Law of Equipartition of Energy

Building the law from a single direction of motion. Section 9.2.2 showed that the average kinetic energy associated with motion along the x-direction alone is $\tfrac12 m\overline{v_x^2}=\tfrac12 kT$. Exactly the same reasoning applies independently to motion along the y-direction, $\tfrac12 m\overline{v_y^2}=\tfrac12 kT$, and along the z-direction, $\tfrac12 m\overline{v_z^2}=\tfrac12 kT$ -- each of the three translational degrees of freedom carries, on average, exactly the same share, $\tfrac12 kT$, of the total kinetic energy.

The general statement. According to kinetic theory, this even sharing is completely general: the average kinetic energy of a system of molecules in thermal equilibrium at temperature $T$ is distributed uniformly over every one of its available degrees of freedom, with each individual degree of freedom receiving exactly $\tfrac12 kT$ of energy on average. This statistical rule is called the law of equipartition of energy.

Applying it to different molecule types. Since the total average kinetic energy of a molecule is simply $\tfrac12 kT$ multiplied by however many degrees of freedom $f$ it has:

• Monatomic molecule ($f=3$): average KE $=3\times\tfrac12 kT=\tfrac32 kT$.
• Diatomic molecule at low/normal temperature ($f=5$): average KE $=5\times\tfrac12 kT=\tfrac52 kT$.
• Diatomic molecule at high temperature ($f=7$): average KE $=7\times\tfrac12 kT=\tfrac72 kT$.
• Linear triatomic molecule ($f=7$): average KE $=7\times\tfrac12 kT=\tfrac72 kT$. …