This proof was originally proposed by Maxwell in 1860. He considered a gas to be a collection of molecules and made the following assumptions about these molecules:

molecules behave as if they were hard, smooth, elastic spheres

molecules are in continuous random motion;

the average kinetic energy of the molecules is proportional to the absolute temperature of the gas;

the molecules do not exert any appreciable attraction on each other

the volume of the molecules is infinitesimal when compared with the volume of the gas

the time spent in collisions is small compared with the time between collisions.

Consider a volume of gas V enclosed by a cubical box of sides L. Let the box contain N molecules of gas each of mass m, and let the density of the gas be r. Let the velocities of the molecules be u

Consider a molecule moving in the x-direction towards face A with velocity u

It will then travel back across the box, collide with the opposite face and hit face A again after a time t, where t = 2L/u

The number of impacts per second on face A will therefore be 1/t = u

Therefore rate of change of momentum = [mu

But the area of face A = L

But there are N molecules in the box and if they were all travelling along the x-direction then

total pressure on face A = [m/L

But on average only one-third of the molecules will be travelling along the x-direction.

Therefore: pressure = 1/3 [m/L

If we rewrite Nc

pressure = 1/3 [m/L

and this is the kinetic theory equation.

Now the total mass of the gas M = mN, and since r = M/V we can write

The root mean square velocity or r.m.s. velocity is written as c

r.m.s. velocity = c

We can use this equation to calculate the root mean square velocity of gas molecules at any given temperature and pressure.

Some further values of the root mean square velocity at s.t.p. for other gases are given below.

Hydrogen 18.39 x 10

Helium 13.10 x 10

Oxygen 4.61 x 10

Carbon dioxide 3.92 x 10

Bromine 2.06 X 10