OSMOTIC AND MEMBRANE EQUILIBRIA 187 



and is the characteristic function corresponding to our choice 

 of independent variables t, p, rrii, rih, ... Wn. The dependence 

 of variations of f on those of the independent variables is 

 given by 



d^ = —r]dt-\- vdp + tildmi + )U2C?W2 . . . + Undnin. (19) [92] 



From (19) [92] we see that 



drtih 

 and 



dp 



= MA, 



(20) 



= V. 



(21) 



Substituting from (20) and (21) into (17) we obtain 



dnh dv 



dp drrih 



= vh, (22) 



where Vh denotes the increase in volume of a very large phase 

 when one adds to it unit quantity of the species Sh, keeping the 

 temperature and pressure constant. The volume Vh may be 

 called the "partial volume" of the species Sh. 



5. Mols and Mol Fractions. Up to this point we have 

 purposely referred to nih as denoting the number of "units of 

 quantity" of the species Sh without specifying what is this 

 "unit of quantity." Willard Gibbs, living at a time when the 

 molecular theory was less firmly established than at present, 

 chose the same unit of mass for the unit of quantity of each 

 species. In a letter to W. D. Bancroft (Gibbs, I, 434) he 

 agrees, however, that "one might easily economise in letters 

 in the formulae by referring densities (7) and potentials (n) to 

 equivalent or molecular weights." We shall therefore adopt 

 this procedure and take as unit quantity of each species the 

 gram-molecule or mol in the highly dilute vapor state. None 

 of the formulae so far given are affected, but the potentials 

 fi now have the dimensions calories per mol instead of calories 

 per gram, and the formulae expressing the dependence of the 



