--- title: "Equipartition and the Heat Capacities of Gases" description: "Each way a molecule can store energy holds an average of half kT, so monatomic, diatomic and polyatomic gases have different heat capacities. Freezing out of modes at low temperature is the first hint" canonical: https://lightmysky.com/learn/science/equipartition-and-the-heat-capacities-of-gases-mt_3OjJwVWlYD source: https://lightmysky.com/learn/science/equipartition-and-the-heat-capacities-of-gases-mt_3OjJwVWlYD.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Equipartition and the Heat Capacities of Gases Each way a molecule can store energy holds an average of half kT, so monatomic, diatomic and polyatomic gases have different heat capacities. Freezing out of modes at low temperature is the first hint that classical physics is incomplete. Subject: Science · Area: Thermal & Statistical Physics · Ages 18 to 19 Page: https://lightmysky.com/learn/science/equipartition-and-the-heat-capacities-of-gases-mt_3OjJwVWlYD ## Ready when they can - Counts the degrees of freedom of a monatomic and a diatomic molecule and predicts the heat capacity - Explains the difference between heat capacity at constant volume and at constant pressure - Describes why measured heat capacities fall below the classical prediction at low temperature ## Lesson: Counting the ways a molecule stores energy Equipartition says each independent way a molecule stores energy holds an average of one half kT per molecule, or one half RT per mole. Count those ways, called degrees of freedom, and the heat capacity at constant volume is that count times R over 2. A lone atom drifts in three directions, so a monatomic gas has Cv of 3R over 2. A two atom molecule at room temperature also spins about two axes, adding two more ways, so a diatomic gas has Cv of 5R over 2. Nitrogen and oxygen follow this rule. **Example.** Heat a gas in a rigid box and every joule stays inside as internal energy. Heat it while it pushes a piston and part of the heat leaves as expansion work. That is why Cp beats Cv by exactly R per mole. **Tip.** Cool a gas far down and some spins and shivers freeze out, so fewer ways share the energy. Measured heat capacities then fall below the classical count, which was the first hint that classical physics is incomplete. **Recap.** Count the active ways to store energy, halve R times that count for Cv, and add R for Cp. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Thermodynamic State Variables and PV Diagrams](https://lightmysky.com/learn/science/thermodynamic-state-variables-and-pv-diagrams-mt_G3EYx84GZK) - [The Maxwell-Boltzmann Distribution of Molecular Speeds](https://lightmysky.com/learn/science/the-maxwell-boltzmann-distribution-of-molecular-speeds-mt_njr5r5YtuT) ## Opens up - [Phonons and the Heat Capacity of Solids](https://lightmysky.com/learn/science/phonons-and-the-heat-capacity-of-solids-mt_WXZPdz_JTd)