Conduction-electron count in the molar electronic heat capacity ($C_{V,m}\simeq ({\pi }^2/2)zRT/T_F$)

The low-temperature heat capacity of a three-dimensional free electron gas is $({\pi }^2/2)N_{e}kT/T_F$. Expressed per mole of metal atoms with $z$ model conduction electrons each, this becomes the displayed result. Expressed per mole of conduction electrons, set $z=1$. A thermal-window counting argument predicts the $T/T_F$ scaling but gives only an order-one coefficient; it must still retain the number of electrons when changing the molar basis.