58 HYDROGEN ION CONCENTRATION 



values of both dissociation constants of the acid; in fact it depends 

 upon the ratio ki:k2. If ki is much greater than k2, then a wide 

 rather flat maximum is formed in which ai is almost = 1 as in 

 figure 5. When ki is but a little greater than ko, as in figure 6, 

 then the maximum has a sharper angle and it does not quite reach 

 the value 1 ; and when the values of ki and k2 are still closer together 

 then these two characteristics are even more pronounced as shown 

 in figure 7. The limiting case is where ki = k2, i.e., when the acid 

 is capable of dissociation to the same extent in both stages. In 

 this case ai is always = 0, which means that only undissociated 

 acid molecules and divalent acid ions are formed and no univalent 

 ions. This last case is not met with in weak electrolytes. The 

 two dissociation constants have more or less differing values. They 

 approach each other quite closely in the case of succinic and fumaric 

 acids, while the two constants differ very widely in maleic and 

 carbonic acids. Wherever the two acid (COOH) groups are spa- 

 tially close to each other in the molecule, as in maleic acid, then the 

 acid nature is quite pronounced and the acid is strong in the primary 

 stage, while in its second stage only a small residue of acid is present 

 in excess. But if the two COOH — ^groups are situated far apart, 

 then each behaves in a manner more independent of each and 

 exert a more nearly equal effect. 



The position of the maximum in the ai-curve is of particular 

 interest. In order to evaluate the maximum it is necessary to 

 differentiate the function ai = f ([H+]) with respect to [H+] and to 

 set the derivative = 0. It is much easier to calculate the minimum 

 or the inverse function which is 



and whose derivative is 



OCl_ _ J^ _ k-2 



d [H+] ~ ki [H+l^ 



By setting this derivative = we obtain the minimum for J_ 

 or the maximum for ai ai 



[H"] = Vki X k2 



