Gases consist of atoms and molecules moving ceaselessly and randomly with all possible velocities in all directions. Each of these gas particles has certain amount of kinetic energy that depends upon its temperature. Lighter particles move faster than heavier ones at all temperatures. Careful observations show that smaller the particles, more rapid is their motion at a given temperature.

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Consider some amount of gas in a container, these gases will be moving randomly in all the direction with all possible velocities, also these gas molecules acquire kinetic energy based on their velocities. In order to explain this observed behavior of gases, Bernoulli proposed a model called **Kinetic Theory of Gas**.

The gas molecules acquire some kinetic energy if they are made to move freely in a container. The kinetic energy possessed them due to their motion randomly in all directions is given by Bernoulli is called **Kinetic Molecular Theory**.

The following are the assumptions of kinetic theory of gas which are also called postulates of Kinetic theory of gas:

(i)**Assumptions Regarding the Molecules**

**Assumptions Regarding Volume**

(iii)**Assumptions Regarding Motion**:

(iv)**Assumptions Regarding Collision**:

**Assumptions Regarding the Force**:

**Assumptions Regarding Pressure**:

**Assumptions Regarding the Density**:

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Let there be an ideal gas enclosed in a cubical vessel of side (i)

- Every gas consists of extremely small particles known as molecules. The molecules of a given gas are all identical but are different than those another gas.
- The molecules of a gas are identical, spherical, rigid and perfectly elastic point masses.
- Their size is negligible in comparison to inter molecular distance (10
^{-9}m).

- The Volume of molecules is negligible in comparison to the volume of gas. (The volume of molecules is only 0.014 % of the volume of gas).

(iii)

- Molecules of a gas keep on moving randomly in all possible directions with all possible velocities.
- The speed of a gas molecule lies between zero and infinity (very high speed)
- The number of molecules moving with most probable speed is maximum.

(iv)

- The gas molecules keep on colliding among themselves as well with the walls of the containing vessel. These collisions are perfectly elastic. (That is the total energy before the collision is equal to the total energy after the collision).
- Molecules move in a straight line with constant speed during successive collisions.
- The distance covered by the molecules between two successive collisions is known as free path and mean of all the free paths is known as
**Mean Free Path.** - The time spent in a collision between two molecules is negligible in comparison to time between two successive collisions.
- If we consider the number of collisions present in one unit volume then the number remains constant.

- No attractive or repulsive force acts between gas molecules.
- Gravitational attraction among the molecules is ineffective due to extremely small mass and very high speed of molecules.

- Molecules constantly collide with the walls of container due to which their momentum changes.
- This change in momentum is transferred to the walls of the container.
- Consequently pressure is exerted by gas molecules on the walls of container.

- The density of a gas is constant at all points of the container.

So, there Velocity is given by

or

The momentum of the molecule in X-direction before collision is

Let the number of molecules present in the vessel be

a) Momentum before the collision = m V

b) Momentum after collision = - m V

$\Delta$ P = Change in momentum of the molecules

= - m V

After the collision the molecule moves towards opposite face

The time taken between two successive collisions in one second

n = $\frac{V_{X}}{2L}$.

The rate of change of momentum = Change of momentum in one Collision $\times$ Number of Collisions per second.

The rate of change of momentum = $\Delta$ P $times$ n

= 2 m V

= $\frac{m V_{X}^{2}}{L}$.

According to Newtons second law of motion, the rate of change of momentum is Force.

Hence the Total force on the wall is

F = $\frac{mV_{x}^{2}}{L}$.

= $\frac{m}{L}$ $\sum$ V

Assuming average velocity in all direction to be equal, we have

$\sum$ V

Average Velocity, V = $\frac{1}{3}$ $\sum$ (V

= $\frac{1}{3}$ $\sum$ V

F = $\frac{1}{3}$ $\frac{M}{L}$ $\sum$ v

L= length of the cube

M = mass of gas = m $\times$ n

Here m= mass of each particle and

n = no of molecules present in the gas

Velocity

The Force exerted by the molecule on the wall ABCD. Thus Force in X-direction is

F

V

L = Length of the cube.

and the Pressure on the wall ABCD by a particle in X-direction is given by

P

Pressure in X-direction is given by

P

Since Pressure exerted is same in all directions, we have

P

$\therefore$ 3P = $\frac{M}{L^{3}}$ $\sum$V

Pressure, P = $\frac{M}{3L^{3}}$ $\sum$ V

since L

M = mass of gas = m $\times$ n

Here m= mass of each particle and

n = Total no of molecules present in the gas.

Total Pressure P is given by,

P = $\frac{M}{3v}$ $\sum$ V

= $\frac{mn}{3v}$ $\sum$ V

$\rho$ = density of the gas = $\frac{mn}{v}$

v = Volume of gas,

V = Velocity of the gas.

The kinetic theory of matter explains the motion and kinetic energy observed in the three states of matter.

In Solids, the particles are closely packed so the motion of the particles will not be observed.

In Liquids, the particles are free to move about but their exists force between the neighboring molecule so the motion will be less.

In gases the particles are free to move as they have force to overcome the intermolecular force of attraction and hence they can move about freely. So here comes the Kinetic energy for gases.

In Solids, the particles are closely packed so the motion of the particles will not be observed.

In Liquids, the particles are free to move about but their exists force between the neighboring molecule so the motion will be less.

In gases the particles are free to move as they have force to overcome the intermolecular force of attraction and hence they can move about freely. So here comes the Kinetic energy for gases.

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Kinetic Theory of Gases | |

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