(i) The first law of thermodynamics is $d E=T d S-p d V+\mu d N$, where $\mu$ is the chemical potential. Briefly describe its meaning.

(ii) What is equipartition of energy? Under which conditions is it valid? Write down the heat capacity $C_{V}$ at constant volume for a monatomic ideal gas.

(iii) Starting from the first law of thermodynamics, and using the fact that for an ideal gas $(\partial E / \partial V)_{T}=0$, show that the entropy of an ideal gas containing $N$ particles can be written as

$$S(T, V)=N\left(\int \frac{c_{V}(T)}{T} d T+k_{\mathrm{B}} \ln \frac{V}{N}+\mathrm{const}\right)$$

where $T$ and $V$ are temperature and volume of the gas, $k_{\mathrm{B}}$ is the Boltzmann constant, and we define the heat capacity per particle as $c_{V}=C_{V} / N$.

(iv) The Gibbs free energy $G$ is defined as $G=E+p V-T S$. Verify that it is a function of temperature $T$, pressure $p$ and particle number $N$. Explain why $G$ depends on the particle number $N$ through $G=\mu(T, p) N$.

(v) Calculate the chemical potential $\mu$ for an ideal gas with heat capacity per particle $c_{V}(T)$. Calculate $\mu$ for the special case of a monatomic gas.