James Clerk Maxwell and Modern Physics

by Richard Glazebrook

From these results we obtain as before the laws of Boyle, Charles and Avrogadro.

Again if σ be the specific heat of the gas at constant volume, the quantity of heat required to raise a single molecule of mass m one degree will be σ m.

Thus, when a molecule is heated, the kinetic energy must increase by this amount. But the increase of temperature, which in this case is 1°, is measured by the increase of kinetic energy of the single molecule. Hence the amount of heat required to raise the temperature of a single molecule of all gases 1° is the same. Thus the quantity σ m is the same for all gases; or, in other words, the specific heat of a gas is inversely proportional to the mass of its individual molecules. The density of a gas—since the number of molecules per unit volume at a given pressure and temperature is the same for all gases—is also proportional to the mass of each individual molecule. Thus the specific heats of all gases are inversely proportional to their densities. This is the law discovered experimentally by Dulong and Petit to be approximately true for a large number of substances.

* * * * *

In the next part of the paper Maxwell proceeded to determine the average number of collisions in a given time, and hence, knowing the velocities, to determine, in terms of the size of the particles and their numbers, the mean free path of a particle; the result so found differed somewhat from that already obtained by Clausius.

Having done this he showed how, by means of experiments on the viscosity of gases, the length of the mean free path could be determined.

An illustration due to Professor Balfour Stewart will perhaps make this clear. Let us suppose we have two trains running with uniform speed in opposite directions on parallel lines, and, further, that the engines continue to work at the same rate, developing just sufficient energy to overcome the resistance of the line, etc., and to maintain the speed constant. Now suppose passengers commence to jump across from one train to the other. Each man carries with him his own momentum, which is in the opposite direction to that of the train into which he jumps; the result is that the momentum of each train is reduced by the process; the velocities of the two decrease; it appears as though a frictional force were acting between the two. Maxwell suggests that a similar process will account for the apparent viscosity of gases.

Consider two streams of gas, moving in opposite directions one over the other; it is found that in each case the layers of gas near the separating surface move more slowly than those in the interior of the streams; there is apparently a frictional force between the two streams along this surface, tending to reduce their relative velocity. Maxwell’s explanation of this is that at the common surface particles from the one stream enter the other, and carry with them their own momentum; thus near this surface the momentum of each stream is reduced, just as the momentum of the trains is reduced by the people jumping across. Internal friction or viscosity is due to the diffusion of momentum across this common surface. The effect does not penetrate far into the gas, for the particles soon acquire the velocity of the stream to which they have come.

Now, the rate at which the momentum is diffused will measure the frictional force, and will depend on the mean free path of the particles. If this is considerable, so that on the average a particle can penetrate a considerable distance into the second gas before a collision takes place and its motion is changed, the viscosity will be considerable; if, on the other hand, the mean free path is small, the reverse will be true. Thus it is possible to obtain a relation between the mean free path and the coefficient of viscosity, and from this, if the coefficient of viscosity be known, a value for the mean free path can be found.

Maxwell, in the paper under discussion, was the first to do this, and, using a value found by Professor Stokes for the coefficient of viscosity, obtained as the length of the mean free path of molecules of air 1/447000 of an inch, while the number of collisions per second experienced by each molecule is found to be about 8,077,200,000.