热力学第二定律习题(5)

(a) (b) 联立求解得 n乙醇 = 167885 mol， n 水 = 17864 mol 配成 56% 的酒，总重量为 13791 kg，其中水重 6068 kg，应加水重 5746kg，其体积是 5751 dm (2) 56% 的酒总体积 V = ∑ nBV'B,m = 15270 dm

24. [答] P = (RT/V) -a/V

V2

3

3

W=∫PdV =∫V12[(RT/V)-a/V]dV = RT ln(V2/V1)+ a(1/V2-1/V1) (?U/?V)T = T(?P/?T)V- P = a/V

UV2

2

2

∫U12dU=∫V12(a/V)dV △U = a(1/V1- 1/V2) Q R = △U + W = RTLn(V2/V1) △H = △U + △PV = △U + (P2V2- P1V1) = △U + [(RT - a/V2) - (RT - a/V1)] = a[(1/V1)-(1/V2)] + a[(1/V1)-(1/V2)]

= 2a[(1/V1)-(1/V2)] △S = QR/T = RLn(V2/V1) △G = △H - T△S = 2a[(1/V1- 1/V2)] - RTLn(V2/V1)

五. 证明题：

1. [答] 终态温度为 T = (T1+ T2)/2 △S1= mCpLn[(T1+ T2)/2T1]，△S2= mCpLn[(T1+ T2)/2T2] △S = △S1+ △S2= 2mCpLn(T1+ T2)/2(T1T2)? 当 T1≠ T2，(T1-T2)> 0 则 T1+ T2> 2(T1T2)? ∴ △S > 0

2. [答] 设 S = f(T,V)

ds = (?S/?T )V dT + (?S/?V )T dV ...... (1) dU = Tds - pdv ∴ (?U/?T)V = CV = T(?S/?T )V 又 (?S/?T)V = CV/T .............(2)

(?S/?V )T = (?P/?T )V ......(3) 将 (2) (3) 代入 (1) 得： dS = nCV,m(dT/T) + (?P/?T)VdV

= nCV,m(dT/T) + [nR/(V-b)]dV △S = nCV,m Ln(T2/T1) + nRLn[(V2- b)/(V1- b)]

3. [答] 令 S = f(P,V)

dS = (?S/?P)VdP + (?S/?V)pdV = (?S/?T)V(?T/?P)VdP + (?S/?T)p(?T/?V)pdV

41

2

= (CV/T)( ?T/?P)VdP + (Cp/T)( ?T/?V)pdV 对于理气 (?T/?P)V = T/P， (?T/?V)p = T/V

dS = CV(dp/P) + Cp(dv/V) △S = CV Ln(P2/P1) + Cp Ln(V2/V1)

4. [答] △S =∫δQ R /T =∫(dU+pdV)/T =∫CVdT/T +∫{[RT/(v-?)]/T}dV = ∫T12[(a/T) + b + cT]dT + ∫V12[R/(v-?)]dV

2 = aLn(T2/T1) + b(T2-T=1) + (c/2)(T2- T21) + RLn(V2-?)/(V1-?)

TV

5. [答] (?U/?V)T = T(?P/?T)V - P = f(v)T - P = 0

6. [答] (?H/?G)T = [(?H/?p)T]/( ?G/?p)T = [-T(?V/?T)p+ V]/V = (1/V)[-T(?V/?T)p+ V] 7. [答] (?U/?V)T = T(?p/?T)V - p

由状态方程算出 (?p/?T)V ，代入上式得 (?U/?V)T = [RT/(Vm- B)](dB/dT)

(?S/?V)T = (?p/?T)V =[R/(Vm- B)] + [RT/(Vm- B)](dB/dT) (?S/?p)T = -(?V/?T)p = (-R/P) - (dB/dT)

(?H/?p)T = V - T(?V/?T)p = B - T(dB/dT)

8. [答] dS = (?S/?U)V(?U/?T)VdT + (?S/?V)TdV

= (CV/T) dT + (nR/V) dV [理想气体 (?S/?V)T = nR/V] △S = CV ln(T2/T1) + nR ln(V2/V1) = 0

T2/T1= (V1/V2)? 即 TV ? = 常数

-1

-1

2

2

2

9. [答] (?V/?T)F = -S/p , (?G/?T)p = -S , (?S/?V)u = p/T (?V/?T)F(?T/?G)p(?S/?V)u = 1/T = R/[p(Vm- b) + a/Vm- ab/Vm]

10. [答] Cp = (?H/?T)p = [?(G + TS)/ ?T]p = (?G/?T)p+ T(?S/?T)p+ S

= -S - T(?G/?T)p+ S = -T(? G/?T)p

11. [答] dS = (?S/?T)VdT + (?S/?V)TdV (?S/?T)V(?T/?V)s+ (?S/?V)T = 0 (?S/?T)v = Cv/T (?S/?V)T = (?p/?T)v 得 (?T/?V)s = (-T/Cv)( ?P/?T)v

理想气体 (?p/?T)v= p/T , (?T/?V)s= -p/Cv

12. [答] dU = CvdT + (?U/?V)TdV = CvdT + [T(?p/?T)v- p]dV

2

2

2

2

2

42

?Q = CvdT + [T(?p/?T)v- p]dV + pdV

范氏气体等温过程 ?Q = T(?p/T)v dV = {nRT/(V - nb)}dV Q = nRT ln(V2- nb)/(V1- nb)

13. [答] ∵ ?/? = -(?V/?p)T(?T/?V)p = (?T/?p)v ? = (1/V)( ?V/?T)p

? = -(1/V)( ?V/?P)T (?T/?V)p= (?T/?p)v ∴ (?S/?p)v = (?S/?T)v(?T/?p)v = (Cv/T)(?/?) = ?Cv/?T

14. [答] (?p/?V)s = -(?S/?V)p(?p/?S)v (?p/?V)T = -(?T/?V)p(?p/?T)v

(?p/?V)s/(?p/?V)T = (?S/?T)p(?T/?S)v 则 (?p/?V)s = (?p/?V)T(?S/?T)p(?T/?S)v

15. [答] ? J-T = (?T/?p)H = (-1/Cp)( ?H/?p)T (∵ dH = CpdT + (?H/?p)TdP)

? J-T = (-1/Cp)[V - T(?V/?T)p] = (1/Cp)[T(?V/?T)p- V]

16. [答] (?U/?V)T = T(?P/?T)v - p = 0 dU = (?U/?T)vdT + (?U/?V)TdV

= Cv dT = - pdV = -[RT/(Vm- B)]dV ∫T12(Cv/T)dT = -∫V12[R/(Vm- B)]dV Cv ln(T2/T1) = -R ln[(Vm,2- B)/(Vm,1- B)] T2(Vm,2- B) T(Vm - B)

17. [答] (?T/?p)H = [T(?V/?T)p- Vm]/Cp 而 (?V/?T)p = R/(P- a/Vm)

得 (?T/?p)H = [ RT/(p - a/Vm) - RT/(p + a/Vm]/Cp ∵ a/Vm > 0 ∴ (?T/?p)H > 0

18.[答] SB,m = (?S/?nB)T,v,nc + (?S/?V)T,n (?V/?n?)T,p,nc = (?S/? nB)T,v,nc + VB,m(?p/?T)v,n ∴ (?S/? nB)T,p,nc = SB,m - VB,m(?p/?T)v,B

2

2

2

2

R/Cv

TV = T1(Vm,1-B)

R/Cv

R/Cv

= 常数

43

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