Question

The hydrogen ion concentration in a solution of weak acid of dissociation constant \(K_{a}\) and concentration \(C\) is nearly equal to (1) \(\sqrt{\frac{K_{\mathrm{u}}}{C}}\) (2) \(\frac{C}{K_{\mathrm{a}}}\) (3) \(K_{\mathrm{a}} \cdot C\) (4) \(\sqrt{K_{\mathrm{a}} \cdot \mathrm{C}}\)

Short answer

The hydrogen ion concentration is \sqrt{K_a \cdot C}\, corresponding to option (4).

Step-by-step solution

Understand the Given Problem

The problem requires determining the hydrogen ion concentration in a solution of a weak acid. The given parameters are the dissociation constant \(K_a\) and the concentration of the acid \(C\).

Write the Dissociation Equation

For a weak acid \(HA\) dissociating in water, the reaction is: \[HA \rightleftharpoons H^+ + A^-\]

Write the Expression for the Equilibrium Constant

The equilibrium constant expression is: \[K_a = \frac{[H^+][A^-]}{[HA]}\]

Assume Degree of Dissociation is Small

Since it is a weak acid, the degree of dissociation is small. Hence, the concentration of \(HA\) remains nearly equal to the initial concentration \(C\).

Simplify the Expression

Assuming \([H^+] = [A^-]\), the expression simplifies to: \[K_a = \frac{[H^+]^2}{[HA]} \approx \frac{[H^+]^2}{C}\]

Solve for Hydrogen Ion Concentration

Rearranging the above equation, we get: \[ [H^+] = \sqrt{K_a C} \]

Compare with Given Choices

Comparing \([H^+] = \sqrt{K_a C} \) with the given options, the correct option is (4) \(\sqrt{K_a \cdot C}\).

Key concepts

Weak Acid Dissociation

In chemistry, a weak acid only partially dissociates in water. When a weak acid such as HA is added to water, it splits into hydrogen ions (H+) and its conjugate base (A-). However, unlike strong acids, weak acids do not ionize completely. Most of the acid molecules remain undissociated in the solution.
For a weak acid dissociation, the reaction can be represented as:
\[HA \rightleftharpoons H^+ + A^-\]
Weak acid dissociation is influenced by several factors including the strength of the acid, the initial concentration, and the temperature. The degree of dissociation, which is the fraction of the original acid that has ionized, is typically small for weak acids. Understanding this concept is crucial when calculating the pH of weak acid solutions because it affects the concentration of hydrogen ions in the solution. If you assume the degree of dissociation is small, the acid's concentration doesn't significantly decrease from its initial value.

Equilibrium Constant

The concept of the equilibrium constant (K) is fundamental in chemistry. It helps predict the extent to which a reaction will proceed. For weak acids, the equilibrium constant is referred to as the acid dissociation constant (K_a).
In the case of a weak acid HA dissociating into H+ and A-, the equilibrium constant expression is:
\[K_a = \frac{[H^+][A^-]}{[HA]}\]
This equation basically states that at equilibrium, the product of the concentrations of the dissociation products divided by the concentration of the undissociated acid is equal to a constant value, K_a. This constant is unique to each acid and is a measure of its strength - a higher K_a value indicates a stronger acid that dissociates more in water.
In practical terms, for a weak acid with a small degree of dissociation, we assume that the change in concentration of the acid (C) is negligible. Therefore, the equilibrium concentrations of [HA] almost remain the same as the initial concentration.

Acid Dissociation Constant

The acid dissociation constant, denoted as K_a, is a specific type of equilibrium constant that applies to the dissociation of weak acids in aqueous solution.
Mathematically, it is expressed as:
\[K_a = \frac{[H^+][A^-]}{[HA]}\]
This constant provides a measure of the degree of ionization of the acid in water. Strong acids have large K_a values because they ionize completely, whereas weak acids have smaller K_a values due to their partial ionization.
To find the hydrogen ion concentration \([H^+]\) in a weak acid solution, you can use the K_a expression. When you set \([H^+] = [A^-]\), the equation simplifies to:
\[K_a = \frac{[H^+]^2}{C}\]
Rearranging this equation for \([H^+]\) gives:
\[ [H^+] = \sqrt{K_a \times C}\ \]
This formula is crucial for calculating the pH of weak acid solutions because pH is directly related to the hydrogen ion concentration. By taking the negative logarithm of \([H^+]\), you can find the pH, further providing practical insights into the acidity of the solution.