Non-relativistic electrons of mass $m$ are confined to move in a two-dimensional plane of area $A$. Each electron has two spin states. Compute the density of states $g(E)$ and show that it is constant.

Write down expressions for the number of particles $N$ and the average energy $\langle E\rangle$ of a gas of fermions in terms of the temperature $T$ and chemical potential $\mu$. Find an expression for the Fermi Energy $E_{F}$ in terms of $N$.

For $k_{B} T \ll E_{F}$, you may assume that the chemical potential does not change with temperature. Compute the low temperature heat capacity of a gas of fermions. [You may use the approximation that, for large $z$,

$$\left.\int_{0}^{\infty} \frac{x^{n} d x}{z^{-1} e^{x}+1} \approx \frac{1}{n+1}(\log z)^{n+1}+\frac{\pi^{2} n}{6}(\log z)^{n-1} .\right]$$