Law of Equipartition of Energy

The law which states that for any system in thermal equilibrium, the total energy is equally distributed among all its degrees of freedom, with energy \[\frac {1}{2}\]kT associated with each degree of freedom per molecule, is called the Law of Equipartition of Energy.

Statement:
For a gas in thermal equilibrium at temperature TT, the average energy associated with each molecule for each quadratic term (degree of freedom) is:

where k_B = 1.38 × 10⁻²³ J/K and T = absolute temperature.

Energy Expressions for Different Types of Motion:

1. Translational K.E.:
   ​\[\frac{1}{2}mv_x^2+\frac{1}{2}mv_y^2+\frac{1}{2}mv_z^2\] (3 degrees of freedom — along x, y, z axes)

2. Rotational K.E.:
   \[\frac{1}{2}I\omega_x^2+\frac{1}{2}I\omega_y^2+\frac{1}{2}I\omega_z^2\] (up to 3 degrees of freedom — rotation about x, y, z axes)

3. Vibrational K.E.:
   \[\frac{1}{2}m\dot{u}^2+\frac{1}{2}kr^2\] (2 terms — kinetic and potential energy of vibration)

Each quadratic term contributes \[\frac {1}{2}\]k_BT to the total average energy of the molecule.