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Kinetic Theory of Gases

Properties of gases have been studied in detail by many scientists. There are many laws governing each property and also the relationship between these properties. Similarly, the velocity and energy distribution of gases have been studied from the postulates of Kinetic molecular theory of gases.

A more precise distribution was obtained by Maxwell and Boltzmann. The distribution curve obtained by them proved the assumptions of the kinetic theory correctly.

We will study the postulates, the observations, etc made by these scientists on velocity and energy distribution of gases.

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Energy Distribution

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Maxwell and Boltzmann also derived an energy distribution which is based on the same assumptions as velocity distribution,
$\frac{dn}{dE}=\frac{2N E^{\frac{1}{2}}}{\pi ^{\frac{1}{2}}(KT)^{\frac{3}{2}}} e^{\frac{-E}{(KT)}}$

From this expression, the average energy can be calculated as :

and the most probable energy as :

Neither the energy distribution nor the average energy is a function of molecular mass and each is only a function of temperature.

The collision frequency increases with increase in (a) temperature (b) molecular size (c) the number of molecules in unit volume of the gas.

Mean Free Path

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  1. A molecule moves along a straight line with a constant speed before colliding with other molecules.
  2. The distance traversed by a molecule between two successive collisions is referred to as the free path.
  3. Free path varies from time to time. The average distance traveled by a molecule before colliding with other molecules is known as mean free path λ.
  4. This depends on molecular size and concentration of the gas molecules. Larger the gas molecules are in size, more frequent the collisions are and consequently, shorter will be the mean free path.
  5. More the number of molecules, larger will be the number of collisions and hence shorter will be the mean free path.

Distribution Curve

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There were many assumptions made by the kinetic molecular theory on the movement of gases. In order to understand the movement of molecular speeds, a precise distribution was obtained by Maxwell and Boltzmann.
In the case of gases, Maxwell and Boltzmann independently predicted, theoretically, the shape of the distribution curve of molecular speeds in a sample of gas at a given temperature. This is called as Maxwell-Boltzmann distribution law.

Maxwell-Boltzmann Distribution Law

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Maxwell-Boltzmann distribution law assumes that the molecular collisions are completely random and involve all possible values for the exchange of kinetic energy. It is only the total kinetic energy of a gas, but not the individual kinetic energies of the molecules, that stays constant at fixed temperature.

The distribution curve, obtained with the molecular speeds of the gas at a varying temperature, from 500K to 1500K, shows that the molecular motion is high at higher temperature.
Say, the speed of the maximum molecules were at 2 x 104 cm/s at 500K, while at 1500K the speed was, 4 x 104 cm/s. The distribution curve tells us what fraction of the molecules have a particular speed.

By looking at it, it was noted that very few molecules had either very high speed or low speed. Most molecules were at an intermediate speed. The kinetic energy of a molecule is 1/2 mv2, hence the speed distribution can be converted into an energy distribution. The assumptions of the kinetic theory made above, satisfactorily account for the observed behavior of gases.

Velocity and Energy Distribution Laws

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Boyle's Law

P1V1 = P2V2

We have assumed that a gas exerts pressure as a result of collision of the molecules on the walls of the vessel.

The magnitude of the pressure would, therefore, depend on the frequency of collisions which in turn depends on the number of molecules and their average speed. When the amount of gas and temperature are constant, the number of molecules and the average speed remains constant. If the volume is reduced, the molecules do not have much space to move and hence, collide with the walls more frequently. As a result, the walls receive more impacts and the observed pressure would, therefore, be greater in a small volume. If, on the other hand, volume is increased, the number of impacts on the walls decreases and hence, the pressure decreases.

Charles Law

V1/T1 = V2/T2

When a sample of gas in a closed container is heated, the molecules receive energy, move faster and hit the walls of the container harder and more frequently, and the pressure of the gas, therefore, increases. Pressure is a result of two factors: number of collisions and force of collisions.

If the pressure is not allowed to increase, any increase in temperature will cause, in accordance to Charles' law, an expansion of gas. In this case, the increased force of collisions is compensated by the decreased number of collisions per unit area resulting from the large volume, so, the pressure remains unchanged.

Dalton's Law

In a gaseous mixture, the molecules of each gas behave independently because of the absence of attractive forces. Each molecule strikes the walls the same number of times per second and with the same force as if no other molecules are present, provided the temperature and pressure are constant.
The partial pressure of the gas depends only on the number of its molecules striking the walls and is unaffected by the presence of molecules of other gases.
The total pressure exerted is due to the impact of molecules of all gases present in the mixture.

Graham's Law

This law follows from the assumption that the average kinetic energy of molecules is fixed for a given temperature. This means at the same temperature, the average kinetic energy of molecules of one gas is equal to the average kinetic energy of molecules of another gas.

If m1 and m2 are the masses of the two gases and u1 and u2 are the average speeds, then,



If we assume that the rates of diffusion are directly proportional to the average speeds of their molecules, we may write this as:

$\frac{Rate \; of \; diffusion \; of \; Gas1}{Rate\;  of \; diffusion\; of \; Gas2}\;=\;\frac{r_{1}}{r_{2}}\;=\;\frac{r_{1}}{r_{2}}\;=\;\left [ \frac{m_{2}}{m_{1}} \right ]^{\frac{1}{2}}$
This is Grahams law of diffusion.
More topics in The Kinetic Theory of Gases
Collision Theory
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