= Thermal-shell estimate of electronic heat capacity
For a three-dimensional <free electron model>, the spin-summed <density of states> at the <Fermi energy> is $g(E_F)=3N/(2E_F)$. A shell estimate treats $N_{\rm act}\sim g(E_F)k_BT$ electrons as thermally active and assigns each an excitation energy of order $k_BT$. For the illustrative choice $3k_BT/2$, $\Delta U\sim(9/4)Nk_B^2T^2/E_F$ and differentiation gives $C_V\sim(9/2)Nk_B T/T_F$. The numerical coefficient is heuristic: the controlled <Sommerfeld expansion> instead gives
$$
C_V=\frac{\pi^2}{2}Nk_B\frac{T}{T_F}+O\!\left(Nk_B(T/T_F)^3\right).
$$
Both estimates express <Pauli exclusion principle> suppression of the classical <heat capacity> when $T\ll T_F$. A shell argument establishes scaling rather than a universal prefactor.
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