Now Let S Continue Our Discussion of Titrations

Now Let S Continue Our Discussion of Titrations


Lecture 13Analytical Chemistry Nov. 20, 2002

Now let’s continue our discussion of titrations.


Almost any chemical reaction can serve as the basis for a titrimetric method provided that three conditions are met.

The first condition is that all reactions involving the titrant and analyte must be of known stoichiometry.

Second, the titration reaction must occur rapidly. If we add titrant at a rate that is faster than the reaction's rate, then the end point will exceed the equivalence point by a significant amount.

Finally, a suitable method must be available for determining the end point with an acceptable level of accuracy.

Even when these conditions are not satisfied there are often ways around them that make a titration analysis feasible.


Suppose you wish to determine the concentration of formaldehyde, H2CO, in an aqueous solution. The oxidation of H2CO by I3 in a basic solution

H2CO(aq) + 3 OH (aq) + I3 (aq) <= => HCO2 (aq) + 3 I (aq)

is a potentially useful reaction because the triiodide ion forms a colored solution in the presence of starch. So we could titrate with triiodide until all the formaldehyde reacts and then the first drop of excess triiodide will color the solution and its appearance serve as the end point.

However, the rate of this reaction is far too slow for a direct titration.

Instead of a direct titration, we add a known amount of I3- that is in excess of the amount needed to titrate the formaldehyde present in the unknown sample. We take the time to allow this initial reaction to go to completion.

H2CO(aq) + 3 OH (aq) + I3 (aq) <= => HCO2 (aq) + 3 I (aq)

initial x molesY

react completely

final 0 Y-x

The I3- remaining can then be titrated with thiosulfate, S2O32- which is a rapid reaction.

I3-(aq) + 2 S2O32 (aq) <= => S4O62 (aq) + 3 I (aq)

This type of titration is called a back titration. We take the total number of moles of triiodide, subtract the number that react with thiosulfate and we have the number of moles that reacted with the analyte, formaldehyde.

Another approach is illustrated by this reaction of calcium with EDTA. A direct analysis of calcium ion in water could be performed using the reaction

Ca2+(aq) + Y4 (aq) <= => CaY2 (aq)

Where Y4 represents the EDTA ion that has a minus 4 charge.

Unfortunately, there is no suitable indicator for this direct titration.

However, if we mix Ca2+ with an excess of the Mg2+ EDTA complex

Ca2+(aq) + MgY2 (aq) <= => CaY2 (aq) + Mg2+(aq)

We find that calcium ion forms a stronger complex with EDTA than magnesium and we release an amount of magnesium ion into solution that is equivalent to the initial amount of calcium ion.

We can now titrate the released Mg2+ with EDTA

Mg2+(aq) + Y4 (aq) <= => MgY2 (aq)

Because this reaction does have an end point indicator.

The moles of Mg2+ titrated in this last reaction provides a measure of the moles of Ca2+ in the original sample.

Since the analyte displaces a species that is then titrated, this is called a displacement titration.

Here is a third example of a way to get around the three conditions for developing a successful titration method.


When a suitable reaction involving the analyte does not exist it may be possible to generate a species that is stoichiometric with the analyte but is easily titrated.

For example, the sulfur content of coal can be determined by using a combustion reaction to convert sulfur to sulfur dioxide.

S(s) + O2(g) <= => SO2(g)

Passing the SO2 through an aqueous solution of hydrogen peroxide, H2O2, produces sulfuric acid

SO2(g) + H2O2(aq) <= => H2SO4(aq),

which we can titrate with NaOH,

H2SO4(aq) + 2 OH (aq) <= => SO4 2 ( aq) + 2 H2O

providing an indirect determination of sulfur.

Monday we went over how to determine the titration curves for a strong acid/strong base titration and a weak acid/strong base titration.

You should be able to see how to do similar calculations for a strong base analyte using a strong acid titration and a weak base analyte using a strong acid titration. The same algebra can be used and it usually involves exchange Ka’s with Kb’s.

In addition, 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.

Section 12-4 describes Harris's approach to generating diprotic and polyprotic acid titration curves. Again the approach is to use the methods we worked out last week except that one also has to use the equations we developed for intermediate forms of these acids. [those big square root equations]

Section 12-9 and Table 12-6 discuss how to set up spreadsheet titration equations to solve any kind of acid-base titration problem. You should look this section over so you will know how to use them if you have any really complex system that must be solved rigorously.

To evaluate the relationship between an equivalence point and an end point, often we only need to construct a reasonable approximation to the titration curve.

Generally we would like to sketch the titration curve quickly, using as few calculations as possible.

To quickly sketch a titration curve we take advantage of the following observation. 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.

We can easily calculate the pH of a buffer using the Henderson-Hasselbalch equation.

But we can avoid even this calculation if we make the following assumption.

You will recall that a buffer operates over a pH range extending approximately ±1 pH units on either side of the buffer's pKa.

The pH is at the lower end of this range, pH = pKa - 1, when the weak acid's concentration is approximately ten times greater than that of its conjugate weak base. Conversely, the buffer's pH is at its upper limit, pH = pKa + 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.

This is not exactly true since at 10% were have a ratio of 1/9 not 1/10 and at 90% the ratio is 9/1 not 10/1. The difference is a volume of 4.55 mL instead of 5.0 mL but for sketching purposes this works okay.


Our strategy for sketching a titration curve is simple. We begin by drawing our axes, placing pH on the y-axis and volume of titrant on the x-axis. After calculating 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 or three volumes before the equivalence point and for two or three 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 and the pKa to estimate the three points.

Straight lines are drawn through each set 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.

As an example, let's 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 calculated the titration curve on Monday.

We begin by drawing the axes for the titration curve. 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.

Before the equivalence point the titrant is the limiting reagent, and the pH is controlled by a buffer consisting of unreacted acetic acid and its conjugate weak base, acetate.

First the pKa is 4.757, which must be the pH at the half way point of the titration because then [HA] = [A-] and that will happen at 25.0 mL of titrant added.

Next the limits for the buffer region are plotted by using a point of pKa -1 at 10% of the equivalence point volume (5.0 mL NaOH) and pKa+1 at 90% (45.0 mL) of the equivalence point volume. [actually 4.55 mL and 45.5 mL]

After the equivalence point the pH is controlled by the concentration of excess NaOH. Again, we have already done this calculation and can use values we calculated for 75.0 mL and 100.0 mL which were 12.3 and 12.52.


An approximate sketch of the titration curve is completed by drawing separate straight lines through the points in the buffer region and the two points in the excess titrant region.

Finally, a smooth curve is drawn connecting the three straight-line segments.

This approach can be used to sketch titration curves for other acid-base titrations including those involving polyprotic weak acids and bases or mixtures of weak acids and bases.


The first figure here is of the titration of a diprotic acid with a strong base.

Since the analyte is 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 of H2A and HA-, and the HA-/A2- buffer determines the pH between the two equivalence points.

After the second equivalence point, the pH reflects the concentration of the excess strong base titrant.

This last sketch 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 concentration 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.

Now let’s go back to the distinction between an end point and an equivalence point. And how to detect end points.

Remember 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.

One of our goals is to minimize the error between the end point and the equivalence point volumes.

It is possible that an equivalence point will not have an associated end point.

We have already learned how to calculate the equivalence point for different types of acid/base titrations. We also have learned to sketch a titration curve with a minimum of calculations.

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 dissociation constants smaller than 10-9, or for very dilute solutions.

The principal limitation to using a titration curve to locate the equivalence point is that an inflection point must be present. Sometimes, however, an inflection point may be missing or difficult to detect.


This figure, demonstrates the influence of the acid dissociation constant, Ka, on the titration curve for a weak acid with a strong base titrant. The pKa’s are 1,3,5,7,9, and 11. The inflection point is visible, even if barely so, for acid dissociation constants larger than 10 9, but is missing when Ka is 10 11.

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 dissociation constants are similar in magnitude. To see why this is true let's consider the titration of a diprotic weak acid, H2A, with NaOH. During the titration the following two reactions occur.


H2A(aq) + OH (aq) > HA (aq) + H2O

HA (aq) + OH (aq) > A2 (aq) + H2O

Two distinct inflection points are seen if the first reaction is essentially complete before the second reaction begins.


This figure shows titration curves for three diprotic weak acids found in biological systems. The titration curve for maleic acid, for which Ka1 is approximately 20,000 times larger than Ka2, shows two very distinct inflection points.

Malonic acid, on the other hand, has acid dissociation constants that differ by a factor of approximately 690. Although malonic acid's titration curve shows two inflection points, the first is not as distinct as that for maleic acid.

This is because the titration of HA- begins before the titration of H2A has been completed.

Finally, the titration curve for succinic acid, for which the two Ka values differ by a factor of only 27, has only a single inflection point corresponding to the completion of the second reaction.

In general separate inflection points are seen when successive acid dissociation constants differ by a factor of at least 500 (a pKa of at least 2.7).


Citric acid is an example of a triprotic acid. First we look at the ladder diagram. We can see that the distance in pH units between each pair of pKa’s is less than 2. Thus we expect to see indistinct equivalence points. What will the curve look like?


Here is the actual titration curve which confirms our expectations.

In order to use inflection accurately, you need a large number of experimental measurments right near each equivalence point and should use one or both of the deriviative methods we discussed Monday.

How do we use a visual indicator? Acid/base indicators are organic dyes that are weak acids and bases. Because such compounds have at least one conjugate acid base species that is highly colored, their titration results in a change in both pH and color.

This change in color can serve as a useful means for determining the end point of a titration, provided that it occurs at the titration's equivalence point.

The pH at which an acid base indicator changes color is determined by its acid dissociation constant. For an indicator that is a monoprotic weak acid, HIn, the following dissociation reaction occurs

HIn(aq) <= => H+(aq) + In (aq)

Since this is a weak acid we can write

pH = pKa + log [In ]


The two forms of the indicator, HIn and In , have different colors. The color of a solution containing an indicator, therefore, continuously changes as the concentration of HIn decreases and the concentration of In increases.

If we assume that both HIn and In can be detected with equal ease, then the transition between the two colors reaches its midpoint when their concentrations are identical or when the pH is equal to the indicator's pKa.

Unfortunately, detecting the point where the concentrations of HIn and In are equal may be difficult if the change in color is subtle.

So our first problem with an indicator is that its pKa may not coincide with the equivalence point.

Second, your eye may register that a color change has happened at a ratio other than 1-to-1 which will shift the pH of the end point.

Most approaches to the use of indicators involve establishing a range of pHs over which the average analyst will observe a change in color.


This is typically assumed to be a change from 10/1 acid/base to 1/10 acid to base. I.e. this means a range of pKa  1.

Actually the pH range of an indicator does not have to be equally distributed on either side of the indicator's pKa. For some indicators only the weak acid or weak base is


For other indicators both the weak acid and weak base are colored, but one form may be easier to see.


Here is a list of some common acid/base indicators, their color changes and the pH range for their end point color transition.

It has been found that by mixing some indicators that the mixture has a narrower range of pH over which the color change occurs.

Also adding a neutral screening dye, such as methylene blue, also has been found to narrow the pH range over which an indicator changes color. In this case, the neutral dye provides a gray color at the midpoint of the indicator's color transition.

The relatively broad range of pHs over which any indicator changes color places additional limitations on the feasibility of a titration. To minimize a systematic titration error, an indicator's entire color transition must lie within the sharp transition in pH occurring near the equivalence point.


In this figure we see that phenolphthalein is an appropriate indicator for the titration of 0.1 M acetic acid with 0. 1 M NaOH.

Bromothymol blue, on the other hand, is an inappropriate indicator since its change in color begins before the initial sharp rise in pH and, as a result, spans a relatively large range of volumes.

The early change in color increases the probability of obtaining inaccurate results, and the range of possible end point volumes increases the probability of obtaining imprecise results.