Law of Equipartition of Energy

Law of equipartition of energy: Statement

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 degree of freedom.

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

, along the x-axis
, along the y-axis
, along the z-axis

When the gas is at thermal equilibrium, the average kinetic energy is denoted as

,along the x-axis
, along the y-axis
, along the z-axis

According to the kinetic theory of gases, the average kinetic energy of a molecule is given by,

, where V_rms is the root mean square velocity of the molecules, K_b is the Boltzmann constant and T is the temperature of the gas.

The mono-atomic gas has three degrees of freedom, so the average kinetic energy per degree of freedom is given by

 

If a molecule is free to move in space, it needs three coordinates to specify its location, thus, it possesses three translational degrees of freedom. Similarly, if it is constrained to move in a plane, it possesses two translational degrees of freedom and if it is a straight line, it possesses one translational degree of freedom. In case of a tri-atomic molecule, the degree of freedom is 6. And the kinetic energy of the per molecule of the gas is given as,

Molecules of a mono-atomic gas like argon and helium has only one translational degree of freedom. The kinetic energy per molecule of the gas is given by

 

The diatomic gases such as O₂ and N₂ which have three translational degrees of freedom can rotate only about their center of mass. Since, they have only two independent axis of rotation, as the third rotation is negligible, due to its 2-D structure. Thus, only two rotational degrees of freedom are considered. The molecule thus has two rotational degrees of freedom, each of which contributes a term to the total energy consisting of transnational energy t and rotational energy.