In this chapter we shall first describe a rather more sophisticated kinetic theory of perfect gases. Part of the exercise is purely computational and, although it looks more impressive, adds little to our physical understanding. There are, however, a number of points which emerge which are interesting and useful and which shed new light on some of the assumptions made in the simpler forms of the kinetic theory. We shall also discuss the velocity distribution in a gas and the thermal energy of its molecules.

A better kinetic theory

Assumptions

First we recapitulate our basic assumptions. Let us assume that we are dealing with a very large number of molecules uniformly distributed in density; that they have complete randomness of direction and velocity; that the collisions are perfectly elastic; that there are no intermolecular forces; and finally that the molecules have zero volume.

We now consider a way of describing their distribution in space. Thus to each molecule we attach a vector representing its velocity in magnitude and direction (figure 4.1 (a)). We then transfer these vectors (not the molecules) to a common origin (figure 4.1 (b)) and construct a sphere of arbitrary radius r, allowing the vectors to cut the sphere (if necessary by extending their length). Then the velocity vectors intersect the sphere in as many points as there are molecules.

If we postulate randomness of molecular motion all directions are equally probable, so that these points will be uniformly distributed over the surface of the sphere.