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.

Energy equipartition theory 1

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 which cause 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.

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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?

Energy equipartition theory 2There are three rotation axes, namely the x, y, and Z-axes. Rotational motion on the x-axis does not counts 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

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The molecular heat capacity obtained theoretically is larger than the heat capacity of the diatomic gas molecules obtained through experiments.

Energy equipartition theory 3When 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.

Energy equipartition theory 4Hydrogen (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.

Based on this fact, we can say that at low temperatures, gas molecules only make a translation motion. After the temperature rises, new gas molecules make a rotational movement. At high temperatures, gas molecules collide with each other so that the atoms are vibrating. So these three types of motion are carried out in stages, first only translational motion (low temperature), after that translation + rotation (medium temperature) and the last translational + rotation + vibration (high temperature). Vibration motion only occurs if gas molecules collide with each other.

Cases like this not only happen to hydrogen gas but other gases too. From experiments conducted by scientists, the heat capacity of gas molecules also tends to change with temperature. Changes that occur are similar to those experienced by hydrogen gas, but because the structure in each gas is different (the number and type of atoms are different), the change in heat capacity also occurs at different temperature ranges.

The energy equipartition theorem states that total energy must be divided equally for each degree of freedom. The additional energy obtained by gas molecules is not evenly distributed for each degree of freedom but is divided gradually. Also, the equation of the molecular heat capacity of the gas that we have theoretically derived based on the kinetic theory of gas states that the heat capacity of the molecule depends only on R (1/2 R for each degree of freedom). The molecular heat capacity is also affected by temperature (T).

Can be concluded, first, the equipartition theorem is derived from classical statistical mechanics, which is based on the laws of Newtonian mechanics. Second, the kinetic theory of gas that we use in explaining the movements of gas molecules is also based on the laws of Newtonian mechanics. Because the energy equipartition theorem and the kinetic theory of gases have been violated, it can be concluded that Newton’s laws of mechanics are unable to explain the movements that occur at the atomic or molecular level. In other words, Newtonian mechanics or classical mechanics can only describe the movement of large matter.

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