A simple model of conduction electrons treats them as nearly free fermions moving through a volume $V$. The Pauli exclusion principle fills one-particle momentum states up to a characteristic energy even at zero temperature.

For electrons with dispersion $$E=\frac{\hbar^2k^2}{2m},$$ each wavevector state can hold two electrons with opposite spin. At $T=0$, all states with $k\le k_F$ are occupied and those above are empty. Counting states in $k$-space gives $$k_F=(3\pi^2n)^{1/3},$$ where $n=N/V$ is electron number density. The corresponding Fermi energy is $$E_F=\frac{\hbar^2k_F^2}{2m}.$$

The Fermi temperature $$T_F=\frac{E_F}{k_B}$$ is typically tens of thousands of kelvin in metals, so ordinary room temperature is small compared with $T_F$. Only electrons within an energy range of order $k_BT$ around $E_F$ can change occupancy significantly.

This explains why the electronic heat capacity of a metal is much smaller than a classical equipartition estimate: most electrons are blocked by filled neighboring states and cannot absorb small thermal energies.

The free-electron model neglects the crystal's periodic potential and electron interactions, yet it introduces the Fermi surface, degeneracy and quantum origin of many metallic properties. Band theory refines the dispersion while retaining the same fermionic state filling.