Problem 160 - A Complete Resource Book in Chemistry for JEE Main

Question

A solution of benzoic acid (a weak monobasic acid) is titrated with \(\mathrm{NaOH}\). The \(\mathrm{pH}\) of the solution is \(4.2\) when half of the acid is neutralized. Dissociation constant of the acid is (a) \(3.2 \times 10^{-5}\) (b) \(6.42 \times 10^{-4}\) (c) \(6.31 \times 10^{-5}\) (c) \(8.7 \times 10^{-8}\)

Step-by-Step Solution

Answer

The dissociation constant is 6.31 × 10⁻⁵ (option c).

Step 1: Understanding Equivalence Point and Henderson-Hasselbalch Equation

When a weak acid is half neutralized during a titration, the concentration of the acid \(\text{[HA]}\) is equal to that of its conjugate base \(\text{[A}^-\text{]}\). This allows us to use the Henderson-Hasselbalch equation: \(\text{pH} = \text{p}K_a + \log\left(\frac{[A^-]}{[HA]}\right)\). Since \(\text{[A}^-\text{]} = \text{[HA]}\) at half-neutralization, the equation simplifies to \text{pH} = \text{p}K_a\.

Step 2: Equating pH and pKa to Find Ka

Given that the \(\text{pH}\) at half-neutralization is 4.2, we can equate this to the \(\text{p}K_a\) of the acid because \(\text{pH} = \text{p}K_a\) at this point. Thus, \(\text{p}K_a = 4.2\).

Step 3: Converting pKa to Ka

Convert the \(\text{p}K_a\) to the dissociation constant \(K_a\) using the formula \(K_a = 10^{-\text{p}K_a}\). Plug in the given value: \(K_a = 10^{-4.2}\).

Step 4: Calculating the Value of Ka

Calculate \(K_a\) using the expression from the previous step: \(K_a = 10^{-4.2} \approx 6.31 \times 10^{-5}\).

Step 5: Identifying the Correct Answer Option

Match the calculated \(K_a\) value of \(6.31 \times 10^{-5}\) with the given options. This corresponds to option (c).

Key Concepts

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is incredibly useful for calculating the pH of a buffer solution. This equation is:\[\text{pH} = \text{p}K_a + \log\left( \frac{[\text{A}^-]}{[\text{HA}]} \right)\]Here, \([\text{A}^-]\) represents the concentration of the conjugate base, and \([\text{HA}]\) represents the concentration of the acid. This equation stems from the rearrangement of the equilibrium expression for a weak acid and its conjugate base.
When we titrate a weak acid with a strong base, we reach halfway to the equivalence point once half the acid has been converted to its conjugate base. At this point, the concentrations of the acid and its conjugate base are equal, simplifying the equation to:\[\text{pH} = \text{p}K_a\]This relation is key when determining the \(\text{p}K_a\) of a weak acid during its titration process. By knowing the pH at half-neutralization, one can directly find the \(\text{p}K_a\) without the need for complex calculations.

pH at half-neutralization

Half-neutralization is a significant point during the titration of a weak acid with a base, where half of the acid has been converted into its conjugate base. At this juncture, the pH of the solution equals the \(\text{p}K_a\) of the acid.
This happens because the concentrations of the acid (\([\text{HA}]\)) and its conjugate base (\([\text{A}^-]\)) are the same. Simplifying the Henderson-Hasselbalch equation gives:- \(\text{pH} = \text{p}K_a\)Understanding this concept is essential because it provides a simple method to determine the \(\text{p}K_a\) of a weak acid by simply measuring the pH at the half-neutralization point, eliminating the need for detailed calculations.

Dissociation constant \(K_a\)

The dissociation constant, \(K_a\), is a measure of the strength of an acid in solution. It reflects the degree to which an acid dissociates into its ions. In mathematical terms, it is expressed as:\[K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}\]In this equation, \([\text{H}^+]\), \([\text{A}^-]\), and \([\text{HA}]\) represent the equilibrium concentrations of the hydrogen ion, the conjugate base, and the acid, respectively.
To find \(K_a\) from \(\text{p}K_a\), use the formula:\[K_a = 10^{-\text{p}K_a}\]This inverse logarithmic relationship allows us to compute \(K_a\) if the \(\text{p}K_a\) is known. A smaller \(K_a\) value indicates a weaker acid, meaning it does not easily dissociate in solution. This concept is crucial in understanding acid strength and reactivity.