Problem 205 - A Complete Resource Book in Chemistry for JEE Main

Question

The concentration of acetic acid, which can be added to \(\mathrm{N} / 2\) formic acid so that the percentage dissociation of both acids is unchanged, would be \(\left(\mathrm{K}_{\mathrm{a}}\right.\) of \(\mathrm{HCO}_{2} \mathrm{H}\) and \(\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}\) are \(2.4 \times 10^{-4} \mathrm{M}\) and \(1.8 \times 10^{-5} \mathrm{M}\) respectively) (a) \(20 \mathrm{~N} / 2\) (b) \(\mathrm{N} / 2\) (c) N/4 (d) \(10 \mathrm{~N} / 3\)

Step-by-Step Solution

Answer

The answer is (c) N/4.

Step 1: Understand the Problem

We are asked to find the concentration of acetic acid that when added to a solution of N/2 formic acid, the percentage dissociation of both acids remains unchanged. We know the dissociation constants \(K_a\) for formic acid \((\text{HCO}_2\text{H})\) and acetic acid \((\text{CH}_3\text{CO}_2\text{H})\) are given as \(2.4 \times 10^{-4} \) and \(1.8 \times 10^{-5} \) M, respectively.

Step 2: Set up the Equation for Dissociation

For an acid \(HA\) dissociation is given by \(HA \rightleftharpoons H^+ + A^-\). The percentage dissociation \( \alpha \) can be expressed in terms of \(K_a\) as \( \alpha = \sqrt{\frac{K_a}{C}}\), where \(C\) is the molarity of the acid. To maintain the same percentage dissociation, both acids must satisfy this relation.

Step 3: Equate Percentage Dissociation of Both Acids

Since the percentage dissociation is to remain the same, equate the percentage dissociation formula for both:\[\alpha_{\text{formic}} = \alpha_{\text{acetic}}\]\[\sqrt{\frac{2.4 \times 10^{-4}}{0.5}} = \sqrt{\frac{1.8 \times 10^{-5}}{C_{\text{acetic}}}}\]

Step 4: Solve the Equation

Squaring both sides of the equation, we get:\[\frac{2.4 \times 10^{-4}}{0.5} = \frac{1.8 \times 10^{-5}}{C_{\text{acetic}}}\]Multiply both sides by \(C_{\text{acetic}}\) and solve for it:\[C_{\text{acetic}} = \frac{0.5 \times 1.8 \times 10^{-5}}{2.4 \times 10^{-4}}\]Calculate the result:\[C_{\text{acetic}} = \frac{0.9 \times 10^{-5}}{2.4 \times 10^{-4}} = \frac{9 \times 10^{-6}}{2.4 \times 10^{-4}} = 0.0375 \text{ M}\]This is equivalent to \( \frac{N}{4} \).

Step 5: Choose the Correct Answer

Given the options, the concentration of acetic acid that results in unchanged percentage dissociation for both acids is (c) N/4.

Key Concepts

Equilibrium Constant (K_a)

When dealing with acid dissociation, the equilibrium constant, denoted as \( K_a \), plays a crucial role. It quantifies the extent to which a weak acid dissociates in solution. For a generic weak acid \( HA \), the dissociation can be represented as:

• \( HA \rightleftharpoons H^+ + A^- \)

The equilibrium constant expression for this dissociation is given by:

• \( K_a = \frac{[H^+][A^-]}{[HA]} \)

This equation shows the ratio of the concentrations of the products to the reactants at equilibrium. A higher \( K_a \) value implies a stronger acid, as it dissociates more completely into ions in solution. The given exercise involves two acids with known \( K_a \) values: formic acid \( (2.4 \times 10^{-4} \text{ M}) \) and acetic acid \( (1.8 \times 10^{-5} \text{ M}) \). Understanding these constants helps us predict and calculate how these acids will behave when mixed.

Percentage Dissociation

Percentage dissociation is a measure of how much an acid dissociates in a solution, expressed as a percentage. It can be calculated using the formula:

• \( \alpha = \sqrt{\frac{K_a}{C}} \times 100 \% \)

Where:

• \( \alpha \) is the degree of dissociation.
• \( K_a \) is the acid dissociation constant.
• \( C \) is the molarity of the acid.

This formula implies that the extent of dissociation diminishes with increasing acid concentration, which aligns with Le Châtelier's principle. In the context of the problem, the aim was to find a concentration of acetic acid that would ensure the percentage dissociation remains unchanged when mixed with formic acid. By equating the percentage dissociations of both acids, we solve for the necessary concentration of acetic acid.

Concentration of Acids

The concentration of an acid significantly impacts its behavior in solution. Acids can be classified as strong or weak based on their dissociation in water. In the given exercise, concentration plays a pivotal role in determining the percentage dissociation of the acids involved. When an acid is diluted, its concentration decreases, thus affecting the dissociation.Finding the Desired Concentration:In the step-by-step solution, the concentration of acetic acid was calculated to ensure that the dissociation percentages of both acids remain identical. This is determined by:

• Setting up an equation involving the dissociation expressions of both acids.
• Solving for the concentration that satisfies the equal percentage dissociation condition.

The final calculation showed that a concentration of \( \frac{N}{4} \) for acetic acid was required to maintain unchanged percentage dissociation when added to \( N/2 \) formic acid.