Problem 58 - Chemistry: The Central Science in SI Units, Global Edition

Question

The acid-dissociation constant for chlorous acid \(\left(\mathrm{HClO}_{2}\right)\) is \(1.1 \times 10^{-2}\). Calculate the concentrations of \(\mathrm{H}_{3} \mathrm{O}^{+}, \mathrm{ClO}_{2}^{-}\), and \(\mathrm{HClO}_{2}\) at equilibrium if the initial concentration of \(\mathrm{HClO}_{2}\) is \(0.0200 \mathrm{M}\)

Step-by-Step Solution

Answer

At equilibrium, the concentrations of \(H_3O^+\), \(ClO_2^-\), and \(HClO_2\) are approximately 0.0149 M, 0.0149 M, and 0.0051 M, respectively.

Step 1: Write the balanced chemical equation

The dissociation of chlorous acid HClO₂ in water is represented by the following balanced chemical equation: \( HClO_2 (aq) \rightleftharpoons H_3O^+(aq) + ClO_2^-(aq) \)

Step 2: Set up an ICE table

An ICE table (Initial, Change, Equilibrium) helps organize the initial concentrations and changes in concentrations as the reaction progresses to equilibrium. For HClO₂, H₃O⁺, and ClO₂⁻ set up the table: ══════╤═══════╤═══════╤═════════╗ │ Initial │ Change │ Equilibrium ══════╪═══════╪═══════╪═════════╣ HClO₂ │ 0.020 │ -x │ 0.020-x H₃O⁺ │ 0 │ +x │ x ClO₂⁻ │ 0 │ +x │ x ╚═════╝ Note: x represents the change in concentration as the reaction approaches equilibrium.

Step 3: Write the expression of the acid-dissociation constant

Using the balanced chemical equation, we can write the acid-dissociation constant (Kₐ) expression: \[ K_a = \frac{[H_3O^+][ClO_2^-]}{[HClO_2]} \]

Step 4: Plug in the values from the ICE table into the Kₐ expression

Substitute the equilibrium concentrations from the ICE table into the Kₐ expression: \[ 1.1 \times 10^{-2} = \frac{x \cdot x}{0.020 - x} \]

Step 5: Solve for x

To find the value of x, we have a quadratic equation. In this case, because the acid-dissociation constant is much greater than the initial concentration, we can approximate the solution by assuming x << 0.020. This makes our equation simpler: \[ 1.1 \times 10^{-2} \approx \frac{x^2}{0.020} \] Solve for x: \[ x \approx \sqrt{(1.1 \times 10^{-2})(0.020)} \] \[ x \approx 0.0149 \, M \]

Step 6: Calculate the equilibrium concentrations

Now that we have the value of x, we can find the equilibrium concentrations of the species: \[ [H_3O^+]_{eq} = x \approx 0.0149 \, M \] \[ [ClO_2^-]_{eq} = x \approx 0.0149 \, M \] \[ [HClO_2]_{eq} = 0.020 - x \approx 0.020 - 0.0149 \approx 0.0051 \, M \] At equilibrium, the concentrations of H₃O⁺, ClO₂⁻, and HClO₂ are approximately 0.0149 M, 0.0149 M, and 0.0051 M, respectively.

Key Concepts

Equilibrium Concentrations

In chemical reactions, equilibrium concentrations are the amounts of each chemical species present when the reaction has reached a state where the rate of the forward reaction equals the rate of the reverse reaction.
This concept is essential when dealing with dissociation reactions like the one involving chlorous acid (\( \text{HClO}_2 \)).
To find equilibrium concentrations, we typically follow these steps:

• Identify and write down the balanced chemical equation for the reaction.
• Use an ICE table to organize initial concentrations, changes in concentrations, and equilibrium concentrations.
• Apply the acid-dissociation constant (\( K_a \)) in an equation to solve for unknowns.
• Make reasonable approximations if necessary to simplify calculations—especially when \( K_a \) is much smaller than the initial concentration.

By understanding and applying these steps, you can accurately determine how a substance like chlorous acid dissociates in a solution.

Chlorous Acid

Chlorous acid, with the chemical formula \( \text{HClO}_2 \), is a weak acid that partially dissociates in water to produce hydronium ions (\( \text{H}_3\text{O}^+ \)) and chlorite ions (\( \text{ClO}_2^- \)).
Its dissociation can be represented by the equation:\[\text{HClO}_2 (aq) \rightleftharpoons \text{H}_3\text{O}^+(aq) + \text{ClO}_2^- (aq)\]
The acid-dissociation constant \( (K_a = 1.1 \times 10^{-2}) \) quantifies the acid's ability to lose a proton in solution.A larger \( K_a \) value indicates a stronger tendency to dissociate, forming more hydronium and chlorite ions.
Chlorous acid is involved in chemical equilibrium, where the extent of dissociation is determined by the equilibrium constant, initial concentration, and the reaction conditions.
Since \( \text{HClO}_2 \) is a weak acid, only a fraction of the acid molecules will dissociate in solution, defining the reaction's equilibrium state.

ICE Table

The ICE table is a valuable tool used in chemistry to track changes and calculate equilibrium concentrations in reactions. ICE stands for Initial, Change, and Equilibrium.
This table helps visualize how the concentrations of various species evolve during a reaction.Here's how an ICE table works:

• Initial: Start with the initial concentrations of reactants and products before any reaction occurs.
• Change: Define variables to represent the change in concentration as the reaction progresses. Typically, these are expressed in terms of \( x \) or another variable.
• Equilibrium: Calculate the final concentrations at equilibrium by applying the changes to the initial amounts.

When solving an acid dissociation problem like chlorous acid, the ICE table simplifies the process.
The table allows you to substitute these equilibrium concentrations into the expression of \( K_a \), making it easier to solve for unknowns.
With practice, using an ICE table becomes a powerful method for solving equilibrium problems in chemistry.