= Constant-pressure gas versus electronic metal heat capacity
{title2=$C_{p,\rm gas}/C_{e,\rm metal}=5E_F/(\pi^2k_BT)$}
For equal atom counts, a classical ideal monatomic gas has $C_p=5Nk_B/2$. A monovalent three-dimensional free-electron metal has electronic contribution $C_e=(\pi^2/2)Nk_Bk_BT/E_F$. Their ratio is the displayed expression. Replacing the gas's constant-pressure heat capacity with its $C_V=3Nk_B/2$ changes the ratio by3/5. Neither ratio includes the metal's lattice heat capacity or gas nonideality.
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