In this explainer, we will learn how to write equations for the dissociation constants of acids and bases and calculate their values.

When comparing the strengths of acids and bases, we often use qualitative terms like “stronger” and “weaker.” A stronger acid will be able to protonate the conjugate base of a weaker acid.

We also have descriptions of acids and bases like “weak” and “strong,” which relate to their reactions with water. A strong acid dissociates completely in water, while a weak acid dissociates only partially.

These terms are useful for making simple judgements, but they are not that useful in calculations. Such terms also fail to distinguish between similar types of acids. Are all strong acids equally acidic? And are all strong bases equally basic? It is hard to tell without an objective measurement of acidity or basicity.

We can use acid dissociation constants and base dissociation constants to provide a quantitative measurement of the strength of an acid or a base.

A dissociation constant is a type of equilibrium constant. An equilibrium constant is a number that expresses the natural balance point of an equilibrium under certain conditions, even when the initial concentrations are different.

In the case of acids and bases, we often use water as our reference substance. A substance that gives protons to water is an acid.

We can write a chemical reaction to show the dissociation of an acid in water:

We can simplify this reaction by substituting the hydrogen ion in place of the hydronium ion while also removing water:

The reaction quotient for this process can be expressed in terms of the concentrations of , , and :

Once at equilibrium, the reaction quotient is the same as the acid dissociation constant:

### Definition: Acid Dissociation Constant, 𝐾_{𝑎}

The acid dissociation constant is the equilibrium constant for the reaction of an acid with water, where the acid, , dissociates into and ions.

The preceding equation includes three concentration terms, but it can be simplified to include just two. We can make the assumption that because any one molecule has to make one anion as it dissociates and makes one hydrogen ion . The previous reaction quotient can be simplified to the following equation:

We will now use one of these equations to determine for one representative weak acid. We will consider ethanoic acid here because it is a relatively simple molecule that has the chemical formula. We will consider the situation where a 0.1 M solution of ethanoic acid has a value of M.

The following chemical equation describes the equilibrium that is established when ethanoic acid partially dissociates in water. The chemical equation has ethanoic acid on its left-hand side and and ethanoate ions on its right-hand side.

The acid dissociation constant for this equilibrium reaction can be expressed in terms of reactant and product concentrations:

The equation has three concentration terms, but it can be simplified to include just two. We can assume that at the point of equilibrium. This enables us to state that is equal to divided by . The value of for ethanoic acid can be calculated with the following equation:

We now have to insert the and values to determine . It was previously stated that . It was also stated that . The value for this reaction can therefore be calculated if we make the denominator equal to 0.1 M and the numerator equal to the square of M:

The result of this calculation is

It is important to realize here that we made an unstated assumption while we were determining the value of for ethanoic acid. We assumed that the equilibrium concentration of a weak acid is equal to its starting concentration. This assumption is generally assumed to be fair because weak acids undergo relatively little dissociation when they are immersed in water.

As with any equilibrium constant, the greater the value of , the closer the position of equilibrium to the products. We also consider the strength of an acid based on the concentration of hydrogen ions in solutions of the acid and express this using pH.

### Equation: pH in terms of Hydrogen Ion Concentration

The equation for pH in terms of hydrogen ion concentration is

The higher the value of for an acid, the greater the hydrogen ion concentration in solutions of the acid. If one acid has a higher than another, it is considered more acidic.

This result might seem strange as a pure acid will not have any free ions. However, we can use as a measure of how acidic a solution an acid would produce. This allows us to talk about the acidity of an acid without talking about a solution.

The following table compares the values of some representative weak acids. The acids are listed in terms of decreasing acid dissociation value. This sentence could be rephrased to state that the acids are listed in terms of decreasing acidity. It is clear that sulfurous acid tends to be more acidic than lots of other weak acids like carbonic acid and boric acid .

Acid | Chemical Formula | Acid Dissociation Constant (M) |
---|---|---|

Sulfurous acid | ||

Hydrofluoric acid | ||

Nitrous acid | ||

Benzoic acid | ||

Carbonic acid | ||

Boric acid |

### Example 1: Identifying the Strongest Acid of a Set Using 𝐾_{𝑎}
Values

The given table shows the values for a selection of acids. Which acid is the strongest?

Acid | Chloroethanoic acid | Benzoic acid | Lactic acid | Hydrofluoric acid |
---|---|---|---|---|

(mol/L) |

### Answer

The acid dissociation constant for a given acid is the equilibrium constant for the reaction of the acid with water. This can be expressed as the dissociation of the acid in aqueous solution:

The table shows four acids and their values in units of moles per litre (mol/L or M). An acid that dissociates more than another will have a higher value.

The more an acid dissociates, the higher the amount of produced when it is added to water.

The higher the concentration of in a solution of acid, the greater the acidity of the solution.

Step by step, this leads to the conclusion that acids with higher values are those that produce the most-acidic solutions (at the same total concentration of acid).

We can put the acids in order of their values:

Of these four acids, the one with the highest value is chloroethanoic acid. This means that chloroethanoic acid is the strongest acid of the set.

We can do a similar thing with bases. A substance that removes protons from water is a base:

We cannot simplify this reaction in the same way we did with an acid, so we can move onto the reaction quotient:

Remember that we do not include water in the reaction quotient because it is a liquid.

At equilibrium, the reaction quotient is the same as the base dissociation constant:

### Definition: Base Dissociation Constant, 𝐾_{𝑏}

The base dissociation constant is the equilibrium constant for the reaction of a base with water, where the base, , reacts with to produce and ions.

It is appropriate to determine for an ethanoate ion solution here because we have already determined for an ethanoic acid solution. We will consider the situation where a 0.1 M solution of ethanoate ions has a of M.

The following chemical equation describes the equilibrium that is established between ethanoate ions and water:

It is important to note here that one ethanoic acid molecule is generated every time a hydrogen ion breaks from a water molecule and pairs with one ethanoate ion. This inference allows us to determine using just two concentration values.

The base dissociation constant for this equilibrium reaction can be expressed in terms of reactant and product concentrations:

This equation has three concentration terms, but it can be simplified to include just two. We can assume here that because one water molecule produces one hydroxide ion when it donates a hydrogen ion to one ethanoate ion. The simplified equation has the term as its denominator and the square of the term as its numerator:

We can use the known and values to determine for the ethanoate ion solution. The following equation shows the result of putting the and values into the simplified equation:

The result of this calculation is

We made an unstated assumption here while we were determining for the ethanoate ion solution. We assumed the spontaneous dissociation of water molecules is negligible. This is considered to be a reasonable assumption. At least a few water molecules in any aqueous solution should spontaneously dissociate, but the vast majority should not.

In a similar way to acid dissociation constants, the greater the value of a base dissociation constant, , the closer the position of equilibrium to the products. We also consider the strength of a base by the concentration of hydroxide ions in solutions of the base, and we express this using pOH (or using pH, where at ).

The higher the value of for a base, the greater the hydroxide ion concentration in solutions of the base. If one base has a higher than another, it is considered more basic.

We use as a measure of how basic a solution a base would produce. This allows us to talk about the basicity of a base without talking about a solution.

### Example 2: Calculating the Concentration of OH⁻ Ions in a Solution of Pyridine given
Its 𝐾_{𝑏}

What is the concentration of ions in a 0.15 mol/L solution of pyridine? The value for pyridine is mol/L. Give your answer to 2 decimal places.

### Answer

Pyridine is an organic compound with a chemical formula of . Its structure is shown below:

Pyridine is basic. It can react with water to produce ions:

is the symbol for a base dissociation constant. The expression for of pyridine can be written as shown:

In a solution of pyridine, water is a liquid (it is the solvent), so we do not include it in the expression.

We do not actually need to know anything about pyridine to answer this question. All we need to know is the general expression for :

When a base reacts with water, one ion is produced for each produced. Therefore, and

A possible mistake here is to assume that refers to the starting concentration of base (in this case, 0.15 M); however, this is actually the concentration of unreacted base at equilibrium.

On the other hand, the value of of pyridine is very low ( M), indicating that the reaction favors the reactants highly.

The equilibrium concentration of base and the starting concentration of base are going to be almost identical, so we can make the approximation

Therefore,

We can rearrange this to give

We can insert the values from the question to get our final answer:

In scientific notation, to 2 decimal places, the concentration of ions in a 0.15 M solution of pyridine is M.

### Example 3: Identifying the Weakest Base in a Set from Their 𝐾_{𝑏}
Values

The ability of a base to accept protons could be deduced from its base dissociation constant . The base dissociation constant of some bases is listed in the table below.

Base Name | A | B | C | D | E |
---|---|---|---|---|---|

Which of the bases listed in the table is the weakest?

### Answer

A base dissociation constant describes this equilibrium:

We ignore water when constructing the expression because it is a liquid.

A base with a high value will produce more basic solutions than a base with a low value (at the same total concentration).

Therefore, a weak base will have a low value.

We can put the bases in order:

The weakest base of the set is the one with the lowest value, which is E.

Since the expressions for and
both have two concentration terms in the numerator and one in the denominator,
we can assign them units of M
(or mol/L or
mol/dm^{3}).
Formally, equilibrium constants do not have units (they are dimensionless); however,
they are often presented as if they have units because of the simplifications made
at this level. It is therefore common to see values of
given with either no units or units of
M:

### Key Points

- Acid dissociation constants and base dissociation constants are equilibrium constants for the reactions of acids and bases with water.
- The greater the value of an acid’s , the stronger the acid.
- The greater the value of a base’s , the stronger the base.
- The general form of is
- The general form of is