Consider a molecule A with an effective diameter d travelling with a velocity v (see Figure 1). In a time t the molecule will travel a distance vt and collide with pd

The mean free path is the total distance covered (vt) divided by the number of collisions. Therefore the mean free path us given by:

This proof assumes that all the other molecules remain at rest. If we consider the case of moving molecules the formula must be modified and becomes:

For air at 0

At an altitude of 100 km the mean free path is 1 metre and at 300 km it is nearly 10 km.

At a depth of 5 000 m in a mine the density of air has risen to 2.32 kg m

A further practical study of the mean free path in a gas may be made with the bromine diffusion experiment. In this experiment bromine vapour is allowed to diffuse through air at atmospheric pressure in a closed tube. The average distance d that the bromine diffuses in a certain time t is found by measuring the progress of the 'middle brown' colour of the bromine gas.

Now it can be shown statistically that d is related to the mean free path (L) of the gas by the formula:

d = L[N]

where N is the number of collisions made while the bromine is diffusing through the distance d.

However, the total distance travelled by an individual molecule of bromine is D, where D = NL, and therefore:

Using the kinetic theory formula the velocity of the gas molecules may be found and hence the total distance D travelled in a certain time T. The mean free path can then be found.