 # Theorem of equipartition of energy

The energy equipartition theorem was derived theoretically by Clerk Maxwell using statistical mechanics. It is called a theorem because there is no proof through experimentation. The energy partition means equal distribution of energy. KE = average translational kinetic energy of gas molecules (Joule)

k = Boltzmann’s constant = 1.38 x 10-23 J/K

T = absolute temperature of the ideal gas molecule (Kelvin)

Translational kinetic energy is derived from translational motion which has three velocity components, namely the velocity component on the x-axis, y-axis, and z-axis. There are three components of this speed, in which case there is a number 3 in the equation above. Each component of speed is called the degree of freedom. Because it has three components of speed, translational kinetic energy has 3 degrees of freedom.

The energy equipartition theorem states that the real energy must be evenly divided in all degrees of freedom. Thus, the average energy for each degree of freedom is 1/2 kT.

Monatomic gas molecules

Monatomic gas molecules only make translation motion so that monatomic gas molecules have 3 degrees of freedom.

The average kinetic energy for each monatomic gas molecule is:

3 (1/2 kT) = 3/2 kT = 3/2 nRT.

The heat capacity of monatomic gas molecules:

C = 3/2 R = 3/2 (8.315 J / mol. K) = 12.47 J/Kg. K

Diatomic gas molecules

In addition to translation motion, diatomic gas molecules also perform rotation and vibration. The number of degrees of freedom for translational motion = 3. What is the degree of freedom for rotational motion and vibration? There are three rotation axes, namely the x, y, and Z-axes. Rotational motion on the x-axis does not count because the two atoms coincide with the axis of rotation. When corresponding with the x-axis, the moment of inertia of the two atoms = 0. Thus, the number of degrees of freedom for rotational motion = 2.

The average energy for each diatomic gas molecule is:

3 (1/2 kT) + 2 (1/2 kT) = 5/2 kT = 5/2 n R T.

The heat capacity of diatomic gas molecules:

C = 5/2 R = 5/2 (8.315 J / mol. K) = 20.79 J / Kg. K

The molecular heat capacity obtained theoretically is larger than the heat capacity of the diatomic gas molecules obtained through experiments. When doing vibrational motion, diatomic gas molecules have two types of energy, namely kinetic energy and elastic potential energy. Thus, the number of degrees of freedom for vibrational motion = 2.
The average energy for each diatomic gas molecule is:

3 (1/2 kT) + 2 (1/2 kT) + 2 (1/2 kT) = 7/2 kT = 7/2 n R T.

The heat capacity of diatomic gas molecules:

C = 7/2 R = 7/2 (8,315 J / mol. K) = 29.1 J / Kg. K

The effect of vibrational motion on the value of heat capacity of diatomic gas molecules also depends on the temperature range (T). Experiments that have been done before occur at a temperature range that is not too wide. Recent experiments carried out at a wide temperature range show that the value of the gas molecular heat capacity also depends on the temperature range. To better understand this problem, let’s review the variation of the heat capacity of hydrogen gas molecules at different temperatures. Hydrogen (H₂) includes diatomic gas. The figure on the side shows the variation of the heat capacity of hydrogen gas molecules at different temperatures. The molecular heat capacity value is 5/2 R = 20.79 J/Kg. K is only in the temperature range of around 250 K to 750 K. Below 250 K, the heat capacity of hydrogen gas molecules regularly decreases until they reach 3/2 R = 12.47 J/Kg. K. In contrast, above 750 K, the heat capacity of gas molecules periodically increases until they reach 7/2 R = 29.1 J/Kg. K.