Law of Equipartition of Energy

The law of Equipartition of energy is the partition or distribution of energy equally among the degrees of freedom. A particle moving has three degrees of freedom, i.e., the coordinates describing its position. For instance, an atom of gas shall have three degrees of freedom, i.e., spatial, position, or coordinates of the atom. An atomic particle moving in a translator motion shall have x, y, and z coordinates, specifying its location. If we consider a diatomic particle such as O2 and N2, they rotate freely around the centre of Mass.

Thus, we can say that diatomic molecules will possess three coordinates due to translator motion. Moreover, they have two rotatory movements in addition. The brilliant works of James Clerk Maxwell of Scotland and Ludwig Boltzmann of Germany in the field of thermodynamics are the principle of the law of Equipartition of energy. 

Thermal Equilibrium

Thermal equilibrium is the condition in which two objects in contact do not transfer heat to one another. They attained the state of equilibrium, i.e., they do not require doing any further transfer of matter or heat. The law of Equipartition of heat depends upon the thermal equilibrium to tell about the distribution of energy among the degrees of freedom. 

Law of Equipartition of Energy

As per this law, for every degree of freedom, an equal amount of energy is associated and this energy is equal to ½ kbT, where kb is the Boltzmann constant and T is the temperature.

The Kinetic energy of any particle or object about the coordinates (x, y, z) axis are:

½ mvx2 (along x-axis)

½ mvy2, (along y-axis)

½ mvz2, (along z-axis)

We have studied the kinetic theory of gases, which states that the average kinetic energy of any object or body is directly proportional to its temperature. It is represented as:

½ mvrms 2 = (3/2) kbT

Where vrms = root mean square velocity of molecules

kb = Boltzmann constant and

T = temperature of the gas

Degrees of Freedom

The average kinetic energy for each of the three translational degrees of freedom for a monatomic particle is-

K Ex = ½ kbT

The average energy for a gas at the thermal equilibrium under the temperature T is-

Eavg = ½ mvx 2+ ½ mvy 2+ ½ mvz2

= ½ kbT + ½ kbT + ½ kbT = 3/2 kbT

Here, kb is the Boltzmann’s constant

As already mentioned, the monatomic molecule undergoes only translational motion. Thus, the energy for each motion shall be equal to ½ kbT. 

(3/2) kbT ÷ 3 = ½ kbT

In the case of a diatomic molecule, they possess translational, vibrational, and rotational motion. The vibrational motion possesses both kinetic and potential energies. Therefore, the energy component of it is as follows:

In case of translational motion = ½ mvx 2+ ½ mvy 2+ ½ mvz2

In case of rotational motion = ½ (l1 w12) + ½ (l2 w22)

Where,

I1 & I2= moments of inertia,

w1 & w2= angular speeds of rotation.

In case of vibrational motion = ½ m (dy/dt)2 + ½ k y2

Where k = force constant of the oscillator,

y = vibrational coordinate.

Heat Capacity

Heat capacity at constant volume Cv is:

Cv= (∂U/∂T)v

According to the Equipartition theorem, each degree of freedom that appears only quadratically in the total energy possesses an average energy of ½kbT in thermal equilibrium. Thus, it contributes ½kb to the system’s total heat capacity.

So, we can say that all three translational degrees of freedom contribute ½R to (3/2 R). For the rotational kinetic energy, the contribution will be R for the linear and 3/2R for the nonlinear molecules. Whereas in the case of vibration, an oscillator has quadratic kinetic and potential terms. It makes the contribution of each vibration as R. 

The heat capacity will be reduced and drop to zero at low temperatures if the kbT exceeds the spacing between the quantum energies.

This corresponding degree of freedom will be known as frozen out.

Significance of Law of Equipartition of Energy

The law of Equipartition of energy tells about the distribution of energy in a thermal equilibrium state of complex molecular systems. It tells us that the specific heat of a complex gas is directly proportional to the number of atoms per molecule. Thus, an increase in the number of atoms shall lead to the increase of the specific heat of a complex gas. The reason is that these diatomic molecules have higher internal energies and specific heat content than monatomic gas particles. And we know that the diatomic particles have 5 degrees of freedom, whereas monatomic particles have only three translational degrees of freedom.

Conclusion

Now, we know that the law of Equipartition of energy is the partition or distribution of energy in a thermal equilibrium system equally among the degrees of freedom. Thermal equilibrium is the condition in which two objects in contact do not transfer heat to one another. A particle moving has three degrees of freedom, i.e., the coordinates describing its position.

There are three rotational degrees of freedom along the x, y, and z-axis. The three translational degrees of freedom along the three-axis can move in different directions. Such as forward, backward, up, down, and left or right. The law of Equipartition of energy tells us that the specific heat of a complex gas is directly proportional to the number of atoms per molecule.