State and prove law of equipartition of energy.

Explanation

Law of Equipartition of Energy: According to this law, for any system in thermal equilibrium the total energy of the system is equality distributed amoug its various degrees of freedom. The energy associated with each degree of freedom per molecule is `1/2 kT`
Proof: COnsider a molecule of mass m, moving with velocity v. THe kinetic energy of molecule is given by
`E=1/2 mv^2=1/2 m (v_2^3+v_y^2+v_z^2)`
`=1/2 mv_2^2+1/2 mv_y^2+1/2 mv_z^2`
For a gas in thermal equilibrium, the average value of kinetic energy is given by
`gt E lt =lt 1/2 mv_z^2 gt + lt 1/2 mv_y^2 gt + lt 1/2 mv_z^2 gt =3/2 kT`
Since all the directions are equally preferred so average kinetic energy along all direction is equal
SO `lt 1/2 mv_z^2 = lt 1/2 mv_y^2 gt = lt 1/2 mv_z^2 gt =3/2 kT`
Also each molecule during its motion in space has three degrees of freedom.
Therefore, average kinetic energy per molecule per degree of freedom is `1/2 kT`