= Solution
With periodic <boundary conditions>, each $k$ state occupies volume $(2\pi)^3/V$ and has two spin states. Filling a sphere gives $n=k_F^3/(3\pi^2)$ and
$$
\boxed{E_F=\frac{\hbar^2}{2m_e}(3\pi^2n)^{2/3}},\qquad
D(E)=\frac1{2\pi^2}\left(\frac{2m_e}{\hbar^2}\right)^{3/2}\sqrt E,
\quad \boxed{D(E_F)=\frac{3n}{2E_F}}.
$$
Here $D$ is <density of states> per volume and per <energy>; the extensive $g(E)$ is $VD(E)$. Both spin states are already included.
For divalent magnesium, $n=8.6\times10^{28}\ \mathrm{m^{-3}}$, giving
$$
\boxed{E_F=7.10\ \mathrm{eV},\qquad
D(E_F)=1.13\times10^{47}\ \mathrm{J^{-1}m^{-3}}
=1.82\times10^{28}\ \mathrm{eV^{-1}m^{-3}}}.
$$
At $T\ll T_F$, only <electrons> within about $k_BT$ of the <Fermi surface> change occupancy. The Sommerfeld <heat capacity> is
$$
\boxed{\frac{C_e}{V}=\frac{\pi^2}{3}k_B^2D(E_F)T
=71.1\,T\ \mathrm{J\,m^{-3}\,K^{-1}}}.
$$
Equivalently $C_{e,m}=0.995\,T\ \mathrm{mJ\,mol^{-1}\,K^{-1}}$, with $T$ in kelvin, so at room <temperature> it is about $0.299\ \mathrm{J\,mol^{-1}\,K^{-1}}$. A <temperature> was not specified for the estimate; the linear coefficient is the general result.
At sufficiently low <temperature>, an ordinary nonsuperconducting metal has $C\simeq\gamma T+\beta T^3$: <electrons> dominate at the lowest <temperatures>, acoustic <phonons> produce the cubic contribution, and the lattice approaches $3R$ per mole as <temperature> exceeds its characteristic Debye scale. Anharmonic expansion can distinguish measured $C_p$ from $C_V$. In the room-temperature regime, $T/T_F\simeq0.0036$ since $T_F\simeq8.24\times10^4\ \mathrm K$, so <electron> excitation is strongly Pauli-limited. The free-electron estimate is only about $1.2\%$ of the classical $3R$ lattice scale. Calling it negligible means small relative to the lattice contribution, not identically zero or negligible as $T\to0$. Real band structure and interactions can alter the coefficient.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2012/paper-2-metal-heat-capacity.png]
{title=Original illustrative metal heat-capacity regimes: Sommerfeld linear electrons and Debye phonons, with explicit dimensionless model parameters}
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