Mean free path

It is interesting for students to see how far physicists can push ‘back of the envelope’ calculations.

Diffusing bromine into air can be used to estimate the mean free path of bromine molecules. See the experiment:

Diffusion of bromine vapour

The clock is started as soon as the bromine is released into the tube of air. After 500 seconds the average distance which the bromine has diffused up the tube is measured (the half-brown position.). The bromine molecules are used as markers to show how one gas travels through another making many collisions on the way. This experiment shows that molecules have ‘size’ as they find targets with which to collide.

A discussion might go something like this:

Put on your super-microscopic spectacles and have another look at this bromine wandering through the air.

Can you see the bromine molecules travelling very fast for a short distance, colliding with another molecule, rebounding in a different direction, then striking another...?

A quick look at the two-dimensional model with marbles in a tray, with one distinctly coloured marble to focus on, will reinforce the idea of random motion here, if wanted.

How far can a molecule go with that sort of movement? The molecule’s progress is rather like yours if you tried to work your way through the rush-hour crowd in a busy station blindfolded so that you had no memory of direction. Scientists call that a random walk.

Imagine you take one stride from your starting point. Then choose a new direction, any direction at random, and take one stride in that direction. Again choose a new direction at random and take one stride in that direction. Go on like that until you have taken a large number of equal strides, say 100. How far are you from your starting point, as the crow flies?

A bromine molecule makes a wandering path from collision to collision, starting out in a new direction every time. The more time a molecule has to make strides, the farther from its start it is likely to end up. But the direction of motion after each collision is random, so you cannot just add up all the steps in a straight line. All the same, it is still possible to make predictions about the distance travelled in a random walk. After a given time, one molecule will have got to a place a long way away from the start. Another will have got back near its start. Many will have got to some middling distance.

You could even make a chart of molecules at various distances. You could make the chart by imagining a random walk or sketching it on paper. If you made hundreds or even thousands of trials of that and took an average, you could find a definite average "crow-flies" distance from start to finish.

There is a definite rule for predicting that average distance. Suppose a molecule takes a 100 steps, bouncing away from each collision in a new direction (sketch the motion). Catalogue its net progress from start to finish and take the average of many trials of 100 steps. The result is not 100 steps (the maximum) or no steps (the minimum) but 10 steps. This is because 10 is the √ 100 and the general rule says that for N steps the average distance travelled is √ N steps.

You need quite a lot of algebra, adding and averaging all those wild wanderings, to predict this strange rule with the √N in it. So instead of trying the algebra, play the game yourself and see if you can test the rule.