Principle

Equipartition theorem

(T3.2.1) T3.2 · Equipartition of Energy →

E_{k} = \frac{1}{2} k_{B} T per degree of freedom

Statement

Each independent quadratic degree of freedom (translational, rotational, vibrational) carries the same average thermal energy.

Symbols and units

$E_{k}$	average kinetic energy per degree of freedom	J
$k_{B}$	Boltzmann constant	1.381×10⁻²³ J/K
$T$	absolute temperature	K

When it applies

Classical (non-quantum) treatment of thermal energy storage, and a temperature high enough to activate the degree of freedom in question.

Don't use when

A classical result — breaks down at low temperature where quantum effects freeze out a degree of freedom (e.g. vibrational modes in a diatomic gas at room temperature).

Derivation

A statistical mechanics result: at thermal equilibrium, energy is shared equally among all accessible, independent degrees of freedom, each contributing ½k_BT on average.

E_{k} = \frac{1}{2} k_{B} T

Worked examples

Internal energy of a monatomic gas

With 3 translational degrees of freedom:

E_{k} = 3 (\frac{1}{2} k_{B} T) = \frac{3}{2} k_{B} T

The origin of the familiar (3/2)nRT formula for the internal energy of an ideal monatomic gas.

Watch out for

A diatomic gas at room temperature has 5 active degrees of freedom (3 translational + 2 rotational), not 3 — rotation stores kinetic energy too, though vibration typically does not activate until much higher temperature.