of Chemical Potential to the Theory of Solutions, 41 



The pressures under which two solutions of different con- 

 centrations coexist in osmotic equilibrium are connected with 

 the vapour pressures of the two solutions by the equation 



rw 



J VO, 6)clv=(p^U)F (s 1 , , 2 , . . . , s M U^p, 6) 



-Q/-.n')P (V, *', . .. , ^,ny, $), (24) 



where, as in equation (9) of Part I., quantities relating to 

 one of the solutions are distinguished by accents from the 

 corresponding quantities relating to the other solution. 

 Since any of the concentrations may have the value zero, 

 this equation is applicable to two solutions in the same solvent 

 of two totally distinct sets of solutes. 



The theory of the coexistence of solution and solvent 

 vapour under different pressures is easily generalized. The 

 necessary alterations in equations (2) and (4) of Part II., 

 and in the theorem following this latter equation, are 

 obvious. 



It will be seen that the extension of the theory to solutions 

 containing any number of involatile solutes is very simple. 

 The extension may be made more simply still, however, by 

 means of an important principle in the theory of chemical 

 potential. This principle is that the choice of the components 

 of a system may be made quite arbitrarily so long as every 

 possible variation in the composition of any part of the system 

 may be specified by variations of the masses of the com- 

 ponents chosen *. Thus in the case of a solution containing 

 two actual substances S and Si, we may choose any imaginary 

 compound of S and S x in sufficient amount to use up the 

 whole of S (or S x ) as one of the components, and the 

 additional amount (positive or negative) of Si (or S ) as the 

 other compouent. If we choose a compound containing a 

 mass r of Si per unit mass of S , then a solution containing 

 masses M of S and M x of S x may be regarded as containing 

 either masses M (l + r) of the compound and Mi — rM of 



Si, or masses M 1 (l + -)of the compound and M Mx 



■of S . V rJ { r 



In the case of a solution of n solutes Si, S 2 , ... S ;i we can 

 choose an imaginary compound S consisting of all the solutes 

 in a certain proportion, making the amount of the compound 

 sufficient to use up the whole of any one solute. The com- 

 position of any solution may be specified by the masses, per 

 unit mass of solvent, of this compound and of the remaining 

 {n— 1) solutes. The masses of the solutes may, of course, 

 * Gibbs, Collected Works, vol. i. p. 63. 



