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

2. 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)

3. 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)

4. 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

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

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

7. 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)

8. 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)

9. 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/VS] ; (P/S) V = - [ 2 U/SV]  Then, (T/V) S = – (P/S) V ;

10. 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.

11. Some definition  Some more useful and interested definition Cp = (H/T)P; Cv = (U/T)V;  = [(V/T)P]/V;  = – [(V/P)T]/V

12. 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)

13. 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)

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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