Consider an ideal Bose gas in an external potential such that the resulting density of single particle states is given by

$$g(\varepsilon)=B \varepsilon^{7 / 2}$$

where $B$ is a positive constant.

(i) Derive an expression for the critical temperature for Bose-Einstein condensation of a gas of $N$ of these atoms.

[Recall

$$\left.\frac{1}{\Gamma(n)} \int_{0}^{\infty} \frac{x^{n-1} \mathrm{~d} x}{z^{-1} e^{x}-1}=\sum_{\ell=1}^{\infty} \frac{z^{\ell}}{\ell^{n}}\right]$$

(ii) What is the internal energy $E$ of the gas in the condensed state as a function of $N$ and $T$ ?

(iii) Now consider the high temperature, classical limit instead. How does the internal energy $E$ depend on $N$ and $T$ ?