Law of Equipartition of Energy

# Law of Equipartition of Energy

## Introduction:

The law of energy equipment states that the total energy of a dynamical system in thermal equilibrium is shared equally by all degrees of freedom. Each degree of freedom per molecule has an energy associated with it of 1/2kT, where k is Boltzmann’s constant. For example, each molecule in a monatomic molecule has three degrees of freedom. The mean kinetic energy of a molecule, according to gas kinetic theory, is 3/2kT.

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The equipartition theorem connects a system’s temperature to its average energies. The original concept of equipartition was that in thermal equilibrium, energy is distributed equally among all of its forms; for example, the average kinetic energy per degree of freedom in a molecule’s translational motion should equal that of its rotational motions.

## Overview:

For any dynamic system in thermal equilibrium, the law of energy equipartition is defined as the equal distribution of energy among the degrees of freedom. Equipartition of energy is a statistical mechanics law that states that in a system in thermal equilibrium, an equal amount of energy will be associated with each degree of freedom on average. (A particle moving through space has three degrees of freedom because its position requires three coordinates.) This law, based on the work of Scottish physicist James Clerk Maxwell and German physicist Ludwig Boltzmann, states specifically that a system of particles in equilibrium at absolute temperature T will have an average energy of 1/2 k T associated with each degree of freedom, where k is the Boltzmann constant. Furthermore, any degree of freedom contributing potential energy will be associated with another 1/2 k T.

Quantitative predictions are made by the equipartition theorem. It, like the virial theorem, gives the total average kinetic and potential energies of a system at a given temperature, from which the heat capacity of the system can be calculated. Equipartition, on the other hand, gives the average values of individual components of energy, such as the kinetic energy of a specific particle or the potential energy of a single spring. In thermal equilibrium, for example, it predicts that every atom in a monatomic ideal gas has an average kinetic energy of, 3/2 K B T where KB is the Boltzmann constant and T is the (thermodynamic) temperature.

### Principle of equipartition of energy:

The Principle of Energy Equipartition is significant because it explains why gas systems composed of more complex molecules have more internal energy than less complex systems. It also explains why the measured molar specific heats for more complex gases increase as the number of atoms per molecule increases. Diatomic ideal gases, in other words, have higher internal energy and molar specific heat values than monatomic ideal gas molecules. As a result, more complex molecules can have a large amount of energy. The number of independent ways molecules can possess energy is defined as degrees of freedom. A monatomic gas molecule has three degrees of freedom, whereas a diatomic molecule has five degrees of freedom. The principle of energy equipartition states that energy is distributed equally among the various degrees of freedom, and each degree of freedom of a molecule has an average energy of 0.5kT, where k is the Boltzmann constant and T is the temperature in Kelvins.

The equipartition theorem is useful in the case of ideal gases. Equipartition is used to derive Newtonian mechanics’ classical ideal gas law. The equipartition theorem makes it simple to derive the corresponding laws for an extreme relativistic System. The equipartition theorem and its related virial theorem have long been used in astrophysics as a tool. The equipartition theorem can be used to calculate a particle’s Brownian motion from the Langevin equation. The same equations can be used to calculate the conditions for star formation in giant molecular clouds.

### Law of equipartition of energy formula:

The total energy for any dynamic system in thermal equilibrium is equally divided among the degrees of freedom, according to the law of energy equipartition.

The kinetic energy of a single molecule along the x, y, and z axes is given as

Along x-axis → ½m v x²

Along y-axis → ½m v y²

Along z-axis→ ½m v z²

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

½ m v r ms 1²=(3/2) K b T

where v rms denotes the root-mean-square velocity of the molecules, Kb denotes the Boltzmann constant, and T denotes the gas temperature.

Because there are three degrees of freedom in a monatomic gas, the average kinetic energy per degree of freedom is given by

K E x=½ K b T

If a molecule is free to move in space, it requires three coordinates to specify its location, implying that it has three translational degrees of freedom. Similarly, if it is constrained to move in a plane, it has two translational degrees of freedom, and if it is constrained to move in a straight line, it has one translational degree of freedom. The degree of freedom for a triatomic molecule is 6. And the kinetic energy of the gas per molecule is given as,

6×N×½ K b T=3×(R/N)N K b T=3 RT

### State the principle of equipartition of energy:

The total internal energy of complex molecular systems is described by the law of energy equipment. It helps to explain why the specific heat of complex gases increases as the number of atoms per molecule increases. When compared to monatomic gas molecules, diatomic gas molecules have higher internal energy and a higher molar specific heat content. This is due to the fact that the diatomic gas molecule has five degrees of freedom, whereas the monatomic gas molecule only has three degrees of freedom.

Question 1: What does it mean to have three degrees of freedom?

Answer: There are six degrees of freedom in total, three of which correspond to rotational movement and three of which correspond to translational movement. Pitch, yaw, and roll are the rotational degrees of freedom along the x, y, and z axes. The translational degrees of freedom along the x, y, and z axes, on the other hand, can be moved forward or backward, up or down, and left or right.

Question 2: What is the value of energy according to the law of energy equipartition?

Answer: In thermal equilibrium, the total energy has an average energy of ½ K b T.

Question 3: In the case of a diatomic gas, how many rotational degrees of freedom are there?

Answer: The total number of rotational degrees of freedom in the case of diatomic gases, which include nitrogen and oxygen, is two. Because the structure is 2-D, they only have two independent axes of rotation, so the third rotation is negligible. As a result, only two rotational degrees are considered, and each rotational degree contributes a term to the total energy, which is made up of rotational energy and transnational energy t.

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