Maxwell's distribution of molecular speeds

Maxwell showed on the basis of statistical mechanics that the actual distribution of molecular speeds within a gas was as shown in Figure 1.


The number of molecules N_c that have velocities between c and (c + Δc) is plotted against the velocity (c). It can be shown that

c:c_m:c_r.m.s = 1:1.13:1.23

where c is the most probable speed, c_m is the mean speed and c_r.m.s is the root mean square speed.

The distribution can be investigated experimentally by passing a stream of gaseous molecules through three rotating discs with a slot in each. The discs are rotated on a common shaft separated by a known distance, the slots being at an angle θ to each other. Only those molecules with the correct velocity will be able to pass through all the slits.

A simplified diagram of the apparatus is shown Figure 2.

Consider a molecule that passes through S₁. If it is to pass through S₂ then the second disc must have rotated through an angle θ during the time that the molecule was travelling between the two discs, where θ is given by

θ = 360nt = 360L/v

where n is the number of revolutions of the shaft per second, L the distance between the discs and v the velocity of the molecule. Hence v can be found.