Oliveira Capitulo 4 Exercicios

UNIVERSIDADE FEDERAL DE MATO GROSSO CAMPUS UNIVERSITÁRIO DO ARAGUAIA MESTRADO NACIONAL PROFISSIONAL EM ENSINO DE FÍSICA TERMODINÂMICA E MECÂNICA ESTATÍSTICA EXERCÍCIOS CAPÍTULO 4 - OLIVEIRA Docente: Prof. Dr. Adellane Araújo Souza Discente: Jefferson Margraf Lopes Barra do Garças – MT – 28 de junho de 2020 OLIVEIRA - CAPITULO 4 - EXERCICIOS 1) PARA CADA UMA DAS RELAÇÕES FUNDAMENTAIS ABAIXO, INDIQUE QUAIS S SÃO EXTENSIVAS. DETERMINE AS EQUAÇÕES DE ESTADO PARA CADA CASO. ACHA A RELAÇÃO FUNDAMENTAL NA REPRESENTAÇÃO DA ENERGIA. a) S = A(UVN)^{1/3} b) S = NclnU + NRlnV + Na N N c) S = B(U^2V)^{1/4} a) S = A(λUλVλN)^{1/3} Temos A(λUλVλN)^{1/3} = A(UVN)^{1/3} = λS ds = 1 dU + pdV − μ dN T T T 1 = 1 A(VN)^{1/3} U^{1/3−1} T 3 1 = 1 A(VN)^{1/3} U^{−2/3} T 3 1 = 1 A(VN)^{1/3} U^{−2/3} T 3 p = 1 ∂S T ∂V p = 1 A(λUN)^{1/3} V^{1/3−1} T 3 p = 1 A(VN)^{1/3} V^{1/3−1} T 3 p = 1 A(UN)^{1/3} V^{−2/3} T 3 p = 1 A(λUN)^{1/3} V^{−2/3} T 3 μ = 1 ∂S T ∂N μ = 1 A(λUV)^{1/3} N^{1/3−1} T 3 μ = 1 A(λUV)^{1/3} N^{−2/3} T 3 μ = 1 A(UV)^{1/3} N^{−2/3} T 3 S = A(UV)^{1/3} N^{1/3} × (1)^{3} (1)^{3}= (A(UV)N)^{1/3} N^{1/3} S^{3} = A^{3} (UVN)^{3/3} S^{3} = A^{3} UVN U = S^{3} A^{3}VN b) S = Ne ln U + NR ln V + Na N N temos = λNcln λU + λNRln λV + λNa = λS λN λN ds = 1 dU + p dV − μ dN T T T OLIVEIRA - CAPITULO 4 - EXERCICIOS 1 - \frac{\partial S}{T \partial U}\quad p/\ S= N c lnU - N c lnN + N R ln\frac{V}{N} + Na \frac{\mu}{T} = \frac{Nc}{U}\quad \frac{p}{T} = \frac{NR}{V} \mu = \frac{\partial S}{T \partial N}\quad p/\ S=N c lnU - N c lnN + N R ln\frac{V}{N}-N R lnN + Na \quad \mu = \frac{c ln \frac{U}{N} + R ln \frac{V}{N} + a - c - R}{T}\quad (x - 1) \frac{\mu}{T} = -c ln \Big(\frac{U}{N}\Big) - R ln\Big(\frac{V}{N}\Big) - a + c + R DEMONSTRACAO I\quad N c lnU = c ln U II\quad - N c lnN = - c lnN - \frac{N c}{N} III\quad N R lnV = R lnV IV\quad - N R lnN = - R lnN - \frac{pR}{NR} Na = a S = N c ln \frac{V}{N} + N R ln \frac{U}{N} + Na N c ln \frac{U}{N} = S - N R ln \frac{V}{N} - Na ln \frac{U}{N} = \frac{S - NR ln \frac{V}{N} - Na}{Nc} = \frac{S - R ln \frac{V}{N} - a}{c} x e^* \frac{ln \frac{V}{N}}{N} = \frac{e^{\frac{S - R ln \frac{V}{N} - a}{Nc}}}{\big(N e^{\frac{\big(S - R ln \frac{V}{N} - a \big)}{Nc}}\big)} \ln \frac{U}{N} = \frac{S - R ln \frac{V}{N} - a}{c}\quad x\ e^* OLIVEIRA CAPITULO 4 - EXERCICIO \frac{1}{T\partial V}\quad = B\Big(\frac{U^{3/4}}{V^{1/4}}\Big)\quad \frac{1}{T\partial U}\quad ?\quad S = B\Big(\frac{U^{3/4}}{V^{1/4}}\Big)\ \text{termos} B \lambda U^{3/4} V^{1/4} = \lambda S S = B\Big(U^{3/4} V^{1/4}\Big)^{\lambda} U = B^{4/3}\quad B^{4/3}\quad \Big(v^{1/4}\Big)^{-4/3} \text{Cada uma das relacoes fundamentais acima, nome} \text{que U e' extensiva, determine as equacoes de} Estudo para cada caso a) U = aS^3 / VN b) U = bNe^{\frac{S}{Nc}-\frac{R}{c} \ln \frac{V}{N}} c) U = c \left(\frac{S^4}{V}\right)^{1/3} a) U = aS^3 / VN a,\(S)^3 = \lambda N \lambda\wedge\lambda N du = TdS - pdV + \mu dN T = \frac{\partial U}{\partial S}|\quad U = \frac{aS^3}{VN} T = \frac{3aS^2}{VN} p = -\frac{\partial U}{\partial V}|\quad N = \frac{aS^3 V^{-1}}{N} p = -(-aS^3 V^{-1-1}){N} p = \frac{aS^3 V^{-2}}{N} p = \frac{aS^3}{V^2 N} OLIVEIRA CAPITULO 4 - EXERCICIO \mu = \frac{\partial U}{\partial N}|\quad U = \frac{aS^3N^{-1}}{V} \mu = -aS^3 N^{-1-1}{V} \mu = -aS^3 N^{-2}{V} \mu = \frac{-aS^3}{VN^2} b) U = bN e^{\frac{S}{Nc} -\frac{R}{c} \ln \frac{V}{N}} bxNe^{\frac{S}{Nc} -\frac{R}{c} \ln \frac{V}{N}} = \lambda U T = \frac{\partial U}{\partial S} T= \frac{1}{Nc} bx e^{\frac{S}{Nc} -\frac{R}{c} \ln \frac{V}{N}} T = \frac{b}{c} e^{\frac{S}{Nc} -\frac{R}{c} \ln \frac{V}{N}} p = -\frac{\partial U}{\partial V}| p = -\left(\frac{R}{cV} \frac{1}{bN} e^{\frac{S}{Nc} -\frac{R}{c} \ln \frac{V}{N}} ight) p = \frac{bRN}{cV} e^{\frac{S}{Nc} -\frac{R}{c} \ln \frac{V}{N}} \mu = \frac{\partial U}{\partial N}|\quad U = bNe^{\frac{S}{Nc} -\frac{R}{c} \ln \frac{V}{N}} \mu = bN\left(-\frac{S}{Nc^2} - \frac{R}{cV N}\right) e^{\frac{S}{Nc} -\frac{R}{c} \ln \frac{V}{N}} \mu = b\rho\left(-\frac{S}{Nc^2} + \frac{R}{cV} \right) e^{\frac{S}{Nc} -\frac{R}{c} \ln \frac{V}{N}} U = \left(-\frac{S}{Nc} + \frac{R}{c} \right) be^{\frac{S}{Nc} -\frac{R}{c} \ln \frac{V}{N}} c) U = c\left(\frac{S^4}{V}\right)^{1/3} U = c \frac{S^{4/3}}{V^{1/3}}\nc(\lambda S)^{4/3} = \lambda U (\lambda V)^{1/3} du = TdS - pdV + \cancel{\mu dN} T = \frac{\partial U}{\partial S}||\quad U = c \frac{S^{4/3}}{V^{1/3}} T= \frac{4}{3} c \frac{S^{1/3}}{V^{1/3}} OLIVEIRA CAPITULO 4-EXERCICIOS T = \frac{4}{3} c S^{4/3} V^{1/3} F=\frac{4}{3} c S^{v3} V_{v3} p=-\frac{\partial F}{\partial V} \, \, \, \, U=cS^{4/3} \cdot V^{v3} p=-\left (\frac{1}{3} c S^{4/3} V^{-1/3} \right ) p=\frac{1}{3} c S^{4/3} V ^{1/3} p=\frac{-1}{3} c S^{4/3} V^{1/3} \large \textcircled{6}\small A PARTIR DAS EQUASOES p=\frac{NRT}{V} \, \, e \, \, C_{v} = Nc \text{ VÁLIDAS PARA UM GÁS IDEAL, OBTENHA A RELAÇÃO} \text{ FUNDAMENTAL NA REPRESENTAÇÃO DA ENERGIA LIVRE} \text{ de HELMHOLTZ} \text{USANDO A MECÂNICA} p=-\left (\frac{\partial F}{\partial V} \right )_{T,N} \, \, \text{como} \, \, p=\frac{NRT}{V}\, \, \text{temos} \left (\frac{\partial F}{\partial V} \right )_{T} = -\frac{NRT}{V} \, \, \text{INTEGRANDO EM V} FONONI