Consider the air surrounding us. The air is composed of many molecules. Consider some considerable volume of air, where there is same temperature everywhere. We cannot still say that all molecules of air in the considered volume travel at the same speed. Some molecules move fast, some move at moderate speed, some move slowly while some do not move at all.

Maxwell-Boltzmann distribution  shows how the speeds of molecules are distributed for an ideal gas. The Maxwell-Boltzmann distribution is represented in the form of a graph.

The graph represents the number of molecules on the Y-axis and speed along the X-axis.

We know that PV = 1/3* mnu² 

where

P = Pressure of gas

V = Volume of gas

m = mass of one molecule of gas

n = number of molecules of gas

u = root mean square velocity of molecules

For 1 mole n = N (Avogadro number)

m × N = Molecular mass M.

Hence PV = 1/3 M u²

u² = 3PV / M

As PV = RT,  u² = 3RT / M

u = √ (3 RT/M) = √ (3 PV/M) 

Average speed: As per kinetic theory of gases, each molecule is moving with altogether different speed. Let n molecules be present in a given mass of gas, each one moving with speed u₁,u₂, u_3, …,u_n.

Average speed = (u₁ + u₂ + ….u_n) / n 

Root Mean Square Velocity: Maxwell proposed the term as the square root of means of square of all such velocities.

u²_rms = (u₁² + u₂² + ….u_n²) / n 

Also, u_rms = √ (3RT/M)

Most probable velocity: It is the velocity possessed by maximum no. of molecules.

Most probable velocity u_mpv = √ (2RT/M)

Kinetic Energy of Gas: As per kinetic equation PV = .1/3 Mu ²_rms

For 1 mole m × n = Molecular Mass (M)

PV = 1/3 Mu²_rms

Multiply and divide LHS by 2

PV = 2/3 * ½ *Mu²_rms

      = 2/3 * KE/mole

Hence, KE/mole = 3/2 PV = 3/2 RT

KE / molecule = 3/2 RT/n = 3/2 kT where k is Boltzmann constant