= Solution
Assume <sodium> supplies one conduction <Electron> per atom. The supplied <Fermi energy> gives
$$
T_F=\frac{3.24(1.602176634\times10^{-19})}{1.380649\times10^{-23}}
=3.760\times10^4\,\mathrm K,\qquad T/T_F=7.98\times10^{-3}.
$$
The gas is strongly degenerate at room <temperature>. For one <mole>, $N=N_A$ and $N_Ak_B=R=8.31446\,\mathrm{J\,mol^{-1}\,K^{-1}}$. Using the requested estimate,
$$
\boxed{C_{V,\mathrm m}^{\rm el}\simeq\frac92R\frac{300}{T_F}=0.299\,\mathrm{J\,mol^{-1}\,K^{-1}}}.
$$
The supplied <mass density> determines the answer per unit <volume>: $N/V=(970/0.023)N_A=2.54\times10^{28}\,\mathrm{m^{-3}}$. Therefore
$$
\boxed{c_V^{\rm el}\simeq1.26\times10^4\,\mathrm{J\,m^{-3}\,K^{-1}}},\qquad
\boxed{c_{V,\rm mass}^{\rm el}\simeq13.0\,\mathrm{J\,kg^{-1}\,K^{-1}}}.
$$
The density is not needed for the <molar heat capacity> once $E_F$ is supplied. Using the precise <Sommerfeld expansion> coefficient instead gives $0.327\,\mathrm{J\,mol^{-1}\,K^{-1}}$.
Either electronic estimate is only about $1\%$ of the stated measured total. Most of the total <heat capacity> comes from lattice <phonons>, whose high-temperature <Dulong-Petit law> limit is $3R=24.94\,\mathrm{J\,mol^{-1}\,K^{-1}}$. The remaining difference from $28\,\mathrm{J\,mol^{-1}\,K^{-1}}$ need not be electronic: anharmonic lattice effects and the difference between <heat capacity at constant pressure> and <heat capacity at constant volume> also contribute. The paper does not specify the measurement constraint, so the two totals should not be equated exactly.
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