---
title: "The Free Electron Gas and the Fermi Energy"
description: "Electrons in a metal fill states up to the Fermi energy even at absolute zero, which explains why they contribute so little to heat capacity. Only the few near the top of the stack can be excited."
canonical: https://lightmysky.com/learn/science/the-free-electron-gas-and-the-fermi-energy-mt_4oTalBcKpm
source: https://lightmysky.com/learn/science/the-free-electron-gas-and-the-fermi-energy-mt_4oTalBcKpm.md
retrieved: 2026-09-12
---

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# The Free Electron Gas and the Fermi Energy

Electrons in a metal fill states up to the Fermi energy even at absolute zero, which explains why they contribute so little to heat capacity. Only the few near the top of the stack can be excited.

Subject: Science · Area: Thermal & Statistical Physics · Ages 21 to 22
Page: https://lightmysky.com/learn/science/the-free-electron-gas-and-the-fermi-energy-mt_4oTalBcKpm

## Ready when they can

- Estimates the Fermi energy of a metal from its electron density
- Explains why the electronic heat capacity is far smaller than equipartition predicts
- Connects the filled band picture to electrical conduction in metals

## Lesson: The electron stack inside a metal

Crowd atoms into a crystal and their sharp levels smear into bands. A filled band cannot carry current, because every moving electron is cancelled by another. Conduction needs a partly filled band, where free slots let electrons shift under a field.

**Example.** Model that partly filled band as particles in a box and fill the lowest states, two electrons per level. The topmost energy at absolute zero is the Fermi energy. It depends only on electron density, so denser metals pack higher.

Heat can only excite electrons near the top of the stack, since deeper ones have no empty states nearby. Only that thin skin of electrons joins in, so the electronic heat capacity is far smaller than old rules predict. Most of the stack sits out.

**Tip.** To judge a metal, ask two questions: is the top band partly filled, and how dense are its electrons. Partly filled means it conducts, and higher density means a higher Fermi energy.

**Recap.** Electrons stack two per level to the Fermi energy, and only the top skin can move or take heat.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Fermi-Dirac and Bose-Einstein Statistics](https://lightmysky.com/learn/science/fermi-dirac-and-bose-einstein-statistics-mt_rU5uPV4SB8)

## Opens up

- [Band Structure: Metals, Insulators and the Gap](https://lightmysky.com/learn/science/band-structure-metals-insulators-and-the-gap-mt_9f74Vu6qnW)
- [Crystal Lattices, the Reciprocal Lattice and Bloch's Theorem](https://lightmysky.com/learn/science/crystal-lattices-the-reciprocal-lattice-and-blochs-theorem-mt_nbNnHpGnZI)