In the classical canonical ensemble, each independent quadratic term in the energy contributes an average $$\frac12k_BT$$ to the thermal energy. This is the equipartition theorem.

If one momentum component appears as $$E_x=\frac{p_x^2}{2m},$$ then $$\langle E_x\rangle=\frac12k_BT.$$ A monatomic ideal-gas particle has three translational quadratic terms, so $$\langle E\rangle=\frac32k_BT.$$ For $N$ particles, $$U=\frac32Nk_BT,$$ which gives a constant-volume heat capacity $$C_V=\frac{\partial U}{\partial T}=\frac32Nk_B.$$

A one-dimensional harmonic oscillator has both a quadratic kinetic term $p^2/(2m)$ and a quadratic potential term $kx^2/2$. Each contributes $k_BT/2$, giving $$\langle E\rangle=k_BT.$$

Equipartition is a classical high-temperature result, not a universal rule. Quantum energy levels can be too widely spaced for some degrees of freedom to be thermally excited. Molecular rotations and vibrations therefore 'freeze out' as temperature falls, and solid heat capacities deviate strongly from the classical prediction at low temperature.

The theorem is valuable because it converts the algebraic structure of a Hamiltonian into immediate thermal predictions. Its failures are equally informative: they signal that the classical continuum of accessible energies is no longer a good approximation.