Polybasic Weak Acids Indistinguishable from Monobasic by Titrimetry
Louis Meites Clarkson College of Technology Potsdam, New York 13676
AS
has been shown by Adams ( I ) and Bjerrum (Z), the statistical lower limit of the ratio K I / K2 of the first and second dissociation constants of a dibasic acid is equal to 4. Ricci (3) showed that the potentiometric titration curve obtained on titrating a dibasic acid with base is exactly identical with that ohtained for a monobasic acid if Kl/K, for the dibasic acid has the statistical lower limit of 4, if K. for the monobasic acid is given by K. = K1/2 = 2K2, and if the initial concentration of the monobasic acid is twice that of the dibasic one. Ricci (4) further showed that the statistical lower limits of the ratios KJKz and Kz/K3 for a tribasic acid are both equal to 3. Sturrock (5), on the basis of the simple form of the Henderson cquation (6), showed that the difference A(pH) between the pH-values a t the points three-quarters and one-quarter of the way from the start of the titration to the equivalence point is equal to 0.954 unit for a monobasic acid, for a dibasic acid with K1/K2= 4, or for a tribasic acid with K1/K, = K2/K3= 3, the last equivalence point being used in each of the latter cases. On this basis he suggested that monobasic and polybanic acids could be distinguished by evaluating A(pH), which he considered to be equal to 0.954 for mooobasic acids but to exceed this for polybasic acids having dissociation constants that do not conform to the statistical ratios. Lomax and Bard (7) showed this prediction to be erroneous for very dilute solutions of fairly strong monobasic acids, and Hamann (8) showed it to be erroneous for extremely weak acids as well, but neithcr of these papers took the effects of dilution into account. The present manuscript generalizes these results. It presents a general equation describing the statistical ratio K,/Kc,+lj of the ith and (i 1)th dissociation constants of a j-basic acid. It shows bhat any j-basic acid whosc dissociation constants conform to these ratios will give a titration curve indistinguishable from that obtained for a monobasic acid a t an appropriate concentration. Finally, it shows that Sturrock's criterion for distinguishing "non-statistical" polybasic acids (i.e., those having dissociation constants whose ratios differ from the statistical ones) from monobasic ones is, like the Hcndcrson equation itself, applicable only over a restricted range of conditions; though it is convenient and useful within that range, it is in principle neither necessary nor sufficient.
j-basic acid H,A. The assumptions leading to the statistical ratios of two dissociation constants are (1) that the ratio of the rate constants for the dissociations of any two acidic speoies is equal to the ratio of the numbers of their protonated sites, so that kilk~i,,, = ( j - i
+ l ) / [ j - ( i + 1) + 11 =
j
-i + 1
-i
(2)
(2) that the ratio of the rate constants for the recombination of hydrogen ion with the corresponding basic species is equal to the ratio of their numbers of unprotonsted sites, so that
k-i/k-~i+,>= lj - (3 - i)l/lj - lj
- ( i + 1)Il =
i
1
(3)
Since
it is easily shown by combining eqns. (2), (3), and (4) that K
