electronic heat capacity
Heat capacity is how much heat something must soak up to warm by one degree. You might expect a metal's huge swarm of free electrons to store a lot of heat, the way a big crowd absorbs a lot of noise. Surprisingly, they store almost none — and the reason is one of the free-electron model's quiet triumphs.
Because electrons can't share states, nearly all of them are jammed deep below the Fermi energy with no empty seat to move to, so they simply can't accept any heat. Only the thin sliver near the top — the ones the Fermi-Dirac blur reaches — can absorb energy. The result is a small contribution that grows in direct proportion to temperature, gentler than you'd ever guess from counting electrons.
It matters because this prediction matches experiment beautifully and was a key early win for quantum theory. The honest nuance: at everyday temperatures the electrons' share of a metal's heat capacity is dwarfed by the atoms' vibrations; you only see the electronic part clearly at very low temperatures, where the vibrations fall silent first.
Cool a metal to within a few degrees of absolute zero and measure its heat capacity: the atomic vibrations have nearly stopped contributing, and the leftover, which grows straight in step with temperature, is the electrons' share — exactly as the Sommerfeld model predicts.
Near absolute zero, the part of heat capacity that rises straight with temperature is the electrons'.
Classical physics predicted electrons should store about a hundred times more heat than they actually do — a glaring failure of the Drude picture. The fix wasn't a new force, just the rule that electrons can't share states.