SEMI-PEEMEABLE FILMS AND OSMOTIC PRESSURE. 417 



This equation, in which a single constant may evidently take the 

 place of B and C, may be regarded as expressing the property of the 

 solution implied in van't HofFs law. For we have the general thermo- 

 dynamic relation (ibid. p. 143) [this vol., p. 88]. 



where v and r\ denote the volume and entropy of the mass considered, 

 and mj and m 2 the quantities of its components. Applied to this 

 case, since t and fa are constant, this becomes 



Substituting the value of d/u. 2 ", derived from the last finite equation, 

 we have 



whence, integrating from y 2 " = and p" =p', we get 



p"-p' = a 2 ty 2 ", 



which evidently expresses van't Hoff's law. 



We may extend this proof to cases in which the solutum is not 

 volatile by supposing that we give to its molecules mutually repulsive 

 molecular forces, which, however, are entirely inoperative with respect 

 to any other kind of molecules. In this way we may make the 

 solutum capable of the ideal gaseous state. But the relations per- 

 taining to the contents of R" will not be affected by these new forces, 

 since we suppose that only a negligible part of the molecules of the 

 solutum are within the range of such forces. Therefore these relations 

 cannot depend on the new forces, and must exist without them. 



To give up the condition that the molecules of the solutum shall 

 not be broken up in the solution, nor united to one another in more 

 complex molecules, would involve the consideration of a good many 

 cases, which it would be difficult to unite in a brief demonstration. 

 The result, however, seems to be that the increase of pressure is to be 

 estimated by Avogadro's law from the number of molecules in the 

 solution which contain any part of the solutum, without reference to 

 the quantity in each. J. WILLARD GIBBS. 



New Haven, Connecticut, February 18. 



G.I. 2D 



