# Kinetic Theory Class 11 Notes Physics Chapter 13 - CBSE

## What Are Kinetic Theory ?

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## Ideal Gas Equation

For n moles of a gas,

$$\text{PV = nRT or}\\\frac{\text{P}_{1}\text{V}_{1}}{\text{T}_{1}} =\frac{\text{P}_{2}\text{V}_{2}}{\text{T}_{2}}$$

For 1 mole of a gas, PV = RT

## Ideal Or Perfect Gas

A gas which obeys gas laws strictly is an ideal or perfect gas. The molecules of such a gas are of point size and there is no force of attraction between them.

## Assumption Of Kinetic Theory Of Gases

• All gases consist of molecules. The molecules are rigid, elastic spheres identical in all respects for a given gas
and differnt for different gases.
• The size of a molecule is negligible compared with the average distance between two molecules.
• The molecules are in a state of continuous random motion, moving in all directions with all possible velocities.
• During the random motion, the molecules collide with one another and with the walls of the vessel.
• Between two collisions, a molecule moves in a straight path with a unifrom velocity.
• The collisions are almost instantaneous.
• The molecular density remains uniform throughout the gas.

## Pressure Exerted By A Gas

According to kinetic theory of gases, the pressure exerted by a gas of mass M and volume V or density r is given by

$$\text{P} =\frac{1}{3}\frac{\text{M}}{\text{V}}\bar{v}^{2}=\frac{1}{3}\rho\bar{v}^{2}\\=\frac{1}{3}mn\bar{v}^{2}$$

## Average Kinetic Energy Of A Gas

Let m be the molecular mass and V the molar volume of a gas. Let m be the mass of each molecule. then

Mean K.E. per mole of a gas

$$\text{E}=\frac{1}{2}\text{M}\bar{v}^{2}=\frac{3}{2}\text{PV}\\=\frac{3}{2}\text{RT}=\frac{3}{2}K_{B}N_{A}\text{T}$$

Mean K.E. per molecule of a gas

$$\bar{E} =\frac{1}{2}m\bar{v}^{2}=\frac{3}{2}k_{B}\text{T}$$

## Graham’s Law Of Diffusion

It states that the rate of diffusion of a gas is inversely proportional to the square root of its density

$$\frac{r_{1}}{r_{2}} =\sqrt{\frac{\rho_{1}}{\rho_{2}}}$$

## Average Speed

$$\bar{v} =\frac{v_{1} +v_{2} +v_{3} +.... + v_{n}}{n}\\\bar{v} =\sqrt{\frac{8k_{8}\text{T}}{\pi m}} =\sqrt{\frac{8\text{RT}}{\pi\text{M}}}\\=\sqrt{\frac{8\text{PV}}{\pi \text{M}}}$$

## Root Mean Square Speed

$$v_{\text{rms}} =\sqrt{\frac{v_{1}^{2} + v_{2}^{2} +v_{3}^{2} + ... + v_{n}^{2}}{n}}\\v_{rms} =\sqrt{\frac{3k_{8}\text{T}}{m}} =\sqrt{\frac{\text{3RT}}{\text{M}}}\\=\sqrt{\bigg(\frac{3\text{PV}}{\text{M}}\bigg)}$$

## Most Probable Speed

$$v_{mp}=\sqrt{\frac{2k_{B}\text{T}}{m}} =\sqrt{\frac{2\text{RT}}{\text{M}}}\\=\sqrt{\frac{2\text{PV}}{\text{M}}}$$

## Relation Between

$$\bar{\textbf{V}},\textbf{V}_{\textbf{rms}}\space\textbf{and}\space\textbf{V}_{\textbf{mp}}$$

$$\bar{v} = 0.92\space v_{rms}\\v_{mp} = 0.816\space v_{rms}\\v_{rms} :\bar{v}:v_{mp}= \\1.73:1.60:1.41\\\text{Clearly,}\space V_{\text{rms}}\gt\bar{\text{V}}\gt V_{\text{mp}}$$

## Degrees Of Freedom

The d.o.f. of a dynamical system are defined as the total number of coordinates or independent quantities required to describe completely the position and configuration of the system.

If N = number of particles in the system
k = Number of independent relations between the particles

∴ f = 2N – k.

## Law Of Equipartition Of Energy

If states that in any dynamical system in thermal equilibrium, the energy of the system is equally divided amongst its various degrees of freedom and the energy associated with each d.o.f. is

$$\frac{1}{2}k_{\text{B}}\text{T}$$

## Mean Free Path

It is the average distance covered by a molecule between two successive collisions. It is given by

$$\bar{\lambda} =\frac{1}{\sqrt{2}\pi nd^{2}}$$

where n is the number density and d is the diameter of the molecule.