The degree of freedom of gases is defined as the number of ways a particle or an atom can move. Thus, the degree of freedom of gases is based on the law of equipartition of energy.
Consider a single atomic particle that moves freely in a three-dimensional space. When that particle moves, it undergoes a translational motion, and if it moves along the axis, then translational movement. The three coordinate systems (x, y, z) specify the location of that atomic particle.
This article has three degrees of freedom because it can move freely in three directions or three axes. Most of the monatomic particles that exist freely possess three degrees of freedom.
Now, let us consider a diatomic particle, such as O2, N2, etc. These particles can rotate freely around their center of mass. But, in addition, they can also undergo translational movement along the three axes. Thus, rotation about the center of mass gives these particles two additional rotations besides the three translational movements they already possessed. As a result, we can say diatomic particles have three translational movements and two rotational movements.
These two rotational movements are vibration and rotation of a diatomic particle. Hence, to locate the position of a diatomic molecule, we need three translational coordinates and a vibrational and rotational coordinate.
According to the law of equipartition of energy, when a particle or atom is kept under constant thermal conditions, the complete molecule splits in the direction of their degrees of freedom. This means a particle can move freely when subjected to external pressure in their respective degrees of freedom or the directions of their free space.
For example, as soon as the school gets over, students move freely toward their respective homes. Here, their homes are the degrees of freedom.
The law of equipartition of energy deals with the total internal energy of complex molecular systems. It answers questions like why the specific heat of a complex gas increases with an increase in the number of atoms per molecule, etc. The diatomic molecules have higher internal energy and high specific heat content than monatomic gas particles. This is because diatomic particles have five degrees of freedom, i.e., three translational and two rotational, while monatomic particles have only three translational degrees of freedom.
The kinetic energy of an object about x, y and z axes are given by-
½ mvx2, along x-axis
½ mvy2, along y-axis
½ mvz2, along z-axis
According to the kinetic theory of gases, the average kinetic energy of an object, body or molecule is directly proportional to the temperature of that particular object/body/molecule. It can be denoted as-
½ mvrms2 = 3/2 kb T
Here, vrms = root mean square velocity of molecules
kb = Boltzmann constant and
T = Temperature of the gas
The average kinetic energy for each degree of freedom of a monatomic particle with three translational degrees of freedom is given by-
K Ex = ½ kb T
The average energy for a gas under a thermal equilibrium kept at a temperature T is given by-
Eavg = ½ mx2 + ½ my2 + ½ mz2
= ½ k T + ½ k T + ½ k T = 3/2 k T
Here, k is the 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 k T ÷ 3 = ½ k T
A diatomic molecule possesses translational, vibrational and rotational motion. The energy component of a diatomic molecule is given by:
For translational motion = ½ mx2 + ½ my2 + ½ mz2
For rotational motion = ½ (l1 w1) + ½ (l2 w2)
Where,
I1 & I2 are moments of inertia,
w1 & w2 are angular speeds of rotation.
For vibrational motion = ½ m (dy / dt)2 + ½ k y2
Where,
k = force constant of the oscillator,
y = vibrational coordinate.
One must note that vibrational motion possesses both kinetic and potential energies.
Under thermal equilibrium conditions, according to the law of equipartition of energy, the system's total energy is distributed equally among various energy modes present in the system. 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.