Each quadratic degree of freedom contributes one half kT to the average energy of a molecule.
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The equipartition theorem states that, in thermal equilibrium at high temperature, every quadratic degree of freedom in a molecule's energy contributes one half of kT to its average energy, where k is the Boltzmann constant. A quadratic degree of freedom is a term that is squared in the energy expression, such as a translational momentum component or a rotational angular momentum. Using this rule, the molar heat capacities of gases and solids can be predicted from the number of active modes. The theorem works well for many gases at room temperature but fails when quantum effects freeze out vibrations at low temperature. Its breakdown was an important clue that quantum mechanics is required.
Key ideas: Each quadratic term gives kT over 2 per molecule or RT over 2 per mole. Count translational, rotational and active vibrational terms, remembering each vibration gives two. Multiply by the number of terms and differentiate to get Cv.
Common mistakes and connections: The rule fails when thermal energy is small relative to level spacing, so vibrations of light diatomics are frozen at room temperature. This failure at low temperature, and the ultraviolet catastrophe, motivated quantum theory. It predicts Dulong-Petit as well.
High-temperature Cv in units of R equals half the number of active quadratic degrees of freedom: 3/2 R for monatomic, 5/2 for diatomic at room temperature. The interactive lesson shows this as a live animation or chart.
A diatomic molecule has 3 translational and 2 rotational quadratic degrees of freedom active at room temperature, giving Cv equal to 5/2 R.
Each quadratic degree of freedom contributes one half kT to the average energy of a molecule.
Equipartition Theorem is a advanced topic in Thermodynamics & Energetics. It builds on beginner and intermediate topics, so review those first if needed.
Study these earlier topics in Thermodynamics & Energetics first: Thermal Equilibrium, Temperature Scales, Latent Heat.
The next topics in Thermodynamics & Energetics are Degrees of Freedom and Statistical Thermodynamics.
Yes. The interactive lesson includes a self-check question on Equipartition Theorem, and the Thermodynamics & Energetics quiz draws on every topic in this area.
Equipartition Theorem is topic 60 of 91 in Thermodynamics & Energetics.