416 SEMI-PERMEABLE FILMS AND OSMOTIC PRESSURE. 



to that component, the entropy and volume of the mass and the 

 quantities of its other components remaining constant. The advantage 

 of using such potentials in the theory of semi-permeable diaphragms 

 consists pirtly in the convenient form of the conditions of equilibrium, 

 the potential for any substance to which a diaphragm is freely per- 

 meable having the same value on both sides of the diaphragm, and 

 partly in our ability to express van't Hoff's law as a relation between 

 the quantities characterising the state of the solution, without reference 

 to any experimental arrangement (see Transactions of the Connecticut 

 Academy, vol. iii, pp. 116, 138, 148, 194) [this vol., pp. 63, 83, 92, 135]. 

 Let there be three reservoirs, R', R", R'", of which the first contains 

 the solvent alone, maintained in a constant state of temperature and 

 pressure, the second the solution, and the third the solutum alone. 

 Let R' and R" be connected through a diaphragm freely permeable 

 to the solvent, but impermeable to the solutum, and let R" and R'" 

 be connected through a diaphragm impermeable to the solvent, but 

 freely permeable to the solutum. We have then, if we write fa and 

 yu 2 for the potentials of the solvent and the solutum, and distinguish 

 by accents quantities relating to the several reservoirs, 



f*i" = Pi = const., fa" = jjL 2 '". 



Now if the quantity of the solutum in the apparatus be varied, the 

 ratio in which it is divided in equilibrium between the reservoirs R" 

 and R" 7 will be constant, so long as its densities in the two reservoirs, 

 y. 2 ", y 2 '", are small. For let us suppose that there is only a single 

 molecule of the solutum. It will wander through R" and R'", and in 

 a time sufficiently long the parts of the time spent respectively in 

 R" and R'", which for convenience we may suppose of equal volume, 

 will approach a constant ratio, say 1:B. isow if we put in the 

 apparatus a considerable number of molecules, they will divide them- 

 selves between R' and R" sensibly in the ratio 1 : B, so long as they 

 do not sensibly interfere with one another, i.e., so long as the number 

 of molecules of the solutum which are within the spheres of action of 

 other molecules of the solutum is a negligible part of the whole, both 

 in R" and R'". With this limitation we have, therefore, 



72 = 



Now in R'" let the solutum have the properties of an ideal gas, which 

 give for any constant temperature (ibid. p. 212) [this vol., p. 152] 



where Oj is the constant of the law of Boyle and Charles, and C 

 another constant. Therefore, 



