106 HYDROGEN ION CONCENTRATION 



disappears. Let us now designate a very small amount of added 



dL 

 alkali by dL^ then t-^ becomes the direct measure of the "buffer 



value" of the solution. Let us now calculate this value for certain 

 cases. We can do this graphically by making use of the titration 

 curves shown in figs. 12 and 13 page 99). Since in these curves pH 

 is represented by the ordinates and L by the abscissae, it follows that 

 dL 



T~^ at each point is equal to the cotangent of the angle of inclina- 

 tion of the curve. As it will be readily seen the buffer value is 

 greater for every given amount of alkali with any acid the greater the 

 equivalent concentration of free acid. If this concentration should 

 approach zero then the buffer value also approaches zero. This will 

 mean then that in very dilute acid solutions small amounts of alkali 

 will effect very large changes in the pH. But not only the concen- 

 tration of free acid, but also the nature of the acid itself has an effect 

 on the buffer value. In comparing the curve for HCl and for a weak 

 acid, such as acetic, we observe only in the curve of the latter a flat 

 or wide point of inflection indicating the formation of the maximum 

 of the buffer value, while the minimum is shown by a steep point of 

 inflection. The flat point of inflection or the maximum occurs at 

 half of the equivalent neutralization, i.e., in the presence of equal 

 concentrations of acetic acid and of sodium acetate. • 



This may be demonstrated analytically in the following manner: 

 Let the total amount of acetic acid in the solution be designated by 

 S, the amount of added alkah by L, then S — L = the amount of 

 free acetic acid and L = the amount of sodium acetate. Now in 

 any mixture of acetic acid and sodium acetate we have 



TTT-n 1 ff^ee acid] , S — L 



[H+J = k -— = k 



INa acetate] L 



<-(!-) 



therefore, pH = — log k — I 7 — 1 J and the buffer value v 



d L d L L (S - L) 



V = ■ = — ; r- = — • • (I) 



d pH ^ , /S \ S log e ^ ^ 



d lc~ 



log (I - 1) 



