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 \(6.31 \times 10^{-5}\) (option c).

Step 1: Understanding the Titration Problem

In this titration problem, benzoic acid, a weak monobasic acid, is being titrated with sodium hydroxide (NaOH). When the pH of the solution is 4.2, half of the acid has been neutralized. This situation occurs at the half-equivalence point where the concentration of the acid equals the concentration of its conjugate base.

Step 2: Applying the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is used to find the dissociation constant. It is given by:\[ \text{pH} = \text{p}K_a + \log\left(\frac{[A^-]}{[HA]}\right) \]At the half-neutralization point, \([A^-] = [HA]\), so \(\log\left(\frac{[A^-]}{[HA]}\right) = \log(1) = 0\). Thus, \( \text{pH} = \text{p}K_a \).

Step 3: Calculating the Dissociation Constant

Since the pH at the half-equivalence point is given as 4.2, and from Henderson-Hasselbalch we have \( \text{pH} = \text{p}K_a \), it follows that \( \text{p}K_a = 4.2 \). To find \(K_a\), convert \(\text{p}K_a\) to \(K_a\) using the formula:\[ K_a = 10^{-\text{p}K_a} \]Substituting \( \text{p}K_a = 4.2 \), we have:\[ K_a = 10^{-4.2} \approx 6.31 \times 10^{-5} \].

Step 4: Confirming the Correct Answer

Based on the values calculated, \(K_a = 6.31 \times 10^{-5}\) which matches option (c) from the given list. Thus, the correct dissociation constant is \(6.31 \times 10^{-5}\).

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a fundamental equation used in acid-base chemistry to estimate the pH of a solution containing a weak acid and its conjugate base. It presents a straightforward relationship between pH, pKa (the dissociation constant, K_a, expressed as its negative logarithm), and the ratio of the concentrations of the conjugate base

• = ext{p}K_a + ext{log} ([A^-]/[HA])

The equation simplifies the process of connecting pH with concentrations of the acid ( [HA] ) and its conjugate base ( [A^-] ). An important outcome of this equation is that at the half-equivalence point of a titration, where exactly half of the acid is neutralized by the base, the concentrations of [HA] and [A^-] are equal. log([A^-]/[HA]) equals zero, simplifying to: = ext{p}K_a. This means you can directly determine by knowing the pH at this point, making it a powerful tool in titration problems. Using Henderson-Hasselbalch, you can easily identify or calculate unknowns in titration scenarios, providing clarity to the behavior of weak acids and bases as they interact in solutions.
The equation also shows the influence of the acid's dissociation constant and concentration changes, helping in understanding buffer solutions and their capacity to resist pH changes. This is crucial in many biochemical processes where maintaining a consistent pH is vital.

Dissociation Constant

The dissociation constant ( K_a ) is a critical measure of a weak acid's strength in solution. It quantitatively reveals how extensively an acid dissociates into its constituent ions (hydrogen ions and its conjugate base) when in solution. This value is crucial in determining the behavior of acids in different chemical environments.
For weak acids, which do not dissociate completely, K_a provides insight into the extent of this dissociation: [A^-] and [H^+] are present in much lower concentrations compared to [HA] . The formula for the dissociation constant is:

• K_a = rac{[H^+][A^-]}{[HA]}

The magnitude of K_a indicates the acid's strength: a smaller K_a suggests weaker acid strength, as fewer hydrogen ions are released. Understanding K_a is essential for predicting and explaining the pH of weak acid solutions, buffer capacities, and acid-base equilibrium.
In titration, knowing the K_a allows you to apply the Henderson-Hasselbalch equation effectively, as well as to predict the pH at various points in the titration process. Knowing these dissociation constants, especially at the half-equivalence point, can help calculate accurate pK_a , which in turn enhances the understanding of the weak acid properties and their role in chemical reactions.

Half-equivalence Point

The half-equivalence point in a titration is a pivotal moment where exactly half of the acid in a solution has reacted with the titrant (the base, in many cases NaOH). This means that the amount of the acid equals the amount of its conjugate base, resulting in special importance for pH calculation.

• At the half-equivalence point, [HA] = [A^-]

This equality simplifies the Henderson-Hasselbalch equation: = ext{p}K_a since log([A^-]/[HA]) becomes zero. This point is insightful for determining the dissociation constant ( K_a ) of the weak acid through its logarithmic form, ( ext{p}K_a ) .
Reaching this point in a titration represents a balance in the acid-base reaction, providing crucial data to determine weak acid properties without requiring precise base endpoint detection. By studying the half-equivalence point, students and chemists can effectively calculate the pK_a of an acid, which in turn enhances understanding of its reactive behavior when titrated.
Identifying the half-equivalence point methodically, often through the use of pH meters or indicators, paints a clearer picture of the acid's characteristics and assists in preparing tactics for various analytical and preparative applications.