Law of equipartition of energy

Interpretation of Temperature in Kinetic Theory:

The quantity in the bracket is the average translational KE of a molecule. In an ideal gas, the molecules are noninteracting, and hence there is no PE term. Thus, the internal energy of an ideal gas is purely kinetic. The average total energy E, therefore, is 

E = N 1/2 mv2

From the equation above,

This means that the average energy per molecule is proportional to the absolute temperature T of the gas. This equation relates the macroscopic parameter of the gas, T, to the KE of a molecule. 

Law of equipartition of energy:

Thus the mean energy associated with every component of translational kinetic energy which is quadratic in the velocity components in x, y and z directions is 1/2 kBT and therefore the total translational energy contribution of the molecule is (3/2) kBT. 

Degrees of freedom:

Degrees of freedom of a system are defined as the total number of coordinates or independent quantities required to describe the position and configuration of the system completely. 

Monatomic gas like helium contains He atoms. An He atom has 3 translational degrees of freedom (dof). Consider for example, O2 or N2 molecule with the two atoms lying along the x-axis. The molecule has 3 translational dof. In addition, it can rotate around z-axis and y-axis. In general, a diatomic molecule can rotate about its centre of mass in two directions that are perpendicular to its molecular axis. The molecules like O2 , are therefore, said to possess 2 additional dof namely 2 rotational dof. Each of these 2 dof contribute to rotational kinetic energy. 

To learn about more about law of equipartition of energy and DOF, watch these videos.

Specific Heat Capacity:

When the temperature of a gas is increased, even a small rise causes considerable change in volume and pressure. Therefore two specific heats are defined for gases, namely specific heat at constant volume Cv and specific heat at constant pressure Cp. Mayer’s relation gives an expression that connects the two specific heats. 

Cp – Cv = R

(c)  Polyatomic Gases: