Law of Equipartition of Energy

The law of equipartition of energy is a result in statistical thermodynamics that states that the mean thermal energy associated with each independent quadratic (i.e., proportional to the square of a coordinate or a momentum component) contribution to the total energy of a system consisting of many particles is $(1/2)\,k_B\,T$, where $T$ is the temperature of the system. (See Section 5.5.5.) It turns out, however, that this law only applies if the contribution in question is governed by classical (as opposed to quantum mechanical) physics. (See Section 5.5.6.)

Consider a particular constituent molecule of an ideal gas whose mass is $m$, and whose velocity is ${\bf v}$. The contribution of molecule's translational kinetic energy to the total energy of the whole gas is

$$\frac{1}{2}\,m\,v_x^{\,2} + \frac{1}{2}\,m\,v_y^{\,2} + \frac{1}{... ...,2} = \frac{\,p_x^{\,2}}{2\,m} +\frac{p_y^{\,2}}{2\,m}+ \frac{p_z^{\,2}}{2\,m},$$ (5.181)

where ${\bf p} = m\,{\bf v}$ is the molecular momentum. It can be seen that the contribution of the molecules's translational kinetic energy to the total energy consists of three terms that are quadratic in a momentum component. Hence, according to the law of equipartition of energy, the mean thermal energy associated with the molecules translational kinetic energy is

$$\langle {\cal K}_{\rm trans}\rangle = \frac{1}{2}\,k_B\,T + \frac{1}{2}\,k_B\,T+\frac{1}{2}\,k_B\,T=\frac{3}{2}\,k_B\,T,$$ (5.182)

in accordance with Equation (5.180).