Law of equipartition of energy

The law of equipartition of energy asserts that the total energy of any dynamical system in thermal equilibrium is evenly distributed among the degrees of freedom. “In thermal equilibrium, the entire energy of the molecule is shared evenly among all Degrees of Freedom of motion,” according to the law. For one degree of freedom, the energy associated with a molecule is ½KT, where k is the Boltzmann’s constant and T is the temperature.

Degree of freedom

The normal components (sq. value) of velocity of a gas molecule c are equally distributed in thermal equilibrium, i.e., along three axes X, Y, and Z include u² = v² = w² as per the kinetic theory of gases. The average value of the components of velocities along the X, Y, and Z axes is u, v, and w, respectively.

½ mu2 = ½ mv2 = ½ mw²

But c² = u² + v² + w² and u² = v²= w²

so, ½ mu² = ½ mv² = ½ mw² = (⅓) x  ½ mc²

Again, we know the average kinetic energy of each molecule.

½ mc² = 3/2 KT

Then, ½ mu2 = ½ mv2 = ½ mw2 = ½ x (3/2) KT = ½ KT

As a result, the average energy for each degree of freedom is calculated which is ½ KT.

Again, in the case of an oscillating particle, kinetic energy makes up half of the total energy and potential energy makes up the other half. As a result, the total energy per degree of freedom equals kinetic energy plus potential energy, which equals ½ KT + ½ KT = KT.

As a result, each component of velocity’s translational kinetic energy accounts for one-third of the overall energy.

The entire energy available is distributed equally among the components as various independent energies. Let’s take a closer look at the law before we go into the computations. If a molecule has 1000 units of energy and 5 degrees of freedom (which includes translational, rotational, and vibrational motions), each motion is given 200 units of energy.

Molecules aren’t geometrical points, although they do have an indefinitely small size. Because molecules have a moment of inertia and mass, rotating motion occurs in addition to translational motion. The sizes of the molecules are not rigid, and as a result of collisions with other molecules, it is no surprise to expect oscillation in them. As a result, they may have a greater degree of independence.

When the energy associated with a degree of freedom becomes a function of an assigned two-dimensional variable of degrees of freedom, the associated average value of energy equals ½ KT, according to Maxwell statistics. Boltzmann’s If total energy is distributed evenly across all degrees of freedom, total energy = f x ½ KT = f/2 KT for a molecule with degrees of freedom. Instead, we’ll look at a less-than-rigorous but plausible derivation based on elementary mechanical models. It doesn’t use the distribution function and indicates how long it takes to reach thermal equilibrium on a rough scale.

Law of equipartition of energy formula:

According to the law of equipartition of energy, the total energy of any dynamic system in thermal equilibrium is evenly distributed among the degrees of freedom.

A single molecule’s kinetic energy along the x, y, and z axes is given as

Along with the x-axis → ½mvx²

Along with the y-axis → ½mvy²

Along with the z-axis→ ½mvz²

The average kinetic energy of a molecule, according to the kinetic theory of gases, is given by

½ mvrms ²=(3/2)Kb T

where vrms is the root-mean-square velocity of the molecules, Kb is the Boltzmann constant, and T is the temperature of the gas

Because a monatomic gas has three degrees of freedom, the average kinetic energy per degree of freedom is

KEx= ½KbT

If a molecule is free to move in space, it needs three coordinates to define its location, meaning that it has three degrees of freedom in translation. It has two translational degrees of freedom if it is confined to travel in a plane, and one translational degree of freedom if it is constrained to move in a straight line. A triatomic molecule has six degrees of freedom. The kinetic energy of the gas per molecule is calculated as follows:

6×N×½ KbT = 3×(R/N)NKbT = 3RT

State the principle of equipartition of energy:

The principle of equipartition of energy describes the total internal energy of complex molecular systems. It explains why, as the number of atoms per molecule grows, the specific heat of complicated gases increases. Monatomic gas molecules have lower internal energy and a lower molar specific heat content.

Conclusion

The entire energy of the system is distributed equally among the many energy modes present in the system under thermal equilibrium circumstances, according to the law of equipartition of energy. The vibrational motion contributes ½kT of energy to the total energy of the motion, whereas the translational and rotational motions each contribute (½)kT of energy to the total energy of the motion.