A gas container of fixed volume V is divided into two compartments by an impermeable fixed wall. One compartment contains ideal gas 1, the other ideal gas 2. Both sides are at pressure p and temperature T. When the wall is removed, the system reaches equilibrium. During this process, going from an

Question

A gas container of fixed volume V is divided into two compartments by an impermeable fixed wall. One compartment contains ideal gas 1, the other ideal gas 2. Both sides are at pressure p and temperature T. When the wall is removed, the system reaches equilibrium.
During this process, going from an initial state i to a final state f, the system is held at constant temperature T. There is no chemical interaction between the two gases. Therefore, the mixture is an ideal gas also.

a) Determine the internal energy variation [latex] ΔU_{if}[/latex] during this process.
b) Show that the total entropy variation [latex] ΔS_{if}[/latex] is given by (Fig. 8.1),

[latex] \Delta S_{if} = -(N_1 + N_2) R \sum\limits_{A=1}^{2}{c_A} \ln (c_A) [/latex].

Accepted Answer

a) The internal energy remains constant during this process, because the temperature of the two ideal gases is constant and the internal energy of an ideal gas is proportional to its temperature. Note that the mixture of two non-interacting ideal gases is also an ideal gas. Thus,

[latex] ΔU_{if} = U_f − U_i = 0 [/latex]

c) According to the Euler equation (4.7), we express the initial internal energy [latex] U_i (S_1, S_2,N_1,N_2) [/latex] and the final internal energy [latex] U_f (S,N_1,N_2) [/latex] in terms of their state variables,

[latex] U = T S − p V + \sum\limits_{A=1}^{r}{μ_A N_A} [/latex] (4.7)

[latex] U_i (S_1, S_2,N_1,N_2) = T (S_1 + S_2) + \sum\limits_{A=1}^{2}{μ_A (T, p) N_A} [/latex]

[latex] U_f (S,N_1,N_2) = T S + \sum\limits_{A=1}^{2}{μ_A (T, p, c_A)N_A} [/latex]

Since the internal energy remains constant,

[latex] ΔU_{if} = T (S − S_1 − S_2) + \sum\limits_{A=1}^{2} {\Bigl(μ_A (T, p, c_A) − μ_A (T, p)\Bigr) N_A} = 0 [/latex]

Using the expression for the entropy variation,

[latex] ΔS_{if} = S_f − S_i = S − S_1 − S_2 [/latex]

and the chemical potential of an ideal gas mixture (8.68),

[latex] μ_A (T, p, c_A) = μ_A (T, p) + R T \ln (c_A) [/latex] (8.68)

the previous relation can be recast as,

[latex] T ΔS_{if} + R T \sum\limits_{A=1}^{2}{\ln (c_A) N_A} = 0 [/latex]

Using expression (8.35) for the concentration [latex] c_A = N_A/ (N_1 + N_2) [/latex], the entropy of mixing [latex] ΔS_{if} [/latex] is found to be,

[latex] c_A = \frac{N_A}{\sum\limits_{B=1}^{r}{N_B}} [/latex] (8.35)

[latex] ΔS_{if} = − (N_1 + N_2) R \sum\limits_{A=1}^{2}{c_A \ln (c_A)} [/latex]

where [latex] ΔS_{if} ≥ 0 [/latex] since 0 ≤ [latex] c_A ≤ 1 [/latex]. This expression for the entropy variation is analogous to the von Neuman entropy.

Question 8.9: A gas container of fixed volume V is divided into two compar...

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