Law of equipartition of energy

According to the law of equipartition of energy, for any dynamic system in thermal equilibrium, the total energy for the system is equally divided among the degrees of freedom.

The kinetic energy of a single molecule along the x- axis, y- axis and z- axis is

Kinetic energy of single molecule is E_t = ½ m v²_x + ½ m v²_y + ½ m v²_z

|E_t| = 3/2 K_BT

Let us take

|½ m v²_x |= ½ K_BT

|½ m v²_y |= ½ K_BT

|½ m v²_z |= ½ K_BT

Motion of a body from one point to another is called translation.

The number of independent ways. By which a molecule can move, without violating any constraint imposed unit, is called number of degrees of freedom.

Molecules of a monatomic gates have only translational degrees of freedom. Diatomic gas has both translational degrees of freedom as well as rotational degrees of freedom around their centre of mass

So, E = E_t + E_r = ½ m v²_x + ½ m v²_y + ½ m v²_z + ½ I₁ ω²₁ + ½ I₁ ω²₂

E_t = translational energy

E_r = rotational energy

I₁ & I₂ are moment of interior ω₁ & ω₂ are angular velocities.

Some molecules even have vibrational energy [E_v] also.

So, E = E_T + E_R + E_V

E_V can written as E_v = ½ m (dy/dt)² + ½ KY²

Where K = constant

            Y = vibrational coordinate [Y]

In equilibrium, the total energy is equally is tribute in all possible energy modes with each mode having an average energy equal to ½ K_BT this is known as “law of equipartition of energy”.