= Solution
For a classical ideal monatomic gas, $C_{V,\rm He}=3Nk_B/2$ and $C_{p,\rm He}=5Nk_B/2$. Treat copper as having one conduction electron per atom. Its degenerate-electron contribution is
$$
C_{e,\rm Cu}=\frac{\pi^2}{2}Nk_B\frac{k_BT}{E_F}.
$$
Consequently the \b[electron-only] estimates at5K are
$$
\boxed{\frac{C_{V,\rm He}}{C_{e,\rm Cu}}=4.94\times10^3,\qquad
\frac{C_{p,\rm He}}{C_{e,\rm Cu}}=8.23\times10^3.}
$$
The <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>,
$$
\frac{C_{p,\rm He}}{C_{V,\rm Cu}}\simeq
\frac{5/2}{(\pi^2/2)k_BT/E_F+(12\pi^4/5)(T/\Theta_D)^3}.
$$
The supplied $E_F$ alone therefore does not determine the ratio to copper's total <heat capacity>; the electronic-only result is conditional.<Statistical Physics by David Tong>[Chapter3, §3.3.1, Debye lattice heat capacity; printedpp.96–97, distinct electronic-linear and phonon-cubic contributions in copper]{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 $c_{p,\rm Cu}(5\,\mathrm K)\simeq0.155\,\mathrm{J\,kg^{-1}K^{-1}}$. Using copper's rounded <molar mass> $0.0635\,\mathrm{kg\,mol^{-1}}$ gives $C_{p,\rm Cu}\simeq9.85\times10^{-3}\,\mathrm{J\,mol^{-1}K^{-1}}$, and hence
$$
\boxed{\frac{C_{p,\rm He}}{C_{p,\rm Cu}}\simeq\frac{5R/2}{9.85\times10^{-3}\,\mathrm{J\,mol^{-1}K^{-1}}}\simeq2.1\times10^3.}
$$
This 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.<Material Properties: OFHC Copper by National Institute of Standards and Technology>[Specific heat: 4–300 K fit, coefficient column a–i, base-ten logarithmic polynomial; stated 10% fit deviation for temperatures below 15 K]{ref}
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