from the geometrical point of view it is clear that knowledge of Y as a function of the slope dY dX would not permit us to reconstruct the curve Y Y X In fact each of the displaced curves shown in Fig 54 corresponds equally well to the relation Y Y P From the analytical point of view the relation Y Y P is a firstorder differential equation and its integration gives Y Y X only to within an undetermined integration constant Therefore we see that acceptance of Y YP as a basic equation in place of Y Y X would involve the sacrifice of some information originally contained in the fundamental relation Despite the y X FIGURE 54 140 Alternatwe Formulations and Legendre Transformations X FIGURE 55 desirability of having P as a mathematically independent variable this sacrifice of the informational content of the formalism would be com pletely unacceptable The practicable solution to the problem is supplied by the duality between conventional point geometry and the Pluecker line geometry The essential concept in line geometry is that a given curve can be represented equally well either a as the envelope of a family of tangent lines Fig 55 or b as the locus of points satisfying the relation Y Y X Any equation that enables us to construct the family of tangent lines therefore determines the curve equally as well as the relation Y Y X Just as every point in the plane is described by the two numbers X and Y so every straight line in the plane can be described by the two numbers P and where P is the slope of the line and is its intercept along the Yaxis Then just as a relation Y Y X selects a subset of all possible points X Y a relation P selects a subset of all possible lines P A knowledge of the intercepts of the tangent lines as a function of the slopes P enables us to construct the family of tangent lines and thence the curve of which they are the envelope Thus the relation lP 518 is completely equivalent to the fundamental relation Y Y X In this Legendre Tramformatwns 141 relation the independent variable is P so that equation 518 provides a complete and satisfactory solution to the problem As the relation iJ J P is mathematically equivalent to the relation Y Y X it can also be considered a fundamental relation Y Y X is a fundamental rela tion in the representation whereas iJ JP is a fundamental relation in the Jrepresentation The reader is urged at this point actually to draw a reasonable number of straight lines of various slopes P and of various intercepts iJ P 2 The relation iJ P 2 thereby will be seen to characterize a parabola which is more conventionally described as Y i X 2 In Jrepresentation the fundamental equation of the parabola is iJ P 2 whereas in rep resentation the fundamental equation of this same parabola is Y iX 2 The question now arises as to how we can compute the relation iJ J P if we are given the relation Y Y X The appropriate mathematical operation is known as a Legendre transformation We consider a tangent line that goes through the point X Y and has a slope P If the intercept is J we have see Fig 56 or t y y iJ p X0 Let us now suppose that we are given the equation Y YX 0IJ x FIGURE 56 519 520 521 142 Alternative Formulatons and Legendre Tramformattom and by differentiation we find P PX 522 Then by elimination 1 of X and Y among equations 520 521 and 522 we obtain the desired relation between and P The basic identity of the Legendre transformation is equation 520 and this equation can be taken as the analytic definition of the function f The function is referred to as a Legendre transform of Y The inverse problem is that of recovering the relation Y Y X if the relation P is given We shall see here that the relationship between X Y and P is symmetrical with its inverse except for a sign in the equation of the Legendre transformation Taking the differen tial of equation 520 and recalling that dY P dX we find or df dY PdX XdP XdP x df dP 523 524 If the two variables and P are eliminated 2 from the given equation P and from equations 524 and 520 we recover the relation Y Y X The symmetry between the Legendre transformation and its inverse is indicated by the following schematic comparison Y YX p dY dX fPXY Elimination of X and Y yields P fP x df dP Y XP Elimination of P and yields Y YX The generalization of the Legendre transformation to functions of more than a single independent variable is simple and straightforward In three dimensions Y is a function of X0 and X1 and the fundamental equation represents a surface This surface can be considered as the locus of points 1TJus ehmmat10n 1s po1blc 1f P 1s not independent of X that 1s 1f d 2 YdX 2 0 In the thermodynamic application this cntenon will tum out to be 1den1Ical to the cntenon of tab1hty The en tenon fuls only at the cntical pomt whJCh arc dcuscd m detail m Chapter IO 2 The cond1t10n that th1 be possible 1s that d 2JiP 2 4 0 which will m the thermodynamic application be guaranteed by the stab1hty of the system under cons1derallon Legendre Transformattom 143 satisfying the fundamental equation Y YX 0 X1 or it can be consid ered as the envelope of tangent planes A plane can be characterized by its intercept 1 on the Yaxis and by the slopes P0 and P1 of its traces on the Y X 0 and Y X1 planes The fundamental equation then selects from all possible planes a subset described by 1 vP 0 P1 In general the given fundamental relation Y YX 0 X1 X 525 represents a hypersurface in a t 2dimensional space with cartesian coordinates Y X 0 X1 X The derivative 526 is the partial slope of this hypersurface The hypersurface may be equally well represented as the locus of points satisfying equation 525 or as the envelope of the tangent hyperplanes The family of tangent hyperplanes can be characterized by giving the intercept of a hyperplane 1 as a function of the slopes P0 P1 P Then 527 Taking the differential of this equation we find 528 whence 529 A Legendre transformation is effected by eliminating Y and the X1 from Y Y X0 X1 X the set of equations 526 and equation 527 The inverse transformation is effected by eliminating 1 and the P1 from 1 vP 1 P2 Pr the set of equations 529 and equation 527 Finally a Legendre transformation may be made only in some n 2 dimensional subspace of the full t 2dimensional space of the relation Y Y X 0 X1 X Of course the subspace must contain the Ycoor dinate but may involve any choice of n 1 coordinates from the set X 0 X1 X For convenience of notation we order the coordmates so that the Legendre transformation is made in the subspace of the first n 1 coordinates and of Y the coordinates Xni X X trf Jpft 144 Alternative Formulations and Legendre Transformattons untransformed Such a partial Legendre transformation is effected merely by considering the variables Xn 1 Xn1 2 X as constants in the trans formation The resulting Legendre transform must be denoted by some explicit notation that indicates which of the independent variables have participated in the transformation We employ the notation YP 0 Pi Pn to denote the function obtained by making a Leg endre transformation with respect to X 0 X1 X 11 on the function Y X0 Xi X Thus Y P0 P 1 Pn is a function of the independent variables P0 Pi Pn Xn 1 X The various relations involved in a partial Legendre transformation and its inverse are indicated in the following table Y YX 0 Xi X The partial differentiation denotes constancy of all the natural varia bles of Yother than Xk ie of all X1 with j k n YP 0 Pn Y LPkXk 0 Elimination of Y and X 0 Xi xn from equations 530 533 and the first n 1 equations of 531 yields the transformed fundamental relation YP 0 P 1 P11 function of Po P1 Pn X11 f I X 530 ay Po p11 Xk aP k ksn 531 aYP 0 Pn P axk kn The partial differentiation denotes constancy of all the natural varia bles of YP0 Pn other than that with respect to which the differentiation is being carried out dYP 0 P II t XkdPk L PkdX 0 nl 532 II 0 533 Elimination of YP 0 P and P 0 Pi Pn from equations 530 533 and the first n 1 equations of 531 yields the origi nal fundamental relation In this section we have divorced the mathematical aspects of Legendre transformations from the physical applications Before proceeding to the Problems 145 thermodynamic applications in the succeeding sections of this chapter it may be of interest to indicate very briefly the application of the formalism to Lagrangian and Hamiltonian mechanics which perhaps may be a more familiar field of physics than thermodynamics The Lagrangian principle guarantees that a particular function the Lagrangian completely char acterizes the dynamics of a mechanical system The Lagrangian is a function of 2r variables r of which are generaltzed coordinates and r of which are generalized velocities Thus the equation 534 plays the role of a fundamental relation The generalized momenta are defined as derivatives of the Lagrangian function p aL k av k 535 If it is desired to replace the velocities by the momenta as independent variables we must make a partial Legendre transformation with respect to the velocities We thereby introduce a new function called the Hamilto nian defined by 3 r 536 A complete dynamical formalism can then be based on the new funda mental relation 537 Furthermore by equation 531 the derivative of H with respect to Pk is the velocity vk which is one of the Hamiltonian dynamical equations Thus if an equation of the form 534 is considered as a dynamical fundamental equation in the Lagrangian representation the Hamiltonian equation 537 is the equivalent fundamental equation expressed in the Hamiltonian representation PROBLEMS 521 The equation y x 210 describes a parabola a Find the equation of this parabola in the line geometry representation if ifP b On a sheet of graph paper covering the range roughly from x 15 to x 15 and from y 25 to y 25 draw straight lines with slopes P 0 3 1n our Ulage the Legendre transform of the Lagrangian u the neputwe H1m1ltoman Actually the accepted mathemahcal convenhon agrees with the usage m mechamc and the function J would be called the Legendre transform of Y 146 Alternatve Formulattons and Legendre Transformatwns 05 I 2 3 and with intercepts i satisfying the relationship i iP as found in part a Drawing each straight line is facilitated by calculating its intercepts on the xaxis and on the yaxis 522 Let y Ae 8 x a Find iP b Calculate the inverse Legendre transform of i P and corroborate that this result is yx c Taking A 2 and B 05 draw a family of tangent lines in accordance with the result found in a and check that the tangent curve goes through the expected points at x 0 1 and 2 53 THERMODYNAMIC POTENTIALS The application of the preceding formalism to thermodynamics is selfevident The fundamental relation Y Y X0 X1 can be inter preted as the energylanguage fundamental relation U US X1 X 2 X or U US V N1 N2 The derivatives P0 P 1 correspond to the intensive parameters T P µ 1 µ 2 The Legendre transformed functions are called thermodynamic potentials and we now specifically define several of the most common of them In Chapter 6 we continue the discussion of these functions by deriving extremum princi ples for each potential indicating the intuitive significance of each and discussing its particular role in thermodynamic theory But for the mo ment we concern ourselves merely with the formal aspects of the defini tions of the several particular functions The Helmholtz potential or the Helmholtz free energy is the partial Legendre transform of U that replaces the entropy by the temperature as the independent variable The internationally adopted symbol for the Helmholtz potential is F The natural variables of the Helmholtz potential are T V N 1 N2 That is the functional relation F F T V N1 N2 constitutes a fundamental relation In the systematic notation introduced in Section 52 F UT 538 The full relationship between the energy representation and the Helmholtz representation is summarized in the following schematic com panson U US V N1 N2 T auas F U TS Elimination of U and S yields F FT V N 1 N2 F FTVN 1N 2 s aFaT U F TS Elimination of F and T yields U US V N 1 N2 539 540 541 Thermodynamic Potenttals 147 The complete differential dF is 542 The enthalpy is that partial Legendre transform of U that replaces the volume by the pressure as an independent variable Following the recom mendations of the International Unions of Physics and of Chemistry and in agreement with ahnost universal usage we adopt the symbol H for the enthalpy The natural variables of this potential are S P Ni N2 and H UP 543 The schematic representation of the relationship of the energy and en thalpy representations is as follows U U S V N 1 N2 P auav H U PV Elimination of U and V yields H HSPNiN 2 H HS P Ni N2 V aHaP U H PV Elimination of H and P yields U USVNiN 2 544 545 546 Particular attention is called to the inversion of the signs in equations 545 and 546 resulting from the fact that P is the intensive parameter associated with V The complete differential dH is dH TdS VdP PidN 1 µ 2 dN2 547 The third of the common Legendre transforms of the energy is the Gibbs potential or Gibbs free energy This potential is the Legendre transform that simultaneously replaces the entropy by the temperature and the volume by the pressure as independent variables The standard notation is G and the natural variables are T P N1 N2 We thus have and G UTP U US V N1 N2 T auas P auav G U TS PV G GTPN 1N 2 s aGar v aGaP U G TS PV 548 549 550 551 552 Elimination of U S and V yields Elimination of G T and P yields G GT P N1 N2 U US V N 1 N2 J 48 Aternatwe Formulatwns and Legendre Transformatwns The complete differential dG is dG S dT V dP µ1 dN 1 µ 2 dN 2 553 A thermodynamic potential which arises naturally in statistical me chanics is the grand canonical potential UT µ For this potential we have U US V N T auas µ auaN U T µ U TS µN Elimmation of U S and N yields U T µ as a function of T V µ and UT µ function of T V andµ 554 s oUT µ oT 555 N oUT µ oµ 556 U UT µ TS µN 557 Elimination of U T µ T and µ yields U US V N dUTµ SdT PdV Ndµ 558 Other possible transforms of the energy for a simple system which are used only infrequently and which consequently are unnamed are UµiJ UP µi UT µ1 µ2 and so forth The complete Legendre transform is UT P µ1 µ2 Pr1 The fact that US V N1 N2 N is a homoge neous firstorder function of its arguments causes this latter function to vanish identically For 559 which by the Euler relation 36 is identically zero 560 PROlJLEMS 531 Find the fundamental equation of a monatomic ideal gas in the Helmholtz representation in the enthalpy representation and in the Gibbs representation Assume the fundamental equation computed in Section 34 In each case find the equations of state by differentiation of the fundamental equation 532 Find the fundamental equation of the ideal van der Waals fluid Section 35 in the Helmholtz representation Perform an inverse Legendre transform on the Helmholtz potential and show that the fundamental equation in the energy representation is recovered Problems 149 533 Find the fundamental equation of electromagnetic radiation in the Helm holtz representation Calculate the thermal and mechanical equations of state and corroborate that they agree with those given in Section 36 534 4 Justify the following recipe for obtaining a plot of FV from a plot of GP the common dependent variables T and N being notationally suppressed for convenience t G A t F l 1 I I I I p D fB I c FV V 1 At a chosen value of P draw the tangent line A 2 Draw horizontal lines B and C through the intersections of A with P 1 and p 0 3 Draw the 45 line D as shown and project the intersection of B and D onto the line C to obtain the point F V Hint Identify the magnitude of the two vertical distances indicated in the G versus P diagram and also the vertical separation of lines B and C Note that the units of F and V are determined by the chosen units of G and P Explain Give the analogous construction for at least one other pair of potentials Note that G P is drawn as a concave function ie negative curvature and show that this is equivalent to the statement that Ky 0 535 From the first acceptable fundamental equation in Problem 1101 calcu late the fundamental equation in Gihbs representation Calculate aT P K 7 T P and cPT P by differentiation of G 536 From the second acceptable fundamental equation in Problem 1101 calculate the fundamental equation in enthalpy representation Calculate VS P N by differentiation 537 The enthalpy of a particular system is H AS 2N 11n 4Adapted from H E Stanley lntroductwn to Phase Transctwns and Crttllal Phenomena Oxford Umvcrs1ty Press 1971 150 Alternative Formulatons and Legendre Transformations where A is a positive constant Calculate the molar heat capacity at constant volume cv as a function of T and P 538 In Chapter 15 it is shown by a statistical mechanical calculation that the fundamental equation of a system of N atoms each of which can exist in an atomic state with energy Eu or in an atomic state with energy Ed and in no other state is F Nk 8 T0eJ eJd Here k 8 is Boltzmanns constant and 1 lk 8 T Show that the fundamental equation of this system in entropy representation is where UNE Y u NEdu Hint Introduce 1 k 8 T 1 and show first that U F 1aF ap iJ1Fa3 Also for definiteness assume E Ed and note that Nkn NR where N is the number of atoms and N is the number of moles 539 Show for the twolevel system of Problem 538 that as the temperature increases from zero to infinity the energy increases from NEu to NEu Ed2 Thus at zero temperature all atoms are in their ground state with energy Eu and at infinite temperature the atoms are equally likely to be in either state Energies higher than NEu Ed2 are inaccessible in thermal equilibrium This upper bound on the energy is a consequence of the unphysical oversimplification of the model it will be discussed again in Section 153 Show that the Helmholtz potential of a mixture of simple ideal gases is the sum of the Helmholtz potentials of each individual gas 5310 a Show that the Helmholtz potential of a mixture of simple ideal gases is the sum of the Helmholtz potentials of each individual gas FTVN 1 NFTVN 1 FTVN Recall the fundamental equation of the mixture as given in equation 340 An analogous additivity does not hold for any other potential expressed in terms of its natural variables 5311 A mixture of two monatomic ideal gases is contained in a volume Vat temperature T The mole numbers are N1 and N2 Calculate the chemical potentials µ1 and µ2 Recall Problems 531 and 5310 Assuming the system to be in contact with a reservoir of given T and µ1 through a diathermal wall permeable to the first component but not to the second calculate the pressure in the system Generalized Masseu Functwns 151 5312 A system obeys the fundamental relation s s04 Avu 2 Calculate the Gibbs potential GT P N 5313 For a particular system it is found that u Pv and P AvT 4 Find a fundamental equation the molar Gibbs potential and the Helmholtz potential for this system 5314 For a particular system of 1 mole the quantity v af is known to be a function of the temperature only YT Here v is the molar volume f is the molar Helmholtz potential a is a constant and YT denotes an unspecified function of temperature It is also known that the molar heat capacity cv is cvbvTi where b v is an unspecified function of v a Evaluate YT and bv b The system is to be taken from an initial state T0 v0 to a final state v1 A thermal reservoir of temperature T is available as is a reversible work source What is the maximum work that can be delivered to the reversible work source Note that the answer may involve constants unevaluated by the stated condi tions but that the answer should be fully explicit otherwise 54 GENERALIZED MASSIEU FUNCTIONS Whereas the most common functions definable in terms of Legendre transformations are those mentioned in Section 53 another set can be defined by performing the Legendre transformation on the entropy rather than on the energy That is the fundamental relation in the form S S U V N 1 N2 can be taken as the relation on which the transforma tion is performed Such Legendre transforms of the entropy were invented by Massieu in 1869 and actually predated the transforms of the energy introduced by Gibbs in 1875 We refer to the transforms of the entropy as Massieu functions as distinguished from the thermodynamic potentials transformed from the energy The Massieu functions will tum out to be particularly useful in the theory of irreversible thermodynamics and they also arise naturally in statistical mechanics and in the theory of thermal fluctuations Three representative Massieu functions are SlT in which the internal energy is replaced by the reciprocal temperature as indepen dent variable S P T in which the volume is replaced by P T as independent variable and SlT P T in which both replacements are 152 Alternative Formulations and Legendre Transformations made simultaneously Clearly ssu ssv and s PsuPV TT T T G T 561 562 563 Thus of the three only SP T is not trivially related to one of the previously introduced thermodynamic potentials For this function S S U V N1 N2 PT asav SP T S P TV Elimination of S and V yields S P T as a function of U P T N1 N2 and SP T function of U P T N1 N2 564 v as P T1 ac P T 565 S SPT PTV566 Elimination of SP T and P T yields S SUVN 1N 2 dSPT 1TdU VdPTµ 1TdN 1 7 dN2 567 Other Massieu functions may be invented and analyzed by the reader as a particular need for them arises PROBLEMS 541 Find the fundamental equation of a monatomic ideal gas in the representa tion s Find the equations of state by differentiation of this fundamental equation 542 Find the fundamental equation of electromagnetic radiation Section 36 a in the representation SlT b in the representation S P T 543 Find the fundamental equation of the ideal van der Waals fluid in the representation SlT Show that SlT is equal to FIT recall that F was computed in Problem 532 6 THE EXTREMUM PRINCIPLE IN THE LEGENDRE TRANSFORMED REPRESENTATIONS 61 THE MINIMUM PRINCIPLES FOR THE POTENTIAlS We have seen that the Legendre transformation permits expression of the fundamental equation in terms of a set of independent variables chosen to be particularly convenient for a given problem Clearly how ever the advantage of being able to write the fundamental equation in various representations would be lost if the extremum principle were not itself expressible in those representations We are concerned therefore with the reformulation of the basic extremum principle in forms ap propriate to the Legendre transformed representations For definiteness consider a composite system in contact with a thermal reservoir Suppose further that some internal constraint has been removed We seek the mathematical condition that will permit us to predict the equilibrium state For this purpose we first review the solution of the problem by the energy minimum principle In the equilibrium state the total energy of the composite systemplus reservmr 1s minimum dU U 0 61 and 62 subject to the isentropic condition dS S 0 63 A mum rnnople m the Legendre Transformed Repreentatwns The quantity d 2U has been put equal to zero in equation 62 because d 2 U is a sum of products of the form which vanish for a reservoir the coefficient varying as the reciprocal of the mole number of the reservoir The other closure conditions depend upon the particular form of the internal constraints in the composite system If the internal wall is movable and impermeable we have dN dN2 dv 1 v2 0 for all 1 64 whereas if the internal wall is rigid and permeable to the k th component we have dN N2 dN dN 12 dv1 dv2i 0 k k J J These equations suffice to determine the equilibrium state 1 k 65 The differential dU in equation 61 involves the terms T 11Jds0 T2ds12i which arise from heat flux among the subsystems and the reservoir and terms such as poidvo p2idv 2 and µldNl 1 µ2 dNf which arise from processes within the composite system The terms T1ids 1 T2ldS 12 combine with the term dU TdS in equa tion 61 to yield 0 66 whence Tl T2 T 67 Thus one evident aspect of the final equilibrium state is the fact that the reservoir maintains a constancy of temperature throughout the system The remaining conditions of equilibrium naturally depend upon the specific form of the internal constraint in the composite system To this point we have merely reviewed the application of the energy minimum principle to the composite system the subsystem plus the reservoir We are finally ready to recast equations 61 and 62 into the The M1mmum Pnnuplel for the Potentials 155 language of another representation We rewrite equation 61 dU U dU TdS 0 68 or by equation 63 dU TdS 0 69 or further since T is a constant dU TS 0 610 Similarly since T is a constant and S is an independent variable equation 62 implies 1 611 Thus the quantity U TS is minimum in the equilibrium state Now the quantity U TS is suggestive by its form of the Helmholtz potential U TS We are therefore led to examme further the extremum properties of the quantity U TS and to ask how these may be related to the extremum properties of the Helmholtz potential We have seen that an evident feature of the equilibrium is that the temperature of the composite system ie of each of its subsystems is equal to T If we accept that part of the solution we can 1mmed1ately restrict our search for the equilibrium state among the manifold of states for which T T But over this manifold of states U TS is identical to U TS Then we can write equation 610 as dF dU TS 0 612 subject to the auxiliary condition that T T 613 That 1s the equilibrium state mm1m1zes the Helmholtz potential not absolutely but over the manifold of states for which T T We thus arrive at the equilibrium condition in the Helmholtz potential representa tion Helmholtz Potential Minimum Principle The equilihnum value of any unconstrained mternal parameter tn a system tn diathermal contact vlfh a heat reservmr minimizes the Helmholtz potential over the mamfold of states for which T T 1d 2 U represents the secondorder terms m the expansion of l m powers of dS the hnear term Trs m equauon 6 11 contnbutc to the expansion only m tirt order 5ec cquitton A Q of Appendix A 156 The Extremum Prmc1ple m the Legendre Transformed Representations The intuitive significance of this principle is clearly evident in equations 68 through 610 The energy of the system plus the reservoir is of course minimum But the statement that the Helmholtz potential of the system alone is minimum is just another way of saying this for dF d V TS and the term d TS actually represents the change in energy of the reservoir since T Tr and dS dSr It is now a simple matter to extend the foregoing considerations to the other common representations Consider a composite system in which all subsystems are in contact with a common pressure reservoir through walls nonrestrictive with re spect to volume We further assume that some internal constraint within the composite system has been removed The first condition of equi librium can be written dU U dV PdV dU PdV 0 614 or dU PV 0 615 Accepting the evident condition that P P we can write dH d U PV 0 616 subject to the auxiliary restriction p pr 617 Furthermore since P is a constant and Vis an independent variable 618 so that the extremum is a minimum Enthalpy Minimum Principle The equilibrium value of any unconstrained internal parameter in a system in contact with a pressure reservoir minimizes the enthalpy over the manifold of states of constant pressure equal to that of the pressure eservoir Finally consider a system in simultaneous contact with a thermal and a pressure reservoir Again d V U dU T dS P dV 0 619 Accepting the evident conditions that T T and P P we can write dG dV TS PV 0 620 The Helmholtz Potential 157 subject to the auxiliary restrictions T T p pr 621 Again 622 We thus obtain the equilibrium condition in the Gibbs representation Gibbs Potential Minimum Principle The equilibrium value of any uncon strained internal parameter in a system in contact with a thermal and a pressure reservoir minimizes the Gibbs potential at constant temperature and pressure equal to those of the respective reservoirs If the system is characterized by other extensive parameters in addition to the volume and the mole numbers the analysis is identical in form and the general result is now clear The General Minimum Principle for Legendre Transforms of the Energy The equilibrium value of any unconstrained internal parameter in a system in contact with a set of reservoirs with intensive parameters P P minimizes the thermodynamic potential UP 1 P2 at constant Pi P2 equal t P P 62 THE HELMHOLTZ POTENTIAL For a composite system in thermal contact with a thermal reservoir the equilibrium state minimizes the Helmholtz potential over the manifold of states of constant temperature equal to that of the reservoir In practice many processes are carried out in rigid vessels with diathermal walls so that the ambient atmosphere acts as a thermal reservoir for these the Helmholtz potential representation is admirably suited The Helmholtz potential is a natural function of the variables T V Ni N2 The condition that T is constant reduces the number of variables in the problem and F effectively becomes a function only of the variables V and N1 N2 This is in marked contrast to the manner in which constancy of T would have to be handled in the energy representa tion there U would be a function of S V N1 N2 but the auxiliary condition T T would imply a relation among these variables Particu larly in the absence of explicit knowledge of the equation of state T TS V N this auxiliary restriction would lead to considerable awk wardness in the analytic procedures in the energy representation As an illustration of the use of the Helmholtz potential we first consider a composite system composed of two simple systems separated by a J 58 The Extremum Prmc1ple m the Legendre Transformed Representatons Hotplate T FIGURE6l movable adiabatic impermeable wall such as a solid insulating piston The subsystems are each in thermal contact with a thermal reservoir of temperature T Fig 61 The problem then is to predict the volumes y and V2 of the two subsystems We write This is one equation involving the two variables v1 and v2 all other arguments are constant The closure condition v1 v2 V a constant 624 provides the other required equation permitting explicit solution for v1 and v2 In the energy representation we would also have found equality of the pressures as in equation 623 but the pressures would be functions of the entropies volumes and mole numbers We would then require the equa tions of state to relate the entropies to the temperature and the volumes the two simultaneous equations 623 and 624 would be replaced by four Although this reduction of four equations to two may seem to be a modest achievement such a reduction is a very great convenience in more complex situations Perhaps of even greater conceptual value is ihe fact that the Helmholtz representation permits us to focus our thought processes exclusively on the subsystem of interest relegating the reservoir only to an implicit role And finally for technical mathematical reasons to be elaborated in Chapter 16 statistical mechanical calculations are enor mously simpler in Helmholtz representations permitting calculations that would otherwise be totally intractable For a system in contact with a thermal reservoir the Helmholtz poten tial can be interpreted as the available work at constant temperature The Helmholtz Potential 159 Consider a system that interacts with a reversible work source while being in thermal contact with a thermal reservoir In a reversible process the work input to the reversible work source is equal to the decrease in energy of the system and the reservoir dWRws du dU du TdS dU TdS dU TS dF 625 626 627 Thus the work delivered in a reversible process by a system in contact with a thermal reservoir is equal to the decrease in the Helmholtz potential of the systet The Helmholtz potential is often referred to as the Helmholtz free energy though the term available work at constant temperature would be less subject to misinterpretation Example 1 A cylinder contains an internal piston on each side of which is one mole of a monatomic ideal gas The walls of the cylinder are diathermal and the system is immersed in a large bath of liquid a heat reservoir at temperature 0C The initial volumes of the two gaseous subsystems on either side of the piston are 10 liters and 1 liter respectively The piston is now moved reversibly so that the final volumes are 6 liters and 5 liters respectively How much work is delivered Solution As the reader has shown in Problem 531 the fundamental equation of a monatomic ideal gas in the Helmholtz potential representation is F0 T 312 V N t F NRT NoRTo In To Vo No At constant T and N this is simply F constant NRT In V The change in Helmholtz potential is AF NRTln6ln5ln10lnl NRTln3 25kJ Thus 25 kJ of work are delivered in this process It is interesting to note that all of the energy comes from the thermal reservoir The energy of a monatomic ideal gas is simply fNRT and therefore it is constant at constant temperature The fact that we withdraw heat from the temperature reservoir and deliver it entirely as work to the reversible work source does not however violate the Carnot efficiency principle because the gaseous subsystems are not left in their initial state Despite the fact that the energy of these subsystems remains constant their entropy increases 160 The Extremum Prmc1ple m the Legendre Transformed Representatwns PROBLEMS 621 Calculate the pressure on each side of the internal piston in Example 1 for arbitrary position of the piston By integration then calculate the work done in Example 1 and corroborate the result there obtained 622 Two ideal van der Waals fluids are contained in a cylinder separated by an internal moveable piston There is one mole of each fluid and the two fluids have the same values of the van der Waals constants b and c the respective values of the van der Waals constant a are a 1 and a2 The entire system is in contact with a thermal reservoir of temperature T Calculate the Helmholtz potential of the composite system as a function of T and of the total volume V If the total volume is doubled while allowing the internal piston to adjust what is the work done by the system Recall Problem 532 623 Two subsystems are contained within a cylinder and are separated by an internal piston Each subsystem is a mixture of one mole of helium gas and one mole of neon gas each to be considered as a monatomic ideal gas The piston is in the center of the cylinder each subsystem occupying a volume of 10 liters The walls of the cylinder are diathermal and the system is in contact with a thermal reservoir at a temperature of 100C The piston is permeable to helium but impermeable to neon Recalling from Problem 5310 that the Helmholtz potential of a mixture of simple ideal gases is the sum of the individual Helmholtz potentials each expressed as a function of temperature and volume show that in the present case T 3 T V N0 F Nfc NRTln N RTln To o 2 To i Vo Ni voN v2N N 1RTln 0 N2RTln 0 2 V No 2 V N2 0 2 0 2 where T0 0 V0 and N0 are attributes of a standard state recall Problem 531 N is the total mole number Np is the mole number of neon component 2 in subsystem 1 and vo and VC2 are the volumes of subsystems 1 and 2 respec tively How much work is required to push the piston to such a position that the volumes of the subsystems are 5 liters and 15 liters Carry out the calculation both by calculating the change in F and by a direct integration as in Problem 621 63 THE ENTHALPY THE Answer work RT lnn 893 J JOULETHOMSON OR THROTILING PROCESS For a composite system in interaction with a pressure reservoir the equilibrium state minimizes the enthalpy over the manifold of states of constant pressure The enthalpy representation would be appropriate to The Enthalpy The Joule Thomson or Throttbng Process 161 processes carried out in adiabatically insulated cylinders fitted with adia batically insulated pistons subject externally to atmospheric pressure but this is not a very common experimental design In processes carried out in open vessels such as in the exercises commonly performed in an elemen tary chemistry laboratory the ambient atmosphere acts as a pressure reservoir but it also acts as a thermal reservoir for the analysis of such processes only the Gibbs representation invokes the full power of Legendre transformations Nevertheless there are particular situations uniquely adapted to the enthalpy representation as we shall see shortly More immediately evident is the interpretation of the enthalpy as a potential for heat From the diffeeritial form 628 it is evident that for a system in contact with a pressure reservoir and enclosed by impermeable walls dHdQ where P N1 Ni are constant 629 That is heat added to a system at constant pressure and at constant values of all the remaining extensive parameters other than S and V appears as an increase in the enthalpy This statement may be compared to an analogous relation for the energy dV dQ where V N1 Ni are constant 630 and similar results for any Legendre transform in which the entropy is not among the transformed variables Because heating of a system is so frequently done while the system is maintained at constant pressure by the ambient atmosphere the enthalpy is generally useful in discussion of heat transfers The enthalpy accord ingly is sometimes referred to as the heat content of the system but it should be stressed again that heat refers to a mode of energy flux rather than to an attribute of a state of a thermodynamic system To illustrate the significance of the enthalpy as a potential for heat suppose that a system is to be maintained at constant pressure and its volume is to be changed from V to Jr We desire to compute the heat absorbed by the system As the pressure is constant the heat flux is equal to the change in the enthalpy 631 If we were to know the fundamental equation H HSPN 632 162 The Extremum Principle m the Legendre Transformed Representations then by differentiation aH V a p V S P N 633 and we could eliminate the entropy to find H as a function of V P and N Then Q 1 HV 1PNHVPN 634 A process of great practical importance for which an enthalpy repre sentation is extremely convenient is the JouleThomson or throttling process This process is commonly used to cool and liquify gases and as a secondstage refrigerator in cryogenic lowtemperature laboratories In the Joule Thomson process or Joule Kelvin process William Thomson was only later granted peerage as Lord Kelvin a gas is allowed to seep through a porous barrier from a region of high pressure to a region of low pressure Fig 62 The porous barrier or throttling valve was originally a wad of cotton tamped into a pipe in a laboratory demonstra tion it is now more apt to be glass fibers and in industrial practice it is generally a porous ceramic termination to a pipe Fig 63 The process can be made continuous by using a mechanical pump to return the gas from the region of low pressure to the region of high pressure Depending on certain conditions to be developed in a moment the gas is either heated or cooled in passing through the throttling valve FIGURE62 Piston mamtammg high pressure v T Porous plug Piston mamtammg low pressure SchematiF representation of the JouleThomson process I For real gases and for given initial and final pressures the change in temperature is generally positive down to a particular temperature and it is negative below that temperature The temperature at which the process changes from a heating to a cooling process is called the inversion temperature it depends upon the particular gas and upon both the initial and final pressures In order that the throttling process operate as an effective cooling process the gas must first be precooled below its inversion temperature To show that the JouleThomson process occurs at constant enthalpy consider one mole of the gas undergoing a throttling process The piston The Enthalpy The JouleThomon or Throttling Process 163 Pump Gas FIGURE63 Schematic apparatus for liquefaction of a gas by throttling process The pump maintains the pressure difference Prugh P10w The spherical termination of the high pressure pipe is a porous ceramic shell through which the gas expands in the throttling process Fig 62 that pushes this quantity of gas through the plug does an amount of work Pv in which v is the molar volume of the gas on the high pressure side of the plug As the gas emerges from the plug it does work on the piston that maintains the low pressure P1 and this amount of work is P1vf Thus the conservation of energy determines the final molar energy of tlie gas it is the initial molar energy plus the work Pv done on the gas minus the work P1v1 done by the gas 635 or 636 which can be written in terms of the molar enthalpy h as 637 Although on the basis of equation 637 we say that the Joule Thomson process occurs at constant enthalpy we stress that this simply implies that the final enthalpy is equal to the initial enthalpy We do not imply anything about the enthalpy during the process the intermediate states of the gas are nonequilibrium states for which the enthalpy is not defined The isenthalpic curves isenthalps of nitrogen are shown in Fig 64 The initial temperature and pressure in a throttling process determine a particular isenthalp The final pressure then determines a point on this same isenthalp thereby determining the final temperature 164 The Extremum Prmc1ple m the Legendre Transformed Representatwns t 4001ltl Q h FIGURE64 01 02 03 04 05 Pressure MPa lsenthalps solid inversion temperature dark and coexistence curve for nitrogen semiquantitative The isenthalps in Fig 64 are concave with maxima If the initial temperature and pressure lie to the left of the maximum the throttling process necessarily cools the gas If the initial temperature lies to the right of the maximum a small pressure drop heats the gas though a large pressure drop may cross the maximum and can either heat or cool the gas The maximum of the isenthalp therefore determines the inversion temperature at which a small pressure change neither heats nor cools the gas The dark curve in Fig 64 is a plot of inversion temperature as a function of pressure obtained by connecting the maxima of the isenthalpic curves Also shown on the figure is the curve of liquidgas equilibrium Points below the curve are in the liquid phase and those above are in the gaseous phase This coexistence curve terminates in the critical point In the region of this point the gas and the liquid phases lose their distinguishability as we shall study in some detail in Chapter 9 If the change in pressure in a throttling process is sufficiently small we can employ the usual differential analysis 638 The derivative can be expressed in terms of standard measurable quanti ties c P a KT by a procedure that may appear somewhat complicated on The Enthalpy The JouleThomson or Throttlmg Process 165 first reading but that will be shown in Chapter 7 to follow a routine and straightforward recipe By a now familiar mathematical identity A22 639 where we suppress the subscripts N1 N2 for simplicity noting that the mole numbers remain constant throughout However dH T dS V dP at constant mole numbers so that d T TiJSjiJPT V dP fl asaTp 640 The denominator is Ncp The derivative iJSiJPh is equal to av iJTp by one of the class of Maxwell relations analogous to equations 362 or 365 in the present case the two derivatives can be corroborated to be the two mixed second derivatives of the Gibbs potential Identifying iJSiJPh iJViJTp Va equation 367 we finally find dT Ta 1 dP 641 Cp This is a fundamental equation of the JouleThomson effect As the change in pressure dP is negative the sign of dT is opposite that of the quantity in parentheses Thus if Ta 1 a small decrease in pressure in transiting the throttling valve cools the gas The inversion temperature is determined by aTmvers1on 1 642 For an ideal gas the coefficient of thermal expansion a is equal to lT so that there is no change in temperature in a JouleThomson expansion All gases approach ideal behavior at high temperature and low or mod erate pressure and the isenthalps correspondingly become flat as seen in Fig 64 It is left to Example 2 to show that for real gases the temperature change is negative below the inversion temperature and positive above and to evaluate the inversion temperature Example 2 Compute the inversion temperature of common gases assuming them to be described by the van der Waals equation of state 341 Solution We must first evaluate the coefficient of expansion a Differentiating the van der Waals equation of state 341 with respect to T at constant P a E I 2au b I u aT P u b Ru 2 166 The Extremum Prmcple in the Legendre Transformed Representatwns To express the righthand side as a function of T and P is analytically difficult An approximate solution follows from the recognition that molar volumes are on the order of002 m31 whence bvis on the order of 103 and aRT vis on the order of 10 3 10 4 see Table 31 HenceaseriesexpansioninbvandaRTvcan reasonably be terminated at the lowest order term Let b a E1 v Ez RTv Then T 2T 1 a vbE 2 1 E1 V i 1 El 21 E1E2JI Returning to equation 641 from which we recall that dT J Ta 1 dP cP Tmv a 1 It then follows that at the inversion temperature 1 E1 22 1 or The inversion temperature is now determined by 2a Tmv bR with cooling of the gas for temperature below Tinv and heating above From Table 31 we compute the inversion temperature of several gases TmH2 224 K TinvNe 302 K TmvN2 850 K Tinv02 1020 K TmvC02 2260 K In fact the inversion temperature empirically depends strongly on the pressure a dependence lost in our calculation by the neglect of higherorder terms The observed inversion temperature at zero pressure for H 2 is 204 K and for neon it is 228 Kin fair agreement with our crude calculation For polyatomic gases the agreement is less satisfactory the observed value for CO2 is 1275 K whereas we have computed 2260 K PROBLEMS 631 A hole is opened in the wall separating two chemically identical single component subsystems Each of the subsystems is also in interaction with a The Gibbs Potential Chemical Reactions 167 pressure reservoir of pressure pr Use the enthalpy minimum principle to show that the conditions of equilibrium are T1 T2 and p1 p2 632 A gas has the following equations of state u P V T3B ui I3 NV where B is a positive constant The system obeys the Nernst postulate S O as T 0 The gas at an initial teperature T and initial pressure P is passed through a porous plug in a Joul Thomson process The final pressure is P1 Calculate the final temperature 7t l 633 Show that for an ideal van der Waals fluid h 20 RTc v V vb where h is the molar enthalpy Assuming such a fluid to be passed through a porous plug and thereby expanded from v to v1 with v1 v find the final temperature 7t in terms of the initial temperature T and the given data Evaluate the temperature change if the gas is CO2 the mean temperature is 0C the mean pressure is 107 Pa and the change in pressure is 106 Pa The molar heat capacity cp of CO2 at the relevant temperature and pressure is 295 JmoleK Carry calculation only to first order in blv and aRTv 634 One mole of a monatomic ideal gas is in a cylinder with a movable piston on the other side of which is a pressure reservoir with P 1 atm How much heat must be added to the gas to increase its volume from 20 to 50 liters 635 Assume that the gas of Problem 634 is an ideal van der Waals fluid with the van der Waals constants of argon Table 31 and again calculate the heat required Recall Problem 633 64 THE GIBBS POTENTIAL CHEMICAL REACTIONS For a composite system in interaction with both thermal and pressure reservoirs the equilibrium state minimizes the Gibbs potential over the manifold of states of constant temperature and pressure equal to those of the reservoirs The Gibbs potential is a natural function of the variables T P N 1 N2 and it is particularly convenient to use in the analysis of problems involving constant T and P Innumerable processes of common experience occur in systems exposed to the atmosphere and thereby maintained at constant temperature and pressure And frequently a pro cess of interest occurs in a small subsystem of a larger system that acts as both a thermal and a pressure reservoir as in the fermentation of a grape in a large wine vat The Gibbs potential of a multicomponent system is related to the chemical potentials of the individual components for G V TS PV 168 The Extremum Prmc1ple m the Legendre Transformed Represenrarwns and inserting the Euler relation U TS PV µ 1N1 µ 2N2 we find 643 Thus for a single component system the molar Gibbs potential is identi cal with µ but for a multicomponent system G N µ 644 645 where x 1 is the mole fraction N of the jth component Accordingly the chemical potential is often ref erred to as the molar Gibbs potential in single component systems or as the partial molar Gibbs potentwl in multicomponent systems The thermodynamics of chemical reactions is a particularly important application of the Gibbs potential Consider the chemical reaction 646 where the v are the stoichiometric coefficients defined in Section 29 The change in Gibbs potential associated with virtual changes d in the mole numbers is dG SdT VdP Lµ1 dN1 J 647 However the changes in the mole numbers must be in proportion to the stoichiometric coefficients so that 648 or equivalently 649 where dN is simply a proportionality factor defined by equation 648 If The Ghhs Potentwl Chenuc al Reaawns J 69 the chemical reaction is carried out at constant temperature and pressure as in an open vessel the condition of equilibrium then implies or dG dNLllfL 0 J 650 651 If the initial quantities of each of the chemical components is N the chemical reaction proceeds to some extent and the mole numbers asume the new values 652 where N is the factor of proportionality The chemical potentials in equation 651 are functions of T P and the mole numbers and hence of the single unknown parameter N Solution of equation 651 for N determines the equilibrium composition of the system The solution described is appropriate only providing that there is a sufficient quantity of each component present so that none is depleted before equilibrium is reached That is none of the quantities N in equation 652 can become negative This consideration is most conveni ently expressed in terms of the degree of reaction The maximum value of N for which all N1 remain positive in equation 652 defines the maximum permissible extent of the reaction Similarly the minimum value of N for which all remain positive defines the maximum permissible extent of the reverse reaction The actual value of N in equilibrium may be anywhere between these two extremes The degree of reaction 1 is defined as iii iilmm f Nmax Nmm 653 It is possible that a straightforward solution of the equation of chemical equilibrium 651 may yield a value of iii that is larger than iilmax or smaller than Nmin In such a case the process is terminated by the depletion of one of its components The physically relevant value of N is then iilmax or Nrrun Although L 1v1t does not attain the value zero it does attain the smallest absolute value accessible to the system Whereas the partial molar Gibbs potentials characterize the equilibrium condition the enthalpy finds its expression in the heat of reaction This 170 The Extremm Prmople m the Legendre Transformed Representatwns fact follows from the general significance of the enthalpy as a potential for heat flux at constant pressure equation 629 That is the flux of heat from the surroundings to the system during the chemical reaction is equal to the change in the enthalpy This change in enthalpy in turn can be related to the chemical potentials for H G TS G T ac aT PNzN2 654 If an infinitesimal chemical reaction dN occurs both Hand G change and dH dN dNT dH dG a dG dN dN aT dN PN1N2 But the change in Gibbs function is whence dG dN dN 655 656 657 At equilibrium dG dN vanishes but the temperature derivative of dG dN does not so that in the vicinity of the equilibrium state equation 655 becomes 658 The quantity dHdN is known as the heat of reaction it is the heat absorbed per unit reaction in the vicinity of the equilibrium state It is positive for endothermic reactions and negative for exothermic reactions We have assumed that the reaction considered is not one that goes to completion If the reaction does go to completion the summation in equation 657 does not vanish in the equilibrium state and this summa tion appears as an additional term in equation 658 As the summation in equation 658 vanishes at the equilibrium com position it is intuitively evident that the temperature derivative of this quantity is related to the temperature dependence of the equilibrium concentrations We shall find it convenient to develop this connection explicitly only in the special case of ideal gases in Section 134 However it is of interest here to note the plausibility of the relationship and to The G1hhs Potential Chemcal Reactwns 171 recognize that such a relationship permits the heat of reaction to be measured by determinations of equilibrium compositions at various tem peratures rather than by relatively difficult calorimetric experiments The general methodology for the analysis of chemical reactions becomes specific and definite when applied to particular sytems To anchor the foregoing treatment in a fully explicit and practica ly important special case the reader may well wish here to interpol te Chapter 13and particularly Section 132 on chemical reactions in ideal gases Example 3 Five moles of H2 1 mole of CO 2 1 mole of CH 4 and 3 moles of H20 are allowed to react in a vessel maintained at a temperature T0 and pressure P0 The relevant reaction is Solution of the equilibrium condition gives the nominal solution 6N t What are the mole numbers of each of the components If the pressure is then increased to P1 P 1 P0 and the temperature is maintained constant T0 the equilibrium condition gives a new nominal solution of 6N 1 2 What are the mole numbers of each of the components Solution We first write the analogue of equation 652 for each component NH 2 5 4 6N Nc0 1 6N Nrn 1 6N NH O 3 2 6N Setting each of 2 4 2 l these mole numbers equal to zero successively we find four roots for 6N 4 1 1 and J The positive and negative roots of smallest absolute values are respectively 6N 1 mm These two bounds on 6N correspond to depletion of CO 2 if the reaction proceeds too far in the forward direction and to depletion of CH 4 if the reaction proceeds too far in the reverse direction The degree of reaction is now by equation 653 e6f11 6f11 1 1 2 If the nominal solution of the equilibrium condition gives 6N then a and NH 3 Nco 1 NcH and NHo 2 If the increase in pressure shifts the nominal solution for tN to 1 2 we reject this value as outside the acceptable range of 6N ie greater than 6N01ax it would lead to the nonphysical value of E 11 whereas E must be between zero and unity Hence the reaction is terminated at 6N 6Nmax or at E 1 by the depletion of CO 2 The final mole numbers are Nu2 1 N co 2 N cu4 2 and NH20 5 172 The Extremum Pnnnple in the Legendre Transformed Representatwns PROBLEMS 641 One half mole of H 2S l mole of H 20 2 moles of H 2 and 1 mole of S0 2 are allowed to react in a vessel maintained at a temperature of 300 K and a pressure of 104 Pa The components can react by the chemical reaction 3H 2 S0 2 H 2S 2H 20 a Write the potentials condition of equilibrium in terms of the partial molar Gibbs b Show that NH 2 36N and similarly for the other components For what value of 6N does each vanish c Show that 6Nmax f and 6Nrrun f Which components are depleted in each of these cases d Assume that the nominal solution of the equilibrium condition gives lN i What is the degree of reaction E What are the mole fractions of each of the components in the equilibrium mixture e Assume that the pressure is raised and that the nominal solution of the equilibrium condition now yields the value 6N 08 What is the degree of reaction What are the mole fractions of each of the components in the final state d e 65 OTHER POTENTIALS Answers c H 2 and H 20 depleted E J 5 6N t Various other potentials may occasionally become useful in particular applications One such application will suffice to illustrate the general method Example4 A bottle of volume V contains Ns moles of sugar and it is filled with water and capped by a rigid lid The lid though rigid is permeable to water but not to sugar The bottle is immersed in a large vat of water The pressure in the vat at the position of the bottle is Pv and the temperature is T We seek the pressure P and the mole number N of water in the bottle Solution We suppose that we are given the fundamental equation of a twocomponent mixture of sugar and water Most conveniently this fundamental equation will be Complatwns of Empmcal Data the Enthalpy of Formation 173 cast in the representation UT V µ NsJ that is in the representation in which S and N are replaced by their corresponding intensive parameters but the volume V and the mole number of sugar N remain untransformed The diathermal wall ensures that T has the value established by the vat a thermal reservoir and the semipermeable lid ensures hat µ has the value established by the vat a water reservoir No problem remains We know all the independent variables of the generalized potential UT V JLw N To find the pressure in the bottle we merely differentiate the potential p BUT V JLw N av 659 It is left to the reader to compare this approach to the solution of the same problem in energy or entropy representations Various unsought for variables enter into the analysissuch as the entropy of the contents of the bottle or the entropy energy and mole number of the contents of the vat And for each such extraneous variable an additional equation is needed for its elimination The choice of the appropriate representation clearly is the key to simplicity and indeed to practicality in thermody namic calculations 66 COMPILATIONS OF EMPIRICAL DATA THE ENTHALPY OF FORMATION In principle thermodynamic data on specific systems would be most succinctly and conveniently given by a tabulation of the Gibbs potential as a function of temperature pressure and composition mole fractions of the individual components Such a tabulation would provide a fundamen tal equation in the representation most convenient to the experimentalist In practice it is customary to compile data on hT P sT P and vT P from which the molar Gibbs potential can be obtained g h Ts The tabulation of h s and v is redundant but convenient For multicomponent systems analogous compilations must be made for each composition of interest Differences in the molar enthalpies of two states of a system can be evaluated experimentally by numerical integration of dh d QIN v dP ford Q as well as P and v can be measured along the path of integration The absolute scale of the enthalpy h like that of the energy or of any other thermodynamic potential is arbitrary undetermined within an additive constant For purposes of compilation of data the scale of enthalpy is made definite by assigning the value zero to the molar enthalpy of each chemical element in its most stable form at a standard temperature and pressure generally taken as To 29815 K 25C P0 01 MPa 1 atm 17 4 The Extremum Pnncple in the Legendre Transformed Representatwns The enthalpy defined by this choice of scale is called the enthalpy of formation The reference to the most stable state in the definition of the enthalpy of formation implies for instance that the value zero is assigned to the molecular form of oxygen 0 2 rather than to the atomic form O the molecular form is the most stable form at standard temperature and pressure If 1 mole of carbon and 1 mole of 0 2 are chemically reacted to form 1 mole of CO 2 the reaction being carried out at standard temperature and pressure it is observed that 39352 X 103 J of heat are emitted Hence the enthalpy of formation of CO2 is taken as 39352 X 103 J mole in the standard state This is the standard enthalpy of formation of CO 2 The enthalpy of formation of CO 2 at any other temperature and pressure is obtained by integration of dh dQN v dP The standard molar enthalpy of formation the corresponding tandard molar Gibbs potential and the molar entropy in the standard state are tabulated for a wide range of compounds in the JANAF Thermochemical Tables Dow Chemical Company Midland Michigan and in various other similar compilations Tables of thermodynamic properties of a particular material can be come very voluminous indeed if several properties such as h s and v or even a single property are to be tabulated over wide ranges of the independent variables T and P Nevertheless for common materials such as water very extensive tabulations are readily available In the case of water the tabulations are referred to as Steam Tables One form of steam table ref erred to as a superheated steam table gives values of the molar volume v energy u enthalpy h and entropy s as a function of temperature for various values of pressure An excerpt from such a table by Sonntag and van Wilen for a few values of the pressure is given in Table 61 Another form referred to as a saturated steam table gives values of the properties of the liquid and of the gaseous phases of water for values of P and T which lie on the gasliquid coexistence curve Such a saturated steam table will be given in Table 91 Another very common technique for representation of thermodynamic data consists of thermodynamic charts or graphs Such charts neces sarily sacrifice precision but they allow a large amount of data to be summarized succinctly and compactly Conceptually the simplest such chart would label the two coordinate axes by T and P Then for a singlecomponent system one would draw families of curves of constant molar Gibbs potential µ In principle that would permit evaluation of all desired data Determination of the molar volume for instance would require reading the values ofµ for two nearby pressures at the tempera ture of interest this would permit numerical evaluation of the derivative lµ lP 7 and thence of the molar volume Instead a family of iso chores is overlaid on the graph with each isochore labeled by v Similarly TABLE6l Superheated Steam Table The quantities u h and s are per unit mass rather than molar the units of u and h are Jouleskilogram of v are m3 kilogram and of s are JouleskilogramKelvin Temperatures are in degrees Celsius The notation Sat under T refers to the temperature on the liquidgas coexistence curve this temperature is given in parentheses following each pressure value From R E Sonntag and G Van Wylen Introduction to Thermodynamics Classical and Statistical John Wiley Sons New York 1982 P 010 MPa 4581 P 050 MPa 8133 P 10 MPa 9963 T h h h V u s V u s V u s 5 Sat 14674 24379 25847 81502 3240 24839 26459 75939 16940 25061 26755 73594 i 50 14869 24439 25926 81749 a 100 17196 25155 26875 84479 3418 25116 26825 76947 16958 25067 26762 73614 150 19512 25879 27830 86882 3889 25856 27801 79401 19364 25828 27764 76134 l 200 21825 26613 28795 89038 4356 26599 28777 81580 2172 26581 28753 78343 250 24136 27360 29773 91002 4820 27350 29760 83556 2406 27337 29743 80333 300 26445 28121 30765 92813 5284 28113 30755 85373 2639 28104 30743 82158 s 400 31063 29689 32796 96077 6209 29685 32789 88642 3103 29679 32782 85435 500 35679 31323 34891 98978 7134 31320 34887 91546 3565 31316 34881 88342 s 600 40295 33025 37054 101608 8057 33022 37051 94178 4028 33019 37047 90976 C 700 44911 34796 39287 104028 8981 34794 39285 9f599 4490 34792 39282 93398 a 800 49526 36638 41590 106281 9904 36636 41589 98852 4952 36635 41586 95652 21 900 54141 38550 43964 108396 10828 38549 43963 100967 5414 38548 43961 97767 i 1000 58757 40530 46406 110393 11751 40529 46405 102964 5875 40528 46403 99764 g 1100 63372 42575 48912 112287 12674 42574 48911 104859 6337 42573 48910 101659 s 1200 67987 44679 51478 114091 13597 44678 51477 106662 6799 44677 51476 103463 1300 72602 46837 5409 7 115811 14521 46836 54096 108382 7260 46835 54095 105183 0l TABLE61 Continued P 20 MPa 12023 P 30 MPa 13355 P 40 MPa 14363 I Sat 8857 25295 27067 71272 6058 25436 27253 69919 4625 25536 27386 68959 150 9596 25769 27688 72795 6339 25708 27610 70778 4708 25645 27528 69299 l 200 10803 26544 28705 7 5066 7163 26507 28656 73115 5342 26468 28605 71706 250 11988 27312 29710 7 7086 7964 27287 29676 75166 5951 27261 29642 73789 C1 300 13162 28086 30718 78926 8753 28067 30693 77022 6548 28048 30668 75662 400 15493 29667 32766 82218 10315 29656 32750 80330 7726 29644 32734 78985 i 500 17814 31308 34871 85133 11867 31300 34860 83251 8893 31292 34849 81913 i s 600 2013 33014 37040 87770 13414 33008 37032 85892 10055 33002 37024 84558 700 2244 34788 39276 90194 14957 34784 39271 88319 11215 34779 39265 86987 800 2475 36631 41582 92449 16499 36629 41578 90576 12372 36624 41573 89244 900 2706 38545 43958 94566 18041 38542 43954 92692 13529 3853 9 43951 91362 s 1000 2937 40525 46400 96563 19581 40523 46397 94690 14685 40520 46394 93360 1100 3168 42570 48907 98458 21121 42568 48904 96585 15840 42565 48902 95256 1200 3399 44675 51473 100262 22661 44672 51471 98389 16996 44670 51468 97060 1300 3630 46832 54093 101982 24201 46830 54090 100110 18151 46828 54088 98780 P SO MPa 15186 P 60 MPa 15885 P 80 MPa 17043 0 Sat 3749 25612 27487 68213 3157 25674 27568 67600 2404 25768 27691 66628 a 200 4249 26429 28554 70592 3520 26389 28501 69665 2608 26306 28393 68158 250 4744 27235 29607 72709 3938 27209 29572 71816 2931 27155 29500 70384 300 5226 28029 30642 74599 4344 28010 30616 73724 3241 27972 30565 72328 350 5701 28826 31677 76329 4742 28812 31657 75464 3544 2878 2 31617 74089 s 400 6173 29632 32719 77938 5137 29621 32703 77079 3843 29597 32671 75716 g 500 7109 31284 34839 80873 5920 31276 34828 80021 4433 31260 34806 78673 600 8041 32996 37017 73522 6697 32991 37009 82674 5018 32979 36994 81333 700 8969 34775 39259 85952 7472 34770 39253 85107 5601 34762 39242 83770 800 9896 36621 4156 9 88211 8245 36618 41565 87367 6181 36611 41556 86033 900 10822 38536 43947 90329 9017 38534 43944 89486 6761 38528 43937 88153 1000 11747 40518 46391 92328 9788 40515 46388 91485 7340 40510 46382 90153 1100 12672 42563 4889 9 94224 10559 42561 48896 93381 7919 42556 48891 92050 1200 13596 44668 51466 96029 11330 44665 51463 95185 8497 44661 51459 93855 1300 1 4521 4682 5 54086 97749 12101 4682 3 54083 96906 9076 46818 54079 95575 Comp1at1ons of Empmwl Data the Enthalpy of Formatwn 177 families of constant molar entropy s of constant molar enthalpy h of constant coefficient of thermal expansion a of constant KT and the like are also overlaid The limit is set by readability of the chart It will be recognized that there is nothing unique about the variables assigned to the cartesian axes Each family of curves serves as a curvilinear coordinate system Thus a point of given v and s can be located as the intersection of the corresponding isochore and adiabat and the value of any other plotted variable can then be read In practice there are many variants of thermodynamic charts in use A popular type of chart is known as a Mollier chart it assigns the molar enthalpy h and the molar entropy s to the cartesian axes whereas the isochores and isobars appear as families of curves overlaid on the di agram Another frequently used form of chart a temperatureentropy chart assigns the temperature and the entropy to the coordinate axes and overlays the molar enthalpy h and various other thermodynamic functions the number again being limited mainly by readability Figure 65 Such full thermodynamic data is available for only a few systems of relatively simple composition For most systems only partial thermody namic data are available A very large scale international program on data compilation exists The International Journal of Thermophysics Plenum Press New York and London provides current reports of thermophysical measurements The Center for Information and Numerical Data Analysis and Synthesis CINDAS located at Purdue University publishes several series of data collections of particular note is the Thermophysical Properties Research Literature Retrieval Guide 19001980 seven volumes edited by J F Chancy and V Ramdas Plenum Publishing Corp New York 1982 Finally we briefly recall the procedure by which a fundamental equa tion for a singlecomponent system can be constructed from minimal tabulated or measured data The minimal information required is aT P cpT P and KTT P plus the values of v0 s0 in one reference state and perhaps the enthalpy of formation Given these data the molar Gibbs potential can be obtained by numerical integration of the GibbsDuhem relation dGN sdT vdPbut only after pre liminary evaluations of sT P and vT P by numerical integration of the equations as as cp ds ar p dT a p T dP T dT va dP and dv vadT vKTdP FIGURE65 Temperatureentropy chart for water vapor steam From Keenan Keyes Hill and Moore Steam Tables copyright 1969 John Wiley and Sons Inc Note that quality is defined as the mole fraction in the gaseous state m thl tWODhae rP11inn nf fl 1 The Maximum Prmuplesfor the Mascu Functwm J 79 Each of these integrations must be carried out over a network of paths covering the entire T P planeoften a gigantic numerical undertaking 67 THE MAXIMUM PRINCIPLES FOR THE MASSIEU FUNCTIONS In the energy representation the energy is m1mmum for constant entropy and from this it follows that each Legendre transform of the energy is minimum for constant values of the transformed intensive variables Similarly in the entropy representation the entropy is maximum for constant energy and from this it follows that each Legendre transform of the entropy is maximum for constant values of the transformed intensive variables For two of the three common Massieu functions the maximum princi ples can be very easily obtained for these functions are directly related to potentials ie to transforms of the energy By equation 561 we have s 660 and as F is minimum at constant temperature SlT is clearly maxi mum Again by equation 563 G T 661 and as G is minimum at constant pressure and temperature SlT P T is clearly maximum For the remaining common Massieu function SP T we can repeat the logic of Section 61 We are concerned with a system in contact with a reservoir that maintains P T constant but permits 1T to vary It is readily recognized that such a reservoir is more of a mathematical fiction than a physically practical device and the extremum principle for the function S P T is correspondingly artificial Nevertheless the derivation of this principle along the lines of Section 61 is an interesting exercise that I leave to the curious reader No text found in the image 7 MAXWELL RELATIONS 71 THE MAXWELL RELATIONS In Section 36 we observed that quantities such as the isothermal compressibility the coefficient of thermal expansion and the molar heat capacities describe properties of physical interest Each of these is essentially a derivative ax aYh w in which the variables are either extensive or intensive thermodynamic parameters With a wide range of extensive and intensive parameters from which to choose in general systems the number of such possible derivatives is immense But there are relations among such derivatives so that a relatively small number of them can be considered as independent all others can be expressed in terms of these few Needless to say such relationships enormously simplify thermodynamic analyses Nevertheless the relationships need not be mem orized There is a simple straightforward procedure for producing the appropriate relationships as needed in the course of a thermodynamic calculation That procedure is the subject of this chapter As an illustration of the existence of such relationships we recall equations 370 to 371 a2u a2u 71 asav avas or VN 1N 1 tN1N2 72 This relation is the prototype of a whole class of similar equalities known as the Maxwell relations These relations arise from the equality of the mixed partial derivatives of the fundamental relation expressed in any of the various possible alternative representations 182 Mawe1 Relatwns Given a particular thermodynamic potential expressed in terms of its t 1 natural variables there are t t 1 2 separate pairs of mixed second derivatives Thus each potential yields tt 12 Maxwell rela tions For a singlecomponent simple system the internal energy is a function of three variables t 2 and the three 2 32 pairs of mixed second derivatives are a2uas av a2uav as a2uas aN a2uaN as and a2uavaN a2uaN av The complete set of Maxwell relations for a singlecomponent simple system is given in the following listing in which the first column states the potential from which the relation derives the second column states the pair of independent varia bles with respect to which the mixed partial derivatives are taken and the last column states the Maxwell relations themselves A mnemonic diagram to be described in Section 72 provides a mental device for recalling relations of this form In Section 73 we present a procedure for utilizing these relations in the solution of thermodynamic problems u sv L N LN 73 dU TdS PdV pdN SN L LN 74 VN Jf Lv LN 75 UT F TV itN LN 76 dF SdT PdV pdN TN n 1i N 77 VN if t LN 7 8 UP H SP LN L V 79 dl TdS t VdP pdN SN H L N 710 PN L p o 7 11 Up sv s v 1 7 12 dU p TdS PdV Ndp S p t V L µ 713 Vµ t I ttµ 7 14 A Thermodynam1 Mnemonic Diagram 183 UT P G TP tN PN 715 dG SdT VdP pdN TN t p LN 716 PN itp N 717 UTp TV tµ Lµ 7 18 dUTp SdT PdV Tp Lv Lµ 719 Ndp Vp L itµ 7 20 UPp SP tµ av as P 721 dUPp TdS VdP Ndp Sp L p L µ 722 Pp Lp L µ 723 72 A THERMODYNAMIC MNEMONIC DIAGRAM A number of the most useful Maxwell relations can be remembered conveniently in terms of a simple mnemonic diagram 1 This diagram given in Fig 7 1 consists of a square with arrows pointing upward along the two diagonals The sides are labeled with the four common thermody namic potentials F G H and U in alphabetical order clockwise around the diagram the Helmholtz potential Fat the top The two corners at the left are labeled with the extensive parameters V and S and the two corners at the right are labeled with the intensive parameters T and P Valid Facts and Theoretical Understanding Generate Solutions to Hard Problems suggests the sequence of the labels Each of the four thermodynamic potentials appearing on the square is flanked by its natural independent variables Thus U is a natural function of V and S F is a natural function of V and T and G is a natural function of T and P Each of the potentials also depends on the mole numbers which are not indicated explicitly on the diagram 111us diagram was presented by Professor Max Born in 1929 in a lecture heard by Professor T1sza It appeared in the literature in a paper by F 0 Koenig J Chem Phys 3 29 1935 and 56 4556 1972 See also L T Klauder Am Journ Phys 36 556 1968 and a number o other vanants presented by a succession of authors in this journal 184 Maxwell Relations F T FIGURE 71 s H P The thermodynamic quare In the differential expression for each of the potenhals in terms of the differentials of its natural flanking variables the associated algebraic sign is indicated by the diagonal arrow An arrow pointing away from a natural variable implies a positive coefficient whereas an arrow pointing toward a natural variable implies a negative coefficient This scheme becomes evident by inspection of the diagram and of each of the following equations dV TdSPdVµkdNk k dF SdT PdV µA dNk k dG SdT VdP µkdN k dH TdS VdP µkdN k 724 725 7 26 7 27 Finally the Maxwell relations can be read from the diagram We then deal only with the corners of the diagram The labeling of the four corners of the square can easily be seen to be suggestive of the relationship V r I I l s L i p r T I I l SL J P constant N1 N2 728 By mentally rotating the square on its side we find by exactly the same construction Sr I I l PL J T r V I I i PL J T constant N1 N2 7 29 Prohems 185 The minus sign in this equation is to be inf erred from the unsymmetrical placement of the arrows in this case The two remaining rotations of the square give the two additional Maxwell relations L t constant N1 N2 730 and L L constant N1 N2 731 These are the four most useful Maxwell relations in the conventional applications of thermodynamics The mnemonic diagram can be adapted to pairs of variables other than S and V If we are interested in Legendre transformations dealing with S and the diagram takes the form shown in Fig 72a The arrow connecting N1 and µ1 has been reversed in relation to that which previ ously connected V and P to ake into account the fact that µ 1 is analogous to P Equations 74 77 713 and 719 can be read directly from this diagram Other diagrams can be constructed in a similar fashion as indicated in the general case in Fig 72b UP 2 X1 P2 u FIGURE 72 PROBLEMS 721 In the immediate vicinity of the state T0 v0 the volume of a particular system of 1 mole is observed to vary according to the relationship v v0 aT T0 bP P0 Calculate the transfer of heat dQ to the system if the molar volume is changed by a small increment dv v v0 at constant temperature T0 Answer aQ T as dV T aP dV abT dV av T ar v 186 Maxwell Relations 722 For a particular system of 1 mole in the vicinity of a particular state a change of pressure dP at constant T is observed to be accompanied by a heat flux dQ A dP What is the value of the coefficient of thermal expansion of this system in the same state 723 Show that the relation 1 a T implies that cp is independent of the pressure acP 0 aP r 73 A PROCEDURE FOR THE REDUCTION OF DERIVATIVES IN SINGLECOMPONENT SYSTEMS In the practical applications of thermodynamics the experimental situa tion to be analyzed frequently dictates a partial derivative to be evaluated For instance we may be concerned with the analysis of the temperature change that is required to maintain the volume of a singlecomponent system constant if the pressure is increased slightly This temperature change is evidently dT aT aP vN dP 732 and consequently we are interested in an evaluation of the derivative ar aPvN A number of similar problems will be considered in Section 74 A general feature of the derivatives that arise in this way is that they are likely to involve constant mole numbers and that they generally involve both intensive and extensive parameters Of all such derivatives only three can be independent and any given derivative can be expressed in terms of an arbitrarily chosen set of three basic derivatives This set is conventionally chosen as cp a and Kr The choice of cP a and Kr is an implicit transformation to the Gibbs representation for the three second derivatives in this representation are a2g aT 2 a2g aTaP and a2g aP 2 these derivatives are equal respec tively to cpT va and VKr For constant mole numbers these are the only independent second derivatives All first derivatives involving both extensive and intensive parameters can be written in terms of second derivatives of the Gibbs potential of which we have now seen that cp a and Kr constitute a complete independent set at constant mole numbers The procedure to be followed in this reduction of derivatives is straightforward in principle the entropy S need only be replaced by A Procedure for the Reducton of Derwatwes m SmgleComponent Systems 187 aGaT and V must be replaced by acaP thereby expressing the original derivative in terms of second derivatives of G with respect to T and P In practice this procedure can become somewhat involved It is essential that the student of thermodynamics become thoroughly proficient in the reduction of derivatives To that purpose we present a procedure based upon the mnemonic square and organized in a step by step recipe that accomplishes the reduction of any given derivative Students are urged to do enough exercises of this type so that the procedure becomes automatic Consider a partial derivative involving constant mole numbers It is desired to express this derivative in terms of cp a and KT We first recall the following identities which are to be employed in the mathematical manipulations see Appendix A 733 and z z z 734 z xi L 735 The following steps are then to be taken in order 1 If the derivative contains any potentials bring them one by one to the numerator and eliminate by the thermodynamic square equations 724 to 727 Example Reduce the derivative ap aucN LN LJl by733 r LN P LJ 1 by 124 r tN LN P LN iLJ 1 by 735 saraPsN v P saraPvN v 1 T saraspN saravPN by 726 188 Maxwell Reanons The remammg expression does not contain any potentials but may involve a number of derivatives Choose these one by one and treat each according to the following procedure 2 If the derivative contains the chemical potential bring it to the numerator and eliminate by means of the GibbsDuhem relation dµ sdT vdP Example Reduce aµavsN LN s LN v L N 3 If the derivative contains the entropy bring it to the numerator If one of the four Maxwell relations of the thermodynamic square now eliminates the entropy invoke it If the Maxwell relations do not eliminate the entropy put a ar under as employ equation 734 with w T The numerator will then be expressible as one of the specific heats either cl or cp Example Consider the derivative aTaPsN appearing in the example of step I Example itN tN LN LN cp by 735 by 7 29 Consider the derivative as av P N The Maxwell relation would give asavPN aParsN equation 728 which would not eliminate the entropy We therefore do not invoke the Maxwell relation but write as asarPN NTcp av PN avaTPN avarPN by 734 The derivative now contains neither any potential nor the entropy It consequently contains only V P T and N 4 Bring the volume to the numerator The remaining derivative will be expressible in terms of a and KT Example Given aTaPvN by 735 Problems 189 5 The originally given derivative has now been expressed in terms of the four quantities cv cp a and K 7 The specific heat at constant volume is eliminated by the equation 7 36 This useful relation which should be committed to memory was alluded to in equation 375 The reader should be able to derive it as an exercise see Problem 732 This method of reduction of derivatives can be applied to multicompo nent systems as well as to singlecomponent systems provided that the chemical potentials µ1 do not appear in the derivative for the GibbsDuhem relation which eliminates the chemical potential for singlecomponent systems merely introduces the chemical potentials of other components in multicomponent systems PROBLEMS 731 Thermodynamicists sometimes refer to the first T dS equation and the second T dS equation TdS NcvdTTaKrdV TdS NcpdT TVadP Derive these equations N constant N constant 732 Show that the second equation in the preceding problem leads directly to the relation and so validates equation 736 733 Calculate 8HaVhNin terms of the standard quantities cp a Kr T and P 734 Reduce the derivative av as P 735 Reduce the derivative asa fv 736 Reduce the derivative as a fp 737 Reduce the derivative 8s8uh Answer T Ta IKr 190 Maxwell Relations 74 SOME SIMPLE APPLICATIONS In this section we indicate several representative applications of the manipulations described in Section 73 In each case to be considered we first pose a problem Typically we are asked to find the change in one parameter when some other parameter is changed Thus in the simplest case we might be asked to find the increase in the pressure of a system if its temperature is increased by T its volume being kept constant In the examples to be given we consider two types of solutions First the straightforward solution that assumes complete knowledge of the fundamental equation and second the solution that can be obtained if c P a and K 7 are assumed known and if the changes in parameters are small Adiabatic Compression Consider a singlecomponent system of some definite quantity of matter characterized by the mole number N enclosed within an adiabatic wall The initial temperature and pressure of the system are known The system is compressed quasistatically so that the pressure increases from its initial value P to some definite final value P1 We attempt to predict the changes in the various thermodynamic parameters eg in the volume tempera ture internal energy and chemical potential of the system The essential key to the analysis of the problem is the fact that for a quasistatic process the adiabatic constraint implies constancy of the entropy This fact follows of course from the quasistatic correspondence dQ TdS We consider in particular the change in temperature First we assume the fundamental equation to be known By differentiation we can find the two equations of state T TS V N and P PS V N By knowing the initial temperature and pressure we can thereby find the initial volume and entropy Elimination of V between the two equations of state gives the temperature as a function of S P and N Then obviously T TS P1 N TS P N 7 37 If the fundamental equation is not known but cp a and KT are given and if the pressure change is small we have dT aT dP aP sN 7 38 By the method of Section 7 3 we then obtain dT Tua dP Cp 739 Some Simple Applrcatrons J 9 J The change in chemical potential can be found similarly Thus for a small pressure change dµ aµ dP ap SN 740 741 The fractional change in volume associated with an infinitesimal adiabatic compression is characterized by the adiabatic compressibility Ks previously defined in equation 373 It was there stated that Ks can be related to K 7 cP and a equation 376 and see also Problem 395 an exercise that is now left to the reader in Problem 748 Isothermal Compression We now consider a system maintained at constant temperature and mole number and quasistatically compressed from an initial pressure P to a final pressure Pr We may be interested in the prediction of the changes in the values of U S V and µ By appropriate elimination of variables among the fundamental equation and the equations of state any such parameter can be expressed in terms of T P and N and the change in that parameter can then be computed directly For small changes in pressure we find ds as dP ap TN 742 aVdP 743 also dU au dP ap TN 744 TaV PVK7 dP 745 and similar equations exist for the other parameters One may inquire about the total quantity of heat that must be extracted from the system by the heat reservoir in order to keep the system at constant temperature during the isothermal compression First assume that the fundamental equation is known Then DtQ TDtS TST P1 N TST P N 746 192 Maxwell Relations where S U V N is reexpressed as a function of T P and N in standard fashion If the fundamental equation is not known we consider an infinitesimal isothermal compression for which we have from equation 743 dQ TaVdP 747 Finally suppose that the pressure change is large but that the fundamen tal equation is not known so that the solution 746 is not available Then if a and V are known as functions of T and P we integrate equation 747 at constant temperature J P Q T aVdP P 748 This solution must be equivalent to that given in equation 746 Free Expansion The third process we shall consider is a free expansion recall Problems 348 and 423 The constraints that require the system to have a volume V are suddenly relaxed allowing the system to expand to a volume If the system is a gas which of course does not have to be the case the expansion may be accomplished conveniently by confining the gas in one section of a rigid container the other section of which is evacuated If the septum separating the sections is suddenly fractured the gas sponta neously expands to the volume of the whole container We seek to predict the change in the temperature and in the various other parameters of the system The total internal energy of the system remains constant during the free expansion Neither heat nor work are transferred to the system by any external agency If the temperature is expressed in terms of U V and N we find T TU N TU V N 749 If the volume change is small dT aT av uN dV 750 Ne dv NcKT 751 Some Simple Applrcatwns 19 3 This process unlike the two previously treated is essentially irreversible and is not quasistatic Problem 423 Example In practice the processes of interest rarely are so neatly defined as those just considered No single thermodynamic parameter is apt to be constant in the process More typically measurements might be made of the temperature during the expansion stroke in the cylinder of an engine The expansion is neither isothermal nor isentropic for heat tends to flow uncontrolled through the cylinder walls Nevertheless the temperature can be evaluated empirically as a function of the volume and this defines the process Various other characterizabons of real processes will occur readily to the reader but the general methodology is well represented by the following particular example N moles of a material are expanded from V1 to Vi and the temperature is observed to decrease from T1 to Ti the temperature falling linearly with volume Calculate the work done on the system and the heat transfer expressing each result in terms of definite integrals of the tabulated functions cP a and K 7 Solution We first note that the tabulated functions cpT P aT P KrT P and vT P are redundant The first three functions imply the last as has already been shown in the example of Section 39 Turning to the stated problem the equation of the path in the TV plane is T A BV A T1 V2 T2V1V 2 Vi B T2 T1lV2 V1 Furthermore the pressure is known at each point on the path for the known function vT P can be inverted to express P as a function of T and v and thence of v alone P PT V PA BV V The work done in the process is then W f ViPA BV V dV V1 This integral must be performed numerically but generally it is well within the capabilities of even a modest programmable hand calculator The heat input is calculated by considering S as a function of T and V dS L dT it dV N aP Tc V dT aT v dV NcP Vai dT dV T Kr Kr But on the path dT B dV so that c BVa 2 dS NB dV KT 194 Maxwell Relations Thus the heat input is Q fv 2NBcP A BVBVa laK 7 dV V1 Again the factors in the integral must be evaluated at the appropriate values of P and T corresponding to the point V on the path and the integral over V must then be carried out numerically It is often convenient to approximate the given data by polynomial expressions in the region of interest numerous packaged computer programs for such fits are available Then the integrals can be evaluated either numerically or analyti cally Example In the Pv plane of a particular substance two states A and D are defined by P0 104 Pa and it is also ascertained that TA 3509 K If 1 mole of this substance is initially in the state A and if a thermal reservoir at temperature 150 K is available how much work can be delivered to a reversible work source in a process that leaves the system in the state D The following data are available The adiabats of the system are of the form Pv2 constant for s constant Measurements of cP and a are known only at the pressure of 105 Pa c Bv 2l 3 p a 3T for P 105 Pa B 10813 4642 Jm 2K for P 105 Pa and no measurements of Kr are available The reader is strongly urged to analyze this problem independently before reading the following solution Solution In order to assess the maximum work that can be delivered in a reversible process A D it is necessary only to know u0 uA and s0 sA The adiabat that passes through the state D is described by Pv2 10 2 Pa m6 it intersects the isobar P 105 Pa at a point C for which Pc 105 Pa vc 10 312 m3 316 X 10 2 m3 Some Simple Apphcat1ons 195 As a twostep quasistatic process joining A and D we choose the isobaric process A C followed by the 1sentropic process C D By considering these two processes in turn we seek to evaluate first Uc uA and sc sA and then un u and Sn Sc yielding finally Un uA and sn sA c We first consider the isobaric process A C du TdsPdv Pdv tBv 113TPAdv We cannot integrate this directly for we do not yet know Tv along the isobar To calculate T v we write ar 1 r av P va 3v or integrating and T 3509 X 50v 113 on P 105 Pa isobar Returning now to the calculation of Uc uA du B X 3509 X 50 113 105 dv 105 dv or Uc uA 105 xvc vA 116 X 103 J We now require the difference Un uc Along the adiabat we have Un Uc DPdv 10 2fD 102v01 Vc 1 216 X 103 J ve ve i Finally then we have the required energy difference Un UA 103 J We now tum our attention to the entropy difference sn sA Sc sA Along the isobar AC ds dv dv Bv dv as Cp 1 13 av P Tva 3 and Sn sA Sc sA iBvf 3 vY 3 61 JK Knowing Au and As for the process we turn to the problem of delivering maximum work The increase in entropy of the system permits us to extract energy from the thermal reservoir Qres TresAS 150 X 61 916 J The total energy that can then be delivered to the reversible work source is Au Q res or work delivered 192 X 10 3 J I 96 Maxwell Relatons PROBLEMS 741 In the analysis of a JouleThomson experiment we may be given the initial and final molar volumes of the gas rather than the initial and final pressures Express the derivativearavh in terms of cp a and KT 742 The adiabatic bulk modulus is defined by f3s v aP v aP av s av SN Express this quantity in terms of cP c a and KT do not eliminate cp What is the relation of your result to the identity KsKT cjcP recall Problem 395 743 Evaluate the change in temperature in an infinitesimal free expansion of a simple ideal gas equation 751 Does this result also hold if the change in volume is comparable to the initial volume Can you give a more general argument for a simple ideal gas not based on equation 751 744 Show that equation 746 can be written as Q V1Pµ UPµ so that UP µ can be interpreted as a potential for heat at constant T and N 745 A 1 decrease in volume of a system is carried out adiabatically Find the change in the chemical potential in terms of cP a and KT and the state functions P T u v s etc 746 Two moles of an imperfect gas occupy a volume of 1 liter and are at a temperature of 100 K and a pressure of 2 MPa The gas is allowed to expand freely into an additional volume initially evacuated of 10 cm3 Find the change in enthalpy At the initial conditions cP 08 Jmole K KT 3 X 106 Pa 1 and a 0002 K 1 Answer AH Av 15 J p cP Pva l cpKT Tva2 747 Show that acjavr Ta 2PaT 2v and evaluate this quantity for a system obeying the van der Waals equation of state 748 Show that n T Tv a 2 J Evaluate this quantity for a system obeying the equation of state P v 2 RT Problems 197 749 One mole of the system of Problem 748 is expanded isothermally from an initial pressure P0 to a final pressure P1 Calculate the heat flux to the system in this process Answer Q RT1n2AP 1 PT 2 7410 A system obeys the van der Waals equation of state One mole of this system is expanded isothermally at temperature T from an initial volume v0 to a final volume v1 Find the heat transfer to the system in this expansion 7411 Two moles of Oz are initially at a pressure of 105 Pa and a temperature of 0C An adiabatic compression is carried out to a final temperature of 300C Find the final pressure by integration of equation 739 Assume that Oz is a simple ideal gas with a molar heat capacity c P which can be represented by cp 2620 1149 X 10 3T 3223 X 10 6Tz where cP is in Jmole and Tis in kelvins Answer P1 15 X 105 Pa 7412 A ball bearing of mass 10 g just fits in a vertical glass tube of crosssec tional area 2 cmz The bottom of the tube is connected to a vessel of volume 5 liters filled with oxygen at a temperature of 30C The top of the tube is open to the atmosphere which is at a pressure of 105 Pa and a temperature of 30C What is the period of vertical oscillation of the ball Assume that the compres sions and expansions of the oxygen are slow enough to be essentially quasistatic but fast enough to be adiabatic Assume that 0 2 is a simple ideal gas with a molar heat capacity as given in Problem 7 411 7413 Calculate the change in the molar internal energy in a throttling process in which the pressure change is dP expressing the result in terms of standard parameters 7414 Assuming that a gas undergoes a free expansion and that the temperature is found to change by dT calculate the difference dP between the mitial and final pressure 7415 One mole of an ideal van der Waals fluid is contained in a vessel of volume V at temperature T A valve is opened permitting the fluid to expand into an initially evacuated vessel so that the final volume is The walls of the vessels are adiabatic Find the final temperature 7t Evaluate your result for V 2 x 10 3 m3 5 x 10 3 m3 N 1 T 300 K and the van der Waals constants are those of argon Table 31 What was the initial pressure of the gas 198 Maxwell Reauons 7416 Assuming the expansion of the ideal van der Waals fluid of Problem 7415 to be carried out quasistatically and adiabatically again find the final temperature T1 Evaluate your result with the numerical data specified in Problem 7415 7417 It is observed that an adiabatic decrease in molar volume of 1 produces a particular change in the chemical potential µ What percentage change in molar volume carried out isothermally produces the same change in µ 7418 A cylinder is fitted with a piston and the cylinder contains helium gas The sides of the cylinder are adiabatic impermeable and rigid but the bottom of the cylinder is thermally conductive permeable to helium and rigid Through this permeable wall the system is in contact with a reservoir of constant T and µHe the chemical potential of He Calculate the compressibility of the system 1VdVdP in terms of the properties of helium cp v a Kr etc and thereby demonstrate that this compressibility diverges Discuss the physical reason for this divergence 7419 The cylinder in Problem 7418 is initially filled with lo mole of Ne Assume both He and Ne to be monatomic ideal gases The bottom of the cylinder is again permeable to He but not to Ne Calculate the pressure in the cylmder and the compressibility 1V dV dP as functions of T V and µHe Hint Recall Problems 531 5310 and 623 7420 A system is composed of I mole of a particular substance In the Pv plane two states A and B lie on the locus Pv2 constant so that PAvJ PBv1 The following properties of the system have been measured along this locus cP Cv2 a Dv and Kr Ev where C D and E are constants Calculate the temperature TB in terms of TA PA vA vB and the constants C D and E Answer TB TA vB vAD 2EPAvJD 1lnvBfva 7421 A system is composed of I mole of a particular substance Two thermody namic states designated as A and B lie on the locus Pv constant The following properties of the system have been measured along this locus cP Cv a D v 2 and Kr Ev where C D and E are constants Calculate the difference in molar energies uB uA in terms of TA PA vA vB and the con stants C D and E 7422 The constantvolume heat capacity of a particular simple system is A constant In addition the equation of state is known to be of the form v v0 P BT where BT is an unspecified function of T Evaluate the permissible functional form of BT Generahzations Magnetic Systems 99 In terms of the undetermined constants appearing in your functional represen tation of BT evaluate o cP and Kr as functions of T and v Hint Examine the derivative a2saTav Answer cP AT 3 T 3 DT where D and E are constants 7423 A system is expanded along a straight line in the Pv plane from the initial state P0 v0 to the final state P1 v1 Calculate the heat transfer per mole to the system in this process It is to be assumed that o Kr and cP are known only along the isochore v v0 and the isobar P P1 in fact it is sufficient to specify that the quantity cvKra has the value AP on the isochore v v0 and the quantity cpva has the value Bv on the isobar P P1 where A and Bare known constants That is for 1 v0 for P P1 Answer Q fAPj Pl fBv vJ P 0 P1v1 v0 7424 A nonideal gas undergoes a throttling process ie a JouleThomson expansion from an initial pressure P0 to a final pressure P1 The initial tempera ture is T0 and the initial molar volume is v0 Calculate the final temperature 7t if it is given that and Kr A2 along the T T0 isotherm A 0 V a o0 along the T T0 isotherm cP c along the P P1 isobar What is the condition on T0 in order that the temperature be lowered by the expansion 75 GENERALIZATIONS MAGNETIC SYSTEMS For systems other than simple systems there exists a complete paralle lism to the formalism of Legendre transformation of Maxwell relations and of reduction of derivatives by the mnemonic square The fundamental equation of a magnetic system is of the form recall Section 38 and Appendix B U US V I N 752 Legendre transformations with respect to S V and N simply retain the magnetic moment I as a parameter Thus the enthalpy is a function of S 200 Maxwell Relatwns P I and N H UP UPVHSPlN 753 An analogous transformation can be made with respect to the magnetic coordinate 754 and this potential is a function of S V Be and N The condition of equilibrium for a system at constant external field is that this potential be minimum Various other potentials result from multiple Legendre transformations as depicted in the mnemonic squares of Fig 73 Maxwell relations and the relationships between potentials can be read from these squares in a completely straightforward fashion av aBe a1 sP aP s UP B sB LP UP UT B V B av aBe TI TP aP T1 UT tB LP UT PB UT as a Be TI VT ar Vl V UTB aT aBe a1 vs as v FIGURE 73 Problems 20 The magnetic enthalpy UP Be U PV Bel is an interesting and useful potential It is minimum for systems maintained at constant pressure and constant external field Furthermore as in equation 629 for the enthalpy dUP Be T dS dQ at constant P Be and N Thus the magnetic enthalpy U P Be acts as a potential for heat for systems maintained at constant pressure and magnetic field Example A particular material obeys the fundamental equation of the paramagnetic model equation 366 with T0 200 K and If2R 10 Tesla2 Km2J Two moles of this material are maintained at constant pressure in an external field of B 02 Tesla or 2000 gauss and the system is heated from an imtial tempera ture of 5 K to a final temperature of 10 K What is the heat input to the system Solution The heat input is the change in the magnetic enthalpy UP Be For a system in which the fundamental relation is independent of volume P au av o so that UP Be degenerates to U Bel UBel Furthermore for the para magnetic model equat10n 366 U NRT and I N1i12RTB so that UPB UBJ NRT Nll12RTB Thus Q N RAT 1 BA 28314 X 5 10 X 004 X 0lJ 83lSJ Note that the magnetic contribution arising from the second term is small compared to the nonmagnetic firstterm contribution in reality the nonmagnetic contribution to the heat capacity of real solids falls rapidly at low temperatures and would be comparably small Recall Problem 396 PROBLEMS 1751 Calculate the magnetic Gibbs potential UT B for the paramagnetic model of equation 366 Corroborate that the derivative of this potential with respect to B at Clntant T has its proper value 752 Repeat Problem 751 for the system with the fundamental equation given in Problem 382 Answer UT Bel 1NAB 1 NRT1nk 8 T21o µo 753 Calculate a I aT s for the paramagnetic model of equation 366 Also calculate asaBe What 1s the relationship between these derivauves as read from the mnemonic square 202 Maxwell Relatwns 754 Show that and 11ira1 cB c 2 ar Xr B CB Xr C Xs where CB and C1 are heat capacities and Xr and Xs are susceptibilities Xr 11ofHaBeh 8 STABILITY OF THERMODYNAMIC SYSTEMS 81 INTRINSIC STABILITY OF THERMODYNAMIC SYSTEMS The basic extremum principle of thermodynamics implies both that dS 0 and that d 2S 0 the first of these conditions stating that the entropy is an extremum and the second stating that the extremum is in particular a maximum We have not yet fully exploited the second condition which determines the stability of predicted equilibrium states Similarly in classical mechanics the stable equilibrium of a rigid pendu lum is at the position of minimum potential energy A socalled unstable equilibrium exists at the inverted point where the potential energy is maximum Considerations of stability lead to some of the most interesting and significant predictions of thermodynamics In this chapter we investigate the conditions under which a system is stable In Chapter 9 we consider phase transitions which are the consequences of instability Consider two identical subsystems each with a fundamental equation S S U V N separated by a totally restrictive wall Suppose the de pendence of Son U to be qualitatively as sketched in Fig 81 If we were to remove an amount of energy tU from the first subsystem and transfer it to the second subsystem the total entropy would change from its initial value of 2S U V N to S U tU V N S U tJU V N With the shape of the curve shown in the figure the resultant entropy would be larger than the initial entropy If the adiabatic restraint were removed in such a system energy would flow spontaneously across the wall one subsystem thereby would increase its energy and its temperature at the expense of the other Even within one subsystem the system would find it advantageous to transfer energy from one region to another developing internal inhomogeneities Such a loss of homogeneity is the hallmark of a phase transition 204 Stabibty of Thermodynamic System SUAU SU AU SJAU FIGURE 81 SU SUAU UAU u UAU For a convex fundamental relation as shown the average entropy is increased by transfer of energy between two subsystems such a system is unstable It is evident from Fig 81 that the condition of stability is the concavity of the entropy 1 su U V N SU U V N 2SU V N For U 0 this condition reduces to its differential form 0 iJ2S au2 vN for all 81 82 However this differential form is less restrictive than the concavity condi tion 81 which must hold for all U rather than for U 0 only It is evident that the same considerations apply to a transfer of volume SU V V N SU V V N 2SU V N 83 or in differential form 0 a 2s av2 uN 84 A fundamental equation that does not satisfy the concavity conditions might be obtained from a statistical mechanical calculation or from 1R B Griffiths J Math Phys S 1215 1964 L Galgani and A Scolll Physca 40 1501968 42 242 1969 Pure and Appl Chem 22 229 1970 t s x i lntrms1c Stah11ty of Thermodynamic Systems 205 FIGURE 82 The underlying fundamental relation ABCDEFG is unstable The stable fundamental relation is ABHFG Points on the straight line BHF correspond to inhomogeneous combinations of the two phases at B and F extrapolation of experimental data The stable thermodynamic fundamen tal equation is then obtained from this underlying fundamental equa tion by the construction shown in Fig 82 The family of tangent lines that lie everywhere above the curve the superior tangents are drawn the thermodynamic fundamental equation is the envelope of these superior tan gent lines In Fig 82 the portion BCDEF of the underlying fundamental relation is unstable and is replaced by the straight line BHF It should be noted that only the portion CDE fails to satisfy the differential or local form of the stability condition 82 whereas the entire portion BCDEF violates the global form 81 The portions of the curve BC and EF are said to be locally stable but globally unstable A point on a straight portion BHF in Fig 82 of the fundamental relation corresponds to a phase separation in which part of the system is in state B and part in state F as we shall see in some detail in Chapter 9 In the threedimensional SVV subspace the global condition of stability requires that the entropy surface S V V lie everywhere below its tangent planes That is for arbitrary AV and AV SV AV V AV N SV AV V AV N 2SV V N 85 from which equations 82 and 84 again follow as well as the additional 206 Stablry of Thermodynamic Systems requirement see Problem 811 that J2s a2s 2 au2 av2 au av 0 86 We shall soon obtain this equation by an alternative method by applying the analogue of the simple curvature condition 82 to the Legendre transforms of the entropy To recapitulate stability requires that the entropy surface lie every where below its family of tangent planes The local conditions of stability are weaker conditions They require not only that a2s au2 v N and a 2SjaV 2 uN be negative but that a 2SaU 2a 2SaV 2 a2S au aV 2must be positive The condition a2S I au2 0 ensures that the curve of intersection of the entropy surface with the plane of constant V passing through the equilibrium point have negative curvature The condition a2s av2 0 similarly ensures that the curve of intersection of the entropy surface with the plane of constant U have negative curvature These two partial curvatures are not sufficient to ensure concavity for the surface could be fluted curving downward along the four directions U and V but curving upward along the four diagonal directions between the U and V axes It is this fluted structure that is forbidden by the third differential stability criterion 86 In physical terms the local stability conditions ensure that inhomogenei ties of either u or v separately do not increase the entropy and also that a coupled inhomogeneity of u and v together does not increase the entropy For magnetic systems analogous relations hold with the magnetic moment replacing the volume 2 Before turning to the full physical implications of these stability condi tions it is useful first Section 82 to consider their analogues for other thermodynamic potentials We here take note only of the most easily interpreted inequality equation 83 which suggests the type of informa tion later to be inferred from all the stability conditions Equation 82 requires that a 2s l ar 1 0 87 au2 VN T2 au VN NT 2c whence the molar heat capacity must be positive in a stable system The remaining stability conditions will place analogous restrictions on other physically significant observables Finally and in summary in an r 2 dimensional thermodynamic space S X 0 Xi Xr stability requires that the entropy hypersurface lie everywhere below its family of tangent hyperplanes 2 R B Gnffiths J Math Phys 5 121 1964 Stab1bty Conditwns for Thermodynamic Potentials 207 PROBLEMS 811 To establish the inequality 86 expand the lefthand side of 85 in a Taylor series to second order in llU and llV Show that this leads to the condition SuullV2 2SuvllUllV SvvllV 2 0 Recalling that Suu a2sau2 c 0 show that this can be written in the form SuullU SuvllV 2 SuuSvv Slv llV 2 O and that this condition in turn leads to equation 86 812 Consider the fundamental equation of a monatomic ideal gas and show that S is a concave function of V and V and also of N 82 STABILITY CONDITIONS FOR THERMODYNAMIC POTENTIALS The reformulation of the stability criteria in energy representation requires only a straightforward transcription of language Whereas the entropy is maximum the energy is minimum thus the concavity of the entropy surface is replaced by convexity of the energy surface The stable energy surface lies above its tangent planes US 1S V 1V N US 1S V 1V N 2US V N 88 The local conditions of convexity become av2 and for cooperative variations of S and V a2u a2u a2u 2 0 as2 av2 as av 89 810 This result can be extended easily to the Legendre transforms of the energy or of the entropy We first recall the properties of Legendre transformations equation 531 P au iJX and X iJU P aP 811 208 Stab1ty of Thermodynamic Systems whence ax oP aP2 a2u 812 ax2 Hence the sign of o2UPoP 2 is the negative of the sign of o2UoX 2 If U is a convex function of X then UP is a concave function of P It follows that the Helmholtz potential is a concave function of the tempera ture and a convex function of the volume 02F 0 ar2 vN o2F 0 av2 TN 813 The enthalpy is a convex function of the entropy and a concave function of the pressure 02H 0 as2 rN 02H 0 oP2 SN 814 The Gibbs potential is a concave function of both temperature and pressure 02G O ar2 rN 02G 0 oP2 TN 815 In summary for constant N the thermodynamic potentials the energy and its Legendre transforms are convex functions of their extensive varia bles and concave functions of their intensive variables Similarly for constant N the Massieu functions the entropy and its Legendre transforms are concave functions of their extensive variables and convex functions of their intensive variables PROBLEMS 821 a Show that in the region X 0 the function Y X is concave for 0 n I and convex for n 0 or n 1 The following four equations are asserted to be fundamental equations of physical systems b F A N1 cs2pl d H l e U D si4 r Physical Consequences of Stal1ty 209 Which of these equations violate the criteria of stability Assume A B C and D to be positive constants Recall the fluting condition equation 810 822 Prove that i 2F av2 T a2u a2u a2u 2 as2 av2 asav a2u as2 Hint Note that iJ 2FiJV 2h iJPiJVh and consider P formally to be a function of S and V This identity casts an interesting perspective on the formalism The quantity in square brackets measures the curvature of the energy along a direction inter mediate between the S and V axes recall the discussion of fluting after equation 86 The primary curvature condition on F along the V axis is redundant with the fluting condition on U Only primary curvature conditions need be invoked if all potentials are considered 823 Show that stability requires equations 815 and a2G a2G 2 O ar 2 ap2 araP Recall Problem 811 83 PHYSICAL CONSEQUENCES OF STABILITY We turn finally to a direct interpretation of the local stability criteria in terms of limitations on the signs of quantities such as c s a and r The first such inference was obtained in equations 82 or tJ7 where we found that c 0 Similarly the convexity of the Helmholtz potential with respect to the volume gives 816 or 817 The fact that both c and KT are positive equations 87 and 817 has further implications which become evident when we recall the identities of 210 Stabhty of Thermodynamic Systems Problem 395 818 and c 819 From these it follows that stability requires 820 and 821 Thus both heat capacities and both compressibilities must be positive in a stable system Addition of heat either at constant pressure or at constant volume necessarily increases the temperature of a stable systemthe more so at constant volume than at constant pressure And decreasing the volume either isothermally or isentropicaly necessarily increases the pressure of a stable systemthe more so isothermally than isentropically PROBLEMS 831 Explain on intuitive grounds why cP c and why Kr s Hint Consider the energy input and the energy output during constantpressure and constantvolume heating processes 832 Show that the fundamental equation of a monatomic ideal gas satisfies the criteria of intrinsic stability 833 Show that the van der Waals equation of state does not satisfy the criteria of intrinsic stability for all values of the parameters Sketch the curves of P versus V for constant T the isotherms of the gas and show the region of local instability 84 LE CHATELIERS PRINCIPLE THE QUALITATIVE EFFECT OF FLUCTUATIONS The physical content of the stability criteria is known at Le Chatelier s Principle According to this principle the criterion for stability is that any Le Chateliers Pnnc1ple The Qualitative Effect of Fluctuations 21 J inhomogeneity that somehow develops in a system should induce a process that tends to eradicate the inhomogeneity As an example suppose that a container of fluid is in equilibrium and an incident photon is suddently absorbed at some point within it locally heating the fluid slightly Heat flows away from this heated region and by the stability condition that the specific heat is positive this flow of heat tends to lower the local temperature toward the ambient value The initial homogeneity of the system thereby is restored Similarly a longitudinal vibrational wave in a fluid system induces local regions of alternately high and low density The regions of increased density and hence of increased pressure tend to expand and the regions of low density contract The stability condition that the compressibility is positive ensures that these responses tend to restore the local pressure toward homogeneity In fact local inhomogeneities always occur in physical systems even in the absence of incident photons or of externally induced vibrations In a gas for instance the individual molecules move at random and by pure chance this motion produces regions of high density and other regions of low density From the perspective of statistical mechanics all systems undergo continual local fluctuations The equilibrium state static from the view point of classical thermodynamics is incessantly dynamic Local inhomo geneities continually and spontaneously generate only to be attenuated and dissipated in accordance with the Le Chatelier principle An informative analogy exists between a thermodynamic system and a model of a marble rolling within a potential well The stable state is at the minimu1n of the surface The criterion of stability is that the surface be convex In a slightly more sophisticated viewpoint we can conceive of the marble as being subject to Brownian motionperhaps being buffeted by some type of random collisions These are the mechanical analogues of the spontaneous fluctuations that occur in all real systems The potential minimum does not necessarily coincide with the instantaneous position of the system but rather with its expected value it is this expected value that enters thermodynamic descriptions The curvature of the potential well then plays a crucial and continual role restoring the system toward the expected state after each Brownian impact fluctuation This induced restoring force is the content of the Le Chatelier principle We note in passing that in the atypical but important case in which the potential well is both shallow and asymmetric the timeaveraged position may deviate measurably from the expected state at the potential mini mum In such a case classical thermodynamics makes spurious predic tions which deviate from observational data for thermodynamic measure ments yield average values recall Chapter 1 Such a pathological case 212 StalJ1litr of Thermodvnamc Systems arises at higherorder phase transitionsthe correct theory of which was developed in the 1970s We shall explore that area in Chapter 11 85 THE LE CHA TELIERBRAUN PRlNCIPLE Returning to the physical interpretation of the stability criteria a more subtle insight than that given by the Le Chatelier principle is formulated in the Le ChatelierBraun principle Consider a system that is taken out of equilibrium by some action or fluctuation According to the Le Chatelier principle the perturbation directly induces a process that attenuates the perturbation But various other secondary processes are also induced indirectly The content of the Le ChatelierBraun principle is that these indirectly induced processes also act to attenuate the initial perturbation A simple example may clarify the principle Consider a subsystem contained within a cylinder with diathermal walls and a loosely fitting piston all immersed within a bath a thermal and pressure reservoir The piston is moved outward slightly either by an external agent or by a fluctuation The primary effect is that the internal pressure is decreasedthe pressure difference across the piston then acts to push it inward this is the Le Chatelier principle A second effect is that the initial expansion dV alters the temperature of the subsystem dT iJT iJVs dV TaNcKT dV This change of temperature may have either sign depending on the sign of a Consequently there is a flow of heat through the cylinder walls inward if a is positive and outward if a is negative sign dQ sign a This flow of heat in turn tends to change the pressure of the system dP 1TaPasvdQ aNT 2cllKTdQ The pressure is increased for either sign of a Thus a secondary induced process heat flow also acts to diminish the initial perturbation This is the Le ChatelierBraun principle To demonstrate both the Le Chatelier and the Le ChatelierBraun principles formally let a spontaneous fluctuation dX occur in a com posite system This fluctuation is accompanied by a change in the inten sive parameter P1 of the subsystem I aP1 i dP dX i ax i l 822 The fluctuation dX also alters the intensive parameter P2 I iJP1 dP dX i ax1 i 823 The Le ChatelerBraun Prmcple 213 Now we can inquire as to the changes in X1 and X2 which are driven by these two deviations dP and dP We designate the driven change in dX1 by dX the superscript indicating response The signs of dX and dX2 are determined by the minimization of the total energy at constant total entropy dP dX dP dX2 s 0 825 Hence since dX and dX are independent dPfdXr 0 1 1 826 and dPf dXr 0 2 2 827 From the first of these and equation 822 828 and similarly 829 We examine these two results in turn The first equation 828 is the formal statement of the Le Chatelier principle For multiplying by d1dX 1 which is positive by virtue of the convexity criterion of stability 830 or dpt dPrl 0 I 1 831 That is the response dX produces a change dPO in the intensive parameter P 1 that is opposite in sign to the change dP induced by the initial fluctuation 214 Stabilty of Thermodyrramc Systems The second inequality 829 can be rewritten by the Maxwell relation in the form oP 2 oP1 ax1 ax2 Then multiplying by the positive quantity dP1dX 1 or 832 833 834 835 That is the response dX produces a change dP12 in the intensive parameter P1 which is opposite in sign to the change in P1 directly induced by the initial fluctuation This is the Le ChatelierBraun princi ple Finally it is of some interest to note that equation 833 is subject to another closely correlated interpretation Multiplying by the positive quantity dP2dX 2 836 or 837 That is the response in X 2 produces a change in P2 opposite in sign to the change induced by the initial fluctuation in X1 PROBLEMS 851 A system is in equilibrium with its environment at a common temperature and a common pressure The entropy of the system is increased slightly by a fluctuation in which heat flows into the system or by the purposeful injection of heat into the system Explain the implications of both the Le Chatelier and the Le ChatelierBraun principles to the ensuing processes proving your assertions in detail 9 FIRSTORDER PHASE TRANSITIONS 91 FIRSTORDER PHASE TRANSITIONS IN SINGLE COMPONENT SYSTEMS Ordinary water is liquid at room temperature and atmospheric pressure but if cooled below 27315 Kit solidifies and if heated above 37315 Kit vaporizes At each of these temperatures the material undergoes a pre cipitous change of propertiesa phase transition At high pressures water undergoes several additional phase transitions from one solid form to another These distinguishable solid phases designated as ice I ice 11 ice III differ in crystal structure and in essentially all thermo dynamic properties such as compressibility molar heat capacity and various molar potentials such as u or The phase diagram of water is shown in Fig 91 Each transition is associated with a linear region in the thermodynamic fundamental relation such as BHF in Fig 82 and each can be viewed as the result of failure of the stability criteria convexity or concavity in the underlying fundamental relation In this section we shall consider systems for which the underlying fundamental relation is unstable By a qualitative consideration of fluctua tions in such systems we shall see that the fluctuations are profoundly influcrzced by the details of the underlying fundamental relation In contrast the average values of the extensive parameters reflect only the stable thermo dynamic fundamental relation Consideration of the manner in which the form of the underlying fundamental relation influences the thermodynamic fluctuations will pro vide a physical interpretation of the stability considerations of Chapter 8 and of the construction of Fig 82 in which the thermodynamic funda mental relation is constructed as the envelope of tangent planes A simple mechanical model illustrates the considerations to follow by an intuitively transparent analogy Consider a semicircular section of pipe closed at both ends The pipe stands vertically on a table in the form of I 216 30 28 26 24 22 t 20 18 6 16 c 14 12 10 8 6 4 2 0 FirstOrder Phase Transitwns 20 Ice I 0 Temperature 0 C 100 FIGURE91 200 TC Cntcal prnnt T 37414C P 22 09 MPa Gas 300 Phase diagram of water The region of gasphrue stab1hty is reprecnted by an mdicerni bly narrow horizontal strip above the positive temperature axis in the phae diagram small figure The background graph is a magnification of the vertical scale to show the gas phase and the gas liquid coexistence curve an inverted U Fig 92 The pipe contains a freelysliding internal piston separating the pipe into two sections each of which contains one mole of a gas The symmetry of the system will prove to have important conse quences and to break this symmetry we consider that each section of the pipe contains a small metallic ball bearing ie a small metallic sphere The two ball bearings are of dissimilar metals with different coefficients of thermal expansion At some particular temperature which we designate as T the two spheres have equal radii at temperatures above T the righthand sphere is the larger The piston momentarily brought to the apex of the pipe can fall into either of the two legs compressing the gas in that leg and expanding the gas in the other leg In either of these competing equilibrium states the pressure difference exactly compensates the effect of the weight of the piston In the absence of the two ball bearings the two competing equilibnum states would be fully eqmvaltnt But with the ball bearings present the FIGURE 92 A simple mecharucal model FirstOrder Phase Trans1twns n Smgle Component vYtems 217 Metallic sphere Freesliding piston Cylinder or pipe more stable equilibrium position is that to the left if T T and it is that to the right if T Tc From a thermodynamic viewpoint the Helmholtz potential of the sys tem is F U TS and the energy U contains the gravitational potential energy of the piston as well as the familiar thermodynamic energies of the two gases and of course the thermodynamic energies of the two ball bearings which we assume to be small andor equal Thus the Helmholtz potential of the system has two local minima the lower minimum corre sponding to the piston being on the side of the smaller sphere As the temperature is lowered through T the two minima of the Helmholtz potential shift the absolute minimum changing from the lefthand to the righthand side A similar shift of the equilibrium position of the piston from one side to the other can be induced at a given temperature by tilting the tableor in the thermodynamic analogue by adjustment of some thermodynamic parameter other than the temperature The shift of the equilibrium state from one local minimum to the other constitutes a firstorder phase transition induced either by a change in temperature or by a change in some other thermodynamic parameter The two states between which a firstorder phase transition occurs are distinct occurring at separate regions of the thermodynamic configuration space To anticipate critical phenomena and secondorder phase transi tions Chapter 10 it is useful briefly to consider the case in which the ball bearings are identical or absent Then at low temperatures the two competing minima are equivalent However as the temperature is in creased the two equilibrium positions of the piston rise in the pipe approaching the apex Above a particular temperature T there is only one equilibrium position with the piston at the apex of the pipe In versely lowering the temperature from T T to T T the single equilibrium state bifurcates into two symmetric eqmlibrium states The 2 J 8 hrstOrder Phase Transitwns temperature Tcr is the critical tempe1 uure and the transition at Tcr is a secondorder phase transition The states between which a secondorder phase transition occurs are contiguous states in the thermodynamic configuration space In this chapter we consider firstorder phase transitions Secondorder transitions will be discussed in Chapter 10 We shall there also consider the mechanical model in quantitative detail whereas we here discuss it only qualitatively Returning to the case of dissimilar spheres consider the piston residing in the higher minimumthat is in the same side of the pipe as the larger ball bearing Finding itself in such a minimum of the Helmholtz potentia the piston will remain temporarily in that minimum though undergoing thermodynamic fluctuations Brownian motion After a sufficiently long time a giant fluctuation will carry the piston over the top and into the stable minimum It then will remain in this deeper minimum until an even larger and enormously less probable fluctuation takes it back to the less stable minimum after which the entire scenario is repeated The probability of fluctuations falls so rapidly with increasing amplitude as we shall see in Chapter 19 that the system spends almost all of its time in the more stable minimum All of this dynamics is ignored by macroscopic thermodynamics which concerns itself only with the stable equilibrium state To discuss the dynamics of the transition in a more thermodynamic context it is convenient to shift our attention to a familiar thermodynamic system that again has a thermodynamic potential with two local minimum separated by an unstable intermediate region of concavity Specifically we consider a vessel of water vapor at a pressure of 1 atm and at a temperature somewhat above 37315 K ie above the normal boiling point of water We focus our attention on a small subsystema spherical region of such a variable radius that at any instant it contains one milligram of water This subsystem is effectively in contact with a thermal reservoir and a pressure reservoir and the condition of equi librium is that the Gibbs potential GT P N of the small subsystem be minimum The two independent variables which are determined by the equilibrium conditions are the energy U and the volume V of the subsys tem If the Gibbs potential has the form shown in Fig 93 where X1 is the volume the system is stable in the lower minimum This minimum corresponds to a considerably larger volume or a smaller density than does the secondary local minimum Consider the behavior of a fluctuation in volume Such fluctuations occur continually and spontaneously The slope of the curve in Fig 93 represents an intensive parameter in the present case a difference in pressure which acts as a restoring force driving the system back toward density homogeneity in accordance with Le Chateliers principle Occa x J FrrstOrder Phase Transltums in Single Component Systems 219 FIGURE93 Thermodynamic potential with multiple minima sionally a fluctuation may be so large that it takes the system over the naximum to the region of the secondary minimum The system then settles in the region of this secondary minimumbut only for an instant A relatively small and therefore much more frequent fluctuation is all that is required to overcome the more shallow barrier at the secondary minimum The system quickly returns to its stable state Thus very small droplets of high density liquid phase occasionally form in the gas live briefly and evanesce If the secondary minimum were far removed from the absolute mini mum with a very high intermediate barrier the fluctuations from one minimum to another would be very improbable In Chapter 19 it will be shown that the probability of such fluctuations decreases exponentially with the height of the intermediate freeenergy barrier In solid systems in which interaction energies are high it is not uncommon for multiple minima to exist with intermediate barriers so high that transitions from one minimum to another take times on the order of the age of the universe Systems trapped in such secondary metastable minima are effectively in stable equilibrium as if the deeper minimum did not exist at all Returning to the case of water vapor at temperatures somewhat above the boiling point let us suppose that we lower the temperature of the entire system The form of the Gibbs potential varies as shown schemati cally in Fig 94 At the temperature T4 the two minima become equal and below this temperature the high density liquid phase becomes absolutely stable Thus T4 is the temperature of the phase transition at the pre scribed pressure If the vapor is cooled very gently through the transition temperature the system finds itself in a state that had been absolutely stable but that is now metastable Sooner or later a fluctuation within the system will discover the truly stable state forming a nucleus of condensed liquid This nucleus then grows rapidly and the entire system suddenly under goes the transition In fact the time required for the system to discover the 220 FintOrder Phme Tramllwm L FIGURE 94 Schematic vanation of Gibbs potential with volume or reciprocal density for various temperatures Ti T2 7 T4 J The temperature T4 i the transition temperature The high density phase is stable below the transition tem perature preferable state by an exploratory fluctuation is unobservably short in the case of the vapor to liquid condensation But in the transition from liqmd to ice the delay time is easily observed in a pure sample The liquid so cooled below its solidification freezing temperature is said to be supercooled A shght tap on the container however sets up longitudi nal waves with alternating regions of condensation and rarefaction and these externally induced fluctuations substitute for spontaneous fluctuations to initiate a precipitous transition A useful perspective emerges when the values of the Gibbs potential at each of its minima are plotted against temperature The result is as shown schematically in Fig 95 If these minimum values were taken from Fig 94 there would be only two such curves but any number is possible At equilibrium the smallest minimum is stable so the true Gibbs potential is the lower envelope of the curves shown in Fig 95 The discontinuities in the entropy and hence the latent heat correspond to the discontinuities in slope of this envelope function Figure 95 should be extended into an additional dimension the ad ditional coordinate P playing a role analogous to T The Gibbs potential is then represented by the lower envelope surface as each of the three t c T flGURE 9 5 Minima or the Gibbs potential as a function of T fln10rder Pha1e Tranvlllons n Single Component Srstenu 22 J singlephase surfaces intersect The projection of these curves of intersec tion onto the PT plane is the now familiar phase diagram eg Fig 91 A phase transition occurs as the state of the system passes from one envelope surface across an intersection curve to another envelope surface The variable X or V in Fig 94 can be any extensive parameter In a transition from paramagnetic to ferromagnetic phases X1 is the magnetic moment In transitions from one crystal form to another eg from cubic to hexagonal the relevant parameter X1 is a crystal symmetry variable In a solubility transition it may be the mole number of one component We shall see examples of such transitions subsequently All conform to the general pattern described At a firstorder phase transition the molar Gibbs potential of the two phases are equal but other molar potentials u f h etc are discontinu ous across the transition as are the molar volume and the molar entropy The two phases inhabit different regions in thermodynamic space and equality of any property other than the Gibbs potential would be a pure coincidence The discontinuity in the molar potentials is the defining property of a firstorder transition As shown in Fig 96 as one moves along the hquidgas coexistence curve away from the solid phase ie toward higher temperature the discontinuities in molar volume and molar energy become progressively smaller The two phases become more nearly alike Finally at the terminus of the liquidgas coexistence curve the two phases become indistinguish able The firstorder transition degenerates into a more subtle transition a secondorder transition to which we shall return in Chapter 10 The terminus of the coexistence curve is called a cntical point The existence of the cntical point precludes the possibility of a sharp distinction between the generic term tqwd and the generic term gas In crossing the liquidgas coexistence curve in a firstorder transition we distinguish two phases one of which is clearly a gas and one of which is D t cl T v FIGURE 96 The two minima of G correspondmg to four points on the coexistence curve The mm1ma coalesce at the critical point D 2 2 2 First Order Phase TranYitOns clearly a liquid But starting at one of these say the liquid immediately above the coexistence curve we can trace an alternate path that skirts around the critical point and arrives at the other state the gas without ever encountering a phase transition Thus the terms gas and liquid have more intuitive connotation than strictly defined denotation Together liquids and gases constitute the fluid phase Despite this we shall follow the standard usage and refer to the liquid phase and the gaseous phase in a liquidgas firstorder transition There is another point of great interest in Fig 91 the opposite terminus of the liquidgas coexistence curve This point is the coterminus of three coexistence curves and it is a unique point at which gaseous liquid and solid phases coexist Such a state of threephase compatibility is a triple pointin this case the triple point of water The uniquely defined temperature of the triple point of water is assigned the arbitrary value of 27316 K to define the Kelvin scale of temperature recall Section 26 PROBLEM 911 The slopes of all three curves in Fig 95 are shown as negative Is this necessary Is there a restriction on the curvature of these curves 92 THE DISCONTINUITY IN THE ENTROPY LA TENT HEAT Phase diagrams such as Fig 91 are divided by coexistence curves into regions in which one or another phase is stable At any point on such a curve the two phases have precisely equal molar Gibbs potentials and both phases can coexist Consider a sample of water at such a pressure and temperature that it is in the ice region of Fig 9la To increase the temperature of the ice one must supply roughly 21 kJkg for every kelvin of temperature increase the specific heat capacity of ice If heat is supplied at a constant rate the temperature increases at an approximately constant rate But when the temperature reaches the melting temperature on the solidliquid coexistence line the temperature ceases to rise As additional heat is supplied ice melts forming liquid water at the same temperature It requires roughly 335 kJ to melt each kg of ice At any moment the amount of liquid water in the container depends on the quantity of heat that has entered the container since the arrival of the system at the coexistence curve ie at the melting temperature When finally the requisite amount of heat has been supplied and the ice has been entirely melted continued heat input again results in an increase in temperaturenow at a The Dscontmuiv m rhe Entropy Latent Heat 113 rate determined by the specific heat capacity of liquid water 42 kJ kgK The quantity of heat required to melt one mole of solid is the heat of fusion or the latent heat of fusion It is related to the difference in molar entropies of the liquid and the solid phase by 91 where T is the melting temperature at the given pressure More generally the latent heat in any firstorder transition is t Ts 92 where T is the temperature of the transition and s is the difference in molar entropies of the two phases Alternatively the latent heat can be written as the difference in the molar enthalpies of the two phases t h 93 which follows immediately from the identity h Ts µ and the fact that µ the molar Gibbs function is equal in each phase The molar enthalpies of each phase are tabulated for very many substances If the phase transition is between liquid and gaseous phases the latent heat is called the heat of vaporization and if it is between solid and gaseous phases it is called the heat of sublimation At a pressure of one atmosphere the liquidgas transition boiling of water occurs at 37315 K and the latent heat of vaporization is then 407 kJmole 540 caljg In each case the latent heat must be put into the system as it makes a transition from the lowtemperature phase to the hightemperature phase Both the molar entropy and the molar enthalpy are greater in the hightemperature phase than in the lowtemperature phase It should be noted that the method by which the transition is induced is irrelevantthe latent heat is independent thereof Instead of heating the ice at constant pressure crossing the coexistence curve of Fig 9la horizontally the pressure could be increased at constant temperature crossing the coexistence curve vertically In either case the same latent heat would be drawn from the thermal reservoir The functional form of the liquidgas coexistence curve for water is given in saturated steam tables the designation saturated denoting that the steam is in equilibrium with the liquid phase Superheated steam tables denote compilations of the properties of the vapor phase alone at temperatures above that on the coexistence curve at the given pressure An example of such a saturated steam table is given in Table 91 from Sonntag and Van Wylen The properties s u v and h of each BLE 91 team Table Properties of the Gaseous and Liquid Phases on the Coexistence Curve of Water 0 Specific Volume Internal Energy Enthalpy Entropy l emp Press Sat Sat Sat Sat Sat Sat Sat Sat i C kPa Liquid Vapor Liquid Evap Vapor Liquid Evap Vapor Liquid Evap Vapor C p v vg u Ufg Ug hf hfg hg sf si Si 01 06ll3 0001 000 20614 00 23753 23753 01 25013 25014 0000 91562 91562 5 08721 0001 000 14712 2097 23613 23823 2098 2489 6 25106 0761 89496 9 0257 c 10 12276 0001000 106 38 4200 23472 23892 4201 24777 25198 1510 87498 89008 E 15 17051 0001 001 7793 6299 23331 23961 6299 24659 25289 2245 85569 87814 a 20 2339 0001 002 5779 8395 23190 24029 8396 24541 25381 2966 83706 86672 25 3169 0001 003 4336 10488 23049 24098 10489 24423 25472 3674 81905 85580 30 4246 0001004 3289 12578 22908 24166 12579 24305 25563 4369 80164 84533 35 5628 0001006 2522 14667 22767 24234 14668 24186 25653 5053 78478 83531 40 7384 0001008 1952 16756 22626 24301 16757 24067 25743 5725 76845 8 2570 45 9593 0001 010 1526 188 44 22484 2436 8 18845 23948 25832 6387 75261 81648 50 12349 0 001 012 1203 20932 22342 24435 20933 2382 7 25921 7038 7 3725 80763 55 15758 0001 015 9 568 23021 22199 24501 23023 23707 26009 7679 72234 7 9913 60 19 940 0001 017 7671 25111 22055 24566 25113 23585 26096 8312 70784 7 9096 65 2503 0001 020 6197 27202 21911 2463 1 27206 23462 26183 8935 6 9375 78310 70 3119 0001 023 5 042 292 95 21766 24696 29298 23338 26268 9549 6 8004 77553 75 3858 0001 026 4131 313 90 21620 2475 9 31393 23214 26353 10155 6 6669 76824 80 4739 0 001 029 3407 33486 21474 24822 33491 23088 26437 10753 65369 7 6122 85 57 83 0001 033 2 828 355 84 21326 24884 35590 22960 26519 11343 64102 75445 90 7014 0001 036 2361 376 85 2117 7 2494 5 37692 22832 26601 11925 62866 74791 il cc 001 4 l I 7 RR 2102 7 2i00 39 2270 2 26R l 1 iO f J6i9 7 4 i9 Press in MPa 100 0101 35 0001 044 16729 41894 20876 2506 5 41904 22570 26761 13069 60480 7 3549 105 0120 82 0001048 14194 44002 20723 25124 44015 22437 26838 13630 5 9328 72958 110 0143 27 0001 052 12102 46114 20570 25181 46130 22302 26915 14185 5 8202 72387 115 0169 06 0 001 056 10366 482 30 20414 25237 48248 22165 26990 14734 57100 71833 120 0198 53 0001 060 08919 503 50 20258 2593 50371 22026 27063 15276 56020 71296 125 02321 0001 065 07706 524 74 20099 2634 6 524 99 21885 27135 15813 54962 7 0775 130 02701 0001 070 06685 54602 19939 25399 546 31 21742 27205 16344 53925 70269 135 03130 0001 075 05822 56735 19777 2545 0 56769 21596 27273 16870 52907 6 9777 140 03613 0001 080 05089 58874 19613 25500 58913 21447 27339 17191 51908 69299 145 04154 0001 085 04463 61018 19447 25549 61063 21296 27403 17907 50926 68833 150 04758 0001 091 03928 63168 19279 25595 63220 21143 27465 18418 49960 68379 l 155 05431 0001 096 03468 65324 19108 2564l 65384 20986 27524 18925 49010 6 7935 t 160 06178 0001102 03071 67487 18935 25684 67555 20826 2758l 19427 48075 6 7502 165 07005 0001108 02727 69656 18760 25725 69734 20662 27635 19925 47153 67078 l 170 07917 0001114 02428 71833 18581 25765 71921 2049 5 27687 20419 46244 66663 s 175 08920 0001121 02168 74017 18400 2580 2 74117 20324 27736 20909 45347 66256 s 180 10021 0001127 0194 05 76209 18216 25837 76322 20150 27782 21396 44461 65857 185 11227 0001134 0179 09 78410 18029 25870 78537 19971 27824 21879 43586 65465 l 190 12544 0001141 0156 54 80619 17838 25900 80762 19788 27864 22359 42720 65079 I b 195 13978 0001 149 0141 05 82837 17644 25928 82998 19600 27900 22835 41863 64698 200 15538 0001157 0127 36 85065 17447 25953 85245 19407 2793 2 2 3309 41014 64323 r 205 0001164 63952 1 7230 0115 21 87304 1724 5 25975 87504 19210 27960 23780 40172 210 1 9062 0001173 0104 41 89553 1703 9 25995 89776 19007 27985 24248 39337 6 3585 215 2104 0001 181 0094 79 91814 16829 26011 92062 1879 9 28005 24714 38507 6 3221 t vi TABLE 91 continued Specific Volume Internal Energy Enthalpy Entropy Temp Press Sat Sat Sat Sat Sat Sat Sat Sat i oc MP a Liquid Vapor Liquid Evap Vapor Liquid Evap Vapor Liquid Evap Vapor i T p Vt Vg uf Ufg Ug h1 hfg hg Sf sfg Sg i 220 2318 0001190 0086 19 94087 16615 26024 94362 18585 2802l 25178 37683 62861 225 2548 0001199 0078 49 96373 16396 26033 96678 18365 28033 25639 36863 62503 230 2795 0001209 0071 58 98674 16172 2609 99012 18138 28040 26099 36047 62146 i t 235 3060 0001219 0065 37 100989 15942 2604l 101362 17905 28042 26558 35233 61791 240 3344 0001229 0059 76 103321 15708 26040 103732 17665 28038 27015 34422 61437 245 3648 0001 240 0054 71 105671 15467 26034 106123 17417 28030 27472 33612 61083 250 3973 0001 251 0050 13 108039 15220 26024 108536 17162 28015 27927 32802 60730 255 4319 0001 263 0045 98 110428 14967 26009 110973 16898 27995 28383 31992 60375 260 4688 0001 276 0042 21 112839 14706 25990 113437 1662 5 27969 28838 31181 60019 265 5081 0001 289 0038 77 115274 14439 25966 115928 16344 27936 29294 30368 59662 270 5499 0001 302 0035 64 117736 14163 25937 118451 16052 2789 7 29751 2 9551 59301 275 5942 0001 317 0032 79 120225 13879 25902 121007 15749 27850 30208 28730 5 8938 280 6412 0001 332 003017 122746 13587 2586l 123599 15436 27796 30668 27903 58571 285 6909 0001 348 0027 77 125300 13284 25814 126231 15110 27733 31130 27070 58199 290 7436 0001 366 0025 57 127892 1297l 25760 128907 1477l 27662 31594 2 6227 57821 295 7993 0001 384 0023 54 13052 12647 25699 13163 14418 27581 32062 25375 57437 300 8581 0001404 0021 67 13320 12310 25630 13440 14049 27490 32534 24511 57045 305 9202 0001425 0019 948 13593 11959 25552 13724 1366 4 27387 33010 23633 56643 310 9856 0001 447 0018 350 13871 11594 25464 14013 13260 27273 33493 22737 56230 315 10547 0001 472 0 016 867 14155 11211 25366 14310 1283 5 27145 3 3982 21821 55804 320 11274 0001 499 OQ15 488 14446 10809 25255 14615 12386 27001 34480 20882 55362 330 12845 0001 561 0012 996 15053 9937 24989 15253 11406 26659 35507 18909 54417 340 14586 0001 638 0010 797 15703 8943 24646 15942 10279 26220 36594 1 6763 53357 350 16513 0001 740 0008 813 16419 7766 24184 16706 8934 25639 37777 14335 52112 360 18651 0001 893 0006 945 17252 6263 23515 17605 7205 24810 39147 11379 50526 370 2103 0002 213 0004 925 18440 3845 22285 18905 4416 23321 41106 6865 47971 37414 2209 0003 155 0003 155 20296 0 20296 20993 0 20993 44298 0 44298 From R E Sonntag and G J Van Wylen lntroduct1on to Thermodvnamcs Cass1cal and Stattsttcal John Wiley Sons New York 1982 adapted from J H Keenan F G Keyes P G Hill and J G Moore Steam Tables John Wiley Som New York 1978 f 3 E i I i I 228 FirstOrder Phase Transitions phase are conventionally listed in such tables the latent heat of the transition is the difference in the molar enthalpies of the two phases or it can also be obtained as T s Similar data are compiled in the thermophysical data literature for a wide variety of other materials The molar volume like the molar entropy and the molar energy is discontinuous across the coexistence curve For water this is particularly interesting in the case of the solidliquid coexistence curve It is common experience that ice floats in liquid water The molar volume of the solid ice phase accordingly is greater than the molar volume of the liquid phasean uncommon attribute of H 20 The much more common situa tion is that in which the solid phase is more compact with a smaller molar volume One mundane consequence of this peculiar property of H 20 is the proclivity of frozen plumbing to burst A compensating consequence to which we shall return in Section 93 is the possibility of ice skating And underlying all this peculiar property of water is essential to the very possibility of life on earth If ice were more dense than liquid water the frozen winter surfaces of lakes and oceans would sink to the bottom new surface liquid unprotected by an ice layer would again freeze and sink until the entire body of water would be frozen solid frozen under instead of frozen over PROBLEMS 921 In a particular solidliquid phase transition the point P0 T0 lies on the coexistence curve The latent heat of vaporization at this point is t 0 A nearby point on the coexistence curve has pressure P0 p and temperature T0 t the local slope of the coexistence curve in the PT plane is pt Assuming v cp o and T to be known in each phase in the vicinity of the states of interest find the latent heat at the point P0 p T0 t 922 Discuss the equilibrium that eventually results if a solid is placed in an initially evacuated closed container and is maintained at a given temperature Explain why the solidgas coexistence curve is said to define the vapor pressure of the solid at the given temperature 93 THE SLOPE OF COEXISTENCE CURVES THE CLAPEYRON EQUATION The coexistence curves illustrated in Fig 91 are less arbitrary than is immediately evident the slope dP dT of a coexistence curve is fully determined by the properties of the two coexisting phases The Slope of Coexistence Curoes The Clapeyron Equotwn 229 The slope of a coexistence curve is of direct physical interest Consider cubes of ice at equilibrium in a glass of water Given the ambient pressure the temperature of the mixed system is determined by the liquidsolid coexistence curve of water if the temperature were not on the coexistence curve some ice would melt or some liquid would freeze until the temperature would again lie on the coexistence curve or one phase would become depleted At 1 atm of pressure the temperature would be 27315 K If the ambient pressure were to decreaseperhaps by virtue of a change in altitude the glass of water is to be served by the flight attendant in an airplane or by a variation in atmospheric conditions approach of a stormthen the temperature of the glass of water would appropriately adjust to a new point on the coexistence curve If AP were the change in pressure then the change in temperature would be AT APdPdTcc where the derivative in the denominator is the slope of the coexistence curve Ice skating to which we have made an earlier allusion presents another interesting example The pressure applied to the ice directly beneath the blade of the skate shifts the ice across the solidliqujd coexistence curve vertically upward in Fig 9la providing a lubricating film of liquid on which the skate slides The possibility of ice skating depends on the negative slope of the liquidsolid coexistence curve of water The existence of the ice on the upper surface of the lake rather than on the bottom reflects the larger molar volume of the solid phase of water as compared to that of the liquid phase The connection of these two facts which are not independent lies in the Clapeyron equation to which we now turn Consider the four states shown in Fig 97 States A and A are on the coexistence curve but they correspond to different phases to the lefthand and righthand regions respectively Similarly for the states B and B The pressure difference PB PA or equivalently PB PA is assumed to be infinitesimal dP and similarly for the temperature difference TB TA dT The slope of the curve is dP dT t p FIGURE97 r Four coexistence states 230 FirstOrder Phase Tranntwns Phase equilibrium requires that 94 and 95 whence 96 But J B J A s dT V dP 97 and J8 JA s dT v dP 98 in which s and s are the molar entropies and v and v are the molar volumes in each of the phases By inserting equations 97 and 98 in equation 96 and rearranging the terms we easily find dP s s 99 dT ti V dP As 910 dT Av in which As and Av are the discontinuities in molar entropy and molar volume associated with the phase transition According to equation 92 the latent heat is t TAs 911 whence dP t 912 dT TAv This is the Clapeyron equation The Clapeyron equation embodies the Le Chatelier principle Consider a solidliquid transition with a positive latent heat st sJ and a positive difference of molar volumes vt vJ The slope of the phase curve is correspondingly positive Then an increase in pressure at constant temper ature tends to drive the system to the more dense solid phase alleviating The Slope of Coexistence Curves The Clapeyron Equalon 23 J the pressure increase and an increase in temperature tends to drive the system to the more entropic liquid phase Conversely if st s5 but v1 V5 then the slope of the coexistence curve is negative and an increase of the pressure at constant T tends to drive the system to the liquid phaseagain the more dense phase In practical problems in which the Clapeyron equation is applied it is often sufficient to neglect the molar volume of the liquid phase relative to the molar volume of the gaseous phase vg v1 vg and to approximate the molar volume of the gas by the ideal gas equation vg e RT P This ClapeyronClausius approximation may be used where appropriate in the problems at the end of this section Example A light rigid metallic bar of rectangular cross sectmn lies on a block of ice extend ing slightly over each end The width of the bar is 2 mm and the length of the bar in contact with the ice is 25 cm Two equal masses each of mass M are hung from the extending ends of the bar The entire system is at atmosphenc pressure and is maintained at a temperature of T 2C What is the minimum value of M for which the bar will pass through the block of ice by regelation The given data are that the latent heat of fusion of water is 80 calgram that the density ofliquid water is 1 gramcm3 and that ice cubes float with 45 of their volume submerged Solution The Clapeyron equation permits us to find the pressure at which the solidliquid transition occurs at T 2C However we must first use the ice cube data to obtain the difference Av m molar volumes of liquid and solid phases The data given imply that the density of ice is 08gcm 3 Furthermore v11q 18 cm3mole and therefore Vsohd 225 x 10 0 m3mole Thus dP 80 X 42 X 18 Jmole 5 X 100 PaK dT cc TAv 271 x 45 x 10 0 Km 3mole so that the pressure difference required is P 5 X 106 X 2 107 Pa This pressure is to be obtained by a weight 2Mg acting on the area A 5 x 10 5 m2 M AP g 107 Pa 5 x 10 5 m1 98 z 26 Kg 232 FirstOrder Phase Transitwns PROBLEMS 931 A particular liquid boils at 127C at a pressure of 800 mm Hg It has a heat of vaporization of 1000 caljmole At what temperature will it boil if the pressure is raised to 810 mm Hg 932 A long vertical column is closed at the bottom and open at the top it is partially filled with a particular liquid and cooled to 5 C At this temperature the fluid solidifies below a particular level remaining liquid above this level If the temperature is further lowered to 52C the solidliquid interface moves upward by 40 cm The latent heat per unit mass is 2 caljg and the density of the liquid phase is 1 gcm 3 Find the density of the solid phase Neglect thermal expansion of all materials Hint Note that the pressure at the original position of the interface remains constant Answer 26 gcm 3 933 It is found that a certain liquid boils at a temperature of 95C at the top of a hill whereas it boils at a temperature of 105C at the bottom The latent heat is 1000 caljmole What is the approximate height of the hill 934 Two weights are hung on the ends of a wire which passes over a block of ice The wire gradually passes through the block of ice but the block remains intact even after the wire has passed completely through it Explain why less mass is required if a semiflexible wire is used rather than a rigid bar as in the Example 935 In the vicinity of the triple point the vapor pressure of liquid ammonia in Pascals is represented by In P 2438 3o 3 This is the equation of the liquidvapor boundary curve in a PT diagram Similarly the vapor pressure of solid ammonia is 3754 In P 2792 y What are the temperature and pressure at the triple point What are the latent heats of sublimation and vaporization What is the latent heat of fusion at the triple point 936 Let x be the mole fraction of solid phase in a solidliquid twophase system If the temperature is changed at constant total volume find the rate of change of x that is find dxdT Assume that the standard parameters u a Kr cp are known for each phase Problems 233 937 A particular material has a latent heat of vaporization of 5 X 103 Jmole constant along the coexistence curve One mole of this material exists in twophase liquidvapor equilibrium in a container of volume V 10 2 m3 at a tempera ture of 300 Kand a pressure of 105 Pa The system is heated at constant volume increasing the pressure to 20 X 105 Pa Note that this is not a small fjP The vapor phase can be treated as a monatomic ideal gas and the molar volume of the liquid can be neglected relative to that of the gas Find the initial and final mole fractions of the vapor phase x NgNg N 938 Draw the phase diagram in the BeT plane for a simple ferromagnet assume no magnetocrystalline anisotropy and assume the external field Be to be always parallel to a fixed axis in space What is the slope of the coexistence curve Explain this slope in terms of the Clapeyron equation 939 A system has coexistence curves similar to those shown in Fig 96a but with the liquidsolid coexistence curve having a positive slope Sketch the isotherms in the Pu plane for temperature T such that a T T b T T c T S T I01 d T T S Int e T Int f T Int Here T and Int denote the triple point and critical temperatures respectively 94 UNSTABLE ISOTHERMS AND FIRSTORDER PHASE TRANSITIONS Our discussion of the origin of firstorder phase transitions has focused quite properly on the multiple minima of the Gibbs potential But although the Gibbs potential may be the fundamental entity at play a more common description of a thermodynamic system is in terms of the form of its isotherms For many gases the shape of the isotherms is well represented at least semiquantitatively by the van der Waals equation of state recall Section 35 p RT v b a v2 913 The shape of such van der Waals isotherms is shown schematically in the Pv diagram of Fig 98 As pointed out in Section 35 the van der Waals equation of state can be viewed as an underlying equation of state obtained by curve fitting by inference based on plausible heuristic reasoning or by statistical mechanical calculations based on a simple molecular model Other em pirical or semiempirical equations of state exist and they all have iso therms that are similar to those shown in Fig 98 We now explore the manner in which isotherms of the general form shown reveal and define a phase transition 234 FirstOrder Phase Translions t p V FIGURE 98 van der Waals isotherms schematic T1 T2 T It should be noted immediately that the isotherms of Fig 98 do not satisfy the criteria of intrinsic stability everywhere for one of these criteria equation 821 is r 0 or 914 This condition clearly is violated over the portion FKM of a typical isotherm which for clanty is shown separately in Fig 99 Because of this violation of the stability condition a portion of the isotherm must be unphysical superseded by a phase transition in a manner which will be explored shortly The molar Gibbs potential is essentially determined by the form of the isotherm From the GibbsDuhem relation we recall that dµ sdT vdP 915 whence integrating at constant temperature µ f vdP tT 916 where f T is an undetermined function of the temperature arising as the constant of integration The integrand vP for constant temperature is given by Fig 99 which is most conveniently represented with P as Unstable Jotherms and FirstOrder Phase Tramtwm 235 i p t vP v FIGURE99 A particular isotherm of the van der Waals shape abscissa and v as ordinate By arbitrarily assigning a value to the chemical potential at the point A we can now compute the value ofµ at any other point on the same isotherm such as B for from equation 916 917 In this way we obtain Fig 910 This figure representingµ versus P can be considered as a plane section of a threedimensional representation of µ versus P and T as shown in Fig 911 Four different constanttempera ture sections of the µsurface corresponding to four isotherms are shown It is also noted that the closed loop of the µ versus P curves which results from the fact that v P is triple valued in P see Fig 99 disappears for high temperatures in accordance with Fig 98 Finally we note that the relation µ µT P constitutes a fundamen tal relation for one mole of the material as the chemical potential µ is the Gibbs function per mole It would then appear from Fig 911 that we have almost succeeded in the construction of a fundamental equation from a single given equation of state but it should be recalled that although each of the traces of the µsurface in the various constant temperature planes of Fig 911 has the proper form each contains an additive constant qT which varies from one temperature plane to another Consequently we do not know the complete form of the µ T Psurface although we certainly are able to form a rather good mental picture of its essential topological properties With this qualitative picture of the fundamental relation implied by the van der Waals equation we return to the question of stability 236 FirstOrder Phase Transitions s t µ P FIGURE 910 Isothermal dependence of the molar Gibbs potential on pressure t µ t 7 µ t p µ t p µ I P p FIGURE 911 Unstable Isotherms and FirstOrder Phase Translions 237 Consider a system in the state A of Fig 99 and in contact with thermal and pressure reservoirs Suppose the pressure of the reservoir to be increased quasistatically maintaining the temperature constant The sys tem proceeds along the isotherm in Fig 99 from the point A in the direction of point B For pressures less than PB we see that the volume of the system for given pressure and temperature is single valued and unique As the pressure increases above PB however three states of equal p and T become available to the system as for example the states designated by C L and N Of these three states L is unstable but at both C and N the Gibbs potential is a local minimum These two local minimum values of the Gibbs potential or of µ are indicated by the points C and N in Fig 910 Whether the system actually selects the state C or the state N depends upon which of these two local minima of the Gibbs potential is the lower or absolute minimum It is clear from Fig 910 that the state C is the true physical state for this value of the pressure and temperature As the pressure is further slowly increased the unique point D is reached At this point the µsurface intersects itself as shown in Fig 910 and the absolute minimum of µ or G thereafter comes from the other branch of the curve Thus at the pressure PE PQ which is greater than P0 the physical state is Q Below P0 the righthand branch of the isotherm in Fig 99a is the physically significant branch whereas above PO the lefthand branch is physically significant The physical isotherm thus deduced from the hypothetical isotherm of Fig 9 9 is therefore shown in Fig 912 The isotherm of Fig 9 9 belongs to an underlying fundamental relation that of Fig 912 belongs to the stable thermodynamic funda mental relation p v FIGURE 912 fhe physical van der Waals isotherm The underlying isotherm is SOMKFDA but the equalarea construction converts it to the physical isotherm SOKDA 238 First Order Phase Transtwns The points D and O are determined by the condition that µ 0 µ 0 or from equation 917 f 0 uPdP 0 D 918 where the integral is taken along the hypothetical isotherm Referring to Fig 99 we see that this condition can be given a direct graphical interpretation by breaking the integral into several portions JFudP JdP fMudP f 0 udP 0 D F K M 919 and rearranging as follows 920 Now the integral fudP is the area under the arc DF in Fig 912 and the integral JvdP is the area under the arc KF The difference in these integrals is the area in the closed region DFKD or the area marked I in Fig 912 Similarly the righthand side of equation 920 represents the area II in Fig 912 and the unique points O and D are therefore determined by the graphical condition area I area II 921 It is only after the nominal nonmonotonic isotherm has been truncated by this equal area constructwn that it represents a true physical isotherm Not only is there a nonzero change in the molar volume at the phase transition but there are associated nonzero changes in the molar energy and the molar entropy as well The change in the entropy can be computed by integrating the quantity 922 along the hypothetical isotherm OMKFD Alternatively by the thermody namic mnemonic diagram we can write liss 0 s 0 f du OMKFD 1 923 A geometrical interpretation of this entropy difference in terms of the area between neighboring isotherms is shown in Fig 913 Unstable lwtherms and FirstOrder Phase Tramuons 239 t p As sv s0 lllT f LlPdv ll shaded area FIGURE913 The discontinuity in molar entropy The area between adJacent isotherms is related to the entropy discontinuity and thence to the latent heat As the system is transformed at fixed temperature and pressure from the pure phase O to the pure phase D it absorbs an amount of heat per mole equal to I 00 Ts The volume change per mole is u u0 u0 and this is associated with a transfer of work equal to Pu Consequently the total change in the molar energy is u u0 u0 Ts Pu 924 Each isotherm such as that of Fig 912 has now been classified into three regions The region SO is in the liquid phase The region DA is in the gaseous phase The flat region OKD corresponds to a mixture of the two phases Thereby the entire Pu plane is classified as to phase as shown in Fig 914 The mixed liquidplusgas region is bounded by the inverted parabolalike curve joining the extremities of the flat regions of each isotherm Within the twophase region any given point denotes a mixture of the two phases at the extremities of the flat portion of the isotherm passing through that point The fraction of the system that exists in each of the two phases is governed by the lever rule Let us suppose that the molar volumes at the two extremities of the flat region of the isotherm are u1 and V8 suggesting but not requiring that the two phases are liquid and gas for definiteness Let the molar volume of the mixed system be u V N Then if x 1 and x 8 are the mole fractions of the two phases V Nu Nx 1u1 Nx 8v8 from which one easily finds 925 926 240 FirstOrder Phase Trans1t10ns t p v FIGURE914 Phase classification of the P v plane and V Vt X g V V g t 927 That is an intermediate point on the flat portion of the isotherm implies a mole fraction of each phase that is equal to the fractional distance of the point from the opposite end of the flat region Thus the point Z in Fig 914 denotes a mixed liquidgas system with a mole fraction of liquid phase equal to the length ZD divided by the length OD This is the very convenient and pictorial lever rule The vertex of the twophase region or the point at which 0 and D coincide in Fig 914 corresponds to the critical pointthe termination of the gasliquid coexistence curve in Fig 9la For temperatures above the critical temperature the isotherms are monotonic Fig 914 and the molar Gibbs potential no longer is reentrant Fig 910 Just as a Pv diagram exhibits a twophase region associated with the discontinuity in the molar volume so a Ts diagram exhibits a twophase region associated with the discontinuity in the molar entropy Example 1 Find the critical temperature Tc and critical pressure Pa for a system described by the van der Waals equation of state Write the van der Waals equation of state in terms of the reduced variabks t T Tm P P Per and ii v v Problems 241 Solution The critical state coincides with a point of horizontal inflection of the isotherm or aP a2p 0 av T av2 T Why Solving these two simultaneous equations gives V 3b p a cr 27b 2 8a RT 27b from which we can write the van der Waals equation in reduced variables st 3 P3v 1 v2 Example 2 Calculate the functional form of the boundary of the twophase region in the PT plane for a system described by the van der Waals equation of state Solution We work in reduced variables as defined in the preceding example We consider a fixed temperature and we carry out a Gibbs equal area construction on the corresponding isotherm Let the extremities of the twophase region correspond ing to the reduced temperature t be v8 and vt The equal area construction corresponding to equations 920 and 921 is 8Pdv PAvg Ve r here Pt P8 is the reduced pressure at which the phase transition occurs at the given reduced temperature T The reader should draw the isotherm identify the significance of each side of the preceding equation and reconcile this form of the statement with that in equations 920 and 921 he or she should also Justify the use of reduced variables in the equation Direct evaluation of the integral gives 91 1 91 1 ln3v 8 1 J l ln3vt 1 r l 4T v8 v8 4T Ve v1 Simultaneous solution of this equation and of the van der Waals equations for Bgi T and vrP T gives v8 vt and P for each value of t PROBLEMS 941 Show that the difference in molar volumes across a coexistence curve is given by 6v p 11j 942 Derive the expressions for v Pc and T given in Example 1 242 FirstOrder Phase Transitonf 943 Using the van der Waals constants for H 20 as given in Table 31 calculate the critical temperature and pressure of water How does this compare with the observed value Tc 64705 K Table IOI 944 Show that for sufficiently low temperature the van der Waals isotherm intersects the P 0 axis predicting a region of negative pressure Find the temperature below which the isotherm exhibits this unphysical behavior Hint Let P 0 in the reduced van der Waals equation and consider the condition that the resultant quadratic equation for the variable v1 have two real roots Answer f H o84 945 Is the fundamental equation of an ideal van der Waals fluid as given in Section 35 an underlying fundamental relation or a thermodynamic funda mental relation Why 946 Explicitly derive the relationship among v8 v1 and f as given in Example 2 947 A particular substance satisfies the van der Waals equation of state The coexistence curve is plotted in the P t plane so that the critical point is at I I Calculate the reduced pressure of the transition for t 095 Calculate the reduced molar volumes for the corresponding gas and liquid phases 09 P081 IC 07 06 06 08 10 12 14 16 18 20 22 24 FIGURE915 The T 095 isotherm The t 095 isotherm is shown in Fig 915 Counting squares permits the equal area construction Answer General Attributes of FirstOrder Phase Trans1t1011s 243 shown giving the approximate roots indicated on the figure Refinement of these roots by the analytic method of Example 2 yields J 0814 vg 171 and v1 0683 948 Using the two points at T 095 and T 1 on the coexistence curve of a fluid obeying the van der Waals equation of state Problem 947 calculate the average latent heat of vaporization over this range Specifically apply this result to H 20 949 Plot the van der Waals isotherm in reduced variables for T 09Tc Make an equal area construction by counting squares on the graph paper Corroborate and refine this estimate by the method of Example 2 9410 Repeat problem 948 in the range 090 T 095 using the results of problems 947 and 949 Does the latent heat vary as the temperature ap proaches I What is the expected value of the latent heat precisely at Tc The latent heat of vaporization of water at atmospheric pressure is 540 calories per gram Is this value qualitatively consistent with the trend suggested by your results 9411 Two moles of a van der Waals fluid are maintained at a temperature T 095Tc in a volume of 200 cm3 Find the mole number and volume of each phase Use the van der Waals constants of oxygen 95 GENERAL A TIRIBUTES OF FIRSTORDER PHASE TRANSITIONS Our discussion of firstorder transitions has been based on the general shape of realistic isotherms of which the van der Waals isotherm is a characteristic representative The problem can be viewed in a more general perspective based on the convexity or concavity of thermodynamic poten tials Consider a general thermodynamic potential UP 5 P that is a function of S X 1 X 2 X5 1 P5 Pr The criterion of stability is that UP P must be a convex function of its extensive parameters and a concave function of its intensive parameters Geometrically the function must lie above its tangent hyperplanes in the X1 X5 1 subspace and below its tangent hyperplanes in the Ps P subspace Consider the function UP 5 P as a function of X and suppose it to have the form shown in Fig 916a A tangent line DO is also shown It will be noted that the function lies above this tangent line It also lies above all tangent lines drawn at points to the left of D or to the right of 0 The function does not lie above tangent lines drawn to points inter mediate between D and o The local curvature of the potential is positive for all points except those between points F and M Nevertheless a phase 144 FirstOrder Phase Tra11s1tons t A S FIGURE916 xt xz xo x t A S Stability reconstruction for a general potential P transition occurs from the phase at D to the phase at 0 Global curvature fails becomes negative at D before local curvature fails at F The amended thermodynamic potential UPs P consists of the segment AD in Fig 915a the straight line twophase segment DO and the original segment OR An intermediate point on the straight line segment such as Z corre sponds to a mixture of phases D and 0 The mole fraction of phase D varies linearly from unity to zero as Z moves from point D to point O from which it immediately follows that This is again the lever rule The value of the thermodynamic potential UPs P in the mixed state ie at Z clearly is less than that in the pure state on the initial curve corresponding to X Thus the mixed state given by the straight line construction does mimmize UPs P and does correspond to the physical equilibrium state of the system The dependence of UP5 P on an intensive parameter Ps is subject to similar considerations which should now appear familiar The Gibbs potential UT P NµT P is the particular example studied in the preceding section The local curvature is negative except for the segment MF Fig 916b But the segment MD lies above rather than below the tangent drawn to the segment ADP at D Only the curve ADOR lies everywhere below the tangent lines thereby satisfying the conditions of global stability Thus the particular results of the preceding section are of very general applicability to all thermodynamic potentials FirstOrder Phase Trans1twns n Mu11wmponent SystemsGbhs Phase Rule 245 96 FIRSTORDER PHASE TRANSITIONS lN MULTICOMPONENT SYSTEMSGIBBS PHASE RULE If a system has more than two phases as does water recall Fig 91 the phase diagram can become quite elaborate In multicomponent systems the twodimensional phase diagram is replaced by a multidimensional space and the possible complexity would appear to escalate rapidly fortunately however the permissible complexity is severely limited by the Gibbs phase rule This restriction on the form of the boundaries of phase stability applies to singlecomponent systems as well as to multi component systems but it is convenient to explore it directly in the general case The criteria of stability as developed in Chapter 8 apply to multicom ponent systems as well as to singlecomponent systems It is necessary only to consider the various mole numbers of the components as extensive parameters that are completely analogous to the volume V and the entropy S Specifically for a singlecomponent system the fundamental relation is of the form U USVN 928 or in molar form u usu 929 For a multicomponent system the fundamental relation is U USVNiN 2 N 930 and the molar form is 931 The mole fractions x 1 N sum to unity so that only r 1 of the x 1 are independent and only r 1 of the mole fractions appear as indepen ent variables in equation 931 All of this is or should be familiar but it 1s repeated here to stress that the formalism is completely symmetric in he variables s v Xi xr i and that the stability criteria can be lllterpreted accordingly At the equilibrium state the energy the enthalpy and the Helmholtz and Gibbs potentials are convex functions of the mole fractions xi x 2 xr i see Problems 961 and 962 If the stability criteria are not satisfied in multicomponent systems a Phase transition again occurs The mole fractions like the molar entropies and the molar volumes differ in each phase Thus the phases generally are ditrerent in gross composition A mixture of salt NaCl and water 246 FirstOrder Phase Transrtrons brought to the boiling temperature undergoes a phase transition in which the gaseous phase is almost pure water whereas the coexistent liquid phase contains both constituentsthe difference in composition between the two phases in this case is the basis of purification by distillation Given the fact that a phase transition does occur in either a single or multicomponent system we are faced with the problem of how such a multiphase system can be treated within the framework of thermodynamic theory The solution is simple indeed for we need only consider each separate phase as a simple system and the given system as a composite system The wall between the simple systems or phases is then com pletely nonrestrictive and may be analyzed by the methods appropriate to nonrestrictive walls As an example consider a container maintained at a temperature T and a pressure P and enclosing a mixture of two components The system is observed to contain two phases a liquid phase and a solid phase We wish to find the composition of each phase The chemical potential of the first component in the liquid phase is µLlTPxLl and in the solid phase it is µSlTPxlsi it should be noted that different functional forms for µ 1 are appropriate to each phase The condition of equilibrium with respect to the transfer of the first component from phase to phase is 932 Similarly the chemical potentials of the second component are µ Ll T P xf L and µSl T P xf si we can rite these m terms of x 1 rather than x 2 because x 1 x 2 is unity in each phase Thus equating µS 1 and µ gives a second equallon which with equation 932 determine xf LJ and xfs Let us suppose that three coexistent phases are observed in the forego ing system Denoting these by I II and III we have for the first component µI T p XI µII T p X 11 µlll T p X Ill I I I I I I 933 and a similar pair of equations for the second component Thus we have four equations and only three composition variables x x1 and x 11 This means that we are not free to specify both T and P a priori but if T is specified then the four equations determine P x x1 and xU Although it is possible to select both a temperature and a pressure arbitrarily and then to find a twophase state a threephase state can exist only for one particular pressure if the temperature is specified In the same system we nught inquire about the existence of a state in which four phases coexist Analogous to equation 933 we have three FirstOrder Phase Transuwns m Multicomponent SstemsG1bhs Phase Rule 247 equations for the first component and three for the second Thus we have l T p 1 II Ill d lV Thi h six equations mvo vmg x 1 x 1 x 1 an x 1 s means t at we can have four coexistent phases only for a uniquely defined temperature and pressure neither of which can be arbitrarily preselected by the experimenter but which are unique properties of the system Five phases cannot coexist in a twocomponent system for the eight resultant equations would then overdetermine the seven variables T P x Xi and no solution would be possible in general We can easily repeat the foregoing counting of variables for a multi component multiphase system In a system with r components the chemical otentials in the first phase are functions of the variables T P xf x 2 x1 The chemical potentials in the second phase are functions of TPxflx 1 x 11 If there are M phases the complete set of independent variables thus consists of T P and Mr 1 mole fractions 2 Mr 1 variables in all There are M 1 equations of chemical potential equality for each component or a total of rM 1 equations Therefore the number f of variables which can be arbitrarily assigned is 2 Mr l rM 1 or frM2 934 The fact that r M 2 variables from the set TPxfx1 x11 can be assigned arbitrarily in a system with r components and M phases is the Gibbs phase rule The quantity f can be interpreted alternatively as the number of thermodynamic degrees of freedom previously introduced in Section 32 and defined as the number of intensive parameters capable of independent variation To justify this interpretation we now count the number of thermodynamic degrees of freedom in a straightforward way and we show that this number agrees with equation 934 For a singlecomponent system in a single phase there are two degrees of freedom the GibbsDuhem relation eliminating one of the three variables T P µ For a singlecomponent system with two phases there are three intensive parameters T P and µ each constant from phase to phase and there are two GibbsDuhem relations There is thus one degree of freedom In Fig 91 pairs of phases accordingly coexist over onedimensional regions curves If we have three coexistent phases of a singlecomponent system the three GibbsDuhem relations completely determine the three intensive parameters T P and µ The three phases can coexist only in a unique zerodimensional region or point the several triple points in Fig 91 For a multicomponent multiphase system the number of degrees of freedom can be counted easily in similar fashion If the system has r components there are r 2 intensive parameters T P µ 1 µ 2 Lr Each of these parameters is a constant from phase to phase But in each of 248 FirstOrder Phase Trans1twns the M phases there is a G1bbsDuhem relation These M relations reduce the number of independent parameters to r 2 M The number of degrees of freedotn f is therefore r M 2 as given in equation 934 The Gibbs phase rule therefore can be stated as follows In a system with r components and M coexistent phases It is possible arbitrarily to preassign r M 2 variables from the set T P x x x11 or from the set T P JJ1 JJ2 µ It is now a simple matter to corroborate that the Gibbs phase rule gives the same results for singlecomponent and twocomponent systems as we found in the preceding several paragraphs For singlecomponent systems r 1 and f 0 if M 3 This agrees with our previous conclusion that the triple point is a unique state for a singlecomponent system Similarly for the twocomponent system we saw that four phases coexist in a unique point 0 r 2 M 4 that the temperature could be arbitrarily assigned for the threephase system 1 r 2 M 3 and that both T and P could be arbitrarily assigned for the twophase system 2 r 2 M 2 PROBLEMS 961 In a particular system solute A and solute B are each dissolved m solvent C a What is the dimensionality of the space in which the phase regions exist b What is the dimensionality of the region over which two phases coexist c What is the dimensionality of the region over which three phases coexist d What is the maximum number of phases that can coexist in this system 962 If g the molar Gibbs function is a convex function of x 1 xi x 1 show that a change of variables to xi x 3 x results in g being a convex function of x 2 x 3 x That is show that the convexity condition of the molar Gibbs potential is independent of the choice of the redundant mole fraction 963 Show that the conditions of stability in a multicomponent system reqwre that the partial molar Gibbs potential µ1 of any component be an increasing function of the mole fraction x1 of that component both at constant v and at constant P and both at constant s and at constant T 97 PHASE DIAGRAMS FOR BINARY SYSTEMS The Gibbs phase rule equation 934 provides the basis for the study of the possible forms assumed by phase diagrams These phase diagrams particularly for binary twocomponent or ternary threecomponent systems are of great practical importance in metallurgy and physical chemistry and much work has been done on their classification To Phrue Diagrams for B1narv Systems 249 illustrate the application of the phase rule we shall discuss two typical diagrams for binary systems For a singlecomponent system the Gibbs function per mole is a function of temperature and pressure as in the threedimensional repre sentation in Fig 911 The phase diagram in the twodimensional TP plane such as Fig 91 is a projection of the curve of intersection of the µsurface with itself onto the TP plane For a binary system the molar Gibbs function G N 1 N2 is a function of the three variables T P and x 1 The analogue of Fig 911 is then fourdimensional and the analogue of the TP phase diagram is threedimensional It is obtained by projection of the hypercurve of intersection onto the P T x 1 hyperplane The threedimensional phase diagram for a simple but common type of binary gasliquid system is shown in Fig 917 For obvious reasons of graphic convenience the threedimensional space is represented by a series of twodimensional constantpressure sections At a fixed value of the mole fraction x 1 and fixed pressure the gaseous phase is stable at high temperature and the liquid phase is stable at low temperature At a temperature such as that labeled C in Figure 917 the system separates into two phasesa liquid phase at A and a gaseous phase at B The t t T Gas T 0 XI PP 3 i T t T Liquid L1qu1d 0 Xt Xt FIGURE 917 The threedimensional phase diagram of a typical gasliquid binary system The two dimensional sections arc constant pressure planes with P1 P2 P3 P4 250 FtrstOrder Phase Transitions composition at point C in Figure 917 is analogous to the volume at point Zin Figure 914 and a form of the lever rule is clearly applicable The region marked gas in Figure 917 is a threedimensional region and T P and x 1 can be independently varied within this region This i true also for the region marked liquid In each case r 2 M 1 and f 3 The state represented by point C in Figure 917 is really a twophase state composed of A and B Thus only A and Bare physical points and the shaded region occupied by point C is a sort of nonphysical hole in the diagram The twophase region is the surface enclosing the shaded volume in Figure 9 17 This surface is twodimensional r 2 M 2 f 2 Specifying T and P determines x and xf uniquely If a binary liquid with the mole fraction x is heated at atmospheric pressure it will follow a vertical line in the appropriate diagram in Fig 917 When it reaches point A it will begin to boil The vapor that escapes will have the composition appropriate to point B A common type of phase diagram for a liquid solid twocomponent system is indicated schematically in Fig 918 in which only a single constantpressure section is shown Two distinct solid phases of different crystal structure exist One is labeled a and the other is labeled 3 The curve BDHA is called the liquidus curve and the curves BEL and ACJ are called solidus curves Point G corresponds to a twophase systemsome liquid at H and some solid at F Point K corresponds to asolid at J plus 8solid at L A i T 0 FIGURE918 Typical phase diagram for a binary system at constant pressure If a liquid with composition x II is cooled the first solid to precipitate out has composition xF If it is desired to have the solid precipitate with the same composition as the iiquid it is necessary to start with a liquid of Phase Diagrams for Binary Systems 2 51 composition x 0 A liquid of this composition is called a eutectic solution A eutectic solution freezes sharply and homogeneously producing good alloy castings in metallurgical practice The liquidus and solidus curves are the traces of twodimensional surfaces in the complete Tx 1P space The eutectic point D is the trace of a curve m the full Tx 1P space The eutectic is a threephase region in which liquid at D 3solid at E and asolid at C can coexist The fact that a threephase system can exist over a onedimensional curve follows from the phase rule r 2 M 3 f 1 Suppose we start at a state such as N in the liquid phase Keeping T and x 1 constant we decrease the pressure so that we follow a straight line perpendicular to the plane of Fig 918 in the Tx 1P space We eventu ally come to a twophase surface which represents the liquidgas phase transition This phase transition occurs at a particular pressure for the given temperature and the given composition Similarly there is another particular pressure which corresponds to the temperature and composition of point Q and for which the solid P is in equilibrium with its own vapor To each point T x 1 we can associate a particular pressure P in this way Then a phase diagram can be drawn as shown in Fig 919 This phase diagram differs from that of Fig 918 in that the pressure at each point is different and each point represents at least a twophase system of which one phase is the vapor The curve BD is now a onedimensional curve M 3 f 1 and the eutectic point D is a unique point M 4 f 0 Point B is the triple point of the pure first component and point A is the triple point of the pure second component Although Figs 918 and 919 are very similar in general appearance they are clearly very different in meaning and confusion can easily arise B L1qu1d vapor D Vapor a 13 0 X1 lIGURE919 Vapor liquid a C A 0 a Phase diagram for a binary system in equilibrium with its vapor phase 2 52 FirstOrder Phafe Trans1tw11s from failure to distinguish carefully between these two types of phase diagrams The detailed forms of phase diagrams can take on a myriad of differences in detail but the dimensionality of the intersections of the various multiphase regions is determined entirely by the phase rule PROBLEMS 971 The phase diagram of a solution of A in B at a pressure of 1 atm 1s as shown The upper bounding curve of the twophase region can be represented by P 1 atm i Gas T Liquid 0 The lower bounding curve can be represented by A beaker containing equal mole numbers of A and B is brought to its boiling temperature What is the composition of the vapor as it first begins to boil off7 Does boiling tend to increase or decrease the mole fraction of A in the remaining liquid Answer xAvapor 0866 972 Show that if a small fraction dN N of the material is boiled off the system referred to in Problem 971 the change in the mole fraction in the remaining liquid is dx A 2x A x x A N Problems 253 973 The phase diagram of a solution of A m B at a pressure of 1 atm and in the region of small mole fraction x A 1 is as shown The upper bounding t T curve of the twophase region can be represented by T T0 CxA and the lower bounding curve by in which C and Dare positive constants D C Assume that a liquid of mole fraction x is brought to a boil and kept boiling until only a fraction Ntf N of the material remains derive an expression for the final mole fraction of A Show that if D 3C and if N1N the final mole fraction of component A is one fourth its initial value BANK OF BARODA QCI1997 1 BANK OF BARODA Established under the Bank of Baroda Act 1961 Owned by the Government of India HOME LOAN FORM For Salaried Self Employed Individuals DocNo FIN526 RevNo 10 Rev Date 30052011 Page 12 10 CRITICAL PHENOMENA 101 THERMODYNAMICS IN THE NEIGHBORHOOD OF THE CRITICAL POINT The entire structure of thermodynamics as described in the preceding chapters appeared at midcentury to be logically complete but the structure foundered on one ostensibly minor detail That detail had to do with the properties of systems in the neighborhood of the critical point Classical thermodynamics correctly predicted that various generalized susceptibilities heat capacities compressibilities magnetic susceptibili ties etc should diverge at the critical point and the general structure of classical thermodynamics strongly suggested the analytic form or shape of those divergences The generalized susceptibilities do diverge but the analytic form of the divergences is not as expected In addition the divergences exhibit regularities indicative of an underlying integrative principle inexplicable by classical thermodynamics Observations of the enormous fluctuations at critical points date back to 1869 when T Andrews 1 reported the critical opalescence of fluids The scattering of light by the huge density fluctuations renders water milky and opaque at or very near the critical temperature and pressure 64719 K 2209 MPa Warming or cooling the water a fraction of a Kelvin restores it to its norrnal transparent state Similarly the magnetic susceptibility diverges for a magnetic system near its critical transition and again the fluctuations in the magnetic moment are divergent A variety of other types of systems exhibit critical or secondorder transitions several are listed in Table IOI along with the corresponding order parameter the thermodynamic quantity that exhibits divergent fluctuations analogous to the magnetic moment 1T Andrews Ph Trans Royal Soc 159 575 1869 Vi 256 Cnttcal Phenomena TABLElOl Examples of Critical Points and Their Order Parameters Cnllcal Pomt Order Parameter Example T K Liquidgas Molar volume H 20 64705 Ferromagnetic Magnetic mome11t Fe 10440 Antiferromagnetic Sublattice magnetic FeF2 7826 moment Aline in 4 He 4 He quantum mechanical 4 He 1 821 amplitude Superconductivity Electron pair amplitude Pb 719 Binary fluid mixture Fractional segregation CC14C 7F14 30178 of components Binary alloy Fraction of one atomic CuZn 739 species on one sublatt1ce Ferroelectric Electnc dipole moment Triglycine sulfate 3225 Adapted from ShangKeng Ma Modern Theory of Crmwl Phenomena Add1onWcIey Advanccd Book Program CA 1976 Used by pemuss1on In order to fix these preliminary ideas in a specific way we focus on the gasliquid transition in a fluid Consider first a point P T on the coexistence curve two local minima of the underlying Gibbs potential then compete as illustrated in Fig 101 If the point of interest were to move off the coexistence curve in either direction then one or the other of the two minima would become the lower The two physical states corre sponding to the two minima have very different values of molar volume r FIGURE 101 Competition of two minima of the Gibbs potential near the coexitencc curve Thermodynamics 1n the Neighborhood of the Crmca Point 257 T FIGURE 102 The coalescence of the minima of the Gibbs potential as the critical point is approached molar entropy and so forth These two states correspond of course to the two phases that compete in the firstorder phase transition Suppose the point P T on the coexistence curve to be chosen closer to the critical point As the point approaches the critical T and P the two minima of the Gibbs potential coalesce Fig 102 For all points beyond the critical point on the extended or extrapolated coexistence curve the minimum is single and normal Fig 103 As the critical point is reached moving inward toward the physical coexistence curve the single minimum develops a flat bottom which in turn develops a bump dividing the broadened minimum into two separate minima The single minimum bifurcates at the critical point The flattening of the minimum of the Gibbs potential in the region of the critical state implies the absence of a restoring force for fluctuations away from the critical state at least to leading orderhence the diver gent fluctuations This classical conception of the development of phase transitions was formulated by Lev Landau 2 and extended and generalized by Laszlo Tisza 3 to form the standard classical theory of critical phenomena The essential idea of that theory is to expand the appropriate underlying thermodynamic potential conventionally referred to as the free energy functional in a power series in T Tc the deviation of the temperature from its value TAP on the coexistence curve The qualitative features described here then determine the relative signs of the first several f L D Landau and E M L1fslutz Stat1stual Physics MIT Press Cambridge Massachusetts and London 1966 3cf L T1za Generalzed Thermodynamics MIT Press Cambridge Massachusetts and London 1966 ee particularly papers 3 and 4 2 58 Crmca Phenomena T FIGURE 103 The classical picture of the development of a firstorder phase transition The dotted curve is the extrapolated nonphysical coexistence curve coefficients and these terms in turn permit calculation of the analytic behavior of the susceptibilities as T approaches the critical temperature T r A completely analogous treatment of a simple mechanical analogue model is given in the Example at the end of this section and an explicit thermodynamic calculation will be carried out in Section 104 At this point it is sufficient to recognize that the Landau theory is simple straightforward and deeply rooted in the postulates of macroscopic thermodynamics it is based only on those postulates plus the reasonable assumption of analyticity of the free energy functional However a direct comparison of the theoretical predictions with experimental observations was long bedeviled by the extreme difficulty of accurately measuring and controllmg temperature in systems that are incipiently unstable with gigantic fluctuations In 1944 Lars Onsager 4 produced the first rigorous statistical mechanical solution for a nontrivial model the twodimensional Ising model and it exhibited a type of divergence very different from that expected The scientific community was at first loath to accept this disquieting fact particularly as the model was twodimensional rather than threedimen sional and furthermore as it was a highly idealized construct bearing little resemblance to real physical systems In 1945 E A Guggenheim 4 L Onsager Phys Rev 65 117 1944 5E A Guggenheim J Chem Phys i3 253 1945 Thermodynanucs m the Neighborhood of the Critical Pomt 259 observed that the shape of the coexistence curve of fluid systems also cast doubt on classical predictions but it was not until the early 1960s that precise measurements 6 forced confrontation of the failure of the classical Landau theory and initiated the painful reconstruction 7 that occupied the decades of the 1960s and the 1970s Deeply probing insights into the nature of critical fluctuations were developed by a number of theoreticians including Leo Kadanoff Michael Fischer G S Rushbrooke C Domb B Widom and many others 89 The construction of a powerful analytical theory renormalization theory was accomplished by Kenneth Wilson a highenergy theorist interested in statistical mechanics as a simpler analogue to similar difficulties that plagued quantum field theory The source of the failure of classical Landau theory can be understood relatively easily although it depends upon statistical mechanical concepts yet to be developed in this text Nevertheless we shall be able in Section 105 to anticipate those results sufficiently to describe the origin of the difficulty in pictorial terms The correction of the theory by renormaliza tion theory unfortunately lies beyond the scope of this book and we shall simply describe the general thermodynamic consequences of the Wilson theory But first we must develop a framework for the description of the analytic form of divergent quantities and we must review both the classical expectations and the very different experimental observations To all of this the following mechanical analogue is a simple and explicit introduction Example The mechanical analogue of Section 91 provides instructive insights into the flattening of the minimum of the thermodynamic potential at the critical point as that minimum bifurcates into two competing minima below Tcr We again consider a length of pipe bent into a semicircle closed at both ends standing vertically on a table in the shape of an inverted U containing an internal piston On either side of the piston there is 1 mole of a monatomic ideal gas The metal 1balls that were inserted in Section 91 in order to break the symmetry and thereby to produce a firstorder rather than a secondorder transition are not presen If IJ is the angle of the piston with respect to the vertical R is the radius of curvature of the pipe section and Mg is the weight of the piston we neglect gravitational effects on the gas itself then the potential energy of the piston is 6 P Heller and G B Benedek Phys Rev Let 8 428 1962 7 H E Stanley Introduction to Phase Transitions and Cr111cal Phenomena Oxford Uruv Press New York and Oxford 1971 8f H E Stanley Ibid 9 P Pfeuty and G Toulouse lntroductwn to the Renorma1zatwn Group and Critical Phenomena John Wiley and Sons NY 1977 260 Critical Phenomena MgRcosO and the Helmholtz potential is F U TS MgR cos 8 FL FR The Helmholtz potentials FL and FR of the gases in the lefthand and righthand sections of the pipe are given by recall Problem 531 FLR FT RTln V where FT is a function of Tonly The volumes are determined by the position 0 of the piston where we have taken V0 as half the total volume of the pipe It follows then that for small 8 28 2 1 20 4 2FT RT 2 MgR 2FT 2 RT MgR0 2 1 8 4 MgR RT O 24 IT2 The coefficient of 04 is intrinsically positive but the coefficient of 0 2 changes sign at a temperature Tcr 1T2 Tcr SR MgR For T Tcr there is then only a single minimum the piston resides at the apex of the pipe and the two gases have equal volumes For T Tcr the state O 0 is a maximum of the Helmholtz potential and there are two symmetric minima at T T 0 6 w er 2 24T w Tcr D11ergence and Stab1hty 261 For T Tr the Helmholtz potential has a very flat minimum arising only from the fourthorder terms Spontaneous fluctuations thereby experience only weak restoring forces The Brownian motion fluctuation of the position of the piston is correspondingly large Furthermore even a trivially small force applied to the piston would induce a very large displacement the generalized suscept ibility diverges Although we have now seen the manner in which this model develops a bifurcating Helmholtz functional at the critical temperature it may be instructive also to reflect on the manner in which a firstorder transition occurs at lower temperatures For this purpose some additional parameters must be introduced to bias one minimum of F relative to the other We might simply tilt the table slightly thereby inducing a firstorder transition from one minimum to the other Alternatively and more familiarly a firstorder phase transition can be thermally induced In Section 91 this possibility was built into the model by inclusion of two metal ballbearings of different coefficients of thermal expansion a more appealing model would be one in which the two gases are differently nonideal Although this example employs a rather artificial system the fundamental equation mimics that of homogeneous thermodynamic systems and the analysis given above anticipates many features of the classical Landau theory to be described in Section 104 102 DIVERGENCE AND STAB1Ll1Y The descriptive picture of the origin of divergences at the critical point as alluded to in the preceding section is cast into an illuminating perspective by the stability criteria equation 815 and Problem 823 a2g 0 ar2 p a2g 0 apz T 101 and 102 These stability criteria express the concavity requirements of the Gibbs potential The flattening of the Gibbs potential at the critical point corresponds to a failure of these concavity requirements In fact all three of the stability criteria fail simultaneously and a KT and cP diverge together Further perspective is provided by a physical rather than a formal point of view Consider a particular point P Ton the coexistence 262 Critical Phenomena v FIGURE 104 Schematic isotherms of a twophase sys tem curve of a twophase system The isotherms of the system are qualitatively similar to those shown in Fig 104 recall Fig 912 although the van der Waals equation of state may not be quantitatively relevant In particular the isotherms have a flat portion in the PT plane On this flat portion the system is a mixture of two phases in accordance with the lever rule Section 94 The volume can be increased at constant pressure and temperature the system responding simply by altering the mole fraction in each of the two coexistent phases Thus formally the isothermal compressibility Kr v 18v8Pr diverges Again considering this same system in the mixed twophase state suppose that a small quantity of heat Q T S is injected The heat supplies the heat of transition the heat of vaporization or the heat of melting and a small quantity of matter transforms from one phase to the other The temperature remains constant Thus c P T as 8T P diverges The divergence of Kr and of cp exists formally all along the coexistence locus Across the coexistence locus in the PT plane both KT and cP are discontinuous jumping from one finite value to another by passing through an intermediate infinity in the mixedphase state see Fig 105 As the point of crossing of the coexistence curve is chosen closer to the critical point classical Landau theory predicts that the jump of KT should decrease but that the intermediate infinity should remain This t I t V TorP FIGURE 105 Discontinuity and divergence of gener alized susceptibilities across a coexis tence locus The abscissa can be either T or P along a line crossing the coexis tence locus in the T P plane Order Parameters a11d Cr111cal Expo11ents 263 description is correct except very close to the critical point in which region nonclassical behavior dominated by the fluctuations intervenes Nevertheless the qualitative behavior remains similara divergence of KT at the critical point albeit of an altered functional form The heat capacity behaves somewhat differently As we shall see later Landau theory predicts that as the critical point is approached both the jump in the heat capacity and the intermediate divergence should fade away In fact the divergence remains though it is a weaker divergence than that of KT 103 ORDER PARAMETERS AND CRITICAL EXPONENTS Although Landaus classical theory of critical transitions was not quantitatively successful it did introduce several pivotal concepts A particularly crucial observation of Landau was that in any phase transi tion there exists an order parameter that can be so defined that it is zero in the hightemperature phase and nonzero in the lowtemperature phase Order parameters for various secondorder transitions are listed in Table 101 The simplest case and the prototypical example is provided by the paramagnetic to ferromagnetic transition or its electric analogue An appropriate order parameter is the magnetic moment which measures the cooperative alignment of the atomic or molecular dipole moments Another simple and instructive transition is the binary alloy orderdisorder transition that occurs for instance in copperzinc CuZn alloy The crystal structure of this material is bodycentered cubic which can be visualized as being composed of two interpenetrating simple cubic lattices For convenience we refer to one of the sublattices as the A sublattice and to the other as the B sublattice At high temperatures the Cu and Zn atoms of the alloy are randomly located so that any particular lattice point is equally likely to be populated by a zinc or by a copper atom As the temperature is lowered a phase transition occurs ruch that the copper atoms preferentially populate one sublattice and the zinc atoms preferentially populate the other sublattice Immediately below the transition temperature this preference is very slight but with decreas ing temperature the sublattice segregation increases At zero temperature one of the sublattices is entirely occupied by copper atoms and the other sublattice is entirely occupied by zinc atoms An appropriate order parameter is N 0 NfuNA or the difference between the fraction of A sites occupied by zinc atoms and the fraction occupied by copper atoms Above the transition temperature the order parameter is zero it becomes nonzero at the transition temperature and it becomes either 1 or 1 at T O As in the orderdisorder transition the order parameter can always be chosen to have unit magnitude at zero temperature it is then normal 264 Crtllca Phenomena ized In the ferromagnetic case the normalized order parameter is JT0 whereas the extensive parameter is the magnetic moment lT In passing we recall the discussion in Section 38 on unconstrainable variables As was pointed out it sometimes happens that a formally defined intensive parameter does not have a physical realization The copperzinc alloy system is such a case In contrast to the ferromagnetic case in which the order parameter is the magnetic moment I and the intensive parameter iJUiJI is the magnetic field Be the order parameter for the copperzinc alloy is N1n Ntu but the intensive parameter has no physical reality Thus the thermodynamic treatment of the CuZn system requires that the intensive parameter always be assigned the value zero Similarly the intensive parameter conjugate to the order parameter of the superfluid 4 He transition must be taken as zero Identification of the order parameter and recognition that various generalized susceptibilities diverge at the critical point motivates the definition of a set of critical exponents that describe the behavior of these quantities in the critical region In the thermodynamic context there are four basic critical exponents defined as follows The molar heat capacity c v in the fluid case or c 8 in the magnetic case diverges at the critical point with exponents a above T and a below T T T 103 CV or CB T Ta T TJ 104 The generalized susceptibilities r aviJPrv in the fluid case or Xr µ 081iJBrv in the magnetic case diverge with expo nents y or y r or X r T T Y 105 y Kr or Xr T T T J 106 Along the coexistence curve the order parameter varies as T T J T J 107 and of course the order parameter vanishes for T T Note that a prime indicates T Tc for the exponents a and y whereas 1 can be defined only for T Tc so that a prime is superfluous Class1cal Theory m the Critical Regwn Landau Theory 265 Finally on the critical isotherm ie for T Tcr the order parameter and its corresponding intensive parameter satisfy the relation I B or iv P P 1B 108 which define the exponent 8 In addition there are several critical exponents defined in terms of statistical mechanical concepts lying outside the domain of macroscopic thermodynamics Perhaps the most significant of these additional expo nents describes the range of fluctuations or the size of the correlated regions within the system The long wavelength fluctuations dominate near the critical point and the range of the correlated regions diverges This onset of longrange correlated behavior is the key to the statistical mechanical or renormalization group solution to the problem Because large regions are so closely correlated the details of the particular atomic structure of the specific material become of secondary importance The atomic structure is so masked by the longrange correlation that large families of materials behave similarlya phenomenon known as univer sality to which we shall return subsequently 104 CLASSICAL THEORY IN THE CRITICAL REGION LANDAU THEORY The classical theory of Landau which evaluates the critical exponents provides the standard of expectation to which we can contrast both experimental observations and the results of renormalization group the ory We consider a system in which the unnormalized order parameter is q We have in mind perhaps the magnetization of a uniaxial crystal in which the dipoles are equally probably up or down above the transition temperature or the binary CuZn alloy The Gibbs potential t f IS a unction of T P q N1 Ni Nr G GT P q N1 Ni N 109 In the immediate vicinity of the critical point the order parameter is small suggesting a series expansion in powers of q 1010 Where G0 G1 Gi are functions of T P N1 Nr For the magnetic system or binary alloy the symmetry of the problems immediately pre cludes the odd terms requiring that the Gibbs potential be even in q there is no a priori difference between spin up and spin down or between 166 Critical Phenomena the A and B sublattices This reasoning is a precursor and a prototype of more elaborate symmetry arguments in more complex systems 1011 Each of the expansion coefficients is a function of T P and the s Gn GnT P Ni N We now concentrate our attention on the extrapolated coexistence curvethe dotted curve in Fig 103 Along this locus P is a function of T and all mole numbers are constant so that each of the expansion coefficients Gn is effectively a function of T only Correspondingly G is effectively a function only of T and q The shape of G T q as a function of q for small q is shown in Fig 106 for the four possible combinations of signs of G2 and G4 GTtJ GT tJ 4 4 GT 4 GT 4 4 4 FIGURE 06 Possible shapes of GT p for various signs of the expansion coefficients A point on the extrapolated coexistence curve beyond the critical point is in the singlephase region of stability where the Gibbs potential has a simple minimum From this fact we conclude that GiT is positive Stability to large fluctuations implies also that Gi T is positive As the point of interest approaches and then passes the critical point along the coexistence curve the curvature GiT passes through zero and becomes negative Fig 106 The function GiT normally remains positive The critical temperature is mewed simply as the temperature at which G 1 happens to have a zero The change of sign of G2 at the critical point implies that a serie expansion of G2 in powers of T T has the form G2T PT T TGf terms of orderT Tr1 1012 Class1cul Theory m the Cmcal Regwn Landau Theory 267 Now let the intensive parameter conjugate to cp have the value zero In the magnetic case in which cp is the normalized magnetic moment this implies that there is no external magnetic field whereas in the binary alloy the intensive parameter is automatically zero Then in either type of case aG o 3 2 T T G 4G 0 acp er 21 41 1013 This equation has different solutions above and below Tr For T Tcr the only real solution is cp 0 1014 Below T the solution cp 0 corresponds to a maximum rather than a minimum value of G recall Fig 106 but there are two real solutions corresponding to minima Go 112 cp 2 G T T 1015 This is the basic conclusion of the classical theory of critical points The order parameter magnetic moment difference in zinc and copper occupa tion of the A sublattice etc spontaneously becomes nonzero and grows as Tr T 112 for temperatures below Tr The critical exponent 3 defined in equation I 0 7 thereby is evaluated classically to have the value f3classical 12 1016 In contrast experiment indicates that for various ferromagnets or fluids the value of f3 is in the neighborhood of 03 to 04 In equation 1013 we assumed that the intensive parameter conjugate to 4 is zero this was dictated by our interest in the spontaneous value of cp below T We now seek the behavior of the susceptibility Xr for temperatures just above T x T being defined by 1017 In the magnetic case xi 1 is equal to NaBeaiho so that µ 0xr is the familiar molar magnetic susceptibility but in the present context we shall not be concerned with the constant factor µ 0 Then 1 T o 2 NXr 2 T a G2 12G4qi 1018 268 Cntwl Phenomena or taking q 0 according to the definition 1017 1 l 0 x 2 T T G N 1 er 2 1019 This result evaluates the classical value of the exponent y equation 105 as unity yclassical 1 1020 Again for f erromagnets and for fluids the measured values of y are in the region of 12 to 14 For T Tr the order parameter q becomes nonzero Inserting equa tion 1015 for qT into equation 1018 4T TG 0 r 2 1021 We therefore conclude that the classical value of y is unity recall equation 106 Again this does not agree with experiment which yields values of y in the region of 10 to 12 The values of the critical exponents that follow from the Landau theory are listed for convenience in Table 102 TABLEI02 Critical Exponents Oasiical Values and Approximate Range of Observed Values Approximate range of Exponelll Classical value observed values a 0 02 a 02 a 0 02 a 03 3 l 03 3 04 2 1 1 2 r L4 r 1 1 r 1 2 8 3 485 Example It is instructive to calculate the classical values of the critical exponents for a system with a given definite fundamental equation thereby corroborating the more general Landau analysis Calculate the critical indices for a system de scribed by the van der Waals equation of state Classcal Theory m the Crrtical Regwn Landau Theory 269 Solution From Example 1 of Section 94 the van der Waals equation of state can be written in reduced variables st 3 P3v 1 02 where P P Pc and similarly for f and ii Then defining pP1 Vii1 ET1 and multiplying the van der Waals equation by 1 v 2 we obtain10 2pl fv 4v2 v 3 3v 3 81 2v v2 or p lv 3 E4 6v 9v2 If E 0 that is T I then v is proportional to p so that the critical exponent 8 is identified as 8 3 To evaluate y we calculate K71 vaP va 6vE av T av t whence y y l To calculate 3 we recall that 8vg 8 where 811 is defined by the last equation in Example 2 page 241 Ov ln3v l3v 1 1 94vT ln3v 23v 2r 1 Hv 1 1 1 1 In 2 113 E 1 e i ve ii Then from OiJg Oiit we find t v Df t vl E 2 E vg vt o Also pDg pD1 which gives v Dft vt 4 6Evg Dt o These ltter two equations constitute two equations in the two unknowns vg and I Eliminating Dg Dt we are left with a single equation in vg Dt we find vg vt 4 which identifies the critical exponent p as The remaining critical exponents are a and a referring to the heat capacity The van der Waals equation of state alone does not determine the heat capacity but we can tum to the ideal van der Waals fluid defined in Section 35 For that 10 H Stanley lntroductwn to Phase Trans1t10ns and Critcal Phenomena Oxford Univ Press New York and Oxford 1971 sect 55 270 Cr111cal Phenomena system the heat capacity cv is a constant with no divergence at the critical point and a a 0 105 ROOTS OF THE CRITICAL POINT PROBLEM The reader may well ask how so simple direct and general an argument as that of the preceding section can possibly lead to incorrect results Does the error lie within the argument itself or does it lie deeper at the very foundations of thermodynamics That puzzlement was shared by thermo dynamicists for three decades Although we cannot enter here into the renormalization theory that solved the problem it may be helpful at least to identify the source of the difficulty To do so we return to the most central postulate of thermodynamicsthe entropy maximum postulate In fact that postulate is a somewhat oversimplified transcription of the theorems of statistical mechanics The oversimplification has significant consequences only when fluctuations become dominantthat is in the critical region The crucial theorems of statistical mechanics evaluate the probability of fluctuations in closed composite systems or in systems in contact with appropriate reservoirs In particular for a closed composite system the energy of one of the subsystems fluctuates and the probability that at any given instant it has a value Eis proportional to expSEk 8 where S is the entropy of the composite system The average energy U is to be obtained from this probability density by a standard averaging process Generally the probability density is very sharp or narrow The average energy then is very nearly equal to the most probable energy The latter is the more easily obtainable from the probability distribution for it ie the most probable energy is simply that value of E that maximizes expSk 8 or that maximizes the entropy S The baste postulate of thermodynamics incorrectly identifies the most probable value of the energy as the equilibrium or average value Fortunately the probability density of macroscopic systems is almost always extremely narrow For a narrow probability density the average value and the most probable value coincide and classical thermodynamics then is a valid theory However in the critical region the minimum of the thermodynamic potential becomes very shallow the probability distribu tion becomes very broad and the distinction between average and mot probable states can become significant To illustrate the consequence of this distinction near the critical point Fig 107 shows the Gibbs potential schematically as a function of the order parameter cp for two temperatures very slightly below T with the intensive parameter equal to zero Only the positive branch of cp 1 shown though there is a similar branch for negative cp we assume the system to be in the minimum with cp 0 For T1 the potential is shallow and asymmetric and the probability density for the fluctuating order t G I I I I I I I I FIGURE 107 Roots of the Critical Pomt Problem 271 Probability distributions average and most probable values for the fluctuating order parameter The temperatures are T2 Ti 7 The probability distributions are shown as dotted curves The classical or most probable values are lP and 12P and these coincide with the minima of G The average or observable values are 11 and 12 The rate of change of the average alues is more rapid than the rate of change of the most probable values because of the asymmetry of the curves for T1 This is more coruistent with a cntical index fJ t rather than t as shown in the small figure parameter shown dotted is correspondingly broad and asymmetric The average value q of q is shifted to the left of the most probable value JP For a temperature T2 further removed from the critical temperature the potential well is almost symmetric near its minimum and the probability density is almost symmetric The average value q and the most probable value fP are then almost identical As the temperature changes from T1 to T2 the classically predicted change in the order parameter is jP JP whereas the statistical mechanical prediction is q 1r Thus we see that rlassical thermodynamics incorrectly predicts the temperature dependence of the order parameter as the critical temperature is approached and that this failure is connected with the shallow and asymmetric nature of the minimum of the potential To extend the reasoning slightly further we observe that JP JtP is smaller than q q Fig 107 That is the classical thermodynamic prediction of the shift in q for a given temperature change is smaller than the true shift ie than the shift in the average value of q This is consistent with the classical prediction of f3 t rather than the true value 3 t as indicated in the insert in Fig 107 This discussion provides at best a pictorial insight as to the origin of the failure of classical Landau theory It gives no hint of the incredible depth and beauty of renormalizationgroup theory about which we later shall have only a few observations to make 272 Critical Phenomena 106 SCALING AND UNIVERSALITY As mentioned in the last paragraph of Section 103 the dominant effect that emerges in the renormalization group theory is the onset of longrange correlated behavior in the vicinity of the critical point This occur because the long wave length excitations are most easily excited As fluctuations grow the very long wave length fluctuations grow most rapidly and they dominate the properties in the cntical region Two effects result from the dominance of long range correlated fluctuations The first class of effects is described by the term scaling Specifically the divergence of the susceptibilities and the growth of the order parame ter are linked to the divergence of the range of the correlated fluctuatiom Rather than reflecting the full atomic complexity of the system the diverse critical phenomena all scale to the range of the divergent correla tions and thence to each other This interrelation among the critical exponents is most economically stated in the scaling hypothesis the fundamental result of renormalizationgroup theory That result state that the dominant term in the Gibbs potential or another thermodynamic potential as appropriate to the critical transition considered in the region of the critical point is of the form 1022 We here use the magnetic notation for convenience but B can be interpreted generally as the intensive parameter conJugate to the order parameter q The detailed functional form of the Gibbs potential i discontinuous across the coexistence curve as expected and this discon tinuity in form is indicated by the notation J the function r applies for T Tcr and the different function F applies for T Tr Further more the Gibbs potential may have additional regular terms the term written in equation 1022 being only the dominant part of the Gibbs potential in the limit of approach to the critical point The essential content of equation 1022 is that the quantity Gs T Tcr2 is not a function of both T and Be separately but only of the single variable BIBIT Trl 2 It can equally well be written as a function of the square of this composite variable or of any other power We shall later write it as a function of BeT Tcr 2 aBI Hl The scaling property expressed in equation 1022 relates all other critical exponents by universal relationships to the two exponents a and S as we shall now demonstrate The procedure is straightforward we simpl evaluate each of the critical exponents from the fundamental equation 1022 We first evaluate the critical index o to corroborate that the symbol Y appearing in equation 1022 does have its expected significance For this Swlmg and Umversafn 2 7 3 purpose we take Be 0 The functions f x are assumed to be well behaved in the region of x 0 with f rO being finite constants Then the heat capacity is 1023 Hence the critical index for the heat capacity both above and below T is identified as equal to the parameter a in G8 whence a a 1024 Similarly the equation of state I IT B is obtained from equation 1022 by differentiation 1025 where f x denotes ddxf x Again the functions O are assumed finite and we have therefore corroborated that the symbol S has its expected significance as defined in equation 108 To focus on the temperature dependence of I and of X in order to evaluate the critical exponents 3 and y it is most convenient to rewrite f as a function g of BT ry 2 allo ll G T T 2 a B s I al g IT Tl2alll ll Then I aG 2 alCtll B ITTI g aB lr IT T12alllll Whence 2a 3113 1026 1027 1028 274 Critical Phenomena Also a1 IT T 12al6lll 11 Be X µo a Be er g IT rJ2a6l6 whence 1 8 y y a 21 S 1029 1030 Thus all the critical indices have been evaluated in terms of a and 8 The observed values of the critical indices of various systems are of course consistent with these relationships As has been stated earlier there are two primary consequences of the dominance of long range correlated fluctuations One of these is the scaling of critical properties to the range of the correlations giving rise to the scaling relations among the critical exponents The second conse quence is that the numerical values of the exponents do not depend on the detailed atomic characteristics of the particular material but are again determined by very general properties of the divergent fluctuations Re normalization group theory demonstrates that the numerical values of the exponents of large classes of materials are identical the values are determined primarily by the dimensionality of the system and by the dimensionality of the order parameter The dimensionality of the system is a fairly selfevident concept Most thermodynamic systems are threedimensional However it is possible to study twodimensional systems such as monomolecular layers adsorbed on crystalline substrates Or onedimensional polymer chains can be studied An even greater range of dimensions is available to theorists who can and do construct statistical mechanical model systems in four five or more dimensions and even in fractional numbers of dimensions The dimensionality of the order parameter refers to the scalar vector or tensorial nature of the order parameter The order parameter of the binary alloy discussed in Section 103 is onedimensional scalar The order parameter of a ferromagnet which is the magnetic moment is a vector and is of dimensionality three The order parameter of a supercon ductor or of superfluid 4He is a complex number having independent real and imaginary components it is considered as twodimensional And again theoretical models can be devised with other dimensionalities of the order parameters Problems 275 Systems with the same spatial dimensionality and with the same dimensionality of their order parameters are said to be in the same universality class And systems in the same universality class have the same values of their critical exponents PROBLEMS 1061 Show that the following identities hold among the critical mdices a 23 y 2 Rushbrookes scaling law y 38 1 Widoms scaling law 1062 Are the classical values of the critical exponents consistent with the scaling relations It is assumed that the mteratomic forces in the system are not of mfimte range The Only fool is the one who thinks hes wise Macbeth I VII 1518 11 THE NERNST POSTULATE 111 NERNSTS POSTULATE AND THE PRINCIPLE OF THOMSEN AND BERTHELOT One aspect of classical thermodynamics remains That is the explora tion of the consequences of postulate IV to the effect that the entropy vanishes at zero temperature The postula le as first formulated by Walther N ernst in 1907 was somewhat weaker than our postulate IV stating only that the entropy change in any isothermal process approaches zero as the temperature approaches zero The statement that we have adopted emerged several decades later through the work of Francis Simon and the formulation of Max Planck it is nevertheless referred to as the Nernst postulate It is also frequently called the third law of thermodynamics Unlike the other postulates of the formalism the Nernst postulate is not integral to the overall structure of thermodynamic theory Having developed the theory almost in its entirely we can now simply append the Nernst postulate Its implications refer entirely to the lowtemperature region near T 0 1 The historical origins of the Nernst theorem are informative they lie in the principle of Thomsen and Berthelot an empirical but nonrigor ous rule by which chemists had long predicted the equilibrium state of chemically reactive systems Consider a system maintained at constant temperature and pressure as by contact with the ambient atmosphere and released from constraints as by mixture of two previously separated chemical reactants According to the empirical rule of Thomsen and Berthelot the equilibrium state to which the system proceeds is such that the accompanying process evolves the greatest efflux of heat or in the more usual language the process is realized that is most exothermic The formal statement of this empirical rule is most conveniently put in terms of the enthalpy We recall that in isobaric processes the enthalpy 278 The Nernst Postulate acts as a potential for heat so that the total heat efflux is heat efflux Hrutal Hrma1 111 The statement of Thomsen and Berthelot therefore is equivalent to the statement that the equilibrium state is the one that maximizes H 1rutia1 H rma1 or minimizes H rma1 The proper criterion of equilibrium at constant temperature and pres sure is of course the minimization of the Gibbs potential Why then should these two differing criteria provide similar predictions at low temperatures and in fact sometimes even at or near room temperature In an isothermal process llG tH TtS 112 so that at T 0 the changes in the Gibbs potential and in the enthalpy are equal tS certainly being bounded But that is not sufficient to explain why they remain approximately equal over some nonnegligible temperature range However dividing by T tH tG tS T 113 We have seen from equation 112 that tH tG at T O hence the lefthand side of equation 113 is an indeterminate form as T 0 The limiting value is obtained by differentiating numerator and denominator separately LHospitals rule whence d tH d tG lim ts dT TO dT TO TO 114 By assuming that lim ts 0 TO 115 it was ensured by Nernst that tH and tG have the same initial slope Fig 111 and that therefore the change in enthalpy is very nearly equal to the change in Gibbs potential over a considerable temperature range The Nernst statement that the change in entropy tS vanishes in any reversible isothermal process at zero temperature can be restated The T 0 isotherm is also an isentrope or adiabat This coincidence of isotherm and isentrope is illustrated in Fig 112 The Planck restatement assigns a particular value to the entropy The T 0 isotherm coincides with the S 0 adiabat t p FIGURE 112 1 1 Problems 279 FIGURE 11l Illustrating the principle of Thomsen and Berthelot s 1 v Isotherms and isentropes adiabats near T 0 In the thermodynamic context there is no a pnon meaning to the absolute value of the entropy The Planck restatement has significance bnly in its statistical mechanical interpretation to which we shall turn in Part II We have in fact chosen the Planck form of the postulate rather than the Nemst form largely because of the pithiness of its statemen rather than because of any additional thermodynamic content The absolute entropies tabulated for various gases and other systems in the reference literature fix the scale of entropy by invoking the Planck form of the Nemst postulate PROBLEMS lll1 Does the twolevel system of Problem 538 satisfy the Nernst postulate Prove your assertion 280 The Nernst Postulate 112 HEAT CAPACITIES AND OTHER DERIVATIVES AT LOW TEMPERATURE A number of derivatives vanish at zero temperature for reasons closely associated with the Nernst postulate Consider first a change in pressure at T 0 The change in entropy must vanish as T 0 The immediate consequence is as r 0 116 where we have invoked a familiar Maxwell relation It follows that the coefficient of thermal expansion a vanishes at zero temperature a l av o var p as r 0 117 Replacing the pressure by the volume in equation 116 the vanishing of as I avr implies again by a Maxwell relation aP o ar as r 0 118 The heat capacities are more delicate If the entropy does not only approach zero at zero temperature but if it approaches zero with a bounded derivative ie if as aT is not infinite then as r O 119 and similarly if as I oT p is bounded cP r L o as r 0 1110 Referring back to Fig 111 it will be noted that both bG and bH were drawn with zero slope whereas equations 114 and 115 required only that bG and bH have the same slope The fact that they have zero slope is a consequence of equation 1110 and of the fact that the temperature derivative of b H is just N bc r The vanishing of c and cP and the zero slope of bG or bH appears generally to be true However whereas the vanishing of a and Kr are direct consequences of the Nernst postulate the vanishing of c and cP are observational facts which are suggested by but not absolutely required by the Nernst postulate The UnattamabhtJ of Zero Temperature 281 Finally we note that the pressure in equation 116 can be replaced by other intensive parameters such as Be for the magnetic case leading to general analogues of equation 117 and similarly for equation 118 113 THE UNATIAINABILITY OF ZERO TEMPERATURE It is frequently stated that as a consequence of the Nernst postulate the absolute zero of temperature can never be reached by any physically realizable process Temperatures of 10 3 K are reasonably standard in cryogenic laboratories 10 7 K has been achieved and there is no reason to believe that temperatures of 10 10 Kor less are fundamentally inacces sible The question of whether the state of precisely zero temperature can be realized by any process yet undiscovered may well be an unphysical question raising profound problems of absolute thermal isolation and of infinitely precise temperature measurability The theorem that does follow from the Nernst postulate is more modest It states that no reversible adiabatic process starting at nonzero temperature can possibly bring a system to zero temperature This is in fact no more than a simple restatement of the Nernst postulate that the T 0 isotherm is coincident with the S 0 adiabat As such the T 0 isotherm cannot be intersected by any other adiabat recall Fig 112 Bank of Baroda Welcome to Bank of Baroda Your Perfect Banking Partner wwwbankofbarodacom BANK OF BARODA BANK OF BARODA 12 SUMMARY OF PRINCIPLES FOR GENERAL SYSTEMS 121 GENERAL SYSTEMS Throughout the first eleven chapters the principles of thermodynamics have been so stated that their generalization is evident The fundamental equation of a simple system is of the form 121 The volume and the mole numbers play symmetric roles throughout and we can rewrite equation 121 in the symmetric form 122 where X 0 denotes the entropy Xi the volume and the remaining X1 are the mole numbers For nonsimple systems the formalism need merely be reinterpreted the X1 then representing magnetic electric elastic and other extensive parameters appropriate to the system considered For the convenience of the reader we recapitulate briefly the main theorems of the first eleven chapters using a language appropriate to general systems 122 THE POSTULATES Postulate I There exist particular states called equilibrium states that macroscopically are characterized completely by the specification of the internal energy Vanda set of extensive parameters Xi X 2 X later to be specifically enumerated 2R1 284 Summary of Principles for General Systems Postulate II There exists a function called the entropy of the extensive parameters defined for all equilibrium states and having the following property The values assumed by the extensive parameters in the absence of a constraint are those that maximize the entropy over the manifold of con strained equilibrium states Postulate III The entropy of a composite system is additive over the constituent subsystems whence the entropy of each constituent system 1s a homogeneous firstorder function of the extensive parameters The entropy is continuous and differentiable and is a monotonically increasing function of the energy Postulate IV The entropy of any system vanishes in the state for which T cauasx x o 1 2 123 THE INTENSIVE PARAMETERS The differential form of the fundamental equation is in which t t dU TdS LPJ dXk LPJ dXk 0 123 124 The term T dS is the flux of heat and E Pk dXk is the work The intensive parameters are functions of the extensive parameters the functional relations being the equations of state Furthermore the conditions of equilibrium with respect to a transfer of Xk between two subsystems is the equality of the intensive parameters Pk The Euler relation which follows from the homogeneous firstorder property is and the GibbsDuhem relation is t LXkdPJ 0 0 Similar relations hold in the entropy representation 125 126 Maxwell Relations 285 t24 LEGENDRE TRANSFORMS A partial Legendre transformation can be made by replacing the variables X0 X1 X 2 Xs by P0 Pi Ps The Legendre transformed function is s UP 0 P1 Ps U LPkXk 127 0 The natural variables of this function are P0 Ps Xs 1 X and the natural derivatives are au P0 PJ a Xk kOl s pk 128 aUP 0 PJ ax pk k s 1 I k 129 and consequently s I dUPoPs lXkdPk LPkdXk 1210 0 sl The equilibrium values of any unconstrained extensive parameters in a system in contact with reservoirs of constant P0 P1 Ps minimize UP0 P at constant P0 Ps Xsi X 125 MAXWELL RELATIONS The mixed partial derivatives of the potential UP0 P are equal whence from equation 1210 and ax 1 axk aPk aP1 ax 1 aPk axk aP1 ifjkss 1211 if j s and k s 1212 if j k s 1213 286 Summary of Prmciples for General Systems FIGURE 121 The general thermodynamic mnemonic diagram The potential U is a gen eral Legendre transform of U The potential U is U That is V is transformed with respect to in addition to all the vari ables of U The other functions are similarly defined In each of these partial derivatives the variables to be held constant are all those of the set P0 Ps X 1 X except the variable with respect to which the derivative is taken These relations can be read from the mnemonic diagram of Fig 121 126 STABILITY AND PHASE 1RANSITIONS The criteria of stability are the convexity of the thermodynamic poten tials with respect to their extensive parameters and concavity with respect to their intensive parameters at constant mole numbers Specifically this requires K7 Ks 0 1214 and analogous relations for more general systems If the criteria of stability are not satisfied a system breaks up into two or more phases The molar Gibbs potential of each component j is then equal in each phase 1215 The dimensionality f of the thermodynamic space in which a given number M of phases can exist for a system with r components is given by the Gibbs phase rule frM2 1216 The slope in the PT plane of the coexistence curve of two phases is given by the Clapeyron equation dP tJs t dT tJv Tv 1217 Properties at Zero Temperature 287 127 CRITICAL PHENOMENA Near a critica1 point the minimum of the Gibbs potential becomes shallow and possibly asymmetric Fluctuations diverge and the most probable values which are the subject of thermodynamic theory differ from the average values which are measured by experiment Thermody namic behavior near the critical point is governed by a set of critical exponents These are interrelated by scaling relations The numerical values of the critical exponents are determined by the physical dimen sionality and by the dimensionality of the order parameter these two dimensionalities define universality classes of systems with equal criti cal exponents 128 PROPERTIES AT ZERO TEMPERATURE For a general system the specific heats vanish at zero temperature as C T o X1X2 ar XtX2 as ro 1218 and as r 0 1219 Furthermore the four following types of derivatives vanish at zero temperature J o as ro 1220 Txtxk1Xkt aPk o ar X1X2 as r 0 1221 k TX1 xktXkJ o as r 0 1222 and i x 1 XklPkXkI o asro 1223 10 Reglas para la Respiración Consciente Naturalmente Sana1 Usa una respiración abdominal es decir estira tu abdomen al inhalar y recógelo al exhalar2 Respira lenta y profundamente3 Regula el ritmo y la velocidad de la respiración4 Mantén la respiración fluida y suave5 Haz pausas entre inhalaciones y exhalaciones6 Presta atención al aliento y la respiración7 Utiliza la respiración periódicamente durante el día8 Mantén una postura abdominal correcta9 Combina la respiración con técnicas de relajación10 Practica diariamente para obtener mejores resultados 13 PROPERTIES OF MATERIALS 131 THE GENERAL IDEAL GAS A brief survey of the range of physical properties of gases liquids and solids logically starts with a recapitulation of the simplest of systemsthe ideal gas All gases approach ideal behavior at sufficiently low density and all gases deviate strongly from ideality in the vicinity of their critical points The essence of ideal gas behavior is that the molecules of the gas do not interact This single fact implies by statistical reasoning to be developed in Section 1610 that a The mechanical equation of state is of the form PV NRT b For a singlecomponent ideal gas the temperature is a function only of the molar energy and inversely c The Helmholtz potential FT V Ni N2 NT of a multicompo nent ideal gas is additive over the components Gibbss Theorem FT V Ni NT FiT V Ni F2 T V Ni FT V NJ 131 Considering first a singlecomponent ideal gas of molecular species j property b implies 132 It is generally preferable to express this equation in terms of the heat capacity which is the quantity most directly observable N1u10 N1j 7cvT dT 133 To where T0 is some arbitrarilychosen standard temperature JOO 290 Properties of Materials The entropy of a singlecomponent ideal gas like the energy is de termined by cvlT Integrating cv1 1TdSjdT and determining the constant of integration by the equation of state PV N1RT 134 Finally the Helmholtz potential of a general multicomponent ideal gas is by property c FT V LT TLST V u TS 135 1 1 Thus the most general multicomponent ideal gas is completely characterized by the molar heat capacities ciT of its individual constituents and by the values of u Jo s10 assigned in some arbitrary reference state The first summation in equation 135 is the energy of the multicompo nent gas and the second summation is the entropy The general ideal gas obeys Gibbss theorem recall the discussion following equation 339 Similarly as in equation 340 we can rewrite the entropy of the general ideal gas equation 134 in the form TI V N c T dT N R In N R L x n x 1 T0 T O 1 136 and the last term is again the entropy of mixing We recall that the entropy of mixing is the difference in entropies between that of the mixture of gases and that of a collection of separate gases each at the same temperature and the same density N V as the original mixture and hence at the same pressure as the original mixture It is left to the reader to show that Kr a and the difference cP c have the same values for a general ideal gas as for a monatomic ideal gas recall Section 38 In particular I Kr P I a T C C R p V 137 The molar heat capacity appearing in equation 133 is subject to certain thermostatistical requirements and these correspond to observational regularities One such regularity is that the molar heat capacity cv of real 2 18 16 14 t 12 C R 08 06 04 02 0 V 0 0 N 0 0 FIGURE 131 The General Ideal Gas 191 I 7 00 N 0 The molar heat capacity of a system with two vibrational modes with w2 15w1 gases approaches a constant value at high temperatures but not so high that the molecules ionize or dissociate If the classical energy can be written as a sum of quadratic terms in some generalized coordinates and momenta then the high temperature value of c is simply R2 for each such quadratic term Thus for a monatomic ideal gas the energy of each molecule is p p 2 p2m there are three quadratic terms and hence cv 3R2 at high temperatures In Section 1610 we shall explore the thermostatistical basis for this equipartition value of cv at high temperatures At zero temperature the heat capacities of all materials in thermody namic equilibrium vanish and in particular the heat capacities of gases fall toward zero until the gases condense At high temperatures the heat capacities of ideal gases are essentially temperature independent at the equipartition value described in the preceding paragraph In the inter mediate temperature region the contribution of each quadratic term in the Hamiltonian tends to appear in a restricted temperature range so that c v versus T curves tend to have a roughly steplike form as seen in Fig 131 The temperatures at which the steps occur in the cv versus T curves and the height of each step can be understood in descriptive terms 292 Properties of Materials anticipating the statistical mechanical analysis of Chapter 16 The quadratic terms in the energy represent kinetic or potential energies associated with particular modes of excitation Each such mode contributes additively and independently to the heat capacity and each such mode is responsible for one of the steps in the c versus T curve For a diatomic molecule there is a quadratic term representing the potential energy of stretching of the interatomic bond and there is another quadratic term representing the kinetic energy of vibration together the potential and kinetic energies constitute a harmonic oscillator of frequency w0 The contribution of each mode appears as a step of height R2 for each quadratic term in the energy two terms or jc R for a vibrational mode Tue temperature at which the step occurs is such that k 8 T is of the order of the energy difference of the lowlying energy levels of the mode k 8 T hw0 for a vibrational mode Similar considerations apply to rotational translational and other types of modes A more detailed description of the heat capacity will be developed in Chapter 16 132 CHEMICAL REACTIONS IN IDEAL GASES The chemical reaction properties of ideal gases is of particular interest This reflects the fact that in industrial processes many important chemical reactions actually are carried out in the gaseous phase and the assump tion of ideal behavior permits a simple and explicit solution Furthermore the theory of ideal gas reactions provides the starting point for the theory of more realistic gaseous reaction models It follows directly from the fundamental equation of a general ideal gas mixture as given parametrically in equations 133 to 135 that the partial molar Gibbs potential of the jth component is of the form 138 The quantity ctiT is a function of T only and x 1 is the mole fraction of the jth component The equation of chemical equilibrium is equation 270 or 651 139 whence 1310 1 1 1 Chenucal Reactions m Ideal Gases 293 Defining the equilibrium constant K T for the particular chemical reaction by lnKT LvtT 1311 J we find the mass action law nx pEvKT 1312 J The equilibrium constant KT can be synthesized from the functions cJ T by the definition 1311 and the functions cplT are tabulated for cmmon chemical gaseous components Furthermore the equilibrium con stant KT is itself tabulated for many common chemical reactions In either case the equilibrium constant can be considered as known Thus given the temperature and pressure of the reaction the product CTx is determined by the mass action law 1312 Paired with the condition that the sum of the mole fractions is unity and given the quantities of each atomic constituent in the system the knowledge of n xv determines each of the x r We shall illustrate such a determination in aA example but we first note that tabulations of equilibrium constants for simple reactions can be extended to additional reactions by logarithmic additivity Certain chemical reactions can be considered as the sum of two other chemical reactions As an example consider the reactions 1313 and 1314 Subtracting these two equations in algebraic fashion gives 1315 Or 1316 We now observe that the quantities ln KT of the various reactions can be subtracted in a corresponding fashion Consider two reactions 1317 294 Properties of Materials and 1318 and a third reaction obtained by multiplying the first reaction by a constant B 1 the second reaction by B2 and adding 0 rv3A rB vl B v2A l 2 1319 Assume that the equilibrium constant of the first reaction is K 1T and that of the second reaction is K 2T so that by definition 1320 and 1321 The equilibrium constant for the resultant reaction equation 1319 is defined by an analogous equation from which it follows that 1322 Thus tabulations of equilibrium constants for basic reactions can be extended to additional reactions by the additivity property Finally we recall that in the discussion following equation 658 it was observed that the heat of reaction is plausibly related to the temperature dependence of the equilibrium constant We showed there that in fact dH a T1vµ dN ar J J PN1N2 and inserting equation 138 dH dN 1323 1325 Recognizing that L 1 v1µ 1 vanishes at equilibrium and recalling the defini tion 1311 of the equilibrium constant we find the vant Hoff relation dH RT 2l KT dN dT n 1326 Chemical Reactions m Ideal Gases 295 Thus measurements of the equilibrium constant at various temperatures enable calculation of the heat of reaction without calorimetric methods the equilibrium constant being measurable by direct determination of the concentrations xd Example Two moles of H 20 are enclosed in a rigid vessel and heated to a temperature of 2000 K and a pressure of 1 MPa The equilibrium constant KT for the chemical reaction has the value K2000 00877 Pa112 What is the equilibrium composition of the system What is the composition if the temperature remains constant but the pressure is decreased to 104 Pa The law of mass action states that The mole numbers of each component are given by so that the sum of the mole numbers is 2 6N 2 Consequently 2 6N 6N X HO 2 11N 2 X H2 2 6N The law of mass action accordingly becomes 32 1 6N p12KT fi 2 6N2 6il 112 an with the righthand side known we can solve numerically for 6N We find AN 0005 for P 1 MPa and 6N 0023 for P 104 Pa Thus for a Pressure of 1 MPa the mole fractions of the components are Xtt 2o 09963 XH 2 00025 Xo 2 00012 Whereas for a pressure of 104 Pa the mole fractions are X H 2o 09828 Xtt 2 00114 Xo 2 00057 296 Properties of Matenals PROBLEMS 1321 How is the equilibrium constant of the reaction in the Example related to that for the same reaction when written with stoichiometric coefficients twice as large Note this fact with caution 1322 What are the mole fractions of the constituents in the Example if the pressure is further reduced to 103 Pa 1323 In the Example what would the final mole fractions be at a pressure of 10 Pa if the vessel initially had contained 1 mole of oxygen as well as 2 moles of water 1324 In an ideal gas reaction an increase in pressure at constant temperature increases the degree of reaction if the sum of the stoichiometric coefficients of the reactants is greater in absolute value than the sum of the Ps of the products and vice versa Prove this statement or show it to be false using the law of mass action What is the relation of this statement to the Le ChatelierBraun principle Sect 85 1325 The equilibrium constant of the reaction S0 3 S0 2 f0 2 has the value 1719 Pa112 at T 1000 K Assuming 1 mole of S0 2 and 2 moles of 0 2 are introduced into a vessel and maintained at a pressure of 04 MPa find the number of moles of S0 3 present in equilibrium 1326 At temperatures above 500 K phosphorus pentachloride dissociates according to the reaction PC15 PC13 Cl 2 A PC15 sample of 19 X 10 3 Kg is at a temperature of 593 Kand a pressure of 0314 X 105 Pa After the reaction has come to equilibrium the system is found to have a volume of 24 liters or 24 X 10 3 m3 Determine the equilibrium constant What is the degree of dissociation ie the degree of reaction E for this dissociation reaction recall equation 653 1327 A system containing 002 Kg of CO and 002 Kg of 0 2 is maintained at a temperature of 3200 K and a pressure of 02 MPa At this temperature the equilibrium constant for the reaction 2C0 2 2CO 0 2 is K 0424 MP a What is the mass of CO 2 at equilibrium 1328 Apply equation 138 to a singlecomponent general ideal gas of species J Evaluate p1 for the singlecomponent ideal gas by equation 134 note that by equation 133 constant U implies constant T and in this way obtain an expression for Pr 1329 An experimenter finds that water vapor is 053 dissociated at a temper ature of 2000 Kand a pressure of 10 5 Pa Raising the temperature to 2100 Kand Smnll Deviations from I dealuy The Vi rial Expanswn 297 keeping the pressure constant leads to a dissociation of 088 That is an initial rnole of H 20 remains as 09947 moles at 2000 K or as 09912 moles at 2100 K after the reaction comes to completion Calculate the heat of reaction of the dissociation of water at P 105 Pa and T 2050 K Answer flH 27 X 105 Jmole 133 SMALL DEVIATIONS FROM IDEALITYTHE VIRIAL EXPANSION Although all gases behave ideally at sufficiently large molar volume they exhibit more complicated behavior as the molar volume u is de creased To describe at least the initial deviations from ideal gas behavior the mechanical equation of state can be expanded in inverse powers of u P R l BT CT T u u u2 1327 This expansion is called a virial expansion BT is called the second virial coefficient CT is the third virial coefficient and so forth The forms of these functions depend on the form of the intermolecular forces in the gas The second virial coefficient is shown in Fig 132 as a function of temperature for several simple gases Corresponding to the virial expansion of the mechanical equation of state in inverse powers of u the molar Helmholtz potential can be similarly expanded 1328 The equality of the coefficients B T C T in these expans10ns follows of course from P a11au All thermodynamic quantities thereby are expressible in virialtype xpansions in inverse powers of u The molar heat capacity c v for Instance is c c RT d 2BT 1 d 2CT 1 d 2DT V vdeal u dT2 2u2 dT2 2u3 dT2 1329 and the molar energy is 2 1 dB 1 dC 1 dD u udeal RT v dT 2v2 dT 3v3 dT 1330 298 Properties of Materials 30 20 A i 10 rScZNe He He cu 0 E 0 E 600 700 TK Ql 10 20 30 FIGURE 132 Second virial coefficient as a function of temperature for several gases Measurement by Holbom and Otto Data from Statistical Thermodynamics by R H Fowler and E A Guggenheim Cambridge University Press 1939 PROBLEMS 1331 In a thermostatistical model in which each atom is treated as a small hard sphere of volume r the leading virial coefficients are B 4NAT C 10Nr2 D 1836Nr3 Using the value of B determined from Fig 132 find the approximate radius of a He atom Given Fig 132 what would be a reasonable though fairly crude gues as to the value of the third virial coefficient of He 1332 Expand the mechanical equation of state of a van der Waals gas equation 341 in a viria expansion and express the virial coefficients in terms of the van der Waals constants a and b 1333 Show that the second virial coefficient of gaseous nitrogen Fig 132 can be fit reasonably by an equation of the form B B B1 o T and find the values of B0 and B1 Assume that all higher virial coefficients can be neglected Also take the molar heat capacity cv of the noninteracting gas to be 5R2 a Explain why cv noninteracting reasonably can be taken as SR2 b Evaluate the values of B0 and B 1 from Fig 132 The Law of Correspondmg Statesfor Gases 299 c What is the value of cuT v for N2 to second order in a virial expansion t334 The simplest analytic form suggested by the qualitative shape of BT of ff 2 and Ne in Fig 132 is BT B0 B1T as in Problem 1333 With this assumption calculate cpT u for H 2 and Ne t335 A porous plug experiment is carried out by installing a porous plug in a plastic pipe To the left of the plug the gas is maintained at a pressure slightly higher than atmospheric by a movable piston To the right of the plug there is a freely sliding piston and the righthand end of the pipe is open to the atmo sphere What is the fractional difference of velocities of the pistons a Express the answer in terms of atmospheric pressure P0 the driving pressure Ph and cP a KT and v assuming that the pressure difference is small enough that no distinction need be made between the values of the latter quantities on the two sides of the plug b Evaluate this result for an ideal gas and express the deviation from this result in terms of the second virial coefficient carrying results only to first order in BT or its derivatives the heat capacity cP is to be left as an unspecified quantity in the solution 134 THE LAW OF CORRESPONDING STATES FOR GASES A complete virial expansion can describe the properties of any gas with high precision but only at the cost of introducing an infinite number of expansion constants In contrast the van der Waals equation of state captures the essential features of fluid behavior including the phase transition with only two adjustable constants The question arises as to whether the virial coefficients of real gases are indeed independent or whether there exists some general relationships among them Alternatively stated does there exist a more or less universal form of the equation of state of fluids involving some finite or even small number of indepen dent constants In the equation of state of any fluid there is one unique pointthe critical point characterized by Tr Pa and vcr A dimensionless equation of state would then be most naturally expressed in terms of the reduced temperature T Tr the reduced pressure P Per and the reduced molar volume vva It might be expected that the three parameters Tcr Per and vcr are themselves independent But evaluation of the dimensionless ratio PcvJ RT for various gases reveals a remarkable regularity as shown in Table 131 the ratio is strikingly constant with small deviations to lower values for a few polar fluids such as water or ammonia The dimensionless constant PvcJRTr has a value on the order of 027 for all normal fluids Of the three parameters that characterize the critical point only two are indepen dent in the semiquantitative sense of this section 300 Properties of Materials TABLE13l Critical Constants and the Rafo Pcrv RT of Various Fluids Substance Molecular Weight TK P10 6 Pa iJO I nJ 1 Pcrvr I RT r Hz 2016 333 130 00649 030 He 4003 53 023 00578 030 CH4 16043 191l 464 00993 029 NH 3 1703 4055 1128 00724 024 H20 18015 6473 2209 00568 023 Ne 20183 445 273 00417 031 Nz 28013 1262 339 00899 029 C2H6 30070 3055 488 01480 028 02 31999 1548 508 00780 031 C2H8 44097 370 426 01998 028 C2H50H 4607 516 638 01673 025 S0 2 64063 4307 788 01217 027 C6H6 78115 562 492 02603 027 Kr 8380 2094 550 00924 029 CC14 15382 5564 456 02759 027 Abstracted from K A Kobe and R E Lynn Jr Chem Rec 52 Il 7 1953 Proceeding further then one can plot vv as a function of P P and T Tr for a variety of fluids Again there is a remarkable similarity among all such reduced equations of state There exists at least semiquantitatively a universal equation of state containing no arbitrary constants if expressed in the reduced variables vi P Pr and T Tr This empirical fact is known as the Law of Corre sponding States The universal reduced equation of state can be represented in a con venient twodimensional form as in Fig 133 from Sonntag and Van Wylen 1 The dependent variable the ordinate in the figure is the dimen sionless quantity PvRT or 027 P Pu vvalT Tr The indepen dent variables are P Per and T Tr The reduced pressure P Per is the abscissa in the graph In order to avoid a third dimension the reduced temperature scale is superimposed as a set of constant reduced temper ature loci in the plane To find v vr at a given value of P Pa and T T r one reads P P on the abscissa and locates the appropriate T Tr curve These values de termine a point of which one can read the ordinate The ordinate 1s 021PPuvvJTT so that vvu is thereby evaluated The existence of such an approximate universal equation of state 1s given a rational basis by statistical mechanical models The force betweeJ1 1 R E Sonntag and G J Van Wylcn lntroduc11011 to Thermodynamus Classual and Stamt1t1if 2nd cd Wiley New York 1982 The Law of Correspondmg Statesor Gases 301 02 03 04 05 10 2 0 3 0 4 0 50 20 II I 0 0 OS 0 07 06 E 01 l2 03 0405 10 2 0 10 20 30 reduced pressure P FIGURE 133 Generalized or universal equation of state of gases in terms of reduced variables From R E Sonntag and G van Wylen lntroducton to Thermodynamics Classcal and Stat1sti ca 2nd edition 1982 John Wiley Sons New York molecules is generally repulsive at small distance where the molecules Physically overlap and attractive at larger distances The longrange attraction in nonpolar molecules is due to the polarization of one mole ule by the instantaneous fluctuating dipole moment of the other such a van der Waals force falls as the sixth power of the distance Thus the force between two molecules can be parametrized by the radii of the l1lolecules describing the shortrange repulsion and the strength of the 302 Properties of Matenals longrange attractive force It is this twoparameter characterization of the intermolecular forces that underlies the twoparameter equation of state 135 DILUTE SOLUTIONS OSMOTIC PRESSURE AND VAPOR PRESSURE Whereas the law of corresponding states applies most accurately in the gaseous region of the fluid state with increasing validity as the density decreases below that at the critical point the liquid region is less subject to a generalized treatment There is however a very useful general regularity that applies to dilute solutions of arbitrary density That regu larity consists of the carryover of the entropy of mixing terms recall equation 136 from ideal gas mixtures to general fluid mixtures Consider a singlecomponent fluid system for which the chemical potential is µP T Then let a second component the solute be added in small concentration The Gibbs potential of the dilute solution can be written in the general form 1331 where 1 is an unspecified function of P and T and where the latter two terms are suggested by the entropy of mixing terms equation 136 of an ideal gas From a statistical mechanical perspective 1 represents the effect of the interaction energy between the two types of molecules whereas the entropy of mixing terms arise purely from combinational considerations to be developed in Chapters 15 et seq For our present purposes however equation 1331 is to be viewed as an empirical thermodynamic approximation In the region of validity ie small concentrations or N2 N1 we can expand the third term to first order in N2 N1 and we can neglect N2 relative to N1 in the denominator of the logarithm in the last term obtaining GT P N1 N2 N1µP T N2iJP T N2RT N2RTln Z 2 l 1332 It follows that the partial molar Gibbs potentials of solvent and solute are Dilute Soutwns Osmotic Pressure and Vapor Pressure 303 respectively p1 P T x i µ P T xRT 1 1333 where x is the mole fraction of solute N2N 1 and ac µ 2P T x aN2 P T RTlnx 1334 It is of interest to examine some simple consequences of these results Consider first the case of the osmotic pressure difference across a semi permeable membrane Suppose the membrane to be permeable to a liquid water for instance A small amount of solute such as sugar is intro duced on one side of the membrane Assume that the pressure on the pure solvent side of the membrane is maintained constant P but that the pressure on the solute side can alter as by a change in height of the liquid in a vertical tube Then the condition of equilibrium with respect to diffusion of the solvent across the membrane is 1335 where P is the as yet unknown pressure on the solute side of the membrane Then by equation 1333 1336 where we have altered the notation slightly to write µ1 P T 0 for PP T Then expanding µ1P T0 around the pressure P Bµ1PTO µ 1 PT0µ 1PT0 BP xPP µ1P TO P Pv 1337 or from equation 1318 P Pv xRT 1338 ultiplying by N1 we find the van t Hoff relation for osmotic pressure in tlute solutions 1339 304 Properties of Materials Another interesting effect in liquids is the reduction in the vapor pressure recall Sections 91 to 93 by the addition of a low concentration of nonvolatile solute In the absence of the solute 1340 But with the addition of the solute as in 1336 1341 Expanding the first term around the original pressure P 1342 and similarly for the gaseous phase whence we find P P xRT 1343 Thus the addition of a solute decreases the vapor pressure If we make the further approximation that vg v1 and that vg RT P the ideal gas equation we obtain tlP p x 1344 which is known as Raouts Law PROBLEMS 13Sl Assuming the latent heat of vaporization of a fluid to be constant over the temperature range of interest and assuming that the density of the vapor can be neglected relative to that of the liquid plot the vapor pressure ie the liquidgas coexistence curve as a function of the dimensionless temperature RT t Plot the corresponding graphs for five and ten percent dissolved solute 13S2 One hundred grams of a particular solute are dissolved in one liter of water The vapor pressure of the water is decreased by roughly 6 Is the solute more likely to be sugar C12H 220u table salt NaCl or sodium bicarbonate NaC0 3 2 Ionic solutions double their effective Raoult concentration 13S3 If 20 grams of sugar C12H 220u are dissolved in 250 cm3 of wate1 what is the change in the boiling temperature at atmospheric pressure Solid Systems 305 136 SOLID SYSTEMS The heat capacities and various other properties of a wide variety of solid systems show marked similarities as we shall see in specific detail in Section 166 where we shall carry out an explicit statistical mechanical calculation of the thermal equation of state of a solid Accordingly we def er further description of the properties of solids other than to stress that the thermal properties of solids are not qualitatively different than those of liquids it is the thermomechanical properties of solids that introduce new elements in the theory Whereas the mechanical state of a fluid is adequately characterized by the volume a solid system can be characterized by a set of elastic strain components These describe both the shape and the angular dilatations twists of the system The corresponding intensive parameters are the elastic stress components These conjugate variables follow the structure of the general thermodynamic formalism For specific details the reader is referred to the monograph by Duane C Wallace2 or to references cited therein However it is important to stress that conventional thermodynamic theory in which the volume is the single mechanical parameter fully applies to solids The more detailed analysis in terms of elastic strains gives additional information but it does not invalidate the results obtained by the less specific conventional form of thermodynamics In the full theory the extensive parameters include both the volume the fully symmetric strain and various other strain components The conjugate intensive parameters are the stress components including the pressure the fully symmetric stress component If the walls of the system impose no stress components other than the pressure then these stress components vanish and the formalism reduces to the familiar form in which the volume is the only explicit mechanical parameter Inversely in the more general case the additional strain components can be appended to the simple theory in a manner fully analogous to the addition of any generalized extensive parameter 2 Duane C Wallace Thermodnanucs of Crystals W1ley New York 1972 El poder de la respiración consciente en la salud mental y física Aprende a calmar la mente reducir el estrés y mejorar tu bienestar general mediante técnicas simples y efectivas de respiración consciente 14 IRREVERSIBLE THERMODYNAMICS 141 GENERAL REMARKS As useful as the characterization of equilibrium states by thermostatic theory has proven to be it must be conceded that our primary interest is frequently in processes rather than in states In biology particularly it is the life process that captures our imagination rather than the eventual equilibrium state to which each organism inevitably proceeds Thermostat ics does provide two methods that permit us to inf er some limited information about processes but each of these methods is indirect and each yields only the most meager return First by studying the initial and terminal equilibrium states it is possible to bracket a process and thence to determine the effect of the process in its totality Second if some process occurs extremely slowly we may compare it with an idealized nonphysical quasistatic process But neither of these methods confronts the central problem of rates of real physical processes The extension of thermodynamics that has reference to the rates of physical processes is the theory of irreversible thermodynamics Irreversible thermodynamics is based on the postulates of equilibrium thermostatics plus the additional postulate of time reversal symmetry of physical laws This additional postulate states that the laws of physics remain unchanged if the time t is everywhere replaced by its negative t and if simultaneously the magnetic field Be is replaced by its negative Be and if the process of interest is one involving the transmutation of fundamental particles that the charge and parity of the particles also be reversed in sign For macroscopic processes the parenthetical restric tion has no observable consequences and we shall henceforth refer to time reversal symmetry in its simpler form The thermodynamic theory of irreversible processes is based on the Onsager Reciprocity Theorem formulated by Lars Onsager 1 in brilliant 1Lars Onager Phvscal Rellew 37 405 1931 38 2265 1931 W7 308 lrreuerHbte Thermodynamics pioneering papers published in 1931 but not widely recognized for almost 20 years thereafter Powerful statistical mechanical theorems also exist the fluctuationdissipation theorem 2 the Kubo relations and the formalism of linear response theory based on the foregoing theorems 3 We review only the thermodynamic theory rooted in the Onsager theo rem 142 AFFINITIES AND FLUXES Preparatory to our discussion of the Onsager theorem we define certain quantities that appropriately describe irreversible processes Basically we require two types of parameters one to describe the force that drives a process and one to describe the response to this force The processes of most general interest occur in continuous systems such as the flow of energy in a bar with a continuous temperature gradient However to suggest the proper way to choose parameters 111 such contmuous systems we first consider the relatively simple case of a discrete system A typical process in a discrete system would be the flow of energy from one homogeneous subsystem to another through an infinitely thin diathermal partition Consider a composite system composed of two subsystems An exten sive parameter has values X and X in the two subsystems and the closure condition requires that X X Xf a constant 14 l If X and X are unconstrained their equilibrium values are determincC by the vanishing of the quantity g as0 aS S as as F F ax xt ax xr ax ax 14 2 Thus if F is zero the system ism equilibrium but if ff is nonzero an irreversible process occurs taking the system toward the eqmlibrium state The quantity F which is the difference in the entropyrepresentation intensive parameters acts as a generalized force which drives the process Such generalized forces are called affinilles 1 H C11len and T Welton Phn Rev 83 34 1951 f R Kuhn lectures III Theoretual Ph1wcs vol I lntercencc New York 1959 p 120203 Affimtul and Fluxel 309 For definiteness consider two systems separated by a diathermal wall Jnd let Xk be the energy U Then the affinity is 1 1 k T T 143 No heat flows across the diathermal wall if the difference in inverse temperatures vamshes But a nonzero difference in inverse temperature acting as a generalized force drives a flow of heat between the subsystems Similarly if Xk is the volume the affinity F is P T P T and if Xk is a mole number the associated affinity is µ1jT µ1T We characterize the response to the applied force by the rate of change of the extensive parameter X The flux J is then defined by J dX k dt 144 Therefore the flux vanishes if the affinity vanishes and a nonzero affinity leads to a nonzero flux It is the relationship between fluxes and affinities that characterizes the rates of irreversible processes The identification of the affinities in a particular type of system is frequently rendered more convenient by considering the rate of produc tion of entropy Differentiating the entropy S X0 X1 with respect to the time we have dS L as dX dt k ax dt 145 or s FJ 146 Thus the rate of productwn of entropy 1s the sum of products of each flux with its associated affimty The entropy production equation is particularly useful in extending the definition of affinities to continuous systems rather than to discrete systems If heat flows from one homogeneous subsystem to another through an infinitely thm diathermal partition the generahzcd force is the difference 1T 1T but if heat flows along a metal rod in which the temperature varies in a continuous fashion it is difficult to apply our Previous definition of the affinity Nevertheless we can compute the rate of Production of entropy and thereby we can identify the affinity With the foregoing considerations to guide us we now turn our atten tion to continuous systems We consider a threedimensional system in 310 lrreversble Thermod1 nanuo which energy and matter flow driven by appropriate forces We choose the components of the vector current densities of energy and matter as fluxes Thus associated with the energy U we have the three energy fluxes J0 x J0y Joz These quantities are the x y and z components of the vector current density J0 By definition the magnitude of J0 is the amount of energy that flows across the unit area in unit time and the direction of J is the direction of this energy flow Similarly the current density Jk may describe the flow of a particular chemical component per unit area and per unit time the components Jh 1a and Jkz are fluxes In order to identify the affinities we now seek to write the rate of production of entropy in a form analogous to equation 146 One problem that immediately arises is that of defining entropy in a nonequilibrium system This problem is solved in a formal manner as follows To any infinitesimal region we associate a local entropy S X0 Xi where by definition the functional dependence of S on the local extenswe parameters X0 Xi is taken to be identical to the dependence in eqw librium That is we merely adopt the equilibrium fundamental equation to associate a local entropy with the local parameters X0 Xi Then 147 or taking all quantities per unit volume 4 148 The summation in this equation omits the term for volume and comc quently has one less term than that in equation 147 Agam the local intensive parameter F is taken to be the same functwn of the local extensive parameters as it would be in equtlihrwm It is because of this convention incidentally that we can speak of the temperature varying continuously in a bar despite the fact that thermostatics implies the existence of temperature only in equilibrium systems Equation 14 7 immediately suggests a reasonable definition of the entropy current density Js 149 in which J is the current density of the extensive parameter X The magnitude of the entropy flux Js is the entropy transported through unit area per unit time 4 11 should be noted that in the remainder of tlus chapter we uc Jowercae Jette to ind1ca1c extensive parameters per umt volume rather than per mole Affinwes and Fluxel 311 The rate of local production of entropy is equal to the entropy leaving the region plus the rate of increase of entropy within the region If s denotes the rate of production of entropy per unit volume and as at denotes the increase m entropy per unit volume then as svJ at s 1410 If the extensive parameters of interest are conserved as are the energy and in the absence of chemical reactions the mole numbers the equations of continuity for these parameters become axk 0 Tt v Jk 1411 We are now prepared to compute s explicitly and thence to identify the affinities in continuous systems The first term in equation 1410 is easily computed from equation 148 1412 The sond term in equation 1410 is computed by t1king the divergence of equation 149 v JS v FkJk LvFk Jk LF1v Jk k k k 1413 ThlS equation 1410 becomes 1414 Finally by equation 1411 we observe that the first and third terms cancel giving 1415 1lthough the affinity is defined as the difference in the entropyrepresentation Intensive parameters for discrete systems it is the gradient of the entropy representation intensive parameters in continuous systems If J 0 z denotes the z component of the energy current density the SSociated affinity z is V 21T the z component of the gradient of the lllverse temperature And if J denotes the kth mole number current 312 Irreversible Thermodrnam1cs density the number of moles of the k th component flowing through umt area per second the affinity associated with Jz is v 2 µkT 143 PURELY RESISTIVE AND LINEAR SYSTEMS For certain systems the fluxes at a given instant depend only on the values of the affinities at that instant Such systems are referred to a purely resistive For other than purely resistive systems the fluxes may depend upon the values of the affinities at previous times as well as upon the instantaneou values In the electrical case a resistor is a purely resistive system whereas a circuit containing an inductance or a capacitance is not purel 1 resistive A nonpurelyresistive system has a memory Although it might appear that the restriction to purely resistive systems is very severe it is found in practice that a very large fraction of the systems of interest other than electrical systems are purely resistive Extensions to nonpurelyresistive systems do exist based on the fluctul tiondissipation theorem or Kubo formula referred to in Section 141 For a purely resistive system by definition each local flux depends onlj upon the instantaneous local affinities and upon the local intensive parameters That is dropping the indices denoting vector components 1416 Thus the local mole number current density of the k th component depends on the gradient of the inverse temperature on the gradients of µT for each component and upon the local temperature pressure and so forth It should be noted that we do not assume that each flux depend only on its own affinity but rather that each flux depends on all affinities It is true that each flux tends to depend most strongly on its 011 associated affinity but the dependence of a flux on other affinities as well is the source of some of the most interesting phenomena in the field of irreversibility Each flux J is known to vanish as the affinities vanish so we can expand J in powers of the affinities with no constant term J LL1k L LL1k 1417 J I J where 1418 Purei Res1stwe and Linear Slems 313 and 1419 The functions L1k are called kinetic coefficients They are functions of the local intensive parameters 1420 The functions L 1 k are called secondorder kinetic coefficients and they are also functions of the local intensive parameters Thirdorder and higher order kinetic coefficients are similarly defined For the purposes of the Onsager theorem which we are about to enunciate it is convenient to adopt a notation that exhibits the functional dependence of the kinetic coefficients on an externally applied magnetic field Be suppressing the dependence on the other intensive parameters 1421 The Onsager theorem states that 1422 That is the value of the kinetic coefficient L1k measured in an external magnetic field Be is identical to the value of L11 measured in the reversed magnetic field Be The Onsager theorem states a symmetry between the linear effect of the jth affinity on the k th flux and the linear effect of the k th affinity on the jth flux when these effects are measured in opposite magnetic fields A situation of great practical interest arises if the affinities are so small that all quadratic and higherorder terms in equation 1417 can be neglected A process that can be adequately described by the truncated approximate equations 1423 is called a linear purely resistive process For the analysis of such processes the Onsager theorem is a particularly powerful tool It is perhaps surprising that so many physical processes of interest are linear But the affinities that we commonly encounter in the laboratory are quite small in the sense of equation 1417 and we therefore recognize that we generally deal with systems that deviate only slightly from equilibrium 314 Irreversible Thermodynamics Phenomenologically it is found that the flow of energy in a thermally conducting body is proportional to the gradient of the temperature Denoting the energy current density by J0 we find that experiment yields the linear law 1424 in which K is the thermal conductivity of the body We can rewrite this in the more appropriate form 1425 and similarly for x and y components and we see that KT 2 is the kinetic coefficient The absence of higherorder terms such as VlT 2 and VlT3 in the phenomenological law shows that commonly employed temperature gradients are small in the sense of equation 1417 Ohms law of electrical conduction and Ficks law of diffusion are other linear phenomenological laws which demonstrate that for the common values of the affinities in these processes higherorder terms are negligible On the other hand both the linear region and the nonlinear region can be realized emical systems depending upon the deviations of the molar concentrations from their equilibrium values Although the class of linear processes is sufficiently common to merit special attention it is by no means all inclusive and the Onsager theorem is not restricted to this special class of systems 144 THE THEORETICAL BASIS OF THE ONSAGER RECIPROCITY The Onsager reciprocity theorem has been stated but not proved in the preceding sections Before turning to applications in the following sections we indicate the relationship of the theorem to the underlying principle of time reversal symmetry of physical laws From the purely thermodynamic point of view the extensive parame ters of a system in contact with a reservoir are constants In fact if an extensive parameter such as the energy is permitted to flow to and from a reservoir it does so in continual spontaneous fluctuations These fluctua tions tend to be very rapid and macroscopic observations average over the fluctuations as discussed in some detail in Chapter 1 Occasionally a large fluctuation occurs depleting the energy of the system by a non negligible amount If the system were to be decoupled from the reservoir before this rare large fluctuation were to decay we would then associate a lower temperature to the system But if the system were not decoupled The Theoretical Bass of the Onsager Rec1prociry 3 J 5 the fluctuation would decay by the spontaneous flow of energy from the reservoir to the system Onsager connected the theory of macroscopic processes to thermody namic theory by the assumption that the decay of a spontaneous fluctuatwn is identical to the macroscopic process of flow of energy or other analogous quantity between the reservoir and the system of depleted energy We consider a system in equilibrium with a pair of reservoirs corre sponding to the extensive parameters nd Xk Let the instantaneous values of these paameters be denoted by and XJ and let 8 denote the deviation of from its aveage value Thus 8X describes a fluctua tion ald the average value of 8 is zero Neverthefess the average value of 8X2 denoted by 8X 2 is not zero Nor is the correlation moment 8X 1 8Xk A very slight extension of the thermodynamic for malism invoking only very general features of statistical mechanics permits exact evaluation of the correlation moments of the fluctuations as we shall see in Chapter 19 More general than te C9rrelation moment 8 8Xk is the delayed correlation mpment 8 Xk T which is the average product of the deviations 8X and 8Xk with the latter being observed a time T after the former It is tis delayed correlation moment upon which Onsager focused attention The delayed correlation moment is subject to certain symmetries that follow from the time reversal symmetry of physical laws In particular assuming no magnetic field to be present the delayed correlation moment must be unchanged under the replacement of T by T 1426 or since only the relative times in the two factors are significant 1427 If we now subtract 8 8Xk from each side of the equation and divide by T we find 1428 In the limit as T 0 we can write the foregoing equation in terms of time derivatives 1429 316 I rrePrshlc Thermodynamics Now we assume that the decay of a fluctuation Xk is governed by the same linear dynamical laws as are macroscopic processes 1430 Inserting these equations in equation 1429 gives 1431 The theory of fluctuations reveals Chapter 19 the plausible result that in the absence of a magnetic field the fluctuation of each affinity 1s associated only with the fluctuation of its own extensive parameter there are no crosscorrelation terms of the form Js ff with i J Further more it will be shown that the diagonal correlation function with i j has the value k 8 though the specific value is not of importance for our present purposes if i j if i j B 0 1432 It follows that in the absence of a magnetic field L 1 LJI which is the Onsager reciprocity theorem equation 1422 In the presence of a magnetic field the proof follows in similar fashion depending upon a similar symmetry in the correlation functions of the spontaneous fluctuations Despite this fundamental basis in fluctuation theory the applications of the Onsager theory are purely macroscopic expressed in terms of phe nomenological dynamical equations This thermodynamic emphasis of application has motivated interjection of the subject prior to the statistical mechanical chapters to follow Accordingly we turn to thermoelectric effects as an illustrative application of the Onsager theorem 145 THERMOELECTRIC EFFECTS Thermoelectric effects are phenomena associated with the simultaneous flow d electric current and heat current in a system Relationships among various such phenomena were proposed in 1854 by Lord Kelvin on the basis of empirical observations Kelvin also presented a heuristic argument leading to the relations carefully pointing out however that the argument was not only unJustified but that it could be made to yield Thcrmoelectru I jfn 3 7 incorrect relations as well as correct ones Unfortunately the argument continually resurfaces with renewed claims of rigorof which the reader of the thermodynamic literature should be forewarned To analyze the thermoelectric effects in terms of the Onsager reciprocity we focus attention on a conductor in which both electric current and heat current flow in one dimension and we describe the electric current as being carried by electrons Then if s is the local entropy density ds du L µ dn k k 1433 in which u is the local energy density µ is the electrochemical potential per particle of the electrons n is the number of electrons per unit volume and in which the sum refers to other components These other components are the various types of atomic nuclei that together with the electrons constitute the solid It will be noted that we have taken n as the number of electrons rather than the number of moles of electrons and µ is accordingly the electrochemical potential per particle rather than per mole In this regard we deviate from the more usual parameters merely by multiplication and division by Avogadros number respectively Just as equation 147 led to equation 149 equation 1433 now leads to 1434 in which Js Ju and JN are current densities of entropy energy and number of electrons respectively The other components in equation 1433 are assumed immobile and consequently do not contribute flux terms to equation 1434 Repeating the logic leading to equation 1415 we find 1 µ svJ vJ T u T N 1435 Thus if the components of Ju and J are taken as fluxes the associated affinities are the components of vlT and vµT Assuming for simplicity that all flows and forces are parallel to the xdirection and omitting the subscript x the linear dynamical laws become 1436 1437 318 I rre1erwblc 1hcrmodynan11u and the Onsager theorem gives the relation 1438 Before drawing physical conclusions from equation 1438 we recast the dynamical equations into an equivalent but instructive form Although J 1 is a current density of total internal energy we generally prefer to discus the current density of heat In analogy with the relation dQ T dS we therefore define a heat current density JQ by the relation 1439 or by equation 1434 JQ JU µJN 1440 In a very rough intuitive way we can look on µ as the potential energy per particle and on µJN as a current density of potential energy subtraction of the potential energy current density from the total energy current density yields the heat current density as a sort of kinetic energy current dnsity At any rate eliminating Ju in favor of JQ from equation 1434 gIVes 1441 It follows from this equation that if the components of JQ and of JN are chosen as fluxes the associated affinities are the corresponding compo nents of 1T and of 1Tµ respectively The dynamical equations can then be written in the onedimensional case as 1442 1443 and the Onsager relation is 1444 The reader should verify that the dynamical equations 1442 and 1443 can also be obtained by direct substitution of equation 1440 into the previous pair of dynamical equations 1436 and 1437 without recourse to the entropy production equation 1441 The Co11dut1ll11es 3 I 9 The significance of the heat current can be exhibited in another manner We consider for a moment a steadystate flow Then both Ju and JN are divergenceless and faking the divergence of equation 1440 gives in the steady state 1445 which states that in the steady state the rate of increase in heat current is equal to the rate of decrease in the potential energy current Furthermore the insertion of this equation into equation 1441 gives 1446 which can be interpreted as stating that the production of entropy is due to two causes The first term is the production of entropy due to the flow of heat from high to low temperature and the second term is the increase in entropy due to the appearance of heat current We now accept the dynamical equations 1442 and 1443 and the symmetry condition equation 1444 as the basic equations with which to study the flow of heat and electric current in a system 146 THE CONDUCTIVITIES We consider a system in which an electric current and a heat current flow parallel to the xaxis in a steady state with no applied magnetic field Then omitting the subscript x 1447 1448 where the Onsager theorem has reduced to the simple symmetry 1449 The three kinetic coefficients appearing in the dynamical equations can be related to more familiar quantities such as conductivities In develop ing this connection we first comment briefly on the nature of the electro chemical potential µ of the electrons We can consider µ as being composed of two parts a chemical portion µ c and an electrical portion µ e 1450 320 lrreversble Thermodynamics If the charge on an electron is e then µ e is simply ecp where cp is the ordinary electrostatic potential The chemical potential µ is a function of the temperature and of the electron concentration Restating these facts in terms of gradients the electrochemical potential per unit charge is 1eµ its gradient 1eVµ is the sum of the electric field 1eVµ plus an effective driving force 1eVµc arising from a concentration gradient The electric conductivity o is defined as the electric current density eJ N per unit potential gradient 1evµ in an isothermal system It is easily seen that 1eVµ is actually the emf for in a homogeneous isothermal system Vµc 0 and Vµ Vµe Thus by definition for VT 0 1451 whence equation 1447 gives 1452 Similarly the heat conductivity K is defined as the heat current density per unit temperature gradient for zero electric current 1453 Solving the two kinetic equations simultaneously we find 1454 where D denotes the determinant of the kinetic coefficients 1455 147 THE SEEBECK EFFECT AND THE THERMOELECTRIC POWER The Seebeck effect refers to the production of an electromotive force in a thermocouple under conditions of zero electric current Consider a thermocouple with junctions at temperatures T1 and T T2 T1 as indicated in Fig 141 A voltmeter is inserted in one arm of the thermocouple at a point at which the temperature is T This voltmeter is such that it allows no passage of electric current but offers no resistance to the flow of heat We designate the two materials composing the The Seebeck Effea and the Thermoelearu Pow 321 FIGURE 141 thermocouple by A and B With JN 0 we obtain from the kinetic equations for either conductor Thus 2 L12 µ2 µI TLA dT 1 11 1 2 Lf2 µ2 µ B dT r TL 11 I LB f 12 dT µ µI TLB l 11 Eliminating µ 1 and µ 2 from these equations 2 LA LB 12 12 dT µr µ f TLA TLB l 11 11 1456 1457 1458 1459 1460 But because there is no temperature difference across the voltmeter the Voltage is simply 1 2 LA LB V 12 12 dT µrµf TLA TLB 1e 11 e 11 1461 The thermoelectric power of the thermocouple EAB is defined as the change in voltage per unit change in temperature difference The sign of AB is chosen as positive if the voltage increment is such as to drive the 322 Irreversible Thermodynamics current from A to B at the hot junction Then 1462 Defining the absolute thermoelectric power of a single medium by the relation 1463 the thermoelectric power of the thermocouple is 1464 If we accept the electric conductivity o the heat conductivity K and the absolute thermoelectric power E as the three physically significant dynami cal properties of a medium we can eliminate the three kinetic coefficients in favor of these quantities and rewrite the kinetic equations in the following form 1 To lvµ T 2oE v N e2 T e T 1465 1466 An interesting insight to the physical meaning of the absolute thermo electric power can be obtained by eliminating 1TVµ between the two foregoing dynamical equations and writing JQ in terms of JN and VlT 2 1 JQ TEeJ N T Kv T or recalling that ls JQT 1467 1468 According to this equation each electron involved in the electric current carries with it an entropy of Ee This flow of entropy is in addition to the entropy current TKVlT which is independent of the electronic cur rent The thermoelectric power can be looked on as the entropy trans ported per coulomb by the electron flow The Pelte Ffect 323 148 THE PELTIER EFFECT The Peltier effect refers to the evolution of heat accompanying the flow of an electric current across an isothermal junction of two materials JNJUA FIGURE 142 A B Consider an isothermal junction of two conductors A and B and an electric current eJN to flow as indicated in Fig 142 Then the total energy current will be discontinuous across the junction and the energy dif ference appears as Peltier heat at the junction We have Ju JQ µJN and since both µ and J N are continuous across the junction it follows that the discontinuity in Ju is equal to the discontinuity in JQ 1469 Because of the isothermal condition the dynamical equations 1465 and 1466 give in either conductor 1470 whence 1471 The Peltier coefficient wA 8 is defined as the heat that must be supplied to the junction when unit electric current passes from conductor A to conductor B Thus 1472 Equation 1472 which relates the Peltier coefficient to the absolute thermoelectric powers is one of the relations presented on empirical evidence by Kelvin in 1854 It is called the second Kelvin relation The method by which we have derived equation 1472 is typical of all applications of the Onsager relations so that it may be appropriate to review the procedure We first write the linear dynamical equations reducing the number of kinetic coefficients appearing therein by invoking the Onsager relations We then proceed to analyze various effects ex pressing each in terms of the kinetic coefficients When we have analyzed as many effects as there are kinetic coefficients we rewrite the dynamical equations in terms of those effects rather than in terms of the kinetic coefficients as in equations 1465 and 1466 Thereafter every additional 324 lrreversble Thermodynamics effect analyzed on the basis of the dynamical equations results in a relation analogous to equation 1472 and expresses this new effect in term of the coefficients in the dynamical equation 149 THE THOMSON EFFECT The Thomson effect refers to the evolution of heat as an electric current traverses a temperature gradient in a material Consider a conductor carrying a heat current but no electric current A temperature distribution governed by the temperature dependence of the kinetic coefficients will be set up Let the conductor now be placed in contact at each point with a heat reservoir of the same temperature as that point so that there is no heat interchange between conductor and re servoirs Now let an electric current pass through the conductor An interchange of heat will take place between conductor and reservoirs This heat exchange consists of two partsthe Joule heat and the Thomson heat As the electric current passes along the conductor any change in total energy flow must be supplied by an energy interchange with the reservoirs Thus we must compute v Ju 1473 which can be expressed in terms of JN and vlT by using equatiorn 1467 and 1468 2 1 e 2 2 1 v Ju v TEeJN T Kv T JN T eEv T JN 1474 or 1 e 2 v J TvE eJ T 2Kv J U N T O N 1475 However the temperature distribution is that which is determined by the steady state with no electric current and we know that v Ju vanishes m that state By putting JN 0 and v Ju 0 in equation 1475 we conclude that the temperature distribution is such as to make the second term vanish and consequently 1476 The Thomwn Ejfeu 325 Furthermore noting that the thermoelectric power is a function of the local temperature we write 1477 and df 1 2 v J TvT eJ eJ u dT N J N 1478 The second term is the Joule heat produced by the flow of electric current even in the absence of a temperature gradient The first term represents the Thomson heat absorbed from the heat reservoirs when the current eJN traverses the temperature gradient v T The Thomson coeffi cient T is defined as the Thomson heat absorbed per unit electric current and per unit temperature gradient T Thomson heat T dr vT eJN dT 1479 Thus the coefficient of the Thomson effect is related to the temperature derivative of the thermoelectric power Equations 1472 and 1479 imply the first Kelvin relation 1480 which was obtained by Kelvin on the basis of energy conservation alone Various other thermoelectric effects can be defined and each can be expressed in terms of the three independent coefficients Lu L 12 and L22 or in terms of a K and r In the presence of an orthogonal magnetic field the number of thermo magnetic effects becomes quite large If the field is in the zdirection an xdirected electric current produces a ydirected gradient of the electro chemical potential this is the Hall effect Similarly an xdirected thermal gradient produces a ydirected gradient of the electrochemical Potential the Nernst effect The method of analysis 5 is identical to that of the thermoelectric effects with the addition of the field dependence equation 1422 of the Onsager reciprocity theorem 5H Cdllcn Phs Rev 73 1349 1948 Guía para una respiración consciente y cuidados asociados Adaptado para personas que buscan mejorar su calidad de vida a través de la respiración consciente y métodos naturales de autocuidado PART II STATISTICAL MECHANICS 15 STATISTICAL MECHANICS IN THE ENTROPY REPRESENTATION THE MICROCANONICAL FORMALISM 151 PHYSICAL SIGNIFICANCE OF THE ENTROPY FOR CLOSED SYSTEMS Thermodynamics constitutes a powerful formalism of great generality erected on a basis of a very few very simple hypotheses The central concept introduced through those hypotheses is the entropy It enters the formulation abstractly as the variational function in a mathematical extremum principle determining equilibrium states In the resultant for malism however the entropy is one of a set of extensive parameters together with the energy volume mole numbers and magnetic moment As these latter quantities each have clear and fundamental physical interpretations it would be strange indeed if the entropy alone were to be exempt from physical interpretation The subject of statistical mechanics provides the physical interpretation of the entropy and it accordingly provides a heuristic justification for the extremum principle of thermodynamics For some simple systems for which we have tractable models this interpretation also permits explicit calculation of the entropy and thence of the fundamental equation We focus first on a closed system of given volume and given number of particles For definiteness we may think of a fluid but this is in no way necessary The parameters U V and N are the only constraints on the system Quantum mechanics tells us that if the system is macroscopic there may exist many discrete quantum states consistent with the specified values of U V and N The system may be in any of these permissible states Naively we might expect that the system finding itself in a particular quantum state would remain forever in that state Such in fact is the lore 329 330 Stallsllcal Mechamcs m Entropy Repreentatwn of elementary quantum mechanics the quantum numbers that specify particular quantum state are ostensibly constants of the motion Thii naive fiction relatively harmless to the understanding of atomic system to which quantum mechanics is most commonly applied is flagrantly misleading when applied to macroscopic systems The apparent paradox is seated in the assumption of isolation of a physical system No physical system is or ever can be truly isolated There exist weak longrange random gravitational electromagnetic and other forces that permeate all physical space These forces not only couple spatially separated material systems but the force fields themselves con stitute physical systems in direct interaction with the system of interest The very vacuum is now understood to be a complex fluctuating entity in which occur continual elaborate processes of creation and reabsorption of electrons positrons neutrinos and a myriad of other esoteric subatomic entities All of these events can couple with the system of interest For a simple system such as a hydrogen atom in space the very weak interactions to which we have alluded seldom induce transitions between quantum states This is so because the quantum states of the hydrogen atom are widely spaced in energy and the weak random fields in space cannot easily transfer such large energy differences to or from the atom Even so such interactions occassionally do occur An excited atom may spontaneously emit a photon decaying to a lower energy state Quan tum field theory reveals that such ostensibly spontaneous transitions actually are induced by the interactions between the excited atom and the modes of the vacuum The quantum states of atoms are not infinitely long lived precisely because of their interaction with the random modes of the vacuum For a macroscopic system the energy differences between successive quantum states become minute In a macroscopic assembly of atoms each energy eigenstate of a single atom splits into some 1023 energy eigen states of the assembly so that the average energy difference between successive states is decreased by a factor of 10 23 The slightest random field or the weakest coupling to vacuum fluctuations is then sufficient td buff et the system chaotically from quantum state to quantum state A realistic view of a macroscopic system is one in which the system makes enormously rapid random transitions among its quantum states A macro scopic measurement senses only an average of the properties of myriads of quantum states All statistical mechanicians agree with the preceding paragraph but not all would agree on the dominant mechanism for inducing transitions Various mechanisms compete and others may well dominate in some or even in all systems No matterit is sufficient that any mechanism exists and it is only the conclusion of rapid random transitions that is needed to validate statistical mechanical theory Physical S1gmficance of the Entropy for Closed Systems 33 J Because the transitions are induced by purely random processes it is reasonable to suppose that a macroscopic system samples every permissible quantum state with equal probabilitya permissible quantum state being one consistent with the external constraints The assumption of equal probability of all permissible microstates is the fundamental postulate of statistical mechanics Its justification will be examined more deeply in Part Ill but for now we adopt it on two bases its a priori reasonableness and the success of the theory that flows from it Suppose now that some external constraint is removedsuch as the opening of a valve permitting the system to expand into a larger volume From the microphysical point of view the removal of the constraint activates the possibility of many microstates that previously had been precluded Transitions occur into these newly available states After some time the system will have lost all distinction between the original and the newly available states and the system will thenceforth make random transitions that sample the augmented set of states with equal probability The number of microstates among which the system undergoes transitions and which thereby share uniform probability of occupation increases to the maximum permitted by the imposed constraints This statement is strikingly reminiscent of the entropy postulate of thermodynamics according to which the entropy increases to the maxi mum permitted by the imposed constraints It suggests that the entropy can be identified with the number of microstates consistent with the imposed macroscopic constraints One difficulty arises The entropy is additive extensive whereas the number of microstates is multiplicative The number of microstates availa ble to two systems is the product of the numbers available to each the number of microstates of two dice is 6 X 6 36 To interpret the entropy then we require an additive quantity that measures the number of microstates available to a system The unique answer is to identify the entropy with the logarithm of the number of available microstates the logarithm of a product being the sum of the logarithms Thus 151 where Q is the number of microstates consistent with the macroscopic constraints The constant prefactor merely determines the scale of S it is chosen to obtain agreement with the Kelvin scale of temperature defined by 7 i aSaU We shall see that this agreement is achieved by taking the constant to be Boltzmanns constant k 8 RNA 13807 X 10 l3JK With the definition 15l the basis of statistical mechanics is established Just as the thermodynamic postulates were elaborated through the formalism of Legendre transformations so this single additional postulate will be rendered more powerful by an analogous structure of mathemati cal formalism Nevertheless this single postulate is dramatic in its brevity 332 S1a11s11ca Mechanics m Entropy Representation simplicity and completeness The statistical mechanical formalism that derives directly from it is one in which we simply calculate the loga rithm of the number of states available to the system thereby obtaining S as a function of the constraints U V and N That is it is statistical mechanics in the entropy representation or in the parlance of the field It 1s statistical mechanics in the microcanomcal formalism In the following sections of this chapter we treat a number of systems by this microcanonical formalism as examples of its logical completeness As in thermodynamics the entropy representation is not always the most convenient representation For statistical mechanical calculations it is frequently so inconvenient that it is analytically intractable The Legendre transformed representations are usually far preferable and we shall turn to them in the next chapter Nevertheless the rnicrocanonical formulation establishes the clear and basic logical foundation of statistical mechanics PROBLEMS 1511 A system is composed of two harmonic oscillators each of natural frequency w0 and each having permissible energies n liw 0 where n is any nonnegative mteger The total energy of the system is E nliw 0 where n is a positive integer How many rrucrostates are available to the system What is the entropy of the system A second system is also composed of two harmonic oscillators each of natural frequency 2w 0 The total energy of this system is E nliw 0 where n is an even integer How many microstates are available to this system What is the entropy of this system What is the entropy of the system composed of the two preceding subsystems separated and enclosed by a totally restnctive wall Express the entropy as a function of E and Answer EE S01 k Bin 22 2i Wo 1512 A system is composed of two harmonic oscillators of natural frequencies w0 and 2w0 respectively If the system has total energy E n 11iw0 where n is an odd integer what is the entropy of the system If a composite system is composed of two noninteracting subsystems of the type just described having energies 1 and 2 what is the entropy of the compo site system The Emstem Model of a Crystallme Solid 333 152 THE EINSTEIN MODEL OF A CRYSTALLINE SOLID With a identification of the meaning of the entropy we proceed to calculate the fundamental equation of macroscopic systems We first apply the method to Einsteins simplified model of a nonmetallic crystal line solid It is well to pause immediately and to comment on so early an introduction of a specific model system In the eleven chapters of this book devoted to thermodynamic theory there were few references to specific model systems and those occasional references were kept care fully distinct from the logical flow of the general theory In statistical me chanics we almost immediately introduce a model system and this will be followed by a considerable number of others The difference is partially a matter of convention To some extent it reflects the simplicity of the general formalism of statistical mechanics which merely adds the logical interpretation of the entropy to the formalism of thermodynamics the interest therefore shifts to applications of that formalism which underlies the various material sciences such as solid state physics the theory of liquids polymer physics and the like But most important it reflects the fact that counting the number of states available to physical systems requires computational skills and experience that can be developed only by explicit application to concrete problems To account for the thermal properties of crystals Albert Einstein in 1907 introduced a highly idealized model focusing only on the vibrational modes of the crystal Electronic excitations nuclear modes and various other types of excitations were ignored Nevertheless for temperatures that are neither very close to absolute zero nor very high the model is at least qualitatively successful Einsteins model consists of the assumption that each of the N atoms in the crystal can be considered to be bound to its equilibrium position by a harmonic force Each atom is free to vibrate around its equilibrium position in any of the three coordinate directions with a natural frequency Wo More realistically recall Section 12 the atoms of crystals are harmoni cally bound to their neighboring atoms rather than to fixed point Accordingly the vibrational modes are strongly coupled givmg rise to 3N collective normal modes The frequencies are distributed from zero for Very long wave length modes to some maximum frequency for the modes o minimum permissible wave length comparable to the interatomic distance There are far more high frequency modes than low frequency modes with the consequence that the frequencies tend to cluster mainly in a narrow range of frequencies to which the Einstein frequency w0 is a rough approximation 334 Sta11s11wl Mechamcs m Emrop Reprefenwton In the Einstein model then a crystal of N atoms is replaced by 3N harmonic oscillators all with the same natural frequency w0 For the present purposes it is convenient to choose the zero of energy so that the energy of a harmonic oscillator of natural frequency w0 can take only the discrete values nhw 0 with n 0 I 2 3 Here h h21r 1055 x 10 34 Js h being Plancks constant In the language of quantum mechanics each oscillator can be oc cupied by an integral number of energy quanta each of energy hw 0 The number of possible states of the system and hence the entropy can now be computed easily If the energy of the system is U it can be considered as constituting U hw 0 quanta These quanta are to be distrib uted among 3N vibrational modes The number of ways of distributing the U hw 0 quanta among the 3N modes is the number of states Q available to the system The problem is isomorphic to the calculation of the number of ways of placing U hw 0 identical indistinguishable marbles in 3N numbered distinguishable boxes FlGURE 15 l Illustrating the combinatorial problem of distributing U Ii w0 indistmguishable obJcm marbles in 3 N distinguishable boxes The combinatorial problefi can be visualized as follows Suppose we have U hw 0 marbles and 3N I match sticks We lay these out in a linear array in any order One such array is shown in Fig 151 The interpretation of this array is that three quanta marbles are assigned to the first mode two quanta to the second none to the third and so forth and two quanta are assigned to the last mode the 3Nth Thus the number of ways of distributing the U hw 0 quanta among the 3N modes 1s the number of permutations of 3N I U lzw 0 objects of which U hw 0 are identical marbles or quanta and 3N I are identical match sticks That is Q 31 Uhw 0 3N I U hw 0 3N Uhw 0 3NUhw 0 152 This completes the calculation for the entropy is simply the logarithm of this quantity multiplied by kn To simplify the result we employ the Stirling approximation for the logarithm of the factorial of a large number lnM M In M M if M 1 15 3 The Einstein Model of a Crystalline Solid 335 whence the molar entropy is s 3Rln1 3Rtnl Uo u0 u0 u 154 where 155 This is the fundamental equation of the system It will be left to the problems to show that the fundamental equation implies reasonable thermal behavior The molar heat capacity is zero at zero temperature rises rapidly with increasing temperature and ap proaches a constant value 3R at high temperature in qualitative agreement with experiment The rate of increase of the heat capacity is not quantitatively correct because of the naivete of the model of the vibra tional modes This will be improved subsequently in the Debye model Section 167 in which the vibrational modes are treated more realisti cally The heat capacity of the Einstein model is plotted in Fig 152 The molar heat capacity cv is zero at T 0 and it asymptotes to 3R at high temperature The rise in cv occurs in the region k8 T 11iWo in particular cJ3R i and the point of maximum slope both occur near k8 Tnw0 At low temperature cv rises exponentially whereas experimentally the heat capacity rises approximately as T3 The mechanical implications of the modelthe pressurevolume rela tionship and compressibilityare completely unreasonable The entropy according to equation 155 is independent of the volume whence the pressure TaSaV is identically zero Such a nonphysical result is of course a reflection of the naive omission of volume dependent effects from the model Certain consequences of the model give important general insights CQnsider the thermal equation of state 156 Now noting that there are 3NNA oscillators in the system U hw0 mean energy per oscillator JNN A 7 A eo B1 157 The quantity hwofk 8 is called the Einstein temperature of the crystal and it generally is of the same order of magnitude a thP 1ru1t 336 Statis11ca Mechanus in Entropy Representatwn 4 45 5 09 07 t c 05 3R 03 01 01 02 03 04 05 FIGURE 152 Heat capacity of the Einstein model or of a single harmonic oscillator The upper curve refers to the upper scale of k 8 Tlliw0 and the lower curve to the lower expanded scale The ordinate can be interpreted as the heat capacity of one harmonic oscillator in units of k 8 or as the molar heat capacity in units of 3R temperature of the solid Thus below the melting temperature the mean energy of an oscillator is less than or of the order of hw0 Alternatively stated the solid melts before the Einstein oscillators attain quantum numbers appreciably greater than unity PROBLEMS 1521 Calculate the molar heat capacity of the Einstein model by equation 157 Show that the molar heat capacity approaches 3R at high temperatures Show that the temperature dependence of the molar heat capacity is exponential near zero temperature and calculate the leading exponential term 1522 Obtain an equation for the mean quantum number ii of an Einstem oscillator as a function of the temperature Calculate ii for k BT hw 0 0 1 2 3 4 10 50 100 ignore the physical reality of melting of the crystal The Two State S1Mem 337 1523 Assume that the Einstein frequency w0 for a particular crystal depends upon the molar volume w w0 A In o O Vo a Calculate the isothermal compressibility of tlus crystal b Calculate the heat transfer if a crystal of one mole 1s compressed at constant temperature from v to v1 153 THE TWOSTATE SYSTEM Another model that illustrates the principles of statistical mechanics in a simple and transparent fashion is the twostate model In this model each atom can be either in its ground state with energy zero or in its excited state with energy To avoid conflict with certain general theorems about energy spectra we assume that each atom has additional states but all of such high energy as to exceed the total energy of the system under consideration Such states are then inaccessible to the system and need not be considered further in the calculation If V is the energy of the system then V atoms are in the excited state and N V atoms are in the ground state The number of ways of choosing V atoms from the total number N is 158 The entropy is therefore or invoking Stirlings approximation equation 153 S N k ln 1 k In V V V U B NE B NE 1510 Again because of the artificiality of the model the fundamental equa tion is independent of the volume The thermal equation of state is easily calculated to be kB In NE 1 T U 1511 338 Statistical Mechanics m Entropy Representation Recalling that the calculation is subject to the condition V ih we observe that the temperature is a properly positive number Solving for the energy 1512 The energy approaches ih2 as the temperature approaches infinity in this model although we must recall that additional states of high energy would alter the high temperature properties At infinite temperature half the atoms are excited and half are in their ground state The molar heat capacity is 1513 A graph of this temperature dependence is shown in Fig 153 The molar heat capacity is zero both at very low temperatures and at very high I I I j I I I I I I I I 04 I I 03 t 01 I I 0 I I I I I I I I I I I I 0 01 02 03 04 05 06 OJ 08 09 10 11 12 FIGURE 153 knT Heat capacity of the twostate model the Schottky hump A Polymer ModelThe Rubber Band Revuued 339 temperatures peaking in the region of kBT 42E This behavior is known as a Schottky hump Such a maximum when observed in empirical dita is taken as an indication of a pair of low lying energy states with all other energy states lying at considerably higher energies This is an example of the way in which thermal properties can reveal information about the atomic structure of materials PROBLEMS 1531 In the twostate model system of this section suppose the excited state energy E of an atom depends on its average distance from its neighboring atoms so that V V N where a and y are positive constants This assumption applied to a somewhat more sophisticated model of a solid was introduced by Gruneisen and y is the Gruneisen parameter Calculate the pressure P as a function of v and T Answer p feafk 8 rv 11 ffYl 154 A POLYMER MODELTHE RUBBER BAND REVISITED There exists another model of appealing simplicity that is euphemisti cally referred to as a polymer model Its connection with a real polymer is tenuous but that connection is perhaps close enough to serve the pedagogical purpose of providing some sense of physical reality while again illustrating the basic algorithm of statistical mechanics And in particular the model provides an insight to the behavior of a rubber band as discussed on purely phenomenological grounds in Section 37 As we saw in that section the extensive parameter of interest which replaces the volume is the length the corresponding intensive parameter analogous to the pressure is the tension We are interested in the equation of state relating tension to length and temperature The rubber band can be visualized as a bundle of long chain polymers Each polymer chain is considered to be composed of N mono mer units each of length a and we focus our attention on one particular polymer chain in the bundle One end of the polymer chain is fixed at a point that is taken as the origin of coordinates The other end of the chain is subject to an externally applied tension r parallel to the positive xaxis Fig 154 340 Stahshcal Mechanics m Entropy Representation String Pulley FIGURE 154 Polymer model The string should be much longer than shown so that the end of the polymer 1s free to move in the ydirection and the applied tension 17 is directed along the xdirection In the polymer model each monomer unit of the chain is permitted to lie either parallel or antiparallel to the xaxis and zero energy is associ ated with these two orientations Each monomer unit has the additional possibility of lying perpendicular to the xaxis in the y or y direc tions only Such a perpendicular monomer unit presumably suffers interference with other polymer chains in the bundle we represent this interference by assigning a positive energy E to such a perpendicular monomer A somewhat more reasonable model of the polymer might permit the perpendicular monomers to lie along the z directions as well as along the y directions and more importantly would account for the inter ference of a chain doubling back on itself Such models complicate the analysis without adding to the pedagogic clarity or qualitative content of the result We calculate the entropy S of one polymer chain as a function of the energy U U of the coodinates Lx and LY of the end of the polymer chain and of the number N of monomer units in the chain Let N and Nx be the numbers of monomers along the x and x directions respectively and similarly for N and NY Then 1514 A Polymer Model The Rubber Band Rev1s1ted 34 I from which we find N 1 R U L X 2 X N lR U L X 2 X N 1UL y 2 y N lU L y 2 y 1515 The number of configurations of the polymer consistent with given coordinates Lx and LY of its terminus and with given energy U is 1516 The entropy is then using the Stirling approximation equation 153 S kBlnO RkBlnR NkBlnN NxkBlnNx or S Rk8 lnR HR U Lk 8 lnHR U L HR U Lk 8 lnHR U L Hu Lk8 lntu L Hu LkBlntu L 1517 1518 With the statistical mechanical phase of the calculation completed the thermodynamic formalism comes into play The ycomponent of the tension is conjugate to the extensive coordinate LY see Problem 1541 Setting 9 0 gives r as kB U L In 0 T oLY 2a U L y 1519 from which we conclude as expected that L y L O y 1520 342 Sta11st1cal Mechamcs m Entropy Representation Similarly 1521 and 1 as kB kB kB 1 In NL U In NL U nU r au 2E x 2E x E 15 22 or 1523 This is the thermal equation of state The mechanical equation of state 1521 can be written in an analogous exponential form N U L e2Tak 8 T x N U L 1524 The two preceding equations are the equations of state in the entropy representation and accordingly they involve the energy U That is not generally convenient We proceed then to eliminate U between the two equations With some algebra we find see Problem 1542 that L sinh9ak 8 T N cosh9ak8T eksT 1525 For small 9a relative to k 8 T the equation can be expanded to first order 1526 The modulus of elasticity of the rubber band the analogue of the compressibility 1 V av a Ph is for small 1527 Counting Techniques and their Circumventwn High Dmenswnality 343 The fact that this elastic modulus decreases with increasing temperature or that the stiffnesf increases is in dramatic contrast to the behavior of a spring or of a stretched wire The behavior of the polymer is sometimes compared to the behavior of a snake if we grasp a snake by the head and tail and attempt to stretch it straight the resistance is attributable to the writhing activity of the snake The snake in its writhing assumes all possible configurations and more configurations are accessible if the two ends are not greatly distant from each other At low temperatures the rubber band is like a torpid snake At high temperatures the number of configurations available and the rate of transitions among them is greater resulting in a greater contractive tension It is the entropy of the snake and of the rubber band that is responsible for the tendency of the ends to draw together The behavior described is qualitatively similar to that of the simple phenomenological model of Section 37 But compared to a truly realistic model of a rubber band both models are extremely naive PROBLEMS 1541 Is the sign correct in equation 1519 Explain 1542 Eliminate UE between equations 1523 and 1524 and show that the formal solution is equation 1525 with a sign before the second term in the denominator Consider the qualitative dependence of LxNa on E and show that physical reasoning rejects the negative sign in the denominator thus validating equation 1525 1543 A rubber band consisting of n polymer chains 1s stretched from zero length to its full extension L Na at constant temperature T Does the energy of the system increase or decrease Calculate the work done on the system and the heat transfer to the system 1544 Calculate the heat capacity at constant length for a rubber band consisting of n polymer chains Express the answer in terms of T and Lx 1545 Calculate the coefficient of longitudinal thermal expansion dfined by K 1 aLX LX ar r Express Kr as a function of T and sketch the qualitative behavior Compare this with the behavior of a metallic wire and discuss the result 155 COUNTING TECHNIQUES AND THEIR CIRCUMVENTION HIGH DIMENSIONALITY To repeat the basic algorithm of statistical mechanics consists of counting the number of states consistent with the constraints imposed the 344 Sta11st1cal Mechanics m Entropy Representatron entropy is then the product of Boltzmanns constant and the logarithm of the permissible number of states Unfortunately counting problems tend to require difficult and sophisti cated techniques of combinatorial mathematics if they can be done at all In fact only a few highly artificial idealized models permit explicit solution of the counting problem even with the full armamentarium of combinatorial theory If statistical mechanics is to be a useful and practical science it is necessary that the difficulties of the counting problem somehow be circumvented One method of simplifying the count ing problem is developed in this section It is based on certain rather startling properties of systems of high dimensionality a concept to be defined shortly The method is admittedly more important for the insights it provides to the behavior of complex systems than for the aid it provides in practical calculations More general and powerful methods of circumventing the counting problem are based on a transfer from thermo dynamics to statistical mechanics of the technique of Legendre transfor mations That transfer will be developed in the following chapters For now we turn our attention to the simplifying effects of high dimensionality a concept that can best be introduced in terms of an explicit model We choose the simplest model with which we are already familiar the Einstein model Recall that the Einstein solid is a collection of N atoms each of which is to be associated with three harmonic oscillators corresponding to the oscillations of the atom along the x y and z axes A quantum state of the system is specified by the 3N quantum numbers nl n 2 n 3 n 3 and the energy of the system is 3N Unpn 2 n 3fir L n1hw0 I 1528 Each such state can be represented by a point with coordinates nin 2n 3 n 3fir in a 31Vdimensional state space Only points with positive integral coordinates are permissible corresponding to the dis creteness or quantization of states in quantum mechanics It is to be stressed that a single point represents the quantum state of the entire crystal The locus of states with a given energy U is a diagonal hyperplane with intercepts Uhw 0 on each of the 3N coordinate axes Fig 155 All states lying inside the plane ie closer to the origin have energies less than U and all states lying outside the plane further from the origin have energies greater than U The first critical observation which is called to our attention by Fig 155 is that an arbitrary diagonal plane corresponding to an arbitrary energy U will generally pass through none of the discrete coordinate points in the space That is an arbitrarily selected number U generally Countmg Techniques and their C1rcumvent10n High D1menswnaht 345 nlX FIGURE 155 Quantum state space for the Einstein solid The threedimensional state space shown is for an Einstein solid composed of a single atom Each addittonal atom would increase the dimensionality of the space by three The hyperplane U has intercepts Uhw 0 on all axes There is one state for each unit of hypervolume and neglectmg surface corrections the number of states with energy less than U is equal to the volume inside the diagonal hyperplane U cannot be represented in the form of equation 1528 such a decomposi tion being possible only if U lzWo is an integer More generally if we inquire as to the number of quantum states of a system with an arbitrarily chosen and mathematically precise energy we almost always find zero But such a question is unphysical As we have stressed previously the random interactions of every system with its environment make the energy slightly imprecise Furthermore we never know and cannot measure the energy of any system with absolute precision The entropy is not the logarithm of the number of quantum states that lie on the diagonal hyperplane U of Fig 155 but rather it is the logarithm of the number of quantum states that lie in the close vicinity of the diagonal hyperplane This consideration leads us to study the number of states between two hyperplanes U and U The energy separation is determined by the imprecision of the energy of the macroscopic system That imprecision may be thought of as a consequence either of environmental interactions or of imprecision in the preparation measurement of the system The remarkable consequence of high dimensionality is that the volume between the two planes U and U and hence the entropy is essentwlly independent of the separation of the planes 346 Statistical Mechanics in Entropy Representation This result is at first so startlingly counterintuitive and so fundamen tal that it warrants careful analysis and discussion We shall first corrobo rate the assertion on the basis of the geometrical representation of the states of the Einstein solid Then we shall reexamine the geometrical representation to obtain a heuristic understanding of the general geometri cal basis of the effect The number of states OU with energies less than or equal to a given value U is equal to the hypervolume lying inside the diagonal hyper plane U This hypervolume is see problem 1551 O U number of states with energies less than U 1 U JN 3N hw0 1529 The fact that this result is proportional to U 3iil where 3N is the dimen sionality of the state space is the critical feature of this result The precise form of the coefficient in equation 1529 will prove to be of only secondary importance By subtraction we find the number of states with energies between U fl and U to be 1 U Jit 1 U fl 3iv OU OU fl 3N hw0 3N hWo or 1530 Bll 1 flU is less than unity raising this quantity to an exponent 3N 1023 results in a totally negligible quantity see Problem 1552 so that nu OU OU fl OU 1531 That is the number Q U of states with energies between U fl and U is essentially equal to the total number OU of states with energies less than Uand this result is essentially independent of fl Thus having corroborated the assertion for our particular model let us reexamine the geometry to discern the more general geometrical roots of this strange but enormously useful result The physical volume in Fig 155 can be looked at as one eighth of a regular octahedron but only the portion of the octahedron in the physical Counting Techniques and their Circumvention High Dmenswnality 347 octant of the space has physical meaning With higher dimensionality the regular polyhedron would become more nearly spherical The dimen sionless energy U lzWo is analogous to the radius of the figure being the distance from the origin to any of the corners of the polyhedron This viewpoint makes evident the fact equation 1529 that the volume is proportional to the radius raised to a power equal to the dimensionality of the space r 2 in two dimensions r 3 in three etc The volume between two concentric polyhedra with a difference in radii of dr is dV av ar dr The ratio of the volume of this shell to the total volume is dV av dr V ar V 1532 dV dr n V r 1533 If we taken 1023 we find dVV 01 only if drr 10 24 For drr greater than 10 24 the equation fails telling us that the use of differentials is no longer valid The failure of the differential analysis is evidence that dV V already becomes on the order of unity for values of drr as small as drr 10 23 In an imaginary world of high dimensionality there would be an automatic and perpetual potato famine for the skin of a potato would occupy essentially its entire volume In the eal world in which threedimensional statistical mechanicians calculate entropies as volumes in manydimensional state spaces the properties of high dimensionality are a blessing We need not calculate the number of states in the vicinity of the system energy Uit is quite as satisfactory and frequently easier to calculate the number of states with energies less than or equal to the energy of the physical system Returning to the Einstein solid we can calculate the fundamental equation using the result 1529 for 0 U the Umber of states with energies less than U the entropy is S k 8 ln Q U and it is easily corroborated that this gives the same result as was obtained in equation 154 The two methods that we have used to solve the Einstein model of a solid should be clearly distinguished In Section 152 we assumed that U lzw 0 was an integer and we counted the number of ways of distributing quanta among the modes This was a combinatorial problem albeit a simple and tractable one because of the extreme simplicity of the model The second method in this section involved no combinatorial calculation whatsoever Instead we defined a volume in an abstract state space and the entropy was related to the total volume inside the bounding surface defined by the 348 Statistical Mechanics m Entropy Representatwn energy U The combinatorial approach is not easily transferable to more complicated systems the method of hypervolumes is general and is usually more tractable However the last method is not applicable at very low temperature where only a few states are occupied and where the occu pied volume in state space shrinks toward zero PROBLEMS 1551 To establish equation 1529 let fJn be the hypervolume subtended by the diagonal hyperplane in n dimensions Draw appropriate figures for n 1 2 and 3 and show that if L is the intercept on each of the coordinate axes fJt L l L x L 2 fJ fJ 1 dx 2 i o L 2 L X 2 L 3 fJJ fJz lo 1 L dx 3 and by mathematical mduction 1552 Recalling that fan 1 x11x e x0 2718 show that l for U 1 With this approximation discuss the accuracy of equation 1531 for a range of reasonable values of t U ranging perhaps from 10 3 to 10 10 With what precision tU would the energy have to be known in order that corrections to equation 1531 might become significant Assume a system with fl 1023 1553 Calculate the fraction of the hypervolume between the radii 09r and r for hyperspheres in 1 2 3 4 and 5 dimensions Similarly for 10 30 and 50 dimensions 16 THE CANONICAL FORMALISM STATISTICAL MECHANICS IN HELMHOLTZ REPRESENTATION 161 THE PROBABILITY DISTRIBUTION The microcanonical formalism of the preceding chapter is simple in principle but it is computationally feasible only for a few highly idealized models The combinatorial calculation of the number of ways that a given amount of energy can be distributed in arbitrarily sized boxes is generally beyond our mathematical capabilities The solution is to remove the limitation on the amount of energy availableto consider a system in contact with a thermal reservoir rather than an isolated system The statistical mechanics of a system in contact with a thermal reservoir may be viewed as statistical mechanics in Helmholtz representation or in the parlance of the field in canonical formalism States of all energies from zero to arbitrarily large energies are avail able to a system in contact with a thermal reservoir But in contrast to the state probabilities in a closed system each state does not have the same probability That is the system does not spend the same fraction of time in each state The key to the canonical formalism is the determination of the probability distribution of the system among its microstates And this problem is solved by the realization that the system plus the reservoir constitute a closed system to which the principle of equal probability of microstates again applies A simple analogy is instructive Consider a set of three dice one of which is red the remaining two being white The three dice have been thrown many thousands of times Whenever the sum of the numbers on the three dice has been 12 and only then the number on the red die has been recorded In what fraction of these recorded throws has the red die shown a one a two a six 349 350 The Canomwl Formahsm Stamtual Medumu1 111 Helmholtz Representatron The result left to the reader is that the red die has shown a one in fs of the throws a two m 2 a five in fs and a six in f5 of the recorded throws The probability of a red six in this restricted set of throws is The red die is the analogue of our system of mterest the white dice correspond to the reservoir the numbers shown correspond to the energies of the respective systems and the restriction to throws in which the sum i 12 corresponds to the constancy of the total energy of system plus reservoir The probability of the subsystem being in state j 1s equal to the fraction of the total number of states of systemplusreservmr in which the subsys tem is in the state j with energy E I ores IOI EJ i 0101101 161 Here 101 is the total energy of the systemplusreservoir and 0 101 is the total number of states of the systemplusreservoir The quantity in the numerator OresC101 E is the number of states available to the reservoir when the subsystem is in the state j leaving energy 101 E1 in the reservoir This is the seminal relation in the canonical formalism but it can be reexpressed in a far more convenient form The denominator is related to the entropy of the composite system by equation 151 The numerator is similarly related to the entropy of the reservoir so that 162 If U is the average value of the energy of the subsystem then the additivity of the entropy implies 163 Furthermore expanding Sre 101 E around the equilibrium point 101 U Sre101 U U ET 164 No additional terms in the expansion appear this being the very defini tion of a reservoir Inserting these latter two equations in the expression for 165 The Probabtty Dsmhutwn 351 The quantity 1k 8 T appears so pervasively throughout the theory that it is standard practice to adopt the notation 166 Furthermore U TS U is the Helmholtz potential of the system so that we finally achieve the fundamental result for the probability f of the subsystem being in the state j 1 167 Of course the Helmholtz potential is not known it is in fact our task to compute it The key to its evaluation is the observation that eJF plays the role of a stateindependent normalization factor in equation 167 Li eJFfeJ I J J or where Z the canonical partition sum is defined by Z fe J J 168 169 16JO We have now formulated a complete algorithm for the calculation of a fundamental relation in the canonical formalism Given a list of all states j of the system and their energies EJ we calculate the partitwn sum 1610 The partition sum is thus obtained as a function of temperature or 3 and of the parameters V Ni N2 that determine the energy levels Equation 169 in turn determines the Helmholtz potential as a function also of T V Ni N This is the sought for fundamental relation The entire algorithm is summarized in the relation 3F In eJE In Z J which should be committed to memory A corroboration of the consistency of the formalism follows from recalling that Ji is the probability of occupation of the jth state which from equations 167 169 and 1610 can be written in the very useful form 1611 352 The Canonual Formalm StallIUal Mehamcs m Helmholtz Rep1e1entatum The average energy is then expected to be U fE 1 E1ePEjlePL 1612 1 1 or U dd3lnZ 1613 Insertion of equation 169 expressing Z in terms of F and recalling that 3 1k 8 T reduces this equation to the familiar thermodynamic relation U F TS F T aF aT and thereby confirms its validity Equa tions 1612 and 1613 are very useful in statistical mechanics but it mmt be stressed that these equations do not constitute a fundamental relation The fundamental relation is given by equations 169 and 1610 giving F rather than U as a function of 3 V N A final observation on units and on formal structure is revealing The quantity 3 is of course merely the reciprocal temperature in natural units The canonical formalism then gives the quantity f3F in terms of 3 V and N That is F Tis given as a function of 1T V and N This is a fundamental equation in the representation SlT recall Section 54 Jmt as the rnicrocanonical formalism is naturally expressed in entropy repre sentation the canonical formalism is naturally expressed in S3 repre sentation The generalized canonical representations to be discussed m Chapter 17 will similarly all be expressed most naturally in terms of Massieu functions Nevertheless we shall conform to universal usage and refer to the canonical formalism as being based on the Helmholtz poten tial No formal difficulties arise from this slight misrepresentation PROBLEMS 1611 Show that equation 1613 is equivalent to U F TS 1612 From the canonical algorithm expressed by equations 169 and 1610 express the pressure in terms of a derivative of the partition sum Further express the pressure in terms of the derivatives aEaV and of T and the E Can you give a heuristic interpretation of this equation 1613 Show that Sk 8 32aF a3 and thereby express Sin terms of Zand its derivatives with respect to 3 1614 Show that cv 3asa3v and thereby express cv in terms of the partition sum and its derivatives with respect to 3 Answer C N ik 132 a21n z B a32 Additive Energies and Factorrzah1bty of the Partttwn Sum 353 162 ADDITIVE ENERGIES AND FACTORIZABILI1Y OF THE PARTITION SUM To illustrate the remarkable simplicity of the canonical formalism we recall the twostate system of Section 153 In that model N distinguish able atoms each were presumed to have two permissible states of energies O and Had we attributed even only three states to each atom the problem would have become so difficult as to be insoluble by the rnicrocanonical formalism at least for general values of the excitation energies By the canonical formalism it i simple indeed We consider a system composed of N distinguishable elements an element being an independent noninteracting excitation mode of the system If the system is composed of noninteracting material constituents s11ch as the molecules of an ideal gas the elements refer to the excitations of the individual molecules In strongly interacting systems the elements may be wavelike collective excitations such as vibrational modes or electromagnetic modes The identifying characteristic of an element is that the energy of the system is a sum over the energies of the elements which are independent and noninteracting Each element can exist in a set of orbital states we henceforth use the term orbital state to distinguish the states of an element from the states of the collective system The energy of the ith element in its jth orbital state is tr Each of the elements need not be the same either in the energies or the number of its possible orbital states The total energy of the system is the sum of the singleelement energies and each element is permitted to occupy any one of its orbital states independently of the orbital states of the other elements Then the partition sum is Z 1614 L e 111e J e 11 1615 JJ 1616 1617 where z the partition sum of the i th element is 1618 The partition sum factors Furthermore the Helmholtz potential is additive 354 The Canonical Formalism Statistical Mechanics in Helmholtz Representation over elements 3F lnZ lnz 1 lnz 2 1619 This result is so remarkably simple powerful and useful that we em phasize again that it applies to any system in which a the energy is additive over elements and b each element is permitted to occupy any of its orbital states independently of the orbital state of any other element The twostate model of Section 153 satisfies the above criteria whence 1620 and 1621 It is left to the reader to demonstrate that this solution is equivalent to that found in Section 153 If the number of orbitals had been three rather than two the partition sum per particle z would merely have contained three terms and the Helmholtz potential would have contained an ad ditional term in the argument of the logarithm The Einstein model of a crystal Section 152 similarly yields to the simplicity of the canonical formalism Here the elements are the vibra tional modes and the partition sum per mode is 1622 This geometric series sums directly to 1 z1ePhwo 1623 There are 3N vibrational modes so that the fundamental equation of the Einstein model in the canonical formalism is 1624 Clearly Einsteins drastic assumption that all modes of vibration of the crystal have the same frequency is no longer necessary in this formalism A more physically reasonable approximation due to P Debye will be discussed in Section 167 Internal Modes ma Gas 355 PROBLEMS 1621 Consider a system of three particles each different The first particle has two orbital states of energies t11 and t12 The second particle has permissible energies t21 and t22 and the third particle has permissible energies t31 and t32 Write the partition sum explicitly in the form of equation 1614 and by explicit algebra factor it in the form of equation 1617 1622 Show that for the twolevel system the Helmholtz potential calculated in equation 1621 is equivalent to the fundamental equation found in Section 153 1623 Is the energy additive over the particles of a gas if the particles are uncharged mass points with negligible gravitational interaction Is the partition sum factorizable if half the particles carry a positive electric charge and half carry a negative electric charge Is the partition sum factorizable if the particles are fermions obeying the Pauli exclusion principle such as neutrinos 1624 Calculate the heat capacity per mode from the fundamental equation 1624 1625 Calculate the energy per mode from equation 1624 What is the leading term in UT in the regions of T 0 and of T large 1626 A binary alloy is composed of NA atoms of type A and of N 8 atoms of type B Each Atype atom can exist in its ground state or in an excited state of energy E all other states are of such high energy that they can be neglected at the temperatures of interest Each Btype atom similarly can exist in its ground state of energy zero or in an excited state of energy 2t The system is in equilibrium at temperature T a Calculate the Helmholtz potential of the system b Calculate the heat capacity of the system 1627 A paramagnetic salt is composed of 1 mole of noninteracting ions each with a magnetic moment of one Bohr magneton µ 8 9274 X 10 24 joulestesla A magnetic field Be is applied along a particular direction the permissible states of the ionic moments are either parallel or antiparallel to this direction a Assuming the system is maintained at a temperature T 4 K and Be is increased from 1 Tesla to 10 Tesla what is the magnitude of the heat transfer from the thermal reservoir b If the system is now thermally isolated and the applied magnetic field Be is decreased from 10 Tesla to 1 Tesla what is the final temperature of the system This process is referred to as cooling by adiabatic demagnetization 163 INTERNAL MODES IN A GAS The excitations of the molecules of a gas include the three translational tnodes of the molecules as a whole vibrational modes rotational modes electronic modes and modes of excitation of the nucleus For simplicity 356 The Canonical Formaltsm Statlst1cal Mechanics m Helmholtz Representatwn we initially assume that each of these modes is independent later return ing to reexamine this assumption Then the partition sum factors with respect to the various modes 1625 and further with respect to the molecules Z ZN VJb vib Z ZN rot rot 1626 and similarly for zelect and znuc The ideality or nonideality of the gas is a property primarily of the translational partition sum The translational modes in any case warrant a separate and careful treatment which we postpone to Section 1610 We now simply assume that any intermolecular collisions do not couple to the internal modes rotation vibration etc The N identical vibrational modes of a given type one centered on each molecule are formally identical to the vibrational modes of the Einstein model of a crystal that is they are just simple harmonic oscillators For a mode of frequency Wo Z N 1 P1rw0 fl vib Z v1b e 1627 and the contribution of this vibrational mode to the Helmholtz potential is as given in equation 1624 with 3N replaced by N The contribution of a vibrational mode to the heat capacity of the gas is then as shown in Fig 152 the ordinate being cR rather than c3R As described in Section 131 the heat capacity rises in a roughly steplike fashion in the vicinity of k 8 T o luu0 and it asymptotes to c R Figure 131 was plotted as the sum of contributions from two vibrational modes with w2 15w1 The characteristic vibrational temperature nwofk 8 ranges from several thousand kelvin for molecules containing very light elements 6300 K for H 2 to several hundred kelvin for molecules containing heavier ele ments 309 K for Br2 To consider the rotational modes of a gas we focus particularly on heteronuclear diatomic molecules such as HCl which require two angu lar coordinates to specify their orientation The rotational energy of such heteronuclear diatomic molecules is quantized with energy eigenvalues given by t 012 1628 Internal Modes ma Gm 357 Each energy level is 2t 1fold degenerate The energy unit Eis equal to Ji2moment of inertia 2 or approximately 2 X 10 21 J for the HCl rnolecule The characteristic separation between levels is of the order of E which corresponds to a temperature EkB 15 K for HCllarger fo tighter molecules and smaller for heavier molecules The rotational partition sum per molecule is 00 zrot L 2t lepttlE tO 1629 If k8 T e the sum can be replaced by an integral Then noting that 21 1 is the derivative of t t 1 and writing x for the quantity tt 1 z Joo efJx dx kBT rot O 3E E 1630 If k BT is less than or of the order of E it may be practical to calculate several terms of the series explicitly to some t such that t t 1 k BT and to integrate over the remaining range from t to infinity see Problem 1632 It is left to the reader to show that for k 8 T E the average energy is kBT The case of homonuclear diatomic molecules such as 0 2 or H 2 is subject to quantum mechanical symmetry conditions into which we shall not enter Only the even terms in the partition sum or only the odd terms are permitted depending upon detailed characteristics of the atoms At high temperatures this restriction merely halves the rotational partition sum per molecule The nuclear and electronic contributions can be computed in similar fashion but generally only the lowest energy levels of each contribute Then znuc is simply the degeneracy multiplicity of th lowest energy configuration Each of these factors simply contributes Nk BT In multi plicity to the Helmholtz potential It is of interest to return to the assumption that the various modes are independent This assumption is generally a good but not a rigorous pproximation Thus the vibrations of a diatomic molecule change the instantaneous interatomic distance and thereby change the instantaneous tnornent of inertia of rotation It is only because the vibrations generally are very fast relative to the rotations that the rotations sense only the average interatomic distance and thereby become effectively independent of the vibrations 358 The Canonical Formalsm Statistical Mechanics m Helmholtz Representation PROBLEMS 1631 Calculate the average rotational energy per molecule and the rotational heat capacity per molecule for heteronuclear diatomic molecules in the region kBT 1632 Calculate the rotational contribution to the Helmholtz potential per molecule by evaluating the first two terms of equation 1629 explicitly and by integratmg over the remaining terms For this purpose note that the leading term in the EulerMcLaurin sum formula are 00 00 1 1 L fj 1 OdO 0 O 10 0 2 12 where f denotes the derivative of 0 1633 A particular heteronuclear diatomic gas has one vibrational mode of frequency w and its characteristic rotational energy parameter is e equation 1628 Assume no intermolecular forces so that the gas is ideal Calculate its full fundamental equation in the temperature region in which T ek B but T lzwkB 164 PROBABILITIES IN FACTORIZABLE SYSTEMS We may inquire as to the physical significance of the factor z associated with a single element in the partition sum of a factorizable macroscopic system Following equation 1617 we referred to z as the partition sum per element And in equation 1619 we saw that kBTlnz is the additive contribution of that element to the Helmholtz potential It is easily shown Problem 1641 that the probability of occupation by the ith element of its jth orbital state in a factorizable system is 1631 In all these respects the statistical mechanics of the single element is closely analogous to that of a macroscopic system The polymer model of Section 154 is particularly instructive Consider a polymer chain with a weight suspended as shown in Fig 154 The magnitude of the weight is equal to the tension Y applied to the chain The length of the chain is equation 1514 1632 and the total energy of chain plus weight in a given configuration is Problems 359 The term YLx is the potential energy of the suspended weight the potential energy being the weight Y multiplied by the height and the height being taken as zero when Lx 0 According to equation 1633 we can associate an energy aY with every monomer unit along x an energy aT with every monomer unit along x and an energy E with every monomer unit along either y or y The partition sum factors and the partition sum per monomer unit is 1634 The Helmholtz potential is given by PF Nlnz 1635 Furthermore the probability that a monomer unit is along x is 1636 and the probability that it is along x is 1637 Consequently the mean length of the chain is LJ NPx Pxa 1638 2Nasinh3az 1639 It is left to the reader to calculate the mean energy U from the f undamen tal equations 1634 and 1635 and to show that both the energy and the length agree with the results of Section 154 PROBLEMS 1641 The probability that the ith element is in its Jth orbital state is the sum of the probabilities of all microstates of the system in which the ith element is in its jth orbital state Use this fact to show that for a factorizable system the Probability of the ith element being in its jth orbital state is as given in equation 1631 1642 Demonstrate the equivalence of the fundamental equations found in this section and in Section 154 360 The Canomcaf Formafrsm Stattsllwf Mehanu m llefmholtz Representmron 165 STATISTICAL MECHANICS OF SMALL SYSTEMS ENSEMBLES The preceding sections have demonstrated a far reaching similanty between the statistical mechanics of a macroscopic system and that of an individual element of a factorizable system The partition sum per element has the same structure as the full partition sum and 1t is subject to the same probability interpretation The logarithm of the partition sum of an element is an additive contribution to the total Helmholtz potential Does this imply that we can simply apply the statistical mechanics to each element We can indeed when the elements satisfy the factorizabilav criteria of Section 162 A further conclusion can be drawn from the preceding observations We can apply the canonical formalism to small nonmacroscopc systems m diathermal contact with a thermal reservoir Suppose that we are given such a small system We can imagine it to be replicated many times over with each replica put into diathermal contact with the reservoir and hence indirectly with all other replicas The ensemble of replicas then constitutes a thermodynamic system to which statistical mechanics and thermodynamics apply Nevertheless no prop erty of the individual element is influenced by its replicas from which it 1s shielded by the intermediate thermal reservoir Application of statistical mechanics to the individual element is isomorphic to its application to the full ensemble Statistical mechanics is fully valid when applied to a single elemenr m diathermal contact with a thermal reseruoir In contrast thermodynamics with its emphasis on extensiuity of potentials applies only to an ensemble of elements or to macroscopic systems Example An atom has energy levels of energies 0 t1 t2 t3 with degeneracie of 1 2 2 l The atom is in equilibrium with electromagnetic radiation which act as a thermal reservoir at temperature T The temperature 1s such that e fl 1s negligible with respect to unity for all energies t1 with j 2 4 Calculate the mean energy and the mean square deviation of the energy from its average value Solution The partition sum is The mean energy is t 2t1ell 2t2ef3 t3ef3z and the mean squared energy is t 2 2tfell 1 2Ee 3 de f3 z Problems 361 fhe mean square deviation is E2 E2 For such a small system the mean square deviation may be very large Only for macroscopic systems are the fluctuations negligible relative to average or observed values It should be noted that an energy level with a twofold degeneracy imphes two states that have the same energy The partition sum is over states not over levels PROBLEMS 1651 The energies of the orbital states of a given molecule are such that r0 0 E1 k 8 200 K E2 k 8 300 K E3 k 8 400 K ad all th orbital states have very high energy Calculate the dispersion a E2 E2 of the energy if the molecule is in equilibrium at T 300 K What is the probability of occupation of each orbital state t652 A hydrogen atom in equilibrium with a radiation field at temperature T can be in its ground orbital level the 1s level which is twofold spin degenerate or it can be in its first excited energy level eightfold degenerate Neglect the probability of higher energy states What is the probability that the atom will be in an orbital pstate 1653 A small system has two normal modes of vibration with natural frequen cies w1 and w2 2w1 What is the probability that at temperature T the system has an energy less than 5w12 The zero of energy is taken as its value at T 0 1 xl x 21 x 2x 2 Answer where X exp 3nw1 1654 DNA the genetic molecule deoxyribonucleic acid exists as a twisted pair of polymer molecules each with N monomer units The two polymer molecules are crosslinked by N base pairs It requires energy f to unlink each base pair and a base pair can be unlinked only if it has a neighboring base palT that is already unlinked or if it is at the end of the molecule Find the probability that n pairs are unlinked at temperature T 1f a one end of the molecule is prevented from unlinking so that the molecule unwinds from one end only b the molecule can unwind from both ends Reference C Kittel Amer J Phys 37 917 1969 1655 Calculate the probability that a harmonic oscillator of natural frequency o is in a state of odd quantum number n 1 3 5 at temperature T To What values do you expect this probability to reduce in the limits of zero and Infinite temperature Show that your result conforms to these limiting values find the dominant behavior of the probability P odd near T 0 and in the high temperature region 362 The Canonical Formahsm StallM1cal MedtanicJ m Helmholtz Representation 1656 A small system has two energy levels of energies O and E and of degeneracies g0 and g1 Find the entropy of this system at temperature T Calculate the energy and the heat capacity of the system at temperature T What is the dominant behavior of the heat capacity at very low and at very high temperature Sketch the heat capacity How would this sketch be affected by an increase in the ratio g1g 0 Explain this effect qualitatively 1657 Two simple harmonic oscillators each of natural frequency w are coupled in such a way that there is no interaction between them if the oscillators have different quantum numbers whereas their combined energy is 2n lfiw 6 1f the oscillators have the same quantum number n The system is in thermal equilibrium at temperatlre T Find the probability that the two oscillators have identical quantum numbers Find and interpret the zerotemperature limit of your result for all values of 6 166 DENSITY OF STA TES AND DENSITY OF ORBITAL STA TES We return to large systems and we shall shortly demonstrate several applications of the canonical formalism to crystals and to electromagnetic radiation These applications and a wide class of other applications call on the concept of a density of states function Because this concept he outside statistical mechanics proper and because we shall find it so pervasively useful it is convenient to discuss it briefly in advance In the canonical formalism we repeatedly are called upon to compute sums of the form sum L eflE1 J 1640 The sum is over all states j of the system and is the energy of the J th state If the quantity in the parenthesis is unity the sum is the partiuon sum Z If the parenthetical quantity is the energy then the sum divided by Z is the average energy U equation 1612 And similar situations hold for other dynamical variables For macroscopic systems the energies 1 are generally but not always closely spaced in the sense that 3 E 1 E 1 Under these Ctf cumstances the sum can be replaced by an integral sum f 00 e prDE dE From 16 41 where min is the energy of the ground state of the system the minimurfl possible energy and D E is the density of states function defined bY number of states in interval dE D E dE 1642 Density of States and Density of Orbital States 363 In many systems the energy eigenstates are combinations of orbital singleelement states the partition sum factors and analogues of equa tions 1641 and 1642 can be applied to single elements The quantity analogous to DE is then a density of orbital states we shall designate it also by D E Further the orbital states are very commonly normal modes that are wavelike in character This is true of the vibrational modes of a crystal and of the electromagnetic modes of a cavity containing electromagnetic radiation From the viewpoint of quantum mechanics it is even the case for the translational modes of a gas the waves being the quantum mechanical wave functions of the molecules The density of orbital states function is then subject to certain general considerations which we briefly review Consider a system in a cubic box of linear dimension L the results are independent of this arbitrary but convenient choice of shape A standing wave parallel to an edge must have a wavelength A such that an integral number of half wavelengths fit in the length L That is the wave vector k 21TA must be of the form nlTL For a wave of general orientation in three dimensions we have similar restrictions on each of the three components of k 1643 n 1 n 2 n 3 integers We consider only isotropic media for which the frequency is a function only of the amplitude k of k w wk or inversely k kw 1644 Then the number of orbital states with frequency less than w is the number of sets of positive integers for which 1645 We can think of n n ni as the radius in an abstract space in Which n 1 n 2 and n 3 are integral distances along the three coordinate axes The number of such integral lattice points with radii less than Vlk J 1T is the volume inside this radius Only one octant of this Spherical volume is physically acceptable because n 1 n 2 and n 3 in equation 1643 must be positive Thus the number of orbital states with 364 The Canomcal Formalism Stat1st1cal Mechamcs n Helmholtz Representatwn frequency less than w is number of orbital states with frequency w k 4t v113 kw r 1646 Differentiating we find the number of orbital states D w dw in the interval dw Dw dw dk 3w dw k2w dkw dw 1647 61T 2 dw 21T2 dw The quantity Dwdw then is analogous to DEdE in the sum equation 1641 see Problem 1661 This is the general result we require Because various models of interest correspond to various functional relations wk we shal1 be able to convert sums to integrals simply by evaluating the density of orbital states function D w by equation 1641 So prepared we proceed to several applications of the canonical formalism PROBLEMS 1661 Show that the number of orbital states in the energy interval de n dw is D E D w h where D w dw is the number of orbital states in the frequency interval dw 1662 For the particles of a gas r p 22m h 22mk 2 or w eh nk2 2m Find the density of orbital states function D w Answer Dw k2 nk mJ1zvw12 21T2 m 2 1121T21312 1663 For excitations obeying the spectral relation w Akn n 0 find the density of orbital states function D w 167 THE DEBYE MODEL OF NONMETALLIC CRYSTALS At the conclusion of Section 162 we reviewed the Einstein model of a crystalline solid and we observed that the canonical formalism makes more sophisticated models practical The Debye model is moderately more sophisticated and enormously more successful It 1a FIGURE 161 The Dehye Model of Nonmetallc Crystals 365 Debye r approx1mat1on I 1a Longitudinal modes Transverse modes Dispersion relation for vibrational modes schematic The shortest wave length is of the order of the interatomic distance There are N longitudinal modes and 2N transverse modes The Debye approximation replaces the physical dispers10n relation with the linear extrapolation of the long wave length region or w vLk and w vk for longitudinal and transverse modes respectively Again consider N atoms on a lattice each atom being bound to its neigpbors by harmonic f rces springs The vibrational modes consist of N longitudinal and 2N transverse normal modes each ofwhich has a sinusoidal or wavelike structure The shortest wave lengths are of the order of twice the interatomic distance The very long wave length longitudinal modes are not sensitive to the crystal structure and they are identical to sound waves in a continuous medium The dispersion curves of w versus k 2wA are accordingly linear in the long wave length limit as shown in Fig 161 For shorter wave lengths the dispersion curves flatten out with a specific structure that reflects the details of the crystal structure P Debye 1 following the lead of Einstein bypassed the mechanical complications and attempted only to capture the general features in a simple tractable approximation The Debye model assumes that the modes all lie on linear dispersion curves Fig 161 as they Would in a continuous medium The slope of the longitudinal dispersion curve is vL the velocity of sound in the medium The slope of the transverse dispersion curve is v1 The thermodynamic implications of the model are obtained by calculat ing the partition sum The energy is additive over the modes so that the Prtition sum factorizes For each mode the possible energies are nhw A With n 1 2 3 where wA 2wvA is given by the dotted linear 1P Debye Ann Phys 39 789 1912 366 The Canonica Formafrsm Stallsllwf Mechamcs m Helmholtz Represeuatron curves in Fig 161 As in the Einstein model equations 1622 and 1623 and 1 zI 1 e Jhw z n zI n 1 eJhwl1 modes modes 1648 1649 where nmodes denotes a product over all 3N modes The Helmholtz potential is F k 8 T L lnl e Jhw 1650 modes It is left to the reader to show that the molar heat capacity is 1651 The summation over the modes is best carried out by replacing the sum by an integral 1z2 lmax w2eJhw c D w dw v k T2 Jhw 2 B O e 1652 where Dw dw is the number of modes in the interval dw To evaluate D w we turn to equation 1647 For the longitudinal modes the func tional relation k w is Fig 161 1653 and similarly for the two polarizations of transverse modes It follows from equation 1647 that D w 1 w2 2w2 vf v 1654 The maximum frequency 2 wrna is determined by the condition that the 2 In the literature max often speched in term of the Dde temperature dchnc t hwu and convcnllonall dcSJgnatcd hy 6 The Dehye Model of Nonmetal1c Crystals 367 10 08 t 06 a 2 04 u 02 I V I J J 02 04 06 08 10 12 14 16 18 20 TBv FIGURE 162 Vibrational heat capacity of a crystal according to the Debye approximation total number of modes be 3NA from which it follows that 1655 1656 Inserting D w in the integral 1652 and changing the integration variable from w to u 3 h w 1657 The molar heat capacity computed from this equation is shown schemati cally in Fig 162 At high temperature kBT hwma1J the behavior of c is best explored by examining equation 1651 In this limit u 2eu eu 12 1 Hence each mode contributes kB to the molar heat capacity a result of much more general validity as we shall see subsequently The molar heat capacity in the high temperature limit is 3NAkB or 3R At low temperature where 3hwm um 1 the upper limit in the integral in equation 1657 can be replaced by infinity the integral is then simply a constant and the temperature dependence of c arises from the u in the denominator Hence c T 3 in the low temperature region a result in excellent agreement with observed heat capacities of nonmetallic 368 The Canonical Formalism Stallstrcal Mechanrcs in llelmholtz Representatron crystals The detailed shape of the heat capacity curve in the intermediate region is less accurate of course The qualitative shape is similar to that of the Einstein model Fig 152 except that the sharp exponential rise at I01 temperature is replaced by the more gentle T 3 dependence PROBLEMS 1671 Calculate the energy of a crystal in the Debye approximation Show that the expression for U leads in turn to equation 1657 for the molar heat capacity 1672 Calculate the entropy of a crystal in the Debye approximation and show that your expression for S leads to equation 1657 for the molar heat capacity 1673 The frequency wA of the Vibrational mode of wave length A is altered 1f the crystal is mechanically compressed To describe this effect Gruneisen intro duced the Gruneisen parameter V dwA y wA dV Taking y as a constant independent of A V T calculate the mechamcal equation of state PT V N for a DebyeGruneisen crystal Show that for a DebyeGruneisen crystal VO YKTCv 168 ELECTROMAGNETIC RADIATION The derivation of the fundamental equation 357 of electromagnetic radiation is also remarkably simple in the canonical formalism Assume the radiation to be contained within a closed vessel which we may think of as a cubical cavity with perfectly conducting walls Then the energy resides in the resonant electromagnetic modes of the cavity As in the Einstein and Debye models the possible energies of a mode of frequency w are nhw with n 0 1 2 Equations 1648 and 1649 are agam valid and F kBT L ln 1 eflhw 1658 modes The sum can be calculated by replacing the sum by an integral the modes are densely distributed in energy 1659 The sole new feature here is that there is no maximum frequency such as Problems 369 that in the Debye model Whereas the shortest wavelength and therefore the largest frequency of vibrational modes in a solid is determined by the interatomic distance there is no minimum wavelength of electromagnetic waves The dispersion relation is again linear as in the Debye model and as there are two polarization modes 1660 where c is the velocity of light 2998 X 108 ms Then the fundamental equation is 1661 To calculate the energy we use the convenient identity recall equation 1613 from which aF V F TS F Tar a3F ap 1662 1663 The integral fox 3 ex 1 1 dx is 34 11415 where r IS the Rie mann zeta function 3 whence 1664 This is the StefanBoltzmann Law as introduced in equation 352 By a simple statistical mechanical calculation we have evaluated the constant b of equation 352 in terms of fundamental constants PROBLEMS 1681 Show that including the zeropoint energies of the electromagnetic modes ie En n 12iw leads to an infinite energy density UV This infinite energy density is presumably constant and unchangeable and hence Physically unobservable 3cf M Abromowitz and I A Stegun Handbook of Mathemallcal Functwns National Bureau of Standards Applied Mathematic Series No 55 1964 See equation 2327 370 The Canonical Formalism Statistical Mechanics m Helmholtz Representation 1682 Show that the energy per unit volume of electromagnetic radiation in the frequency range dw is given by the Planck Radiation Law U Fw3 fJII 1 dw e 1 dw V Ti2c 3 and that at high temperature k 8 T liw this reduces to the RayleighJeans Law 1683 Evaluating the number of photons per unit volume in the frequency range dw as NVdw UjVdwnw where U is given in problem 1682 calculate the total number of photons per unit volume Show that the average energy per photon UN is approximately 22k 8 T Note that the integral encountered can be written in terms of the Riemann zeta function as in the preceding footnote 1684 Since radiation within a cavity propagates isotropically with velocity c the flux of energy impinging on unit area of the wall or passing in one direction through an imaginary unit surface within the cavity is given by the StefanBoltzmann Law 1 1 Energy flux per unit area 4c U V 4cbT 4 o8 T 4 The factor of c 4 arises as c 2 the factor of selecting only the radiation crossing the imaginary area from right to left or vice versa and the factor of c 2 representing the average component of the velocity normal to the area element The constant 0 8 cb4 is known as the StefanBoltzmann constant As an exercise in elementary kinetic theory derive the StefanBoltzmann law explicitly demonstrating the averages described 169 THE CLASSICAL DENSITY OF STA TES The basic algorithm for the calculation of a fundamental equation in the canonical formalism requires only that we know the energy of each of the discrete states of the system Or if the energy eigenvalues are reasona bly densely distributed it is sufficient to know the density of orbital states In either case discreteness and therefore countability of the states is assumed This fact raises two questions First how can we apply statistical mechanics to classical systems Second how did Willard Gibbs invent statistical mechanics in the nineteenth century long before the birth of quantum mechanics and the concept of discrete states The Class1cal Density of States 371 As a clue we return to the central equation of the formalismthe equation for the partition sum which for a wavelike mode is equation 1647 z ePi JePDwdw JePk 2wdkw 1665 2772 We seek to write this equation in a form compatible with classical mechanics for which purpose we identify ik with the generalized momentum hk p 1666 whence z 1JePVp 2 dp 27T2h3 1667 To treat the coordinates and momenta on an equal footing the volume can be written as an integral over the spatial coordinates Furthermore the role of the energy E in classical mechanics is played by the Hamiltonian function Yix y z Px Pv Pz And finally we shift from 47Tp2 dp to dpxdpvdp 2 as the volume element in the momentum subspace whence the partition function becomes 1 f JJf z h 3 e dxdydzdpdPvdp 2 1668 Except for the appearance of the classically inexplicable prefactor lh 3 this representation of the partition sum per mode is fully classical It was in this form that statistical mechanics was devised by Josiah Willard Gibbs in a series of papers in the Journal of the Connecticut Academy between 1875 and 1878 Gibbs postulate of equation 1668 with the introduction of the quantity h for which there was no a priori classical justification must stand as one of the most inspired insights in the history of physics To Gibbs the numerical value of h was simply to be de termined by comparison with empirical thermophysical data The expression 1668 is written as if for a single particle with three µosition coordinates and three momentum coordinates This is purely symbolic The x y and z can be any generalized coordinates q1 q2 and the momenta Px Pv and Pz are then the conjugate momenta The number of coordinates and momenta is dictated by the structure of the system and more generally we can write z JeJXn dqj dpj J hl2 hl2 1669 this is the basic equation of the statistical mechanics of classical systems 372 The Canonical Formalism Statistical Mecl1amcs m Helmholtz Representaton Finally we take note of a simple heuristic interpretation of the classical density of orbital states function In the class1cal phase space coordi natemomentum space each hypercube of linear dimension h corre sponds to one quantum mechanical state It is as if the orbital states are squeezed as closely together in phase space as is permitted by the Heisenberg uncertainty principle q p h Whatever the interpretation and quite independently of the plausibility arguments of this section classical statistical mechanics is defined by equation 668 or 669 1610 THE CLASSICAL IDEAL GAS The monatomic classical ideal gas provides a direct and simple applica tion of the classical density of states and of the classical algorithm 1669 for the calculation of the partition function The model of the gas is a collection of N NNA point mass atoms in a container of volume V maintained at a temperature T by diathermal contact with a thermal reservoir The energy of the gas is the sum of the energies of the individual atoms Interactions between molecules are disbarred unless such interactions make no contribution to the energyas for instance the instantaneous collisions of hard mass points The energy is the sum of oneparticle kinetic energies and the partition sum factors We undertake to calculate zranJ the oneparticle translational partition sum and from the classical formulation 1669 we find directly that 1 J J J d d d Joo I Joo d d d Jp tp fp2m zransl h3 X Y z oo oo oo P Pl Pee 1670 It is of interest to note that we could have obtained this result by treating the particle quantum mechamcally by summing over its discrete states and by approximating the summation by an integral This exercise is left to the reader Problem 16104 Having now calculated z we might expect to evaluate Z as z N and thereby to calculate the Helmholtz potential F If we do so we find J Helmholtz potential that is not extensive We could have anticipated this impending catastrophe for the oneparticle partition function z is exten siv equation 1670 whereas we expect it to be intensive F Nk 8 T In z The problem lies not in aQ error of calculation but in a fundamental principle To identify Z as zN is to assume the particles to be The Class1ca Ideal Gas 373 distinguishable as if each bears an identifying label or number like a set of billiard balls Quantum mechanics unlike classical mechanics gives a profound meaning to the concept of indistinguishability Indistinguish ability does not imply merely that the particles are identical it re quires that the identical particles behave under interchange in ways that have no classical analogue Identical particles must obey either Fermi Dirac or BoseEinstein permutational parity concepts with statistical mechanical consequences which we shall study in greater detail in Chapter 17 Now however we seek only a classical solution We do so by recognizing that z N is the partition sum of a set of distinguishable particles We therefore attempt to correct this partition sum by division by N The rationale is that all N permutations of the labels among the N distinguishable particles should be counted as a single state for indis tinguishable particles Thus we finally arrive at the partition sum for a clasical monatornic ideal gas Z 1Nz 1nsi 1671 with z transl as calculated in equation 16 70 The Helmholtz potential is V27Tmk 8T312 F k 8 TlnZ Nk 8 Tln N h2 Nk 8 T 1672 where we have utilizedthe Stirling approximation In N N In N N which holds for large N To compare this equation with the fundamental equation introduced in Chapter 3 we make a Legendre transform to entropy representation finding SNk 8ln37Tli2mNk 8 lnU 312 VN 512 1673 This is precisely the form of the monatornic ideal gas equation with which We have become familiar The constant s0 undetermined in the thermody namic context has now been evaluated in terms of fundamental constants Reflection on the problem of counting states reveals that division by N is a rather crude classical attempt to account for indistinguishability The error can be appreciated by considering a model system of two identical Particles each of which can exist in either of two orbital states Fig 163 Classically we find four states for the distinguishable particles and we then divide by 2 to correct for indistinguishab1lity If the particles are fermions only one particle is permitted in a single oneparticle state so that there is only one permissible state of the system For bosons in 374 The Canomrnl Formalism Statistical Mechanics m Helmholtz Representation Classical counting Fermi particles Bose particles IG 01 I I 0 0 I 1 ol I IG I I 1 ol I CD I I I 0 I 0 I I I CD I Corrected number of states t42 FIGURE 163 States of a twoparticle system according to classical Fermi and Bose counting contrast any number of particles are permitted in a oneparticle state consequently there are three permissible states of the system Fig 163 Corrected classical counting is incorrect for either type of real particle At sufficiently high temperature the particles of a gas are distributed over many orbital states from very low to very high energies The probability of two particles being in the same orbital state becomes very small at high temperature The error of classical counting then becomes insignificant as that error is associated with the occurrence of more than one particle in a oneparticle state All gases approach ideal gas behamor at sufficiently high temperature Consider now a mixture of two monatomic ideal gases The partition sum is factorizable and as in equation 1671 1 1 Z Z Z zN 1 zN 1 12 N1N 1 2 1 2 1674 The Helmholtz potential is the sum of the Helmholtz potentials for the two gases The volume appearing in the Helmholtz potential of each gas is the common volume occupied by both The temperature is of course the common temperature The fundamental equation so obtained is equiv alent to that introduced in Section 34 equation 340 but again we have evaluated the constants that were arbitrary in the thermodynamic context PROBLEMS 16101 Show that the calculation of Z zN with z given by equation 1670 is correct for an ensemble of individual atoms each in a different volume V Show High Temperature Properties The Equipartton Theorem 375 that the fundamental equation obtained from Z zN is properly extensive when so interpreted 16102 Show that the fundamention equation of a multicomponent simple ideal gas which foHows from equation 1674 is identical to that of equation 340 16103 The factors lllJV 2 in equation 1674 give an additive contribu tion to the Helmholtz potential that does not depend in any way on the forms of z1 and z 2 Show that these factors lead to a mixing term in the entropy not in the Helmholtz potential of the form This mixing term appears in fluids as well as in ideal gases It accounts for the fact that the mixing of two fluids is an irreversible process recall Example 2 of Section 45 16104 Consider a particle of mass m in a cubic container of volume V Show that the separation of successive energy levels is given approximately by 6 w2h 2 2m V 213 and roughly evaluate 6 for helium atoms in a container of volume one m 3 Show that for any temperature higher than 10 8 K the quantum mechanical partition sum can be approximated well by an integral Show that this approximation leads to equation 1670 16105 A single particle is contained in a vessel of volume 2V which is divided into two equal subvolumes by a partition with a small hole in it The particle carries an electric charge and the hole in the partition is the site of a localized electric field the net effect 1s that the particle has a potential energy of zero on one side of the partition and of E on the other side What is the probability that the particle will be found in the zeropotential half of the vessel if the system is maintained in equilibrium at temperature T How would this result be affected by internal modes of the particles How would the result be affected if the dispersion relation of the particles were such that the energy was proportional to the momentum rather than to its square If the container were to contain one mole of an ideal gas noninteracting particles despite the electric charge on each What would be the pressure in each subvolume 16U HIGH TEMPERATIJRE PROPERTIES THE EQUIPARTITION THEOREM The evaluation of ztransi in equation 1670 in which z 1ransi was found to be proportional to Ti is but a special case of a general theorem of wide applicability Consider some normal mode of a systemthe mode may be translational vibrational rotational or perhaps of some other more abstract nature Let a generalized coordinate associated with the mode be q and let the associated or conjugate momentum be p Suppose the 376 The Canomwl Formalism Stattcal Mechamo III Helmholtz Representat011 energy Hamiltonian to be of the form E Aq 1 Bp 2 1675 Then the classical prescription for calculating the partition function will contain a factor of the form 1676 or as in equation 1670 if A 0 and B 0 lTkBTl2 lTkBTl2 z hA hB 1677 If either A or B is equal to zero the corresponding integral is a bounded constant determined by the limits on the associated integral The integr1 tion over x in equation 1670 is an example of such a case and the corresponding integral is V The significant result in 1677 is that at sufficiently high temperature o that the classical density of states is applicable every quadratic term in the energy contributes a factor of y to the partition function Equivalently at sufficiently high temperature every quadratic term in the energy contributes a term i N ln T to 3F or a term i Nk 8 T In T to the Helmholtz potential F or a term iNk 8 Tl In T to the entrop Or finally the result in its most immediately significant form is At sufficiently high temperature every quadratic term in the energy contribute a term iNk 8 to the heat capacity This is the equipartition theorem of classical statistical mechanics A gas of point mass particles has three quadratic terms in the energ p p p2m Theheat capacity at constant volume of such a ga at high temperature is f Nk 8 or f R per mole Application of the equipartition theorem to a gas of polyatomic mole cules is best illustrated by several examples Consider first a heteronucle1r diatorruc molecule It has three translational modes each such mode ha a quadratic kinetic energy but no potential energy these three mode contribute ik 8 to the high temperature molar heat capacity In addit10n the molecule has one vibrational mode this mode has both kinetic and potential energy both quadratic and the mode therefore contributes f kw Finally the molecule has two rotational modes ie it requires two angle to specify its orientation These rotational modes have quadratic kinetic energy but no potential energy terms they contribute kw Thus the he1t capacity per molecule rs Jk8 at high temperature or 1R per mole High Temperature Properties The I q111par111w11 TheoTlm 377 In general the total number of modes must be three times the number of atoms in the molecule This is true because the mode amplitudes are a substitute set of coordinates that can replace the set of cartesian coordi nates of each atom in the molecule The number of the latter clearly is triple the number of atoms Consider a heteronuclear triatomic molecule There are nine modes Of these three are translational modes each contributes k 8 to the heat capacity There are three rotational modes corresponding to the three angles required to orient a general obJect in space Each rotational mode has only a kinetic energy term and each contributes f k 8 to the heat capacity By subtraction there remain three vibrational modes each with kinetic and potential energy and each contributing k 8 Thus the high temperature heat capacity is 6k 8 per molecule If the triatomic molecule is linear there is one less rotational mode and therefore one additional vibrational mode The high temperature heat capacity is increased to k 8 Note that the shape of the molecule can be discerned by measurement of the heat capacity of the gas In all of the preceding discussion we have neglected contributions that may arise from the internal structure of the atoms These contributions generally have much higher energy and they contribute only at enor mously high temperature If the molecules are homonuclear indistinguishable atoms rather than heteronuclear additional quantum mechanical symmetry requirements again complicate the counting of states Nevertheless the analogous form of the equipartition theorem emerges at high temperature The classical partition function simply contains a factor of O t to account for the indistinguishability of the Jwo atoms within each of the N molecules and i contains a factor of 1 N to account for the indistinguishability of the N molecules 17 ENTROPY AND DISORDER GENERALIZED CANONICAL FORMULATIONS 171 ENTROPY AS A MEASURE OF DISORDER In the two preceding chapters we have considered two types of physical situations In one the system of interest is isolated in the other the system is in diatherrnal contact with a thermal reservoir Two very different expressions for the entropy in terms of the state probabilities l result If the system is isolated it spends equal time in each of the permissible states the number of which is 0 171 and the entropy is 17 2 If the system is in diathermal contact with a thermal reservoir the fraction of time that it spends in the state j is 173 and the entropy is U T F T which we write in the form S k 83LfiE1 k 8 lnZ 174 J We now pause to inquire as to whether these results reveal some underlying significance of the entropy Are they to be taken purely 380 E11tropv and Disorder Ge11cra1zed anonual Formufllcon formally as particular computational results or can we infer from them some intuitively revealing insights to the significance of the entropy concept In fact the conceptual framework of information theory erected hy Claude Shannon1 in the late 1940s provides a basis for interpretation of the entropy in terms of Shannons measure of disorder The concept of order or its negation disorder is qualitatively familiar A neatly built brick wall is evidently more ordered than a heap of bricks Or a hand of four playing cards is considered to be more ordered if it consists of four aces than if it contains for instance neither pairs nor a straight A succession of groups of letters from the alphabet is recognized as more ordered if each group concords with a word listed in the dictionary rather than resembling the creation of a monkey playing with a typewriter Unfortunately the heap of bricks may be the prized creation of a modern artist who would be outraged by the displacement of a single brick Or the hand of cards may be a winning hand in some unfamiliar game The apparently disordered text may be a perfectly ordered but coded message The order that we seek to quantify must be an order with respect to some prescribed criteria the standards of architecture the rules of poker or the corpus of officially recognized English words Disorder within one set of criteria may be order within another set In statistical mechanics we are interested in the disorder in the distribu tion of the system over the permissible microstates Again we attempt to clarify the problem with an analogy Let us suppose that a child is told to settle down in any room of his choice and to wait in that room until his parents return this is the rule defining order But of course the child does not stay in a single roomhe wanders restlessly throughout the house spending a fraction of time f in the th room The problem solved by Shannon is the definition of a quantitative measure of the disorder associated with a given distribution Several requirements of the measure of disorder reflect our qualitative concepts a b fc The measure of disorder should be defined entirely in terms of the set of numbers f If any one of the f is unity and all the rest consequently are zero the system is completely ordered The quantitative measure of disorder should then be zero The maximum disorder corresponds to each f being equal to 1Q that is to the child showing no preference for any of the rooms in the house among which he wanders totally randomly 1 C E Shannon and W Weaver The Mathematual Theo1r of ommumcatwm Uruv of Illmrn Prloi I Jrhn I Q4Q Entropy as a Measure of Duorder 381 d The maximum disorder should be an increasing function of 0 being greater for a child wandering randomly through a large house rather than through a small house e The disorder should compound additively over partial disorders That is let fl be the fraction of time the child spends on the first floor and let Disorder1 be the disorder of his distribution over the first floor rooms Similarly for J2 and Disorder2 Then the total disorder should be Disorder J0 X Disorder1 J2 X Disorder2 175 These qualitatively reasonable attributes uniquely determine the mea sure of disorder 2 Specifically Disorder k fi In 17 6 where k is an arbitrary positive constant We can easily verify that the disorder vanishes as required if one of the l is unity and all others are zero Also the maximum value of the disorder when each 1 10 is klnO see Problem 1711 and this does increase monotonically with O as required in d above The maximum value of the disorder k In 0 is precisely the result equation 171 previously found for the entropy of a closed system Complete concurrence requires only that we choose the constant k to be Boltzmanns constant kB For a closed system the entropy corresponds to Shannons quantitative measure of the maximum possible disorder in the distribution of the system over its permissible microstates We then turn our attention to systems in diathermal contact with a thermal reservoir for which f exp f3EZ equation 173 Inserting this value of the into the definition of the disorder equation 176 we find the disorder o be Disorder kB3 fiE1 kB In Z 17 7 Again the disorder of the distribution is precisely equal to the entropy recall equation 174 This agreement between entropy and disorder is preserved for all other boundary conditionsthat is for systems in contact with pressure reservoirs with particle reservoirs and so forth Thus we recognize that the physical interpretation of the entropy is that the entropy is the quantitative measure of the disorder in the relevant distribution of the system over Us permissible microstates 2 For a proof see A I Khinclun Matlrema11cal Fou11datwns of 111ormatwn Theory Dover Publications New York 1957 382 Entropy and Disorder Generahzed anoniwl Formulations It should not be surprising that this result emerges Our basic assump tion in statistical mechanics was that the random perturbations of the environment assure equal fractional occupation of all microstates of a closed systemthat is maximum disorder In thermodynamics the en tropy enters as a quantity that is maximum in equilibrium Identification of the entropy as the disorder simply brings these two viewpoints into concurrence for closed systems PROBLEMS 1711 Consider the quantity x In x in the limit x 0 Show by LHopitals rule that x ln x vanishes in this limit How is this related to the assertion after equation 176 that the disorder vanishes when one of the f is equal to unity 1712 Prove that the disorder defined in equation 176 is nonnegative for all physical distributions 1713 Prove that the quanbty k 1f In f is maximum if all the are equal by applying the mathematical inequality valid for any continuous convex function lx I AI ak kI Iak Give a graphical interpretation of the inequality 172 DISTRIBUTIONS OF MAXIMAL DISORDER The interpretation of the entropy as the quantitative measure of dis order suggests an alternate perspective in which to view the canonical distribution This alternative viewpoint is both simple and heuristically appealing and it establishes an approach that will be useful in discussions of other distributions We temporarily put aside the perspective of Legendre transformations and even of temperature returning to the most primitive level at which a thermodynamic system is described by its extensive parameters U V Ni Nr We then consider a system within walls restrictive with respect to V N1 Nr but nonrestrictive with respect to the energy U The values of V Ni Nr restrict the possible microstates of the system but it is evident that states of any energy consistent with V Ni Nr are permitted Nevertheless a thermodynamic measurement of the energy yields a value U This observed value is the average energy weighted by the as yet unknown probability factors 17 8 D1str1but1ons of Maximal Disorder 383 As a matter of curiosity let us explore the following question What distribution fJ maximizes the disorder subject only to the reqwrement that it yields the obseroed value of U equation 17 8 The disorder is Disorder k f ln f 17 9 J and if this is to be maximum BDisorder k 8 Lln l 1 Bf 0 1710 J Now if the f were independent variables we could equate each term in the sum separately to zero But the factors f are not independent They are subject to the auxiliary condition 178 and to the normalization condition Li 1 1711 J The mathematical technique for coping with these auxiliary conditions is the method of Lagrange multipliers 3 The prescription is to calculate the differentials of each of the auxiliary conditions L Bf 0 17 12 J 17 13 to multiply each by a variational parameter 1 and 2 and to add these to equation 1710 k 8 Llnf 1 1 2Ef 0 1714 J The method of Lagrange multipliers guarantees that each term in equation 1714 then can be put individually and independently equal to zero providing that the variational parameters are finally chosen so as to satisfy the two auxiliary conditions 17 8 and 1711 Thus for each j In f 1 1 Ji 2 E1 0 1715 3cf G Arfken Matlrema11cal Methods for Plrysmsts Academic Press New York 1960 or any similar reference on mathemaucal methods for sc1enllsts 384 Entropy and Dorder Generalzed Canomwl Formulations or 1716 We now must determine A1 and A2 so as to satisfy the auxiliary condi tions That is from 1711 and from 17 8 e11Ee2E 1 1 e1 1LE1eE U 1 1717 17 18 These are identical in form with the equations of the canonical distribu tion The quantity A2 is merely a different notation for and then from 1718 and 1612 J 1 z 1719 1720 That is except for a change in notation we have rediscovered the canonical distribution The canonical distribution is the distribution over the states of fixed V N1 Nr that maximizes the disorder subject to the condition that the average energy has its observed value This conditional maximum of the disorder is the entropy of the canonical distribution Before we turn to the generalization of these results it may be well to note that we refer to the l as probabilities The concept of probability has two distinct interpretations in common usage Objective probability refers to a frequency or a fractional occurrence the assertion that the probability of newborn infants being male is slightly less than one half is a statement about census data Subjective probability is a measure of expectation based on less than optimum information The subjective prob ability of a particular yet unborn child being male as assessed by a physician depends upon that physicians knowledge of the parents family histories upon accumulating data on maternal hormone levels upon the increasing clarity of ultrasound images and finally upon an educated but still subjective guess The Grand Canonical Formalism 385 The disorder a function of the probabilities has two corresponding interpretations The very term disorder reflects an objective interpretation based upon objective fractional occurrences The same quantity based on the subjective interpretation of the fs is a measure of the uncertainty of a prediction that may be based upon the fs If one f is unity the uncertainty is zero and a perfect prediction is possible If all the f are equal the uncertainty is maximum and no reliable prediction can be made There is a school of thermodynamicists 4 who view thermodynamics as a subjective science of prediction If the energy is known it constrains our guess of any other property of the system If only the energy is known the most valid guess as to other properties is based on a set of probabilities that maximize the residual uncertainty In this interpretation the maximi zation of the entropy is a strategy of optimal prediction To repeat we view the probabilities f as objective fractional occur rences The entropy is a measure of the objective disorder of the distribu tion of the system among its microstates That disorder arises by virtue of random interactions with the surroundings or by other random processes which may be dominant PROBLEMS 1721 Show that the maximum value of the disorder as calculated in this section does agree with the entropy of the canonical distribution equation 174 1722 Given the identification of the disorder as the entropy and of as given in equation 1716 prove that A2 1k 8 T equation 1719 173 THE GRAND CANONICAL FORMALISM Generalization of the canonical formalism is straightforward merely substituting other extensive parameters in place of the energy We il 1ustrate by focusing on a particularly powerful and widely used formalism known as the grand canonical formalism Consider a system of fixed volume in contact with both energy and particle reservoirs The system might be a layer of molecules adsorbed on a surface bathed by a gas Or it may be the contents of a narrow necked but open bottle lying on the sea floor Considering the system plus the reservoir as a closed system for which every state is equally probable we conclude as in equation 161 that the fractional occupation of a state of the system of given energy E1 and mole 4cf M Tribus Thermostatstcs and Thermodynamus D Van Nostrand and Co New York 1961 E T Jaynes Papers 011 Probability Statistics and Stattstcal Phscs Edited by R D Rosenkrantz D Reidel Dordrecht and Boston 1983 386 Entropy and Disorder Generalized Canonical Formulations number is 1721 But again expressing O in terms of the entropy h exp 8 sres Etotal E1 NtotaI 8 s 101 Etotal Nto1a1 l 1722 Expanding as in equations 163 to 165 where 1t is the grand canonical potential 1t U TS µN U T µ The factor eP plays the role of a normalizing factor where Z the grand canonical partition sum is LePE1µN J 1723 1724 17 25 1726 The algorithm for calculating a fundamental equation consists of evaluating the grand canonical partition sum Z as a function of T and µ and implicitly as a function also of V Then 111 is simply the logarithm of Z This functional relationship can be viewed in two ways summarized in the mnemonic squares of Fig 171 The conventional view is that ltT V µ is the Legendre transform of U or 11 T V µ U T µ The thermodynamics of this Legendre trans formation is exhibited in the first mnemonic square of Fig 171 It is evident that this square is isomorphic with the familiar square merely replacing the extensive parameter V by N and reversing the correspond ing arrow The more fundamental and far more convenient view is based on Massieu functions or transforms of the entropy Section 54 The second and third squares exhibit this transform the third square merely alters the scale of temperature from T to k 8 T or from 1T to 3 The logarithm of the grand canonical partition sum Z is the Massieu transform 311 N T u s J µ Uµ u µIT SµIT llF l pIJI Slµ lµ FIGURE 171 The Grand Canonical Formalism 387 i UTµJ U TS µN ai ai aT s aµ N a 11TL U alT aiT N aµT 31 S33µ S 3U 3µN api u ap a31 N a3µ Mnemonic squares of the grand canonical potential A particularly useful identity which follows from these relationships is U a 311 a In Z a13 a13 lµ 1727 This relationship also follows directly from the probability interpretation of the see Problem 1731 In carrying out the indicated differentiation after liaving calculated 2 or 311 we must pair a factor 3 with every factor µ and we then maintain all such 3µ products constant as we differentiate with respect to the remaining 3 s Before illustrating the application of the grand canonical formalism it is interesting to corroborate that it too can be obtained as a distribution of 388 Entropy and Disorder Generalized Canomcal Fornrulatwm maximal disorder We maximize the disorder entropy 1728 subject to the auxiliary conditions that 17 29 1730 and 17 31 Then BS k 8 Llnl 1 Bl 0 17 32 J Taking differentials of equations 1729 to 1731 multiplying by Lagrange multipliers Ai A 2 and A3 and adding L0nl 1 A1 2 1 A3 0 J 1733 Each term then may be equated separately to zero as in equation 1715 and 1734 The Lagrange multipliers must now be evaluated by equations 1729 to 1731 Doing so identifies them in terms of 3 A2 3µ A 3 and 31r 1 A1 again establishing equation 1723 It should be nted that he mole number can be replaced by the particle number N1 where X Avagadros number In that caseµ the Gibbs potential per mole is replaced by the Gibbs potential per particle Although a rational notation for the latter quantity would be µ we shall henceforth write µ for either the Gibbs potential per mole or the Gibbs potential per particle permitting the distinction to be established bl the context Example Molecular Adsorption on a Surface Consider a gas in contact with a solid surface The molecules of the gas can adsorb on specific sites on the surface the sites bemg determined by the The Grand Canonual Formahsm 389 molecular structure of the surface We assume for simplicity that the sites are sparsely enough distributed over the surface that they do not directly interact There are N such sites and each can adsorb zero one or two molecules Each site has an energy that we take as zero if the site is empty as e1 if the site is singly occupied and as e2 if the site is doubly occupied The energies e1 and e2 may be either positive or negative positive adsorption energies favor empty sites and negative adsorption energies favor adsorption The surface is bathed by a gas of temperature T and pressure P and of sufficiently large mole number that it acts as a reservoir with respect to energy and particle number We seek the fractional coverage of the surface or the ratio of the number of adsorbed molecules to the number of adsorption sites The solution of this problem by the grand canonical potential permits us to focus our attention entirely on the surface sites These sites can be populated by both energy and particles which play completely analogous roles in the for malism The gaseous phase which bathes the surface establishes the values of T and p being both a thermal and a particle reservoir The given data may be and generally is unsymmetric specifying T and P of the gas rather than T and p In such a case p the Gibbs potential per particle of the gas must first be evaluated from the fundamental equation of the gas if known or from integration of the GibbsDuhem relation if the equations of state are known We assume that this preliminary thermodynamic calculation has been carried out and that T and p of the gas are specified Thenceforth the analysis is completely symmetric between energy and particles Because the surface sites do not interact the grand partition sum factors The grand partition sum for a single site contains just three terms correspond ing to the empty the single occupied and the doubly occupied states Each of the three term in z divided by z is the probability of the corresponding state Thus the mean number of molecules adsorbed per site is e fJ µJ 2eJu 2µl n z and the mean energy per site is e1e J µ e2ef3 µ e z An alternative route to these latter two results and to the general thermody namics of the system is via calculation of the grand canonical potential 1t 390 Entropy and Disorder Generaflzed Canonical Formulations k 8 TlogZequation 1725 qr Nk8 Tlog 1 eJ eP 2 The number N of adsorbed atoms on the N sites is obtained thermodynamically by differentiation of 1 air N aµ and of course such a differentiation is equal to Nn with ii as previously found Similarly the energy of the surface system is found by equation 1727 and this gives a result identical to NE The reader is strongly urged to do Problem 17 34 PROBLEMS 1731 Calculate alogaP 13 directly from equation 1726 and show that the result is consistent with equation 1727 1732 A system is contained in a cylinder with diathermal impermeable walls fitted with a freely moveable piston The external temperature and pressure are constant Derive an appropriate canonical formalism for this system Identify the logarithm of the corresponding partition sum 1733 For the surface adsorption model of the preceding Example investigate the mean number of molecules adsorbed per site ii in the limit T 0 for all combinations of signs and relative magnitudes of E 1 p0 and E 2 2µ 0 where p0 is the value of the µ of the gas at T 0 Explain these results heuristically 1734 Suppose the adsorption model to be augmented by assuming that two adsorbed molecules on the same site interact in a vibrational mode of frequency w Thus the energy of an empty site is zero the energy of a singly occupied site is E1 and the energy of a doubly occupied site can take any of the values E2 nhw with n 0 1 2 Calculate a The grand canonical partition sum b The grand canonical potential c The mean occupation number as computed directly from a d The mean occupation number as computed directly from b e The probability that the system is in the state with n 2 and n 3 Answer Denoting E1 µ by E b 1 Nk8 Tln 1 eP 1 c d 1 e 3hwef3 2ef3 n 1 e 3llwl e3 efl e Problems 39 l 1735 Calculate the fundamental equation of the polymer model of Section 154 in a formalism canonical with respect to length and energy Note that the weight in Fig 154 plays the role of a tension reservoir Also recall Problem 1732 the results of which may be helpful if the volume there is replaced by the length as an extensive parameter as if the two transverse dimensions of the system are formally taken as constant 1736 A system contains N sites and N electrons At a given site there is only one accessible orbital state but that orbital state can be occupied by zero one or two electrons of opposite spin The site energy is zero if the site is either empty or singly occupied and it is e if the site is doubly occupied In addition there is an externally applied magnetic field which acts only on the spin coordinates a Calculate the chemical potential µ as a function of the temperature and the magnetic field b Calculate the heat capacity of the system c Calculate the initial magnetic susceptibility of the system ie the magnetic susceptibility in small magnetic field 1737 Carbon monoxide molecules CO can be adsorbed at specific sites on a solid surface The oxygen atom of an adsorbed molecule is immobilized on the adsorption site the axis of the adsorbed molecule thereby is fixed perpendicular to the surface so that the rotational degree of freedom of the adsorbed molecule is suppressed In addition the vibrational frequency of the molecule is altered the effective mass changing from the reduced mass mcm 0 mc m 0 to me Only one molecule can be adsorbed at a given site The binding energy of an adsorbed molecule is Eb The surface is bathed by CO gas at temperature T and pressure P Calculate the fraction of occupied adsorption sites if the system is in equilibrium Assume the temperature to be of the order of one or two hundred Kelvin and assume the pressure to be sufficiently low that the CO vapor can be regarded as an ideal diatomic gas Hint Recall the magnitudes of characteristic rotational and vibrational frequen cies as expressed in equivalent temperatures in Section 163 ARABESQUE 18 QUANTUM FLUIDS 181 QUANTUM PARTICLES A FERMION PREGAS MODEL At this point we might be tempted to test the grand canonical for malism on the ideal gas not to obtain new results of course but to compare the analytic convenience and power of the various formalisms Remarkably the grand canonical formalism proves to be extremely uncongenial to the classical ideal gas model The catastrophe of nonexten sivity that plagued the calculation in the canonical formalism becomes even more awkward in the grand canonical formalism 1 As so often happens in physics the formalism points the way to reality The awkwardness of the formalism is a signal that the model is unphysical that there are no classical particles in nature There are only fermions and bosons two types of quantum mechanical particles For these the grand canonical formalism becomes extremely simple Fermions are the quantum analogues of the material particles of classi cal physics Electrons protons neutrons and a panoply of more esoteric particles are fermions The nineteenth century law of impenetrability of matter is replaced by an antisymmetry condition on the quantum mecha nical wave function 2 This condition implies as the only consequence of which we shall have need that only a single fermion can occupy a given orbital state Bosons are the quantum analogues of the waves of classical physics Photons the quanta of light are typical bosons Just as waves can be freely superposed classically so an arbitrary number of bosons can occupy a single orbital state Furthermore there exist bosons with zero rest masssuch bosons like classical waves can be freely created or annihi 1The root of the difficulty lies in the fact that the grand canonical formalism focusses not on the particles but on the orbital states There is then no natural way to count the states as if the particles had labels later to be corrected by division by lV 2 The wave function must be antisymmetric under interchange of two fenmons thereby mterposmg a node between the fermions and preventing two fermions of the same pm state from occupying the same spatial position 394 Quantum Fluids lated The radiation of electromagnetic waves by a hot body is described in quantum terminology as the creation and emission of photons The fundamental particles in nature possess intrinsic angular momen tum or spin The immutable magnitude of this intrinsic angular momentum 1s necessarily a multiple of h2 those particles with odd multiples of h2 are fermions and those with even multiples of h2 are bosons The orientation of the intrinsic angular momentum is also quantized For fermions of spin angular momentum h12 the angular momentum can have either of two orientations along any arbitrarily designated axis These two orientations are designated by up and down or by the two values ms and ms of the magnetic quantum number ms Finally an orbital state of a quantum particle is labeled by the quantum numbers of its spatial wave function and by the magnetic quantum number ms of its spin orientation For a particle in a cubic container the three spatial quantum numbers are the three components of the wave vector k recall equation 1637 so that an orbital state is completely labeled by k and ms Preparatory to the application of the grand canonical formalism to Fermi and Bose ideal gases it is instructive to consider a simpler model that exhibits the physics in greater clarity This model has only three energy levels so that all summations over states can be exhibited ex plicitly Except for this simplification the analysis stands in strict step by step correspondence with the analysis of quantum gases to be developed in the following sections hence the name pregas model We consider first the spin fermion pregas model The model system is such that only three spatial orbits are permitted particles in these spatial orbits have energies e e2 and e3 The model system is in contact with a thermal reservoir and with a reservoir of spin Fermi particles the reservoirs impose fixed values of the temperature T and of the molar Gibbs potential µ which for fermion systems is also known as the Fermi level Each spatial orbit corresponds to two orbital states one of spin up and one of spin down There are therefore six orbital states which can be numbered n ms with n 1 2 3 and ms The grand canonial partition sum factors with respect to the six orbital states 181 and each orbital state partition sum has two terms corresponding to the state being either empty or occupied In the absence of a magnetic field 182 Quantum Particles A Fermwn PreGas Model 395 Alternatively we can pair the two orbital states with the same n but with ms t 183 This product can be interpreted in terms of the four states of given n the empty state two singly occupied states and one doubly occupied state The probability that the orbital state n ms is empty is lznm and the probability that it is occupied is 184 The fundamental equation follows directly from equations 181 to 183 We can find the mean number of particles in the system by differentia tion N BY aµ Alternatively we can sum the probability of oc cupation In m over all six orbital states The entropy of the system can be obtained by differentiation of the fundamental equation S BY BT Alternatively it can be calculated from the occupation probabilities Problem 1811 The energy is found thermodynamically by differentiation U BJY BJp equation 1727 Alternatively from the probability inter pretation or1 m 2E1 2E2 2E3 U LE mm fll11 efll21 e31ll ntn e 187 If the system of interest is actually in contact with T and µ reservoirs these results are in convenient form But it may happen that the physical system that we wish to describe is enclosed in nonpermeable walls that impose constancy of the particle number N rather than of µ Nevertheless the fundamental equation is an attribute of the thermodynamic system independent of boundary conditions so that the preceding formalism re 396 Quantum Ffwds t f 09 08 07 06 05 04 03 02 II 1111 1111 1111 1111 IIII llllillll 1111 1111 11111111 1111 1111 1111 1111 1111 1111 1111 II I I Ffttttr1 f f f f t t r T µ 01 r Jl 1111111111111111 11111111 llll llll 1111 llllllllr111111111111 I O O O 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 1 1 1 2 1 3 1 4 15 1 6 1 7 1 8 1 9 2 0 tiµ FIGURE 181 The probability of occupation by a fermion of an orbital state of energy Eat temperature T mains valid However the Fermi level µ is not a known quantity Instead the value of µ adjusts to a change in temperature in such a way as to maintain N constanta response governed by equation 186 Unfortunately equation 186 does not lend itself easily to explicit solution for µ as a funchon of T and N However the solution can be obtained numerically or by series expansions in certain temperature regions as we shall soon see It is instructive first to reconider the preceding analysis in more pictorial terms The occupation probability f of an orbital state of energy as given by equation 184 is shown in Fig 181 This occupation probability is more general than the present model of course It applies to any orbital state of a fermion In the limit of zero temperature any state of energy µ is occupied and any state of energy µ is empty As the temperature is raised the states with energies slightly less than µ become partially depopulated and the states with energies slightly greater than µ become populated The range of energies within which this population transfer occurs is of the order of 4k 8 T see Problems 1814 1815 1816 The probability of occupation of a state wllh energy equal to µ is always one half and a plot off t as a function of such as in Fig 181 is symmetnc under mversion through the point µ f see Problem 1816 Ill 2 3µ Quantum Particles A Fermion PreGa Model 397 3 FIGURE 18 2 The Bose mean occupation number n of an orbital state of energy f at given T and µ The insert is schematic for T2 T1 and µ2 µ1 With these pictorial insights we can explore the dependence ofµ on T for the fermion pregas model For defimteness suppose the system to contain four fermions Furthermore suppose that two of the energy levels coincide with 11 12 and with 13 12 At T 0 the four fermions fill the four orbital states of energy 11 12 and the two states of energy 1o3 are empty The Fermi level must lie somewhere between 12 and 13 but the precise value of µ must be found by considering the limiting value as T 0 For very low T f I ef3µ ef3 µ I I ef3µ for 1 µ and T 0 for E µ and T 0 188 398 Quanrum Fluids Thus if 1 2 3 and N 4 equation 186 becomes for T 0 189 or 1810 In this case µ is midway between 1 and 3 at T 0 and µ increases linearly as T increases It is instructive to compare this result with another special case in which 1 2 3 If we were to have four fermions in the system the Fermi level µ would coincide with 2 at T 0 More interesting is the case in which there are only two fermions Then at T 0 the Fermi level lies between 1 and 2 3 We proceed as previously Equation 189 is replaced for T 0 by 1811 and 1812 In each of the cases the Fermi level moves away from the doubly degenerate energy level The reader should visualize this effect in the pictorial terms of Fig 181 recognizing the centrality of the inversion symmetry off relative to the point at µ From these several special cases it now should be clear that the general principles that govern the temperature dependence of µ for a system of constant N are a The occupation probability departs from zero or unity over a region of J 2k 8 Taround µ b As T increases the Fermi level µ is repelled by high densities of states within this region PROBLEMS 1811 Obtain the mean number of particles in the fermion pregas model by differentiating I as given in equation 185 Show that the result agrees with N as given in equation 186 1812 The entropy of a system is given by S k 8L 1f Inf where f is the probability of a microstate of the system Each microstate of the fermion prega The Ideal Ferm Fluid 399 model is described by specifying the occupation of all six orbital states a Show that there are 26 64 possible microstates of the model system and that there are therefore 64 terms in the expression for the entropy b Show that this expression reduces to S kBLfnmlnfnm nm and that this equation contains only six terms What special properties of the model effect this drastic reduction 1813 Apply equation 1727 for U to the fundamental equation of the fermion pregas model and show that this gives the same result for U as in equation 187 1814 Show that dde 34 ateµ With this result show that f falls to f 025 at approximately E µ k BT and that f rises to f 075 at approxi mately e µ k BT check this result by Fig 181 This rule of thumb gives a qualitative and useful picture of the range of e over which f changes rapidly 1815 Show that Fig 172 of T as a function of e is symmetric under inversion through the point E µ f That is show that e T is subject to the symmetry relation µ 6 T 1 µ 6 T or E T 2µ E T and explain why this equation expresses the symmetry alluded to 1816 Suppose f e T is to be approximated as a function of E by three linear regions as follows In the vicinity of E µ f e µ is to be approximated by a straight line going through the point E µ f and havmg the correct slope at that point For low E e µ is to be taken as unity And at high e e µ is to be taken as zero What is the slope of the central straight line section What is the width in energy units of the central straight line section Compare this result with the rule of thumb given in Problem 1814 182 THE IDEAL FERMI FLUID We tum our attention to the ideal Fermi fluid a model system of Wide applicability and deep significance The ideal Fermi fluid is a quantum analogue of the classical ideal gas it is a system of fermion Particles between which there are no or negligibly small interaction forces Conceptually the simplest ideal Fermi fluid is a collection of neutrons and such a fluid is realized in neutron stars and in the nucleus of heavy atoms as one component of the neutronproton twocomponent fluid 400 Quantum Fluuls Composite particles such as atoms behave as fermion particles if they contain an odd number of fermion constituents Thus heliumthree 3He atoms containing two protons one neutron and two electrons behave as fermions Accordingly a gas of 3He atoms can be treated as an ideal Fermi fluid In contrast 4He atoms containing an additional neutron behave as bosons The spectacular difference between the proper ties of 3He and 4He fluids at low temperatures despite the fact that the two types of atoms are chemically indistinguishable is a striking con firmation of the statistical mechanics of these quantum fluids Electrons in a metal are another Fermi fluid of great interest to which we shall address our attention in Section 184 We first consider the statistical mechanics of a general idea Fermi fluid The analysis will follow the pattern of the fermion pregas model of the preceding section Since the number of orbital states of the fluid is very large rather than being the mere six orbital states of the pregas model summations will be replaced by integrals But otherwise the analyses stand in strict step by step correspondence To calculate the fundamental relation of an ideal fermion fluid we choose to consider it as being in interaction with a thermal and a particle reservoir of temperature T and electrochemical potential µ We stress again that the particular system being studied in the laboratory may have different boundary conditionsit may be closed or it may be in di athermal contact only with a thermal reservoir and so forth But thermo dynamic fundamental relations do not refer to any particular boundary condition and we are free to choose any convenient boundary condition that facilitates the calculation We choose the boundary conditions ap propriate to the grand canonical formalism The orbital states available to the fermions are specified by the wave vector k of the wave function recall equation 1643 and by the orienta tion of the spin up or down for a spin fermion The partition sum factors over the possible orbital states 1813 where ms can take two values ms implying spin up and m implying spin down Each orbital state can be either empty or singly occupied The energy of an empty orbital state is zero and the energy of an occupied orbital state k ms is p2 1i2k2 ekm 2m 2m independent of mJ 1814 so that the partition sum of the orbital state k ms is 1815 It is conventional to refer to the product zk 112 zk 112 as zk the The Ideal Fermi Fluid 401 partition sum of the mode k ni 2eJh2k22mµ eJ212k22m2µ1 1816 The three terms refer then to the totally empty mode to the singly occupied mode with two possible spin orientations and to the doubly occupied mode with one spin up and one down Each orbital state k m 5 is independent and the probability of occupa tion is 1817 This function is shown in Fig 181 At this point we can proceed by either of two routes The fundamental algorithm instructs us to calculate the grand canonical potential k 8 T In Z thereby obtaining a fundamental relation Alternatively we can calculate all physical quantities of interest directly from equation 1817 We shall first calculate the fundamental relation and then return to explore the parallel information available from knowledge of the orbitalstate distribution function f k The grand canonical potential is Y k 8 TLzk k 8 TL In 1 efJ 2 2 2mµ 2 1818 k k The density of orbital states of a single spin orientation is De de which has been calculated in Equation 1647 V dk V 2m 312 De de k 2 de e112 de 2w2 de 4iT2 tz2 1819 Inserting a factor of 2 to account for the two possible spin orientations can then be written as kBTf2m32i12Jnl efJeµldE 2T2 h 2 0 1820 Unfortunately the integral cannot be evaluated in closed form Quantities of direct physical interest obtained by differentiation of must also be 402 Quantum Fluids expressed in terms of integrals Such quantities can be calculated to any desired accuracy by numerical quadrature or by various approximation schemes In principle the statistical mechanical phase of the problem is completed with equation 1820 It is of interest to calculate the number of particles N in the gas By differentiation of 11 aq 100 1 N a 2 fJ DE dE µ 0 e Ep 1 V 2m 32 oo E12 dE 27T 2 lz2 i efJµ 1 1821 The first form of this equation reveals most clearly that it is identical to a summation of occupation probabilities over all states Similarly the energy obtained by differentiation is identical to a summation of Ej over all states ua311 21 00 E DEdE a3 Jp Q JEµ 1 V 2m 32 oo E32 27T 2 2 fo ePµI 1 dE 1822 A flowchart for the statistical mechanics of quantum fluids is shown in Table 181 Bose fluids are included although we shall consider them explicitly only in later sections The analysis differs only in several changes in sign as will emerge in Section 185 Before exploring these general results in specific detail it is wise to corroborate that for high temperature they do reduce to the classical ideal gas and to explore the criterion that separates the classical from the quantum mechanical regime PROBLEMS 1821 Prove equations c g h i and j of Table 181 for fermions only 183 THE CLASSICAL LIMIT AND THE QUANTUM CRITERION The hallmark of the quantum regime is that a fermion particle is net free to occupy any arbitrarily chosen orbital state for some states may already be filled However at low density or high temperature the prob ability of occupation of each orbital state is small thereby minimizing the The Classical Limit and the Quantum Criterion 403 TABLE 181 Statiftical Mechanics of Quantum Fluids The upper sign refers to fermions and the lower to bosons a The partition sum factors The number of spin orientations is g0 2S 1 g 0 1 for bosons of spin zero g0 2 for fermions of spin etc b zk is the partition sum of a single orbital state of definite k and ms c km is the mean occupation number or occupation probability of the orbital state k ms d e and f DE is the density of orbital states of a single spin orientation g lrTµ is a fundamental relation h i and j P P U V is an equation of state common to both fermion and bosons a b c 31 lnZ g0 lnzk g 0 lnl ell 11 d k k e DE v 2m i2 32 2w2 n2 f Integrating by parts 2 goV 2m 1121 f12 dE 3 2w 2 li 2 o ellµJ 1 Fundamental Equation g Note 21 00 2 11 3 0 EEgoDede 3v h Also 11PV for simple systems i P2 U 3 V equation of state j effect of the fermion prohibition against multiple occupancy All gases become classical at low density or high temperature in which conditions relatively few particles are distributed over many states The probability of occupancy of a state of energy E is ellµ 1r1 and this is small for all E if eflµ is large or if the fugacity ellµ is small ell 1 classical regime 1823 404 Quanrum Fluids In this classical regime the occupation probability reduces to 1824 In terms of Fig 181 the classical region corresponds to the recession of the Fermi level µ to such large negative values that all physical orbitals lie on the tail of the E T curve We first corroborate that the occupation probability of equation 1824 does reproduce classical results and we then explore the physical condi tion that leads to a small fugacity The number of particles N is expressed by equation 1821 which for small fugacity becomes 1825 where X T a quantity to be given a physical interpretation momentarily is defined by 1826 and where g0 2S I is the number of permissible spin orientations equal to two for the spin t case Similarly the energy as expressed in equation 1762 becomes 1827 Dividing U 3 Nk T 2 B 1828 This is the wellknown equation of state of the classical ideal gas In addition the individual equations 1825 and 1827 can be corroborated as valid for the classical ideal gas With the reassurance that the Fermi gas does behave appropriately in the classical limit we may inquire as to the criterion that divides the quantum and classical regimes It follows from our discussion that this division occurs when the fugacity is of the order of unity classicalquantum boundary 1829 The Strong Quantum Regm Elellrons 111 a Metal 405 or from equation 1825 classicalquantum boundary 1830 This quantum criterion acquires a revealing pictorial interpretation when we explore the significance of Ar In fact Ar is the quantum mechanical wave length of a particle with kinetic energy k 8 T see Problem 1832 whence Ar is known as the thermal wave length From equa tion 1825 we see that in the classical limit the fugacity is the ratio of the thermal volme A3r to the volume per particle of a single spin orien tation V N g 0 The system is in the quantum regime if the thermal volume is larger than the actual volume per partzcle of a single spin orientation either by virtue of large N or by virtue of low T and conse quently of large Ar PROBLEMS 1831 Calculate the definite integrals appearing in equations 1825 and 1826 by letting E x 2 and noting that each of the resulting integrals is the derivative with respect to f3 of a simpler integral 1832 Validate the interpretation of A7 as the thermal wavelength by identi fying the wavelength with the momentum p by the quantum mechanical defini tion p hA and by comparing the energy p 22m to k 8 T 184 THE STRONG QUANTUM REGIME ELECTRONS IN A METAL The electrons in a metal would appear at first thought to be a very poor example of an ideal Fermi fluid for the charges on the electrons ostensibly imply strong interparticle forces However the background positive charges of the fixed ions tend to neutralize the negative charges of the electrons at least on the average And the very long range of the Coulomb force ensures that the average effect is the dominant effect for the potential at any point is the resultant of contributions from enor mously many electrons and positive ionssome nearby and many further removed in space All of this can be made quantitative and the accuracy of the approximation can be estimated and controlled by the methodology of solid state physics We proceed by simply accepting the model of electrons in a metal as an ideal fermion ga on the basis of the slender plausibility of these remarks 406 Quantum Flwds An estimate of the Fermi level to be made shortly will reveal that for all reasonable temperatures µ k 8 T Thus electrons in a metal are an example of an ideal Fermi gas in the strong quantum regime The analysis of this section is simply an examination of the Fermi gas in this strong quantum regime with the allusion to electrons in a metal only to provide a physical context for the more general discussion Consider first the state of the electrons at zero temperature and denote the value of the Fermi level at T 0 as µ 0 the Fermi energy The occupation probability f is unity for µ 0 and is zero for µ 0 so that from equation 1821 1831 or f 2 fe23 Jo 2m 3w V 1832 The number of conduction electrons per unit volume in metals is of the order of 1022 to 1023 electronscm 3 corresponding to one or two elec trons per ion and an interionic distance of 5 A Consequently for electrons in metals the Fermi energy µ0 or the Fermi temperature µ0k 8 is of the magnitude 1833 For other previously cited Fermi fluids the Fermi temperature may be even higherof the order of 109 K for the electrons in white dwarf stars or 1012 K for the nucleons in heavy atomic nuclei and in neutron stars The enormously high Fermi temperature implies that the energy of the electron gas is correspondingly high The energy at zero temperature is 1834 Thus the energy per particle is iµ0 or approximately 104 Kin equivalent temperature units As the temperature rises the Fermi level decreases being repelled by the higher density of states at high energy as we observed in the fetmion pregas model of Section 181 Furthermore some electrons are pro moted from orbitals below µ to orbitals above µ increasing the energy of the system To explore these effects quantitatively it is convenient to The Strot1g Quat1tum Regime Electrom ma Metal 407 invoke a general result for integrals of the form I T dE where f1 is an arbitrary function and T is the Fermi occupation prob ability This integral can be expanded in a power series in the temperature by invoking the stepfunction shape of T at low temperatures Problem 1842 giving 7714 k T4 360 B p µ 1835 where q and q are the first and third derivatives of q with respect to evaluated at E µ It should be noted that µ is the temperature depen dent Fermi level not the zerotemperature Fermi energy µ 0 We first find the dependence of the Fermi energy on the temperature The Fermi energy is determined by equation 1821 oo V 2m11 oo N 2 i TDE dE 2 2 i 1111 T dE 0 271 1z 0 1836 Then taking qE 12 in equation 1835 V2 32 2kT2 l m 31 71 B N µII 3712 1z2 8 µ 1837 At zero temperature we recover equation 1832 for µ 0 To carry the solution to second order in T it is sufficient to replace µ by µ 0 in the secondorder term whence 1838 This result corroborates our expectation that the Fermi level decreases with increasing temperature But for a typical value of µ 0k 8 on the order of 104 K the Fermi level at room temperature is decreased by only around 01 from its zerotemperature value The energy is given in an identical fashion merely replacing 112 by 3 2 giving 2 32 k T1 l V m s 2 5 2 B U p I 1 71 571 2 h 2 8 µ 1839 408 Quantum Fuuls Comparison with equation 1832 corroborates that at T 0 we recover the relationship U iNµ 0 equation 1834 This suggests dividing equa tion 1839 by equation 1837 giving 1840 Replacingµ T by equation 1838 we finally find 3 5 2k8T 2 l U Nµ 1 w 5 12 µo 1841 and the heat capacity is C INk 2 kBT OT3 2 B 3 Jo 1842 The prefactor tNk 8 is the classical result and the factor in parentheses is the quantum correction factor due to the quantum properties of the fermions The quantum correction factor is of the order of fo at room temperature for µ 0k 8 104 K This drastic reduction of the heat capacity from its classically expected value is in excellent agreement with experiment for essentially all metals In order to compare the observed heat capacity of metals with theory it must be recalled Section 166 that the lattice vibrations also contribute a term proportional to T3 in addition to the linear and cubic terms contributed by the electrons C AT BT 3 1843 The coefficient A is equal to the coefficient in equation 1842 whereas B arises both from the cubic tem1s in equation 1842 and predominately from the coefficient in the Debye theory It is conventional to plot experimental data in the form CT versus T2 so that the coefficient A i obtained as the T 0 intercept and the coefficient B is the slope of the straight line In fact such plots of experimental data do give excellent straight lines with values of A and B in excellent agreement with equation 1842 and the Debye theory 1651 The heat capacity 18 42 can be understood semiquantitatively and intuitively As the temperature rises from T 0 electrons are promoted from energies just below µ 0 to energies just above µ 0 This population Prohlems 409 transfer occurs primarily within a range of energies of the order of 2 k 8 T recall Fig 181 and Problem 1817 The number of electrons so pro moted is then of the order of Dµ 02k 8 T and each increases its energy by roughly k 8 T Thus the increase in energy is of the order of 1844 But Dµ 0 3N 2µ 0 so that 1845 and 1846 This rough estimate is quite close to the quantitative result calculated in equation 1842 which merely substitutes 1T23 for the factor 2 in the parentheses of equation 1846 PROBLEMS 1841 Show that equation 1832 can be interpreted as µ 0 li2k2m where kF is the radius of the sphere in kspace such that one octant contains 2N particles recall Section 166 Why 2N rather than N particles 1842 Derive equation 1835 by the following sequence of operations a Denoting the integral in equation 1835 by first integrate by parts and let tl fcqE de Then expanding tl E in a power series in E µ to third order show that with I 1ooE µdf dE 13mJoo e xdx o dE Jµex12 b Show that only an exponentially small error is made by taking the lower limit of integration as oo and that then all terms with m odd vanish c Evaluate the first two nonvanishing terms and show that these agree with equation 1835 40 Quantum Fluids 185 THE IDEAL BOSE FLUID The formalism for the ideal Bose fluid bears a strikingly close similarity to that for the ideal Fermi fluid As was anticipated in Table 181 and as we shall validate here the formalisms differ only in several changes in sign But the consequences are dramatically different Whereas fermions at low temperatures tend to saturate orbital states up to some specific Fermi energy bosons all tend to condense into the single lowest orbital state This condensation happens precipitously at and below a sharply defined condensation temperature The resultant phase transition leads to superfluidity in 4He a phenomenon not seen in 3He which is a fermion fluid and it leads to superconductivity in lead and in various other metals We consider an ideal Bose fluid composed of particles of integral spin The number of spin orientations is then g0 2S 1 where S is the magnitude of the spin The possible orbital states of the bosons in the flmd are labeled by k and m precisely as in the fermion case and again the grand canonical partition sum factors with respect to the orbital states as in line a of Table 181 The partition sum of a single orbital state is independent of ms and is for each value of ms 1 1847 1 efiµ This validates line b of Table 181 The average number of bosons in the orbital state k m 1s ii e fl µ 2ei2A 2µ 3e 13A 3µ z kmi km 18 48 which is just the analogue of the relation f3N ooµ In Z but is now applied to a single orbital state Carrying out the differentiation we find n l km km ef3 µ l 1849 and this is the result listed in line c of Table 181 It is important to note that in contrast to the fermion case k is not necessarily less than or The Ideal Bose Fluid 41 J equal to unity The quantity f k is frequently referred to as an occupa tion probability but it is more properly identified as a mean occupation number nk m A moments reflection on the form of nk m reveals that for a gas of material Bose particles the molar Gibbs function must be negative For if µ were positive the orbital state with Ek equal toµ would have an infinite occupation number We thus conclude that for a gas with a bounded number of particles and with a choice of energy scale in which the lowest energy orbital has zero energy the molar Gibbs potential µ is always negative The form of n as a function of 3 E µ is shown in Fig 182 The occupation number falls from an infinite value at r µ to unity at E µ 0693k 8 T In the insert of Fig 182 the orbital occupation number is shown schematically as a function of r for two different temperatures T2 T1 and for two choices of µ If the system of interest is in contact with a particle reservoir so that µ is constant then the curve oi n E T2 in the insert should be shifted to the right The number of particles in such a system increases with tempera ture If the system of interest is maintained at constant particle number the integral of nr TDr is conserved As is evident from the figure the molar Gibbs potential µ then must decrease with increasing temperature just at it does in the Fermi gas The grand canonical potential II is the logarithm of Z which in turn is the product of the zk m given in equation 1847 Thus as in Table 181 lines d to g 1850 or integrating by parts 11 dr 2 goV 2m 32ioo 32 3 2w 2 112 o ePpl 1 1851 and again the mechanical equation of state is P 2U 3V lines i and j of Table 181 For a system of particles maintained at constant µ by a particle reservoir the thermodynamics follows in a straightforward fashion But for a system at constant N the apparently innocuous formalism conceals some startling and dramatic consequences with no analogues in either fermion or classical systems As a preliminary to such considerations it is useful to turn our attention to systems in which the particle number is physically nonconserved 41 l Quantum Fluids 186 NONCONSERVED IDEAL BOSON FLUIDS ELECTROMAGNETIC RADIATION REVISITED As we observed in Section 181 bosons are the quantum analogues of the waves of classical physics A residue of this classical significance is that unlike fermions bosons need not be conserved In some cases as in a fluid of 4He atoms the boson particles are conserved in other cases as in a photon gas recall Section 36 the bosons are not conserved There exist processes for instance in which two photons interact through a nonlinear coupling to produce three photons How then are we to adapt the formalism of the ideal Bose fluid to this possibility of nonconserva tion We recall the reasoning in Sections 17 2 and 17 3 leading to the grand canonical formalism We there maximized the disorder subject to auxiliary constraints on the energy equation 1730 and on the number of particles equation 1731 These constraints introduced Lagrange parameters A2 and X3 equation 1733 which were then physically identified as 2 3 and as 3 3µ Treatment of nonconserved particles simply requires that we omit the constraint equation on particle number Omission of the parameter X3 is equivalent to taking X3 0 or to takingµ 0 We thus arrive at the conclusion that the molar Gibbs potential of a nonconserved Bose gas is zero For µ 0 the grand canonical formalism becomes identical to the canonical formalism Hence the grand canonical analysis of the photon gas simply reiterates the canonical treatment of electromagnetic radiation as developed in Section 167 The reader should trace this parallelism through in step by step detail referring to Table 181 and Section 167 see also Problem 1862 It is instructive to reflect on the different viewpoints taken in Section 167 and in this section In the previous analysis our focus was on the normal modes of the electromagnetic field and this led us to the canonical formalism In this section our focus shifted to the quanta of the field or the photons for which the grand canonical formalism is the more natural But the nonconservation of the particles requires µ to vanish and thereby achieves exact equivalence between the two formalisms Only the language changes The number of photons of energy E is e 3 1 1 where the permitted energies are given by 27T he E hw he 1852 Here c is the velocity of light and is the quantum mechanical wave length of the photon or the wavelength of the normal mode in the mode language of Section 167 The population of bosons of infinitely long Bose Condensaton 413 wavelength is unbounded 3 The energy of these long wavelength photons vanishes so that no divergence of the energy is associated with the formal divergence of the boson number To recapitulate electromagnetic radiation can be conceptualized either in terms of the normal modes or in terms of the quanta of excitation of these modes The former view leads to a canonical formalism The latter leads to the concept of a nonconserved Bose gas to the conclusion that the molar Gibbs potential of the gas is zero and to an unbounded population of unobservable zero energy bosons in the lowest orbital state All of this might appear to be highly contrived and formally baroque were it not to have a direct analogy in conserved boson systems giving rise to such startling physical effects as superfluidity in 4 He and supercon ductivity in metals to which we now tum PROBLEMS 1861 Calculate the number of photons in the lowest orbital state in a cubic vessel of volume 1 m3 at a temperature of 300 K What is the total energy of these photons What is the number of photons in a single orbital state with a wavelength of 5000 A and what is the total energy of these photons 1862 a In applying the grand canonical formalism to the photon gas can we use the density of orbital states function D E as in equation f of Table 181 Explain b Denoting the velocity of light by c show that writing c wavelengthperiod implies w ck From this relation and from Section 165 find the density of orbital states D E c Show that the grand canonical analysis of the photon gas corresponds precisely with the theory given in Section 167 187 BOSE CONDENSATION Having the interlude of Section 186 to provide perspective we focus on a system of conserved particles enclosed in impermeable walls Then as we saw in Fig 182 and the related discussion the molar Gibbs potential µ must increase as the temperature decreases just as in the fermion case Assuming the bosons to be material particles of which the kinetic energy is E p 22m the density of orbital states is proportional to 1112 3 0f course such infimtewavelength photons can be accommodated only in a infinitely large container but the number of photons can be increased beyond any preassigned bound 1n a finite container of sufficiently large size 414 Quantum Flwd1 equation f of Table 18l and the number of particles i 1853 where is the fugacity 1854 and where the subscript e is affixed to Ne for reasons that will become undestandable only later for the moment Ne is simply another notation for N The molar Gibbs potential is always negative for conserved particles so that the fugacity lies between zero and unity 1855 This observation encourages us to expand the integral in equation 1853 in powers of the fugacity giving where AT is the thermal wavelength equation 1826 and 00 e e e F 32 rl r32 2fi 33 1857 At high temperature the fugacity is small and F3i can be replaced by its leading term in which case equation 1856 reduces to its classical form 1825 Similarly U g 0V 2m 32 3 k T52 F ik TgoV F 2w 2 12 4 B 52 2 B A3T 52 1858 where 00 e e e F E 512 1 r 512 4fi 93 1859 Bose Condensation 415 2612 FIGURE 183 The functions fj 12 and fs12n that characterize the particle number and the energy equations 18571860 of a gas of conserved bosons Again the equation for U reduces to its classical form 1827 if F512 fl is replaced by the leading term in the series Dividing 1858 by 1856 1860 so that the ratio t12flF 312fl measures the deviation from the classical equation of state For both F312 fl and t12 fl all the coefficients in their defining series are positive so that both functions are monotonically increasing functions of t as shown in Fig 183 Each function has a slope of unity at 0 At 1 the functions F312 and F512 have the value 2612 and 134 respec tively The two functions satisfy the relation 1861 from which it follows that the slope of F512 at 1 is equal to F3121 or 2612 The slope of F31i at 1 is infinite Problem 1872 416 Quantum Fluids The formal Procedure in analyzing a given gas is now exelict Let s suppose that Ne V and T are known Then F312 fl N1Tg0 V 1s known and the fugacity can be determined directly from Fig 183 Given the fugacity all thermodynamic functions are determined in the grand canonical formalism The energy for example can be evaluated by Fig 183 and equations 1858 or 1860 All of the previous discussion seems to be reasonable and straightfor ward until one suddenly recognizes that given values of Ne V and T may result in the quantity NeA3Tgr being greater than 2612 Then Fig 183 permits no solution for the fugacity The analysis fails in this extreme quantum limit A moments reflection reveals the source of the problem As N13Tg 0V F31i approaches 2612 the fugacity approaches unity or the molar Gibbs potential µ approaches zero But we have noted earlier that at µ 0 the occupation number ii of the orbital state of zero energy diverges This pathological behavior of the groundstate orbital was lost in the transition from a sum over orbital states to an integral weighted by the density of orbital states that vanishes at µ 0 This formalism is acceptable for g0V N13T 2612 but if this quantity is greater than 2612 we must treat the replacement of a sum over states by an integral with greater care and delicacy We postpone briefly the corrections to the analysis that are required if g0V N13T 2612 to first evaluate the temperature at which the failure of the integral analysis as opposed to the summation analysis occurs Setting g0V N13T 2612 we find 2wh2 1 N 213 k BT 2612 gr 1862 where T is called the Bose condensation temperature For temperature greater than T the integral analysis is valid At and below T a Bose condensation occurs associated with an anomalous population of the orbital ground state If the atomic mass m and the observed number density N giV of liquified 4He are inserted in equation 1862 one finds a condensation temperature reasonably close 3 K to the temperature 217 K at which superfluidity and other nonclassical effects occur This agreement is reasonable in light of the gross approximation involved in treating 4 He liquid as an ideal noninteracting gas To explore the population of the orbital ground state and of other lowlying excited orbital states we recall that the total number of particles is 1863 Bose Condensa1011 417 and the allowed values of rk are r n2 n2 n 2 p 2 h 2 1 1 1 h 2 nnn 2m 2m 2 A A 8mV 213 X 1864 where we have again invoked the quantum mechanical relationship be tween momentum and wavelength p hA assumed a cubic box of length v 113 and required that an integral number of half wavelengths fit along each axis dnxAx v 113 etc The energies of the discrete quantum mechanical states are precisely those from which we inferred the density of orbital states function in Section 165 The ground state energy is that in which n x n 1 n z 1 and we normally choose the energy scale relative to this state The first excited state has two of the ns equal to unity and one equal to twothis state is threefold degenerate The difference in energy is r 211 Em 6h 2mV 213 For a container of volume 1 liter V 10 3m3 and with m taken as the atomic mass of 4 He 66 X 10 27 Kg the energy of the first excited state relative to the ground state energy is Em flll 6h 2 m V 213 25 X 10 37 J or 1865 Thus the discrete states are indeed very closely spaced in energyfar closer than k 8 T at my reasonable temperature We might well have felt confident in replacing the sum by an integral But let us examine more closely the population of each state as the chemical potential approaches rw from below In particular we inquire as to the value of µ for which the population of the orbital ground state alone is comparable to the entire number of particles in the gas Let n 0 be the number of particles in the ground state orbital so that expr 111 µ 11 1 n 0 Then if n 0 1 it follows that 3r111 µ 1 and we can expand the exponential to first order so that n 0 k 8 T rlll µ Thus the population of the orbital ground state becomes comparable to the entire number of particles in the system say n0 1022 if 3 r111 µ 1022 What then is the population of the first excited orbital state The energy difference rill µk 8 is 10 21 K for T 10 K whereas r 211 r111k 8 10 14 K equation 1865 It follows that nmn 0 10 1 The population of higher states continues to fall extremely rapidly 418 Quantum Flwds As the temperature decreases in a Bose gas the molar Gibbs potential increases and approaches the energy of the ground state orbital The population of the ground state orbital increases becoming a nonnegligible fraction of the total number of bosons in the gas at the critical tempera ture T The occupation number of any individual other state is relatively negligible As the temperature decreases further µ cannot approach closer to the ground state energy than 3 µ rm 1 N 10 23 at which value the ground state alone would host all N particles in the gas Hence the ground state shields all other states from too close an approach ofµ and each other state individually can host only a relatively small number of particles Together of course the remaining states host all the particles not in the ground state With this understanding of the mechanism of the Bose condensation it is a simple matter to correct the analysis All orbital states other than the ground state are adequately represented by the integral over the density of orbital states function The ground state energy must be separately and explicitly listed in the sum over states The number of particles is then N n 0 Ne 1866 where n O is the number of particles in the ground state orbital no e Jµ 1 i 1 1867 and where Ne is the number of particles in excited states ie in all orbital states other than the ground orbital state The number of excited particles Ne is as given in equation 1854 The expression 1859 for the energy remains correct since the popula tion of the zero energy orbitals makes no contribution to the energy Thus the entire correction to the theory consists of the reinterpretation of Ne as the number of excited particles and the adjuncture of the two additional equations 1867 and 1868 Equivalently we can simply add the ground state term to our previous expression for the grand canonical potential equation 1851 giving the fundamental relation 1868 where of course is the fugacity ePP With equations 1856 to 1860 and 1866 to 1867 we can explore a variety of observable properties of Bose fluids These properties are summarized in Table 182 and illustrated schematically in Fig 184 BoseCondensatwn 419 TABLE 182 Properties of the Ideal Bose Fluid Fundamental equation Condensation temperature Condensed and excited bosons T T T T T 32 n0N 1 NN 1 T Energy T T T T 3 fs2 I T 32 T 52 U 2NkBT T 076NkBT T 2 1 Heat capacity c per particle T T T T T 32 c 1 9k8 T Entropy T T T I 420 Quantum Flutd1 t 05 075 05 025 2 I 2 FIGURE IR 4 05 Classical value uNkB Classical value UlNk T 2 B J 0 25Tc O 5T 0 75T T l 25T l 50Tc 1 75T T Properties of an ideal Bose flwd The energy and heat capacity for T T are schcmallc First consider the temperature dependence of the number of bosons in the orbital ground state For T T the maximum number of bosons that can be accommodated in excited states is T T 1869 and in particular as T T Ne N so that 1870 Roe Condenslllton 421 where X is the value of AT at T T Dividing 3 32 N AT T 1871 The number of particles in the ground state is then no 1 e 1 32 N N T 1872 This dependence is sketched in Fig 184 The energy of the system is also of great interest as its derivative is the heat capacity an easily observable quantity For T T the energy is given by equation 1860 For T T equation 1858 can be written in the form T 52 076NkBT T T 1873 For T T the energy is given by equation 1860 or U 1NkBTF52aF 312fl so that the energy is always less than its classical value The fugacity is determined as a function of T by Fig 182 Calculation of the molar heat capacity for T T follows directly by differentiation of equation 18 73 T l2 cv 19Nk 8 T T T 1874 It is of particular interest that c 19Nk 8 at T T a value well above the classical value 15Nk 8 which is approached m the classical regime at high temperature Calculation of the heat cpacity at T T requires differentiation of equation 1860 at constant N and elimination of d dT by equation 1856 The results are indicated schematically in Fig 184 and given in Table 182 The unique cusp in the heat capacity at T T is a signature of the Bose condensation A strikingly similar discontinuity 1s observed in 4 He 42 2 Quantum Fluids fluids its detailed shape appears to be in agreement with the renormaliza tion group predictions for the universality class of a twodimensional order parameter recall the penultimate paragraph of Chapter 12 Finally we note that the Bose condensation in 4 He is accompanied by striking physical properties of the fluid Below T the fluid flows freely through the finest capillary tubes It runs up and over the side of breakers It is as its name denotes superfluid The explanation of these proper ties lies outside the scope of statistical mechanics It is sufficient to say that it is the condensed phase or the ground state component that alone flows so freely through narrow tubes This component cannot easily dissipate energy through friction as it is already in the ground state More significantly the condensed phase has a quantum coherence with no classical analogue the bosons that share a single state are correlated in a fashion totally different from the excited particles which are randomly distributed over enormously many states A similar Bose condensation occurs in the electron fluid in certain metals By an interaction involving phonons pairs of electrons bind together in correlated motion These electron pairs then act as bosons The Bose condensation of the pairs leads to superconductivity the analogue of the superfluidity of 4 He PROBLEMS 1871 Show that equations 1856 and 1858 for N and U respectively ap proach their proper classical limits in the classical regime 1872 Show that F312 1 F512 1 and F12 1 are all finite whereas F12 1 1s infinite Here F12 1 denotes the derivative of F312 x evaluated at x 1 Hint Use the integral test of convergence of infinite series whereby fifn converges or diverges with frfx dx if O fn 1 fn for all n 1873 Show that the explicit inclusion of the orbital ground state contributes g0k 8 T In 1 to the grand canonical partition sum thereby validating equa tion 1868 19 FLUCTUATIONS 191 THE PROBABILITY DISTRIBUTION OF FLUCTUATIONS A thermodynamic system undergoes continual random transitions among its microstates If the system is composed of a subsystem in diathermal contact with a thermal reservoir the subsystem and the reservoir together undergo incessant and rapid transitions among their joint microstates These transitions lead sometimes to states of high subsystem energy and sometimes to states of low subsystem energy as the constant total energy is shared in different proportions between the subsystem and the reservoir The subsystem energy thereby fluctuates around its equilibrium value Similarly there are fluctuations of the volume of a system in contact with a pressure reservoir The subsystem may in fact be a small portion of a larger system the remainder of the system then constituting the reservoir In that case the fluctuations are local fiuctuations within a nominally homogeneous system Both the volume and the energy simultaneously fluctuate in a system that is in open contact with pressure and thermal reseroirs If the microstates of small volume tend to have relatively large or small energy the fluctuations of volume and energy will be negatively or positively correlated Gross macroscopic observations of an open system generally reveal only the thermodynamic values of the extensive parameters Only near the critical point do the fluctuations become so large that they become evident to simple macroscopic observations as by the critical opalescence alluded to in Section 101 Farther from the critical point the fluctuations can be observed with increasing difficulty using increasingly sophisti cated instruments of high temporal and spatial resolving power Further more as we shall see shortly theory reveals interesting relationships between the fluctuations and thermodynamic quantities such as the heat capacities These relationships are exploited by materials scientists to 423 42 4 Fluctuattonr provide a convenient method of calculation of the heat capacities and of similar properties The statistical mechanical form of the probability distribution for a fluctuating extensive parameter is now familiar If the subsystem is m diathermal contact with a thermal reservoir the probatility that the system occupies a particular microstate of energy E is ef1F f3E If the subsystem is in contact with both a pressure and a thermal reservoir the probability that the system occupies a particular microstate of energy E and volume V is exp JG J E PV And more generally for a system m contact with reservoirs corresponding to the extensive parameters X 0 X1 Xs the probability that the system occupies a particular microstate with parameters X0 X1 is 191 Here SF 0 F is the Massieu function the Legendre transform of the entropy and F0 F are the entropic intensive parameters with values equal to those of the reservoirs 192 MOMENTS OF THE ENERGY FLUCTUATIONS Let us suppose temporarily that the energy E is the only fluctuating variable all other extensive parameters being constrained by restrictive walls The deviations E U of E from its avrage value U is itself a fluctuating variable of average value zero The mean square deviatwn E U 2 or the second central moment is a measure of the width of the energy fluctuations A full description of the energy fluctuations requires knowledge of all the central moments E U with n 234 The second central moment of the fluctuations follows directly from the form of the canonical probability distribution for But we recall that E u 2 L El u 2e 3 F EJ J a 3F U a13 192 193 Moments of the Energ1 fluauutwm 425 so that equation 192 can be written as U2 LEJ u aePFf J 194 195 The first central moment vanishes and the derivative au a13 is related to the heat capacity whence 2 au 2 E u a3 kBT Ne 196 There are several attributes of this result that should be noted Most important is the fact that the mean square energy fluctuations are propor tional to the size of the system Therefore the relative root mean square dispersion U 2 112U which measures the amplitude of the fluctuations relative to the mean energy1 is proportional to N 112 For large systems N oo the fluctuation amplitudes become negligible relatwe to the mean values and thermodynamics becomes precise For systems in which a large amount of heat is required to produce an appreciable change in temperature c large the fluctuations in energy are correspondingly large Furthermore the energy fluctuations in all systems become very small at low temperatures where c 0 Finally we recall that both the heat capacity and the fluctuation amplitudes diverge at the critical point consistent with equation 196 Calculation of higher moments of the energy fluctuations recapitulates equations 194 to 196 u i I1 u aePF f J 3 IEJ uePF f IePF E 3 u J J aE U nE u 1 197 1 For definiteness the energy U here taken as zero m the T 0 tale of the system 426 Fluctuatwns The higherorder moments of the energy fluctuations can be generated from the lowerorder moments by the recursion relation 197 In particular the third moment is 198 PROBLEMS 1921 A molecule has a vibrational mode of natural frequency w The molecule is embedded in a macroscopic system of temperature T Calculate the second central moment 2 2 of the energy of the vibrational mode as a function of w T and fundamental constants 1922 Calculate the third central moment for the molecule in the precedmg problem 1923 Calculate the mean square deviation of the energy contained within a fixed volume V in a radiation field recall Section 36 Assume the volume V to be small compared to the volume V of the radiation and assume the radiation to be in equilibnum at temperature T Note that the product Nev in equation 196 is the total heat capacity of the sample com1dered 193 GENERAL MOMENTS AND CORRELATION MOMENTS 2 In the general case we are interested not only in the fluctuation moment of variables other than the energy but also in combined moments that measure the correlation of two or more fluctuating variables We consider first the general second moments of the form XJXk L x1xJxkXJfxoX1 199 states where fx0x1 x is given by equation 191 This second m9ment mfasures the correlation of the fluctuations of the two variables X and XA in a system in contact with reservoirs of constant F0 F1 F To carry out the summation over the rnicrostates we first observe that because of the form of f equation 191 at 1 A a i 1 A iJF k xk iJF s FoF f k xk xk I k 8 k 8 1910 2 0n the Formalism of Thermodynamic Fluctuation Theory R F Greene and H B Callen Phrs Rev 85 16 1956 General Moments and Correlation MomentJ 427 so that 1911 1913 The first term vanishes because tiX vanishes independently of the value of Fk so that 1914 This equation is the most significant general result of the theory of thermodynamic fluctuations Particular note should be taken of the variables to be held constant in the derivative in equation 1914 These are precisely the variables held constant in the physical system the intensive parameters F0 of the reservoirs except for FJ and the extensive parameters Xs 1 which are constrained by the walls It should also be noted that the righthand side of equation 1914 is symmetric in j and k by virtue of a Maxwell relation If X1 and X in equation 1914 are each taken as the energy we recover equation 196 for the fluctuations of energy in a system in contact with a thermal reservoir But consider the same system in contact simultaneously with thermal and pressure reservoirs so that both the energy and the volume can fluctuate Then 2 au ti kB a1T PTN 1 N2 1915 av tiEtiV kB aIT PTN 1 1916 428 f1utuatwm and A 2 av LlV kB iJPT ITN 1 k T av B ip TN 1 1917 The energy fluctuations are indeed quite different from those given in equation 196 Furthermore the energy and the volume fluctuations are correlated as expected Finally we can obtain a recursion relation relating higher order correla tion moments to lower order moments fully analogous to equation 197 Cnsidr the moment cjLlXk where cp is a product of the form Ll X Ll X1 Then equations 19 9 to 1912 can be repeated with replacing LlX1 so that 1918 which permits generation of successively higher correlation moments As an example of this procedure take cp as LlX LlX1 to obtain the third moment but LlX LlX 0 so that A A A a A A LlXLlX 1LlX1J k 8 iJF LlXLlX 2 a2x 8 aFaF I 1919 1920 Again the variables to be held constant in the differentiation reflect the boundary conditions of the fluctuating system Finally it should be noted that the fluctuations we have calculated are thermodynanuc fluctuations There are additional quantum mecharncal Problems 429 fluctuations that can be nonzero even for a system in a single quantum state For normal macroscopic systems excluding quantum systems such as superconductors or superfluids the thermodynamic fluctuations totally dominate the quantum mechanical fluctuations PROBLEMS 1931 An ideal gas is in contact with a thermal and a pressure reservoir Calculate the correlation moment JEJV of its energy and volume fluctuations 1932 Repeat Problem 1931 for a van der Waals gas recall Problem 383 1933 A conceptual subsystem of N moles in a singlecomponent simple ideal gas system undergoes energy and volume fluctuations The total system is at a temperature of 0C and a pressure of 1 atm What must be the size of N for the root mean square deviation in energy to be 1 of the average energy of the subsystem 1934 What is the order of magnitude of the mean square deviation of the volume of a typical metal sample of average volume equal to 1 cm3 The sample is at room temperature and pressure 1935 Consider a small volume V withm a twocomponent simple system Let x 1 N1 N1 N2 in which N1 and N2 are the mole numbers within V Show that 2 A 2 2 A 2 A A 2 A 2 N Ax 1 x 2 JN 2x 1x 2 JN 1 JN 2 x1 6N2 and compute the mean square deviation of concentration Jx12 1936 Consider a small quanllty of matter consisting of a fixed number N moles in a large fluid system Let PN be the average density of these N moles the mass divided by the volume Show that equation 1917 implies that the density fluctuations are in which V is the average volume of the N moles 1937 Show that the density fluctuations of an ideal gas are given by JpN2 frt Pt That is the relative mean square density deviation s the reciprocal of the number of molecules in the subsystem 1938 Show that the relative root mean square deviation in density of 10 g of air at room temperature and pressure is negligible Consider air as an ideal gas Show that the relative rms deviation in density of 1018 g of air at room temperature and pressure is approximately 1 Show that the average volume of the samples is approximately 1mm 1 in the first case and smaller than the cube of the wavelength of visible light in the second case 1939 The dielectric constant E of a fluid varies with the density by the relation E 1 E 2 Ap in which A is a constant Show tliat the fluctuations in dielectric constant of a small quantity of N moles of matter in a large system are M2 kT E 12E 22 in which V is the average volume of N moles 19310 If light of intensity I0 is incident on a region of volume V which has a difference li of dielectric constant from its average surroundings the intensity of light I0 scattered at an angle 8 and at a distance r is I IT2V2i2 I 1 cos28 2Ab o r2 in which AO is the vacuum wavelength of the incident light This is called Rayleigh scattering In a fluid each small volume V scatters incoherently and the total scattered intensity is the same as the scattered intensities from each region From problem 1939 we have v2M 2 k 8 TKrE 12E 22v and summing this quantity over the total fluid we find L V2M 2 k 8 TKrE 12E 22Vtotal where Vtotal is the total volume of the fluid Consequently the total scattered intensity at an angle 8 and at a distance r from the scattering system is I IT 2 k 8 TKr l2 22IV 1cos 28 J 18 4 E E 0 total 2 0 r By integrating over the surface of a sphere show that the total scattered intensity is 81T 3 2 2 Icattered kBTKrE 1 E 2 IoVtotal 27A Discuss the relevance of this result to critical opalescence Section 101 It is interesting to note that because of the 04 dependence of the scattering blue light 1s much more strongly scattered than red The sun appears red when it is low on the horizon because the blue light is selectively cattered leavmg the direct rays fro the sun deficient in blue On the other hand the diffuse light of the daytime sky composed of the indirectly scattered sunlight is predommntlv Problems 4 31 blue The color of the sky accordingly is everyday evidence of the existence of thermodynarruc fluctuations 19311 The classical theory of fluctuations due to Einstein proceeds from equation 19l which in general form is 1S F X k I 1 I eSFo e B J x0x1 Expanding S around its equilibrium value S in powers of the deviations A X1 JS and keeping terms only to second order t s s l fxoX1 A exp 2k L tsjktXJXk B O 0 where k a2sa 0X11 and there A is a normalizing constant This is a multidimensional Gaussian probability distribution By direct integration calcu late the second moments and show that they are correctly given The third and higher moments are not correct BABY DOLL 20 VARIATIONAL PROPERTIES PERTURBATION EXPANSIONS AND MEAN FIELD THEORY 201 THE BOGOLIUBOV VARIATIONAL THEOREM To calculate the fundamental equation for a particular system we must first evaluate the permissible energy levels of the ystem and then given those energies we must sum the partition sum Neither of these steps is simple except for a few textbook models In such models several of which we have studied in preceding chapters the energy eigenvalues follow a simple sequence and the partition sum is an infinite series that can be summed analytically But for most systems both the enumeration of the energy eigenvalues and the summation of the partition sum pose immense computational burdens Approximation techniques are required to make the calculations practical In addition these approximation tech niques provide important heuristic insights to complex systems The strategy followed in the approximation techniques to be described is first to identify a soluble model that is somewhat similar to the model of interest and then to apply a method of controlled corrections to calculate the effect of the difference in the two models Such an approach is a statistical perturbation method Because perturbation methods rest upon the existence of a library of soluble models there is great stress in the statistical mechanical literature on the invention of new soluble models Few of these have direct physical relevance as they generally are devised to exploit some ingenious mathematical trick of solution rather than to mirror real systems thereby giving rise to the rather abstract flavor of some statistical mechanical literature The first step in the approximation strategy is to identify a practical criterion for the choice of a soluble model with which to approximate a given system That criterion is most powerfully formulated in terms of the Bogoliubov variational theorem 4B 434 lanat1onal Properties Perturhatmn Expanswnr and Mean Field Theory Consider a system with a Hamiltonian Yf and a soluble model system with a Hamiltonian Let the difference be Yf1 so that Yf It is then convenient to define 201 where A is a parameter inserted for analytic convenience By permitting I to vary from zero to unity we can smoothly bridge the transition from the soluble model system to the system of interest Yfl Yf1 The Helmholtz potential corresponding to Yf A is F A where 3FA In e PE In trell j 202 Here the symbol treJl to be read as the trace of eBJI is defined by the second equality the trace of any quantity is the sum of lls quantum eigenvalues We use the notation tr simply as a convenience We now study the dependence of the Helmholtz potential on A The first derivative is1 dFA dA and the second derivative is tr Yf e Jl Jr x I tr e Jl u r 32 Jf12 3 i2 203 205 206 where the averages are taken with respect to the canonical weighting factor e JlK The operational meaning of these weighted averages will be clarified by a specific example to follow An immediate and fateful consequence of equation 206 is that d 2F dA 2 is negative or zero for all A for all A 207 1 In the quantum mechanical context the operators 0 and 1 are here asumed to commute The result is independent of this assumption For the noncommutativc cae and for an elegant general discuss10n sec R Feynman Stat1s11cal MechanicsA Set of Lectures W A BcnJarnin Inc Reading Massachusett 1972 The Bogo11101 imwtwnal Theorem 435 Consequently a plot of FA as a function of A is everywhere concave It follows that F A lies below the straight line tangent to F A at A O 208 and specifically taking A 1 209 The quantity 1 0 is as defined in equation 203 but with A O it is the average value of 1 in the soluble model system Equation 209 is the Bogoliubov inequality It states that the Helmholtz potential of a ystem with Hamiltonian is less than or equal to the unperturbed Helmholtz potential corresponding to plus the average value of the perturbation 1 as calculated in the unperturbed or soluble model system Because the quantity on the right of equation 209 is an upper bound to the Helmholtz potential of the perturbed system it clearly is desirable that this bound be as small as possible Consequently any adjustable parameters in the unperturbed system are best chosen so a to minimize the quantity F0 1 0 This is the criterion for the choice of the best soluble model system Then F0 is the Helmholtz potential of the optimum model system and 1 0 is the leading correction to this Helmholtz potential The meaning and the application of this theorem are liest illustrated by a specific example to which we shall turn momentarily However we first recast the Bogoliubov inequality in an alternative form that provides an important insight If we write F0 the Helmholtz potential of the unper turbed system explicitly as 2010 then equation 209 becomes 2011 or 2012 That is the Helmholtz potential of a system with Hamiltonian J 1 is less than or equal to the full energy Yt averaged over the state probabilities of the unperturbed system minus the product of T and the entropy of the unperturbed system 436 Varwtwnal ProperIes Perturbatwn Expanswns and Mean Field Theory Example 1 A particle of mass m 1s constrained to move in one dimension in a quartic potential of the form Vx Dxa4 where D 0 and where a is a measure of the linear extension of the potential The system of interest is composed of N such particles in thermal contact with a reservoir of temperature T An extensive parameter of the system is defined by X Na and the associated intensive parameter is denoted by P Calculate the equations of state U UT X N and P PT X N and the heat capacity cpT X N To solve this problem by the standard algorithm would require first a quantum mechanical calculation of the allowed rnergies of a particle in a quartic potential and then summation of the partition sum Neither of these calculations is analytically tractable We avoid these difficulties by seeking an approximate solution In particular we inquire as to the best quadratic potential ie the best simple harmonic oscillator model with which to approximate the system and we then assess the leading correction to account for the difference in the two models The quadratic potential that together with the kinetic energy defines the unperturbed Hamiltonian is a where Wo is an asydunspecified constant Then the perturbing potential or the difference between the true Hamiltonian and that of the soluble model system is b The Helmholtz potential of the harmonic oscillator model system is recall equations 1622 to 16242 F0 NknTlnz 0 fvp 1 lnellliwo2 eflliw2 and the Bogoliubov inequality states that F fv13lneflliwo2 eflliw112 c d Before we can draw conclusions from this result we must evaluate the second and third terms It is an elementary result of mechamcs the virial theorem that the value of the potential energy mw5x 2 in the nth state of a harmonic oscillator 1s one half the total energy so that I 22 Ji 21WoX nth state 2 n 2 Wo e 2 But note that the zero of energy has been slufted by hw02 the socalled zero point energy The allowed energies are n hw 0 The Bogufuhot iarwnona Theorem 437 and a similar quantum mechanical calculation gives 4 3112 2 I x n1hs1a1c 2m2w6 n n 2 f With the values of these quantities in the nth state we must now average over all states n Averaging equation e in the unperturbed system g and we also find h Inserting these last two results equations g and h into the Bogoliubov inequality equation d F s R13 I ln ePhw2 e Jhwo2 1 eJhw I Nnw 2 u ePhw I i The first term is the Helmholtz potential of the unperturbed harmonic oscilla tor system and the two remaining terms are the leading correction The inequality states that the sum of all higherorder corrections would be positive so that the righthand side of equation 1 is an upper bound to the Helmholtz potential The frequency w0 of the harmonic oscillator system has not yet been chosen Clearly the best approximation is obtained by makmg the upper bound on F as small as possible Thus we choose w0 so as to m1mm1ze the righthand side of equat10n 1 which then becomes the best avalable approx1ma1ton to the Helmholtz potential of the system Denote the value of w0 that minimizes F by wu a function of T X Na and N Then w0 in equation i can be replaced by wu and the less than or equal sign s can be replaced by an approximately equal sign So interpreted equation i is the approximate fundamental equation of the ystem 438 Varwtronal Propertes Perlurbatron Expansions and Mean Field Theory The mechanical equation of state is then At this point the algebra becomes cumbersome though straightforward in princi ple The remaining quantities sought for can be found in similar form Instead we turn our attention to a simpler version of the same problem Example 2 We repeat the preceding Example but we consider the case in which the coefficient D a 4 is small in a sense to be made more quantitative later permitting the use of classical statistics Furthermore we now choose a squarewell potential as the unperturbed potential Vox c L L I 2X 2 c L I jxl 2 The optimum value of L is to be determined by the Bogoliubov criterion The unperturbed Helmholtz potential is determined by 1 JL2 Joo 2 dx dpxeflp2m h L2 po We have here used classical statistics as in Sections 168 and 169 tentatively assuming that L and Tare each sufficiently large that k 8 T is large compared to the energy differences between quantum states The quantity 1 0 is then Furthermore Vox 0 for jxl L2 whereas eJlto 0 for jxj L2 so that the term involving Vox vanishes Then D D f L2 D L 4 o x 4 0 x 4 dx a 4 a 4L L2 80 a Prohlems 439 The Bogoliubov inequality now becomes Minimizing with respect to L This result determines the optimum size of a squarewell potential with which to approximate the thermal properties of the system and it determines the corre sponding approximate Helmholtz potential Finally we return to the criterion for the use of classical statistics In Section 166 we saw that the energy separation of translational states is of the order of h 2 2mL 2 and the criterion of classical statistics is that k 8 T h 2 2mL 2 In terms of D the analogous criterion is For larger values of D the procedure would be similar in principle but the calculation would require summations over the discrete quantum states rather than simple phasespace integration Finally we note that if the temperature is high enough to permit the use of classical statistics the original quartic potential problem is itself soluble Then there is no need to approximate the quartic potential by utilizing a variational theorem It is left to the reader Problem 2012 to solve the original quartic potential problem in the classical domain and to compare that solution with the approximate solution obtained here PROBLEMS 2011 Derive equation h of Example 1 first showing that for a harmonic oscillator n I az z a3hwo and where 00 z efl11012z E efllio no arwuonat rroperte Perturhutum fpansums a11d Mean field 1 heon 2012 Solve the quartic potential problem of Example 2 aiiuming the tempera ture to he sufficiently high that classical statistics can be dpphed Compdre the Helmholtz potential with that calculated m Example 2 by the variational theorem 2013 Complete Example 2 by writmg the Helmholtz potential FT a ex plicitly Calculate the tension T conJugate to the length a Calculate the compliance coefficient a 1 aa aTr 2014 Consider a particle in a quadratic potential Vx Ax 22a 2 Despite the fact that this problem is analytically solvable approximate the problem hy a lquare potential Assume the temperature to be sufficiently high that classical statistics can be used in solvmg the square potential Calculate the teniion Y and the Compliance coeffic1ent a 1 aa aTr 202 MEAN FIELD THEORY The most important application of statistical perturhation theory 1s that in which a system of interacting particles is approximated hy a system of noninteracting particles The optimum noninteracting model system 1s chosen in accordance with the Bogoliuhov inequality which abo yield the firstorder correction to the noninteracting or unperturhed Helmholtz potential Because very few interacting systems are soluhle analytically and hecause virtually all physical systems consist of inter acting particles the mean field theory descrihed here is the hasic tool of practical statistical mechanics It is important to note immediately that the term mean field theorv of ten is used in a less specific way Some of the results of the procedure can he ohtained hy other more ad hoc methods Landautype theones recall Section 114 ohtain a temperature dependence of the order par1m eter that is identical to that obtained hy statistical mean field theory Another approximation known as the random phase approximation also predicts the same equation of state Neither of these provides a full thermodynamic fundamental equation Nevertheless various such ap proximations are referred to generically as mean field theories We use the term in the more restrictive sense Certainly the simplest model of interacting systems and one that his played a key role in the development of the theory of interacting system is the twostate nearestneighhor Ismg model The model consists of a regular crystalline array of particles each of which can exist in either of two orhital states designated as the up and down states Thus the states of the particles can be visualized in terms of classical spins each of which is permitted only to he either up or down a ite variahle a t1ke the value a 1 if the spin at site j is up or a 1 if the spin at site I is down The energies of the two states are B and B for the up and down states respectively In addition nearest neighbor spins hivc in Mean Freid Theory 441 interaction energy 2J if they are both up or both down or of 2J if one spin is up and one spin is down Thus the Hamiltonian is 2013 iJ J where J 0 if i and j are not nearest neighbors whereas J J if i and j are nearest neighbors It should be noted that a specific pair of neighbors say spins 5 and 8 appears twice in the sum i 5 j 8 and i 8 j 5 Quite evidently the problem is an insoluble manybody problem for each spin is coupled indirectly to every other spin in the lattice An approximation scheme is needed and we invoke the Bogoliubov in equality A plausible form of the soluble model system is suggested by focussing on only the jth spin in the Hamiltonian 2013 the Hamilto nian is then simply linear in or We correspondingly choose the un perturbed model Hamiltonian to he 2014 J J where B is to e chosen according to the fogoliybov criterion We anticipate that B will be independent of j B B for all spins are equivalent Thus 2015 J J where we define B iJ B 2016 Accordingly the unperturbed Helmholtz potential is where N is the number of sites in the lattice The Bogoliubov inequality assures us that F F0 Jt Yt0 0 or F F0 LJ oo 0 BN o 0 2018 iJ and we procede to calculate o 0 and 010 0 In the unperturbed system the average of products centered on different sites simply factors oo 0 o 0 o 0 ot 2019 442 Vara10nal Properlles PerlJrhaton Expansions and Mean Field Theory so that F F0 NJznno B BNo 0 2020 where znn is the number of nearest neighbors of a site in the lattice z nn 6 for a simple cubic lattice 8 for a bodycentered cubic lattice etc Furthermore ef1B ePB o 0 pB PB tanh PB e e 2021 We must minimize F with respect to B But from equations 2020 2017 nd 2021 we observe that B appears in F only in the combination B B B Hence we can minimize F with respect to B giving the result that B B B 2znnJo 0 2022 This is a selfconsistent condition as o 0 is expressed in terms of B by equation 2021 Prior to analyzing this selfconsistent solution for o 0 we observe its significance If we were to seek o in mean field theory we might proceed by differentiation with respect to B of the Helmholtz potential F as calculated in mean field theory equation 2020 The applied field B appears explicitly in eqn 2020 but it is also 1mpict in o 0 Fortunately however o 0 depends on B only in the combination B B B and we have imposed the condition that a F a B 0 Thus in diff erentia tion only the expict dependence of F on B need be considered With this extremely convenient simplifying observation we immediately corroh orate that differentiation of F equation 2020 with respect to B does give o 0 The spontaneous moment o in meanfield theory is gwen properv by its zeroorder approximation Returning then to equation 2021 for o 0 and hence for o the solution is best obtained graphically as shown in Fig 201 The abcissa of the graph is PB or from equation 2022 x PB P2zJo B 2023 so that equation 2021 can be written as kBT B o 2 1 x 2 J tanhx zn znn 2024 A plot of o versus x from the first equality is a straight line of slope k 8 T 2zJ and of intercept B2zJ A plot of o versus x from the second equality is the familiar tanh x curve shown in Fig 201 The intersection of these two curves determines o t a 10 09 08 07 06 05 04 03 02 0 1 l V I y a R R 2znnJ ZnnJ Mean Field Theory 443 1 i a tanhPB V V O O O 1 0 2 03 0 4 0 5 0 6 0 7 0 8 0 9 1 0 1 1 1 2 13 14 1 5 1 6 1 7 1 8 1 9 20 1m 3B B FlGURE 201 The qualitative behavior of a T B is evident For B 0 the straight line passes through the origin with a slope k 8 T 2z 1111J The curve of tanhx has an initial slope of unity Hence if k 8 T 2z 1111J 0 the straight line and the tanh x curve have only the trivial intersection at a 0 However if k 8 T 2z 11J 1 there is an intersection at a positive value of a and another at a negative value of a as well as the persistent intersection at a 0 The existence of three formal solutions for a is precisely the result we found in the thermodynamic analysis of firstorder phase transitions in Chapter 9 A stability analysis there revealed the intermediate value a 0 to be intrinsically unstable The positive and negative values of a are equally stable and the choice of one or the other is an accidental event We thus conclude that the system exhibits a firstorder phase transition at low temperatures and that the phase transition ceases to exist above the Curie temperature T given by 2025 We can also find the susceptibility for temperatures above T For small arguments tanh y y so that equation 2024 becomes for T 7 a 32znnJa B 2026 444 Varwllonal Properlles Perturbaton Expansons and Mean Field Theorv or the susceptibility is o B T T T T T 2027 This agrees with the classical value of unity for the critical exponent y as previously found in Section 114 To find the temperature dependence of the spontaneous moment o for temperatures just below Tc we take B 0 in equation 2021 and 2022 and we assume o to be very small Then the hyperbolic tangent can be expanded in series whence or o 3 12 Tc X T i 2 2028 We thereby corroborate the classical value of for the critical exponent a It is a considerable theoretical triumph that a firstorder phase transi tion can be obtained by so simple a theory as mean field theory But it must be stressed that the theory is nevertheless rather primitive In reality the Ising model does not have a phase transition in one dimension though it does in both two and three dimensions Mean field theory in contrast predicts a phase transition without any reference to the dimensionality of the crystalline array And of course the subtle details of the critical transitions as epitomized in the values of the critical exponents are quite incorrect Finally it is instructive to inquire as to the thermal properties of the system In particular we seek the mean field value of the entropy S aF aT i We exploit the stationarity of F with respect to B by rewriting equation 2020 with B rewritten as k 8 T3B F Nk 8Tln ePB epB NJzno 2 Nk 8 T3Bo NBo 2029 Then in differentiating F with respect to T we can treat 3B as a constant S FT Nk 8 In eP80 e PB Nk 8 3B o U pBa 2030 Mean Field Theon 445 The first term is recognized as F 0T from equation 2017 and the second term is simply o T Thus 2031 The mean field value of the entropy like the induced moment o is given correctly in zero order The energy U is given by 2032 The energy is also given correctly in zero order if interpreted as in 2032 but note that this result is quite different from r0 A more general Ising model permits the spin to take the values S S 1 S 2 S 2 S 1 S where S is an integer or half integer the value of the spin The theory is identical in form to that of the twostate Ising model which corresponds to S except that the hyperbolic tangent function appearing in o 0 is replaced by the Brillouin function 0 0 SBdf3B S coth 2s2 1 f3B 1 coth 2033 The analysis follows stepbystep in the pattern of the twostate Ising model considered above merely replacing equation 2021 by 2033 The corroboration of this statement is left to the reader In a further generalization the Heisenberg model of ferromagnetism permits the spins to be quantum mechanical entities and it associates the external field B with an applied magnetic field B Within the mean field theory however only the component of a spin along the external field axis is relevant and the quantum mechanical Heisenberg model reduces directly to the classical Ising model described above Again the reader is urged to corroborate these conclusions and he or she is referred to any introductory text on the theory of solids for a more complete discussion of the details of the calculation and of the consequences of the conclusions The origin of the name mean field theory hes in the heuristic reasoning that led us to a choice of a soluble model Hamiltonian in the Ising or Heisenberg problem above Although each spin interacts with othfr spins the mean field approach effectively replaces the bilinear spm interaction 00 1 by a linear term Bor The quantity B plays the role of an effective magnetic field acting on a and the optimum choice of B is o Equivalently the product 00 1 is 1Jinearized replacing one factor by its 446 Varwtwnal Properties Perturhatwn Expa11swm a11d Mean Field Theo average value A variety of recipes to accomplish this in a consistent manner exist However we caution against such recipes as they generally substitute heuristic appeal for the wellordered rigor of the Bogoliubov inequality and they provide no sequence of successive improvements More immediately the stationarity of F to variations in B greatly simplifies differentiation of F required to evaluate thermodynamic quan tities recall equation 2030 and the analogue of this stationarity has no basis in heuristic formulations But most important there are applications of the mean field formalism as based on the Bogoliubov inequality in which products of operators are not simply linearized For these the very name mean field is a misnomer A simple and instructive case of this type is given in the following Example Example N Ising spins each capable of taking three values o 1 0 1 form a planar triangular array as shown Note that there are 2N triangles for N spins and that each spin is shared by six triangles We assume N to be sufficiently large that edge effects can be ignored The energy associated with each triangle a threebody interaction is E if two spins are up 2e if three spins are up 0 otherwise Calculate approximately the number of spins in each spm state if the system is in equilibrium at temperature T Solution The problem differs from the Ising and Heisenberg prototypes in two respects we are not given an analytic representation of the Hamiltonian though we could devise one with moderate effort and a mean field type of model Hamiltonian of the form B 1a1 would not be reasonable This latter observation follows from the stated condition that the energies of the various possible configurations depend only on the populations of the up states and that there is no Mean Field Theory 447 distinction in energy between the o 0 and the o 1 states The soluble model Hamiltonian should certainly preserve this symmetry which a meanfield type Hamiltonian does not do Accordingly we take as the soluble model Hamiltonian one in which the energy e is associated with each up spin in the lattice the o 0 and 1 states each having zero energy The energy E will be the variational parameter of the problem The unperturbed value of the Helmholtz potential is determined by elJFo ell 2 N and the probability that a spin is up to zero order is whereas for ell 1 2ell1 ell2 I I 1 fo T JOl JO 2 Within each triangle the probability of having all three spins up is Jlr and the probability of having two spins up is 30 1 0 T We can now calculate 0 and 0 0 directly oo Nefo whereas 0 2NE2JJ 1 3fv2t 1 01 2NEM 1 30 The variational condition then is F 5 Nk 8 Tlnell 2 2N EJ1 3EJ0 Nifor It is convenient to express the argument of the logarithm in terms of 0 F 5 NkT In 1 l 2N JJ 3o mt The variational parameter E appears explicitly only in the last term but it is also implicit in 0 T It is somewhat more convenient to minimize F with respect to 0 T inverting the functional relationship 0 r E to consider E as a function of 0 t dF Nk BT 2 A dE 0 d 6NEJ01 l2NEJ 01 NE N 01 d OT 1or Of The last term is easily evaluated to be Nk 8 Tf 0 1 for 1 so that the variational condition becomes 6JJJ 123fo o In l 2 l 1 0 This equation must be solved numerically or graphically Given the solution for 0 T as a function of the temperature the various physical properties of the system can be calculated in a straightforward manner cu neory PROBLEMS 2021 Formulate the exact solution of the twoparticle Ising model with an external field assume that each particle can take only two states o 1 or 1 Find both the magnetization and the energy and show that there is no phase transition in zero external field Solve the problem by mean field theory and show that a transition to a spontaneous magnetization in zero external field is predicted to occur at a nonzero temperature Tc Show that below Tc the spontaneous moment varies as T Tcfl and find Tc and the critical exponent f3 recall Chapter 11 2022 Formulate mean field theory for the three state Ising model in which the variables o1 in equation 2013 can take the three values 1 0 1 Find the Curie temperature T as in equation 2025 2023 For the Heisenberg ferromagnetic model the Hamiltonian is Yt t1SS µBBets lJ J where µ8 is the Bohr magneton and Be is the magnitude of the external field which is assumed to be directed along the z axis The zcomponents of S are quantized taking the permitted values SF S S l S 1 S Show that for S t the mean field theory is identical to the mean field theory for the twostate Ising model if 2S is associated with o and if a suitable change of scale is made in the exchange interaction parameter Jr Are correspondmg changes of scale required for the S 1 case recall Problem 2021 and if so what is the transformation 2024 A metallic surface is covered by a monomolecular layer of N organic molecules in a square array Each adsorbed molecule can exist in two stenc configurations designated as oblate and prolate Both configurations have the same energy However two nearest neighbor molecules mechanically interfere 1f and only if both are oblate The energy associated with such an oblateoblate interference is E a positive quantity Calculate a reasonable estimate of the number of molecules in each configuration at temperature T 2025 Solve the preceding problem if the molecules can exist in three stenc configurations designated as oblate spherical and prolate Again all three con figurations have the same energy And again two nearestneighbor molecules interfere if and only if both are oblate the energy of interaction is E Calculate approximately the number of molecules in each configuration at temperature T N 10 at k 8 T E 0266 N4 at k8 TE 247 N3 at k 8 T oo Anstter N5 at k 8TE 115 3N10 at k 8 TE 778 Mean Freid m Generahzed Representatwn The Bmury Alloy 449 2026 In the classical Heisenberg model each spin can take any orientation in space recall that the classical partition function of a single spin in an external field B is Zc1assical fePBScosB sin8d8dq Show that in mean field theory Sz Scoth 3B BS 3B BS 2027 2N twovalued Ising spins are arranged sequentially on a circle so that the last spin is a neighbor of the first The Hamiltonian is iii Yf 2 L 100 1 BLo 1l j where 1 Je if j is even and 1 10 if j is odd Assume 10 Je There are two options for carrying out a mean field theory for this system The first option is to note that all spins are equivalent Hence one can choose an unperturbed system of 2N single spins each acted on by an effective field to be evaluated variationally The second option is to recognize that we can choose a pair of spins coupled by 10 the larger exchange interaction Each such pair is coupled to two other pairs by the weaker exchange interactions Je The unper turbed system consists of N such pairs Carry out each of the mean field theories described above Discuss the relative merits of these two procedures 2028 Consider a sequence of 2N alternating A sites and B sites the system being arranged in a circle so that the 2N 1h site is the nearest neighbor of the first site Even numbered sites are occupied by twovalued Ising spins with o 1 Odd numbered sites are occupied by threevalued Ising spins with o 1 0 1 The Hamiltonian is Yf 21 L oio 1 BL o j J a Formulate a mean field theory by choosing as a soluble model system a collection of independent A sites and a collection of independent B sites each acted upon by a different mean field b Formulate a mean field theory by choosing as a soluble model system a collection of N independent AB pairs with the Hamiltonian of each pair being air 2Joodd0even BoddOodd Beveneven c Are these two procedures identical If so why If not which procedure would you judge to be superior and why 203 MEAN FIELD IN GENERALIZED REPRESENTATION THE BINARY ALLOY Mean field theory is slightly more general than it might at first appear from the preceding discussion The larger context is clarified by a particu 450 Var1at1onal Properties Perturbatwn Expanswns and Mean Freid Theory lar example We consider a binary alloy recall the discussion of Section 113 in which each site of a crystalline array can be occupied by either an A atom or a B atom The system is in equilibrium with a thermal and particle reservoir of temperature T and of chemical potentials ie partial molar Gibbs potentials µA and µB The energy of an A atom in the crystal is EA and that of a B atom is E8 In addition neighboring A atoms have an interaction energy EAA neighboring B atoms have an interaction energy EBB and neighboring AB pairs have an interaction energy EAB We are interested not only in the number of A atoms in the crystal but in the extent to which the A atoms either segregate separately from the B atoms or intermix regularly in an alternating ABAB pattern That is we seek to find the average numbers NA and NB of each type of atom and the average numbers NAA NAB and NBB of each type of nearest neighbor pair These quantities are to e calculated as a function of T µ A and µ 8 The various numbers NA NAB are not all independent for 2034 and by counting the number of bonds emanating from A atoms 2035 Similarly 2036 where we recall that znn is the number of nearest neighbors of a single site Consequently all five numbers are determined by two which are chosen conveniently to be NA and NAA The energy of the crystal clearly is 2037 If we associate with each site an Ising spin such that the spin is up o 1 if the site is occupied by an A atom and the spin is down o 1 if the spin is occupied by a B atom then C L Ll100 3 Lo I J where 2038 2039 2040 2041 Mean field rn Generahzed Representatron The Binary AUO 45 These values of J B and C can be obtained in a variety of ways One simple approach is to compare the values of E equation 2037 and of Jf equation 2038 in the three configurations in which a all sites are occupied by A atoms b all sites are occupied by B atoms and c equal numbers of A and B atoms are randomly distributed Except for the inconsequential constant C the Hamiltonian is now that of the Ising model However the physical problem is quite different We must recall that the system is in contact with particle reservoirs of chemical potentials µA and µ 8 as well as with a thermal reservoir of temperature T The problem is best solved in a grand canonical for malism The essential procedure in the grand canonical formalism is the calcu lation of the grand canonical potential iTµAµ 8 by the algorithm3 2042 This is isomorphic with the canonical formalism on which the mean field theory of Section 202 was based if we simply replace the Helmholtz potential F by the grand canonical potential ir and replace the Harpiltonian Jf by the grand canonical Hamiltonian Jf µANA jlRNB In the present context we augment the Hamiltonian 2038 by terms of the form rn jl A jl B jl A jl B EI o The grand canonical Hamiltonian is then Jf C L 1 10 101 Bo 2043 I j where 2044 and 2045 The analysis of the Ising model then applies directly to the binary alloy problem with the Helmholtz potential being reinterpreted as the grand canonical potential Again mean field theory predicts an orderdisorder phase transition Again that prediction agrees with more rigorous theory in two and three dimensions whereas a onedimensional binary crystal should not have an orderdisorder phase transition And again the critical exponents are incorrectly predicted More significantly the general approach of mean field theory is appli cable to systems in generalized ensembles requiring only the reinterpreta tion of the thermodynamic potential to be calculated and of the effective Hamiltonian on which the calculation is to be based 3A p4 Avogadros number is the chemical potential per partrcle BLUSH PART III FOUNDATIONS 21 POSTLUDE SYMMETRY AND THE CONCEPTUAL FOUNDATIONS OF THERMOSTATISTICS 211 STATISTICS The overall structure of thermostatistics now has been establishedof thermodynamics in Part I and of statistical mechanics in Part II Although these subjects can be elaborated further the logical basis is essentially complete It is an appropriate time to reconsider and to reflect on the uncommon form of these atypical subjects Unlike mechanics thermostatistics is not a detailed theory of dynamic response to specified forces And unlike electromagnetic theory or the analogous theories of the nuclear strong and weak forces thermosta tistics is not a theory of the forces themselves Instead thermostatistics characterizes the equilibrium state of microscopic systems without ref er ence either to the specific forces or to the laws of mechanical response Instead thermostatistics characterizes the equilibrium state as the state that maximizes the disorder a quantity associated with a conceptual framework information theory outside of conventional physical the ory The question arises as to whether the postulatory basis of thermosta tistics thereby introduces new principles not contained in mechanics electromagnetism and the like or whether it borrows principles in unrec ognized form from that standard body of physical theory In either case what are the implicit principles upon which thermostatistics rests There are in my view two essential bases underlying thermostatistical theory One is rooted in the statistical properties of large complex systems The second rests in the set of symmetries of the fundamental laws of physics The statistical feature veils the incoherent complexity of the atomic dynamics thereby revealing the coherent effects of the underlying physical symmetries 455 456 Postlude Symmetrr and the Conceptual Foundatwns of Themwtatnllcs The relevance of the statistical properties of large complex systems is universally accepted and reasonably evident The essential property is epitomized in the central limit theorem 1 which states roughly that the probability density of a variable assumes the Gaussian form if the variable is itself the resultant of a large number of independent additive subvariables Although one might naively hope that measurements of thermodynamic fluctuation amplitudes could yield detailed information as to the atomic structure of a system the central limit theorem precludes such a possibility It is this insensitivity to specific structural or mechani cal detail that underlies the universality and simplicity of thermostatistics The central limit theorem is illustrated by the following example Example Consider a system composed of N elements each of which can take a value of X in the range t X The value of X for each element is a continuous random variable with a probability density that is uniform over the permitted region The value of X for the system is the sum of the values for each of the elements Calculate the probability density for the system for the cases N 1 2 3 In each case find the standard deviation o defined by where f X is the probability density of X and where we have given the definition of o only for the relevant case in which the mean of Xis zero Plot the probability density for N 1 2 and 3 and in each case plot the Gaussian or normal distribution with the same standard deviation Note that for even so small a number as N 3 the probability distribution X rapidly approaches the Gaussian form It should be stressed that in this example the uniform probability density of X is chosen for ease of calculation a simtlar approach to the Gaussian form would be observed for any initial probabiltty density Solution The probability density for N 1 is 1X 1 for X and zero otherwise This probability density is plotted in Fig 21la The standard devia tion is o1 12 ff The corresponding Gaussian 112 x 2 cX2w o 1 exp 2cr2 with o o1 is also plotted in Fig 21la for comparison 1cf any standard reference on probability such as L G Parratt Probabtlttv and 1penmental Errors in Science Wiley New York 1961 or E Parzen Modem Probabr11 Theon and IH Apprcatwns Wiley New York 1960 3111 45 7 a l 5 I 10 I 1x I 05 I I 00 b 10 fix r 05 V 00 c l 0 3t 05 00 1 l 0 l 3 2 2 F 2 IICURE 21 I Convergence of probab1hty density IO the Gaussian form The probability dcrn1ty for 1y1tem compmed of one two and three elements each vith the probability dcn1ty 5hown m Figure 21la In each case the Gaus1an with the ame standdfd dev1at1on 1 plotted In accordance with the central hm1t theorem the probdb1lity dcrnity become Glll5lan for large N To calculate the probability demity 2 X for N 2 we note problem 2111 that filiX foc fXXf 1XdX oc or with 1 X as given J l2 fNc1X dXXdX I 2 That is f ii 1 X is the average value of f N X over a range of length unity centered at X This geometric interpretation easily permits calculation of 2 X as shown m Fig 21lb From 2 X m turn we find f 1 x 2 1f I XI s 1 3 x i 0 x 1x 2 1f s 1x1 s l if IXI 1 458 Postlude Symmetry and the Conceptual Founda11ons of Thermosta1tst1cs The values of a are calculated to be o1 1 fi o2 1 6 and o3 These values agree with a general theorem that for N identical and independent subsystems afl ii o1 The Gaussian curves of Fig 211 are calculated with these values of the standard deviations For even so small a value of N as 3 the probability distribution is very close to Gaussian losing almost all trace of the irutial shape of the singleelement probability distribution PROBLEMS 2111 The probability of throwing a seven on two dice can be viewed as the sum of a the probability of throwing a one on the first die multiplied by the probability of throwing a six on the second plus b the probability of throwing a two on the first die multiplied by the probability of throwing a five on the second and so forth Explain the relationship of this observation to the expres sion for ffliX in terms of iX X and 1X as given in the Example and derive the latter expression 2112 Associate the value 1 with one side of a coin head and the value 1 with the other side tail Plot the probability of finding a given value when throwing one two three four and five coins Note that the probability 1s discretefor two coins the plot consists of just three points with probability for X 1 and probability for X 0 Calculate a for the case n 5 and roughly sketch the Gaussian distribution for this value of a 212 SYMMETRY2 As a basis of thermostatistics the role of symmetry is less evident than the role of statistics However we first note that a basis in symmetry does rationalize the peculiar nonmetric character of thermodynamics The results of thermodynamics characteristically relate apparently unhke quantities yielding relationships such as aTaPv aVaSr but providing no numerical evaluation of either quantity Such an emphasis on relationships as contrasted with quantitative evaluations is appropri ately to be expected of a subject with roots in symmetry rather than in explicit quantitative laws Although symmetry considerations have been seen as basic in science since the dawn of scientific thought the development of quantum mechan ics in 1925 elevated symmetry considerations to a more profound level of power generality and fundamentality than they had enjoyed in classical physics Rather than merely restricting physical possibilities symmetry was increasingly seen as playing the fundamental role in establishing the 2 H Callen Foundations of Physics 4423 1974 Symmetry 459 form of physical laws Eugene Wigner Nobel laureate and great modern expositor of symmetry laws suggested 3 that the relationship of symmetry properties to the laws of nature is closely analogous to the relationship of the laws of nature to individual events the symmetry principles provide a structure or coherence to the laws of nature just as the laws of nature provide a structure and coherence to the set of events Contemporary grand unified theories conjecture that the very existence and strength of the four basic force fields of physical theory electromagnetic gravita tional strong and weak were determined by a symmetry genesis a mere 10 35 seconds after the Big Bang The simplest and most evident form of symmetry is the geometric symmetry of a physical object Thus a sphere is symmetric under arbitrary rotations around any axis passing through its center under reflections in any plane containing the center and under inversion through the center itself A cube is symmetric under fourfold rotations around axes through the face centers and under various other rotations reflections and inver sion operations Because a sphere is symmetric under rotations through an angle that can take continuous values the rotational symmetry group of a sphere is said to be continuous In contrast the rotational symmetry group of a cube is discrete Each geometrical symmetry operation is described mathematically by a coordinate transformation Reflection in the xy plane corresponds to the transformation x x y y z z whereas fourfold 90 rota tion around the zaxis is described by x y y x z z The symmetry of a sphere under either of these operations corresponds to the fact that the equation of a sphere x 2 y 2 z 2 r 2 is identical in form if reexpressed in the primed coordinates The concept of a geometrical symmetry is easily generalized A transfor mation of variables defines a symmetry operation A function of those variables that is unchanged in form by the transformation is said to be symmetric with respect to the symmetry operation Similarly a law of physics is said to be symmetric under the operation if the functional form of the law is invariant under the transformation Newtons law of dynamics f md 2rdt 2 is symmetric under time inversion r r t t for a system in which the force is a function of position only Physically this timeinversion symmetry implies that a video tape of a ball thrown upward by an astronaut on the moon and falling back to the lunar surface looks identical if projected backward or forward On the earth in the presence of air friction the dynamics of the baJI would not be symmetric under time inversion The symmetry of the dynamical behavior of a particular system is governed by the dynamical equation and by the mechanical potential that 3E Wigner Symmetry and Conservalton Laws Physics Today March 1964 p 34 460 Postlude Symmetry and the Coneptua Foundatwm of Thermostat1stus determines the forces For quantum mechanical problems the dynamical equation is more abstract Schrodingers equation rather than Newtons law but the principles of symmetry are identical 213 NOETHERS THEOREM A far reaching and profound physical consequence of symmetry is formulated in N oether s theorem4 The theorem asserts that every continuous symmetry of the dynamical behavior of a system ie of the dynamical equation and the mechanical potential implies a conservatwn law for that system The dynamical equation for the motion of the center of mass point of any material system is Newtons law If the external force does not depend upon the coordinate x then both the potential and the dynamical equa tion are symmetric under spatial translation parallel to the xaxis The quantity that is conserved as a consequence of this symmetry is the xcomponent of the momentum Similarly the symmetry under translation along they or z axes results in the conservation of they or z components of the momentum Symmetry under rotation around the z axis implies conservation of the zcomponent of the angular momentum Of enormous significance for thermostatistics is the symmetry of dy namical laws under time translation That is the fundamental dynamical laws of physics such as Newtons law Maxwells equations and Schrodingers equation are unchanged by the transformation t t fv ie by a shift of the origin of the scale of time If the external potential is independent of time Noethers theorem predicts the existence of a conserved quantity That conserved quanhty is called the energy Immediately evident is the relevance of timetranslation symmetry to what is often called the first law of thermodynamics the existence of the energy as a conserved state function recall Section 13 and Postulate I It is instructive to reflect on the profundity of Noethers theorem by comparing the conclusion here with the tortuous historical evolution of the energy concept in mechanics recall Section 14 Identification of the conserved energy began in 1693 when Leibniz observed that mv 2 mgh is a conserved quantity for a mass particle in the earths gravitational field As successively more complex systems were studied it was found that additional terms had to be appended to maintain a conservation principle 4 See E Wigner 1b1d The physical content or Noether theorem 1s implicit m Emmy Noether purely mathematical studies A beautrul apprecialion or this bnllant mathcmalicians lire and work m the face of implacable prejudice can be round m the introductory remarks to her collected works Emmy Noether Gesammelte Abha11Jlu11gen Collected Papen Spnnger Verlag BerlmNew Yori 1983 bwrgr Momclllum and Angular Momentum the Gencraized First taw of Thermostatmcs 461 but that in each case such an ad hoc addition was possible The develop ment of electromagnetic theory introduced the potential energy of the interaction of electric charges subsequently to be augmented by the electromagnetic field energy In 1905 Albert Einstein was inspired to alter the expression for the mechanical kinetic energy and even to associate energy with stationary mass in order to maintain the principle of energy conservation In the 1930s Enrico Fermi postulated the existence of the neutrino solely for the purpose of retaining the energy conservation law in nuclear reactions And so the process continues successively accreting additional terms to the abstract concept of energy which is defined by its conservation law That conservation law was evolved historically by a long series of successive rediscoveries It is now based on the assumption of time translation symmetry The evolution of the energy concept for macroscopic thermodynamic systems was even more difficult The pioneers of the subject were guided neither by a general a priori conservation theorem nor by any specific analytic formula for the energy Even empiricism was thwarted by the absence of a method of direct measurement of heat transfer Only inspired insight guided by faith in the simplicity of nature somehow revealed the interplay of the concepts of energy and entropy even in the absence of a priori definitions or of a means of measuring either 214 ENERGY MOMENTUM AND ANGULAR MOMENTUM THE GENERALIZED FIRST LAW OF THERMOSTATISTICS In accepting the existence of a conserved macroscopic energy function as the first postulate of thermodynamics we anchor that postulate directly in Noethers theorem and in the timetranslation symmetry of physical laws An astute reader will perhaps turn the symmetry argument around There are seven first integrals of the motion as the conserved quantities are known in mechanics These seven conserved quantities are the energy the three components of linear momentum and the three components of the angular momentum and they follow in parallel fashion from the translation in spacetime and from rotation Why then does energy appear to play a unique role in thermostatistics Should not momentum and angular momentum play parallel roles with the energy In fact the energy is not unique in thermostahstics The linear momen tum and angular momentum play precisely parallel roles The asymmetry in our account of thermostatistics is a purely conventwna one that obscures the true nature of the sub1ect We have followed the standard convention of restricting attention to systems that are macroscopically stationary in which case the momentum 462 Postlude Symmetry and the Conceptual Foundatwns of Thermostalslcs and angular momentum arbitrarily are required to be zero and do not appear in the analysis But astrophysicists who apply thermostatistics to rotating galaxies are quite familiar with a more complete form of thermo statistics In that formulation the energy linear momentum and angular momentum play fully analogous roles The fully generalized canonical formalism is a straightforward extension of the canonical formalism of Chapters 16 and 17 Consider a subsystem consisting of N moles of stellar atmosphere The stellar atmosphere has a particular mean molar energy UN a particular mean molar momen tum P N and a particular mean molar angular momentum J N The fraction of time that the subsystem spends in a particular microstate i with energy E momentum P and angular moment J is EPJ V N Then is determined by maximizing the disorder or entropy subject to the constraints that the average energy of the subsys tem be the same as that of the stellar atmosphere and similarly for momentum and angular momentum As in Section 172 we quite evi dently find 1 Z exp 31 P P J J 211 The seven constants 3 px PP pz Jx Jy and Jz all arise as Lagrange parameters and they play completely symmetric roles in the theory just as 3µ does in the grand canonical formalism The proper first law of thermodynamics or the first postulate in our formulation is the symmetry of the laws of physics under spacetime translation and rotation and the consequent existence of conserved energy momentum and angular momentum functions 215 BROKEN SYMMETRY AND GOLDSTONES THEOREM As we have seen then the entropy of a thermodynamic system is a function of various coordinates among which the energy is a prominent member The energy is in fact a surrogate for the seven quantities conserved by virtue of spacetime translations and rotations But other independent variables also existthe volume the magnetic moment the mole numbers and other similar variables How do these arise in the theory The operational criterion for the independent variables of thermostatis tics recall Chapter 1 is that they be macroscopically observable The low temporal and spatial resolving powers of macroscopic observations re quire that thermodynamic variables be essentially time independent on the atomic scale of time and spatially homogeneous on the atomic scale of distance The time independence of the energy and of the linear and angular momentum has been rationalized through Noethers theorem Broken Symmetry and Goldstones Theorem 463 The time independence of other variables is based on the concept of broken symmetry and Goldstone s theorem These concepts are best intro duced by a particular case and we focus specifically on the volume For definiteness consider a crystalline solid As we saw in Section 167 the vibrational modes of the crystal are described by a wave number k 217 A where is the wavelength and by an angular frequency wk For very long wavelengths the modes become simple sound waves and in this region the frequency is proportional to the wave number w ck recall Fig 161 The significant feature is that wk vanishes for k 0 ie for oo Thus the very mode that is spatially homogeneous has zero frequency Furthermore as we have seen in Chapter 1 refer also to Problem 2151 the volume of a macroscopic sample is associated with the amplitude of the spatially homogeneous mode Consequently the volume is an acceptably time independent thermodynamic coordinate The vanishing of the frequency of the homogeneous mode is not simply a fortunate accident but rather it is associated with the general concept of broken symmetry The concept of broken symmetry is clarified by reflect ing on the process by which a crystal may be formed Suppose the crystal to be solid carbon dioxide dry ice and suppose the CO2 initially to be in the gaseous state contained in some relatively large vessel infinite in size The gas is slowly cooled At the temperature of the gassolid phase transition a crystalline nucleus forms at some point in the gas The nucleus thereafter grows until the gas pressure falls to that on the gassolid coexistence curve ie to the vapor pressure of the solid From the point of view of symmetry the condensation is a quite remarkable development In the infinite gas the system is symmetric under a continuous translation group but the condensed solid has a lower symme try It is invariant only under a discrete translation group Furthermore the location of the crystal is arbitrary determined by the accident of the first microscopic nucleation In that nucleation process the symmetry of the system suddenly and spontaneously lowers and it does so by a nonpredictable random event The symmetry of the system is broken Macroscopic sciences such as solid state physics or thermodynamics are qualitatively different from microscopic sciences because of the effects of broken symmetry as was pointed out by P W Anderson 5 in an early but profound and easily readable essay which is highly recom mended to the interested reader At sufficiently high temperature systems always exhibit the full symme try of the mechanical potential that is of the Lagrangian or Hamilto nian functions There do exist permissible microstates with lower symmetry but these states are grouped in sets which collectively exhibit the full symmetry Thus the microstates of a gas do include states with crystallike spacing of the moleculesin fact among the microstates all manner of different crystallike spacings are represented so that collec 5P W Anderson pp 175182 in Concepts in Sohds W A Benjamin Inc New York 1964 464 Postlude Symmetry and the Conceptual Foundatom of Thernws1a11s110 tively the states of the gas retain no overall crystallinity whatever How ever as the temperature of the gas is lowered the molecules select that particular crystalline spacing of lowest energy and the gas condenses into the corresponding crystal structure This is a partial breaking of the symmetry Even among the microstates with this crystalline periodicity there are a continuum of possibilities available to the system for the incipient crystal could crystallize with any arbitrary position Given one possible crystal position there exist infinitely many equally possible posi tions slightly displaced by an arbitrary fraction of a lattice constant Among these possibilities all of equal energy the system chooses one position ie a nucleation center for the condensing crystallite arbitrarily and accidentally An important general consequence of broken symmetry is formulated in the Goldstone theorem 6 It asserts that any system with broken symmetry and with certain weak restrictions on the atomic interactwns has a spectrum of excitations for which the frequency approaches zero as the wavelength becomes infinitely large For the crystal discussed here the Goldstone theorem ensures that a phonon excitation spectrum exists and that its frequency vanishes in the long wavelength limit The proof of the Goldstone theorem is beyond the scope of this book but its intuitive basis can be understood readily in terms of the crystal condensation example The vibrational modes of the crystal oscillate with sinusoidal time dependence their frequencies determined by the masses of the atoms and by the restoring forces which resist the crowding together or the separation of those atoms But in a mode of very long wavelength the atoms move very nearly in phase for the infinite wavelength mode the atoms move in unison Such a mode does not call into action any of the interatomic forces The very fact that the original position of the crystal was arbitrarythat a slightly displaced position would have had precisely the same energyguarantees that no restoring forces are called into play by the infinite wavelength mode Thus the vanishing of the frequency in the long wavelength limit is a direct consequence of the broken symmetry The theorem so transparent in this case is true in a far broader context with farreaching and profound consequences In summary then the volume emerges as a thermodynamic coordinate by virtue of a fundamental symmetry principle grounded in the concept of broken symmetry and in Goldstones theorem PROBLEMS 2151 Draw a longitudinal vibrational mode in a onedimensional system with a node at the center of the system and with a wavelength twice the nominal length 6 P W Andcrson hul Other Broken Symmetry CoordmatesElectricand Magnetic Moments 465 of the system Show that the instantaneous length of the system is a linear function of the instantaneous amplitude of this mode What is the order of magnitude of the wavelength if the system is macroscopic and if the wavelength is measured in dimensionless units ie relative to interatomic lengths 216 OTHER BROKEN SYMMETRY COORDINATESELECTRIC AND MAGNETIC MOMENTS In the preceding two sections we have witnessed the role of symmetry in determining several of the independent variables of thermostatistical theory We shall soon explore other ways in which symmetry underlies the bases of thermostatistics but in this section and in the following we continue to explore the nature of the extensive parameters It should perhaps be noted that the choice of the variables in terms of which a given problem is formulated while a seemingly innocuous step is often the most crucial step in the solution In addition to the energy and the volume other common extensive parameters are the magnetic and electric moments These are also prop erly time independent by virtue of broken symmetry and Goldstones theorem For definiteness consider a crystal such as HCl This material crystallizes with an HCl molecule at each lattice site Each hydrogen ion can rotate freely around its relatively massive Cl partner so that each molecule constitutes an electric dipole that is free to point in any arbitrary direction in space At low temperatures the dipoles order all pointing more or less in one common direction and thereby imbueing the crystal with a net dipole moment The direction of the net dipole moment is the residue of a random accident associated with the process of cooling below the ordering temper ature Above that temperature the crystal had a higher symmetry below the ordering temperature it develops one unique axisthe direction of the net dipole moment Below the ordering temperature the dipoles are aligned generally but not precisely along a common direction Around this direction the dipoles undergo small dynamic angular oscillations librations rather like a pendulum The librational oscillations are coupled so that libra tional waves propagate through the crystal These librational waves are the Goldstone excitations The Goldstone theorem implies that the librational modes of infinite wavelength have zero frequency 7 Thus the electric 7In the interests of clarity I have oversimplified slightly The discussion here overlooks the fact that the crystal structure would have already destroyed the spherical symmetry even above the ordering temperature of the dipoles That is the discussion as given would apply to an amorphous sphencally symmetric crystal but not to a cubic crystal In a cubic crystal each electric dipole would be coupled by an anisotropy energy to the cubic crystal structure and this coupling would naively appear to provide a restoring force even to infinite wavelength librational modes However under these circumstances librations and crystal vibrations would couple to form mixed modes and these coupled librationvibration modes would again satisfy the Goldstone theorem 466 Postlude Symmetry and the Conceptual Foundatwns of Thermostatist1cs dipole moment of the crystal qualifies as a time independent thermody namic coordinate Similarly ferromagnetic crystals are characterized by a net magnetic moment arising from the alignment of electron spins These spins par ticipate in collective modes known as spin waves If the spins are not coupled to lattice axes ie in the absence of magnetocrystalline ani sotropy the spin waves are Goldstone modes and the frequency vanishes in the long wavelength limit In the presence of magnetocrystalline ani sotropy the Goldstone modes are coupled phononspinwave excitations In either case the total magnetic moment qualifies as a time independent thermodynamic coordinate 217 MOLE NUMBERS AND GAUGE SYMMETRY We come to the last representative type of thermodynamic coordinate of which the mole numbers are an example Among the symmetry principles of physics perhaps the most abstract is the set of gaug symmetries The representative example is the gauge transformation of Maxwells equations of electromagnetism These equa tions can be written in terms of the observable electric and magnetic fields but a more convenient representation introduces a scalar poten tial and a vector potential The electric and magnetic fields are derivable from these potentials by differentiation However the electric and magnetic potentials are not unique Either can be altered in form providing the other is altered in a compensatory fashion the coupled alterations of the scalar and vector potentials constituting the gauge transformation The fact that the observable electric and magnetic fields are invariant to the gauge transformation is the gauge symmetry of electromagnetic theory The quantity that is conserved by virtue of this symmetry is the electric charge8 Similar gauge symmetries of fundamental particle theory lead to con servation of the numbers of leptons electrons mesons and other particles of small rest mass and of the numbers of baryons protons neutrons and other particles of large rest mass In the thermodynamics of a hot stellar interior where nuclear transfor mations occur sufficiently rapidly to achieve nuclear equilibrium the numbers of leptons and the numbers of baryons would be the appropriate mole numbers qualifying as thermodynamic extensive parameters In common terrestrial experience the baryons form longlived associa tions to constitute quasistable atomic nuclei It is then a reasonable 8The result is a uniquely quantum mechanical result It depends upon the fact that the phase of the quantum mechanical wave function is arbitrary gauge symmetry of the second kind and it is the interplay of the two types of gauge symmetry that leads to charge conservation Tme Reversal the Equal Probab1llfles of Mcrostates and the Entropy Prmcple 467 approximation to consider atomic or even molecular species as being in quasistable equilibrium and to consider the atomic mole numbers as appropriate thermodynamic coordinates 218 TIME REVERSAL THE EQUAL PROBABIUTIES OF MICROSTA TES AND THE ENTROPY PRINCIPLE We come finally to the essence of thermostatistics to the principle that an isolated system spends equal fractions of the time in each of its permissible microstates Given this principle it then follows that the number of occupied microstates is maximum consistent with the external constraints that the logarithm of the number of microstates is also maximum and that it is extensive and that the entropy principle is validated by interpreting the entropy as proportional to ln Q The permissible microstates of a system can be represented in an abstract manydimensional state space recall Section 155 In the state space every permissible microstate is represented by a discrete point The system then follows a random erratic trajectory in the space as it undergoes stochastic transitions among the permissible states These tran sitions are guaranteed by the random external perturbations which act on even a nominally isolated system although other mechanisms may dominate in particular casesrecall Section 151 The evolution of the system in state space is guided by a set of transition probabilities If a system happens at a particular instant to be in a microstate i then it may make a transition to the state j with probability per unit time lr The transition probabilities 1 form a network joining pairs of states throughout the state space The formalism of quantum mechanics establishes that at least in the absence of external magnetic fields9 212 That is a system in the state i will undergo a transition to the state j with the same probability that a system in state j will undergo a transition to the state i The principle of detailed balance equation 212 follows from the symmetry of the relevant laws of quantum mechanics under time inversion ie under the transformation t t 9 The restnct10n that the external magnehc field must be zero can be dealt with most sm1ply by mcludmg the source of the magnetic field as part of the ystem In any case the presence of external magneuc fields complicates intermediate statements but does not alter final conclusions and we shall here ignore such fields in the interests of simphc1ty and danty 468 Pmtlude Srmmetry and the Conuptua Foundatums of ThermostalfllS Although we merely quote the principle of detailed balance as a quantum mechanical theorem it is intuitively reasonable Consider a system in the microstate i and imagine a video tape of the dynamics of the system a hypothetical form of video tape that records the microstate of the system After a brief moment the system makes a transition to the microstate If the video tape were to be played backwards the system would start in the state j and make a transition to the state z Thus the interchangeability of future and past or the time reversibility of physical laws associates the transitions z j and J z and leads to the equality 212 of the transition probabilities The principle of equal probabilities of states in equilibrium 1 Q follows from the principle of detailed balance To see that this is so we first observe that 1 is the conditional piobability that the system will undergo a transition to state j zf it is initially in state i The number of such transitions per unit time is then the product of f1 and the probability that the system is initially in the state z Hence the total number of transitions per unit time out of the state 1 is fJ Similarly the number of transitions per unit time into the state i is L 1 Ip However in equilibrium the occupation probability of the th state must be independent of time or df dt Lf1 LI 0 Fl fPl 213 With the symmetry condition l a general solution of equation 213 is for all i and j That 1s the configuration 1 Q is an eqwltbnum configuration for any set of transition probabilities 1 for which 1 f As the system undergoes random trans1 t1ons among its microstates some states are visited frequently ie L is large and others are visited only infrequently Some states are tenacious of the system once it does arrive ie r 111 is small whereas others permit it to depart rapidly Because of time reversal symmetry however those states that are visited only infrequently are tenacious of the system Those states that are visited frequently host the system only fleetingly By virtue of these compensating attributes the system spends the same fraction of time in each state The equal probab1itzes of permissible states for a closed system m eqw 1brium is a consequence of time reversal symmetry of the relevant quantum mechanical laws 10 1111n fact d weaker condllon L J 0 which follow from a more abrract reqmrcmcnt of cauahty s abo utticent to cnurc that 12 m cquhbnum Tiu fact doe not mvahdatc the prcvou statement Symmetn and Completeness 469 219 SYMMETRY AND COMPLETENESS There is an additional more subtle aspect of the principle of equal a priori probabilities of states Consider the schematic representation of state space in Fig 212 The boundary B separates the permissible states inside from the nonpermissible states outside The transition probabilities J1 are symmetric for all states I and j inside the bound ary B A FlURI 21 2 Suppose now that the pemms1ble region in state space is divided into two subregions denoted by A and A in Fig 212 such that all transition probabilities f vanish if the state i 1s in A and J is in A or vice versa Such a set of transition probabihlles is fully consistent with time reversal symmetry or detailed balance but it does not lead to a probability uniform over the physically permissible region A A If the system were initially in A the probability density would diffuse from the initial state to eventually cover the region A uniformly but it would not cross the internal boundary to the region A The accident of such a zero transition boundary separatmg the perm1ss1ble states into nonconnected subsets would lead to a failure of the assumption of equal probabilities throughout the permissible region of state space It is important to recognize how incredibly stnngent must be the rule of vanishing of the f between subregions if the principle of equal probabilities of states is to be violated It is not sufficient for transition prohab1littes between subregions to be very smallevery such transition probabihty must be absolutely and rigorously zero If even one or a few transition probabilities were merely very small across the internal boundary it would take a very long time for the probability density to fill both A and A uniformly but eventually it would The accident that we feared might vitiate the conclusion of equal probabilities appears less and less likelyunless it is not an accident at all but the consequence of some underlying principle Throuihrn1t rnrnn 470 Postlude Symmetry and the Conceptual Foundatwns of Thermos1allsllcs tum physics the occurrence of outlandish accidents is disbarred physics is neither mystical nor mischievous If a physical quantity has a particular value say 45172 then a second physical quantity will not have precisely that same value unless there is a compelling reason that ensures equality Degeneracy of energy levels is the most familiar examplewhen it occurs it always reflects a symmetry origin Similarly transition prob abilities do not accidentally assume the precise value zero when they do vanish they do so by virtue of an underlying symmetry based reason The vanishing of a transition probability as a consequence of symmetry is called a selection rule Selection rules that divide the state space into disjoint regions do exist They always reflect symmetry origins and they imply conservation princi ples An already familiar example is provided by a ferromagnetic system The states of the system can be classified by the components of the total angular momentum States with different total angular momentum com ponents have different symmetries under rotation and the selection rules of quantum mechanics forbid transitions among such states These selec tion rules give rise to the conservation of angular momentum More generally then the state space can be subdivided into disjoint regions not connected by transition probabilities These regions are never accidental they reflect an underlying symmetry origin Each region can be labeled according to the symmetry of its statessuch labels are called the characters of the group representation The symmetry thereby gives rise to a conserved quantity the possible values of which correspond to the distinguishing labels for the disjoint regions of state space In order that thermodynamics be valid it is necessary that the set of extensive parameters be complete Any conserved quantity such as that labelling a disjuncture of the state space must be included in the set of thermodynamic coordinates Specifying the value of that conserved quan tity then restricts the permissible state space to a single disjoint sector A alone or A alone in Fig 212 The principle of equal probabilities of states is restored only when all such symmetry based thermodynamic coordinates are recognized and included in the theory Occasionally the symmetry that leads to a selection rule is not evident and the selection rule is not suspected in advance Then conventional thermodynamics leads to conclusions discrepant with experiment Puzzle ment and consternation motivate exploration until the missing symmetry principle is recognized Such an event occurred in the exploration of the properties of gaseous hydrogen at low temperatures Hydrogen molecules can have their two nuclear spins parallel or antiparallel the molecules then being designated as orthohydrogen or parahydrogen respec tively The symmetries of the two types of molecules are quite different In one case the molecule is symmetric under reflection in a plane perpendicu lar to the molecular axis in the other case there is symmetry with respect to inversion through the center of the molecule Consequently a select10n Smmetry and Completeness 47 I rule prevents the conversion of one form of molecule to the other This unsuspected selection rule led to spectacularly incorrect predictions of the thermodynamic properties of H 2 gas But when the selection rule was at last recognized the resolution of the difficulty was straightforward Ortho and parahydrogen were simply considered to be two distinct gases and the single mole number of hydrogen was replaced by separate mole numbers With the theory thus extended to include an additional con served coordinate theory and experiment were fully reconciled Interestingly a different operational solution of the orthoH 2 paraH 2 problem was discovered If a minute concentration of oxygen gas or water vapor is added to the hydrogen gas the properties are drastically changed The oxygen atoms are paramagnetic they interact strongly with the nuclear spins of the hydrogen molecules and they destroy the symmetry that generates the selection rule In the presence of a very few atoms of oxygen the ortho and parahydrogen become interconvertible and only a single mole number need be introduced The original naive form of thermodynamics then becomes valid To return to the general formalism we thus recognize that all symme tries must be taken into account in specifying the relevant state space of a system As additional symmetries are discovered in physics the scope of thermo statistics will expand Perhaps all the symmetries of an ideal gas at standard temperatures and pressures are known but the case of ortho and parahydrogen cautions modesty even in familiar cases Moreover thermodynamics has relevance to quasars and black holes and neutron stars and quark matter and gluon gases For each of these there will be random perturbations and symmetry principles conservation laws and Goldstone excitationsand therefore thermostatistics A1 PARTIAL DERIVATIVES APPENDIX A SOME RELATIONS INVOLVING PARTIAL DERIVATIVES In thermodynamics we are interested in continuous functions of three or more variables if ifx y z AI If two independent variables say y and z are held constant if becomes a function of only one independent variable x and the derivative of if with respect to x may be defined and computed in the standard fashion The derivative so obtained is called the partial derivative of if with respect to x and is denoted by the symbol ai I ax y z or simply by iif I ax The derivative depends upon x and upon the values at which y and z are held during the differentiation that is aiax is a function of x y and z The derivatives aiay and aiaz are defined in an identical manner The function ai ax if continuous may itself be differentiated to yield three derivatives which are called the second partial derivatives of if A2 By partial differentiation of the functions aiJay and aiJaz we obtain other second partial derivatives of if a2fl iziJy A7 a2i ax iz 474 Some Relatwns novmg Partwl Der1latwes It may be shown that under the continuity conditions that we have assumed for if and its partial derivatives the order of differentiation is immaterial so that a2f azax A3 There are therefore just six nonequivalent second partial derivatives of a function o three independent variables three for a function of two variables and nn 1 for a function of n variables A2 TAYLORS EXPANSION The relationship between ifxyz and ifx dxy dyz dz where dx dy and dz denote arbitrary increments in x y and z is given by Taylors expansion ifx dx y dy z dz a2f 2 a2f a2f a 21 dz 2 dx dy 2 dx dz 2 dy dz az 2 axay axaz ayaz A4 This expansion can be written in a convenient symbolic form if x dx Y dy z dz edxix tdiJa l tdzllifx y z A5 Expansion of the symbolic exponential according to the usual series 1 1 ex 1 X x 2 x 2 n A6 then reproduces the Taylor expansion equation A4 Composite Functwns 475 A3 DIFFERENTIALS The Taylor expansion equation A4 can also be written in the form l x dx y dy z dz l x y z d 1 d2 1 dn o 2 o n o A7 where dl al dx al dy al dz ax ay az A8 d2l a2 dx 2 a2 dy 2 a2 dz 2 2 aa2ao dx dy ax ay az X Y a2tJ a2f 2 ax az dx dz 2 a ya z dy dz A9 and generally a a a n d no dx a X dy a y dz a z o X y Z AIO These quantities dl d 2l dnl are called the first second and nthorder differentials of l A4 COMPOSITE FUNCTIONS Returning to the firstorder differential A11 an interesting case arises when x y and z are not varied independently but are themselves considered to be functions of some vanable u Then whence dx dx du du dy du du and dz dz du du dtJ atJ dx al dy atJ dz d a X V z du a y z du Tz f du u A12 476 Some Reatons ltwollmg Partial Derwatwes or A13 If x and y are functions of two or more variables say u and v then dx ax du x dv etc dU V UV U and dl al ax al ay al az du ax yz au I ay xz au I au t au tLJL tLJL tLJLdv A14 or dl o au V du al dv av U A15 where l l I t XJ l X J l A16 and similarly for al av u It may happen that u is identical to x itself Then tv 1 Jxi A17 Other special cases can be treated similarly A5 IMPLICIT FUNCTIONS If l is held constant the variations of x y and z are not independent and the relation l x y z constant A18 lmplwt Furutwns 477 gives an implicit functional relation among x y and z This relation may be solved for one variable say z in terms of the other two z zxy A19 This function can then be treated by the techniques previously described to derive certain relations among the partial derivatives However a more direct method of obtaining the appropriate relations among the partial derivatives is merely to put di 0 in equation A8 ai ai ai 0 dx dy dz ax Jl ay XZ az X V A20 If we now put dz 0 and divide through by dx we find 0 ai aif a y ax yz ay xz ax Jz A21 in which the symbol a y ax J 2 appropriately indicates that the implied functional relation between y and x is that determined by the constancy of if and z Equation A21 can be written in the convenient form A22 This equat10n plays a very prominent role in thermodynamic calculations By successively putting dy 0 and dx 0 in equation A 20 we find the two similar relations Ly aiaxJZ ai az xy A23 and Lx aiayxz aiaz Ly A24 Returning to equation A20 we again put dz 0 but we now divide through by dy rather than by dx whence 0 ai ax ai ax z ay Jz ay z ax ay Jz aiay aiaxz A25 A26 478 Some Relations lnvovmg Partwl Deriatwes and on comparison with equation A21 we find the very reasonable result that i Lz 1 A27 From equations A22 to A24 we then find i LJ LJ Ly l A28 Finally we return to our basic equation which defines the differential df and consider the case in which x y and z are themselves functions of a variable u as in equation A12 dfaf dx af dy al dzdu A 29 ax yzdU ay XZdU az XdU If I is to be constant there must be a relation among x y and z hence also among dxdu dydu and dzdu We find O tLL LJL LJL AJO If we further require that z sha11 be a constant independent of u we find or O yz z t xz fz ay au fz axaufz af ax y aiJaYtz Comparison with equation A22 shows that ay ayaufz ax fz axauz A31 A32 A33 Equations A22 A27 and A33 are among the most useful formal manipulations in thermodynamic calculations APPENDIXB MAGNETIC SYSTEMS If matter is acted on by a magnetic field it generally develops a magnetic moment A description of this magnetic property and of its interaction with thermal and mechanical properties requires the adoption of an additional extensive parameter This additional extensive parameter X and its corresponding intensive parameter P are to be chosen so that the magnetic work dWmag IS dWmag PdX Bl where dU dQ dWM dWc dWmag B2 Here dQ is the heat T dS dW is the mechanical work eg P dV and dW is the chemical work µ 1 dNr We consider a specific situation that clearly indicates the appropriate choice of parameters X and P Consider a solenoid or coil as shown in Fig B1 The wire of which the solenoid is wound is assumed to have zero electrical resistance supercon ducting A battery is connected to the solenoid and the electromotive force emf of the battery is adjustable at will The thermodynamic system is inside the solenoid and the solenoid is enclosed within an adiabatic wall If no changes occur within the system and if the current I is constant the battery need supply no emf because of the perfect conductivity of the wire Let the current be I and let the local magnetization of the thermody namic system be Mr The current I can be altered at will by controlling the battery emf The magnetization Mr then will change also We assume that the magnetization at any position r is a singlevalued function of the current Mr Mr I R 1 480 Magnetic Systems FIGURE Bl Battery of adjustable emf Adabatrc wall Wire of zero electrical resistivity Systems for which Mr is not single valued in I are said to ex hibithystereszs most ferromagnetic systems have this property Hysteresis generally is associated with a magnetic heterogeneity of the sample the separate regions being known as domains The analysis we shall develop is generally applicable within a ferromagnetic domain but for simplicity we explicitly exclude all hysteretic systems Paramagnetic diamagnetic and antiferromagnetic systems satisfy the requirement that Mr I is single valued in If the thermodynamic system were not within the solenoid the current I would produce a magnetic field more accurately a magnetic flux densuy Be I This external field may be a function of position within the solenoid but it is linear in That is B4 where b is a vector function of position We suppose that the current is increased thereby increasing the exter nal field Be The magnetic moment changes in response In order to accomplish these changes the battery must deliver work and we seek the relationship between the work done and the changes in Be and M The rate at which work is done by the battery is given by dWmag I xvoltage B5 in which voltage denotes the back emf induced in the solenoid windings by the changes that occur within the coil The induced emf in the solenoid arises from two sources One source is independent of the thermodynamic system and results from a change in Magnetic Systems 481 the flux associated with the field Be Rather than compute this flux and voltage we can write the resultant contribution to dWmag directly For an empty solenoid the work is just the change in the energy of the magnetic field or B6 in which µ0 41T X 10 1T mA and in which the integral is taken over the entire volume of the solenoid The second contribution to dWmag results from the thermodynamic system itself and consequently is of more direct interest to us It is evident that the change of magnetic moment of each infinitesimal element of the system contributes separately and additively to the total induced emf and furthermore that the induced emf produced by any change in dipole moment depends not on the nature of the dipole but only on the rate of change of its moment and on its position in the solenoid Consider then a particular model of an elementary dipole at the position r a small current loop of area a and current i with a magnetic moment of m ia If the current in the solenoid is I the field produced by the solenoid at the point r is Ber br This field produces a flux linkage through the small current loop of magnitude br a Thus the mutual inductance between solenoid and current loop is br a If the current in the current loop changes it consequently induces a voltage in the solenoid given by d voltage br a d dm br dt 1 dm Ber dt Thus the work done by the battery is dWmag dm Ber dt B7 B8 B9 B10 Although this result has been obtained for a particular model of an elementary dipole it holds for any change in elementary dipole moment In particular if Mr is the magnetization or the dipole moment per unit volume in the system at the point r we set m f Mr dV B11 482 Magnetic Systems To obtain the total work we sum over all elementary dipoles or integrate over the volume of the sample dWmag JB dM dV dt e dt B12 Adding the two contributions to the magnetic work we find B13 This is the fundamental result on which the thermodynamics of magnetic systems is based In passing we note that the local field H can be introduced in place of the external field He by noting that the difference H He is just the field produced by the magnetization M r acting as a magnetos ta tic source In this way it can be shown1 that dWmag f H dBdV B14 where H and B are local values However the form of the magnetic work expression we shall find most convenient is the first derived equation B13 In the general case the magnetization Mr will vary from point to point within the system even if the external field Be is constant This variation may arise from inherent inhomogeneities in the properties of the system or it may result from demagnetization effects of the boundaries of the system We wish to develop the theory for homogeneous systems We therefore assume that Be is constant and that the intrinsic properties of the system are homogeneous We further assume that the system is ellipsoidal in shape For such a system the magnetization M is indepen dent of position as shown in any text on magnetostatics The magnetic work equation can now be written as B15 where I is the total magnetic dipole moment of the system I jMdV MV B16 1See V Heine Proc Cambridge Phil Soc 52 546 1956 Magnetic Systems 483 The energy differential is dEnergy TdS PdV df f Bdv Be di tµ 1 d µo 1 B17 The third term on the right of the foregoing equation does not involve the thermodynamic system itself but arises only from the magnetostatic energy of the empty solenoid Consequently it is convenient to absorb this term into the definition of the energy We define the energy U by 1 f 2 U Energy 2µ 0 Be dV B18 so that U is the total energy contained within the solenoid relative to the state in which the system is removed to its field free fiducial state and the solenoid is left with the field Be This redefinition of the internal energy does not alter any of the formalism of thermodynamics Thus we write r dU TdS PdV Bed 8 Lµ1 d B19 I where I 8 is the component of I parallel to B The extensive parameter descriptive of the magnetic properties of a system is l 8 the component of the total magnetic moment parallel to the external field The intensive parameter in the energy representation is Be The fundamental equation is U USV1 8 N 1 NJ B20 and B21 Ketchup Tomato catsup or tomato sauce was probably first made in China as a fermented fish sauce called ketsiap It was brought to Europe in the 17th century by traders GENERAL REFERENCES THERMODYNAMICS R Kubo Thermodynamics Wiley 1960 Concise text with many problems and explicit solutions K J Laidler and J F Meiser Physical Chemistry BenjaminCummings 1982 Chemical applications of thermodynamics A B Pippard Elements of Classical Thermodynamics Cambridge Univer sity Press 1966 A scholarly and rigorous treatment R E Sonntag and G J Van Wylen Introduction to Thermodynamics Classical Statistical 2nd edition Wiley 1982 Very thorough thermo dynamic treatment Engineering viewpoint G Weinreich Fundamental Thermodynamics AddisonWesley 1968 Idi osyncratic insightful and original M W Zemansky and R H Dittman Heat and Thermodynamics An Intermediate Textbook 6th edition McGrawHill 1981 Contains care ful and full discussions of empirical data experimental methods practi cal thermometry and applications STATISTICAL MECHANICS R P Feynman Statistical Mechanics A Set of Lectures W A Benjamin 1972 Advancedlevel notes with the unique Feynman flair Particularly strong emphasis on the Bogoliubov variational theorem R J Finkelstein Thermodynamics and Statistical PhysicsA Short Intro duction W H Freeman and Co 1969 A brief and unconventional formulation of the logic of thermostatistics J W Gibbs The Scientific Papers of J Willard Gibbs Volume 1 Thermo dynamics Dover 1961 Gibbs not only invented modern thermody namics and statistical mechanics but he also anticipated explicitly or implicitly almost every subsequent development His exposition is not noted for its clarity C Huang Statistical Mechanics Wiley 1963 Classic graduate text 485 486 General References C Kittel and H Kroemer Thermal Physics 2nd edition W H Freeman 1980 Introductory treatment Large number of interesting illustrative applications R Kubo Statistical Mechanics Wiley 1965 Concise text with many problems and explicit solutions L D Landau and E M Lifshitz Statistical Physics 3rd edition Part 1 by E M Lifshitz and L P Pitaevskii Pergamon Press 1980 E M Lifshitz and L P Pitaevskii Statistical Physics Part 2 of reference above Pergamon Press 1980 Advanced treatment P T Landsburg Thermodynamics and Statistical Mechanics Oxford Univ Press 1978 Contains many novel observations and 120 fullysolved problems F Reif Fundamentals of Statistical and Thermal Physics McGrawHill 1965 Classic text with an immense collection of excellent problems M Tribus Thermostatics and Thermodynamics Van Nostrand 1961 A development based on the informationtheoretic approach of E T Jaynes CRITICAL PHENOMENA D J Amit Field Theory the Renormalization Group and Critical Phe nomena McGrawHill 1978 Advanced theory ShangKeng Ma Modern Theory of Critical Phenomena Benjamin 1976 P Pfeuty and G Toulouse Introduction to the Renormalization Group and to Critical Phenomena Wiley 1977 H E Stanley Introduction to Phase Transitions and Critical Phenomena Oxford University Press 1971 Excellent introduction Predates Wilson renormalization theory CONCEPTUAL OVERVIEWS P W Anderson Basic Notions of Condensed Matter Physics BenjaminCummings 1984 A profound and pentrating analysis of the role of symmetry in the general theory of properties of matter Al though the level is quite advanced Andersons interest in underlying principles of universal generality rather than in mathematical tech niques of calculation make the book a treasure for the reader at any technical level R D Rosenkrantz editor E T Jaynes Papers on Probability Statistics and Statistical Physics Reidel 1983 An unconventional conceptualiza tion of statistical mechanics as an informationtheoretic exercise in prediction Jaynes point of view is reflected in a rather pale form in Chapter 17 of this text INDEX Adiabat 43 Adiabatic wall 1 7 A Adsorption on surface 388 391 Affinities 308 311 Alloy binary 263 355 449 Anderson P W 463 Atomic mass 9 Auxiliary system 118 Availability 113 Available work at constant temperature 158 Average and most probable values 270 Avogadros number 9 B Baryons conservation of 466 Binary alloy 263 355 449 Bogoliubov variational theorem 433 435 Boltzmanns constant 47 Born Max 183 Bose condensation 413 temperature 416 BoseEinstein permutational parity 373 Bose fluid 403410419 nonconserved 412 Boson 373 393 Brayton cycle 129 Broken symmetry 462 C Canonical formalism 349 Canonical partition sum 3 51 Caratheodory 27 48 Carnot cycle 118 Celsius temperature scale 48 Central limit theorem 456 Chemical potential 36 55 56 417 Chemical reactions 56 167 in ideal gases 292 ClapeyronClausius approximation 23 l Clapeyron equation 228 286 Classical ideal gas 372 Classical limit of quantum fluids 402 Closed systems 17 26 Coexistence curve 221 228 Composite systems 26 Compressibility adiabatic 86 190 isothermal 84 1 91 Concavity condition of stability 204 208 Conductivity electric and thermal 319 Configuration space 9 5 487 488 Index Constraints 15 internal 26 Convexity condition of stability 207 208 Correlation moment delayed 315 of fluctuations 426 Critical exponents 263 opalescence 255 423 430 point 221 240 25 5 transitions 255 Crystal vibrational modes 333 Einstein model of 333 melting temperature 336 Cycle Brayton 129 Carnot 118 Diesel 130 Joule 129 Otto 128 D Debye Peter 88 365 Debye model 364 temperature 366 Degree of reaction 169 Degrees of freedom 62 24 7 Density of states 362 classical 3 70 of orbital states 362 364 Dewar walls 16 Diathermal walls 17 Diatomic molecule equipartition theorem 376 Diesel cycle 130 Differentials imperfect 20 Dilute solutions 302 Disorder 379 455 DNA 361 E Einstein model of crystal 333 354 temperature 335 Elastic strain and stress components 305 Electrochemical potential 35 Electromagnetic radiation 78 368 412 Electrons in metal 405 Element as independent mode 353 Endoreversible engine 125 efficiency of 12 7 Energy conservation of 11 internal 11 minimum principle 131 representation 41 units 21 Engine coefficient of performance 113 125 efficiency 106 114 endoreversible 12 7 thermodynamic 91 113 Ensemble 360 Enthalpy 147 160 of formation 173 magnetic 201 minimum principle 156 standard enthalpy of formation 174 Entropy 27 absolute 279 current density 310 measurability of 123 of mixing 69 108 290 production in irreversible process 309 representation 41 329 statistical mechanical interpretation 331 Equation of state 37 generalized 301 reduced 300 universal 300 Equilibrium 13 metastable 15 stable 31 unstable 31 quilibrium constant for ideal gas reactions 293 logarithmic additivity of 293 Equipartition theorem 375 for polyatomic molecule 376 value of heat capacity 291 Euler equation 59 284 Eutectic solution 251 Exothermic process 277 Expansion coefficient 84 Extensive parameters 10 energetic 42 entropic 42 F Fahrenheit temperature scale 48 FermiDirac permutational parity 373 Fermi fluid ideal 399 403 Fermi gas see Fermi fluid Fermi level 394 404 Fermion 373 393 Fermi temperature for electrons in metal 406 for nucleons 406 for white dwarf stars 406 Ficks law of diffusion 314 First law of thermostatistics generalized 461 First order function 28 First order phase transitions 243 245 Fluctuationdissipation theorem 308 Fluctuations 218 423 Fluxes 308 310 317 Free energy Gibbs 147 Helmholtz 15 146 Free energy functional 257 Free expansion 192 Fugacity 403 414 Fundamental relation 28 energetic 41 entropic 41 underlying 205 G Gas constant 66 Gasoline engine 128 Gauge symmetry 466 Index 489 Gaussian probability density 431 456 Geometry line and point 140 Gibbs Josiah Willard 27 370 371 GibbsDuhem relation 60 284 Gibbs phase rule 245 286 Gibbs potential or free energy 14 7 167 minimum principle for 167 standard molar Gibbs potential 174 Gibbss theorem 69 289 Goldstones theorem 462 Grand canonical formalism 3 85 451 partition sum 386 potential 148386401418 Gruneisen model 339 parameter 368 Hall effect 325 Hamiltonian 145 Heat 8 18 36 H of fusion 223 quasistatic 18 19 ofreaction 169 294 of sublimation 223 of vaporization 223 490 Index Heat capacity at constant pressure 84 at constant volume 86 of electrons in metal 408 of ideal Bose fluid 421 Heat flow IO 1 Heat pump 115 coefficient performance of 116 Heat source reversible 104 Heisenberg model of ferromagnetism 445 Heisenberg uncertainty principle 372 Helium four 4He 400 Helium three 3He 400 Helmholtz potential or free energy 146 157 additivity of 354 minimum principle for 155 Homogeneous firstorder functions 28 Homogeneous zeroorder functions 37 Homonuclear molecules 377 Hydrogen ortho and para 470 I Ice skating 229 Ideal Bose fluid 403 Ideal Fermi fluid 399 403 Ideal gas classical 3 72 general 289 monatomic 66 simple 66 Imperfect differential 20 Impermeable walls 16 Indistinguishability of particles 373 Information theory 380 Intensive parameters or intensive variables 35 38 entropic 40 Inversion temperature 162 Irreversibility 18 Irreversible thermodynamics 307 Isenthalp or isenthalpic process 163 278 Isentrope or isentropic process 43 Ising model 258 440 spins 446 Isobar or isobaric process 42 Isochore 1 77 Isotherm or isothermal process 39 Isotope separation 108 J Joule cycle 129 JouleKelvin process 162 JouleThomson process 160 K Kelvin relations of irreversible response 316 323 325 Kelvin scale of temperature 4 7 Kinetic coefficients 31 3 Kubo relations 308 L Landau L D 257 Landau theory 265 Latent heat 222 of fusion 222 of sublimation 222 of vaporization 223 Law of corresponding states 299 LeChatelierBraun principle 212 LeChateliers principle 210 Legendre transform 142 285 Legendre transformation 1 3 7 Leptons conservation of 466 Lever rule 239 244 Liquidus curve 250 M Macroscopic coordinates 5 Magnetic field 82 quantum number 394 susceptibility 89 systems 81 199479 Mass action law 29 3 Massieu functions 151 423 maximum principles for 179 Materials properties of 289 Maximum work theorem 103 Maxwell relations I 81 285 Mean field theory 440 449 Mean square deviation of fluctuations 424426 Melting temperature 222 Metastable equilibrium 15 Microcanonical formalism 329 332 Mnemonic diagram 183 286 for grand canonical potential 387 Modes of excitation 292 electronic 355 rotational 355 356 translational 355 vibrational 35 5 Molar mass 9 volume 10 Mole 9 fraction 9 number 9 Monier chart I 77 Moments of fluctuating parameters 424 correlation moments 426 Monatomic ideal gas 66 Most probable and average values 270 N Nernst Walter 277 Nernst effect 325 Nemst postulate 30 277 Noether Emily 40 Noethers theorem 460 Normal coordinates 7 modes 7 0 Ohms law of electrical conduction 314 Onsager L 258 307 Onsager reciprocity theorem 307 theoretical basis of 314 Orbital state 353 Order and disorder 380 Order parameter 255 256 263 Otto cycle 128 p Paramagnet 83 355 Partial molar Gibbs potential 168 Partition sum canonical 351 factorizability of 353 rotational 357 translational 356 375 vibrational 356 Peltier coefficient 323 effect 323 heat 323 Permeable walls 16 Perturbation expansion for equation of state 297 for Helmholtz potential 473 Pfaffian forms 49 Phase diagram binary systems 248 water 216 Phase transitions first order 215 second order 21 7 Photons 412 492 Index Planck Max 30 277 278 radiation law 370 Pluecker line geometry 140 Polymer model 339 358 391 Potential for heat enthalpy 161 170 Potentials thermodynamic 146 Pressure partial 73 Probabilities of states 358 Q Quadratic potential 436 Quantum fluids 39 3 403 Quantum regime for gases 405 Quartic potential 436 Quasistatic process 19 95 R Radiation 78 Rankine scale of temperature 4 7 Raoults law 304 RayleighJeans law 370 Rayleigh scattering 430 Reduced variables 301 Reduction procedure for derivatives 186 Refrigerator coefficient of performance 115 Regelation 232 Relaxation time 99 Reservoir thermal 106 Resistive systems in irreversible thermodynamics 312 Reversible process 91 Reversible work source 103 Rotational modes equipartition 376 Rubber band 80 339 Rushbrooks scaling law 275 s Scaling 272 Schottky hump 338 339 Second order phase transitions 255 Seebeck effect 320 Selection rules 4 70 Shannon Claude 380 Simon Francis 277 Simple ideal gas 66 Simple systems 9 Small nonmacroscopic systems 360 Solid systems 305 Solidus curve 250 Solutions dilute 302 Square well potential 438 Stability 44 203 286 convexity condition 207 global 205 local 205 physical consequences 209 State space 344 Steam tables saturated 224 superheated 175 176 StefanBoltzmann Law 78 369 Stellar interiors conservation laws in 466 Stirling approximation 334 Stoichiometric coefficients 56 Superfluidity 422 Surface adsorption 388 391 Susceptibility magnetic 89 90 generalized 25 5 Symmetry 458 and completeness 469 broken 462 gauge 466 T Temperature 43 measurability of 123 negative 29 scales 47 units 46 Tepsion 80 339 Thermal expansion coefficient 84 Thermal reservoir 106 Thermal volume 405 Thermal wave length 405 414 Thermodynamic engine efficiency 105 Thermoelectric effects 316 Thermoelectric power 320 absolute 322 Thermometer ideal gas constant pressure 72 ideal gas constant volume 71 Third Law 30 Thomsen and Bertolot principle of 277 Thomson effect 324 Throttling process 160 Time reversal symmetry 307 467 Tisza L 183257 Trace of quantum operators 434 Triple point 232 247 Twostate model 337 354 u Uncertainty principle 372 Unconstrainable variables 81 Underlying fundamental equation 205 Universality 272 van der Waals constants 77 V equation 74 ideal fluid 74 van der Waals J D 74 vant Hoff relation Index 493 for heat of reaction 294 for osmotic pressure 303 Vapor pressure 232 in dilute solutions 303 Variational principle 433 Vibrational modes of crystal 333 365 Virial coefficient 297 Virial expansion 297 w Walls Dewar 16 diathermal 1 7 impermeable 1 7 permeable 1 7 restrictive 15 Widoms scaling law 275 Wigner Eugene 459 Wilson Kenneth 25 9 Work 19 36 chemical 36 Work source reversible 103 z Zeroorder functions 37 Zeropoint energy 369 Zero temperature properties at 287 unattainability of 281 This original ketchup was dark blackishbrown and did not contain tomatoes In the early 19th century ketchup became very popular in America where tomatoes were added Today ketchup usually refers to tomato ketchup It is used as a condiment on hamburgers hot dogs french fries and many other foods UNITS AND CONVERSION FACTORS Energy 1 Joule Pressure 10 7 ergs 02389 calories 9480 X 10 4 Btu 9869 x 10 3 literatmospheres 07376 footpounds 2778 x 10 4 watthours 3724 X 10 7 horsepowerhours 1 Pascal Volume 1 Newtonm 2 10 dynescm 2 10 baryes 10 5 bars 1450 x 10 4 psi poundsinch 2 0 9869 X 10 5 atmospheres 75006 X 10 3 Torr or mm Hg 1 m3 10 6 cm3 10 3 liters 61024 x 104 inch3 35315 ft 3 26417 US gallons 21997 British Imperial gallons Temperature T 0 C TKelvin 27315 T 0 R 18 x TKelvin 10 7 J erg 4186 Jcal 1055 JBtu 1013 J literatm 13561ftpound 3600 Jwatthr 2685 X 106 J hphr 6897 Papsi l013 X 10 5 Paatm 1333 PaTorr 1639 X 10 5 m3in 3 02832 m3 ft 3 3785 x10 3 m3gal 4546 X 10 3 m3 gal TF T 0 R 45967 18 x T 0 C 32 CONSTANTS R 8314 JoulemoleKelvin 1986 caloriesmoleKelvin k 8 1381 X 10 23 JoulesKelvin NA Rk 8 6022 x 1023 mole h 6626 X 10 34 Joulesec