Proof of Law of Equipartition of Energy
The law of equipartition of energy sates that for any dynamical system in a thermal equilibrium, the total energy is equally divided among the degrees of freedom. The energy associated with a molecule for one degree of freedom is ½ KT where k is the Boltzmann’s constant and T is th temperature.
From kinetic theory of gases we know, in thermal equilibrium average square value of components of velocity of a gas molecule c along three axes X, Y and Z are equally distributed, i.e., u2 = v2 = w2. Here average value of the components of velocities along X, Y and Z axes are respectively u, v and w.
½ mu2 = ½ mv2 = ½ mw2
But c2 = u2 + v2 + w2 and u2 = v2 = w2
so, ½ mu2 = ½ mv2 = ½ mw2 = 1/3 x ½ mc2
Again, we know, average kinetic energy of each molecule.
½ mc2 = 3/2 KT
Then, ½ mu2 = ½ mv2 = ½ mw2 = ½ x 3/2 KT = ½ KT
So, average energy for each degree of freedom = ½ KT.
Again, in case of an oscillating particle half of the total energy is kinetic and rest half is potential energy. So, total energy per degree of freedom = kinetic energy + potential energy = ½ KT + ½ KT = KT.
So, we see that translational kinetic energy associated with each component of velocity is one third at the total energy.
Available total energy is equally divided as different independent energy into the components.
Again molecules are not geometrical points but have infinitely small size. So, molecules have moment of inertia and mass, hence along with translational motion there is also rotational motion. Sizes of the molecules are not rigid and due to collision with other molecules we can expect oscillation in them. As a result their degree of freedom may be more. By Maxwell-Boltzmann statistics it can be shown that energy associated with a degree of freedom becomes function of assigned two-dimensional variable of degrees of freedom, then associated average value of energy is equal to ½ KT. If the total energy is distributed equally to all the degrees freedom, then a molecule having degrees of freedom off total energy = f x ½ KT = f/2 KT.