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Chapter 9 Titrimetric Methods of Analysis 287

diprotic there are two equivalence points, each requiring the same volume of
titrant. Before the first equivalence point the pH is controlled by a buffer consisting
–
–
2–
of H 2 A and HA , and the HA /A buffer determines the pH between the two equiv-
alence points. After the second equivalence point, the pH reflects the concentration
of the excess strong base titrant.
Figure 9.8b shows a titration curve for a mixture consisting of two weak acids:
HA and HB. Again, there are two equivalence points. In this case, however, the
equivalence points do not require the same volume of titrant because the concen-
tration of HA is greater than that for HB. Since HA is the stronger of the two weak
acids, it reacts first; thus, the pH before the first equivalence point is controlled by
–
the HA/A buffer. Between the two equivalence points the pH reflects the titration
–
of HB and is determined by the HB/B buffer. Finally, after the second equivalence
point, the excess strong base titrant is responsible for the pH.

9 B.2 Selecting and Evaluating the End Point
Earlier we made an important distinction between an end point and an equivalence
point. The difference between these two terms is important and deserves repeating.
The equivalence point occurs when stoichiometrically equal amounts of analyte and
titrant react. For example, if the analyte is a triprotic weak acid, a titration with
NaOH will have three equivalence points corresponding to the addition of one, two,
–
and three moles of OH for each mole of the weak acid. An equivalence point,
therefore, is a theoretical not an experimental value.
An end point for a titration is determined experimentally and represents the
analyst’s best estimate of the corresponding equivalence point. Any difference be-
tween an equivalence point and its end point is a source of determinate error. As we
shall see, it is even possible that an equivalence point will not have an associated end
point.

Where Is the Equivalence Point? We have already learned how to calculate the
equivalence point for the titration of a strong acid with a strong base, and for the
titration of a weak acid with a strong base. We also have learned to sketch a titra-
tion curve with a minimum of calculations. Can we also locate the equivalence
point without performing any calculations? The answer, as you may have guessed,
is often yes!
3
It has been shown that for most acid–base titrations the inflection point,
which corresponds to the greatest slope in the titration curve, very nearly coincides
with the equivalence point. The inflection point actually precedes the equivalence
point, with the error approaching 0.1% for weak acids or weak bases with dissocia-
–9
tion constants smaller than 10 , or for very dilute solutions. Equivalence points de- 14.0
12.0 (f)
termined in this fashion are indicated on the titration curves in Figure 9.8.
10.0 (e)
The principal limitation to using a titration curve to locate the equivalence 8.0 (d)
point is that an inflection point must be present. Sometimes, however, an inflection pH 6.0 (c)
point may be missing or difficult to detect. Figure 9.9, for example, demonstrates 4.0 (b) (a)
the influence of the acid dissociation constant, K a , on the titration curve for a weak 2.0
0.0
acid with a strong base titrant. The inflection point is visible, even if barely so, for 0.00 20.00 40.00 60.00
–9
acid dissociation constants larger than 10 , but is missing when K a is 10 –11 . Volume of titrant
Another situation in which an inflection point may be missing or difficult to
detect occurs when the analyte is a multiprotic weak acid or base whose successive Figure 9.9
Titration curves for 50.00 mL of 0.100 M
dissociation constants are similar in magnitude. To see why this is true let’s con-
weak acid with 0.100 M strong base. The
sider the titration of a diprotic weak acid, H 2 A, with NaOH. During the titration the pK a s of the weak acids are (a) 1, (b) 3, (c) 5,
following two reactions occur. (d) 7, (e) 9, (f) 11.