In order to explain the observed behavior of gases, a model was proposed based on the molecular and kinetic concepts of gas molecules.

This model takes into account the particulate nature of matter and the constant movement of particles.
The various gas laws like Boyle’s law, Charles’s law, Dalton’s law of partial pressure, etc have been arrived at by experiment. There was no theoretical explanation for the behavior of gases.
To account for the behavior of gases, a model was proposed which takes into account the absorbed behavior of gases.

A model was proposed which takes into account the molecular concept as well as the kinetic concept of gas molecules. The kinetic theory of gases has been devised to rationalize the behavior of gases.
Bernoulli proposed this kinetic theory in 1738 and it was elaborated and developed by Clausius, Maxwell, Boltzmann, van der Waal and others.

The theory proposed to explain the particle nature of a gas is called kinetic theory of gases.
The kinetic gas theory takes into account the particle nature of matter as well as their constant motion. Since its assumption relates to microscopic particles, this theory is also known as the **‘microscopic model’**.

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The kinetic molecular theory was proposed by Kronig, Clausius, Maxwell and Boltzmann in the nineteenth century. This theory proved to be a sound theoretical base to the various gas laws like Boyle's law, Charles Law, Grahams law, Dalton's law, etc. Kinetic theory is basically a theoretical proof for all the gas laws. Kinetic theory is defined as **"a theory that gases consist of small particles in random motion."**

The assumptions of kinetic theory of gases are as follows.

From the above postulates or assumptions of kinetic molecular theory of gases, it is possible, by applying the laws of classical mechanics, to derive an expression for the pressure of a gas.

Consider a gas enclosed in a cube, the sides of which are I meters long.

Let the number of molecules in the cube be c and let the mass of each molecule be m kg. Let the square mean square velocity of each molecules be I meters.

The molecules present in the cube will be moving in all possible directions. But since they do not prefer any particular direction and also because the pressure exerted on all sides is the same, it may be imagined that at any time one- third of the molecules move perpendicular to each pair of the faces, i.e., n / 3 molecules along x- axis, n / 3 along y-axis and n / 3 along the z-axis.

Now consider the motion of a single molecule striking against face A of the cube. Its momentum = mc. After striking the cube it will rebound with the same speed in the opposite direction.** **Therefore,

**Change in momentum of the molecule colliding **

**= Initial momentum - Final momentum**

= mc - (-mc) = 2mc.

Before another collision can occur with this small wall, the molecule must travel a distance of 2l. Number of collisions per second by a molecule on one face = c/2l.

So, Total change in momentum per second = 2mc x c/2l

Molecule at the face A = mc2/l

For n/3 molecules striking against the face A, the total change of momentum per second

= n / 3 x mc^{2}/l = 1/3 mnc^{2} / l.

But change of momentum per second = force.

Therefore, total force on the face A of area l^{2} = 1/3 mnc^{2} / l.

**Pressure = Force /Area**

**P = 1/3 mnc**^{2} / l.

** = 1/3. mnc**^{2} / l^{3}

**l**^{3} is the volume. So,

**P = 1/3 mnc**^{2} / V

This is called as the kinetic gas equation. This is applicable for ideal gases only.

The three types of molecular velocities are

### 1. Most probable velocity (Cp)

This is defined as M the velocity possessed by maximum fraction of the total number of molecules of the gas at a given temperature.

**Cp =**$\sqrt{\frac{2RT}{M}}$

Where

R = gas constant

T= temperature and

M is the molecular weight

### 2. Average velocity

It is expressed as C. Average velocity is the mean of the velocities of all the molecules. It is calculated using the formula.

### 3. Root mean square velocity

It is defined as the square root of the mean of the square of the velocities of all the molecules of the gas.

**Root mean square velocity = 1.085 x Average velocity**

According to kinetic gas equation,

**PV = $\frac{1}{3}$mn$c^{2}$**

Where M = mn = molar mass of the gas.

Therefore,

**PV = $\frac{2}{3}$ x $\frac{1}{2}$ M$c^{2}$**

½ Mc^{2} is the kinetic energy of the gas.

So, Kinetic energy,

**K.E = $\frac{3RT}{2}$$s^{-1}$**

### Solved Examples

**Question 1: **The mass of a molecule is 4.5 x $10^{-23}$ kg. If its average velocity is 10.0 x $10^{2} ms^{-1}$ , calculate the kinetic energy?

** Solution: **

K.E = $\frac{1}{2}$ $MV^{-2}$^{}

= $\frac{1}{2}$ x 4.5 x $10^{-23}$ x (10.0 x $10^{2}$)^{2}

= 2.25 x $10^{17}$ Joules.

**Question 2: **Calculate the average energy in joules of molecules in 32 grams of oxygen at 27 degree Celsius.

** Solution: **

The assumptions of kinetic theory of gases are as follows.

- All gases are made up of minute particles called molecules. The molecules are so small that their actual volume is a negligible fraction of the total volume (space) occupied by the gas.
- The molecules are separated from one another by large distances. The empty spaces among the molecules are so large that the actual volume occupied by the molecules is negligible as compared to the volume of the gas.
- The molecules are in a state of constant motion in all directions. During their motion, they collide with each other and also with the walls of the container and thus change their path.
- Molecules are assumed to be spherical and perfectly elastic in nature. Hence molecular collisions are perfectly elastic, i.e, no loss of energy occurs when the molecules collide with one another or with the walls of the container. However, there may be redistribution of energy during the collisions.
- There are no forces of attraction or repulsion between molecules. They move completely independent of one another because of the large distance of separation.
- The effect of gravity on the gas molecules is negligible because the gas molecules are continuously moving with high intensity.
- The pressure exerted by the gas is due to the bombardment of its molecules against the walls of the container. However, no energy is lost in these collisions.
- The average kinetic energy of the gas molecules is directly proportional to the absolute temperature.
**K. E ∞ T** - The velocity of the gas molecules is directly proportional to the square root of the absolute temperature.

From the above postulates or assumptions of kinetic molecular theory of gases, it is possible, by applying the laws of classical mechanics, to derive an expression for the pressure of a gas.

Consider a gas enclosed in a cube, the sides of which are I meters long.

The molecules present in the cube will be moving in all possible directions. But since they do not prefer any particular direction and also because the pressure exerted on all sides is the same, it may be imagined that at any time one- third of the molecules move perpendicular to each pair of the faces, i.e., n / 3 molecules along x- axis, n / 3 along y-axis and n / 3 along the z-axis.

Now consider the motion of a single molecule striking against face A of the cube. Its momentum = mc. After striking the cube it will rebound with the same speed in the opposite direction.

= mc - (-mc) = 2mc.

So, Total change in momentum per second = 2mc x c/2l

Molecule at the face A = mc2/l

For n/3 molecules striking against the face A, the total change of momentum per second

= n / 3 x mc

But change of momentum per second = force.

Therefore, total force on the face A of area l

This is called as the kinetic gas equation. This is applicable for ideal gases only.

The three types of molecular velocities are

- Most probable velocity
- Average velocity
- Root mean square velocity

This is defined as M the velocity possessed by maximum fraction of the total number of molecules of the gas at a given temperature.

Where

R = gas constant

T= temperature and

M is the molecular weight

It is expressed as C. Average velocity is the mean of the velocities of all the molecules. It is calculated using the formula.

$\bar{C}= \sqrt{\frac{8RT}{\pi M}}$

It is defined as the square root of the mean of the square of the velocities of all the molecules of the gas.

Therefore,

So, Kinetic energy,

K.E = $\frac{1}{2}$ $MV^{-2}$

= $\frac{1}{2}$ x 4.5 x $10^{-23}$ x (10.0 x $10^{2}$)

= 2.25 x $10^{17}$ Joules.

K. E = 3/2 nRT where n is the number of moles of oxygen gas.

Number of moles = Weight of oxygen / Gram molecular weight

= 32 / 32 = 1

K. E = 3/2 x 1 x 8.314 x 300 = 3741.3 Joules.

Number of moles = Weight of oxygen / Gram molecular weight

= 32 / 32 = 1

K. E = 3/2 x 1 x 8.314 x 300 = 3741.3 Joules.

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