# Law of Equipartition of Energy

Law of Equipartition of Energy is given by Claussius which 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 RMS velocity of gas molecules can be resolved into its components along x, y and z axis as -

c^{2} = c_{x}^{2} + c_{y}^{2} + c_{z}^{2}

multiplying by 1/2m, we get-

1/2mc^{2} = 1/2mc_{x}^{2} + 1/2mc_{y}^{2} + 1/2mc_{z}^{2}

or, E = E_{x} + E_{y} + E_{z} -----(1) (as K.E.(E) = 1/2mc^{2})

We know that gas molecules do not have any preferential direction. So, velocities along all three axes are equally probable-

c_{x} = c_{y} = c_{z}

or, E_{x} = E_{y} = E_{z}

or, E = 3E_{x} -----(2)

Each energy term in component contributes equally to the total energy. This is called the law of equipartition of energy.

We know that-

PV = 1/3 mnc^{2}

PV = 2/3 X 1/2 mnc^{2}= 2/3 K.E.

or, K.E.= 3 X 1/2 PV

or, E = 3 X 1/2 RT (as PV = RT for one mole)

From equation (2), we have-

3 E_{x} = 3 X 1/2 RT

so, **E _{x} = E_{y} = E_{z} = 1/2 RT**

Hence, the average kinetic energy possed by molecules in each component per degree of freedom is 1/2 RT per mole.

### Q. According to the law of equipartition of energy, for a molecule one vibrational mode contributes-

a. Only kinetic energy

b. Only potential energy

*c. Both kinetic and potential energy*

d. None of these