Question

prove that the pressure exerted by the gas is P=1/3pc2

Answer 1

Consider a molecule moving in the x-direction towards face A with velocity u1. On collision with face A the molecule will experience a change of momentum equal to 2mu1. (Figure 3)
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/u1.

The number of impacts per second on face A will therefore be 1/t = u1/2L.

Therefore rate of change of momentum = [mu12]/L = force on face A due to one molecule.

But the area of face A = L2, so pressure on face A = [mu12]/L3

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/L3](u12 + u22 +...+ uN2)

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

Therefore: pressure = 1/3 [m/L3](u12 + u22 +...+ uN2)

If we rewrite Nc2 = [u12 + u22 + …+ uN2 ] where c is the mean square velocity of the molecules: