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The Advanced Chemical Engineering Thermodynamics The thermodynamics properties of fluids (I) Q&A_-7- 10/27/2005(7) Ji-Sheng Chang

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Property relations The primary equation The first law of thermodynamics: d(nU) = dQ rev + dW rev The work of the reversible process for closed system: dW rev = -Pd(nV) The second law of thermodynamics: dQ rev = Td(nS)

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Property relations To combine each terms: d(nU) = Td(nS) – Pd(nV) dU = TdS – PdV For unit mole system For the mathematic view for the internal energy: U = f(S,V) or U = U(S,V)

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Property relations dU = TdS – PdV The total differential of the internal energy: dU = (U/S) V dS + (U/V) S dV The partial differential term relation to the property: T = (U/S) V ; P = - (U/V) S

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Other energy functions The enthalpy: H = U + PV; The Helmholtz free energy: A = U – TS; The Gibbs free energy: G = H – TS;

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Other energy functions The enthalpy: H = U + PV; dH = dU + d(PV); dH = dU + PdV + VdP; dH = TdS + V dP; H = H (S,P)

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Other energy functions The Helmholtz free energy: A = U – TS; dA = dU – d(TS); dA = dU – TdS – SdT; dA = – SdT – PdV; A = A (T,V)

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Other energy functions The Gibbs free energy: G = H – TS; dG = dH – d(TS); dG = dH – TdS – SdT; dG = – SdT + VdP; G = G (T,P)

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The Maxwell’s equations dU = TdS – PdV, dU = (U/S) V dS + (U/V) S dV T = (U/S) V ; P = – (U/V) S Doing the next variable partial differential for each terms (T/V) S = [ 2 U/VS] ; (P/S) V = - [ 2 U/SV] Then, (T/V) S = – (P/S) V ;

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The Maxwell’s equations The Maxwell ’ s equation of each one that derived from internal energy. (T/V) S = – (P/S) V There have four Maxwell ’ s equations for a closed homogeneous system.

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Some definition Some more useful and interested definition Cp = (H/T)P; Cv = (U/T)V; = [(V/T)P]/V; = – [(V/P)T]/V

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Enthalpy and entropy Enthalpy and entropy as functions of T and P H=H(T,P) dH = (H/T) P dT + (H/P) T dP In terms of PVT and Cp result in (6.20) S=S(T,P) dS = (S/T) P dT + (S/P) T dP In terms of PVT and Cp result in (6.21)

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Internal energy and entropy Internal energy and entropy as functions of T and V U=U(T,V) dU = (U/T) V dT + (U/V) T dV In terms of PVT and Cv result in (6.32) S=S(T,V) dS = (S/T) V dT + (S/V) T dV In terms of PVT and Cv result in (6.33)

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Generating function of Gibbs free energy G = G(T,P) In the view of classical mechanics In the view of empirical thermodynamics In the view of traditional thermodynamics

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Generating function of Gibbs free energy G = G(T,P) dG = VdP – SdT d(G/RT) = (1/RT)dG – (G/RT 2 )dT Using G = H – TS and dG = VdP – SdT d(G/RT) = (V/RT)dP – (H/RT 2 )dT

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Generating function of Gibbs free energy G/RT = f(T,P) V/RT= [(G/RT)/P] T H/RT= - T [(G/RT)/T] P S = H/T - G/T U = H - PV

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Generating function of Helmholtz free energy A = A(T,V) In the view of the quantum mechanics In the view of the quantum physics In the view of the statistical thermodynamics

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Generating function of Helmholtz free energy A = A(T,V) dA = – PdV – SdT d(A/RT) = (1/RT)dA – (A/RT 2 )dT Using A = U – TS and dA = – PdV – SdT d(A/RT) = (P/RT)dV – (U/RT 2 )dT

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Generating function of Helmholtz free energy A/RT = f(T,V) P/RT= [(A/RT)/V] T U/RT= - T [(A/RT)/T] V H=U+PV S = U/T - A/T

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