Question

A \(0.15-M\) solution of a weak acid is 3.0\(\%\) dissociated. Calculate \(K_{\mathrm{a}}\) .

Short answer

The acid dissociation constant (\(K_a\)) for the given weak acid solution is calculated as: \[K_a = 1.39 \times 10^{-4}\]

Step-by-step solution

Find the initial concentration of the weak acid

The problem states that we have a 0.15 M solution of the weak acid. This is our initial concentration of the acid \((HA)\). Let it be represented as \(C_{HA}\): \[C_{HA} = 0.15\,\text{M}\]

Find the degree of dissociation and calculate equilibrium concentrations

The degree of dissociation is given as 3.0%. That means 3.0% of the weak acid molecules are dissociated into \(H^+\) and \(A^-\) ions at equilibrium. First, we need to convert the percentage to a decimal by dividing by 100: \[\text{Degree of dissociation (decimal)} = \frac{3.0}{100} = 0.03\] Now, multiply the degree of dissociation (as a decimal) by the initial concentration \(C_{HA}\) to determine the change in concentration of all species during the dissociation: \[\text{Change in concentration } = 0.15\,\text{M} \times 0.03 = 0.0045\,\text{M}\] At equilibrium, the concentrations will be: \[[H^+] = [A^-] = 0.0045\,\text{M}\] \[[HA] = 0.15\,\text{M} - 0.0045\,\text{M} = 0.145\,\text{M}\]

Calculate the acid dissociation constant, \(K_a\)

The equilibrium expression for the dissociation of weak acid is: \[K_a = \frac{[H^+][A^-]}{[HA]}\] Using the concentrations from Step 2: \[K_a = \frac{(0.0045\,\text{M})(0.0045\,\text{M})}{(0.145\,\text{M})} = \frac{(0.0045)^2\,\text{M}^2}{(0.145)\,\text{M}} = 1.39 \times 10^{-4}\]

Write down the final answer

The acid dissociation constant (\(K_a\)) for the given weak acid solution is calculated as: \[K_a = 1.39 \times 10^{-4}\]

Key concepts

Weak Acid Dissociation

When we talk about weak acid dissociation, we are referring to a process where a weak acid partially separates into ions in a solution. Unlike strong acids, which dissociate completely, weak acids only do so partially. This means not all the acid molecules become ions.
The amount of dissociation is usually quite small, which is why they are called 'weak'.

• For a weak acid, which we can represent as \( HA \), the dissociation can be shown as: \( HA \rightleftharpoons H^+ + A^- \).
• This symbol \( \rightleftharpoons \) indicates that the reaction is reversible, which means the formation and breaking of ions occurs continuously until equilibrium is achieved.
• The concentration of the weak acid remains largely in its non-dissociated form.

This concept is important because it helps us understand how weak acids behave in a solution, and it allows us to calculate important properties like equilibrium concentration.

Equilibrium Concentration

Equilibrium concentration refers to the concentrations of reactants and products in a chemical reaction that remain unchanged over time. In the context of weak acid dissociation, it means the concentration levels of the acid and its ions once the system has stabilized.

• When a weak acid reaches equilibrium, the amount of dissociated ions \([H^+]\) and \([A^-]\) is the same, since each \(HA\) molecule dissociates to produce one \(H^+\) and one \(A^-\).
• The remaining concentration of the non-dissociated weak acid is the initial concentration minus the concentration of the dissociated part.

To find these values, you'd calculate the change in concentration due to dissociation and adjust the initial concentration of the weak acid accordingly.
This knowledge allows us to solve for the acid dissociation constant \( K_a \), which is a crucial indicator of the strength of the weak acid.

Degree of Dissociation

The degree of dissociation is a way to express how much of a weak acid has dissociated into ions in a solution. It’s usually given as a percentage, but we often convert it into a decimal for calculations.

• It tells us the fraction of the initial concentration of an acid that has dissociated.
• A higher degree means more dissociation, indicating a relatively stronger weak acid.
• This is calculated by dividing the change in concentration of \(H^+\) ions by the initial concentration of the acid.

For instance, if a 0.15 M solution of an acid is 3% dissociated, then using this percentage as a decimal, we calculate how much the concentrations of ions \( [H^+] \) and \( [A^-] \) increase by 0.0045 M in equilibrium.
Understanding this concept helps in computing \( K_a \), as it directly links to the equilibrium concentrations of the reactants and products in the dissociation process.