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 v_x, v_y, and v_z, then kinetic energy

Along x-axis = ½mv_x²

Along y-axis = ½mv_y²

Along z-axis = ½mv_z²

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

Kinetic Energy = ½mv_x² + ½mv_y² + ½mv_z²

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_rms² = 3/2 k_B T

where,

• v_rms = Root mean square velocity of molecules,
• k_B = Boltzmann constant,
• T = Temperature of gas.

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

<K.E.> = <½mv_x²> + <½mv_y²> + <½mv_z²> = ½k_B T

The overall translational energy contribution of the molecule is consequently 3/2k_B 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 ½k_B 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_x² + ½ m_y² + ½ m_z²

Vibrational motion

K.E. = ½ m (dy / dt)² + ½ k y²

Where,

• k is the oscillator’s force constant,
• y is the vibrational coordinate.

Rotational motion

K.E. = ½ (I₁ω₁) + ½ (l₂ω₂)

Where,

• I₁ and I₂ is the Moments of inertia,
• ω₁ and ω₂ is the angular speeds of rotation.

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.

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.