Kinetic Molecular Theory of Gas: Boyle’s Law, Dalton’s Law of Partial Pressure

Kinetic Molecular Theory of Gas: We have learned about different laws applied to ideal gases. Like any laws, these ideal gas laws are merely summaries of observations, such as how pressure and volume are inversely proportional or the way volume and temperature are directly proportional. We observe these relationships as always true, but it does not explain just like any other law. Why? For that, we are going to need something more powerful; we need a theory. And the theory that explains the behaviour of gases is called a Kinetic Molecular Theory.

What is Kinetic Molecular Theory of Gas?

In all three states of matter, the molecules exhibit some motion. In solid-state only vibratory motion, in the liquid state both vibratory and rotatory. All three types, namely vibratory, rotated, and translatory motion, are conspicuous in the gaseous state. By these different types of motion, the molecules acquire some energy. Such energy is called the energy of motion or kinetic energy. Thus, the theory that explains the behaviour of gases is known as the kinetic theory of gases.

This theory was put forth by Bernoulli in \(1728\) and later on developed by Clausius, Maxwell Boltzmann and Waterson. This theory is applicable to perfect or an ideal gas. Moreover, the theory gives a microscopic picture of the gaseous state.

Postulates of Kinetic Gas Theory

The following are the postulates of the Kinetic gas theory:

1. Gases are made up of many small tiny, and discrete particles called molecules.
2. Molecules of gas are well separated from each other. The volume occupied by molecules of a gas is negligible compared to the volume of gas. The mean path is very high compared to the diameter of the molecule.
3. Molecules of gas are electrically neutral, and they do not have attractions and repulsion between them.
4. Gas molecules move rapidly and randomly in all directions with high velocities.
5. The motion of gas molecules is not affected by the gravitational force.
6. Molecules of gas may collide with each other and also with the inner walls of the container.
7. The pressure exerted by a gas is due to collisions of molecules made on the vessel’s inner walls. Pressure \({\rm{\alpha }}\) is the number of collisions on the inner walls.
8. Collisions among gas molecules are perfectly elastic. It means that there is no change in the total kinetic energy as well as in their momentum.
9. There can be a transfer of energy among the molecules during a collision, but the average kinetic energy of a gas is constant.
10. Different gas molecules may have different velocities, but average kinetic energy is directly proportional to the gas’s absolute temperature. The distribution of molecular speeds remains constant at a particular temperature.
    Kinetic energy \({\rm{\alpha }}\) absolute temperature.

Gases which follow the postulates of the Kinetic molecular theory are ideal gases . Most of the gases do not behave as ideal gases. Such gases are known as real gases. Calculations and predictions based on the kinetic theory of gases agree well with the experimental observations.

Explanation of Postulates

The diameter of the molecule is approximately \({10^{ – 8}}{\rm{cm}}.\) It is very small compared to the distance between molecules (intermolecular distance) at room temperature and pressure. Hence, the volume of molecules is negligible as compared to the total volume of the gas.

Despite greater molecular velocities of nearly \(400\,{\rm{m}}{\rm{.}}{{\rm{s}}^{ – 1}}\) a gas slowly spreads due to random molecular motion as well as random molecular collisions. It is illustrated in the figure.

When molecules collide, energy may be transferred from one molecule to another, but total kinetic energy remains constant. Thus, gas laws like Boyle’s Law, Charle’s Law can be explained based on molecular collisions.

Explanation of Gas Laws

Boyle’s law: According to kinetic molecular theory, the pressure exerted by a gas is due to the molecular collision on the walls of the container. As the number of collisions increases, the gas pressure also increases because the average velocity of molecules is constant at a given temperature. When the volume of a gas is decreased, the space available for the movement of molecules decreases. Then the number of collisions on the walls of the container increases, increasing the pressure of the gas. It means as the volume of a gas is decreased, pressure is increased at a given temperature. This is Boyle’s law.

Deduction from Kinetic Gas Equation.

According to the kinetic gas equation,

\({\rm{PV = }}\frac{{\rm{1}}}{{\rm{3}}}{\rm{mn}}{{\rm{c}}^{\rm{2}}}{\rm{ = }}\frac{{\rm{2}}}{{\rm{3}}}{\rm{.}}\frac{{\rm{1}}}{{\rm{2}}}{\rm{mn}}{{\rm{c}}^{\rm{2}}}{\rm{ = }}\frac{{\rm{2}}}{{\rm{3}}}{\rm{.}}\frac{{\rm{1}}}{{\rm{2}}}{\rm{M}}{{\rm{c}}^{\rm{2}}}{\rm{(Where}}\,{\rm{m \times n}}\,{\rm{ = }}\,{\rm{M}}\,{\rm{is}}\,{\rm{the}}\,{\rm{total}}\,{\rm{mass}}\,{\rm{of}}\,{\rm{the}}\,{\rm{gas)}}\)

But \(\frac{1}{2}{\rm{M}}{{\rm{c}}^{\rm{2}}}{\rm{ = }}{\mkern 1mu}\) Kinetic energy of the gas \(\therefore {\mkern 1mu} {\rm{PV = }}\frac{{\rm{2}}}{{\rm{3}}}{\rm{K}}.{\rm{E}}\,\,\,\,\,\,…\left( 1 \right)\)

Further, according to one of the postulates of the kinetic theory of gases,

\({\rm{K}}{\rm{.E}}{\rm{.\alpha }}\) Absolute Temperature \(({\rm{T}})\) or \({\rm{K}}.{\rm{E}}{\mkern 1mu} {\rm{ = }}{\mkern 1mu} {\rm{kT}}\,\,\,\,\,\,\,\,…\left( 2 \right)\)

where \({\rm{k}}\) is a constant of proportionality.

Putting this value in equation \(\left( 1 \right),\) we get \({\rm{PV = }}\frac{{\rm{2}}}{{\rm{3}}}{\rm{kT}}\)

As \(\frac{{\rm{2}}}{{\rm{3}}}\) is a constant quantity, \({\rm{k}}\) is also a constant, therefore, if \({\rm{T}}\) is kept constant, \(\frac{2}{3}{\rm{kT}}\) will be constant. Hence, \({\rm{PV}}\,{\rm{ = }}\) constant if \({\rm{T}}\) is kept constant, which is Boyle’s law.

Charle’s Law: Considered a given amount of gas in a cylinder fitted with a frictionless piston. When it is heated, according to kinetic molecular theory, the average kinetic energy of molecules increases, then the molecular velocities increase, thereby increasing the number of collisions on the walls of the container as well as the momentum of each molecule. As a result, the pressure exerted by the gas increases, pushing the piston outward. The volume of the gas is then increased. Hence, at constant pressure, when the temperature of a gas is increased, its volume is increased. This is Charle’s Law.

Deduction from Kinetic Gas Equation: As deduced in equation \(\left( 2 \right)\) above from the kinetic gas equation, we have \({\rm{PV}}\,{\rm{ = }}\,\frac{2}{3}\,{\rm{kT}}\)

This may be rewritten as \(\frac{{\rm{V}}}{{\rm{T}}}{\rm{ = }}\frac{{\rm{2}}}{{\rm{3}}}\frac{{\rm{k}}}{{\rm{P}}}\)

\(\frac{{\rm{2}}}{{\rm{3}}}\) is constant, \({\rm{k}}\) is also constant, hence if \({\rm{P}}\) is kept constant, \(\frac{{\rm{V}}}{{\rm{T}}} = \) constant, which is Charles’ law.

Dalton’s law of partial pressure: According to Kinetic molecular theory, the volumes of the molecules is negligible compared to the total volume of the gas. When two gases are mixed, the molecules of the two gases do not affect each other. Then, there is no change in the number of collisions of each gas on the vessel’s walls. Hence, there should be no change in the individual pressure of the two gases in the vessel. Therefore, the total pressure of the mixture should be a sum of the partial pressures of the two gases. This is Dalton’s Law of partial pressures.

Deduction from Kinetic Gas Equation: Let us consider only two gases. According to the kinetic gas equation,

\({\rm{PV = }}\frac{{\rm{1}}}{{\rm{3}}}{\rm{mn}}{{\rm{c}}^{\rm{2}}}\)

Now, if only the first gas is enclosed in the vessel of volume V, the pressure exerted would be:

\({{\rm{P}}_1}{\rm{ = }}\frac{{\rm{1}}}{{\rm{3}}}\frac{{{{\rm{m}}_{\rm{1}}}{{\rm{n}}_{\rm{1}}}{{\rm{c}}^{\rm{2}}}_{\rm{1}}}}{{\rm{V}}}\)

Again, if only the second gas is enclosed in the same vessel (so that V is constant), then the pressure exerted would be:

\({{\rm{P}}_2} = \frac{1}{3}\frac{{{{\rm{m}}_{\rm{2}}}{{\rm{n}}_{\rm{2}}}{{\rm{c}}_{\rm{2}}}^{\rm{2}}}}{{\rm{V}}}\)

Lastly, if both the gases are enclosed together in the same vessel, the gases do not react when their molecules behave independently. Hence, the total pressure exerted would be:

\({\rm{P = }}\frac{{\rm{1}}}{{\rm{3}}}\frac{{{{\rm{m}}_{\rm{1}}}{{\rm{n}}_{\rm{1}}}{{\rm{c}}^{\rm{2}}}_{\rm{1}}}}{{\rm{V}}}{\rm{ + }}\frac{1}{3}\frac{{{{\rm{m}}_{\rm{2}}}{{\rm{n}}_{\rm{2}}}{{\rm{c}}_{\rm{2}}}^{\rm{2}}}}{{\rm{V}}} = {{\rm{P}}_{\rm{1}}}{\rm{ + }}{{\rm{P}}_{\rm{2}}}\)

Similarly, if more than two gases are present, then it can be proved that \({\rm{P}} = {{\rm{P}}_{\rm{1}}}{\rm{ + }}{{\rm{P}}_{\rm{2}}} + {{\rm{P}}_3} + ……\)

Kinetic Gas Equation

Based on the postulates of kinetic molecular theory of gases and root mean square velocity \(‘{\rm{C}}’\) the Kinetic gas equations, \({\rm{PV = }}\frac{{\rm{1}}}{{\rm{3}}}{\rm{mn}}{{\rm{C}}^2}\) is derived, where \({\rm{P}}\) is pressure, \({\rm{V}}\) is volume, \({\rm{m}}\) is mass of gas molecule, \({\rm{n}}\) is number of molecules and \({\rm{C}}\) is \({\rm{RMS}}\) velocity of gas.

Significance of RMS Velocity in Kinetic Gas Equation

Velocity is a vector quantity. It is represented by a positive sign in any direction and a negative sign in the opposite direction. As the gas molecules move in all directions, sometimes the average velocity of the molecules may become negative or zero. To avoid this kind of an anomaly, the velocities of molecules are squared, which will always be positive. The square root of the average squares of velocities thus gives the molecule’s velocity called RMS velocity. Hence, it is used in the kinetic gas equation.

RMS Speed

The root-mean-square speed denoted by \({{\rm{V}}_{{\rm{rms}}}}\) is defined as the average velocity of the molecules in a gas.

\({{\rm{V}}_{{\rm{rms}}}} = \sqrt {\frac{{{\rm{3RT}}}}{{\rm{M}}}} \)

Where: \({\rm{R}}\) is the gas constant

\({\rm{T}}\) is the absolute temperature

\({\rm{M}}\) is molar mass

Summary

This theory helps to understand the behaviour of gases. The particles are constantly moving in a straight line until they collide with other particles or the container’s walls. In this article, we learnt about the importance of the kinetic theory of gases, Boyle’s law, Charles law, Avogadro’s law, Dalton’s law of partial pressure and kinetic gas equation.

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