OSMOTIC AND MEMBRANE EQUILIBRIA 197 



the value of g' being that at an external pressure p' somewhat 

 greater than P. 



Comparing (64) with (45) we see that, when we neglect the 

 compressibihty, the osmotic coefficient is the ratio of the actual 

 osmotic pressure in a non-ideal solution to its value in an ideal 

 solution of the same composition. This is the origin of the name 

 "osmotic coefficient." 



13. Extremely Dilute Solutions. If a solution, whether ideal 

 or non-ideal, is so dUute that the mol fractions N, of all the 

 solute species are extremely small compared with that of the 

 solvent A^o, we may make the three approximations: 



log ^^ = - log (l - S ^•) = S ''•• ^^^'^ 



N. = ^^^ = ^'^ (66) 



mo 



Wo 4- 7 , ms 



s 



V = moVo -\- 2j ^« ^» = ^0 1'o*. (67) 



8 



Formula (45) for ideal solutions then takes the approximate 

 form 



P = ~^rn, = At^ y., (68) 



where 7, denotes volume concentration. Similarly formula (46) 

 for non-ideal solutions takes the approximate form 



P =gAt^y,. (69) 



s 



Formula (68) is contained in some fragmentary material by 

 Willard Gibbs published after his death (Gibbs, I, 421, equation 

 [7]). For its approximate validity it is necessary to assume 

 not merely that the solution is ideal and incompressible, but also 

 that it is extremely dilute. This formula was originally due to 

 van't Hoff, who realised its limitations. It has unfortunately 



