Law of Equipartition of Energy (original) (raw)

Last Updated : 23 Jul, 2025

Law of Equipartition of Energy has many names such as Equipartition Theorem, Equipartition Principle, Law of Equipartition, or simply Equipartition and it describes the distribution of energy among the particles in a system that is at thermal equilibrium. The law of Equipartition of Energy tells us about how each degree of freedom of a particle in a system contributes to the average energy of the system. The Equipartition Theorem holds key significance in a wide range of fields of study, from thermodynamics and statistical mechanics to materials science and chemistry. This article covers the topic of the Law of Equipartition of Energy in varying detail.

What is Law of Equipartition of Energy?

According to the Law of Equipartition of Energy, at thermal equilibrium, the total energy of a particle is equally divided among its direction of movement, which is known as the degree of freedom. This means that the particle can move freely in all these directions, even under external pressure. For better understanding, we can take the analogy of students after school, they can freely go in different directions toward their respective homes, which represents their freedom of movement.

Other than this, the Law of Equipartition also applies to complex systems such as diatomic and triatomic molecules. For example, diatomic molecules have five degrees of freedom including two rotational degrees, which gives them more energy and higher specific heat than the monoatomic particles as monoatomic particles only have three degrees of freedom.

Now, let the mass of a particle be m and the velocity along x, y, and z directions by vx, vy, and vz, then kinetic energy

**Along x-axis = ½mv x 2

**Along y-axis = ½mv y 2

**Along z-axis = ½mv z 2

At thermal equilibrium, total kinetic energy is given as follows:

**Kinetic Energy = ½mv x 2 + ½mv y 2 + ½mv z 2

According to the kinetic theory of gases, an item, body, or molecule's average kinetic energy is directly inversely related to its temperature. You could indicate it as:

**½ mv rms 2 = 3/2 k B T

where,

For a gas at a temperature T, the average kinetic energy per molecule denoted as <K.E.> is

****<K.E.> = <½mv** x** 2 **> + <½mv** **y** **2** **> + <½mv** **z** **2** **> = ½k B T

The overall translational energy contribution of the molecule is consequently 3/2kB T since the mean energy associated with each component of translational kinetic energy, which is quadratic in the velocity components in the x, y, and z directions, is ½kB T.

A monatomic molecule only experiences translational motion, hence each motion requires ½ KT of energy. Divide the molecule's total energy by the number of degrees of freedom to get this value:

**K.E. = 3/2 k T ÷ 3 = ½ k T

Motions in translation, vibration, and rotation are all present in diatomic molecules. A diatomic molecule's energy is represented by the following:

**Translational motion

**K.E. = ½ m x 2 + ½ m y 2 + ½ m z 2

**Vibrational motion

**K.E. = ½ m (dy / dt) 2 + ½ k y 2

Where,

**Rotational motion

**K.E. = ½ (I 1 ω 1 ) + ½ (l 2 ω 2 )

Where,

You should be aware that vibrational motion consists of both kinetic and potential energies.

The entire energy of the system is allocated evenly among the many energy modes present in the system under thermal equilibrium circumstances, in accordance with the law of energy partition. The motion's total energy is contributed by the translational, rotational, and vibrational motions, each of which contributes a ½ k T of energy. The vibrational motion, which possesses both kinetic and potential energy, provides a whole 1 k T of energy.

Degree of Freedom

When a molecule can move around in three dimensions, we say that it has three degrees of freedom. If it can only move on a two-dimensional plane, it has two degrees of freedom, and if it moves in a straight line, it only has one degree of freedom.

To describe the motion of a molecule, we need to use coordinates like x and y, and velocity components like vx and vy. The number of coordinates or independent variables needed to fully describe the position and configuration of a system is known as its degree of freedom.

So, when a molecule has three degrees of freedom, we need three coordinates to describe its position and motion. When it has two degrees of freedom, we only need two coordinates, and when it has one degree of freedom, we only need one coordinate.

Kinetic Energy per Molecule

In this section, we will learn about Kinetic Energy Per Molecule for Triatomic and Monoatomic gaseous molecules.

**Triatomic Molecule: In the case of a Triatomic Molecule the degree of freedom is 6 as given by formula 3N - k where N = 3 which number of atoms and k = 2 which is the number of independent relations between the atoms. Hence, kinetic energy per molecule for a triatomic gas molecule is given by

**6 × N a × 1/2K b T = 3 x (R/N a ) × N a × T = 3RT, {k b = R/N a }

Where,

**Diatomic Molecule: From the formula 3N - k the degree of freedom for the diatomic molecule is 5 as independent relation, k is 1, and N = 2. Hence, kinetic energy per molecule for a diatomic gas molecule is given by

**5 × N a × 1/2KbT = 5/2 x (R/N a ) × N a × T = 5/2RT

where symbols have usual meanings as mentioned above

**Monoatomic Molecule: From the formula 3N - k the degree of freedom for the diatomic molecule is 3 as independent relation, k is 0, and N = 1. Hence, kinetic energy per molecule for a diatomic gas molecule is given by

**3 × N a × 1/2KbT = 3/2 x (R/N a ) × N a × T = 3/2RT

where symbols have usual meanings as mentioned above

Diatomic Molecules

Helium atoms in monoatomic gases, like helium gas, have three degrees of freedom for translation. For molecules like O2 or N2, which have two atoms positioned along the x-axis, they also have three degrees of freedom for translation but can also rotate around the z-axis and y-axis.

A diatomic molecule, like O2, has two additional degrees of freedom due to its two perpendicular rotational orientations around its center of mass. This means it has a total of five degrees of freedom, with two degrees of rotational freedom. The rotating kinetic energy for each rotational degree of freedom is influenced by the moments of inertia about the z and y axes and the corresponding angular speeds. The total energy due to the degrees of freedom for translation and rotation in a diatomic molecule can be calculated accordingly. As a result, the total energy owing to the degrees of freedom for translation and rotation in a diatomic molecule is,

**E = ½mv x 2 + ½mv y 2 + ½mv z 2 + ½I z ω z 2 + ½I y ω y 2

The quadratic terms in the aforementioned expression are related to the several degrees of freedom that a diatomic molecule can have. Each of them adds ½kB T to the molecule's overall energy. It was implied in the explanation above that the rotating molecule is a rigid rotator. Real molecules, on the other hand, have covalent links between their atoms, which allows them to execute extra motion, namely atomic vibrations about their mean locations, similar to a one-dimensional harmonic oscillator. As a result, these molecules have an extra degree of freedom that corresponds to their various vibrational modes. Only along the internuclear axis may the atoms oscillate in diatomic molecules like O2, N2, and CO. The vibrational energy associated with this motion is added to the molecule's overall energy.

**E = E(translational) + E(rotational) + E(vibrational)

Both the kinetic energy term and the potential energy term contribute to the word E(vibrational), which is composed of two components.

**E(vibrational) = ½mu 2 + ½kr 2

Where u denotes the rate of vibration of the molecule's atoms, r denotes the distance between the oscillating atoms, and k is the force constant. Each of the quadratic velocity and position terms in equation 1 will contribute ½kB T. The total internal energy is thus increased by 2 × ½kB T for each mode or degree of freedom for vibrational motion.

Thus, for a non-rigid diatomic gas in thermal equilibrium at a temperature T, the mean kinetic energy associated with molecular translation along three directions is 3 × ½kB T, and the mean kinetic energy associated with molecular rotation about two perpendicular axes is 2 × ½kB T, and the total vibrational energy is 2 × ½kB T, which corresponds to kinetic and potential energy terms. The average energy for each molecule associated with each quadratic term is ½kB T when the law of energy partition is applied to gas in thermal equilibrium at a temperature T. In contrast to extremely low temperatures, where quantum effects are significant, the law of energy equilibria only applies to high temperatures.

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