Chapter 3 The second law

1. Chapter 3 The second law A spontaneous direction of change: the direction of change that does not require work to be done to bring it about. 3.1 The second law Clausius statement:No cyclic process is possible in which the sole result is the transfer energy from a cooler to a hotter body. All real process is irreversible process.

2. Second kind of perpetual motion machine A machine that converts heat into with 100 percent efficiency. It is impossible to built a second kind of perpetual motion machine. Kelven statement :No cyclic process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work.

3. 3.1.1 Carnot principle 1. Efficiency of heat engine 

4. Process AB： (i) CD ： (ii) DA ： (iv) BC ： (iii) (iii),（iv) therefore (v) 2. Carnon cycle

5. Irreversible engine(<) Irreversibleengine(<) Reversible engine (=) Reversible engine (=) 3. Carnot principle No heat engine can be more efficient than a reversible heat engine when both engines work between the same pair of temperature T1 and T2.

6. Q Q + ≤ 0 1 2 T T Q δ å 1 2 0 ≤ T ex Q δ ò ≤ 0 T Irreversible engine(<) Irreversible engine(<) Irreversible engine(<) ex Reversible engine (=) Reversible engine (=) Reversible engine (=) Irreversible process Reversible process 3.1.2 Entropy 1. Definition of entropy Clausius inequality

7. Reversible process Reversible process * Reversible process * (1) S is a state function (2) S is an extensive property (3) unit is J·K－1 Entropy S

8. (Irreversible cycle) therefore Irreversible Reversible Irreversible Reversible 2. Clausius inequality

9. For an adiabatic process ，δQ＝０， Irreversible Reversible Irreversible Reversible 3 . The principle of the increase of entropy The entropy of a closed system must increase in an irreversible adiabatic process.

10. For an isolated system ，δQ＝０， Since all real process is irreversible, when processes are occurring in an isolated system, its entropy is increasing. Thermodynamic equilibrium in an isolated system is reached when the system’s entropy is maximized. 4 . The entropy criterion of equilibrium

11. Tex=constant 5. Calculation of entropy change of surrounding Note: δQsu＝- δQsy

12. Reversible process 3.1.3 Calculation of entropy change of system Reversible adiabatic process. S=0 Reversible isothermal process.

13. (perfect gas) • Isochoric • Isobaric • Isothermal • Adiabatic • reversible • irreversible • liquid and solid 1. p, V, T change (1). p, V, T change

14. n2 T, p,V2 n1+n2 T, p, V1+V2 n1 T, p, V1 at constant T, p (2) mixing of different perfect gas at constant temperature and constant pressure mixing at constant T, p mixS =－(n1Rlny1＋n2Rlny2) y1＜1，y2＜1， so mixS＞０

15. 2. The phase transition (1) The phase transition at transition temperature At constant T, p , the two phase in the system are in equilibrium, the process is reversible, and W′＝０，so Qp＝H， fusHm＞０，vapHm＞０， Sm(s)＜Sm(l)＜Sm(g)

16. S=? B(,T1,p1) B(,T2,p2) Irreversible S1 S3 S2 B(,Teq,peq) B(, Teq,peq) reversible (2) irreversible phase transition S＝S１＋S２＋S３

17. S =? H2O（l, 90℃,100kPa) H2O（g, 90℃, 100kPa) Irreversible S1 S3 S2 H2O（l, 100℃, 100kPa) H2O（g, 100℃, 100kPa) Reversible For example S＝S１＋S２＋S３

18. 3.2 The third law 3.2.1 The third law Nernst heat theorem: The entropy change accompanying any physical or chemical transformation approaches zero as the temperature approaches zero: S0, T0. The entropy of all perfect crystalline substance is zero at 0 K. S *(perfect crystalline，0 K)＝０Ｊ·Ｋ-１

19. Second law 3.2.2 Conventional molar entropy and standard molar entropy Third law S *(０Ｋ)=０ Sm*（B，T)— conventional molar entropy of substance at temperature T。 Sm(B,,T)—standard molar entropy (p＝100kPa)

20. 3.2.3 The entropy change of chemical reaction For reaction aA＋b B →yY＋zZ rS m(T)＝ΣνＢS m（B, ,T ) rS m(298.15K)＝ΣνＢS m（B,  ,298.15K) rS m (T) = yS m (Y, ,T )＋z S m (Z, ,T ) －a S m (A, ,T )－b S m (B, ,T )

21. zZ bB yY aA + + bB aA zZ + yY + r Sym (T1) = Sy1 ＋Sy2 ＋r Sym(T2)＋S y3 ＋S y4

22. 3.3 Helmholtz and Gibbs energies 3.3.1 Helmholtz energy A Tex（S2－S1）≥Q at constant T , T2S2－T1S1＝Δ(TS) Q＝ΔU－W Δ(TS）≥ΔU－W －Δ(U－TS）≥－W

23. irreversible irreversible irreversible  AT≥  W AT ≤ W dAT ≤  W reversible reversible reversible Helmholtz energy A is an extensive state function The maximum work done by a closed system in an isothermal process is obtained when the process is carried out reversible.

24. δW′＝0 irreversible irreversible dAT, V ≤ 0 AT, V ≤ 0 reversible reversible At constant T and V W＝0 Helmholtz energy criterion In a closed system doing volume work only at constant T and V, the Helmholtz energy A decrease in a spontaneous change. In a closed system capable doing only volume work, the constant temperature and volume equilibrium condition is the minimization of the Helmholtz energy A. dAT, V = 0

25. 3.3.2 Gibbs energy G at constant T Δ(TS)≥ΔU－W at constant p p1＝p2＝pex W＝－pex（V2－V1）＋W ′ ＝－p2V2＋p1V1＋W′ ＝－Δ(pV)＋W′ Δ(TS)≥ΔU＋ Δ(pV)－W′ －［ΔU＋Δ(pV)-Δ(TS)］≥－W′ －Δ(U+pV-TS)≥－W′ Δ(H－TS）≤W′

26. G H－TS＝U＋pV－TS＝A＋pV GT ,p≤ W dGT, p ≤ W irreversible irreversible reversible reversible Gibbs energy G is an extensive state function The maximum possible non-volume work done by a closed system in a constant temperature and pressure process is equal to the G

27. GT ,p≤ W GT ,p≤ 0 dGT, p ≤ W dGT, p ≤ 0 irreversible irreversible irreversible irreversible reversible reversible reversible reversible W′＝0, δW′＝0 Gibbs energy criterion In a closed system doing volume work only at constant T and p, the Gibbs energy G decrease in a spontaneous change. In a closed system capable doing only volume work, the constant temperature and pressure equilibrium condition is the minimization of the Gibbs energy G. dGT, p = 0

28. dAT≤  W irreversible reversible For perfect gas 3.3.3 Calculation of A andG (1).change p ,V at constant temperature ΔA＝ΔU－TΔS ΔG＝ΔH－TΔS at constant T, reversible, dAT＝δWr＝－pdV＋δW r ' If δWr '＝0，then dAT＝－pdV closed system, change p ,V at constant T, W′＝0 reversible process

29. For an perfect gas G＝A＋pV, dG＝dA＋pdV＋Vdp at constant T, reversible, δW′r＝0, dA＝－pdV dGT＝Vdp closed system, change p ,V at constant T, W′＝0 reversible process

30. (2) phase transition (i) reversible phase transition G =H - (TS) A = U - (TS) ΔG =ΔH-Δ(TS) = ΔH-TΔS ΔA = ΔU -Δ(TS) = ΔU - TΔS reversible phase transition at constant T and p ΔH =TΔS ΔG= ΔH-TΔS = 0 reversible vaporization or sublimation at constant T and p, and vapor is an perfect gas ΔU = ΔH-Δ(pV) = ΔH- pΔV = ΔH- nRT ΔA = ΔU -Δ(TS) = ΔH - nRT - TΔS = - nRT (ii) irreversible phase transition

31. H U pV TS A pV TS G 3.3.4 Funtdantal relations of thermodynamic functions U, H, S, A, G, p, V and T H = U + pV A = U - TS G = H - TS= U + pV -TS = A +pV

32. For a reversible change dU=δQr+δWr， closed system of constant composition，reversible process，volume work only. 1. Master equation of thermodynamic If δWr ′＝0，then δWr＝－pdV， δQr＝TdS dU = TdS pdV H =U+pV dH = TdS + Vdp A =U-TS dA = SdTpdV G =H-TS dG = SdT + Vdp Master equation of thermodynamic

33. dZ = M dx + N dy dU = TdS－ pdV dH = TdS + Vdp dA = －SdT－ pdV dG = －SdT + Vdp

34. 2. Maxwell’s relation dZ = M dx + N dy dU = TdS－ pdV dH = TdS + Vdp dA = －SdT－ pdV dG = －SdT + Vdp

35. 3. Gibbs - helmholtz equation

36. 3.4 Chemical potential 3.4.1 Chemical potential Consider a system that consists of a single homogeneous phase whose composition can be varied G＝f (T，p，nA，nB……) Total differential

38. Chemical potential is an intensive state function. Bof pure substance G*(T，p，nB) = nBG*m,B(T，p，)

39. Heterogeneous system

40. dnB B() B() T，p Condition of phase equilibrium   spontaneous 3.4.2 Equilibrium criterion of substance (1) Condition of phase equilibrium

41. Chemical reaction (2) Condition of chemical reaction equilibrium Homogeneous system dξ ΣBμB＜0，dξ＞0 spontaneous ΣBμB ＝0， equilibrium For example aA＋bB＝yY＋zZ aμA+bμB＝yμY+zμZ

42. A—potential of chemical reaction aA＋bB＝yY＋zZ A= (aμA+bμB) -( yμY+zμZ ) A＝0 at equilibrium； A＞0 right spontaneous ； A＜0 left spontaneous 。