9.0 Law of Equipartition of Energy

The total kinetic energy of a gas molecule is equally distributed among its all degree of freedom and the energy associated with each degree of freedom at absolute temperature is $$\frac{1}{2}{k_B}T$$

For one molecule of gas,

Energy related with each degree of freedom $ = \frac{1}{2}{k_B}T$

Energy related with $f$ degree of freedom $= \frac{f}{2}{k_B}T$

$$\because \overline {v_x^2} = \overline {v_y^2} = \overline {v_z^2} = \frac{{v_{rms}^2}}{3} $$

$$\therefore \frac{1}{2}mv_{rms}^2 = \frac{3}{2}{k_B}T$$

So energy related with one degree of freedom is $$ = \frac{1}{2}m\frac{{v_{rms}^2}}{3} = \frac{3}{2}\frac{{{k_B}T}}{3} = \frac{1}{2}{k_B}T$$