John Butcher
ond Quintus Fernando University of Arizona Tucson
I
I
Theoretical Error in Acid-Base Titrations
The equations for titration curves of acid-base, precipitation, complex-formation, and redox reactions can be readily derived. For example, the condition of electroneutrality of the system gives the equation for the titration curve of an acid versus a base. In the calculation of a series of points on such a titration curve, the student is forced to realize the relative magnitude of each of the terms in the equation and, consequently, the relative importance of the reactions that govern the shape of the titration curve at the various stages of the titration. The pedagogic value of such calculations is obvious, and analytically important titration curve calculations are carried out with an increasing degree of sophistication from the freshman through the senior levels. We have found that the calculation of theoretical titration errors, before and after the equivalence point in a titration, presents a much more challenging problem to students, and considerably increases their nnderstanding of the chemical behavior of the systems involved. The theoretical titration error is that emor which results when the end point of a titration differs from the equivalence point. This quantity is important in chemical analysis in that it specifies the minimum possible error in a titration procedure, exclusive of any experimental error (1, 9?, 5). Unfortunately, titration error calculations have been totally ignored in most analytical texts. In some instances where such calculations have been described, a nonrigorous approach is more confusing than helpful to the student. In one recent paper, an equation has been derived for the calculation of the error in titration of monobasic acids with monoacidic bases (4). This is the simplest and most rigorous approach to acid-base titration error calculations that has been reported. The purpose of the present report is to describe simple derivations of equations for titration error calculations for a number of practical acid-base titrations, and to show error curves (% error versus pH error or end point pH) for some typical cases. A graphical method for obtaining titration errors is also described. This method emphasizes the relative importance of the terms in the error equations and helps students visualize the relative magnitude of each of these terms a t various end point pH values. For the case of a titration of a monobasic acid with a strong base, the calculations have been extended to include the effect of ionic strength changes during the titration. Monobasic Acid
The following equations are relevant to the description of a titration of monopmtic acid, H A , with a uniunivalent strong base. The acid dissociation constant: 546 / Journal of Chernicol Education
The material balance for the acid: CA = [HA1 [A-I The electroneutrality equation: [Ht1 C s = [OH-] [A-1 The ion product of water: K . = [H+][OH-]
+
+
+
(3)
(4)
Here, CA and Ca are the formal concentrations of acid and base, respectively, in the titration solution. The fraction of the acid titrated, X, is given by
and the fractional titration error, E, is E = -C B - C A = ~ - ~ Cn
(6)
When defined in this manner, the titration error is negative before the equivalence point is reached, and positive after the equivalence point. The per cent titration error is simply 100 X E. Substitutionof eqns. (I), (2), (3), and (5) into equation (6) gives an expression for the fractional titration error as a function of the hydrogen ion concentration a t the end point, E =
[OH-] - [H+] CA
1
KA l+m
(7)
where [OH-] is,. of course, calculated from eqn. (4). It is noteworthy that the pH a t the equivalence point does not enter into this expression. When the end point occurs well before the equivalence point, the titration error is given by the second term in eqn. (7)
The derivation of equation (8) has been described previously (6). Somewhat after the equivalence point, the titration error is a measure of the excess base added to the system and is given by the first term of eqn. (7)
In most cases, however, it is of interest to calculate the titration error in the region where the end point occurs near the equivalence point; when the end point is less than about 0.7 pH units removed from the equivalence point, eqns. (8) and (9) fail, and eqn. ( 7 ) must be used.
The relationship between these three equations is demonstrated by Figure 1.
The approximate equation in this case predicts a much larger error than the actual theoretical error incurred. Some sample titration error calculations, performed using eqn. (7), are presented in Figure 2.
Figure 1. Titration error as colcvloted from equations (71, (81, and (9). C* = 0.1 F; ~ K = A 5.
It is of interest to examine the reasons for the failure of eqns. (8) and (9). The right hand side of eqn. (8) is the expression for aQ,the fraction of the acid present in the undissociated form. Before the equivalence point, this fraction closely approximates the fraction of the untitrated acid in solution. Kear the equivalence point, however, hydrolysis of the relatively large amount of acid anion causes the approximation to fail. I n the alkaline region, the right side of eqn. (9) measures the titration error in terms of the excess base present in the solution. Well after the equivalence point, this quantity sufficiently approximates the error. Near the equivalence point, however, hydrolysis of the acid anion has an important effect upon the pH of the system, so that the approximation fails. Also noteworthy is the fact that when the acid is a strong electrolyte, such as a mineral acid, I
