What is mean energy in Kinetic theory of gases

Gas molecules are far from each other and the interaction force between them is negligible. A given volume of gas consists of a large number of molecules, to the order of Avogadro’s number. These molecules are in constant random motion. The average separation between the molecules is greater than the average size of a molecule. Hence, the interaction force between the molecules is neglected except during the collision when the molecules collide with each other due to their random motion. This type of interaction between the molecules during the collision is considered an elastic type of collision.

The Kinetic Theory of gases is based on the assumption that the molecules move freely and are in random motion, with the molecular force of interaction being negligible and hence can be neglected. The Kinetic Theory of gas explains the properties of gases at the molecular level. In this theory, the physical properties of the gas are explained by considering the average over a large number of particles (molecules), and the averages are related to the macroscopic properties of the gas.

Maxwell and Boltzmann are the pioneers who developed the Kinetic Theory of gases.

Properties of Gas

·    Real gas v/s Ideal gas

In ideal gas, interatomic interaction is absent. It obeys the ideal gas equation at all temperatures and pressure. In real gas, atoms or molecules might interact with each other. Hence real gases are not ideal as defined. The Kinetic Theory of gas is generalised for ideal gas, which can be applied to real gas in an idealised state.

·    Pressure of an Ideal Gas – In Kinetic Theory of Gas

Consider n moles of ideal gas enclosed in volume V. The system is kept at constant temperature T. The gas molecules constantly move in random motion. During the motion, the molecules undergo collision. The collision of molecules is regarded as an elastic collision. As per the Kinetic Theory of gas, the interaction force between the molecules can be neglected, and hence we consider the elastic collision of the gas molecules with the walls of the container.

Let vx, vy, vz be the velocity of the gas in the x, y, and z directions, respectively. For elastic collision, the total momentum is conserved. Hence velocity before the collision is equal to the velocity after the collision.

Let the molecule collide with the wall of the container in the yz plane. After the collision, vx changes the sign while vy and  vz remain the same. Hence momentum change is only along the x-direction.

The change in momentum = final momentum – initial momentum

         = (-m vx) – (m vx) = -2 m vx

Therefore the momentum imparted on the wall = 2 m vx

Next to calculate force exerted by the molecule on the wall we need to calculate momentum per unit time.

Let L be the length of the box. Total distance travelled by the molecule = 2 L

Total time taken during the collision is t = 2 L / vx

∴ Force = momentum / time = 2 m vx/ (2 L /vx) = m vx2/ L

Pressure = Force / Area = m vx2 / L x L2= m vx2 /  V

For N molecules Total pressure is obtained by taking an average.

∴ P = N m <vx2> / V

Also, <vx2>  =  <vy2>  =  <vz2>  =  v2

∴ v2 = <vx2> + <vy2> + <vz2>

∴3 <vx2> = v2

∴<vx2> = v2/ 3

Hence Pressure (P) = N m <vx2>/ V = N m v2/ 3V

∴ P =N m v2/ 3V

The average translational kinetic energy of gas molecule = E =mv2/2

For N molecules, E = N mv2/2

∴ P = (2E) / 3

∴ PV = V (2E) / 3

From Ideal gas equation, P V= n R T

∴ V (2E) / 3 = n R T

∴ E = (3/2) N KB T

∴ E / N = (3/2) KB T

This is the expression for the mean energy of a gas molecule in the kinetic theory of gas

Law of equipartition of energy (Kinetic Theory of Gases)

·    MEAN ENERGY BY KINETIC THEORY OF GASES

According to the kinetic theory of gas, the average kinetic energy for a single molecule is proportional to the temperature of the gas.

Thus according to the kinetic theory of gas, the expression for the mean energy per molecule of the gas for each degree of freedom is given as follows-

         <E> = (1/2) KB T

           Therefore for an ideal gas in 3D (degree of freedom =3)

           Mean Kinetic Energy per molecule = (3/2) KB T

Examples

1.     Monoatomic Gases –

Molecules of monoatomic gas have a degree of freedom equal to 3.

          ∴<E> = (1/2) KB T x 3

2.     Diatomic Gas –

Molecules of diatomic gas have a degree of freedom equal to 5.

          ∴ <E> = (1/2) KB T x 5

Conclusion

The Kinetic Theory of gases is based on the assumption that the molecules move freely and are in random motion, with the molecular force of interaction being negligible and hence can be neglected.

Thus as per the Kinetic theory of gas, the mean energy per molecule of gas is given as-

∴ <E> = (1/2) KB T

This energy is for the degree of freedom.

For the degree of freedom – f,

<E> = (1/2) KB T x f