Law of equipartition of energy

Law of equipartition of energy

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 moelcule is ? kT.  where k is the Boltzmann?s constant.

Let us consider one mole of a monoatomic gas in thermal equilibrium at temperature T. Each molecule has 3 degrees of freedom due to translatory motion. According to kinetic theory of gases, the mean kinetic energy of a molecule is 3 /2 kT.

Let C_x , C_y and C_z be the components of RMS velocity of a molecule along the three axes. Then the average energy of a gas molecule is given by

? mC² = ? mC_x² +  ? mC_y² +  ? mC_z²

? mC_x² +  ? mC_y² +  ? mC_z² =3/2 kT

Since molecules move at random, the average kinetic energy corresponding to each degree of freedom is the same.

? mC_x² +  ? mC_y² +  ? mC_z²

? mC_x² +  ? mC_y² +  ? mC_z²= 1/2 kT

∴ Mean kinetic energy per molecule per degree of freedom is 1/2 kT