(i) Define the Gibbs free energy for a gas of $N$ particles with pressure $p$ at a temperature $T$. Explain why it is necessarily proportional to the number of particles $N$ in the system. Given volume $V$ and chemical potential $\mu$, prove that

$$\left.\frac{\partial \mu}{\partial p}\right|_{T}=\frac{V}{N} .$$

(ii) The van der Waals equation of state is

$$\left(p+\frac{a N^{2}}{V^{2}}\right)(V-N b)=N k_{B} T$$

Explain the physical significance of the terms with constants $a$ and $b$. Sketch the isotherms of the van der Waals equation. Show that the critical point lies at

$$k_{B} T_{c}=\frac{8 a}{27 b}, \quad V_{c}=3 b N, \quad p_{c}=\frac{a}{27 b^{2}} .$$

(iii) Describe the Maxwell construction to determine the condition for phase equilibrium. Hence sketch the regions of the van der Waals isotherm at $T<T_{c}$ that correspond to metastable and unstable states. Sketch those regions that correspond to stable liquids and stable gases.

(iv) Show that, as the critical point is approached along the co-existence curve,

$$V_{\text {gas }}-V_{\text {liquid }} \sim\left(T_{c}-T\right)^{1 / 2} .$$

Show that, as the critical point is approached along an isotherm,

$$p-p_{c} \sim\left(V-V_{c}\right)^{3} .$$