The 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 molecule is $\frac{1}{2}\mathrm{kT}$, where k is the Boltzmann’s constant.

For example, for a monoatomic molecule, each molecule has 3 degrees of freedom. According to the kinetic theory of gases, the mean kinetic energy of a molecule is $\frac{3}{2}\mathrm{kT}$.

Specific heat capacity of Monatomic gas:

The molecules of a monatomic gas have 3 degrees of freedom.

The average energy of a molecule at temperature T is $\frac{3\mathrm{KBT}}{2}$

The total internal energy of a mole is: $\frac{3\mathrm{KBT}}{2}\times {\mathrm{N}}_{\mathrm{A}}$

The molar specific heat at constant volume C_V is

For an ideal gas,

C_V (monatomic gas) $=\frac{\mathrm{dU}}{\mathrm{dT}}=\frac{3\mathrm{RT}}{2}$

For an ideal gas, C_P – C_V = R

where C_P is molar specific heat at constant pressure.

Thus, ${\mathrm{C}}_{\mathrm{P}}=\frac{5\mathrm{R}}{2}$

Specific heat capacity of Diatomic gas:

The molecules of a monatomic gas have 5 degrees of freedom, 3 translational, and 2 rotational.

The average energy of a molecule at temperature T is $\frac{5{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}{2}$

The total internal energy of a mole is: $\frac{5{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}{2}\times {\mathrm{N}}_{\mathrm{A}}$

The molar specific heat at constant volume C_V is

For an ideal gas,

C_V (monatomic gas) $=\frac{\mathrm{dU}}{\mathrm{dT}}=\frac{5\mathrm{RT}}{2}$

For an ideal gas, C_P – C_V = R

where C_P is molar specific heat at constant pressure.

Thus, ${\mathrm{C}}_{\mathrm{P}}=\frac{7\mathrm{R}}{2}$