Distribution e~ 2h x and van der Waal's Theorem. 415 



art. 13, when the density is so great that no distribution 

 except a perfectly uniform distribution can exist without 

 making ^ infinite. In every other case local deviations from 

 the mean density mu«t occur. These deviations do not affect 

 Clausius' equation (7), but they do affect the value of U 

 (art. 16). As a> changes the law of the deviations changes, 



and therefore U cannot generally vary as — 2 . The law e~ 2Av , 



and van der Waals' formula, each true in its own place, 

 cannot coexist. 



In the second factor (g> — b) of van der Waals' formula, 

 b denotes the diminution of effective volume due to the 

 molecules being elastic spheres. As stated, ante art. 15, the 

 collisions of these elastic spheres (if such existed) would 

 make for uniformity of distribution, but could not produce 

 complete uniformity. In like manner if the molecules be, as 

 Boltzmann suggests, centres ot repulsive force varying as the 

 fifth or other high inverse power of the distance, these re- 

 pulsive forces make for uniformity of distribution, but cannot 

 produce complete uniformity. The attractive force assumed 

 by van der Waals, as the only finite intermolecular force, 

 not only does not make for uniformity, but has the opposite 

 effect. With attractive forces the perfectly uniform distri- 

 bution assumed to exist always and everywhere is (if Boltz- 

 mann' ] s law €~ 2h * applies to the substance in question) the most 

 improbable of all possible distributions. 



20. Boltzmann's law, then, and van der Waals' equation 

 (8), though both true, are not both true of the same substance 

 in the same state. Now Boltzmann's law is true for gases. 

 If van der Waals' theorem is also true for gases, the term 

 gas is ambiguous. It represents two different states sepa- 

 rated, at least mathematically, by a sharp dividing line, on 

 one side of which the law e~ 2h x prevails, on the other van 

 der Waals'. 



But Boltzmann's analysis, involving € -hzm(v*+vi+n*) f or tne 

 law of distribution of velocities, admits of no abrupt change. 

 There is no point, i. e. no state, at which you can with any 

 reason draw the line, and say that the theorem ceases at that 

 point to hold. It is true, indeed, that the gas being com- 

 pressed at constant temperature, an abrupt change must, from 



physical reasons, take place when -^-=0, or shortly before 



that point is reached. But the change in question is lique- 

 faction. And if the law e~ 2 >* prevails in all non-liquid 

 states, then it would seem that van der Waals' theory must 

 be confined to liquids, if in that case it could exist at all, 



