FOUNDATIONS OF THE KINETIC THEORY OF GASES. 263 



^ v-v(l {V ~^ \ 1 ( m I a V- g (V>\ 



In these equations j5, v, £ belong to the critical point, determined by the conditions that 

 at such a point p is a minimax in terms of v. The special advantage of this mode of 

 representing the isothermals depends on the fact that the first part of the value of p 

 belongs to the critical isothermal ; so that by comparing, at any one volume, the 

 pressures in different isothermals (as given experimentally) we have a comparatively 

 simple numerical method of calculating the values of some of the constants in the 

 equation. 



67. But, even if we were to regard the formula of Van der Waals as a purely 

 empirical one, there is a fatal objection to it in the fact that it contains only two dispos- 

 able constants. Thus, if it were correct, the extraordinary consequence would follow that 

 there is a necessary relation among the three quantities, pressure, volume, and tempera- 

 ture, at the critical point :— so that, no matter what the substance, when two of these 

 are given the third can be calculated from them. I do not see any grounds on which 

 we are justified in assuming that this can be the case. Certainly, if it were established 

 as a physical truth, it would give us views of a much stronger kind than any we yet 

 have as to the essential unity of all kinds of matter. Van der Waals seems to have 

 taken his idea in this matter from one of Andrews' papers, in which there is a 

 hazardous, and therefore unfortunate, speculation of a somewhat similar character. Any- 

 how, it would seem that, at least until experiment proves the contrary, we are bound to 

 provide, in our theoretical work, for the mutual independence of at least the three follow- 

 ing quantities : — 



1. The diameters of the particles. 



2. The range of sensible molecular force. 



3. The maximum relative potential energy of two particles. 



Besides these, there is the question of the law of molecular force, which we are certainly 

 not entitled to assume as necessarily the same in all bodies. This has most important 

 bearings on the formation of doublets, triplets, &c, at lower temperatures. 



The modified formula of Clausius has one additional constant, and is therefore not 

 so much exposed to the above objections as is that of Van der Waals. Still I think it 

 has at least one too few. 



XX. — The Virial Equation for attracting Spherical Particles. 



68. What is required is not an exact equation, for this is probably unattainable even 

 when we limit ourselves to hard spherical particles. To be of practical value the equation 

 must (while presenting a fair approximation to the truth) be characterised by simplicity. 

 And, should the experimental data require it, we must be prepared to give the equation 

 of any one isothermal in two or more forms, corresponding to various ranges of volume. 



