(i) Given the following density of states for a particle in 3 dimensions

$$g(\varepsilon)=K V \varepsilon^{1 / 2}$$

write down the partition function for a gas of $N$ such non-interacting particles, assuming they can be treated classically. From this expression, calculate the energy $E$ of the system and the heat capacities $C_{V}$ and $C_{P}$. You may take it as given that $P V=\frac{2}{3} E$.

[Hint: The formula $\int_{0}^{\infty} d y y^{2} e^{-y^{2}}=\sqrt{\pi} / 4$ may be useful.]

(ii) Using thermodynamic relations obtain the relation between heat capacities and compressibilities

$$\frac{C_{P}}{C_{V}}=\frac{\kappa_{T}}{\kappa_{S}}$$

where the isothermal and adiabatic compressibilities are given by

$$\kappa=-\frac{1}{V} \frac{\partial V}{\partial P},$$

derivatives taken at constant temperature and entropy, respectively.

(iii) Find $\kappa_{T}$ and $\kappa_{S}$ for the ideal gas considered above.