Question 8.14: An isolated system consists of two rigid subsystems of volum...

a) Applying Gibbs’ relation (2.12) to the helium in each subsystem yields,

\dot{U} = T \dot{S} – p \dot{V} + \sum\limits_{A=1}^{r}{\mu _A \dot{N}_A }            (2.12)

\dot{U}_1 = T \dot{S}_1 + μ_1 \dot{N}_1       and    \dot{U}_2 = T \dot{S}_2 + μ_2 \dot{N}_2 .

which implies that,

\dot{S} = \dot{S}_1 + \dot{S}_2 = \frac{1}{T} (\dot{U}_1 +\dot{U}_2) – \frac{μ_1}{T}\dot{N}_1 – \frac{μ_2}{T}\dot{N}_2 .

Since the system is isolated \dot{U} = 0 , which implies that \dot{U}_1 = − \dot{U}_2 . The helium conservation law implies that \dot{N}_1 = −\dot{N}_2 . Thus,

\dot{S} = \frac{μ_1 − μ_2}{T} \dot{N}_1 .

and equivalently,

\frac{∂S}{∂N_1} = \frac{μ_1 − μ_2}{T} .

According to the second law, the total entropy S of the system reaches a maximum at equilibrium. Thus, at equilibrium,

\frac{∂S}{∂N_1} = 0 .    (equilibrium)

which implies that μ_1 ≡ μ_A (T, p_1) is equal to μ_2 ≡ μ-A (T, p_2, c_A) ,

μ_A (T, p_1) = μ_A (T, p_2, c_A)          (equilibrium)

b) Using the ideal mixture relation (8.68),

μ_A (T, p, c_A) = μ_A (T, p) + R T \ln (c_A)                 (8.68)

μ_A (T, p_2, c_A) = μ_A (T, p_2) + R T \ln (c_A) .

which implies that at chemical equilibrium,

μ_A (T, p_1) = μ_A (T, p_2) + RT \ln (c_A) .

Moreover, according to relation (8.58),

μ (T, p) = μ (T, p_0) + R T \ln \Bigl(\frac{p}{p_0}\Bigr) .       (8.58)

μ_A (T, p_1) = μ_A (T, p_2) + R T \ln \Bigl(\frac{p_1}{p_2}\Bigr) .

Comparing the two previous relations and using the definition of the concentration c_A ,we obtain,

c_A = \frac{p_1}{p_2}       and      c_A = \frac{N_2}{N_2 + N_0} .

The ideal gas equation of state implies that,

p_1 = \frac{(N_0 − N_2) R T}{V_0}       and    p_2 = \frac{(N_0 + N_2) R T}{V_0} .

Using the last four equations, we obtain,

N_2=\frac{N_0}{2}.

Thus,

p_1 = \frac{N_0 R T}{2 V_0}        and    p_2 = \frac{3 N_0 R T}{2 V_0} .

Moreover, since,

p_i = \frac{N_0 R T}{ V_0} .

we have that,

p_1 = \frac{1}{2} p_i         and      p_2 = \frac{3}{2} p_i .