Kinetic Theory

Consider a cubical box, L on each side. The box contains N molecules of ideal gas, each of mass m.

All collisions are elastic. The force exerted by one molecule when it collides with a wall of the box that is perpendicular to the x-axis is, from the impulse equation:
F =
2mvx
Dt

Our single molecule collides with this wall once every:
Dt =
2L
vx

This gives an average force of:
F =
mvx2
L

This is the force from a single molecule. The total force on the wall is the sum over all the molecules:
F = S
mvx2
L
=
mN
L
S
vx2
N

The sum represents the average value of vx2. The square root of this average is known as the root-mean-square (rms) average of vx, so:
F =
mN
L
vx2rms

By symmetry, all directions in the box are equivalent, so:

v2 = vx2 + vy2 + vz2 = 3vx2

This gives:
F =
mN
3L
vrms2

Dividing by the area, L2, of the wall gives the pressure:
P =
mN
3L3
vrms2

which is:
PV =
N
3
mvrms2 =
2N
3
(mvrms2)

The term in brackets is Kav, the average translational kinetic energy of the molecules in the box.
Therefore   PV =
2N
3
Kav