30 PROCEEDINGS OF THE AMERICAN ACADEMY, 



where tt is the electromotive force, Vi and v^ are the moleciiLar volumes 



of the metal iu tlie two amalgams, and q is the heat of the process in 



electrical units. From this, 



v 

 neoTT = B Tin ~ + U, 



but n Pq -t is the electrical work per gram-molecule, which is equal to 

 the change of free energy, since the cell is a reversible one. Therefore 

 yl =r n Co TT, or 



^ = ^rin^+ u, 



which is identical with (43 ci). 



When we consider solutions of all concentrations, varying from the 

 state where one of the constituents of the phase is in great excess to the 

 state where the other constituent is in great excess, as, for example, 

 when water is added continuously to a definite amount of alcohol, then 

 the form which the osmotic pressure curve assumes is very complicated. 



Here equation (41 ) must be used, and -— ^ and — — will both be com- 



d i\ a Vi 



plex functions of i\. may be looked upon as the sum of two quanti- 



ties, one due to the attraction of unlike, the other to the attraction of like 

 molecules. Concerning the manner in which the former 'will change 

 we are ignorant. The latter, however, according to reasoning exactly 



similar to that which led van der Waals to the term — , may be 

 shown to be inversely proportional to the square of the volume, or equal 



to -^ — 7, — ^ . We see from this that equation (41) is at least of the 



'11- 



third degree in v\. Bredig * and Noyes f have each proposed a general 

 formula for osmotic pressure based upon kinetic reasoning. Both these 

 equations are of the third degree in v. The osmotic pressure curve 

 represented by equation (41) is not necessarily, therefore, single valued. 

 There may be more than one volume corresponding to one osmotic pres- 

 sure. This is a further analogy between solutions and gases. In fact, 

 a luimber of cases are known in which the osmotic pressure may be 

 shown to be the same at two different concentrations, namely, the cases 

 of liquids that are mutually soluble to a limited extent, thus forming two 

 phases in equilibrium with each other. It is evident that in order to 



* Zeit. Phys. Chem., IV. 444. t Zeit. Pliys. Chem., V. 53. 



