Constant-pressure gas versus electronic metal heat capacity ($C_{p,\mathrm{gas}}/C_{e,\mathrm{metal}}=5E_F/({\pi }^2{k}_{B}T)$)

For equal atom counts, a classical ideal monatomic gas has $C_p=5N{k}_B/2$. A monovalent three-dimensional free-electron metal has electronic contribution $C_e=({\pi }^2/2)N{k}_B{k}_{B}T/E_F$. Their ratio is the displayed expression. Replacing the gas's constant-pressure heat capacity with its $C_V=3N{k}_B/2$ changes the ratio by3/5. Neither ratio includes the metal's lattice heat capacity or gas nonideality.