Using Ka to Calculate pH of Weak Acids

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Calculating the pH of Acids

pH is the most common way to represent how acidic something is. The pH scale is a handy way of making comparisons of how much acidic solutions are, for example. So how does the scale work?

The pH Equation

To calculate pH all you need is the H⁺ ion concentration and a basic calculator, because it is a very straightforward calculation. The H⁺ ion concentration must be in mol dm^-3 (moles per dm³).

pH = -log [H⁺]

The key is knowing the concentration of H⁺ ions, and that is easier with strong acids than it is with weak acids. Have another read of our previous article if you need a reminder of how to tell the difference between strong and weak acids.

By the way, you can work out the H⁺ ion concentration if you already know the pH. This is something you will also need to do when carrying out weak acid calculations. Just use this simple equation:

[H⁺] = 10^-pH

Calculating the pH of Strong Acids

Strong acids dissociate completely. Every molecule dissociates, so if you know the concentration of the acid then it is very straightforward to calculate the concentration of H⁺ ions.

Then just use the pH equation above.

Calculating the pH of Weak Acids

Calculating the pH of weak acids is not straightforward because calculating the H⁺ ion concentration is not straightforward.

It’s not straightforward because weak acids only dissociate partially. A relatively small proportion of the acid molecules dissociate, meaning the H⁺ ion concentration is much smaller than the acid concentration.

In fact the dissociation is a reversible reaction that establishes an equilibrium.

To illustrate, let’s consider a generic acid with the formula HA. The acid dissociates into H⁺ ions and A^– ions in a reversible reaction, which can be represented with this equation:

So how do we work out the H⁺ ion concentration? We need to use the fact that, as a reversible reaction, we can construct an equilibrium constant for the reaction. We even give this equilibrium constant a name: the acid dissociation constant, and a symbol, K_a.

The Acid Dissociation Constant, K_a

The acid dissociation constant is just an equilibrium constant. Chemists give it a special name and symbol just because we use it specifically for weak acids. It makes it more memorable and saves you from having to construct a new equation for the equilibrium constant each time.

There are only four terms in the equation, and we will simplify it further later in this article. The equation for our generic weak acid HA is represented as:

Where K_a is the acid dissociation constant,

[H⁺] is the hydrogen ion concentration in mol dm^-3 ,

[A-] is the concentration of the acid’s anion in mol dm^-3 ,

and [HA] is the concentration of the undissociated acid mol dm^-3 .

How to Use K_a for Weak Acids

Although the equation looks straight forward there are still some ways we can simplify the equation. This is by making two assumptions.

It’s important to note that we should use these assumptions when making calculations involving solutions of only a weak acid.

When you make calculations for acid buffers these assumptions do not make sense. And it is easy to become confused when to use which assumptions. The assumptions we look at here apply only when calculations are related to a weak acid in water, with no other reagent added.

Assumptions for Weak Acid Calculations

The first assumption is that the concentration of hydrogen ions is exactly equal to the concentration of the anions.

Why is that an assumption, and not an absolute fact? After all, each molecule of acid that dissociates produces one hydrogen ion and one anion.

It’s because there is another source of H⁺ ions. Water also dissociates, and one of the products of that dissociation is also H⁺ ions.

However, the proportion of water molecules that dissociate is very small. So the “extra” H⁺ ions are negligible and we can comfortably ignore them in all the calculations we will be asked to do with weak acids.

The second assumption we make is about the concentration of undissociated acid, HA, at equilibrium. We make the assumption that the acid concentration [HA] is unchanged from the initial concentration.

But we know that some of that acid has dissociated, so we know that this isn’t the true concentration. So why can we make this assumption?

It’s because the proportion of molecules that dissociate in aqueous solution is small, typically less than 1%. That means that using the original acid concentration is a reasonable approximation, so our assumption is a fair one.

Calculating [H⁺] and pH for Weak Acids

Typically you will be asked to find the pH for a weak acid solution, and you will be given the acid concentration and the K_a value.

Using our assumption that [H⁺] = [A^–]. we can re-write the equation for the acid dissociation:

To calculate pH we need to know the concentration of hydrogen ions. So we need to rearrange the simplified equation to make [H⁺] the subject of the equation:

Now you have the equation in this format, calculating [H⁺] is as easy as using the values of K_a and [HA].

And once you have the [H⁺], calculating the pH value is straightforward too – see “the pH equation” section above.

Calculating [HA] for Weak Acids

You may also be asked to find the concentration of the acid. This is another favourite question of examiners.

The question won’t spell out that they want you to calculate [HA], but that’s what you need to do.

To make the calculation you need to make a simple rearrangement of the acid dissociation constant again, this time to make [HA] the subject. When you have done this you should get:

Once again, you only need to put in the value for Ka and the H⁺ ion concentration.

Sometimes you are given the pH instead of the hydrogen ion concentration. You can easily calculate the H⁺ ion concentration using the formula [H⁺] = 10^-pH.

Calculating K_a for Weak Acids

Naturally, you may be asked to calculate the value of the acid dissociation constant. And some students find that prospect intimidating, but it shouldn’t be. It is no more difficult than the calculations we have already covered in this article.

To start with we need to use the equation with K_a as the subject. We already have derived this simplified version:

We merely need to use the values for [H⁺] and [HA] to solve the equation. As previously, you can easily calculate the H⁺ ion concentration using the formula [H⁺] = 10^-pH.

Calculations Involving Acid Buffers

All the above assumptions and calculation methods and apply to weak acids, but not to acid buffers.

That may seem strange when you consider that the formulation of an acid buffer includes a weak acid.

So why must we be careful about the calculations we carry out with buffers? It’s because the assumptions we made earlier in this article do not apply for buffers.

We will cover calculation techniques involving acid buffers in another article.