Page:Elementary Text-book of Physics (Anthony, 1897).djvu/252

238 exhibited by gases from the gaseous laws can be accounted for by extending the theory so as to include the consideration of the size of the molecules and of their mutual attractions. In the elementary theory the molecules were assumed to be points or particles of negligible magnitude, but if we assume them to have volumes which, though small, are appreciable, it is plain that the effective volume within which the molecules have free motion is reduced by an amount dependent on the molecular volumes. It was furthermore assumed in the elementary theory that the time of encounter is negligible in comparison with the time during which the molecule is free from the action of other molecules; but if we assume that the time of an encounter, though small, is not negligible, it is plain that the molecular attractions will tend to hold together the mass of gas or will be equivalent to an addition to the pressure upon the gas. From these considerations van der Waals expressed the relations among the pressure, volume, and temperature of a gas by the formula where $$a$$ is a constant depending upon the molecular attractions, and $$b$$ is four times the sum of the volumes of the molecules. This formula, when tested by experiment, represents the behavior of gases far more accurately than the simpler form; it is not, however, exact, and various others, constructed empirically, have been proposed which give even a better representation of the facts. It is as yet the only formula for which a theoretical demonstration has been given. This formula possesses the great advantage that it can represent the behavior of a body, at least in certain cases, not only in the gaseous but in the liquid state; that is, it exhibits the continuity which we have every reason to think exists between those states. In particular it gives an explanation of critical temperature and a determination of it in terms of the molecular constants $$a$$ and $$b.$$ If the formula be expanded and arranged in the order of the descending powers of $$v,$$ it becomes $$v^3 - v^2 \left( b + \frac{RT}{p} \right) + v\frac{a}{p} - \frac{ab}{p} = 0.$$