Pressures in Liquid Mixtures. 107 



Therefore, by (27) and (23), 



d r =Y s dx s for x=l (5=1, 2, 3, ... , «), (30) 

 that is, by (4), 



a*." 1 - 



for x = 1 (r, s = 1, 2, 3, . . . , *— 1 and 



3 ,=0j K40) 



^=-P>r*. = l (r = l,2,3,...,«-l). 



Equations (40) express Raoult's law, which is thus seen to 

 hold for a mixture of any number of components and to be 

 independent of any assumptions excepting those made in 

 conditions a—g *. 



5. Going back to equation (35), we assume that the func- 

 tions xl can be developed according to positive integral powers 

 of the s's ; this is equivalent, by (33), to the assumption that 

 the u's can be developed according to positive integral 

 powers of the #'s. Then, by (34), co can be similarly 

 developed. Any term of such a development is of the form 



Jr jg* & ^k-i 



*1 *2 ~3 »""'*«-l 



multiplied by a constant coefficient, where each of the 

 exponents <7i, g<& #3* •••>#«-! is any positive integer or 0. 

 Let the coefficient of the product of powers of the z's just 



written in u be denoted by a and the coefficient 



r J 9v9v9v-i9 K -\ 



* Gahl (Zeitschr . fiir physikalische Chemie, vol. xxxiii. pp. 192-195) 

 lias considered what might happen if the last part of our condition e were 

 not satisfied, — but his cases are purely hypothetical. Considering only a 

 binary mixture, he assumes that the partial pressure p of one component 

 is proportional to a power of the corresponding molar fraction x whose 

 exponent is an integer as great as 2, — whereas it is not certain even that 

 this pressure can be developed according to integral poM ers of x. He 



savs that it often happens that ~ = for x = 0, but cites no specific 



case. It may be well doubted whether a mixture can have any com- 

 ponent for which this condition is satisfied unless p = for every value 

 of .r, in which case the component has no pressure of its own and the 

 Duhem-Margules equation is not proved for any mixture that contains it. 



