pH and pKa relationship for buffers | Chemistry | Khan Academy

We're going to talk about the relationship between pH and pKa and buffers. Specifically, we're going to be talking mostly about this in terms of the Henderson-Hasselbalch equation. But before we go to the Henderson-Hasselbalch equation, which I'm going to assume you've seen before—if not, we have some other videos introducing it and also deriving it—let's do a quick review of what exactly is a buffer.

So, a buffer is something that contains an aqueous solution. It's something that contains both a weak acid, which generically we write as HA, and it also contains the conjugate base of our acid, A-. This is as written; it's the acid dissociation reaction for HA. Since it's an equilibrium, we can write an expression called Ka, which is just the equilibrium constant for this equation. It just has a special name because it happens to be for the dissociation of an acid.

Ka is just equal to [H₃O⁺] * [A⁻] / [HA], and we don't include water because it's a pure liquid, so we assume that the concentration is always one. Based on this expression for Ka, we can—and do in a separate video—derive the Henderson-Hasselbalch equation. The Henderson-Hasselbalch equation just says that the pH is equal to the pKa plus the log of [A⁻] / [HA], where HA is our weak acid and A⁻ is its conjugate base.

As you can see up here, an acid and its conjugate base are just related by the fact that the acid has an extra H. We can rearrange the Henderson-Hasselbalch equation to get a lot of different kinds of information. One kind of problem you see a lot is for some buffer; you know they might ask you, “Oh, what's the pH?” And then that means you probably know the pKa, and you know the concentrations of A⁻ and HA.

The other thing that you can use the Henderson-Hasselbalch equation to tell you is the relationship between A⁻ and HA, which is something you might want to know. A lot of times, you just want to know, you know, what's in your solution. Depending on what you want to do to your solution—if you want to add things to it, maybe you want to add some acid, you want to add some base—you want to know what's going on. The Henderson-Hasselbalch equation gives you a really quick and easy way of doing that.

So, what we're going to do is we're going to rearrange this equation to solve for this ratio that we might be interested in. We're going to subtract pKa from both sides, and that gives us the log of [A⁻] / [HA]. I don't know about you, but I actually find logs not super intuitive sometimes. So, I'm actually going to get rid of the log by raising both sides to the 10th power.

So, that gives us 10^(pH - pKa) = [A⁻] / [HA]. So, what does this tell us? It might not look like it tells us a whole lot more, but actually, it tells us a lot. It tells us about the relative relationship in size between [A⁻] and [HA] concentrations, and it's saying that these two things are related to the relative size of pH and pKa.

If we look at this, we can derive a couple relationships. We're saying that pH versus pKa; this relationship can tell us about [A⁻] / [HA], which is a ratio, but that in turn, we can relate to just the relative size of HA versus A⁻. So, let's go ahead and look at all the possible scenarios for these three things. We're going to start with the simplest possible scenario, which is that pH is equal to pKa.

When pH is equal to pKa, we're raising 10 to the zeroth power. So anything to the zeroth power is equal to 1, which tells us that this ratio is equal to 1. If [A⁻] concentration over [HA] concentration is equal to 1, that means that they have the same concentration. I forgot a minus sign there. This is a really helpful thing to remember. Anytime you have a buffer and the pH of your solution is equal to the pKa of your buffer, you immediately know that the concentration of your acid and its conjugate base are the same.

This comes up a lot—not just when you're talking about buffers by themselves, but also when you're doing titrations. The point in your titration where [HA] is equal to [A⁻] is called the half equivalence point. If you haven't learned about buffers, that's okay; or sorry, if you haven't learned about titrations yet, that's totally fine. Just ignore what I just said. But if you have, the moral is just that this is a really, really important relationship that it's really helpful to remember.

I said "really" a lot there. There are two other possibilities for pH and pKa. We can have a pH that's greater than pKa for your buffer, and you can have a pH that is less than your pKa for your buffer. So, if your pH is bigger than your pKa, then this term up here, 10^(pH - pKa), is going to be positive. And when you raise 10 to a positive number, you get a ratio that is greater than one.

So, if our ratio [A⁻] / [HA] is greater than 1, that tells us that [A⁻], the numerator, is actually greater than the denominator [HA]. So, if you know the pH and you know it's bigger than the pKa of your buffer—buffer's acid, to be more specific—then you immediately know that you have more conjugate base around than your acid.

The last scenario is when pH is less than pKa. Well, in that case, we're raising 10 to a negative number because we're subtracting a bigger number from a smaller number. And that means that our ratio [A⁻] / [HA] is actually less than one. So that tells us that our denominator [HA] is actually bigger than our numerator [A⁻].

So, just to wrap up, we can look at the Henderson-Hasselbalch equation, and we can just look at the relationship between pKa and pH. Depending on whether they're equal to each other or one is bigger than the other, we can immediately know what the relationship is between our acid and its conjugate base in our solution.

It's very easy to derive; we did it in just a few minutes. So, it's okay if you don't remember this all the time. I usually just remember that pH equals pKa means they have the same concentration, and then if I forget, I will derive the other relationship.