Consider a classical gas of $N$ particles in volume $V$, where the total energy is the standard kinetic energy plus a potential $U\left(\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{N}\right)$ depending on the relative locations of the particles $\left\{\mathbf{x}_{i}: 1 \leqslant i \leqslant N\right\}$.

(i) Starting from the partition function, show that the free energy of the gas is

$$F=F_{\text {ideal }}-T \log \left\{1+\frac{1}{V^{N}} \int\left(e^{-U / T}-1\right) d^{3 N} x\right\}$$

where $F_{\text {ideal }}$ is the free energy when $U \equiv 0$.

(ii) Suppose now that the gas is fairly dilute and that the integral in $(*)$ is small compared to $V^{N}$ and is dominated by two-particle interactions. Show that the free energy simplifies to the form

$$F=F_{\text {ideal }}+\frac{N^{2} T}{V} B(T)$$

and find an integral expression for $B(T)$. Using ( $\dagger$ ) find the equation of state of the gas, and verify that $B(T)$ is the second virial coefficient.

(iii) The equation of state for a Clausius gas is

$$P(V-N b)=N T$$

for some constant $b$. Find the second virial coefficient for this gas. Evaluate $b$ for a gas of hard sphere atoms of radius $r_{0}$.