186 GUGGENHEIM art. e 



where p/q is the ratio in which methylene and water combine 

 to form methyl alcohol. Substituting (14) into C13) we obtain 



PMcH, + ^Mh,o = P^CH. + e^ao. (15) 



But according to [121] and the definition of the ratio it follows 

 that (15) is equivalent to 



MCH4O = MCH«0, (16) 



the same as (12). We see then that the complications discussed 

 by Gibbs in the paragraph preceding [78] can be avoided if we 

 always include among the independent components all those 

 species which can pass through the membrane independently. 



4. Choice of Independent Variables. Although the conditions 

 for any membrane equilibria are completely contained in 

 Gibbs' formula [77] it is advantageous from a practical point of 

 view to transform this into a form involving quantities more 

 directly measurable than the potential n. For this purpose it is 

 most convenient to choose as independent variables the tem- 

 perature t, the pressure p and the number mi, nh, . . . nin oi 

 units of quantity of the various species *Si, S2, ... Sn. The 

 potentials m, m, ... /in in each phase will then be regarded as 

 functions of t, p, Wi, nh, • . . Mn. 



The manner of dependence of the potentials mij M2, • • • Mn 

 on the temperature t need concern us very httle as we shall 

 always deal with systems maintained at a given constant tem- 

 perature throughout and shall not need to consider tempera- 

 ture variations. The manner of dependence of the potentials 

 Mi> ^2, • • • Mn on the pressure p is, on the other hand, of funda- 

 mental importance in the treatment of membrane equiUbria 

 because in general the pressures of two phases in membrane 

 equihbrium will be unequal. The required relation is obtained 

 by making use of the mathematical identity 



dp dnih dvih dp 

 where ^ is defined by 



^ = e-tv + pv, (18) [91] 



