A gas in equilibrium at temperature $T$ and pressure $P$ has quantum stationary states $i$ with energies $E_{i}(V)$ in volume $V$. What does it mean to say that a change in volume from $V$ to $V+d V$ is reversible?

Write down an expression for the probability that the gas is in state $i$. How is the entropy $S$ defined in terms of these probabilities? Write down an expression for the energy $E$ of the gas, and establish the relation

$$d E=T d S-P d V$$

for reversible changes.

By considering the quantity $F=E-T S$, derive the Maxwell relation

$$\left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial P}{\partial T}\right)_{V}$$

A gas obeys the equation of state

$$P V=R T+\frac{B(T)}{V}$$

where $R$ is a constant and $B(T)$ is a function of $T$ only. The gas is expanded isothermally, at temperature $T$, from volume $V_{0}$ to volume $2 V_{0}$. Find the work $\Delta W$ done on the gas. Show that the heat $\Delta Q$ absorbed by the gas is given by

$$\Delta Q=R T \log 2+\frac{T}{2 V_{0}} \frac{d B}{d T}$$