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 is \(1.39 \times 10^{-5}\).

Step-by-step solution

Write down the given information

We are given: 1. A 0.15 M solution of a weak acid. 2. 3.0% dissociated

Set up the reaction of the weak acid dissociation

Let the weak acid be represented by HA. The dissociation reaction of the weak acid (HA) can be written as: \(HA <=> H^+ + A^-\) We know that the initial concentration of the weak acid is 0.15 M, and 3.0% of it dissociates. Therefore, we can calculate the change in concentration of each of the species in the equilibrium.

Calculate the change in concentration

- HA dissociates by 3.0%, so \(0.15 \times 0.03 = 0.0045\, M\). - The change in concentration of \(H^+\) and \(A^-\) is equal to that of HA, so they increase by 0.0045 M.

Write the equilibrium concentrations of each species

In the equilibrium we have: - \(HA: 0.15\, M - 0.0045\, M = 0.1455\, M\) - \(H^+: 0.0045\, M\) - \(A^-: 0.0045\, M\)

Set up the Ka expression

The expression for the acid dissociation constant (Ka) is: \(K_a = \dfrac{[H^+][A^-]}{[HA]}\)

Plug in the equilibrium concentrations and calculate Ka

Using the calculated equilibrium concentrations from step 4, the Ka expression becomes: \(K_a = \dfrac{(0.0045)(0.0045)}{0.1455}\) Now, calculate the value of K_a: \(K_a = 1.39 \times 10^{-5}\)

Writing the final answer

The acid dissociation constant \(K_a\) for the given weak acid is \(1.39 \times 10^{-5}\).

Key concepts

Weak Acid

A weak acid is a type of acid that only partially dissociates into its ions in a solution. This means that when a weak acid is dissolved in water, only a small fraction of its molecules release hydrogen ions (\(H^+\)).
This partial dissociation is what distinguishes weak acids from strong acids, which dissociate completely.Some characteristics to remember about weak acids include:

• They have equilibrium between the undissociated acid and its ions.
• The percentage of dissociation is usually low, as seen in our example where only 3% dissociates.

Understanding weak acids is crucial as they are often involved in various chemical reactions and solution chemistry problems.

Equilibrium Concentration

When dealing with weak acids, we often discuss equilibrium concentrations. These are the concentrations of the reactants and products at equilibrium, the point at which the forward and reverse reactions occur at the same rate.
In the given exercise, we start with an initial concentration of the weak acid, HA, at 0.15 M. Once the acid establishes equilibrium, it partially dissociates into ions, resulting in new equilibrium concentrations for each component.To find the equilibrium concentration:

• Calculate the change in concentration for each species based on the percentage dissociation.
• Subtract this change from the initial concentration for the weak acid, and add this change to 0 for the ions, because they start with a concentration of zero.

In our example, we find that at equilibrium, HA has a concentration of 0.1455 M, both \(H^+\) and \(A^-\) have concentrations of 0.0045 M.

Percentage Dissociation

Percentage dissociation is a measure of how much an acid dissociates in a given solution. It's usually expressed as a percentage of the initial concentration of the acid that turns into ions.
This value is important because it tells us how strong or weak an acid is in terms of dissociation. To calculate percentage dissociation:

• Take the amount of acid that has dissociated into ions.
• Divide it by the initial concentration of the acid.
• Multiply by 100 to get the percentage.

In our specific case, a 3.0% dissociation means that 3.0% of the initial 0.15 M of HA converts into hydrogen and conjugate base ions. This is typically small for weak acids, reflecting their inability to dissociate fully compared to stronger acids.

Ka Calculation

The acid dissociation constant, \(K_a\), is a fundamental aspect of weak acids. It quantifies the extent of dissociation of the acid in solution.
A smaller \(K_a\) value suggests a weaker acid, as it indicates less dissociation.The \(K_a\) expression is set up based on the equilibrium concentrations of the dissociated ions and the undissociated acid:\[K_a = \dfrac{[H^+][A^-]}{[HA]}\]To calculate \(K_a\):

• Substitute the equilibrium concentrations of \(H^+\), \(A^-\), and HA into the \(K_a\) expression.
• Solve the expression to find the \(K_a\) value.

In the given problem, substituting the values gives us:\(K_a = \dfrac{(0.0045)(0.0045)}{0.1455} = 1.39 \times 10^{-5}\).This result confirms the weak nature of the acid as it illustrates a low but definite dissociation into ions.