 # Law of Equipartition of Energy

The equipartition theorem is related to the temperature of the system and its average kinetic and potential energy. This theorem is also called the law of equipartition of energy or just equipartition. The concept behind equipartition is that, under thermal equilibrium, the energy of a system is equally shared among all the forms of energy. This means that the average kinetic energy for each degree of freedom will be equal in both translational motion and rotational motion.

A single molecule can freely move along the three-dimensional spaces i.e in X, Y, and Z axes. For this movement, a particle requires some energy which is generated by the energy stored in an atom. The Law of equipartition of energy describes the amount of energies stored inside the molecule is used for each mode of energy (translational, vibrational, and rotational) of the molecule.

### Degree of Freedom of Gases

Let us consider a single particle of an atom which is free to move in the three-dimensional space. When a particle moves from one point to another point it is called translational motion and its movement along the axis is called translational movement. Three coordinates (x, y, and z) are needed to specify the location of the atom.

In other words, it can be said that a single atom has 3 degrees of freedom, i.e it can move along any of the three axes. Most of the monatomic molecules (molecules consisting only one atom) usually possess three degrees of freedom.

Let us now consider a diatomic molecule (molecules add off 2 atoms like O₂ or N₂). These molecules can undergo translational movement and possess three degrees of freedom. In addition to that, they can also rotate around the centre of their mass. Two possible rotations can occur by having an axis normal to the axis joining the two atoms.

Hence, these molecules possess 2 extra rotational degrees of freedom. In other words, in order to specify the location of the atom, three translational X, Y and Z coordinates are needed in addition to two rotational coordinates of the atoms of the molecule.

In order to describe the location of a diatomic molecule, three translational coordinates along with the vibrational and rotational coordinates are required. Degree of freedom is described as the number of ways in which a molecule or an atom can move. This concept is the basis of the Law of Equipartition of energy.

### State the Law of Equipartition

The law of equipartition states that when an atom is kept under constant thermal condition, the sum total of the molecule gets splitted up all throughout consistently where the amount of degree of freedom is free from opposition. Let us consider a molecule that has 1000 units of energy and 5 degrees of freedom. In this case, each degree of freedom will save 200 units of energy.

The kinetic energy of a molecule in the x-axis, y-axis, and z-axis is given by:

$\frac{1}{2}mv^{2}$, along the axis

$\frac{1}{2}$mx2, along the x-axis

$\frac{1}{2}$my2, along the y-axis

$\frac{1}{2}$mz2, along the z-axis

The kinetic theory of gases states that the average kinetic energy of a molecule is directly proportional to the temperature of the molecule. So based on the kinetic theory of gases, the average kinetic energy of a molecule becomes,

$\frac{1}{2} mv_{rms^{2}} = \frac{3}{2} k_{b} T$

Where,

Vrms = Root mean square velocity of molecules,

Kb = Boltzmann constant and

T = Temperature of the gas

A mono - atomic gas has three translational degrees of freedom, and so the average kinetic energy for each degree of freedom of the gas is given by,

KEx=$\frac{1}{2}$KbT

The Kinetic energy of a single gas molecule is given by:

KE = $\frac{1}{2}$ mv².

For a gas under thermal equilibrium condition at a temperature T, the average Energy is given by:

Eavg= $\frac{1}{2}$mx2+$\frac{1}{2}$my2+$\frac{1}{2}$mz2= $\frac{1}{2}$KT+$\frac{1}{2}$KT+$\frac{1}{2}$KT=3/2KT

Where

K = Boltzmann’s constant

Since a monatomic molecule undergoes only translational motion, the energy for each motion is equal to ½ KT. This value is obtained by dividing the total energy of the molecule by the number of degrees of freedom:

3/2 KT ÷ 3 = ½ KT

A diatomic molecule possesses translational, vibrational and rotational motion. The energy component of a diatomic molecule is given by:

For translational motion = $\frac{1}{2}$mx2 + $\frac{1}{2}$my2 + $\frac{1}{2}$mz2.

For rotational motion = ½(I1w1) + $\frac{1}{2}$ (I2w2),

Where,

I1 &I2 are moments of inertia,

w1 & w2 are angular speeds of rotation.

For vibrational motion = $\frac{1}{2}$ m (dy/dt)²+ $\frac{1}{2}$ ky².

Where,

k = force constant of the oscillator,

y = vibrational coordinate.

It should be noted that the vibrational motion possesses both kinetic and potential energies.

Based on the Law of Equipartition of Energy, under thermal equilibrium condition, the total energy of the system is equally distributed among the energy modes. The translational and rotational motion, each contributes ½KT energy to the total energy of the motion, and the vibrational motion contributes 1KT of energy as it possesses both kinetic and potential energy.

1. What is the Principle of Equipartition of Energy.

Ans - The law of equipartition of energy describes the total internal energy of complex molecular systems. It helps to explain the concept, why the specific heat of complex gases increases with an increase in the number of atoms per molecule.

The diatomic gas molecules have high internal energy and high molar specific heat content as compared to the monatomic gas molecules. This is because the diatomic gas molecule has five degrees of freedom while the monatomic gas molecule has only three degrees of freedom.

2. What is the Application of Law of Equipartition Energy in Specific Heat of a Gas.

Ans - One of the major applications of the law of equipartition of energy is in Meyer’s relation (empirical relation between the size of a hardness test indentation & the load needed to leave the indentation)

This shows:

Cp − Cv = R,

Where,

Cp is the molar specific heat capacity of an ideal gas at constant pressure, and

Cv is the molar specific heat at constant volume,

R is the gas constant.

This equation connects the two specific heats of an ideal gas of one mole to the ideal gas.

The law of Equipartition of energy is also used to calculate the value of CP − CV, and also to calculate the ratio between them, which is given by,

γ = CP / CV.

Where

γ = adiabatic exponent of the gas molecule.

Let’s take the example of a monatomic molecule.

Average kinetic energy of a molecule

=[3/2kT]

Total energy of a mole of gas

=3/2kT*NA=3/2RT

For one mole, the molar specific heat at constant volume

Cv=dU/dT=d/dT[3/2RT]

Cv=[3/2R]

Cp=Cv+R=3/2R+R=5/2R

The ratio of specific heats,

γ= Cp/Cv=(5/2R)/(3/2R)=5/3=1.67