55 



All of their calculated curves coincided completely with Proctor's 

 experimental ones, and for this reason the writer considers the essential 

 part of the theory as proved. The importance of the subject warrants 

 our giving a review of the mathematical deductions, which include an 

 adequate explanation of Procter's theory. 



Consider the purely hypothetical substance O which is a colloid 

 jelly completely permeable to water and all dissolved electrolytes, 

 is elastic and under all conditions under consideration follows Hooke's 

 law, and which combines chemically with the positive, but not the 

 negative ion of the electrolyte MN, according to the equation — ■ 



[G]x[M+]= K[OM+] (1) 



In other words, the compound OMN is completely ionized into 

 0M+ and N^. The brackets indicate that concentration is meant 

 and all concentrations are in moles per litre. The electrolyte MN is 

 also considered totally ionized. 



Now take one miUimole of G and immerse it in an aqueous solution 

 of MN . The solution penetrates O, which thereupon combines mth 

 some of the positive ions, removing them from solution, and conse- 

 quently the solution within the jelly will have a greater concentration 

 of iV~ than of JIf + , while in the external solution [ilf+] is necessarily 

 equal to [iV^~]. The solution thus becomes separated into two phases, 

 that within and that surrounding the jeUy, and the ions of one phase 

 must finally reach equilibrium with those of the other phase. 



At equilibrium, in the external solution, let — 



x= [if+]= [N-] 

 and in the jelly phase let — 



2/=[M+] 

 and— 



z= [QM+] 

 whence — 



The relation existmg between the concentrations of diffusible ions 

 of the two phases at equilibrium can be derived from the consideration 

 of the transfer of an infinitesimaUy small amount, dn moles, ot M + 

 and N~ from the outer solution to the jelly phase, in which case, 

 since no work is done — 



dnRT log xjy + dnRT log xj{y -\- z) = Q, 

 whence — 



x^ = y{y + z). (2) 



But in this equation, the product of equals is equated to the product 

 of unequals, from which it follows that the sum of those unequals is 

 greater than the sum of the equals, or — 



2y-{-z> 2x. 



This is a mathematical proof that the concentration of diffusible 

 ions of the jelly phase is greater than that of the external solution, 

 and makes possible the derivation of a second equation involving e, 



