8 A SYSTEM OF PHYSICAL CHEMISTRY This is known as the normal law of errors. The curve is shown in Fig. I. X denotes the error and y the probability of its occurrence. As X increases numerically, positively, or negatively, decreases jj' rapidly, and when x becomes large y becomes vanishingly small. It will be observed that the curve is symmetrical. Maxwell's Law.— Maxwell has applied the principle of probability to the problem of the distribution of velocities among the molecules of a gas, the gas being in a condition of statistical equilibrium at a uniform temperature throughout. A gas is to be regarded as a molecular chaos, the speed of any molecule varying from zero to infinity, its instantaneous value being the result of chance collisions with its neighbours. Although all values are theoretically possible for the speed of a molecule, it is found that in a system containing a large number of molecules, there are very few possessing either very great or very small speeds. Negative Errors. Positive Errors. Fig, I. The majority of the molecules possess speeds which lie within relatively restricted limits. A numerical illustration taken from Meyer's Kinetic Theory of Gases is given in Appendix I. There are, as a matter of fact, two ways of expressing Maxwell's law of distribution. One of these ways has already been stated in Chap. I., Vol. I. This way of expressing the law may be form— put in the dn — constant x N x e^^'^'/''^ ' c^ • dc where N is the total number of molecules in the system, and dn is the number whose speeds lie between the limits c and c + dc. It is to be clearly understood that the speed here referred to is simply a velocity magnitude, and no restriction has been introduced as to direction of motion. The term ~c is known as the root-mean-square speed (or r-m-s speed). At a given temperature the system is characterised by a certain mean or average kinetic energy of its molecules, which is
MAXWELL'S LAW 9 maintained as long as the temperature is maintained constant. This average kinetic energy may be written as ^J^mc", where c is the sum of the squares of the velocities of all the molecules at any given instant divided by N. This r-?n-s speed is not the same thing as the mean or average speed, which is simply the sum of all the velocities divided by N, though the two quantities are not very different numerically. The above way of expressing Maxwell's law, i.e. the above expression, is of the general form : y = x'^e'^^ which, it is to be observed, is not identical with the normal law of errors. The resulting curve is, in fact, not The alternative mode of expressing Maxwell's law where- symmetrical. in the curve is symmetrical, and the distinction between the two modes, is considered in Appendix I. Maxwell's law expresses the distribution of velocities as a continuous function of the number of molecules present. That is, the speed of one molecule may differ by any amount (down to the infinitesimally small) from the speed of any other molecule. When we come to consider the quantum theory we shall find continuous functions replaced by discontinuous ones, i.e. abrupt changes in finite small steps in place of gradual change in infinitely small steps. ^ It will be observed that Maxwell's expression involves the squares of velocities. Since the kinetic energy of a molecule depends upon the square of its velocity, it should be possible, on similar lines, to express the distribution of kinetic energy amongst the molecules constituting a gas system. If we denote by dn the number of molecules which possess kinetic energy lying between the limits E and E + ^, it can be shown that at a given temperature— dn = constant x N x ^"^/At , E"^ .