Finding Ka

How many weak acids can you think of in your everyday life? We can name a few. Citric acid is an obvious choice - as the name suggests, it is mainly found in citrus fruits like lemons, grapefruits, kumquats, and satsumas. Lactic acid is produced in your cells when there isn't enough oxygen present to fully oxidise a molecule called pyruvate into carbon dioxide; it causes the burning ache you associate with a hard workout. However, there is also chlorogenic acid (found in coffee and green tea), salicylic acid (an acid derived from willow trees, used in acne treatments and anti-dandruff shampoos), and glutamic acid (the amino acid responsible for the 'umami' taste of certain foods, such as soy sauce and Parmesan cheese). To easily compare the strength of weak acids, we use their Ka value. It tells us the extent of their dissociation in water. In this article, we'll explore different methods of finding Ka.

• This article is about finding Ka in physical chemistry.
• We'll define Ka before looking at the expression for Ka, and its relationship to pKa.
• We'll then explore different ways of finding Ka.
• This will include finding Ka from pH, titration curves and equivalence point, and Kb.
• We'll also take a deep dive into Ka and percentage ionisation.

Ka definition

If you've read a little about acids and bases before, you might know what a weak acid is.

We can represent the dissociation (also known as ionisation) of the weak acid HA with the following equilibrium reaction:

$$HA\rightleftharpoons H^++A^-$$

Here, HA dissociates into H⁺ and A^- ions in a reversible reaction. This reversible reaction forms a dynamic equilibrium, and like all chemical equilibria, it can be represented by an equilibrium constant. The equilibrium constant for the dissociation of a weak acid is known as Ka.

For the reaction that we saw above, the Ka expression looks like this:

$$Ka=\frac{[H^+]\space [A^-]}{HA}$$

You might also come across pKa. pKa is simply an alternative way of measuring the dissociation of a weak acid. You see, Ka is often very small and this makes it tricky to work with. For example, it is easy to accidently omit a decimal place in chemical calculations. pKa, on the other hand, is a typically larger number. Here's how Ka and pKa are related:

$$pKa=-\log_{10}Ka$$ $$Ka=10^{-pKa}$$

Finding Ka

Ka is useful because it tells us about the strength of the acid:

• The higher the value of Ka, the stronger the acid, and the greater the extent of its dissociation.
• The lower the value of Ka, the weaker the acid, and the lesser the extent of its dissociation.

Therefore, with Ka, it is easy to compare different acids and find out which is the strongest, and which is the weakest. But how do we actually calculate Ka? Well, there are a few different ways, which we'll spend the rest of the article exploring. These include finding Ka from:

• pH. This involves working with the Ka expression.
• Titration curves. This involves looking at equivalence point, half-neutralisation, and pKa.
• From Kb. This involves conjugate acid-base pairs.
• We'll also briefly consider finding Ka from percentage ionisation.

Finding Ka from pH

The first way of finding Ka that you could be tested upon in your exams is finding Ka from pH. You'll be given information about a weak acid dissolved in solution and told the resulting pH. You must then use your knowledge of how weak acids behave in solution, alongside a few basic assumptions and the expression for Ka, to calculate Ka.

Method

To find Ka from pH:

1. Write an expression for Ka.
2. Find [H⁺] using the pH given in the question. Hence, find [A^-].
3. Find [HA] using other information given in the question.
4. Substitute values for [H⁺], [A^-], and [HA] into the expression for Ka and solve to get your final answer.

Here are the assumptions that we make. They make our calculations a lot simpler:

• Firstly, we know that weak acids (HA) only partially ionise in solution at equilibrium. This means that only a minimal proportion of the acid dissociates into H⁺ and A^- ions - the vast majority remains as HA molecules. Therefore, for our first assumption, we assume that the concentration of HA molecules at equilibrium is the same as the original amount of acid added to the solution.
• Secondly, we know that water molecules also partially ionise in solution, producing H⁺ and OH^- ions. But once again, this dissociation only affects a tiny proportion of water molecules, and so the amount of H⁺ ions in solution from water molecules is minimal. Therefore, we assume that the entire equilibrium concentration of H⁺ is due to the dissociation of the weak acid, HA.

Example

Still confused about finding Ka from pH? Read through our worked example to get a feel for the method, then try the problem by yourself. This is a useful way of practising chemical calculations.

Finding Ka from equivalence point

Do you want to calculate Ka in a more hands-on way? You could carry out a titration reaction and find Ka using the equivalence point and the half-neutralisation point.

Half-neutralisation is significant for one particular reason. Here, the value of Ka equals the concentration of H⁺ ions in solution, and so pKa is identical to the value of pH. At half-neutralisation, pKa = pH. Therefore, if you know the pH of a solution at half-neutralisation, you know the acid's pKa. Once you know pKa, you can easily find Ka, simply by applying the formula we learned earlier that relates the two values: \(pKa=-\log_{10}Ka\)

Method

Here's how you find Ka from titration curves, equivalence point, and half-neutralisation:

1. Find the pH of the solution at half-neutralisation using either a titration experiment or titration curve.
2. Hence, infer pKa.
3. Use pKa to calculate pKa.

Titration experiment

We gave you two ways of finding the pH at half-neutralisation. This first way is more practical - it involves a titration reaction. Let's now look at how you carry out the experiment and use the results to find the pH at half-neutralisation.

1. Set up a titration experiment in the standard way by suspending a burette above a conical flask using a stand and clamp. In addition, prepare and calibrate a pH probe.
2. Use a volumetric flask to measure 25 cm³ of a weak acid into the conical flask. Add a few drops of a suitable indicator.
3. Fill the burette with a strong base. This is your titrant.
4. Add the base titrant to the acid in 1 cm³ intervals. As you approach the equivalence point, add the base drop-wise.
5. Stop adding the base when the indicator just changes colour. Then, immediately another 25 cm³ of the weak acid to the conical flask.
6. Measure the pH of the solution.

See what we've done here? At the equivalence point, all of the acid has been neutralised. By adding the same volume of unreacted acid to the solution, we end up at half-neutralisation - the concentration of A^- ions equals the concentration of HA molecules. Notice that this means the volume of titrant at half-neutralisation is half the volume of titrant at the equivalence point. Once at half-neutralisation, we can then use a pH probe to accurately find the solution's pH.

Finding Ka from titration curves

Don't have the time or facilities to carry out a full-blown titration? Not to worry! You can also use existing data to find the pH of a weak acid at half-neutralisation, in the form of a titration curve.

Here's how you identify the equivalence point and the pH at half-neutralisation on a weak acid-strong base titration curve:

• When you first add the base to the acid, the pH of the solution rises slowly. This creates a curve with an initial shallow gradient.
• However, once you've added a certain amount of base, the pH increases rapidly. This results in a steep vertical section.
• The curve then levels off again, creating another section with a shallow gradient.
• We first locate the equivalence point. This point lies in the middle of the steep vertical section of the curve.
• We use the equivalence point to find the equivalence point volume - the volume of titrant needed to fully neutralise the acid.
• We then find the half-neutralisation point volume. This is precisely half of the equivalence point volume.
• Finally, we use our graph to identify the half-neutralisation pH. This is simply the pH of the solution when we've added the half-neutralisation point volume of titrant.

A titration curve with the equivalence point volume and half-neutralisation point volume labelled. You use the half-neutralisation point volume to find the pH at half-neutralisation, and thus Ka.StudySmarter Originals

Example

It's your turn: use the graph above to find Ka, with the help of the following worked example.

Finding Ka from Kb

For our last way of calculating Ka, let's explore how we can find Ka from Kb. This involves looking at conjugate acid-base pairs.

When a weak acid (HA) dissociates in solution, it splits up into a proton (H⁺) and a negative ion (A^-). However, as we learned at the start of the article, weak acids only partially dissociate in solution, and so the reaction is reversible. This means that the negative A^- ion can recombine with a positive H⁺ ion to form HA once again. In other words, it gains a proton, and so acts as a base. Therefore, A^- is HA's conjugate base.

An equation showing the dissociation of a weak acid, HA, into a proton and a negative ion. The negative ion is the weak acid's conjugate base and can combine with the proton to reform HA. Together, HA and the conjugate base make a conjugate acid-base pair.StudySmarter Originals

Just like Ka measures the extent of dissociation of a weak acid, the equilibrium constant Kb measures the extent of dissociation of a weak base. Kb applies to conjugate bases, too. In fact, the value of Ka for a weak acid and the value of Kb for its conjugate base partner have a special relationship:

$$Ka\times Kb=Kw$$

Note that Kw is a fixed value at a certain temperature. This means that Ka and Kb are inversely proportional:

• The higher the value of Ka, the stronger the acid. This results in a weaker conjugate base with a lower Kb value.
• The lower the value of Kb, the weaker the acid. This results in a stronger conjugate base with a higher Kb value.

Method

Finding Ka from Kb is probably the simplest type of Ka calculation. Here's the method, if you can even call it a method - it is just one step!

1. Divide Kw by Kb to find Ka.

We told you it was easy!

Example

Now, give the following question a go. If you get stuck, our worked example will guide you through the problem.