The Dieterici equation of state of a gas is

$$P=\frac{k_{B} T}{v-b} \exp \left(-\frac{a}{k_{B} T v}\right)$$

where $P$ is the pressure, $v=V / N$ is the volume divided by the number of particles, $T$ is the temperature, and $k_{B}$ is the Boltzmann constant. Provide a physical interpretation for the constants $a$ and $b$.

Briefly explain how the Dieterici equation captures the liquid-gas phase transition. What is the maximum temperature at which such a phase transition can occur?

The Gibbs free energy is given by

$$G=E+P V-T S$$

where $E$ is the energy and $S$ is the entropy. Explain why the Gibbs free energy is proportional to the number of particles in the system.

On either side of a first-order phase transition the Gibbs free energies are equal. Use this fact to derive the Clausius-Clapeyron equation for a line along which there is a first-order liquid-gas phase transition,

$$\frac{d P}{d T}=\frac{L}{T\left(V_{\text {gas }}-V_{\text {liquid }}\right)}$$

where $L$ is the latent heat which you should define.

Assume that the volume of liquid is negligible compared to the volume of gas and that the latent heat is constant. Further assume that the gas can be well approximated by the ideal gas law. Solve $(*)$ to obtain an equation for the phase-transition line in the $(P, T)$ plane.