(a) Consider an ideal gas consisting of $N$ identical classical particles of mass $m$ moving freely in a volume $V$ with Hamiltonian $H=|\mathbf{p}|^{2} / 2 m$. Show that the partition function of the gas has the form

$$Z_{\text {ideal }}=\frac{V^{N}}{\lambda^{3 N} N !}$$

and find $\lambda$ as a function of the temperature $T$.

[You may assume that $\int_{-\infty}^{\infty} e^{-a x^{2}} d x=\sqrt{\pi / a}$ for $a>0$.]

(b) A monatomic gas of interacting particles is a modification of an ideal gas where any pair of particles with separation $r$ interact through a potential energy $U(r)$. The partition function for this gas can be written as

$$Z=Z_{\text {ideal }}\left[1+\frac{2 \pi N}{V} \int_{0}^{\infty} f(r) r^{2} d r\right]^{N}$$

where $f(r)=e^{-\beta U(r)}-1, \quad \beta=1 /\left(k_{B} T\right)$. The virial expansion of the equation of state for small densities $N / V$ is

$$\frac{p}{k_{B} T}=\frac{N}{V}+B_{2}(T) \frac{N^{2}}{V^{2}}+\mathcal{O}\left(\frac{N^{3}}{V^{3}}\right)$$

Using the free energy, show that

$$B_{2}(T)=-2 \pi \int_{0}^{\infty} f(r) r^{2} d r$$

(c) The Lennard-Jones potential is

$$U(r)=\epsilon\left(\frac{r_{0}^{12}}{r^{12}}-2 \frac{r_{0}^{6}}{r^{6}}\right)$$

where $\epsilon$ and $r_{0}$ are positive constants. Find the separation $\sigma$ where $U(\sigma)=0$ and the separation $r_{\min }$ where $U(r)$ has its minimum. Sketch the graph of $U(r)$. Calculate $B_{2}(T)$ for this potential using the approximations

$$f(r)=e^{-\beta U(r)}-1 \simeq \begin{cases}-1 & \text { for } \quad r<\sigma \\ -\beta U(r) & \text { for } r \geqslant \sigma\end{cases}$$