Discussion
An approximate partition function \(Q(N, V, T)\) of a gas is given below.
\[ \begin{aligned} Q(N,V,T) = &\frac{1}{N!} \left( \frac{2\pi m k_B T}{h^2} \right)^{3N/2} \\ &\times (V - Nb)^{N} \end{aligned} \]
The equation of state(s) for this gas is/are
[Note: b is a parameter independent of volume.]
Answer: A
The equation of state is \(P(V - Nb) = Nk_BT\).
Pressure follows from the canonical partition function by
\[ P = k_B T \left( \frac{\partial \ln Q}{\partial V} \right) \]
at constant N and T.
Only the factor \((V - Nb)^N\) depends on V, so
\[ \begin{aligned} \ln Q &= N \ln(V - Nb) + \ldots \\ P &= \frac{N k_B T}{V - Nb} \end{aligned} \]
This is the van der Waals equation without the attraction term: b is the excluded volume per molecule. \(PV = Nk_BT\) ignores that excluded volume, \(P(V - Nb) = Nk_B\) has lost the temperature and is dimensionally wrong, and \(PV^{(N-b)} = k_BT\) is not what differentiation gives. Of the four, only one equation follows, so this multiple-select question has a single correct choice.
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