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) \(16: 1\)

Short answer

The relative strengths of the acids is \( 16: 1 \), option (d).

Step-by-step solution

Understand Dissociation Constant

The dissociation constant, denoted as \( K_a \), is a measure of the strength of an acid in solution. The higher the \( K_a \) value, the stronger the acid is, which means it dissociates more in solution.

Compare the Given Constants

You are given that the dissociation constants for \( \mathrm{HA}_1 \) and \( \mathrm{HA}_2 \) are \( 3.0 \times 10^{-4} \) and \( 1.8 \times 10^{-5} \) respectively. Compare the two values by dividing them to find the relative strength.

Calculate Relative Strength

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

Simplify the Ratio

Perform the division: \[ \frac{3.0 \times 10^{-4}}{1.8 \times 10^{-5}} = \frac{3.0}{1.8} \times \frac{10^{-4}}{10^{-5}} = \frac{3.0}{1.8} \times 10 \].

Calculate the Final Value

Using the above simplification \( \frac{3.0}{1.8} = 1.6667 \approx 1.67 \). Then multiply by 10, yielding a ratio of \( 16.7/1 \) which approximately corresponds to \( 16/1 \).

Choose the Closest Answer

The closest answer to 16:1 is option (d) \( 16: 1 \).

Key concepts

Dissociation Constant

The dissociation constant, often symbolized as \( K_a \), is a fundamental concept in understanding acid strength. It quantifies how completely an acid ionizes in a solution. When an acid (\( \text{HA} \)) dissolves in water, it releases hydrogen ions (\( \text{H}^+ \)) and conjugate base ions (\( \text{A}^- \)).

The dissociation constant is defined by the equation:\[K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}\]Where:

• \([\text{H}^+]\): Concentration of hydrogen ions
• \([\text{A}^-] \): Concentration of the conjugate base
• \([\text{HA}]\): Concentration of the undissociated acid

The larger the \( K_a \), the greater the degree of dissociation, meaning the acid is stronger.

Understanding \( K_a \) helps compare acids and predict their behavior in chemical reactions.

Relative Strengths

The relative strength of acids is determined by comparing their dissociation constants. An acid with a higher \( K_a \) value will be stronger because it dissociates more in solution, releasing more \( \text{H}^+ \) ions.

To find the relative strengths of two acids, the \( K_a \) values are compared by forming a ratio:\[\text{Relative Strength} = \frac{K_{a1}}{K_{a2}}\] This ratio shows how many times stronger one acid is compared to the other. For example, if \( K_{a1} = 3.0 \times 10^{-4} \) and \( K_{a2} = 1.8 \times 10^{-5} \), calculate:\[\frac{3.0 \times 10^{-4}}{1.8 \times 10^{-5}} = 16.67 \]This indicates that HA1 is approximately 16 times stronger than HA2, highlighting the practical application of comparing \( K_a \) values.

Chemical Equilibrium

Chemical equilibrium is a dynamic state in a chemical reaction where the concentrations of reactants and products remain constant over time. In the context of acid dissociation, equilibrium is established between the undissociated acid molecules and the ions formed when the acid dissociates.

Written for an acid \( \text{HA} \) in water, the equilibrium can be represented as:\[\text{HA} \rightleftharpoons \text{H}^+ + \text{A}^-\]At equilibrium, the rates of the forward and reverse reactions are equal, making the concentration ratios described by the dissociation constant \( K_a \) persist.

This equilibrium can be affected by changes in concentration, temperature, and pressure. Le Chatelier's principle aids in predicting how the equilibrium position shifts with such changes.