## Kinetic Molecular Theory of Gases

### Table of contents

- Introduction to Kinetic Molecular Theory of Gases
- Postulates or Assumptions of Kinetic Theory of Gases
- Maxwell’s Distribution of Molecular Speeds/Energies
- Kinetic Gas Equation
- Type of Speeds
- Relationship between different types of speeds
- Explanation/deduction of Gas Laws from Kinetic Theory

- Relation between Average Kinetic Energy and Absolute Temperature-Deduction from Kinetic Theory

## Introduction to Kinetic Molecular Theory of Gases

The general properties of gases such as expansibility, compressibility, liquefaction, low densities, diffusion or effusion etc. have been observed experimentally. Similarly, various gas laws such as Boyle’s Law, Charles’ Law etc. have been obtained on the basis of experimental studies. To explain these general properties of the gases and the gas laws theoretically, various workers, from time to time, have tried to give a theoretical model of the gas. The first such model was put forward by Bernoulli in 1738 and later improved upon by various workers. Finally in the complete form, it has been put forward by Clausius in 1857 and is known as ‘**Kinetic-molecular theory of gases**’. It was so called because it assumes the gas to be made up of a large number of molecules which were in ceaseless motion. For the same reason, it is also called a ‘**Dynamic particle model**’ of the gas. The name ‘**Microscopic model**’ is also used because the model assumes the gas to be made up of molecules which cannot be seen.

## Postulates or Assumptions of Kinetic Theory of Gases

The Microscopic Model of a Gas. The main assumptions of kinetic theory of gases are as follows:

(i) Large number of extremely small particles called **Molecules** come up together to form gas. In a particular gas all the molecules are identical in mass and size but differ from gas to gas.

(ii) The actual volume of the molecules is negligible when compared to that of total volume of the gas as the molecules of a gas are separated from each other by large distances

(iii) Forces of attraction or repulsion between the molecules are negligible because of large distance of separation between them.

(iv) The force of gravitation on the molecules is also supposed to be negligible.

(v) The molecules are supposed to be moving continuously in different directions with different velocities.

Hence, they keep on colliding with one another (called **Molecular Collisions**) as well as on the walls of the containing vessel.

(vi) Bombardment of the molecules on the walls of the containing vessel exerts pressure on the walls of the containing vessel.

(vii) The molecules are supposed to be perfectly elastic hard spheres so that no energy is wasted when the molecules collide with one another or with the walls of the vessel. The energy may, however, be transferred from some molecules to the other on collision.

(viii) As the molecules move with different velocities, they possess different kinetic energies. However, the average kinetic energy of the molecules of a gas is directly proportional to the absolute temperature of the gas.

## Maxwell’s Distribution of Molecular Speeds/Energies

At a particular temperature, the different molecules of a gas possess different speeds. Further, due to continuous collisions of the molecules among themselves as well as against the walls of the container, their speeds keep on changing. Maxwell and Boltzmann, however, showed that as a result of collisions, though some molecules are speeded up, some others are slowed down and hence the fractions of molecules possessing particular speed remains constant at constant temperature. This is called **Maxwell-Boltzmann** distribution and is shown by the curves in figure no. 1 for two different temperatures.

**Figure No. 1 Maxwell’s distribution of speed at different temperatures**

The peak of the curve corresponds to a speed possessed by the maximum fraction of the molecules and is called **the most probable speed**. We observe that with rise in temperature, the most probable speed increases. This is expected because with rise in temperature, the average speed of the gas molecules increases. However, it may be noted that the fraction of molecules possessing most probable speed decreases with increase in temperature.

## Kinetic Gas Equation

On the basis of the various assumption made in kinetic theory of gases, a mathematical equation has been derived from which all the gas laws can be deduced. This equation is known as ‘**kinetic gas equation**’. It is usually written in the form

Where P = Pressure exerted by the gas, V = Volume of the gas, m = Mass of each molecule of the gas, n = Total no. of molecules present in the volume V, u = Root mean Square speed of gas

For 1 mole, m × n = mass of 1 mole = M, molar mass in grams.

Hence,

## Types of Speeds

It may be noted that there are three types of speeds of gaseous molecules which are commonly used.

**Most Probable Speed (α)**. This is the speed possessed by the maximum fraction of molecules.**Root Mean Square (RMS) Speed (u).**It is the square root of the mean of the squares of the speeds of the molecules. Thus if v_{1}, v_{2}, v_{3}……v_{n}are the speeds of the molecules, then

**Average Speed (v).**It is the average of the different speeds of all the molecules.

Instead of symbols, α, v and u, symbols u_{mp}, u_{av}, and u_{rms}, are often used.

## Relationship between different types of speeds

As derived above, for root mean square speed,

Similarly, for most probable speed (α) and average speed (v), we have

Thus there are three type of speed related to each other as follow:

## Explanation/deduction of Gas Laws from Kinetic Theory

### Boyle‘s Law

Explanation on the Basis of Kinetic Theory. At constant temperature the average kinetic energy and the average speed of the molecules is constant,(according to Kinetic theory of gases). Further the number of molecules present in a given mass of a gas is also constant.

Let the volume of a given mass of a gas be reduced to one half of its original volume. The same number of molecules with their same average speed will now have half the original space to move about. As a result, the number of molecules striking the unit area of the walls of the container in a given time will get doubled and consequently the pressure is also doubled. Conversely if the volume of a given mass of a gas is doubled at constant temperature figure no. 2 the same number of molecules with their same average speed will now have double the space to move about.

**Fig. No. 2 Rate of molecular collisions increases by decrease in volume at constant temperature.**

Consequently, the volume of number of molecules at constant temperature striking the unit area of the walls of the container in a given time will now become one half of the original number. As a result, the pressure of the gas will be reduced to one half of its original value. Thus, it is obvious from the above discussion that pressure increases accordingly as the volume decreases or vice versa at constant temperature. This statement represents Boyle’s Law.

**Deduction from Kinetic Gas Equation**.

According to kinetic gas equation,

(where m × n is total mass of the gas)

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

KE ∝ Absolute Temperature (T)

Or KE = kT

Where k is proportionality constant.

Putting this value in equation no. 1, we get

As 2/3 is a constant quantity, k is also a constant, therefore if T is kept constant, 2/3 KT will be constant. Hence PV = constant if T is kept constant, which is Boyle‘s law.

### Charles’ Law

**Explanation on the Basis of Kinetic Theory**

According to the kinetic theory of gases, the average kinetic energy and hence the average speed of the gas molecules is directly proportional to its absolute temperature. Thus, it follows that when the temperature of a gas is increased at constant volume, the average kinetic energy of its molecules increases and hence the molecules would move faster as in figure no. 3. As a result, the molecules of the gas will strike the unit area of the walls of the container more frequently and vigorously. Consequently, the pressure of the gas will increase accordingly. Thus, at constant volume, the pressure increases with rise in temperature. This explains the Pressure Law.

**Fig. No. 3 Increase of pressure due to faster collisions with increase of temperature at constant volume**.

On the other hand, if the pressure of the gas is to be maintained constant, the force per unit area on the walls of the container in a given time must be kept the same. This can, however, be achieved by increasing the volume proportionately. Thus, at constant pressure, the volume of a given mass of a gas increases with increase in temperature. This explains Charles’ Law.

**Deduction from Kinetic Gas Equation.**

As deduced in equation no. 2 above from the kinetic gas

This may be rewritten as:

As 2/3 is a constant quantity, k is also a constant, hence if P is kept constant V/T = constant, which is Charles law.

### Dalton’s Law of Partial Pressures

**Explanation on the Basis of Kinetic Theory.**

According to the Kinetic theory of gases, the attractive forces between the molecules of the same or different gases are very weak under ordinary conditions of temperature and pressure. Therefore, the molecules of a gaseous mixture move completely independent of one another. As a result, each molecule of the gaseous mixture would strike the unit area of the walls of the container the same number of times per second as if no other molecules were present. Therefore, the pressure due to a particular gas is not changed by the presence of other gases in the container. Consequently, the total pressure exerted by a gaseous mixture must be equal to the sum of the partial pressure of each gas when present alone in that space. Hence, the Kinetic theory explains Dalton’s Law of Partial Pressures.

**Deduction from Kinetic Gas Equation**.

If we consider only two gases, then according to kinetic gas equation,

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

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

And when in the same vessel, both the gases are enclosed together then since the gases do not react with each other, their molecules behave independent of each other. Hence, the total pressure exerted would be

Similarly, if more than two gases are present, then it can be proved that P = P_{1} + P_{2} + P_{3} + …..

## Relation between Average Kinetic Energy and Absolute Temperature-Deduction from Kinetic Theory

According to Kinetic gas equation,

where P is the pressure, V is the volume of the gas, m is the mass of each molecule, n is the number of molecules present and u is the root mean square speed of the molecules

If 1 mole of the gas is taken, then the total mass of the gas, m x n = M, the molar mass of the gas. Hence, equation no. 1 becomes

It can be re-written as

But 1/2 × Mu^{2} represents the kinetic energy per mole of the gas.

Hence,

for 1 mole of the gas. (Equation no. 3)

But for 1 mol of gas, PV = RT

Putting this value in equation no. 3 we get:

K.E. = 3/2 RT for 1 mole of the gas (Equation No. 4)

To get average kinetic energy per molecule, divide both sides of equation no. 4 by the Avogadro’s number. N_{A}, that is, the number of molecules present in one mole of the gas. Thus, we have

Where k = R/N_{a }is called **Boltzmann Constant**

From equation no. 5, we observe that

Average K.E. ∝ Absolute temperature of the gas irrespective of nature of the gas. This is one of the postulates of kinetic theory of gases.

As, Average

that is, Average K.E. ∝ u^{2 }and also

Average K.E. ∝ T, this means that u^{2 }∝ T

Thus, when T = 0 K, u = 0, that is, molecular motion in a gas should become zero at absolute zero. In fact, the gas liquefies before this temperature is attained. **The motion of the gas molecules due to temperature ****is called Thermal Motion**

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Kinetic Molecular Theory of Gases