Define the notions of entropy $S$ and thermodynamic temperature $T$ for a gas of particles in a variable volume $V$. Derive the fundamental relation

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

The free energy of the gas is defined as $F=E-T S$. Why is it convenient to regard $F$ as a function of $T$ and $V$ ? By considering $F$, or otherwise, show that

$$\left.\frac{\partial S}{\partial V}\right|_{T}=\left.\frac{\partial P}{\partial T}\right|_{V}$$

Deduce that the entropy of an ideal gas, whose equation of state is $P V=N T$ (using energy units), has the form

$$S=N \log \left(\frac{V}{N}\right)+N c(T)$$

where $c(T)$ is independent of $N$ and $V$.

Show that if the gas is in contact with a heat bath at temperature $T$, then the probability of finding the gas in a particular quantum microstate of energy $E_{r}$ is

$$P_{r}=e^{\left(F-E_{r}\right) / T}$$