276 Mr Vegard, On some general Properties 



pressure on the solution and solvent respectively. The relations 

 (1, 2, 3) are quite general for binary solutions. They hold for any 

 concentration without regard to volatility of the compounds, and 

 the fluids may be compressible. 



(6) As a corollary from the three equations it was found that 



in sfeneral ( ^r- 1 !^wd 1 ^r- ) must be different, or the variation 

 ^ _ \dcjp \dcJp, 



of osmotic pressure with concentration will be different according 



as the pressure during the variation is kept constant on the 



solution or on the solvent. 



The problem we shall deal with in the following is to find the 

 distribution in a solution containing an arbitrary number of sub- 

 stances acted on by any field of gravity, provided equilibrium has 

 set in and the tempeiature is constant all through the system. 



We shall treat the problem in two ways: in one we shall use 

 the thermodynamic potentials, in the other we shall make use of 

 the conception of osmotic pressure. But as the osmotic pressure 

 up to the present has only been defined for binary solutions, it 

 will be necessary to extend it to solutions in general. 



Part I. — Solution of the problem by means of 

 thermodynamic potentials. 



1 2. Some properties of thermody mimic potential functions. — Let 

 the solution be composed of {r + 1) substances 0, 1, 2 ... /•, with mo- 

 lecular masses i)/o, i/j ... M,., and let the concentration be measured 

 by the number of gram-molecules in unit volume ??„, "i, »2 ••• >h-, 

 which in the case considered will vary from one point to another. 

 In the followins; the concentrations as well as their first derivates 

 will be considered as continuous functions of the coordinates. In 

 the case of varying concentration, the thermodynamic potentials 

 of the solution will no longer be homogeneous functions with 

 respect to the total mass of each component ; but we must be able 

 to assume, considering the thermodynamic potential Aco for a 

 volume element Av, that when Av approaches zero Aco will 

 approach the value it would have if the element had contained a 

 homogeneous solution with concentrations equal to those at any 

 point inside the element. Thus we can put 



Ao) = If {Anio, Anil , Am^ . . . Am,., p, T), 



where Am^,Am-^ ... Am,, are the masses inside the element and H 

 is the same function as in the case of a homogeneous solution. T is 

 the temperature, which is supposed constant, p the pressure on 

 the element. Putting A»i, = il7,»(Ai' : 



Lim -r— = 0)' = H (MqUo, il/i?2i . . . M,.n,., p, T) ; 



Av 



