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Law of Equipartition of Energy

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Last updated date: 23rd May 2024
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What is the Law of Equipartition of Energy?



The Law of Equipartition of Energy states that the total energy in thermal equilibrium for any dynamic system gets divided equally among the degrees of freedom.


The kinetic energy along the x-axis, the y-axis, and the z-axis for a single molecule is given by-


Along x-axis  \[\frac {1} {2}\]  mvx2


Along y-axis   \[\frac {1} {2}\]  mvy2


Along z-axis   \[\frac {1} {2}\]  mvz2


In the case of thermal equilibrium the average kinetic energy of the gas is given by-


Along x-axis ( \[\frac {1} {2}\]  mvx2)


Along y-axis ( \[\frac {1} {2}\]  mvy2)


Along z-axis ( \[\frac {1} {2}\]  mvz2)


Now, the average kinetic energy of a molecule according to the kinetic theory of gases is represented as-


\[\frac {1} {2}\]mvrms2=32KbT


where the root-mean-square velocity is represented by vrms


&


the Boltzmann constant is represented by Kb


&


T is temperature

Degree of Freedom

There are three degrees of freedom in the case of the monoatomic gas. Thus, the average kinetic energy per degree of freedom is represented as-


KEx= \[\frac {1} {2}\] KbT


A molecule possesses three translational degrees of freedom, which is free to move in space and hence needs three coordinates in order to specify its location. The molecule possesses two degrees of freedom if it is constrained to move in a plane and if it is in a straight line, there is one translational degree of freedom. On the contrary in the case of a molecule that is triatomic, the translational degree of freedom is 6 and in this case, the kinetic energy of the per molecule is given by-


6 x N  x \[\frac {1} {2}\] x KbT= 3 x  \[\frac {R} {N}\] x NKbT=3 RT


The translational degree of freedom in the case of the molecules of mono-atomic gases such as helium and argon is one. Then, the kinetic energy per molecule is given by-


3 x N x  \[\frac {1} {2}\] x KbT=3 x \[\frac {R} {N}\] x NKbT= \[\frac {3} {2}\]   RT


The translational degree of freedom in the case of diatomic gases such as oxygen and nitrogen is 3.

FAQs on Law of Equipartition of Energy

1. What is the Principle of Equipartition of Energy.

The law of equipartition of energy describes the total internal energy of complex molecular systems. It helps to explain the concept, why the specific heat of complex gases increases with an increase in the number of atoms per molecule.

 

The diatomic gas molecules have high internal energy and high molar specific heat content as compared to the monatomic gas molecules. This is because the diatomic gas molecule has five degrees of freedom while the monatomic gas molecule has only three degrees of freedom.

2. What is the Application of Law of Equipartition Energy in Specific Heat of a Gas.

 One of the major applications of the law of equipartition of energy is in Meyer’s relation (empirical relation between the size of a hardness test indentation & the load needed to leave the indentation)

This shows:

Cp − Cv = R,


Where, 


Cp is the molar specific heat capacity of an ideal gas at constant pressure, and


Cv is the molar specific heat at constant volume,


R is the gas constant.


This equation connects the two specific heats of an ideal gas of one mole to the ideal gas.


The law of Equipartition of energy is also used to calculate the value of CP − CV, and also to calculate the ratio between them, which is given by,


γ = CP / CV.

Where

γ = adiabatic exponent of the gas molecule.


Let’s take the example of a monatomic molecule.


Average kinetic energy of a molecule

=[3/2kT]


Total energy of a mole of gas

=3/2kT*NA=3/2RT


For one mole, the molar specific heat at constant volume


Cv=dU/dT=d/dT[3/2RT]


Cv=[3/2R]


Cp=Cv+R=3/2R+R=5/2R


The ratio of specific heats,


γ= Cp/Cv=(5/2R)/(3/2R)=5/3=1.67

3. What is meant by the three degrees of freedom?

In total there are six degrees of freedom in which three degrees of freedom correspond to the rotational movement while the other three correspond to the translational movement. The rotational degrees of freedom along the x,y, and z axes are referred to as pitch, yaw, and roll. On the other hand, the translational degrees of freedom along the x, y, and z axes can be moved forward or backwards, up or down, and left or right.

4. What is meant by translational motion?

The motion in which there is shifting of a body from one point to another in space is referred to as the translational motion of the body. The motion of all the points of the body which is moving is uniform in the same line or direction. Examples of translational motion include a train moving in the train track, a man walking on a road, flying of the birds in the sky, and so on. The orientation relative to the fixed point does not change if translational motion is being executed in a body.

5. What is meant by rotational motion?

When a body is rotating about an axis that is fixed, it is said to have a rotational motion in it. When a rigid body is undergoing rotational motion it means all the particles of the rigid body undergo circular motion about a common axis. Some of the examples of rotational motion include the motion of the wheel, rotation of the earth, a spinning top, and so on. To learn more about the system of particles and the rotational motion, students can visit Vedantu for better learning and understanding of the concept.

6. What is the number of rotational degrees of freedom in the case of a diatomic gas?

In the case of diatomic gases, which include nitrogen and oxygen, the total number of rotational degrees of freedom is two. This is because they have only two axes of rotation which are independent as the structure is 2-D and hence the third rotation is negligible. Therefore, only two rotational degrees are considered where each rotational degree contributes a term to the total energy which consists of rotational emergy & transnational energy t.

7. What is meant by the Maxwell-Boltzmann theory?

The distribution of various speeds for gas at a certain temperature is represented by an equation known as the Maxwell-Boltzmann equation which forms the basis of the kinetic theory of gases. This equation states that in a particular energy state the probability of finding the molecule varies exponentially with the energy that is divided by Kb. With the help of this equation the most probable speed, the distribution function, the average speed, along the root-mean-square speed can be derived. In order to learn the derivation of the Maxwell-Boltzmann equation students can visit Vedantu.

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