Question

The acid-dissociation constant for hypochlorous acid (HClO) is \(3.0 \times 10^{-8}\). Calculate the concentrations of \(\mathrm{H}_{3} \mathrm{O}^{+}, \mathrm{ClO}^{-}\), and \(\mathrm{HClO}\) at equilibrium if the initial concentration of \(\mathrm{HClO}\) is \(0.0090 \mathrm{M}\).

Short answer

The equilibrium concentrations for hypochlorous acid (HClO), hydronium ion (H3O⁺), and hypochlorite ion (ClO⁻) are calculated as follows: [HClO] = 0.00895 M, [H3O⁺] = 5.2 × 10⁻⁵ M, and [ClO⁻] = 5.2 × 10⁻⁵ M.

Step-by-step solution

Write the equilibrium expression for the ionization of HClO

The ionization of hypochlorous acid (HClO) can be represented by the following equilibrium equation: \[ HClO(aq) \rightleftharpoons H_{3}O^{+}(aq) + ClO^{-}(aq) \]

Set up an ICE table

To set up an ICE table, we list the initial concentrations of all species, the change in their concentrations, and their equilibrium concentrations. Initial Concentrations: [HClO] = 0.0090 M [H3O⁺] = 0 [ClO⁻] = 0 Change in Concentrations: [HClO] = -x [H3O⁺] = +x [ClO⁻] = +x Equilibrium Concentrations: [HClO] = 0.0090 - x [H3O⁺] = x [ClO⁻] = x

Write the expression for Ka using the concentrations from the ICE table

Given that the Ka for HClO is 3.0 × 10⁻⁸, we can write the equilibrium expression: \[ Ka = \frac{[H_{3}O^{+}][ClO^{-}]}{[HClO]} \] Substituting the equilibrium concentrations from the ICE table: \[ 3.0 \times 10^{-8} = \frac{x^2}{0.0090 - x} \]

Solve for the unknown values by quadratic formula or a simplifying approximation

Given the small value of Ka, we can assume that the change in concentration (x) will be much smaller than the initial concentration of HClO. Therefore, we can approximate 0.0090 - x ≈ 0.0090. Now, we can solve for x: \[ 3.0 \times 10^{-8} = \frac{x^2}{0.0090} \] \[ x^2 = 3.0 \times 10^{-8} \times 0.0090 \] \[ x = \sqrt{2.7 \times 10^{-9}} = 5.2 \times 10^{-5} \]

Calculate the final concentrations of each species

Now, we can calculate the equilibrium concentrations for each species: [HClO] = 0.0090 - x ≈ 0.0090 - (5.2 × 10⁻⁵) = 0.00895 M [H3O⁺] = x = 5.2 × 10⁻⁵ M [ClO⁻] = x = 5.2 × 10⁻⁵ M The final concentrations at equilibrium are: [HClO] = 0.00895 M [H3O⁺] = 5.2 × 10⁻⁵ M [ClO⁻] = 5.2 × 10⁻⁵ M

Key concepts

Hypochlorous Acid

Hypochlorous acid, represented by the formula HClO, is a weak acid and acts as an important agent in various chemical processes. Understanding this compound's behavior in aqueous solutions requires grasping its ability to dissociate into ions.

This compound primarily remains undissociated due to its weak nature, meaning it doesn't completely break apart into ions when dissolved in water. However, when it does ionize, it results in the production of hydronium ions \(H_{3}O^{+}\) and hypochlorite ions \(ClO^-\). This ability to generate ions is crucial for its roles in cleaning and disinfection protocols, as well as various industrial applications.

Consider the equation for its dissociation:
\[ HClO(aq) ightleftharpoons H_{3}O^{+}(aq) + ClO^{-}(aq) \]

This reversible reaction highlights hypochlorous acid's equilibrium between its molecular and ionic forms. Knowing how much of HClO ionizes in solution is key in calculating the concentrations of \(H_{3}O^{+}\), \(ClO^{-}\), and the remaining HClO in solution.

Acid-Dissociation Constant

The acid-dissociation constant, often seen as \(K_a\), is an essential parameter in chemistry for evaluating acid strength in a solution. It helps determine the extent to which an acid can donate a proton in water. For weak acids like hypochlorous acid (HClO), the \(K_a\) value is very small, highlighting that only a tiny fraction of the acid molecules dissociate.

Given as \(3.0 \times 10^{-8}\) for HClO, the \(K_a\) indicates that this acid behaves mildly in water, producing only small quantities of \(H_{3}O^{+}\) and \(ClO^{-}\) ions. The equilibrium expression derived from the dissociation reaction is:
\[ K_a = \frac{[H_{3}O^{+}][ClO^{-}]}{[HClO]} \]

This equation illustrates the relationship between the concentrations of the ions and the undissociated acid at equilibrium. When solving problems involving weak acids, using this constant helps predict how much the acid will dissociate, which is fundamental in numerous chemical calculations and applications. Understanding \(K_a\) simplifies decisions around reactions happening in aqueous environments.

ICE Table Calculation

An ICE (Initial, Change, Equilibrium) table is a systematic method often used to track concentration changes in chemical equilibria. It helps visualize the dissociation process of acids like HClO by clearly laying out initial concentrations, changes during dissociation, and resulting equilibrium concentrations.

Here's how it works for the dissociation of hypochlorous acid:

• Initial: Start with the initial concentration of HClO, which is given as 0.0090 M, while assuming zero initial molarity for both \(H_{3}O^{+}\) and \(ClO^-\).
• Change: As dissociation occurs, HClO decreases by a certain amount (-x), and each ion increases by the same amount (+x for both).
• Equilibrium: Calculate the concentrations at equilibrium as \(0.0090 - x\) for HClO, and \(x\) for both ions.

This setup enables solving for the variable \(x\) using the \(K_a\) expression. By substituting the values from the ICE table into the \(K_a\) equation, and assuming \(x\) is negligible compared to the initial concentration, we can simplify calculations, allowing quick assessment of equilibrium concentrations. This approach is especially handy for weak acids, where changes in concentration are minor compared to the initial values, ensuring accuracy without needing complex calculations.