Page 301 - Modern Analytical Chemistry

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284 Modern Analytical Chemistry

The approach that we have worked out for the titration of a monoprotic weak
acid with a strong base can be extended to reactions involving multiprotic acids or
bases and mixtures of acids or bases. As the complexity of the titration increases,
however, the necessary calculations become more time-consuming. Not surpris-
2
1
ingly, a variety of algebraic and computer spreadsheet approaches have been de-
scribed to aid in constructing titration curves.
Sketching an Acid–Base Titration Curve To evaluate the relationship between an
equivalence point and an end point, we only need to construct a reasonable approx-
imation to the titration curve. In this section we demonstrate a simple method for
sketching any acid–base titration curve. Our goal is to sketch the titration curve
quickly, using as few calculations as possible.
To quickly sketch a titration curve we take advantage of the following observa-
tion. Except for the initial pH and the pH at the equivalence point, the pH at any
point of a titration curve is determined by either an excess of strong acid or strong
base, or by a buffer consisting of a weak acid and its conjugate weak base. As we
have seen in the preceding sections, calculating the pH of a solution containing ex-
cess strong acid or strong base is straightforward.
We can easily calculate the pH of a buffer using the Henderson–Hasselbalch equa-
tion. We can avoid this calculation, however, if we make the following assumption.
You may recall that in Chapter 6 we stated that a buffer operates over a pH range ex-
tending approximately ±1 pH units on either side of the buffer’s pK a . The pH is at the
lower end of this range, pH = pK a – 1, when the weak acid’s concentration is approxi-
mately ten times greater than that of its conjugate weak base. Conversely, the buffer’s
pH is at its upper limit, pH = pK a + 1, when the concentration of weak acid is ten
times less than that of its conjugate weak base. When titrating a weak acid or weak
base, therefore, the buffer region spans a range of volumes from approximately 10% of
the equivalence point volume to approximately 90% of the equivalence point volume.*
Our strategy for quickly sketching a titration curve is simple. We begin by draw-
ing our axes, placing pH on the y-axis and volume of titrant on the x-axis. After calcu-
lating the volume of titrant needed to reach the equivalence point, we draw a vertical
line that intersects the x-axis at this volume. Next, we determine the pH for two vol-
umes before the equivalence point and for two volumes after the equivalence point. To
save time we only calculate pH values when the pH is determined by excess strong acid
or strong base. For weak acids or bases we use the limits of their buffer region to esti-
mate the two points. Straight lines are drawn through each pair of points, with each
line intersecting the vertical line representing the equivalence point volume. Finally, a
smooth curve is drawn connecting the three straight-line segments. Example 9.1 illus-
trates this approach for the titration of a weak acid with a strong base.

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EXAMPLE .1
Sketch the titration curve for the titration of 50.0 mL of 0.100 M acetic acid
with 0.100 M NaOH. This is the same titration for which we previously
calculated the titration curve (Table 9.3 and Figure 9.6).

SOLUTION
We begin by drawing the axes for the titration curve (Figure 9.7a). We have already
shown that the volume of NaOH needed to reach the equivalence point is 50 mL,
so we draw a vertical line intersecting the x-axis at this volume (Figure 9.7b).

*Question 4 in the end-of-chapter problems asks you to consider why these pH limits correspond to approximately
10% and 90% of the equivalence point volume.