Law of Equipartition of Energy: Definition, Degree of Freedom and Sample Questions

Equipartition of energy states that in a thermal equilibrium system, equal distribution of energy will be connected with each degree of freedom on average. For example, a molecule that is moving through space will have three degrees of freedom as its location will be defined through the three coordinates.

• The degree of freedom of a system is defined as the total number of coordinates or independent quantities required to describe the position and configuration of the system completely.
• The scientific definition of this law is ‘The Equal distribution of energy among the degrees of freedom for any dynamic system in thermal equilibrium’.
• According to the law of Equipartition of energy, in equilibrium the energy is equally distributed in all possible energy modes i.e. translational, rotational, and vibrational with each mode having an average energy equal to 1/2KT.
• This law can be applied to predict the specific heat of gases.

Key Terms: Degree of freedom, energy, rms velocity, Equipartition of energy, Boltzmann constant, Vibrational motion, Gas constant, Temperature

Read More: Elastic Limit

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Law of Equipartition of Energy

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The invention of this law was done by the physicists named Ludwig Boltzmann from Germany & James Clerk Maxwell from Scotland. If we apply formulas to define this law then the law states:

“A system of equilibrium molecules at established Temperature (T) will consist of a standard 1/2KT energy correlated with each degree of freedom where ‘K’ is seen as the Boltzmann constant”.

Additionally, if there is any other degree of freedom providing energy which is quite prospective, then it will have another 1/2KT energy correlated with it.

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In a system where ‘s’ is the degree of freedom & ‘t’ has prospective energies. The average energy in totality will be

E= 1/2(s+t)kT

Let us take an example:

• A particle of gas contains three degrees of freedom (coordinates of the particle).
• Therefore, it will have an average energy of 3/2 kT in totality.
• Hence, for the particle in a solid form, the vibrational movement consists of possible energy and also kinetic energy.
• With both modes contributing to a term 1/2 kT, it will result in an average energy of 3 kT in totality.

Among the kinetic theory of gases, the average thermal equilibrium square value of elements of the velocity of a ‘C’ gas molecule across three axes X, Y, and Z are uniformly dispersed, i.e., u² = v² = w². The average total value of the elements of velocities across the X, Y, and Z axes are respectively u, v, and w.

The kinetic energy is distributed uniformly in a single particle along the x-axis, y-axis, and z-axis.

1. X-Axis = 1/2 m(v_x)²
2. Y-Axis = 1/2 m(v_y​)²
3. Z-Axis = 1/2 m(v_z​)²

The average kinetic energy when gas is at thermal equilibrium is constituted as

1. X-Axis = 1/2K_bT
2. Y-Axis = 1/2K_bT
3. Z-Axis = 1/2K_bT

The average kinetic energy according to the kinetic theory of gases is constituted as

E= 3/2m(v_rms)² = 3/2 KbT

The v_rms is represented as the square root of the velocity of the particles. The Boltzmann constant is indicated by Kb and T is the temperature of the gas.

Read More: Mean Free Path

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Degree of Freedom

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As we have discussed above, every particle consists of three translational degrees of freedom to allow the particle to freely move through space. For specifying the location in orbit, the particle has three coordinates. Similarly, if required to move it to a place then the molecule is supposed to have two translational degrees of freedom & one translational degree of freedom is required if there is a straight line. Therefore, it varies from one atom to another.

Likewise, if we discuss the triatomic atom, it should contain 6 degrees of freedom. So, the per molecule kinetic energy of the gas will be:

6 x N x ½ KbT = 3 x R / N N KbT = 3RT

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Let us take another different example for a deeper understanding of the degree of freedom. The gases, helium and argon contain only one translational degree of freedom, the equation of per particle kinetic energy of the gas will be:

3 x N x ½ KbT = 3 x R / N N KbT = 3 / 2 RT

Thirdly, if we consider a diatomic molecule (2 atoms like O₂ or N₂). These molecules go through translational movement and contain three degrees of freedom. Additionally, they can also revolve around the center of the mass.

The degree of freedom can be comprehensively detailed as the number of criteria where a molecule or an atom has space to move. This concept is the foundation of the Law of Equipartition of energy. 

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Things to Remember

•  Equipartition of energy states that in a thermal equilibrium system, equal distribution of energy will be connected with each degree of freedom on average.
• A system of equilibrium molecules at an established Temperature (T) will consist of a standard ½ kT energy correlated with each degree of freedom where ‘k’ is seen as the Boltzmann constant.
• The kinetic energy is distributed uniformly in a single particle along the x-axis, y-axis, and z-axis.
• The v_rms is represented as the square root of the velocity of the particles. The Boltzmann constant is indicated by Kb and T is the temperature of the gas.
• The degree of freedom can be comprehensively detailed as the number of criteria where a molecule or an atom has space to move. This concept is the foundation of the Law of Equipartition of energy.