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Question: According to Maxwell-Boltzmann distribution law of molecular velocities of gases, the average transl...

According to Maxwell-Boltzmann distribution law of molecular velocities of gases, the average translational kinetic energy is:
(A) KT2\dfrac{KT}{2} per molecule
(B) KTKT per molecule
(C) RTRT per molecule
(D) 32KT\dfrac{3}{2}KT per molecule

Explanation

Solution

Recollect the concept of kinetic theory of gases and its postulates. Write the law of Maxwell-Boltzmann distribution for molecular velocities of gases. Take a look at all the options and then tick the correct option for translational kinetic energy of one molecule of gas.

Complete step by step solution:
- Kinetic energy is given as, K.E. = 12mv2\dfrac{1}{2}m{{v}^{2}} where m is the mass and v is the velocity.
- According to kinetic theory of gases, the kinetic energy of a gas is, K.E. = 12mvrms2\dfrac{1}{2}mv_{rms}^{2} where m is the mass of a molecule of gas and vrms{{v}_{rms}} is the root mean square velocity of the gas which is equal to 3KTm\sqrt{\dfrac{3KT}{m}} where K is the Boltzmann constant, m is the mass of a molecule of gas and T is the absolute temperature.
- If we substitute the value of root mean square velocity of a gas in kinetic energy equation we get, K.E.=12mvrms2=12m×3KTm=32KTK.E.=\dfrac{1}{2}mv_{rms}^{2}=\dfrac{1}{2}m\times \dfrac{3KT}{m}=\dfrac{3}{2}KT.
- According to Maxwell-Boltzmann distribution law of molecular velocities of gases, at a given temperature, all the gas molecules irrespective of their masses, have the same average translational kinetic energy.
- Therefore, the average translational kinetic energy for a molecule is equal to 32KT\dfrac{3}{2}KT where K is the Boltzmann constant and T is the absolute temperature.

Therefore, the correct answer is option (D).

Note: Remember according to Maxwell-Boltzmann distribution law of molecular velocities of gases, at a given temperature, all the gas molecules irrespective of their masses, have the same average translational kinetic energy which is equal to 32KT\dfrac{3}{2}KT per molecule.