Random walk experiment 2
Finding displacement after ‘random walk’ using squared paper.
Apparatus and materials
Sheets of paper ruled in squares
Logs or bags of balls (see technical note)
Health & Safety and Technical notes
If the pin-through-straw method (teaching note 3) is used, plain steel dressmaking pins would be safer than the larger 'optical pins'.
The ‘logs’ are small rectangular blocks of wood. Cheap wood strips of square cross-section, say 1 cm x 1 cm, can be cut into 5 cm lengths. Before cutting up the strip, the four faces could be painted four different colours to signal a move UP, DOWN, LEFT or RIGHT (or, after cutting, the faces could be marked U, D, L and R).
The bag of balls (or beads) may be more expensive but could get a stronger feeling of chances in a concealed lottery. The balls will need to be of four colours and the bag opaque.
Paper ruled in centimetre squares may be available inexpensively for primary school and maths classes.
a Mark a starting point on the paper, roll the log (or pick a ball from the bag) and draw a mark 1 square distance. That is one ‘stride length’.
b Repeat this 25 times.
c Measure the distance from start to finish and write it on the squared paper.
d When all students have completed steps a–c, preferably several times, calculate a class average of all the results.
1 A simple analysis of ‘random walk’ shows that the most likely displacement R from the starting point for N steps of length s is about s times √N (here 5). That is R = s √N.
With only the relatively small number of trials obtained, even using the whole class several times, the average of all the results is unlikely to give good agreement with that predicted. However, it is very unlikely to be zero displacement and probably a long way from 25 times the stride length for 25 throws. It is more likely to be near enough to the predicted result to lend some support.
(Since we really should have taken the root-mean-square of the displacement results, the ideal result should be about 0.8 times the arithmetical mean. That is about 4 times s, the stride length.)
2 Students who have looked at Diffusion of bromine vapour may well grasp that there are millions upon millions of bromine molecules carrying out random walks on a profuse scale. So the statistical prediction will therefore work much better.
Indeed, if studentsfound the average displacement in 500 seconds – the ‘half-brown distance’ – to be about 10 cm, then they are in a position to find a value for the average distance the molecules move between collisions (the mean-free-path). Appealing to measurements of density and pressure of bromine vapour gives the average speed of molecules as 200 metres/second.
Density of bromine vapour, ρ = 7.5 kg/m3
Pressure of bromine vapour, P = 105 N/m2(at normal pressure)
Using the relationship PV = 1/3 (mnv2),
Speed of bromine molecules, v = √(3 P / ρ)
= √(3 x 105 / 7.5)
= 200 m/s
So in 500 seconds the molecules will have travelled a ‘straightened out’ path of 200 x 500 m which is equal to the path between collisions, s, multiplied by the number of collisions, N.
200 x 500 = sN (1)
The random path (the half brown distance) = s√N
0.1 = s√N (2)
From equations (1) and (2) then s = 10-7 m and N = 1012
The distance which a molecule travels between collisions is known as its ‘mean free path’.
Some teachers may then go on to get a rough value for the size of a molecule.
3 It is also possible to estimate the time for an air particle to cross a room. See the relevant guidance pages linked below.
4 Another version of this investigation, which does not have any restriction on direction, uses a piece of drinking straw 4 cm long, marked at one end, and with a pin through its middle vertically into a small cork. A cross is marked on a sheet of plain paper and the centre of the cork is placed on the cross. The cork is held with one hand and the other hand is used to flick the straw to spin it. When it comes to rest, a mark is made on the paper under the marked end. The cork is moved to the new mark and the procedure repeated, until 25 steps are obtained.
This experiment was safety-checked in August 2006