What is meant by the heat capacity $C_{V}$ of a thermodynamic system? By establishing a suitable Maxwell identity, show that

$$\left.\frac{\partial C_{V}}{\partial V}\right|_{T}=\left.T \frac{\partial^{2} P}{\partial T^{2}}\right|_{V} .$$

In a certain model of $N$ interacting particles in a volume $V$ and at temperature $T$, the partition function is

$$Z=\frac{1}{N !}(V-a N)^{N}(b T)^{3 N / 2}$$

where $a$ and $b$ are constants. Find the equation of state and the entropy for this gas of particles. Find the energy and hence the heat capacity $C_{V}$ of the gas, and verify that the relation $(*)$ is satisfied.