Topic 8.5: Acid-Base Titrations

Acid-base titrations: the curve

Every mixture so far was one snapshot. Add the base a little at a time instead, and write down the pH after every few drops, and all those snapshots join up into one curve, which is what a titration is.

Here’s a lab on the shore of the lake. The beaker holds twenty milliliters of acetic acid whose concentration we don’t know, and finding it is the whole point. The buret holds sodium hydroxide whose concentration we do know, point one zero zero zero molar, and a pH probe reads the solution the whole time. The unknown is called the analyte, and what we add from the buret is the titrant.

Watch the curve as the base goes in. The pH jumps a little at first, then it climbs slowly for a long stretch, because the acid and its conjugate base are holding it steady, then, near twenty-four milliliters, it shoots almost straight up, and after that it levels off high.

And look, we’ve already calculated three points on this curve, four point two seven at six milliliters, eight point seven four at twenty-four, and twelve point oh eight at thirty. A titration curve is every one of those mixture problems lined up in a row. The middle of that steep climb is the equivalence point, where the base we’ve added has used up exactly all the acid that was there at the start.

Now you run one. Start with a fresh sample, add the base, and try to stop right at the equivalence point. So how does that give us the concentration? At the equivalence point, the moles of hydroxide added equal the moles of acid we started with, because the two react one to one.

Twenty-four milliliters of point one molar base is two point four millimoles of hydroxide. So the sample held two point four millimoles of acid. Divide by its twenty milliliters, and you get point one two molar. And this is the line that earns the point on the AP exam, because a lot of students divide by forty-four milliliters, everything in the beaker at the end, when the concentration belongs to the original twenty-milliliter sample.

Here’s a different unknown. Twenty-five milliliters of a monoprotic weak acid needs eighteen point four milliliters of point one molar sodium hydroxide to reach equivalence. What was the acid’s concentration?

Half-equivalence: reading pKₐ

There’s a second point on this curve that’s worth even more than the jump. Go back to twelve milliliters, exactly halfway to the equivalence point. Halfway there, exactly half of the acid has turned into acetate, so the beaker holds just as much acetic acid as acetate, twelve of each in our counting.

Put that into Henderson-Hasselbalch. The base over the acid is one, and the log of one is zero, so the pH equals the pKa. Which means you can read the acid’s pKa straight off the graph, right here at halfway, four point seven four.

Notice that it’s half the volume that counts, because cutting the equivalence pH in half gives a number that means nothing. You take half the volume, go up to the curve, and read the pH there. And for a weak base like ammonia, halfway gives you the pKa of ammonium, nine point two six.

Here’s the whole curve for that mystery acid from a minute ago. Tap the point where you can read its pKa, and then use the table to tell me which acid it is.

What sets the pH at equivalence

Now let’s put three titrations side by side. Each one starts with twenty milliliters of a point one two molar solution and uses point one molar titrant, so all three reach equivalence at the same place, twenty-four milliliters.

Hydrochloric acid titrated with sodium hydroxide reaches equivalence at exactly seven. Acetic acid reaches it at eight point seven four. And ammonia, titrated with hydrochloric acid, comes down to equivalence at five point two six. Same volume, three different pH values.

The difference is what’s left in the beaker at that moment. With acetic acid, what’s left is acetate, which pulls protons off water and makes hydroxide. With ammonia, what’s left is ammonium, which hands protons to water and makes hydronium. And with hydrochloric acid, what’s left is sodium and chloride ions, and neither one does anything to water, so you get exactly seven.

So at equivalence, the pH is set by whatever’s left, and the question to ask is always the same one, whether that leftover species does anything to water. Think of the reason before you look at the choices. Why is the equivalence point of the acetic acid titration above seven?

More than one proton

The limestone that saved the lake has one more thing to show us, because a carbonate ion takes its two protons one at a time, and a titration curve can see each one. Here’s twenty milliliters of point one molar sodium carbonate, the kind of carbonate that dissolves, titrated with point one molar hydrochloric acid.

This time it comes down in two steps. First the acid turns carbonate into bicarbonate, and at twenty milliliters every carbonate has taken one proton, the first equivalence point. Then it turns bicarbonate into carbonic acid, and at forty milliliters that’s done too.

Two jumps mean two protons, and each step has its own halfway point and its own pKa. Halfway to the first jump, at ten milliliters, there’s as much carbonate as bicarbonate, and the pH is about ten point three, the pKa of bicarbonate. The second halfway point is halfway between the two jumps, at thirty milliliters, not at twenty, which is half of forty, and there the pH is about six point four, the pKa of carbonic acid.

On the AP exam you’ll name the species present in large amounts at a point like this, and you’ll never have to calculate every concentration along a curve with more than one proton. At thirty milliliters, which two species are there in large amounts?

Last one, and it takes us back to the lake. By 2000, Lake Orta’s water was back to a pH of about six point eight. Where on this curve does that water belong?

So let’s go back out on the water. When that boat started spraying in 1989, the scientists were doing what you can do now. They measured the acid in the lake. They let a weak base use up a strong acid, and the lake came out of it with a buffer of its own again.

First find what reacted, then ask what’s left, and if you ask that after every single drop, you get a titration curve, which can tell you how much acid was in the water and how strong it is. That’s how a lake that was full of acid in 1985 ended up, fifteen years later, right back where it started.