Law of Equipartition of Energy

The kinetic energy of a single molecule, expressed component-wise, is

Averaged over a gas at temperature $T$, the mean kinetic energy per molecule is (Section 3.7) $\overline{KE} = \tfrac{3}{2}k_BT$. Since no direction ($x$, $y$ or $z$) is preferred over another (the gas has no built-in directionality), each of the three quadratic terms must, by symmetry, contribute equally to this total:

so that the mean energy associated with every quadratic velocity term is $\tfrac{1}{2}k_BT$, and the three translational terms together contribute the full $\tfrac{3}{2}k_BT$.

Generalising this observation -- built up further using diatomic rotational and vibrational terms in Section 3.8.2 -- gives the law of equipartition of energy: for a gas in thermal equilibrium at temperature $T$, the average energy associated with each independent quadratic term in a molecule's total energy (whether translational, rotational or vibrational) is the same, namely $\tfrac{1}{2}k_BT$. …