A plot of  vs v_x at a certain temperature is represented in Fig. (4.15).

The curve is symmetric about v_x = 0 and has a maximum value of 

Fig. 4.15 Maxwell Boltzmann velocity distribution function for x-component velocity

Average, Root-Mean-Square and Most Probable Velocity

The average speed,  of a gas molecules is defined as the sum of the speed of all molecules divided by the number of molecules.

(4.109)

Let  x = β²v², then
And         

(4.110)

(4.111)

or,

(4.112)

The root-mean-square speed is defined as the squares of the velocities of all molecules divided by the number of molecules.

(4.113)

Putting                                

(4.114)