Question

Beginning with the equilibrium equation for the dissociation of a weak acid \(H A\), show that the percent dissociation varies directly as the square root of \(K_{\mathrm{a}}\) and inversely as the square root of the initial concentration of HA when the concentration of HA that dissociates is negligible compared with its initial concentration.

Short answer

Percent dissociation is directly proportional to \( \sqrt{K_a} \) and inversely proportional to \( \sqrt{[HA]_0} \).

Step-by-step solution

Writing the Equilibrium Expression

For a weak acid \( HA \), the dissociation in water can be expressed as \( HA \rightleftharpoons H^+ + A^- \). The equilibrium constant \( K_a \) is given by \( K_a = \frac{{[H^+][A^-]}}{{[HA]}} \).

Expressing Initial and Equilibrium Concentrations

Let the initial concentration of \( HA \) be \( [HA]_0 \) and the degree of dissociation be \( x \). At equilibrium, \([HA] = [HA]_0 - x[HA]_0\), \( [H^+] = x[HA]_0 \) and \([A^-] = x[HA]_0\). Assuming \( x \) is small, \([HA] \approx [HA]_0\).

Substituting into the Equilibrium Constant

Substitute the equilibrium concentrations back into the expression for \( K_a \):\[ K_a = \frac{{(x[HA]_0)(x[HA]_0)}}{{[HA]_0}} = x^2 [HA]_0 \].

Isolating the Degree of Dissociation

Rearranging for \( x \), we find: \[ x = \sqrt{\frac{K_a}{[HA]_0}} \].

Calculating Percent Dissociation

The percent dissociation of \( HA \) is given by \( \, \% \, \text{dissociation} = x \times 100 \% \). Thus, \( \, \% \, \text{dissociation} = 100 \times \sqrt{\frac{K_a}{[HA]_0}} \).

Key concepts

Equilibrium Constant (Ka)

In the context of weak acid dissociation, the equilibrium constant, denoted as \( K_a \), is an important measure. It indicates how well a weak acid can donate protons (\( H^+ \) ions) in an aqueous solution. For a given weak acid \( HA \), which dissociates into \( H^+ \) and \( A^- \), the equilibrium expression is:

• \( HA \rightleftharpoons H^+ + A^- \)
• Equilibrium constant \( K_a = \frac{{[H^+][A^-]}}{{[HA]}} \)

The value of \( K_a \) reflects the extent of dissociation:

• A larger \( K_a \) value means more H\(^+\) ions are produced, indicating a stronger weak acid.
• A smaller \( K_a \) value indicates a less dissociated acid, making it weaker.

Understanding \( K_a \) helps predict how acids will behave in different chemical reactions and solutions.
It also assists in calculating key parameters such as the degree and percent dissociation for acids.

Degree of Dissociation

The degree of dissociation, represented as \( x \), is used to describe what fraction of the initial amount of a weak acid has dissociated into ions. It is a value that ranges from 0 (no dissociation) to 1 (complete dissociation). Let's express it mathematically.Initially, let's take the concentration of acid \( HA \) as \([HA]_0\). At equilibrium, if \( x \) fraction of it dissociates, the concentrations are:

• \( [HA] = [HA]_0 - x[HA]_0 \)
• \( [H^+] = x[HA]_0 \)
• \( [A^-] = x[HA]_0 \)

For weak acids, since \( x \) is typically very small, we often make the approximation \([HA] \approx [HA]_0\) in equations.
By rearranging the equilibrium equation, we find:\[ x = \sqrt{\frac{K_a}{[HA]_0}} \]This equation shows that as \( K_a \) increases, more acid dissociates, raising \( x \).
If the initial concentration \([HA]_0\) increases, \( x \) decreases, reflecting less dissociation per given quantity. The expression is key to predicting and understanding weak acid behavior.

Percent Dissociation

The percent dissociation of a weak acid is a more practical expression of how much of the initial acid has actually dissociated into ions at equilibrium.

• This is especially useful when comparing different acids or concentrations.
• The percent dissociation gives an intuitive grasp of an acid's strength at a given concentration.

This value is derived directly from the degree of dissociation \( x \) through the formula:\[ \, \% \, \text{dissociation} = x \times 100 \% \]Replacing \( x \) with its derived expression:\[ \, \% \, \text{dissociation} = 100 \times \sqrt{\frac{K_a}{[HA]_0}} \]This shows that percent dissociation:

• Increases with larger \( K_a \) - a stronger acid at equilibrium.
• Decreases as the initial concentration \([HA]_0\) increases - an interesting twist on how concentration affects dissociation.

Percent dissociation helps to visualize how weak acids behave under different conditions, providing context for their strength and reactivity in solutions.