( 228 ) 



which have been soh'ed in general by van dkii Waals, the following 

 simplifications have been made : 



r. The first form with two constants a and h, in which van dkr 

 Waals has written the equation of state, is used instead of (he real 

 equation of state ; 



2". The supposition has been made that also the equation of state 

 of each mixture with the proportion x has the same form with 

 two constants ax and 6,, whereby the law of corres})ondin,o: states is 

 rigorously satisfied ; 



3". it has been supposed that the critical quantities determined by 

 ax and h^. are related to those of the simple substances detei-iniucd 

 by rtji and />„, a^^ and h.^„^ hy means of the relations 



a, = «„ .^■^ 4- 2 a,, x (1 - x) + a,^ (1 - ,r)^ 



hx = h,, ^v' + 2 b,, ,v (1 - ,r) + ?>,, (1 - x)\ 



SO that the entire behaviour of the mixtures of two known substances 

 is determined by two additional constants a^^ and h^^ ; 



and 4". it has often been assumed that the vapour phase satisfies 

 the laws of ideal gases. 



In this way van der Waals has obtained important approximation 

 formulae. Though they do not always represent numerically accurately 

 the behaviour of the mixtures, most of the particularities of the con- 

 ditions of coexistence are sufficiently explained by these in general 

 valid formulae ^). 



If in the treatment of these problems we want to use equations 

 of state which over the whole range of temperatures and pressures 

 agree accurately with the observations, if therefore we do not require 

 the simplifications ad 1", 2" and 3°, and if finally we want to 

 consider other than rarefied vapour phases, so that the neglection of 

 the deviations from the law of Boyle-Gay Lussac-Avogadro mentioned 

 ad 4" is not allowed, an analytical treatment of the conditions of 

 coexistence in general becomes impossible. 



Such problems we come across, for instance, when we derive 

 the conditions of coexistence for mixtures of oxygen and nitrogen 

 (critical state of air and relation between composition anil pressure 

 in the boiling off liquid air) from equations of state which also at 

 the ordinary temperature accurately represent the compressibility of 

 these substances and of their mixtures. An instance of an entirely 

 different kind is given by the following group of problems : determine 

 at ordinary temperature the absorption of hydrogen in ether and the 



1) VAN DER Waals, Die Gontinuitüt etc. II p. 52. 



