Answers to exam 170109

Note that some of the solutions are too short for being OK on an exam. Problems 7 and 8 are for the 1.5hp course Topics in thermodynamics.

1. (a) Microstate = a state where all properties of all particles are listed.
   Macrostate = a state specified only by the total properties of the system.
   Multiplicity of a macrostate = the number of microstates that belong to that macrostate.
   (b) Multiplicity tends to increase.
   (c) dU = TdS -pdV + μdN gives p/T=(∂S/∂V)_UN. Write S=N k_B[ln V + const] → p/T= (∂S/∂V)_UN = N k_B/V gives the ideal gas law.
   (d) dU = TdS -pdV + μdN gives 1/T=(∂S/∂U)_VN. With S=N k_B[(3/2)ln U + const] → p/T= (∂S/∂U)_VN = N k_B/V gives U=(3/2) N k_b T.
   
2. (a) T_f = 600K, work performed by the gas = -W = 1400J.
   (b) T_f=227K, work performed by the gas = -W = 852J.
   
3. (a) Q_r=Q_c+Q_f,
   (b) COP = (T_c/T_f)(T_f-T_r)/(T_r-T_c) or any other equivalent expression.
4. (The values below were before incorrectly given with the unit kJ/K - Peter 170223.)
   (a) 36.6 J/K
   (b) 11 J/K
   (c) -44 J/K
   (d) Net change in the entropy: 3.5 J/K. This is as expected since the entropy of a closed system should increase.
5. (a) T₁ = T₂(V₂/V₁)^γ-1
   Q₁₂ = 0, W₁₂ = (f/2) Nk(T₂-T₁),
   Q₂₃ = -W₂₃ = NkT₂ ln(V₁/V₂),
   Q₃₁ = (f/2) Nk(T₁-T₂),
   (b) e = 1-Q_c/Q_h, where Q_h=Q₂₃, Q_c=-Q₃₁,
   We have e=1-(T₂-T₁)/[T₂ ln(V₁/V₂)^(γ-1) = 1-(T₁/T₂)[(V₁/V₂)^(γ-1)-1]/ln(V₁/V₂)^(γ-1).
   With x=(V₁/V₂)^(γ-1)-1 this becomes e=1-(T₁/T₂) x/ln(1+x)≈1-(T₁/T₂) x/(x-x²/2) < 1-T₁/T₂.
6. (a) Δ H=-890kJ, Δ G=-818kJ.
   (b) Optimal: W_other=Δ G. This is work done on the system. The electric work is work done by the system: W_el=-W_other.
   (c) From formulas for G and H: Q = TΔS = Δ H - Δ G = -72 kJ.
   Since positive Q is heat entering the system, this is heat leaving the system=waste heat.
7. (a) T₃<T₂<T₁.
   (b) The Gibbs free energy has its minima away from x=0 and x=1 since the entropy gives sizable contributes at these limits because ∂S/∂x diverges at x=0 and x=1.
   (c) At T₁: x':liquid, x'':liquid, x''':liquid+β. At T₂: x':liquid, x'':liquid+β, x''':liquid+β
   
8. (a) μ_B = f(T,P) + kT ln(N_B/N_A).
   (b) T-T₀= N_BkT²/L,
   (c) T-T₀ = -N_B k T²/L = -n_b R T²/L,
   (d) Use the formula above with n_B=1.2 moles = number of Na ions plus Cl ions. This gives -2.2 K.

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