 Solution

For a classical ideal monatomic gas, $C_{V,\mathrm{He}}=3N{k}_B/2$ and $C_{p,\mathrm{He}}=5N{k}_B/2$. Treat copper as having one conduction electron per atom. Its degenerate-electron contribution isConsequently the electron-only estimates at5K areThe constant-pressure gas versus electronic metal heat capacity convention makes the second numerator appropriate to a constant-pressure gas measurement. For the solid, the low-temperature difference between $C_p$ and $C_V$ is usually small, but its phonon contribution is not fixed by $E_F$. In a Debye model,The supplied $E_F$ alone therefore does not determine the ratio to copper's total heat capacity; the electronic-only result is conditional.^[ref] An estimate using additional material data is possible: the specific-heat fit in Material Properties: OFHC Copper by National Institute of Standards and Technology gives $c_{p,\mathrm{Cu}}(5\mathrm{K})\simeq 0.155\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$. Using copper's rounded molar mass $0.0635\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$ gives $C_{p,\mathrm{Cu}}\simeq 9.85\times 10^{-3}\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, and henceThis supplementary estimate includes copper's lattice contribution and retains the ideal-gas approximation for helium. The NIST fit reports a 10% deviation from its source data below 15 K; neither that fit nor the stated pressure alone supplies helium nonideality corrections.^[ref]