= Solution
{c}
For a nondegenerate two-dimensional <electron gas>, the two quadratic momentum terms supply $k_BT/2$ each by the <equipartition theorem>. There is no lattice restoring <potential energy> for a free <electron>. Thus
$$
U=N_ek_BT,\qquad
\boxed{\frac{C_v}{N_e}\longrightarrow k_B.}
$$
On cooling below the <Fermi temperature>, the <Pauli exclusion principle> restricts the thermally active <electrons> to within approximately $k_BT$ of the <Fermi energy>. The <linear low-temperature heat capacity of a two-dimensional Fermi gas> is
$$
\boxed{\frac{C_v}{N_e}\simeq\frac{\pi^2}{3}k_B\frac{k_BT}{E_F}\longrightarrow0.}
$$
The <heat capacity> therefore decreases from its classical value as <quantum degeneracy> becomes important. The <particle number> of <electrons> and sample area are held fixed when differentiating the <internal energy>.
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