Law of equipartition of energy states that for a dynamical system in thermal equilibrium the total energy of the system is shared equally by all the degrees of freedom. The energy associated with each degree of freedom per molecule is ½ kT, where k is the Boltzmann’s constant.
Let consider one mole of a monoatomic gas in thermal equilibrium at temperature T. Each molecule has 3 degrees of freedom due to the translatory motion. According to the kinetic theory of gases, the 3 mean kinetic energy of a molecule is 3/2 kT.
Let Cx, Cy, and Cz be the components of RMS velocity of a molecule along the three axes. Then the average energy of a gas molecule is given by
½ mC2 = ½ mCx2 + ½ mCy2 + ½ mCz2
So, ½ mCx2 + ½ mCy2 + ½ mCz2 = 3/2 kT
Since molecules move at random, the average kinetic energy corresponding to each degree of freedom is the same.
½ mCx2 = ½ mCy2 = ½ mCz2
So, ½ mCx2 = ½ mCy2 = ½ mCz2 = ½ kT
Mean kinetic energy per molecule per degree of freedom is: ½ kT