In this explainer, we will learn how to define and calculate the degree of dissociation of a weak acid and use it to derive Ostwaldโs law of dilution.

In general, the dissociation of a substance involves the separation of the substance into individual components. We can consider the degree of dissociation in the same way we think about the percentage yield.

Imagine a reversible reaction where a substance dissociates into two pieces.

The degree of dissociation is similar to the percentage of substance that has dissociated.

The degree of dissociation is defined as the proportion of the substance that is dissociated, and it is usually expressed as a decimal. The degree of dissociation is often given the symbol (alpha).

### Equation: Degree of Dissociation

The equation of the degree of dissociation is

When expressing amounts in moles, we can express the degree of dissociation in the following way.

### Equation: Degree of Dissociation Using Amounts in Moles

The equation of the degree of dissociation is

The table below shows how we could analyze the simple systems we have already looked at. It is straightforward to calculate the amount of undissociated substance. The amount of dissociated substance can be determined by counting complete sets of products of dissociation. When we calculate the degree of dissociation, it is essential that we use the total amount of substance (dissociated and undissociated) when we perform the division.

Degrees of dissociation are always in the range of 0 to 1. When performing calculations, care is needed to not count the number of individual pieces. Only the number of units of dissociated substance should be measured regardless of how many pieces the substance dissociates into. We need to count sets of pieces, not the pieces themselves.

Degrees of dissociation greater than 0 and less than 1 are the most interesting. If something completely dissociates, or does not dissociate at all, then we do not need to consider the degree of dissociation at all.

We can apply degrees of dissociation to many different types of chemical systems. Here, we will focus on weak acids.

A weak acid is, by definition, an acid that dissociates incompletely in water. The degree of dissociation is an important characteristic of weak acids. A stronger weak acid will have a higher degree of dissociation than a weaker weak acid.

### Definition: Weak Acid

A weak acid is an acid that dissociates incompletely in water.

When a weak acid dissolves in water, some acid will also dissociate:

Once the system has reached equilibrium, there will be fixed concentration of hydrogen ions (and therefore a fixed pH):

This equation is the same equation we use when discussing the acid dissociation constant, .

### Definition: Acid Dissociation Constant, ๐พ_{๐}

The acid dissociation constant is the equilibrium constant for the dissociation of an acid in water. An acid dissociation constant is a measure of the strength of an acid.

We can express the acid dissociation constant of a weak acid in terms of the concentrations of the products ( and ions) and the reactants :

### Equation: Acid Dissociation Constant, ๐พ_{๐}

We can derive a relationship between an acid dissociation constant and the degree of dissociation using this equation.

To start, letโs look at the dissociation equation again:

If we start with 1 mol of , we will produce mol of and mol of . Letโs go through the initial, change, equilibrium (ICE) table step by step to see how we arrive at this result.

In this scenario, we start with having a concentration of 1 M.

As we are dealing with a weak acid, we know that it will dissociate partially. This will produce some and ions, but there will still be some left. The amount of ions and ions produced will depend on the strength of the acid.

The proportion of that will dissociate is the degree of dissociation, . Therefore, the concentration of will decrease by M. Also, the dissociation will increase the concentration of ions and ions by M:

We can then calculate the final concentrations in terms of the degree of dissociation:

We can now write in terms of at a concentration of 1 M for :

However, there is something wrong here: this equation for applies only when the concentration is exactly 1 M. We can, however, derive an equation for for a general concentration, , in place of 1 M. Starting as before, we calculate it as follows:

When we put these terms into our expression for , this is what we get:

The result is known as Ostwaldโs dilution law, and it applies to any dissociation constant, not just acid dissociation constants.

### Equation: Ostwaldโs Dilution Law

Ostwaldโs dilution law is as follows:

is the acid dissociation constant of the acid.

is the degree of dissociation of the weak acid at equilibrium.

is the total concentration of weak acid.

For many weak acids, the degree of dissociation is close enough to 0; this allows us to use a helpful approximation. If is very small, is very close to 1.

This means that we can derive an approximate relationship between degrees of dissociation and acid dissociation constants: and if is very small,

### Equation: Approximate Relationship Between the Degree of Dissociation, the Acid Dissociation Constant, and the Concentration for a Weak Acid

For acids that are weak enough, the degree of dissociation is proportional to the square root of its acid dissociation constant and inversely proportional to the square root of its concentration. As the concentration goes down, the degree of dissociation goes up, and the larger the acid dissociation constant, the greater the degree of dissociation.

The basic consequence of this is that, as we dilute a solution, we reduce the solute concentration by adding more solvent, and so the degree of dissociation of the solute goes up.

### Example 1: Identifying the Factor That Increases the Degree of Dissociation of a Weak Acid in a Set of Factors

Which of the following factors would increase the degree of dissociation of a weak acid?

- Increasing dilution
- Decreasing volume
- Decreasing dilution
- Increasing acid concentration

### Answer

The degree of dissociation of an acid is the proportion of acid molecules that have dissociated in a given solution. The degree of dissociation will depend on the acid and factors such as the strength of internal bonds or how much energy is released when it dissociates.

A weak acid is an acid that, when mixed with water, will only dissociate a little. Therefore, only a fraction of acid molecules will dissociate, and the rest will be dissolved as whole molecules.

The acid dissociation constant is a fixed property of a weak acid (for a given temperature). Acid dissociation constants are related to degrees of dissociation by Ostwaldโs dilution law, which can be expressed in this simple form for a weak acid: Here, is the degree of dissociation of the weak acid at equilibrium, is the acid dissociation constant, and is the total concentration of the weak acid (including dissociated and undissociated forms).

All four of the potential answers relate in some way to the concentration of the acid. Increasing dilution will decrease the acid concentration, while decreasing volume and decreasing dilution will increase the acid concentration.

If we increase the concentration of the weak acid, the degree of dissociation will decrease as Ostwaldโs dilution law suggests:

The only factor that will produce the opposite effect is โIncreasing dilution.โ This will reduce the concentration of the weak acid and increase the degree of dissociation.

The answer is option A, increasing dilution.

### Example 2: Calculating ๐พ_{๐} given the Concentration and the Percentage of Dissociation
of an Acid

A 0.3 M solution of benzoic acid was found to be dissociated. What is the value of , to 2 decimal places, for this acid? Assume .

- mol/L
- mol/L
- mol/L
- mol/L
- mol/L

### Answer

Benzoic acid is a weak acid. We can guess as much from the low percentage of dissociation (). Fewer than 2 molecules of benzoic acid in this solution are in their dissociated form: a hydrogen ion and a benzoate ion.

To calculate , the acidโs dissociation constant, we need to relate with the percentage of dissociation.

For this, we can use Ostwaldโs dilution law:

The degree of dissociation, , is typically presented as a decimal, so we can convert the percentage of dissociation to a decimal:

We could plug this value into the equation above, but the question includes one extra detail: โAssume .โ This means that the denominator of the equation, , is approximately equal to 1, and so we can restate the equation like this:

Now, all we need to do is insert the values from the question:

As requested, we give our answer to 2 decimal places:

The answer is option A.

The graph below shows the relationship between the concentration and the degree of dissociation for ethanoic acid .

We can zoom in on the interesting area: below 0.1 M.

Here, we can see that the degree of dissociation remains very low until the concentration falls below about 0.01 M.

The relationship between the concentration and the degree of dissociation is of particular importance when dealing with electrolytes.

If we have a weak electrolyte (like a solution of ethanoic acid), we might expect that if we double the concentration, the electrical conductivity will double as well. As the ion concentration increases, so does the electrical conductivity.

However, the electrical conductivity will be less than double its prior value.

The same occurs if we halve the concentration. The electrical conductivity will be a little above half its prior value. This is because, as the concentration increases, the degree of dissociation goes down, and as the concentration decreases, the degree of dissociation goes up.

This graph shows how the concentration of dissociated ethanoic acid changes with the total concentration.

The dissociation of acids is affected by temperature and other conditions, so values of and can change for reasons other than concentration changes.

### Example 3: Identifying the Acid Solution with the Highest Conductivity based on the Degree of Dissociation of Some Acids in Solutions

You are given 5 acidic solutions in a lab experiment, and each acid exhibits a different degree of dissociation, as seen in the table.

Acid | |||||
---|---|---|---|---|---|

Degree of Dissociation |

Which acid has better electrical conductivity? Assume that their concentration is the same and that room temperature is maintained.

### Answer

An acid will dissociate according to this equation:

The higher the degree of dissociation, the more and ions there will be in solution.

The electrical conductivity of a solution is a rough indication of how concentrated ions are in the solution. The greater the concentration of ions, the higher the electrical conductivity of the solution. The question tells us that all the acid solutions have the same temperatures and concentrations, so the only thing we need to know in order to find the most conductive solution is to identify the acid with the highest degree of dissociation.

In order, .

Since has the highest degree of dissociation out of all the acids, and they are otherwise in identical conditions, the electrical conductivity of the solution of will be the highest of all the solutions.

The answer is option E, .

### Key Points

- The degree of dissociation of a substance in a solution is the proportion
of the sample that has dissociated.

The equation of the degree of dissociation is - Weak acids have low degrees of dissociation.
- Acid dissociation constants can be related to the degrees of dissociation and the total concentration by Ostwaldโs dilution law:
- For a weak acid, we can make the approximation that is equal to 1 since will be small:
- We can rearrange this equation to be in terms of the degree of dissociation:
- The higher the concentration of a weak acid, the lower its degree of dissociation.