46 IDEAL AND ACTUAL GASES 107 



real gases and not of ideal gases only. The point in which 

 it most especially needs improvement is the manner in which 

 the law of cohesion is introduced. 



47. Completions of van der Waals's Formula 



Properly recognising this imperfection, Clausius 1 has 

 attempted to improve van der Waals's formula by sub- 

 stituting another expression for the value of the cohesion. 

 His formula, which has the form 



{p + a(v + /3)- 2 } (v - b) = B(l + aS), 



differs from that of van der Waals essentially in this: 

 that the cohesion-pressure is put inversely proportional, 

 not simply to the square of the volume v, but to the square 

 of the volume v increased by a constant /3. By this means 

 Clausius obtains a better agreement of the formula with 

 observations that have been made under high pressures, 

 and therefore with small volumes. 



Earn say and Young 2 think it more correct to sub- 

 stitute the more general expression of an nth power instead 

 of that of the square of the volume v. 



A second difference between the formulae of Clausius 

 and van der Waals consists in this : that the magnitude 

 a is with Clausius not a constant, but a function of the 

 temperature. As he first employed his formula only in the 

 case of carbonic acid, he could be content with the assump- 

 tion of the simple formula 



<-. 



in which c is a constant and <) denotes the absolute tem- 

 perature 



= 3 + a' 1 . 



Later on, when trying to apply the formula to other gases 

 and vapours, he assumed a more general expression, 



a = (A- n - B)R(l + aty, 



1 Wied. Ann. ix. 1880, p. 337; xiv. 1881, pp. 279, 692; Mechanische 

 Warmetheorie, 1889-91, iii. pp. 184, 21,5, 227. 



2 Proc. Boy. Soc. xlii. 1887, p. 5. 



