Problem 96 - A Complete Resource Book in Chemistry for JEE Main

Question

The dissociation constant of two acids \(\mathrm{HA}_{1}\) and \(\mathrm{HA}_{2}\) are \(3.0 \times 10^{-4}\) and \(1.8 \times 10^{-5}\) respectively. The relative strengths of the acids is (a) \(1: 16\) (b) \(1: 4\) (c) \(4: 1\) (d) \(16: 1\)

Step-by-Step Solution

Answer

The relative strength of the acids is (d) \(16:1\).

Step 1: Understanding the Concept

The strength of an acid is indicated by its dissociation constant, denoted as \( K_a \). A higher \( K_a \) value indicates a stronger acid. Therefore, to find the relative strength of the two acids, we need to compare their \( K_a \) values.

Step 2: Identifying Known Values

We are given two dissociation constants: \( K_{a1} = 3.0 \times 10^{-4} \) for acid \( \mathrm{HA}_1 \) and \( K_{a2} = 1.8 \times 10^{-5} \) for acid \( \mathrm{HA}_2 \).

Step 3: Calculating the Ratio

The relative strength of the acids can be expressed as the ratio of their dissociation constants: \(\text{Relative strength} = \frac{K_{a1}}{K_{a2}} = \frac{3.0 \times 10^{-4}}{1.8 \times 10^{-5}} \).

Step 4: Performing the Calculation

Calculate the above fraction: 1. Convert the scientific notation into similar powers: - \(3.0 \times 10^{-4} = 3.0 \times 10^{-4}\) - \(1.8 \times 10^{-5} = 0.18 \times 10^{-4}\)2. Divide \(3.0\) by \(0.18\): \[ \frac{3.0}{0.18} = \frac{300}{18} = \frac{50}{3} \approx 16.67 \]3. Thus, the closest integer ratio is approximately \(16:1\).

Step 5: Choosing the Correct Option

The calculated ratio of \(16:1\) corresponds to option (d). Therefore, the relative strengths of acids \(\mathrm{HA}_1\) and \(\mathrm{HA}_2\) is \(16:1\).

Key Concepts

Dissociation Constant

The dissociation constant, known as the equilibrium constant, is a key indicator in chemistry that measures the extent to which an acid dissociates in water. Represented as \( K_a \), it is a value that helps us understand how completely an acid will release its hydrogen ions in a solution.
This value is crucial because:

• A larger \( K_a \) means the acid dissociates more completely, suggesting a stronger acid since it releases more hydrogen ions, contributing to the acidity of a solution.
• Conversely, a smaller \( K_a \) indicates a weaker acid, as fewer hydrogen ions are released.

The dissociation constant is determined based on the concentration of the products and reactants at equilibrium for the dissociation reaction of the acid in water:

• The general formula for an acid \( HA \) dissociating in water is: \[ HA + H_2O \rightleftharpoons H_3O^+ + A^- \]
• The dissociation constant \( K_a \) is then given by: \[ K_a = \frac{[H_3O^+][A^-]}{[HA]} \]

This relationship helps chemists predict and compare acid behaviors under similar conditions.

Ka Value

The \( K_a \) value is critical in understanding the strength of different acids. It is a numerical representation of an acid's ability to donate protons (H⁺ ions) when in solution.
The \( K_a \) value is:

• A numerical expression denoting how well an acid dissociates in water. The higher the \( K_a \), the stronger and more soluble the acid is in releasing its protons.
• Measured in molarity units (M or mol/L), indicating the concentration of hydrogen ions generated per mole of acid in solution.

The significance of the \( K_a \) value mean:

• Strong acids, which have high \( K_a \) values, fully dissociate in water, providing more H⁺ ions and thus resulting in lower pH values and greater acidity.
• Weak acids, with lower \( K_a \) values, only partially dissociate, releasing fewer H⁺ ions and resulting in higher pH values with less acidity.

By comparing the \( K_a \) values of different acids, we can directly determine which acid is stronger under similar conditions.

Relative Strength of Acids

The relative strength of acids is important when comparing different acids to determine which is stronger in similar environments. The strength is usually inferred from the dissociation constant, \( K_a \) values.
To determine relative strength:

• Compare the \( K_a \) values of acids. Higher \( K_a \) values indicate stronger acids since they dissociate more completely, releasing more hydrogen ions into the solution.
• Calculate the ratio of \( K_a \) values if needed. For instance, to find out how much stronger one acid is than another, simply divide their \( K_a \) values.

With the example from the problem:

• If acid HA\(_1\) has a \( K_a \) of \( 3.0 \times 10^{-4} \) and acid HA\(_2\) has a \( K_a \) of \( 1.8 \times 10^{-5} \), the relative strength can be found by calculating their \( K_a \) ratio: \[ \text{Relative strength} = \frac{3.0 \times 10^{-4}}{1.8 \times 10^{-5}} = 16.67 \]
• The result shows that acid HA\(_1\) is roughly 16 times stronger than acid HA\(_2\), aligning with observable pH changes in solutions of each acid.

Thus, knowing the relative strengths of acids helps predict their effects in different chemical processes and solutions.