Write down an expression for the partition function of a classical particle of mass $m$ moving in three dimensions in a potential $U(\mathbf{x})$ and in equilibrium with a heat bath at temperature $T$.

A system of $N$ non-interacting classical particles is placed in the potential

$$U(\mathbf{x})=\frac{\left(x^{2}+y^{2}+z^{2}\right)^{n}}{V^{2 n / 3}}$$

where $n$ is a positive integer. The gas is in equilibrium at temperature $T$. Using a suitable rescaling of variables, show that the free energy $F$ is given by

$$\frac{F}{N}=-k T\left(\log V+\frac{3}{2} \frac{n+1}{n} \log k T+\log I_{n}\right)$$

where

$$I_{n}=\left(\frac{2 m \pi}{h^{2}}\right)^{3 / 2} \int_{0}^{\infty} 4 \pi u^{2} e^{-u^{2 n}} d u$$

Regarding $V$ as an external parameter, find the thermodynamic force $P$, conjugate to $V$, exerted by this system. Find the equation of state and compare with that of an ideal gas confined in a volume $V$.

Derive expressions for the entropy $S$, the internal energy $E$ and the total heat capacity $C_{V}$ at constant $V$.

Show that for all $n$ the total heat capacity at constant $P$ is given by

$$C_{P}=C_{V}+N k$$

[Note that $\left.\int_{0}^{\infty} u^{2} e^{-u^{2} / 2} d u=\sqrt{\frac{\pi}{2}} .\right]$