= Solution
<Heat capacity> at fixed volume is $C_V=(\partial E/\partial T)_{V,N}=T(\partial S/\partial T)_{V,N}$. From the <Helmholtz free energy> differential $dF=-S\,dT-P\,dV$ at fixed particle number, equality of mixed derivatives gives $(\partial S/\partial V)_T=(\partial P/\partial T)_V$. Differentiating once more gives
$$
\boxed{\left(\frac{\partial C_V}{\partial V}\right)_T=T\left(\frac{\partial^2P}{\partial T^2}\right)_V.}
$$
For the <canonical partition function>, $F=-k_BT\log Z$, $E=k_BT^2\partial_T\log Z$, and $S=k_B(\log Z+T\partial_T\log Z)$. Thus, for $V>aN$,
$$
\boxed{P=\frac{Nk_BT}{V-aN},\quad E=\frac32Nk_BT,\quad C_V=\frac32Nk_B,}
$$
and
$$
\boxed{S=k_B\left[-\log N!+N\log(V-aN)+\frac{3N}{2}\log(bT)+\frac{3N}{2}\right].}
$$
No Stirling approximation is needed. The <heat capacity> is independent of volume, while the <pressure> is linear in <temperature>, so both sides of the Maxwell-derived identity vanish.
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