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# Kinetic Theory

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.

 Related Calculators Calculate Kinetic Energy Calculate Kinetic Friction

## Kinetic Theory Definition

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.

## Kinetic Molecular Theory Definition

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.

## 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
1. 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.
2. The molecules of a gas are identical, spherical, rigid and perfectly elastic point masses.
3. Their size is negligible in comparison to inter molecular distance (10-9 m).
(ii) Assumptions Regarding Volume
1. 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) Assumptions Regarding Motion:
1. Molecules of a gas keep on moving randomly in all possible directions with all possible velocities.
2. The speed of a gas molecule lies between zero and infinity (very high speed)
3. The number of molecules moving with most probable speed is maximum.

(iv) Assumptions Regarding Collision:
1. 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).
2. Molecules move in a straight line with constant speed during successive collisions.
3. 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.
4. The time spent in a collision between two molecules is negligible in comparison to time between two successive collisions.
5. If we consider the number of collisions present in one unit volume then the number remains constant.
(v) Assumptions Regarding the Force:
1. No attractive or repulsive force acts between gas molecules.
2. Gravitational attraction among the molecules is ineffective due to extremely small mass and very high speed of molecules.
(vi) Assumptions Regarding Pressure:
1. Molecules constantly collide with the walls of container due to which their momentum changes.
2. This change in momentum is transferred to the walls of the container.
3. Consequently pressure is exerted by gas molecules on the walls of container.
(vii) Assumptions Regarding the Density:
1. The density of a gas is constant at all points of the container.

## Kinetic Theory of Gases Equation

Let there be an ideal gas enclosed in a cubical vessel of side L with its perfectly elastic walls.

Consider a molecule of gas move with velocity V in any direction. Its Velocity can be resolved into three components Vx, Vy, Vz parallel to three co-ordinate axes X, Y, Z which are assumed parallel to the sides of the cube. The molecule move along X-axis and strikes elastically the face ABCD which is perpendicular to X-axis.
So, there Velocity is given by
V = Vx I + Vy j + Vz k
or
V2 = Vx2 + Vy2 + Vz2 Where, Vx, Vy and Vz represent the Velocity of gas molecules in x, y, z axes respectively. The molecule collides with the one wall of the cube with a velocity of Vx. By this motion, only the Vxcomponent of velocity will be affected while Vy and Vz remains unaltered.
The momentum of the molecule in X-direction before collision is mvx. when it collides with the face ABCD, as the collision is perfectly elastic, it rebounds with the same velocity but reversed in direction. So the momentum of the molecule now becomes -mVx.

Let the number of molecules present in the vessel be n and mass of each molecule be m.
a) Momentum before the collision = m Vx
b) Momentum after collision = - m Vx (assuming elastic collision)
$\Delta$ P = Change in momentum of the molecules
$\Delta$ P = Momentum before collision - Momentum after Collision
= - m Vx - m Vx = -2 m Vx.

After the collision the molecule moves towards opposite face EFHG and collides with it. Here again it rebounds and travels back to face ABCD after covering a distance 2L.
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 Vx $\times$ $\frac{V_{x}}{2L}$.
= $\frac{m V_{X}^{2}}{L}$.

According to Newtons second law of motion, the rate of change of momentum is Force.
Hence the Total force on the wall is
F = $\frac{mV_{x}^{2}}{L}$.
= $\frac{m}{L}$ $\sum$ VX2 ............(a)

Assuming average velocity in all direction to be equal, we have
$\sum$ Vx2= $\sum$ Vy2 = $\sum$ Vz2
Average Velocity, V = $\frac{1}{3}$ $\sum$ (Vx2 + vy2 + Vz2)
= $\frac{1}{3}$ $\sum$ V2.............(b)
Thus the Total force on the wall due to the particles is
F = $\frac{1}{3}$ $\frac{M}{L}$ $\sum$ v2 ..................(c) where,
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, V2 = Vx2 + Vy2 + Vz2

## Kinetic Theory of Gases Pressure

The Force exerted by the molecule on the wall ABCD. Thus Force in X-direction is
Fx = $\frac{M}{L}$ $\sum$ VX2......................(d)where, M= mass of the gas,
Vx = Velocity of the particle in x-direction,
L = Length of the cube.

and the Pressure on the wall ABCD by a particle in X-direction is given by
Px = $\frac{Force}{Area}$ = $\frac{F_{x}}{L^{2}}$.....(e)From equation (d) we have,
Pressure in X-direction is given by
Px = $\frac{M}{L^{3}}$ $\sum$Vx2.............(f)
Since Pressure exerted is same in all directions, we have
Px = Py = Pz = P.
$\therefore$ 3P = $\frac{M}{L^{3}}$ $\sum$Vx2 + $\frac{M}{L^{3}}$ $\sum$Vy2 + $\frac{M}{L^{3}}$ $\sum$Vz2.........(g)or
Pressure, P = $\frac{M}{3L^{3}}$ $\sum$ V2........(h)where Velocity V2 = Vx2 + Vy2 + Vz2
since L3 = v = Volume of the cube
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 = $\frac{M}{3v}$ $\sum$ V2
= $\frac{mn}{3v}$ $\sum$ V2
= $\frac{\rho}{3}$ $\sum$ V2
.............(i)
where,
$\rho$ = density of the gas = $\frac{mn}{v}$
v = Volume of gas,
V = Velocity of the gas.