The Kinetic Theory of Gases

If we fix our attention on any one component of velocity, say u, disregarding the values of v and w entirely, it is found that the number of molecules for which this component lies between u and u+du is This partition of values of u is shown graphically in the thin curve in fig. 1. The numbers along the horizontal axis represent values of u, and the corresponding height of the thin curve is proportional to the total number of molecules which have this particular value of u in Maxwell's steady state. The whole area of this curve is equal to or 177.25, of the small squares. Let us, for instance, suppose that each of these small squares represents a million molecules, then we find that in a gas of 177,250,000 molecules, rather less than 20 million have values of u between o and o.i, about 19 million have values of u between 0.1 and 0.2 and so on, while less than a million have values between 1.7 and 1.8, and less than 40,000 have values between 2.4 and 2.5. These values of u are measured on an arbitrary scale which has not yet been determined, but which, as we shall immediately see, depends on the total amount of heat motion, and so on the temperature of the gas. But whatever the scale on which u is measured, we see that high values of u are very rare, the majority of molecules keep to reasonably small values and the favourite value of all is zero. In many problems the total velocity of a molecule is of greater importance than its individual com ponents. If is written for so that c is the total velocity of a molecule, it can readily be deduced from formula (I) that the total number of molecules for which c lies between c and c+dc is This distribution of velocities is represented graphically by the thick curve in fig. I. We see that both very high and low values of c are now very rare, the favourite value for c being unity. This explains the meaning of the scale on which velocities have been measured. The kinetic energy of a molecule is and on averaging over all molecules it is found, from formula (3), that the average kinetic energy of all the molecules of the gas is 4h. The total heat energy of the gas is accordingly pro

portional to 1/h. The total heat energy of a gas is, however, known to be proportional to its temperature on the absolute scale ; indeed it is through this property that the absolute scale is most simply defined. Denoting the absolute temperature by T, we see that i/h must be proportional to T.

The actual relation is taken to be where R is a constant, generally known as the "universal gas constant," but sometimes called "Boltzmann's constant." The average kinetic energy of a molecule is now -3-R.T. Comparing this with experimental evaluations of this quantity, it is found that in centimetre-gramme-second-centigrade units.

Equipartition of Energy.

If molecules of different kinds are mixed, as for instance oxygen and nitrogen to form air, and the mixture is then allowed to assume its normal state, it can be proved that in the final state h has the same value for both types of molecules. Since h= iRT this is merely equivalent to say ing that when two sorts of gas are mixed they will in the long run assume the same temperature. The average kinetic energy of a molecule of either sort being •iRT, it follows that the process of mixing causes equalization of the kinetic energies of the different types of molecules. Thus in the normal state the lighter molecules move, on the average, faster than the more massive ones, the masses of the different types of molecules being inversely proportional to the average squares of the veloci ties. This provides the simplest instance of the "Theorem of Equipartition of Energy," a general theorem which proves that the total energy in the normal state is divided up between the different capacities for holding energy, in such a way that on the average each has the same amount.