Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 3 / 35D / iii / Solution

In the classical limit the fugacity is small and ${\mathrm{Li}}_s(z)=z+O(z^2)$. The Bose-Einstein distribution approaches the Maxwell-Boltzmann distribution, givingTaking the ratio eliminates both $z$ and $B$, yielding $\boxed{E\simeq \frac{9}{2}N{k}_{B}T}$. The coefficient $9/2$ follows from this density of states, rather than the $3/2$ of a free three-dimensional gas.