Question

The acid-dissociation constant for hypochlorous acid \((\mathrm{HClO})\) 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 of \(\mathrm{H}_{3}\mathrm{O}^{+}, \mathrm{ClO}^{-}\), and \(\mathrm{HClO}\) are approximately \(1.63 \times 10^{-5} \mathrm{M}\), \(1.63 \times 10^{-5} \mathrm{M}\), and \(0.0090 \mathrm{M}\), respectively.

Step-by-step solution

Write the balanced chemical equation

The dissociation of hypochlorous acid \((\mathrm{HClO})\) can be represented by the equation: \[\mathrm{HClO} \rightleftharpoons \mathrm{H}_{3}\mathrm{O}^{+} + \mathrm{ClO}^{-}\]

Write the \(K_{a}\) expression

The equilibrium constant expression for the dissociation of hypochlorous acid \((\mathrm{HClO})\) is: \[K_{a} = \frac{[\mathrm{H}_{3}\mathrm{O}^{+}][\mathrm{ClO}^{-}]}{[\mathrm{HClO}]}\]

Set up the ICE table

Using the initial concentrations, we can set up an ICE table to determine the change in concentrations and find the equilibrium concentrations. Here, ICE stands for Initial, Change, and Equilibrium. Since the initial concentration of \(\mathrm{HClO}\) is given as \(0.0090 \mathrm{M}\), we assume that the initial concentrations of \(\mathrm{H}_{3}\mathrm{O}^{+}\) and \(\mathrm{ClO}^{-}\) are \(0\). We then write the changes in equilibrium concentrations as: \[\begin{array}{c|ccc} & \mathrm{HClO} & \mathrm{H}_{3}\mathrm{O}^{+} & \mathrm{ClO}^{-} \\ \hline \text{Initial} & 0.0090 & 0 & 0 \\ \text{Change} & -x & +x & +x \\ \text{Equilibrium} & 0.0090-x & x & x \\ \end{array}\]

Substitute the equilibrium concentrations into the Ka expression

Substituting the equilibrium concentrations from the ICE table into the \(K_{a}\) expression yields: \[3.0 \times 10^{-8} = \frac{x \cdot x}{0.0090 - x}\] We can simplify this to: \(3.0 \times 10^{-8} = \frac{x^{2}}{0.0090 - x}\)

Solve for x

Because the value of \(K_{a}\) is very small, we can assume that \(x\) is much smaller than \(0.0090 \mathrm{M}\). Therefore, we can approximate the denominator as: \(0.0090-x \approx 0.0090\) Now, we can solve for \(x\): \(3.0 \times 10^{-8} = \frac{x^{2}}{0.0090}\) \(x^{2} = 3.0 \times 10^{-8} \cdot 0.0090\) \(x = \sqrt{3.0 \times 10^{-8} \cdot 0.0090}\) \(x = 1.63 \times 10^{-5}\)

Calculate the equilibrium concentrations

Now that we have calculated the value of \(x\), we can determine the equilibrium concentrations for all species involved: \[[\mathrm{H}_{3}\mathrm{O}^{+}] = x = 1.63 \times 10^{-5} \mathrm{M}\] \[[\mathrm{ClO}^{-}] = x = 1.63 \times 10^{-5} \mathrm{M}\] \[[\mathrm{HClO}] = 0.0090 - x = 0.0090 - 1.63 \times 10^{-5} \approx 0.0090 \mathrm{M}\] The equilibrium concentrations of \(\mathrm{H}_{3}\mathrm{O}^{+}, \mathrm{ClO}^{-}\), and \(\mathrm{HClO}\) are approximately \(1.63 \times 10^{-5} \mathrm{M}\), \(1.63 \times 10^{-5} \mathrm{M}\), and \(0.0090 \mathrm{M}\), respectively.

Key concepts

Equilibrium Concentration

When a chemical reaction reaches a state where the concentrations of the reactants and products no longer change over time, it is said to have reached equilibrium. At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction.
This balance is crucial in acid-base reactions and is described by the equilibrium constant, often expressed as \( K_a \) for acid dissociation.
For the dissociation of hypochlorous acid, we calculate equilibrium concentrations to understand the concentrations of all species involved once equilibrium is achieved. We express these concentrations in the equation:

• \([\mathrm{HClO}] = \text{initial concentration} - \text{change}\)
• \([\mathrm{H}_{3}\mathrm{O}^{+}] = \text{change} \)
• \([\mathrm{ClO}^{-}] = \text{change} \)

In this case, the change in concentration is represented by \( x \), which is solved using the \( K_a \) expression. Understanding how equilibrium concentrations work helps predict the amount of product formed at equilibrium.

ICE Table

An ICE table is a structured method to track changes in concentration throughout a chemical reaction up to equilibrium. "ICE" stands for Initial, Change, and Equilibrium:
- **Initial**: Lists the starting concentrations of all substances involved.- **Change**: Represents the shifts in concentration as the system approaches equilibrium.- **Equilibrium**: Denotes the concentrations of substances once equilibrium is reached.For example, in the dissociation of hypochlorous acid:

• Initially, \([\mathrm{HClO}] = 0.0090 \text{ M}\), and \([\mathrm{H}_{3}\mathrm{O}^{+}] = [\mathrm{ClO}^{-}] = 0 \text{ M}\).
• The system changes by \(-x\) for the reactant and \(+x\) for the products.
• At equilibrium, these changes adjust concentrations to \(([\mathrm{HClO}] = 0.0090 - x)\), \(([\mathrm{H}_{3}\mathrm{O}^{+}] = x)\), and \(([\mathrm{ClO}^{-}] = x)\).

The ICE table helps visualize the relationship between initial amounts and equilibrium concentrations, which is key for solving equilibrium problems.

Acid-Base Equilibrium

In acid-base chemistry, equilibrium describes the balance between an acid and its conjugate base in solution. The dissociation of an acid into its ions demonstrates such equilibrium, which is crucial for understanding chemical reactions.
In our example with hypochlorous acid, the reaction:\[\mathrm{HClO} + \mathrm{H}_{2}\mathrm{O} \rightleftharpoons \mathrm{H}_{3}\mathrm{O}^{+} + \mathrm{ClO}^{-}\]This shows the acid \(\mathrm{HClO}\) dissociating into hydronium \(\mathrm{H}_{3}\mathrm{O}^{+}\) and hypochlorite \(\mathrm{ClO}^{-}\).
The equilibrium constant \( K_a \) provides an insight into the strength of the acid, representing the acid's tendency to donate protons to water. A small \( K_a \) value, like that of \( 3.0 \times 10^{-8} \), indicates a weak acid that dissociates minimally in solution. Understanding this equilibrium helps in predicting the behavior of acids in various chemical contexts, such as buffer solutions and titrations.