Artigo - Thermodynamics Of Fluid-phase

ANNIVERSARY ARTICLE Thermodynamics of Fluid-Phase Equilibria for Standard Chemical Engineering Operations John M. Prausnitz Dept of Chemical Engineering, University of California, Berkeley, CA 94720 and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Frederico W. Tavares Dept of Chemical Engineering, University of California, Berkeley, CA 94720 and Escola de Quı´mica, Universidade Federal do Rio de Janeiro, Caixa Postal 68542, CEP, 21949–900, Rio de Janeiro, RJ, Brazil DOI 10.1002/aic.10069 Published online in Wiley InterScience (www.interscience.wiley.com). Thermodynamics provides one of the scientific cornerstones of chemical engineering. This review considers how thermodynamics is and has been used to provide phase equilibria as required for design of standard chemical engineering processes with emphasis on distillation and other conventional separation operations. While this review does not consider “modern” thermodynamics for high-tech applications, attention is given to 50 years of progress in developing excess-Gibbs-energy models and engineering- oriented equations of state; these developments indicate rising use of molecular physics and statistical mechanics whose application for chemical process design is made possible by increasingly powerful computers. As yet, results from molecular simulations have not had a major influence on thermodynamics for conventional chemical engineering; how- ever, it is likely that molecular simulation methods will become increasingly useful, especially when supported by quantum-mechanical calculations for describing intermo- lecular forces in complex systems. © 2004 American Institute of Chemical Engineers AIChE J, 50: 739–761, 2004 Keywords: fluid-phase equilibria, chemical thermodynamics, equations of state Introduction Separation of fluid mixtures is one of the cornerstones of chemical engineering. For rational design of a typical separa- tion process (for example, distillation), we require thermody- namic properties of mixtures; in particular, for a system that has two or more phases at some temperature and pressure, we require the equilibrium concentrations of all components in all phases. Thermodynamics provides a tool for meeting that re- quirement. For many chemical products (especially commodity chemi- cals), the cost of separation makes a significant contribution to the total cost of production. Therefore, there is a strong eco- nomic incentive to perform separations with optimum effi- ciency. Thermodynamics can contribute toward that optimiza- tion. In this article, we indicate some of the highlights of progress in the thermodynamics of phase equilibria since the AIChE Journal began about 50 years ago. Although it is clear that, in addition to phase equilibria, thermodynamics also provides useful caloric information (en- thalpy balances) that affects separation operations, we do not consider that here. Furthermore, we recognize that in recent years, thermodynamic research has given rising attention to product (as opposed to process) design. There is good reason to Correspondence concerning this article should be addressed to J. M. Prausnitz at prausnit@cchem.berkeley.edu © 2004 American Institute of Chemical Engineers THERMODYNAMICS AIChE Journal 739 April 2004 Vol. 50, No. 4 believe that “modern” chemical engineering will be increas- ingly concerned not with large-scale production of classical (commodity) chemicals, but with new (specialty) materials for application in biotechnology, medicine, pharmaceuticals, elec- tronics, and optics, as well as in the food and personal-care industries. Thermodynamics can contribute to the development of these “new” areas and, indeed, much progress toward that end has been reported in the ever-growing literature. In our short review here, it is not possible to give adequate attention to “new” thermodynamics. This review is essentially confined to progress in thermodynamics for what is often called “tradi- tional” chemical engineering. Furthermore, during the last 50 years, the literature on ap- plied thermodynamics has grown to vast proportions, even when we limit our attention to phase equilibria. Therefore, this review cannot be comprehensive. Indeed, it is unavoidably selective, reflecting the authors’ (right or wrong) opinions and preferences. Space and time are finite, and some subconscious prejudice is always with us. The authors, therefore, apologize to their colleagues whose work is not sufficiently mentioned here or worse, not mentioned at all. Because the editor has necessarily limited the total number of pages available to us, we must, with regret, omit much that may merit inclusion. Fifty years ago, most chemical engineering thermodynamics was based on representation of experimental data in charts, tables, and correlating equations that had little, if any, theoret- ical basis. Fifty years ago, most chemical engineering thermo- dynamics was in what we may call an empirical stage. How- ever, the word “empirical” has several interpretations. When we represent experimental data by a table or diagram (for example, the steam tables or a Mollier diagram), we call such representation empirical. When we fit experimental ideal-gas heat capacities to, say, a quadratic function of temperature,we choose that algebraic function only because it is convenient to do so. However, when, for example, we fit vapor–pressure data as a function of temperature, we inevitably do so by expressing the logarithm of the vapor pressure as a function of the recip- rocal absolute temperature. This expression is also empirical but in this case our choice of dependent and independent variables follows from a theoretical basis, viz. the Clausius– Clapeyron equation. Similarly, when for a binary mixture, we represent vapor-liquid equilibrium data with, say, the Margules equation for activity coefficients, we also call that representa- tion empirical, although it has a theoretical foundation, viz. the Gibbs–Duhem equation. We should distinguish between “blind” empiricism, where we fit experimental data to a totally arbitrary mathematical function, and “thermodynamically- grounded” empiricism where data are expressed in terms of a mathematical function suggested by classical thermodynamics. If now, in addition to thermodynamics, we introduce into our method of representation some more-or-less crude picture of molecular properties, for example, the van der Waals equation of state, we are still in some sense empirical but now we are in another realm of representation that we may call “phenomeno- logical” thermodynamics. The advantage of proceeding from “blind” to “thermodynamically-grounded” to “phenomenolog- ical” is not only economy in the number of adjustable param- eters but also in rising ability to interpolate and (cautiously) extrapolate limited experimental data to new conditions, where experimental data are unavailable. Because “phenomenologi- cal” thermodynamics uses molecular concepts, an alternate designation is to say “molecular” thermodynamics. The most striking engineering-oriented examples of molecular thermo- dynamics are provided by numerous useful correlations, based on the theorem of corresponding states or on the concept of group contributions. In contrast to what we (somewhat carelessly) call “empiri- cal” thermodynamics, we also have “modern theoretical” ther- modynamics that uses statistical mechanics and molecular sim- ulations. In this “theoretical” area, we relate macroscopic thermodynamic properties to microscopic characteristics in a more-or-less rigorous manner. Although “thermodynamically-grounded” thermodynamics was well known fifty years ago, and whereas some correlations based on corresponding states or group contributions have a long history, the last 50 years have provided significant progress in “phenomenological” or “molecular” thermodynam- ics with numerous applications in chemical process design. During the last 25 years, there has also been much progress in statistical thermodynamics and molecular simulation. How- ever, regrettably, with few exceptions, that progress has not yet seen significant application in engineering design of fluid- phase chemical processes. For engineering application, applied thermodynamics is pri- marily a tool for “stretching” experimental data: given some data for limited conditions, thermodynamics provides proce- dures for generating data at other conditions. However, ther- modynamics is not magic. Without some experimental infor- mation, it cannot do anything useful. Therefore, for progress in applied thermodynamics, the role of experiment is essential: there is a pervasive need for ever more experimental results. Anyone who “does” thermodynamics is much indebted to those who work in laboratories to obtain thermodynamic properties. It is impossible here to mention even a small fraction of the vast body of new experimental results obtained over a period of fifty years. However, it is necessary here to thank the hundreds of experimentalists who have provided essential contributions to progress in chemical thermodynamics. Particular recognition must go to the laboratories of Grant Wilson (Provo, Utah) and D. Richon (Fontainebleau, France) who have pioneered in obtaining experimental results for “ difficult” industrial sys- tems. A useful compilation of experimental phase-equilibrium data is provided by the multivolume DECHEMA series (Behrens and Eckermann, 1980–1997). Thermodynamic Properties of Pure Fluids For common fluids (for example, water, ammonia, light hydrocarbons, carbon dioxide, sulfur dioxide, and some freons) we have detailed thermodynamic data conveniently compiled in tables and charts; the outstanding example of such compi- lations is the steam table with periodic improvements and extensions (Harvey et al., 1998; Harvey and Parry, 1999). Although thermodynamic data for less common fluids are often sketchy, a substantial variety of thermodynamic proper- ties for “normal” fluids can be estimated from corresponding- states correlations, especially those based on Pitzer’s use of the acentric factor (1955–1958) for extending and much improving classical (van der Waals) corresponding states (Lee and Kesler, 1975). Here “ normal” applies to fluids whose nonpolar or slightly polar (but not hydrogen-bonded) molecules are not 740 AIChE Journal April 2004 Vol. 50, No. 4 necessarily spherical but may be quasi-elliptical. Although the Lo Lee-Kesler tables are useful for numerous fluids, with few exceptions, correlations that are linear in the acentric factor J cannot be used for strongly polar molecules or for oligomers, Q or other large molecules whose acentric factors exceed O8 + (roughly) 0.4. 4 Two Procedures for Calculating Phase Equilibria 06 ~” For calculating fluid-phase equilibria, it is common practice , , . to use either one of two methods. In Method I, we use fugacity Y, , 4 coefficients for all components in the vapor phase, and activity K coefficients for all components in the liquid phase. In Method 04 yf . II, we use fugacity coefficients for all components in all fluid / | Ov phases. We find fugacity coefficients from an equation of state y © EXPERIMENT (EOS), and activity coefficients from a model for the molar f / cos excess Gibbs energy, as discussed in numerous textbooks (for 02 o 7 — PONS example, Smith et al. 2001; Kyle, 1999; Prausnitz et al., 1999; f oo eer Sandler, 1999; Tester and Modell, 1997). Each method has i’ ~~-—-~ FROM RAQULTS LAW some advantages and some disadvantages. For Method I, for every component i, the essential equation 0 of equilibrium is Q 0.2 0.4 x 0.6 08 LQ : YidiP = xvii (1) Figure 1. Predicted VLE for CgHg(1)/n-C7H,,(2) at 70 °C with the original regular solution theory. Here y and x are mole fractions in the vapor and the liquid, Here y is the vapor-phase mole fraction and x is the liquid- respectively; P is the total pressure; @” is the vapor-phase phase mole fraction. fugacity coefficient, and y is the liquid-phase activity coeffi- cient. Liquid-phase reference fugacity fis typically the pure- —_ hhase mixtures, there was a preference to use well established liquid vapor pressure at system temperature with (usually — activity-coefficient models (typically the Margules and van small) corrections for pure-fluid vapor-phase nonideality, and Laar equations). However, a pioneering application of Method for the effect of total pressure (Poynting factor). ; II was presented by Benedict et al. (1940, 1942) whose Bene- For liquid-liquid equilibria, where the same standard state is dict-Webb-Rubin (BWR) equation of state provided the basis used in all phases, we have for an extensive correlation of high-pressure vapor-liquid equi- libria of paraffin mixtures. However, because computers were (xy)' = (xy)" (2) then in their infancy, the cumbersome BWR equation was not used much. Furthermore, because the BWR equation of state where “and” refer, respectively, to the two liquid phases. for a mixture (at that time) contained only pure-component (no For Method II, for every component i, the essential equation binary) constants, its accuracy was limited. of equilibrium is Activity-Coefficient Models for Liquid Mixtures of yb! = x" (3) Nonelectrolytes More than one-hundred years ago, Margules proposed to for vapor-liquid equilibria or correlate isothermal binary vapor-liquid equilibria (VLE) with a power series in a liquid-phase mole fraction to represent In i bn y,, where y, is the activity coefficient of component 1. The (xii)! = Codi) (4) activity coefficient of component 2, y 5, is then obtained from the Gibbs—Duhem equation without requiring additional pa- for liquid-liquid equilibria. Here superscripts V and L refer, rameters. About 15 years later, van Laar derived equations for respectively, to the vapor phase and liquid phase; while ' and In y , and In y 5 based on the original van der Waals equation " refer, respectively, to the two liquid phases. of state. After introducing a key simplifying assumption for Before 1950, Method I was dominant because there was liquids at modest pressures (no volume change upon isothermal reluctance to use an equation of state for condensed fluids. mixing), van Laar assumed that the isothermal entropy of Although many years before, van der Waals had clearly shown mixing at constant volume is equal to that for an ideal solution. that his (and similar) equations of state are applicable to both Furthermore, upon assuming that the cross coefficient in the gases and liquids, there was little confidence in the ability of van der Waals equation of state a, is given by the geometric such equations to represent the properties of liquids with suf- mean (@,, = Vd,1@2), van Laar obtained expressions for In ficient accuracy. Furthermore, before computers became y , and In y , that require only pure component parameters. readily available, equilibrium calculations were prohibitive if However, regrettably, agreement with experiment was not both phases were described by an equation of state. For liquid- good. AIChE Journal April 2004 Vol. 50, No. 4 741 About 1930, Hildebrand and (independently) Scatchard, pre- 0.02 sented a derivation similar to that of van Laar but, instead of of 2 M4 © Aromatic Components benzene van der Waals constant a, they used the concept of cohesive ook os ® Aromatic Component: toluene energy density, that is, the energy required to vaporize a liquid , ‘6 per unit liquid volume; the square root of this cohesive energy a density is the well-known solubility parameter 6. In the final ° 10 17% Wg pp Hildebrand expressions for In y , and In y 5, the square root of ~ &* 6 os the cohesive energy density appears because of a geometric- “ -0.01 ; o 73 mean assumption similar to that used by van Laar. Because of 2aeos ° this geometric-mean assumption, the original regular-solution -0.02 ‘s theory is predictive, requiring only pure-component experi- ; mental data (vapor pressures, enthalpies of vaporization, and rz umber of CHs groups in saturated component _ . . we . . . 5 -0.03 total number of carbon atoms in saturated component liquid densities). For simple mixtures, Hildebrand’s regular- ah solution theory often gives a good approximation as illustrated Oo Ot 02 03 04 05 O06 O7 O8 in Figure 1. DEGREE OF BRANCHING + Results from regular-solution theory did not have apprecia- : : : ineeri : : Figure 3. Binary parameter ¢'? for aromatic-saturated ble influence in chemical engineering thermodynamics until i) . y P : 3 about 25 years after that theory was published. To make the hydrocarbon mixtures at 50 °C. theory more flexible, the geometric-mean assumption is cor- Binary systems shown are: 1. Benzene(2)-Pentane(1); 2. Ben- d by introduci incle bi fficient /,, that zene(2)-Neopentane(1); 3.Benzene(2)-Cyclopentane(1); 4. recte . y introducing a single inary coe cien 12) at very Benzene(2)-Hexane(1); 5. Benzene(2)-Methylpentane(1); 6. much improves agreement with experiment. Figure 2 illustrates Benzene(2)-2, 2-Dimethylbutane(1); 7. Benzene(2)-2, 3-Dim- . on} . _ ethylbutane(1); 8. Benzene(2)-Cyclohexane(1); 9. Ben- how binary parameter / 12 can significantly increase the accu zene(2)-Methylcyclopentane(1): 10. Benzene(2)-Heptane(1): racy of the regular-solution theory. Numerous efforts have not 11. Benzene(2)-3-Methylhexane(1); 12. Benzene(2)-2, succeeded in correlating /,, in terms of pure-component prop- £Dimethy/pemane()).). menzene (2), 2, O rimethy bu : : : : oo : : t ; . -Met ; . - erties. Such a correlation is possible only for limited situations, vameh -Octanetlyy 1 Bee) > Fa Trimethyipen. as illustrated in Figure 3. tane(1); 17. Toluene(2)-Hexane(1); 18. Toluene(2)-3- Methylpentane(1); 19. Toluene(2)-Cyclohexane(1); 20. Toluene(2)- Methylcyclopentane (1); 21. Toluene(2)-Hep- 7 . . __ . _ . tane(1); 22. Toluene(2)-Methylcyclohexane(1); 23. Tolu- - i ei i ene(2)-2, 2, 4-Trimethylpentane(1). | Pressure = 1.013 bar © Expariment i When the modified regular-solution theory for liquid mix- 6 p - tures was combined with the Redlich-Kwong equation of state ¥ (1- x) for vapor mixtures, it was possible to correlate a large body of Bio? % a ~y,) jf | VLE data for mixed hydrocarbons, including those at high mw & i 3 pressures found in the petroleum and natural-gas industries g (Prausnitz et al., 1960). The resulting Chao—Seader correlation a 1 (Chao and Seader, 1961) was used extensively in industry until - ; it was replaced by other simpler methods, based on a cubic + 4 1.70 ad equation of state applied to all fluid phases. e i2 , Until about 1964, most chemical engineering applications of J i ( activity coefficients were based on either the Margules or the a 3| / DO i. van Laar equations, although in practice, the two binary coef- > PAN ficients in the van Laar equation were not those based on the ud 6 4 | van der Waals equation of state, but instead, those obtained = of” Zio *-O015 from reduction of binary VLE data. y 2 fy In an influential article, Wohl (1946) showed how activity- ad F coefficient equations can be systematically derived from a Ww i phenomenological model where the molar excess Gibbs energy g” is expressed as a function of liquid-phase composition. I | Wohl’s systematic method had two primary effects: first, it encouraged the development of new models (variations on Margules and van Laar), and second, very important for dis- ow . a nad tillation column design, it showed how binary VLE data can be G 0.2 0.4 Os o8 1.0 systematically “scaled up” to predict VLE for ternary (and . higher) liquid mixtures with or without requiring any ternary MOLE FRACTION BENZENE (2) (or higher) VLE data (Severns et al., 1955). Figure 2. VLE with the regular solution theory with bi- For liquid mixtures containing strongly polar or hydrogen- nary parameter (72. bonding components, the van Laar equations for activity coef- Effect of €'2 on calculating relative volatility for the 2, ficients are often not satisfactory, especially when applied to 2-dimethylbutane(1)/benzene(2) system. multicomponent liquid-liquid equilibria (LLE). For LLE, we 742 April 2004 Vol. 50, No. 4 AIChE Journal can use the Margules equations but for good agreement with Ethanol experiment for ternary (and higher) systems, it is often neces- sary to use many empirical coefficients. If ternary (and higher) coefficients are omitted, Wohl’s method for multicomponent liquid mixtures assumes additiv- ity; for a ternary mixture, g* is essentially given by the sum g%, + g®, + g%, where the subscripts denote binary mixtures. However, whereas molecular considerations indicate that ad- ditivity is (approximately) correct for h”, the excess molar enthalpy (heat of mixing), there is no physical basis for the additivity of g”. Wohl’s method, in effect, emphasizes the contribution of ”, while neglecting the contribution of s”, the molar excess entropy (g” = h® — Ts"). With a fundamentally different method that emphasizes s” (rather than h®), Wilson (Wilson, 1964) derived an equation for g” based on a generalization of Flory’s theoretical expression for the entropy of mixing noninteracting spheres and chains of LE spheres (polymers). To take molecular interactions into ac- gee F*N count, Wilson used the concept of local composition that, in Vettes i im’ : ‘ n-Hexane Etliylnitrile turn, is based on Guggenheim’s quasi-chemical theory for nonrandom mixing, that is, the tendency of molecules in a _ Figure 4. Experimental and calculated ternary LLE for liquid mixture to show preferences in choosing their immediate ethanol-ethylnitrile-n-hexane at 40 °C. neighbors. For example, in a mixture of methanol and hexane, Concentrations are in mole fractions. Dashed lines are pre- because of hydrogen bonding between two (or more) methanol dictive calculations with parameters obtained from binary molecules, a methanol molecule prefers to be near another systems; points are experimental data, and full lines are ob- served phase-envelope and experimental tie lines (Nagata and methanol molecule, rather than near a hexane molecule. How- Kawamura, 1979). ever, in a mixture of chloroform and acetone, because of hydrogen bonding between the CH group of chloroform and the C= O group of acetone, a chloroform molecule prefers to the components in the mixture are not identical in size and be near an acetone molecule. shape, extension to multicomponent mixtures (like the Wilson With two binary constants per binary mixture, Wilson’s model) does not rely on simple additivity of binary excess equation for g” is often superior to the older two-constant Gibbs energies. equations, especially for VLE of mixtures where one (or more) NRTL and UNIQUAC have been extensively used for about components can hydrogen bond. Unfortunately, however, Wil- thirty years, largely (but not totally) replacing the equations of son’s equation cannot be used for binary LLE without one Margules, van Laar, and Wilson. additional binary parameter. Finally, for liquid mixtures containing strongly interacting Encouraged by Wilson’s use of the local composition con- molecules (for example, alcohols), some models for g” are cept of 1964, two other models with the same concept were based on chemical equilibria either with or without a contri- proposed: Renon’s nonrandom two-liquid (NRTL) model of bution from “physical” interactions. Because molecules do not 1968, and Abrams’ universal quasi-chemical (UNIQUAC) “know” whether they are “doing” physics or chemistry, any model of 1975. Although the theoretical basis of these local- division between “physical” and “chemical” interactions is composition models is not strong, subsequent to their publica- somewhat arbitrary. Nevertheless, when guided by molecular tion, they obtained some support from molecular simulation physics, such arbitrary division can be useful for correlation of studies (Hu et al., 1983; Nakanishi and Tanaka, 1983; Phillips experimental data. and Brennecke, 1993). Although NRTL uses three adjustable A “chemical” theory for activity coefficients was first intro- parameters per binary, one of these (nonrandomness parameter duced nearly 100 years ago by Dolezalek who claimed that a a>) can often be set a priori; a typical value is a,, =0.3. Both real mixture is an ideal solution provided that we correctly NRTL and UNIQUAC are readily generalized to multicompo- identify the mixture’s molecules. For example, VLE for mix- nent mixtures without additional parameters, and both may be tures of ethanol and heptane can be represented by an ideal- used for VLE and LLE. solution calculation when we consider that some ethanol mol- Modifications of the Wilson equation and the NRTL equa- ecules are dimers, or trimers, and so on, as determined by tion have been used to describe phase equilibria in polymer chemical equilibrium constants. Improvement is obtained solutions (Heil and Prausnitz, 1966; Chen 1993). when, in addition, we allow the various “true” chemical species The UNIQUAC equations use only two adjustable binary to interact with each other through “physical” forces as given parameters per binary and, because the configurational part of by an equation of the Margules, van Laar, NRTL or UNIQUAC the excess entropy is based on Flory’s expression for mixtures form. Because such “chemical” theories necessarily require of noninteracting short and long-chain molecules, UNIQUAC numerous adjustable equilibrium constants, it is customary, in is directly applicable to liquid mixtures that contain polymers. practice, to make simplifying assumptions; a common one is to Furthermore, because UNIQUAC separates the configurational assume that the equilibrium constant for association (for ex- entropy contribution to g* from the residual contribution that is ample, alcohols) is independent of the degree of association. primarily because of attractive intermolecular forces whenever Figure 4 shows calculated and measured liquid-liquid equilib- AIChE Journal April 2004 Vol. 50, No. 4 743 ria for the ethanol—ethylnitrile-n—hexane system at 40°C. Cal- Temperature, K culations for the molar excess Gibbs energy use a chemical- 600 500 400 300 plus-UNIQUAC equation g ~. These calculations use two 4500 — vores binary “physical” (UNIQUAC) parameters and two “chemical” No | parameters (Nagata and Kawamura, 1979). A more detailed ' : “chemical plus physical” model is the ERAS model developed on Og by Heintz and coworkers (Heintz et al., 1986; Letcher et al., | fo 1995); “chemical plus physical” solution theory is frequently P CoM, used for describing the properties of electrolyte solutions (for ‘A | example, Lu and Maurer, 1993). | Although “chemical” theories are often successful for binary . as . . . 3500 mixtures, generalization to ternary (and higher) mixtures with we i CO, more than one associated component is often not possible = without introduction of numerous additional adjustable param- nl eters or additional (doubtful) simplifications. However, for Oo some cases, good results are achieved as indicated by Nagata et x £ al. (2000). & £ HoS ~ . é Effect of Temperature on VLE and LLE 2500 . & The effect of temperature presents a fundamental problem in & . the application of activity-coefficient models because the ad- gS justable binary (or higher) parameters depend on temperature. go . Although thermodynamics provides exact equations that relate cS that temperature dependence to either the excess enthalpy or excess entropy of mixing, such equations are of little use ; i because the required enthalpy or entropy data are only rarely % available. Fortunately, for VLE calculations, the effect of tem- 1500 perature on activity coefficients is often not large; the primary 30 40 50 60 effect of temperature on VLE comes from the (large) known Water Density at Saturation, moi u whe of temp erature on pure-comp onent vapor pressures. For Figure 5. Henry’s constant for several gases (2) in water , It is common practice either to neglect the effect of temperature on In or, as predicted by regular-solution (1) from Japas and Levelt-Sengers (1989). Pp Y 1 OF, Pp y feg . . . . theory, to assume that at constant composition, In y ; is pro- re line is obtained from a linear equation where p is solvent . lensity and parameters A and B are constants for each gas. portional to 1/7. However, for LLE (where pure-component The solvent’s fugacity is f,. vapor pressures play no role), the effect of temperature is likely to be significant. Regrettably, at present, we do not have any consistently reliable molecular thermodynamic methods for from the effect of temperature on solvent density. For many calculating the effect of temperature on LLE. cases, solvent density is the dominant influence on Henry’s constant. An example is shown in Figure 5 that correlates oe . solubilities of five gases in water over a large temperature Solubilities of Gases and Solids range. In Figure sf, is the fugacity of the solvent. , At moderate pressures, the solubility of a gas j in a liquid 7 Some attention has been given to correlating Henry’s con- is given by Henry’s constants H ; ; that depends on temperature. stant for a gas in a mixed solvent. For simple systems, for a gas We have a reasonably large data base for these constants for _j, good approximations can often be made with a volume- common gases in a variety of common liquids. However, the fraction average for In H; that requires knowing only H; for major part of that data base is for temperatures near 25 °C; the every solvent in the mixture. A better approximation is often further we go from 25°C, the smaller the database. For some obtained by assuming additivity of binary interactions. In that nonpolar systems, we can estimate Henry’s constants with case, for a ternary mixture, we need not only H; for gas j in both Hildebrand’s (Hildebrand et al., 1970) or Shair’s correlation pure solvents but, in addition, some information on interactions (Prausnitz and Shair, 1961) based on solubility parameters. For in the binary (gasfree) solvent mixture (Campanella et al., advanced pressures, we can add a correction to Henry’s law 1987; Shulgin and Ruckenstein, 2002). using partial molar volumes of the gaseous solutes; these are In nonelectrolyte systems, the solubilities of solids are com- often not known but for nonpolar systems, we can often esti- monly calculated by referring the solute’s activity coefficient to mate them with a correlation (for example, Lyckman et al., the solute’s subcooled liquid. The ratio of the fictitious vapor 1965; Brelvi and O’Connell, 1972). Some correlations for pressure (or fugacity) of the subcooled liquid to that of the Henry’s constants are based on scaled-particle theory (Pierotti, stable solid at the same temperature is found from knowing 1976, Geller et al., 1976) or on assumptions concerning the primarily the solute’s melting temperature and enthalpy of radial distribution function for a solute molecule completely fusion, and secondary, from the difference in heat capacities of surrounded by solvent molecules (Hu et al., 1985). the solid and subcooled liquid. Correlations of solid solubilities The effect of temperature on gas solubility follows, in part, based on such calculations have been presented by numerous 744 April 2004 Vol. 50, No. 4 AIChE Journal Oo perature and pressure of a pure fluid; as a result, starting in the % o Nophtholene © Biphenyl mid-fifties, the literature is rich in group-contribution methods v Fluorene © Acenaphthene for estimating critical properties; some of these are summarized x o-Terphenyl + Pyrene in “Properties of Gases and Liquids” by Poling et al. (2001). " . © Fluoranthene @ m-Terphenyl It is by no means simple to establish a reliable group- ~0.2 x Phenanthrene . contribution method for pure fluids, however, it is more diffi- ¢ ———ldeol solubility curve cult to establish such a method for fluid mixtures, in particular, g for activity coefficients of all components in that mixture. ‘¥ However, for applications in chemical process design, it is -04 x useful to have such a correlation because, for many (indeed, . \ most) mixtures, activity-coefficient data are at best sketchy and ‘ often nonexistent. About 75 years ago, Langmuir briefly discussed a possible iu ° activity-coefficient correlation based on group contributions. 2-06 yy However, Langmuir’s idea remained dormant for about 40 _ nN years, primarily because the required data base was too small, 4 \ and because the necessary calculations are too tedious without . o\ a computer. Langmuir’s idea was revived by Deal and Derr \ (1969) who presented an early version of their ASOG correla- ~0.8 , \ tion . To use this correlation, we need temperature-dependent \ group— group parameters; because Deal and Derr provided only \ a few of these parameters, the usefulness of ASOG was se- \ verely limited. At present, its usefulness is somewhat larger -1.0 \ thanks to a monograph (Kojima and Tochigi, 1979) and articles . \ by Tochigi et al. (1990, 1998). \ When the UNIQUAC model for activity coefficients is mod- ified toward a group contribution form, it leads to the UNIFAC correlation (Fredenslund et al., 1975), UNIFAC is simpler to 1.2 use than ASOG because (to a rough approximation) its param- 10 Ll 4 1.3 1.4 eters are independent of temperature. UNIFAC was eagerly Tn / T picked up by numerous users because the authors of UNIFAC supplied necessary software; a monograph (Fredenslund et al., Figure 6. Solubility of aromatic solids in benzene at dif- ferent temperatures. The ideal solubility is calculated with the equation: 60 54.4 Th log = (s:) (= ~ i) where 54.4 Jmol"! K"' is an average value for the entropy of , fusion of solids considered. T,,, is the melting temperature, and x, is the mole fraction of the solute in the liquid solution. 40 3 authors, notably by McLaughlin (McLaughlin and Zainal, & 1959, 1960; Choi and Mclaughlin, 1983); an example is shown a in Figure 6. 20 Thermodynamic Properties from Group Contributions Because desired thermodynamic data are frequently in short supply, many efforts have been made to estimate these prop- Ls erties from known molecular structure. If we divide a molecule iP es J into its constituent groups, it then seems reasonable to assume 0 that each group contributes to a particular thermodynamic 0.0 0.5 1.0 property, such as the molar volume, or the normal boiling x point. For example, Seinfeld and coworkers (Asher et al., 2002) uy have used group contributions to estimate thermodynamic Figure 7. VLE calculations for the benzene(1)-sulfo- properties of some oxygen containing liquids and solids that lane(2) system at several temperatures with are required for analysis of air-pollution data. UNIFAC (Wittig et al., 2003). Because corresponding-states-correlation methods are often In the vapor phase, the mole fraction y, is essentially unity for useful, it is particularly important to estimate the critical tem- pressures higher than 5 kPa. AIChE Journal April 2004 Vol. 50, No. 4 745 1977), and a subsequent series of articles provided a large number of group–group interaction parameters (Gmehling et al., 1982; Macedo et al., 1983; Hansen et al., 1991; Freden- slund and Sørensen, 1994; Wittig et al., 2003). UNIFAC is simple to use because it requires no experimental mixture data; as a result, UNIFAC became immensely popular, despite its limitations, especially for dilute solutions. Numerous empirical modifications, primarily by Gmehling and coworkers (includ- ing temperature dependence of some UNIFAC parameters) have improved the ability of UNIFAC to predict activity co- efficients in binary or multicomponent liquid mixtures of typ- ical subcritical liquids, including hydrocarbons, petrochemicals and water. To illustrate, Figure 7 shows vapor-liquid equilibria for the benzene(1)-sulfolane(2) system at several temperatures with a recent set of UNIFAC parameters from Wittig et al. (2003). Figure 8 compares activity coefficients at infinite dilu- tion, predicted by UNIFAC with experiment. Oishi’s extension of UNIFAC to polymer solutions (Oishi and Prausnitz, 1978) has been modified by others (for example, Holten-Anderson et al., 1987; Goydan et al., 1989). A few efforts have been made to extend UNIFAC to solutions con- taining electrolytes; especially for dilute solutions, such exten- sions requires important corrections for the long-range forces between charged particles. Furthermore, such extensions are necessarily limited because the data base is essentially confined to aqueous systems. Efforts to include supercritical components (for example, hydrogen) have not had much success because UNIFAC is based on a lattice model where each molecule is confined to the immediate vicinity of a lattice position. A lattice model is not suitable for a highly mobile gaseous solute. Furthermore, in UNIFAC, the activity coefficient refers to a standard state fugacity of pure liquid at system temperature. For a supercriti- cal component that fugacity is necessarily hypothetical. The primary application of UNIFAC is to estimate VLE for multicomponent mixtures of nonelectrolytes for screening and for preliminary design of distillation or absorption operations. Although UNIFAC can often provide good results, like all group-contribution methods, UNIFAC is not always reliable, especially for liquid mixtures where the molecules of one (or more) components have two or more close-by polar groups (for example, ethylene glycol). As indicated later, some recent promising developments with quantum mechanics are directed at reducing this limitation of UNIFAC. Although, UNIFAC provides an attractive method for estimating phase equilibria, it is important to keep in mind that, as yet, there is no substitute for high-quality experimental data. Because ASOG and UNIFAC parameters are obtained from binary VLE data, predictions from ASOG and UNIFAC are useful only for VLE, not for LLE. In VLE calculations, the primary quantities are pure-component vapor pressures; activ- ity coefficients play only a secondary role. However, in LLE calculations, where activity coefficients are primary, a much higher degree of accuracy is required. UNIFAC correlations for LLE (discussed in Poling et al., 2001) are useful only for semiquantitative predictions. Better results can be achieved when UNIFAC LLE parameters are regressed from (and then applied to) a limited class of mixtures. For example, Hooper et al., (1988) presented a set of UNIFAC LLE parameters for aqueous mixtures, containing hydrocarbons and their deriva- tives for temperatures between ambient and 200°C. The DISQUAC group-contribution correlation is useful for estimating enthalpies of liquid mixtures (Kehiaian, 1983, 1985). A semiempirical class of methods that tries to account for the correlation between close-by polar groups uses descriptors. In these methods, the structure of a molecule is represented by a two-dimensional(2-D) graph, with vertices (atoms) and edges (bonds). The numerical values resulting from the operation of a given descriptor on a graph are related to a physical property, for example, the activity coefficient at infinite dilution (Faulon et al., 2003; He and Zhong, 2003). Because the method has little theoretical basis, the type and the number of descriptors needed is property-dependent, that is, the method is specific for each thermodynamic property. Furthermore, for reliable pre- dictions, any correlation based on descriptors requires a large database. The SPACE model with solvatochromic parameters for es- timating activity coefficients is an extension of regular-solution theory where the cohesive energy density is separated into dispersion forces, dipole forces, and hydrogen bonding. The dipolarity and hydrogen-bond basicity, and acidity parameters were correlated with the activity coefficient database by Hait et al. (1993). Castells et al. (1999) compare different methods (including SPACE) to calculate activity coefficients for dilute systems. A method similar to SPACE is described by Abraham and Platts (2001); their group contribution model is used to calculate solubilities of several pharmaceutical liquids and solids in water at 289 K. A recent method for calculating solubilities of pharmaceuticals is given by Abildskov and O’Connell (2003). Equations of state (EOS) In the period 1950–1975, there were two major develop- ments that persuaded chemical engineers to make more use of Method II that is, to use an EOS for fluid-phase (especially VLE) equilibria. First, in the mid-1950s, several authors sug- gested that successful extension of a pure-component equation Figure 8. Comparison of activity coefficients at infinite dilution predicted by UNIFAC with experiment for benzene, toluene, cyclohexene, hexane, 2, 2, 4-trimethylpentane and undecane in sulfo- lane at several temperatures (Wittig et al., 2003). 746 AIChE Journal April 2004 Vol. 50, No. 4 of state to mixtures could be much improved by introducing 210 one binary constant into the (somewhat arbitrary) mixing rules, a Colcuta hed that relate the constants for a mixture to its composition. For 160 With kyp* 0.163 example, in the van der Waals EOS, parameter a for a mixture 597 0.08 is written in the form r 150 © Experiment a a(mixture) = Ss Ss Zi jij (5) ut 120 nod 2 96 wi . . 2 Yapar Liquid where i and j represent components and z is the mole fraction. a 60 When i = j, van der Waals constant a;,; is that for the pure tl component. When i # j, the common procedure is to calculate 30 a,; as the geometric mean corrected by (1 — k,;) where k;; is a binary parameter 8 0.2 04 0.6 0.8 1.0 — MOLE FRACTION WATER ay = \aidiy (1 ~ ky) (6) Figure 10. Isothermal pressure-composition phase dia- Parameter k,; is obtained from some experimental data for the gram for water-hydrogen sulfide. . . tye . : Calculation with the Redlich-Kwong-Soave equation of i — j binary. It seems strange now, but it was not until about state (Evelein et al., 1976). Parameters k,, and c,, were 1955-1960 that the currently ubiquitous k,, became a common adjusted to give good agreement with experimental data feature of articles in chemical engineering thermodynamics. It from Selleck et al. (1952). was during that period that lists of kj, appeared and that (mostly futile) attempts were made to correlate k,, with prop- erties of pure components | and 2. activity coefficients, it would be attractive to use one EOS for Second, about 1965, there was a growing recognition that all fluid phases. However, to apply that idea, the pure-compo- because an EOS of the van der Waals form can be used to nent constants in the equation of state must be evaluated to fit generate both vapor-phase fugacity coefficients or liquid-phase — what for VLE is the most important quantity, viz. the pure- component vapor pressure. If an EOS can correctly give the vapor pressure of every pure component in the mixture, VLE 120 4 for a mixture can be calculated, essentially, by interpolation as ~—e {Gotculeted dictated by mixing rules. When these mixing rules are made ~~ thy 0.029 sufficiently flexible through one (or sometimes two) adjustable * © Experiment binary parameters, good results for VLE can often be achieved. 1% Observed Critical Point / : . . 100 “ 5 To illustrate, Figure 9 shows experimental and calculated re- sults for methane-propane. Although k,, is very small com- _ ° P pared to unity, it nevertheless has a significant effect. 80 yo 4 Because methane and propane are simple and similar mol- 5 a’ 6 14 ecules, a single binary parameter is sufficient to achieve good a fo » results. To represent phase equilibria for the much more com- ul i & plex system, water-hydrogen sulfide, Evelein et al. (1976) 5 60 a & wp A ? ~ introduced a second parameter c,, in the mixing rule for van & é y j der Waals size-parameter b. With two binary parameters, it is x “if 7 327.6K 277.6 K possible to achieve good agreement with experiment as shown $ ° # ? in Figure 10. 40 / “ As a result of these happy developments, the literature was ad 24 a d soon flooded with proposed EOS where the pure-component Z Age constants were fit to pure-component vapor pressure data. In p fp * ‘ ‘ this flood, a favorite target was to modify the Redlich-Kwong 206-4) ae (RK) EOS, published in 1949, where the authors had intro- J oor? duced a simple but remarkably effective modification of the L, - density dependence in the van der Waals equation. Because Redlich and Kwong were concerned only with dense gases, not oe a2 D4 BB 08 To liquids, their particular temperature dependence was dictated MOLE FRACTION METHANE by second-virial coefficient (not vapor pressure) data. After about 1972, numerous articles reported modifications of the Figure 9. Isothermal pressure-composition phase dia- RK EOS where, for each pure fluid, the characteristic attractive gram for methane-propane. constant a is given as a function of temperature such that good Calculation with Te Pat wong Soave equation of state agreement is obtained with experimental vapor-pressure data. seod pene with S tmenatal dota from Reamer ef al The best known modification of the RK EOS is that by Soave (1950). (1972) who was one of the first to show that a simple EOS of AIChE Journal April 2004 Vol. 50, No. 4 747 the van der Waals form is useful for calculating VLE of a FT variety of mixtures at both moderate and high pressures. 400 -- / _— 4 In 1976, Peng and Robinson (PR) published their modifica- ee pee, 275C | tion of the van der Waals EOS (Peng and Robinson, 1976) that, Te ies a 4 unlike Soave’s modification (SRK), introduces a new density a er, " 4 dependence in addition to a new temperature dependence into Sy a the RK equation. Although Soave‘s equation and the PR equa- ao 4 ' tion necessarily (by design) give good vapor pressures, the PR 2 3 | EOS gives better liquid densities. The PR EOS and the SRK s tres | EOS are now the most common “working horses” for calcu- we 2 foo 200 | lating high-pressure VLE in the natural-gas, petroleum and & : / petrochemical industries. For application of the PR EOS to mixtures containing polar as well as nonpolar components, a | , particularly useful correlation is that given by Vera and Stryjek "op . neti (1986). For mixtures where one (or more) components are well a. re 1 50 : ie below their normal boiling points, a useful modification is that ; } a by Mathias and Copeman (1983). at The van der Waals EOS is a perturbation on a highly ‘ oversimplified model for hard spheres; the perturbation is in- : : tended to account for attractive forces, although, in effect, it 0.0 0.2 0.4 0.6 0.8 1.0 also corrects the oversimplified hard-sphere term. Equations of Mole Fraction 2-Propanol the van der Waals form can be improved with a more realistic hard-sphere model or by changing the density dependence of Figure 11. Isothermal pressure-composition phase dia- the perturbation term or both. Many efforts along these lines gram for 2-propanol/water. have been reported. However, except for the Soave and PR Calculation with the PR EOS as modified by Stryjek and equations, they have not enjoyed much success in chemical gandlon, 1992) with Ree 396 (Orbe end Sandler 4098), process-design calculations essentially for two reasons: first, because for some “improved” EOS, additional constants must be determined from some experimental physical property that Toward further improvement of EOS for VLE, during the may not be readily available, and second, because the calcula- 1980’s significant attention was given to establish better mix- tions are often more tedious if the EOS is not (unlike the PR ing rules. Because the EOS can be used to find the molar excess and Soave EOS) cubic in volume; in that event, iterations are Gibbs energy, Vidal (1978, 1983) suggested that experimental needed to find the fugacity coefficient when temperature, pres- data for liquid-mixture activity coefficients be used to deter- sure and composition are given. Although standard computers mine both mixing rules and binary constants that appear in can easily perform such iterations, when very many VLE those rules. The resulting EOS can then be used to generate calculations are required for a particular design, impatient VLE at temperatures and pressures beyond those used to fix the engineers prefer an EOS where calculations are relatively sim- constants that appear in the mixing rules. ple and fast. Unfortunately, the procedure suggested by Vidal (and oth- Because all analytic EOS based on the van der Waals model ers) leads to a thermodynamic inconsistency because the cor- are poor in the vapor-liquid (VL) critical region, some attempts responding mixing rules, in general, do not give the theoreti- have been reported to obtain improvement by translation cally correct quadratic composition dependence of the second (Peneloux et al., 1982; Mathias et al., 1989), that is, by adding virial coefficient. To avoid this problem, it was suggested that a correction (in the first approximation, a constant) to the the mixing rules should be modified such that they simulta- volume in the EOS; this correction horizontally moves (trans- neously reproduce the activity-coefficient data at high fluid lates) the V-L coexistence curve plotted on P—V coordinates. densities, and at low fluid densities, the correct composition The correction is designed to make the calculated critical dependence of the second virial coefficient. The most success- coordinates (V,, T;, P-) agree with experiment. This procedure ful of these suggestions is the Wong-Sandler (WS) mixing has been used to calculate better saturated liquid volumes, rules (Wong and Sandler, 1992). Subsequent experience with however, because the correction toward that end introduces these rules led to a monograph that gives details and pertinent errors elsewhere in the P—V plane, this excessively empirical computer programs (Orbey and Sandler, 1998). Figure 11 translation procedure has only limited application (Valderrama, shows a successful application of WS mixing rules. 2003). In another example, shown in Figure 12, the PR EOS with Calculation of VLE in the critical region provides a severe WS mixing rules was used to calculate LLE. Here, Escobedo- challenge because any attempt to do so along rigorous lines Alvarado and Sandler (1998) used experimental LLE data at (renormalization group theory) requires numerous approxima- low pressure to determine binary parameters in the WS mixing tions and much computer time. Although some efforts have rules. The PR EOS was then used to predict LLE at higher been reported toward better representation of VLE in the crit- pressures up to 781 bar. ical region (for example, Anisimov and Sengers, 2000; Lue and The phenomenological basis of a traditional EOS of the van Prausnitz, 1998; Jiang and Prausnitz, 1999; Kiselev et al., der Waals form is based on molecules that are spherical (or 2001; Kiselev et al., 2002), as yet, they are not sufficiently globular). That basis is not appropriate for fluids with chain- developed for general engineering use. like molecules, especially polymers, not only because such 748 April 2004 Vol. 50, No. 4 AIChE Journal 700 : ie Ly . a 6004 OO "g a “DoD Roe ' One ~ XY BOO 1 co) : : O | RR L@ & ; OQ oO 2 oer | Oo. 300 . a 0 ah g a ey . QO WS 200 RY ay oY. 0.20 gy 8 cog 0 fee? : 0 0.15 04 : 5 w : So ee ae a0 o oO; ae eo Ay U 3 20 Gop” Oo . 4 She o. p Bete 00” 380 oa™ £0.98 320 300 0.00 TIK) Figure 12. Calculated and experimental LLE for the 2-butoxyethanol (C,E,)/water system, with the Peng-Robinson EOS and Wong-Sandler-NRTL mixing rules where x is the mole fraction. Low-pressure LLE data were used to determine binary parameters in the EOS (Escobedo-Alvarado and Sandler, 1998). molecules are not spherically symmetric in shape, but also (PHCT = perturbed hard chain theory) was proposed in 1975 because such molecules (unlike small spheres) exercise rota- (Beret and Prausnitz, 1975) based on Prigogine’s theory for tions, and vibrations that, because they depend on density, must liquid polymers (Prigogine, 1957); a particularly simple form be included in a suitable EOS (Vera and Prausnitz, 1972). of PHCT is the cubic EOS of Sako et al. (1989). In essence, A phenomenological EOS for chain-like molecules PHCT is similar to the van der Waals model but, unlike that model (in its original form), it allows for contributions from 0.20 so-called external degrees of freedom (density-dependent rota- oO expt 323.2 K v > expt 343.2 100 0.15 \ V_ expt 363.2 ° ol © ° PHSC-WS $ &\ Bo Aer g — SAFT x *\ “A N sarT s J > ® 0.10 piMPa \ 5 \ a : o (Ne 2 ONG 0.05 \ \. \, \, XN YO oO N \ 5,00 0.05 0.10 0.15 0.20 0.25 0.30 NN. ‘oO, o. solubility (weight fraction of hexane) 6. 001 O01 01 10 Figure 13. Calculated and experimental hexane (weight- *H,0 ‘Cee rich phase) fracti lubility i I I ly- ner ion) solubility in a polypropylene copoly Figure 14. Experimental and calculated (mole-fraction) . . . . solubility of water in the ethanerich phase for Here, expt means experimental data. Calculations with the t th ixt H d Rad perturbed-hard-sphere EOS with Wong-Sandler mixing water-ethane mixtures (Huang an adosz, rules or with the SAFT EOS (Feng et al., 2001). 1991, 1993). AIChE Journal April 2004 Vol. 50, No. 4 749 tions and vibrations) in addition to translations to the EOS. An EOS similar to PHCT is the chain-of-rotators EOS (Chien et al., 1983; Kim et al., 1986). Variations of PHCT have been used extensively in representing phase equilibria for vapor- phase polymer-monomer-solvent mixtures at high pressure as used, for example, in the production of polyethylene (Feng et al., 2001), as shown in Figure 13. When attention is limited to the liquid phase, an EOS by Flory (1965, 1970), Eichinger and Flory (1968), and Patterson (1969) can be used to find the so-called equation-of-state contribution to the excess Gibbs energy of a polymer solution. These contributions that arise because of differences in free volume of the polymer solution’s components are neglected in the classical Flory–Huggins lattice theory for polymer solu- tions. These contribution are essential for explaining the often observed high-temperature lower critical-solution temperature of polymer solutions and polymer blends (Olabisi et al., 1979). An alternative method by Sanchez and Lacombe (1976, 1978) proposed a relatively simple EOS for polymers that is an extension of Flory–Huggins lattice theory. The essential con- tribution of this equation is the inclusion of holes (empty sites) into the lattice. This inclusion provides the communal entropy at the ideal gas limit. Smirnova and Victorov (2000) give a comprehensive review of EOS based on lattice models. An elegant theory by Wertheim (1984–1986) has led to SAFT (Chapman et al., 1989 and 1990), a theoretically well- founded EOS for chain-like molecules where SAFT stands for Statistical Association Fluid Theory. Mu¨ller and Gubbins (2001) have given a user-friendly review of the derivation and applications of SAFT. To illustrate, Figure 14 shows the ap- plication of SAFT to represent properties of a solution contain- ing a strongly associating component, and Figures 15 shows calculated and experimental phase equilibria for a solution containing components with large size difference. Figure 15. Experimental and calculated solubility of bi- tumen in compressed carbon dioxide (Huang and Radosz, 1991). Figure 16. Comparison of experimental and calculated cloud points with the perturbedchain-SAFT EOS for mixtures of polyethylene and several solvents: ethylene, ethane, propylene, propane, butane, and 1-butene (Gross et al., 2003). 750 AIChE Journal April 2004 Vol. 50, No. 4 800 [ ret I | if | Og, 0% PB - 16 i! 600 “Sees sa 8 8 8 4 323 K 32 a ST TP o ° zs | ° \goe ee so 14 - | 2 400 | ° | _ 88 if 0. ¢ 2 97 12 J ; / ‘et i 200 | i. . r t t fi | | _ 1 0 a} i oO 4 — a oh 0 oo — o. 8 br tf 0 50 100 150 200 250 e ; iy T/°C Mt , , , 6 i a Figure 17. Comparison of experimental and calculated ' \ f cloud points with the perturbedchain-SAFT 4% EOS for mixtures of poly(ethylene-co-1- A - \ 4 butene) and propane with varying repeat-unit ‘ » . wp composition (from 0 to 97 mass %) of 283K ~“S ap 1-butene in the copolymer (Gross et al., 2 + 2003). x3 % PB represents the percentage of 1-butene in the polymer. On the basis of some clever assumptions concerning the 00 02 04 06 08 1.0 structure of a chain-molecule fluid, Chiew (1990, 1991) has . derived an EOS for polymer and polymer-like fluids that, in mole fraction HF effect, is similar to SAFT. By considering the next-to-nearest- Figure 19. Calculated and experimental Pxy diagram for neighbor cavity correlation function, Hu et al. (1995) obtained the hydrogen fluoride/refrigerant R12 sys- a more accurate EOS for hard-sphere chain fluid mixtures in tem. Lines are for the Association-plus-PR EOS proposed by Visco and Kofke (1999). Solid lines use the van der Waals P/SMPa mixing rules, whereas the dashed lines use the Wong-San- = dler mixing rules (Baburao and Visco, 2002). 1.8 ese good agreement with computer simulation data. In other * “TS SAFT-related studies, a number of successful efforts have \ ray incorporated a more accurate description of the structure and \ ‘A free energy of the monomer reference fluid (Ghonasgi and \ ‘ Chapman, 1994; McCabe et al., 2001, Paredes et al., 2001, 1& ‘ \. Paricaud et al., 2002). For example, Gross and Sadowski \ \ (2001) applied second-order Barker-Henderson perturbation ‘Y \ theory to a hard-chain reference fluid to improve the SAFT “ \ equation of state. To illustrate, Figures 16 and 17 show calcu- “NY \ lated and experimental phase equilibria for some polymer and 1.0 ~ copolymer solutions, respectively. Ss In addition to poor agreement with experiment in the critical region, all currently available EOS for chain-like molecules suffer from one limitation: for long chains, at dilute conditions, ot 9 0.8 the calculated second virial coefficient has an incorrect depen- 0.2 . My “6 . dence on chain length. As a result, SAFT (and similar) EOS give unreliable results for the dilute polymer phase when used Figure 18. Calculated and experimental VLE for sulfur to calculate fluid—fluid equilibria when the polymer is concen- dioxide (1)-propane (2) at 50 °C. trated in one phase and dilute in the other. Full lines are for calculations with an optimized binary For strongly associating components it is tempting to super- interaction parameter (k,), whereas dotted lines are for impose a “chemical” theory onto a “physical” EOS. The gen- k,,=0. Here x is mole fraction, and the points are experi- 1 d for doi h by Heid d mental data.Calculations with the Association-plus-PR EOS era proce ure for ong sO was Ss own y rer emann an of Anderko (1989). Prausnitz (1976) who introduced a chemical association equi- AIChE Journal April 2004 Vol. 50, No. 4 751 librium constant into the van der Waals EOS. Following some reasonable simplifications, an analytic EOS was derived with two physical interaction constants, van der Waals a and b, and one temperature-dependent equilibrium constant, K (T). On the basis of similar ideas, several authors have represented the properties of solutions containing one associating component and one or more normal fluids; an example is shown in Figure 18 by Anderko (1989). Another example for particularly “nasty” mixtures containing hydrogen fluoride is shown in Figure 19. Equations of state have been used extensively to design supercritical-extraction processes (McHugh and Krukonis, 1994). Because supercritical extraction uses high pressures and because of the large difference in size and shape between solute and solvent, these systems present a variety of phase diagrams. For example, binary mixtures of ethane-linalool and ethane- limonene present double retrograde condensation, as discussed by Raeissi and Peters (2002, 2003). For supercritical-extraction conditions, calculation of phase equilibria with an EOS, and the equifugacity criteria may present multiple roots. The “correct” root should be selected by a global phase-stability method. In a series of articles, Brennecke and Stadtherr (Xu et al., 2000; Maier et al., 2000; Xu et al., 2002) used an EOS to obtain high-pressure solid-fluid equilibria. To illustrate, Figures 20 and 21 show the solubility of biphenyl in CO2 at different pressures calculated with two methods: equifugacity and global optimization. Figure 21 shows that, at 333.15K the equifugac- ity calculation gives an unstable root at some pressures. Most supercritical-extraction processes use carbon dioxide as the solvent because of its attractive environmental proper- ties. Because many compounds have very low solubilities in carbon dioxide, even at high pressures, a supercritical-extrac- tion process may require an undesirable high solvent flow rate. To increase solubilities dramatically, it has been suggested to add a surfactant that can form micelles of the extracted com- pounds in the solvent. However, ordinary surfactants (intended for oil-water systems) are not effective in dense carbon diox- ide. In a significant contribution that shows the importance of chemistry in applied thermodynamics, Beckman (Ghenciu et al., 1998; Sarbu et al., 2000) synthesized entirely new surfac- tants suitable for micellization in dense carbon dioxide. Equations of state can be used to describe adsorption of pure gases and their mixtures on solid surfaces. An early discussion was given by Van Ness (1969); a more recent one is by Myers (2002). Here, the parameters of the EOS depend not only on the adsorbate but also on the adsorbent, that is, the solid surface provides an energetic field that affects the forces between the adsorbed molecules. Equations of state are useful for design of crystallization processes where it is important that the precipitated solids have a narrow size distribution (Chang and Randolph, 1989, 1990), and for calculating hydrate formation in moist natural gases as discussed in the monograph by Sloan (1990). Equations of state are also useful for describing particle precipitation from liquid solution. For that case, it is convenient to write the EOS in terms of the McMillan–Mayer framework (1945) where the potentially precipitating particles are dis- solved in a continuous liquid medium. The pressure is replaced by the osmotic pressure; the density now is not the density of the system but that of the particles in the liquid medium. Although this type of EOS has been used for many years in colloid science, it has received only little attention from chem- ical engineers in the conventional chemical industries. Ogston (Laurent and Ogston, 1963; Edmond and Ogston, 1968) has used an EOS in the McMillan-Mayer framework to describe dilute solutions of two (or more) polymers, while Wu et al. (1998, 2000) have shown how an EOS of this type can be used to describe precipitation of asphaltenes from heavy petroleum. Regardless of what EOS is used, perhaps the most important engineering application of an EOS lies in the estimation of Figure 21. PR EOS calculation of the solubility of biphe- nyl in CO2 at 60 °C. At this temperature, there are multiple equifugacity roots for pressures below 160 bar. At low pressure, the lowest solu- bility root corresponds to stable equilibrium, but at higher pressure, it is the highest solubility root that is stable. A three-phase line, indicating solid-liquid-vapor coexistence, occurs at about 45.19 bar (Xu et al., 2000). Here y is mole fraction. Figure 20. PR EOS calculation of the solubility of biphe- nyl in CO2 at 35 °C. At this temperature, there is only one equifugacity root at each pressure, and that root corresponds to stable solid-fluid equilibrium (Xu et al., 2000). Here y is the mole fraction. 752 AIChE Journal April 2004 Vol. 50, No. 4 12 rr ns + ny 100 grocer enema erica yaaa ta ee REE AMOS Detted Symbols: Solid KC! i : : : Zx{NOa), i $ ; Filled Symbots: Solids KCI and Nati : : : : | 200°C 5 Open Symbols. Seid NaCl ° 90 ones muaboas poe ZA(NO3)7 HzO eth ad i £ & —~ Pitzer Model : . : . = o* ; ** 140°C ° : ZANO;)x2H,0 6” : - 5 £0 poo ed ‘9 igo'c At IN x : 3 2 4 NOs) 4HZ0 3 | : 2 64 s0°C . P | 76 sPeeeemeenes courte acione RR nn En fees cen coed 2 #. i i 0 } i ' a > & e = i : YS : i = f osc % : : i ; : : i E - > 6 N 60 4 ooh GF eet ern cae ‘| ~~ E | ZaNOsy 6H, | Xr 2 iP — > a a : ; i : ; SA \ = 40 ¥€-- In(NO3),9H,0 a pent cond 9 2 4 6 8 a“ b i i i : ; a FS 30 be tl Prac, Mol kg | ar’ > =~ —— ; ~~ po wscemener ao i i : . speas Q j : 5 : Figure 22. Calculated and experimental solubilities of 20 +------0) oferta ante a salts in the ternary system NaCl/KCI/H,O at lee : | i Calculated | i several temperatures. 10 |~-—--— + 0 Experimental) | Intersections of isothermal curves represent calculated ter- dé : i : i nary invariant points where three phases are in equilibrium: i a i : r pure NaCl solid, pure KCl solid, and aqueous solution 0: ‘ ° " containing both salts (from Prausnitz et al., 1999). Here, m . : 1 = mobility. 25 6 35 50) mperative co multicomponent VLE with only parameters obtained from sin- Figure 24. Solubility of Zn(NO,), in water as a function of gle-component and binary experimental data (or from correla- temperature. tions based on such data). In an EOS of the van der Waals In addition to the anhydrous salt, five hydrates are formed in . . : the solution containing zinc nitrate: nonahydrate, hexahy- form, we consider only two-body interactions. Therefore, once drate, tetrahydrate, dihydrate, and monohydrate. Calcula- we have properly extended that EOS to a binary mixture, no tions using a g” model that combines UNIQUAC with the further assumptions are required to achieve extension to ternary DebyeHuckel theory are able to describe the five eutectic . , - . . points and one peritectic point (P) of the aqueous Zn(NO;), (and higher) mixtures. In a mixture, two-body interactions are system (from Iliuta et al., 2002). reflected by mixing rules that are quadratic in composition. If mixing rules use higher-order terms, extension from binary to ternary (and higher) mixtures presents significant theoretical problems as noted by Michelsen and Kistenmacher (1990) and by Mollerup and Michelsen (1992). 102 iy Extensive experience in applying an EOS to calculate fluid- phase equilibria has shown that for typical fluid mixtures, the ' daze,» , o'° role of details in the EOS itself or in its mixing rules is less 9.98 “eer Hee important than that of the choice of constants obtained from 0.86 ie mee Fs some experimental source. 2 0.94 SS ° ae oN 2 : Electrolyte Solutions B 0.92 NN a *o7 i . . = aN “O : Because electrostatic forces between ions are long-range, the o8 XN oe mS os physical chemistry of electrolyte solutions is qualitatively dif- 0.88 Q os : ferent from (and more difficult than) that for solutions of 0.86 ' nonelectrolytes. Because electric neutrality must be main- oe ad tained, the concentrations of cations and anions are not inde- v0.00 0.02 0.04 0.08 0.08 o.10 pendent. As a result, conventional experimental thermody- Mole Fraction Electrolyte namic data for electrolyte solutions do not give the activity of ; . _o. the cation and that of the anion but instead, a mean ionic Figure 23. Calculated and experimental water activities activity coefficient. Furthermore, because salts are not volatile at different electrolyte concentrations for at ordinary temperatures, mean ionic activity coefficients refer various aqueous sodium carboxylates: meth- not to the pure electrolyte but to an ideal dilute solution of the anoate (C1), ethanoate (C2), propanoate (C3), electrolyte in the solvent. butanoate (C4), pentanoate (C5), hexanoate For very dilute solutions of strong electrolytes (complete (C6), heptanoate (C7),octanoate (C8), non- dissociation into ions), we have the Debye—Hiickel (DH) the- anoate (C9), and decanoate (C10). ory of 1923; this theory gives the mean ionic activity coeffi- Calculations with a ee model that combines NRL for ionic cient arising from electrostatic ion—ion forces in a medium of systems wl or ougomers are able to describe the . : : : . . . oe abrupt change in the water activity at the critical micelle known dielectric constant. In its rigorous, highly dilute limit, concentration (Chen et al., 2001). DH theory neglects the finite sizes of the ions and van der AIChE Journal April 2004 Vol. 50, No. 4 753 3 - * 20 s oa / 5 Myy,= 3.2 mol kg! ” . g 5 ’ 3 § 15 af | 5 : @ , = a ? S ° o 16 / a c i Bs 2 0 a Ee 2 a 0 10 20 30 40 5 ia Tie-line Length, wt % a 4 _y Muy, = 9.1 mol kg” Figure 27. Calculated and experimental partition coeffi- I “On ae 3 * cients for three dilute proteins in an aqueous Q i tn two-phase system containing PEG 3350, dex- 0 4 8 4 12 tran T-70, and 50 mM KCI (overall) at pH = 7.5 _ Molality SO, mol kg and 25 °C. Figure 25. Simultaneous solubilities of NH, and SO, in molalities, Experimental and calculated results are fr albu water at 100 °C, calculated by Edwards et al. min (circles), chymotrypsin (squares), and lysozyme (trian- (1978); experimental results are from Rumpf gles) (Haynes et al., 1991). et al. (1993). Waals attractive forces between ions. To describe the thermo- and the second part is, essentially, an osmotic virial series in dynamic properties of concentrated electrolyte solutions, nu- electrolyte concentration. Regrettably, this power series re- merous phenomenological extensions of DH theory have been quires several system-specific coefficients that depend on tem- presented. Perhaps the most successful is the one by Pitzer perature. However, because we have a large body of experi- (1973, 1995) which, m effect, expresses the excess Gibbs mental results for aqueous salt solutions over a reasonable energy (relative to an ideal dilute solution) as a sum of two temperature range, we now have a fair inventory of Pitzer parts: the first part is based on a slightly modified DH theory, parameters. To extend Pitzer’s model to multisalt solutions, it is necessary to make some simplifying assumptions or else to introduce one or more ternary parameters. Pitzer’s model has 10 been applied toward optimizing process design for salt recov- ery from Trona mines as discussed by Weare (Harvey et al., og 1984), and for designing a recovery process for radioactive ; salts from aqueous solutions (Felmy and Weare, 1986). 5 oa To illustrate Pitzer’s theory for an aqueous system contain- fe a ing two salts (potassium chloride and sodium chloride), Figure 8 a 22 shows experimental and calculated salt solubilities in the 2 OBS ion 0 to 200 °C E A ; region 0 to . g mi * An alternate model for activity coefficients in aqueous elec- g 06 i eae ey a a”. tt trolyte solutions was developed by Chen et al. (1982, 1986, 5 ed . — 2001) who used the NRTL equation to account for ion—ion and 5 05 4 EA Ee ion—solvent interactions beyond those given by DH theory. An = a M3, ~ advantage of Chen’s model is that, in at least some cases, it 04 requires fewer binary parameters than Pitzer’s model. Figure i 23 shows an application of Chen’s model to aqueous organic 90 05 10 15 20 258 30 325 46 45 80 55 electrolytes. A similar theory, based on the UNIQUAC equa- Molality of KCL/motkg! tion, was presented by Tliuta et al. (2002) who gave particular attention to solubilities of heavy-cation salts. Figure 24 shows Figure 26. Calculated and experimental mean ionic ac- calculated and observed solubilities of various hydrates of tivity coefficients of KCI in mixtures of meth- Zn(NO3)p. anol and water at 25 °C: pure water (L), 90 wt. When augmented by chemical equilibria, a much simplified % (salt-free) water (MM), 80 wt. % water (A), 60 version of Pitzer’s model has been used by Edwards (Edwards wt. % water (A), 40 wt. % water (©), 20 wt. % et al., 1978) to correlate multicomponent vapor-liquid equilib- water (@), 10 wt. % water (V), 0 wt. % water ria for aqueous solutions of volatile electrolytes (NH;, H,S, (¥) (Papaiconomou et al., 2002). CO,, SO,) that are frequently encountered in chemical pro- 754 April 2004 Vol. 50, No. 4 AIChE Journal cesses. To illustrate, Figure 25 shows the total pressure as a function of SO2 concentration for aqueous mixtures of SO2 and NH3 at 100°C. The early theory by Edwards has been much improved by applying the full Pitzer theory. Maurer and coworkers have presented an extensive correlation of VLE for aqueous solu- tions of weak electrolyte gases with or without selected added salts (Rumpf et al., 1993; Bieling et al. 1995, Kamps et al., 2002, Kamps et al., 2003). Integral-equation theory can be used to establish a theoreti- cal basis for describing electrolyte solutions as discussed by Papaiconomou et al. (2002). To account for electrostatic and free-volume interactions of ions in solution, including the con- centrated region, these authors used the integral theory of solution coupled with Blum’s mean spherical approximation; the effects of van der Waals attractive forces are provided by an equation similar to NRTL. This theory is readily applicable to a salt in a solvent mixture; to illustrate, Figure 26 shows calculated and observed mean-ion activity coefficients for KCl in methanol–water mixtures at 25°C. Figure 28. Comparison of Monte Carlo simulations with experimental data for adsorption of binary mixtures of propane (1)- H2 S(2) on H-mordenite at 8.13 kPa and 30°C. Parameters were obtained from each pure-component adsorption isotherm. (a) gives the phase diagram, and (b) gives the total amount of gas adsorbed (Cabral et al., 2003). AIChE Journal 755 April 2004 Vol. 50, No. 4 Integral-equation theory can also be used to establish an EOS for describing electrolyte solutions as discussed by Jin and Donohue (1988, 1988), Wu and Prausnitz, (1998), and Myers et al. (2002). Depending on pH, proteins carry an electric charge. Therefore, a description of the thermodynamic properties of protein solutions must include electrostatic effects in addi- tion to van der Waals forces, as discussed, for example, in Albertsson’s (1986) book on separation of protein mixtures by extraction in aqueous two-phase systems. Such systems are formed upon dissolving two water-soluble polymers, for example, dextran and poly(ethylene glycol) (PEG). A mea- sure of how these two aqueous phases differ is provided by the length of a tie line on a plot where the percent PEG in one phase is plotted against the percent dextran in the other. Figure 27 shows calculated and observed distribution coef- ficients for three dilute proteins in a two-phase aqueous system as a function of the difference between the two phases (Haynes et al., 1991). Because this system also contains a small amount of KCl that partitions unequally between the two aqueous phases, and because the charges on the three proteins are not identical, the distribution coeffi- cients differ widely, facilitating protein separation. Molecular Simulations An alternative to algebraic expressions for activity coeffi- cients or equations of state is provided by molecular simula- tions as discussed in several textbooks, notably that by Sadus (1999) and that by Frenkel and Smit (2002). Molecular simulations are attractive because they require as input only quantitative data for molecular structure, and for the potential of molecule–molecule interaction. The dis- advantage of molecular simulations is that results are re- stricted to a particular case; these results are not easily generalized. The last 15 years have produced a large number of articles showing how molecular simulation can be used to calculate phase equilibria for a large variety of systems based on the Gibbs–ensemble method of Panagiotopoulos (1987). Although numerous authors have contributed to this important develop- ment, particularly noteworthy are the articles by Cummings and coworkers (for example, McCabe et al., 2001; Rivera et al., 2003) and those by de Pablo et al. (for example, Nath et al., 1998; Yan and de Pablo, 2001; de Pablo and Escobedo, 2002; Jendrejack et al., 2002). To illustrate applicability to process design, Monte Carlo simulations can be used to describe the adsorption of pure gases and their mixtures on solid surfaces as discussed, for example, by Smit and Krishna (2003), and by Steele (2002). To illustrate, Figure 28 shows calculated and experimental gas-solid adsorption equilibria for mixtures of propane (1)- H2S (2) on H-mordenite (Cabral et al., 2003). Because the results from molecular simulations are sensitive to the potential function that describes intermolecular forces, it is necessary to obtain that potential function from the reduction of some experimental data. For some cases, the extent of required experimental data can be reduced by quantum me- chanics. Application of Quantum Mechanics One of the most promising recent developments in chemical engineering thermodynamics is provided by applying quantum mechanics for calculating thermodynamic properties, in partic- ular, activity coefficients of components in liquid mixtures. Quoting from Sandler’s review (Sandler, 2003): “In the most direct and computational intensive form, com- putational quantum mechanicsis used to obtain information on the multidimensional potential energy surface between mole- cules, which is then used in computer simulation to predict thermodynamic properties and phase equilibria. At present, this method is limited to the study of small molecules because of the computational resources available. The second method is much less computationally intensive and provides a way to Figure 29. Calculated and experimental VLE phase dia- grams for acetronitrile(1)-methanol(2) at 60.31 °C. Phase diagrams show (a) vapor vs. liquid mole fraction and (b) pressure vs. vapor and liquid mole fraction. Filled circles are experimental data (DECHEMA), and open squares and diamonds are predictions from Gibbs-ensemble Monte Carlo NPT and NVT simulations, respectively; the solid line is the best fit of the experimental data (Sum et al., 2002). Figure 30. Calculated and experimental VLE for ben- zene(1)-N-methyl formamide(2) at 45 and 55 °C; calculation with the COSMO-SAC model (Lin and Sandler, 2002). Circles and triangles are experimental. 756 AIChE Journal April 2004 Vol. 50, No. 4 improve group-contribution methods by introducing correc- tions based on the charge and dipole moment of each functional group that is unique to the molecule in which it appears. The third method is based on the polarizable continuum model, in which the free energy of transferring a molecule from an ideal gas to a liquid solution is computed, leading directly to values of activity coefficients and phase equilibrium calculations.” For typical polar molecules, such as acetonitrile or methyl fluoride, it is now possible to establish a reliable two-body potential that depends on all distances between the atoms of one molecule and those of the other. In some cases, the poten- tial can be simplified by considering only the distance between the center of mass of one molecule and that of the other in addition to angles of orientation. For polar molecules having less than (about) 100 electrons, knowing the geometric and electronic structures of the molecules is sufficient to establish the two-body potential; for more difficult cases (for example, methanol), a well-measured thermodynamic property (typically the second virial coefficient) is used to augment results ob- tained from quantum mechanics. For a binary mixture containing components 1 and 2, we need three two-body potentials: one each for 1–1, 2–2 and 1–2 interactions. Some, or perhaps all of these potentials may be obtained from quantum mechanics. These potentials are then used in a Monte Carlo-simulation program to generate vapor- liquid or other phase equilibria. Although this promising type of calculation is likely to see increasing popularity, at present, for industrial application, it suffers from two disadvantages: typical simulation calculations are limited by the additivity assumption (the total potential energy of a system is given by the sum of all two-body interactions), and by insufficiently powerful computers. Although corrections for nonadditivity are not simple, they are often significant, especially for hydro- gen-bonding systems. A highly computer-intensive method for calculating multibody potentials is provided by Car and Par- rinello (1985), but as yet this method is not sufficiently sensi- tive for application to mixtures of ordinary liquids (Trout, 2001). Figure 29 shows a successful application of quantum me- chanics-plus-Monte Carlo simulation for vapor-liquid equilib- ria for methanol-acetonitrile at 333.46 K (Sum et al., 2002). It is remarkable that, although no mixture data were used to generate Figure 29, the calculations give the correct pressure and composition of the azeotrope. More than a century ago, Mossotti derived an equation for the change in energy experienced by a dipolar molecule when it is transferred from an ideal gas into a continuous liquid medium characterized by its dielectric constant (Israelachvili, 1992). Similarly, more than eighty years ago, Born indicated how the free energy of a charged molecule changes when it goes from one dielectric medium to another (Israelachvili, 1992). In the same spirit, but with more powerful physics, Klamt and coworkers (Klamt, 1995; Klamt and F. Eckert, 2000) have developed a method for calculating the activity coefficient of a solute dissolved in a continuous polarizable medium. This method does not use functional groups but uses surface charges for atoms that depend not only on the particular atom, but also on the identity of other atoms in the same molecule. Thus, Klamt’s method, in effect, overcomes one of the serious limitations of UNIFAC. Klamt’s method is attrac- tive for engineering because computational requirements are relatively low. However, at present this method is limited to activity coefficients of solutes in dense liquids, that is, liquids well below their critical temperatures; it is not (yet) applicable to gaseous mixtures or to low-density liquid mixtures encoun- tered in the vapor-liquid critical region. Figure 30 shows a successful example of Klamt’s COSMOSAC model for vapor- liquid equilibria in the benzene–N–methyl formamide system (Lin and Sandler, 2002) Figure 31. Comparison of calculated and experimental distribution coefficients for a large number of solutes distributed between water (W) and octanol (O) at high dilution, near room-tem- perature. The partition coefficient is defined by the ratio of solute molar concentrations (mol/L). Crosses are for monofunc- tional molecules, and circles are for multifunctional mole- cules (Sandler, 2003). AIChE Journal 757 April 2004 Vol. 50, No. 4 Quantum Mechanics for Group-Contribution authors are grateful to the National Science Foundation, to the Office for Basic Parameters Sciences of the U.S. Dept. of Energy, to the Donors of the Petroleum Research Fund administered by the American Chemical Society, and to the Brazilian The popularity of UNIFAC (and other group-contribution Minister of Education, CAPES/Brazil, for grants BEX 0621/02-1. methods) has encouraged numerous authors toward seeking improvements that overcome some of UNIFAC s well-known Literature Cited limitations. Perhaps the most important limitation of UNIFAC j , ae . is its neglect of neighbor effects; in UNIFAC, the interaction Abildskov, J., and J. P. oO Connell, Predicting the Solubilities of Complex . . . Chemicals I. Solutes in Different Solvents,” Ind. Eng. Chem. Res., 42, between a functional group X and a functional group Y is 5622 (2003) assumed to be independent of the identities of whatever func- Abraham, M. H., and J. A. Platts, “Hydrogen Bond Structural Group tional groups are bonded to X or Y. For example, in UNIFAC, Constants,” J. Org. Chem., 66, 3484 (2001). a chloride group in say, CH; — CH,Cl _ CH;, is equivalent to Abrams, D. S., and J. M. Prausnitz, “Statistical Thermodynamics of Liquid that in say, CH. — CHCl - CH.OH. With quantum mechan- Mixtures: a New Expression for the Excess Gibbs Energy of Partly or . a y 3 2 2 . q . Completely Miscible Systems.” AIChE J., 21, 116 (1975). ics, it is now possible to correct UNIFAC group-group inter- Albertsson, P.-A., Partition of Cell Particles and Macromolecules, 3rd ed., action parameters for the proximity effect because of neigh- Prentice Hall PTR, N.J. (1986). boring bonded groups. For molecules that contain only one Anderko, A., “Calculation of Vapor-Liquid Equilibria at Elevated Pres- polar functional group, proximity corrections are not large. sures by Means of an Equation of State Incorporating Association,” . Chem. Eng. Sci., 44, 713 (1989). However, for molecules that contain two or more P olar func- Anisimov, M. A., and J. V. Sengers, “Critical Region,” Equation of State tional groups, proximity corrections are often significant, es- for Fluids and Fluid Mixtures. Experimental Thermodynamics, Vol. 5. pecially if two polar functional groups are in close proximity as J. V. Sengers, R. F. Kayser, C. J. Peters, and H. J. White Jr., eds., found, for example, in biomolecules and pharmaceuticals. To Elsevier, Amsterdam (2000). ee illustrate, Figure 31 presents calculated and experimental dis- Asher, W. E., J. F. Pankow, G. B. Erdakos and J. HL. Seinfeld, | Estimating . . : . . the Vapor Pressures of Multi-Functional Oxygen-Containing Organic tribution coefficients for a large number of dilute solutes dis- Compounds Using Group Contribution Methods,” Atmospheric Envi- tributed between water (W) and octanol (O) near room-tem- ronment, 36, 1483 (2002). perature (Lin and Sandler, 2000; Sandler, 2003). Part (a) of Baburao, B., and D. P. Visco, “WLE/VLLE/LLE Predictions For Hydrogen Figure 31 shows UNIFAC calculations without proximity cor- Fluoride Mixtures Using An Improved Thermodynamic Equation Of i . . State,” Ind. Eng. Chem. Res., 41, 4863 (2002). rections, whereas Part (b) shows calculations with quantum- Behrens, D. and R. Eckermann, Chemistry Data Series, DECHEMA, Frankfurt mechanical proximity corrections. For monofunctional mole- a.M., Vol. I, (subdivided into nineteen separate volumes) VLE Data Col- cules there is little difference; however, for multifunctional lection by J. Gmehling, U. Onken, W. Arlt, P. Grenzheuser, U. Weidlich, molecules, proximity corrections produce a large improvement and B. Kolbe (1980-1996);Vol II, Critical Data by K. H. Simmrock in agreement with experiment. (1986); Vol II, (subdivided into four volumes) Heats of Mixing Data . . . Collection by C. Christensen, J. Gmehling, P. Rasmussen, and U. Weidlich There is good reason to believe that, as computer speed rises, (1984-1991); Vol V, (subdivided into four volumes) LLE-Data Collection we will see increasing use of molecular simulations, and in- by J. M. Sorensen and W. Arl (1979-1987); Vol VI, (subdivided into four creasing use of quantum mechanics for the calculation of volumes) VLE for Mixtures of Low-Boiling Substances by H. Knapp, R. thermodynamic properties. It is likely that extensive use of Doring, L. Oellrich, U. Plocker, J. M. Prausnitz, R. Langhorst and S. Zeck . or : . “ay: . : (1982-1987); Vol VIII, Solid-Liquid Equilibrium Data Collection by H. rigorous ab initio calculations is still in the indefinite future. Knapp, R. Langhorst and M. Teller, 1987; Vol IX,(subdivided into four However, it is now clear that the time is ripe for with molecular volumes) Activity Coefficients of Infinite Dilution by D. Tiegs, J. Gmehling, simulations and quantum mechanics to extend and improve A. Medina, M. Soares, J. Bastos, P. Alessi, and, I. Kikic, 1986-1994; Vol current methods for calculating phase equilibria. XIL, (subdivided into nine volumes) Electrolyte Data Collection by J. Barthel, R. Neueder, R. Meier et al., 1992-1997. Benedict, M., G. B. 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