Question

Explain why the equilibrium concentration of \(\mathrm{H}_{3} \mathrm{O}^{+}\) is equal to the equilibrium concentration of \(\mathrm{OCl}^{-}\)

Short answer

The equilibrium concentration of \(\mathrm{H}_{3} \mathrm{O}^{+}\) is equal to the equilibrium concentration of \(\mathrm{OCl}^{-}\) because according to the ionization reaction of the hypochlorous acid in water, for each hypochlorous acid molecule that ionizes, one hydronium ion and one hypochlorite ion are produced.

Step-by-step solution

Understand Hypochlorous Acid Ionization in Water

In water, the hypochlorous acid (HOCl) ionizes to produce a hydronium ion (\(\mathrm{H}_{3} \mathrm{O}^{+}\)) and a hypochlorite ion (\(\mathrm{OCl}^{-}\)). The chemical equation for this reaction is: \(HOCl \rightarrow \mathrm{H}_{3} \mathrm{O}^{+} + \mathrm{OCl}^{-}\). Note that for every one molecule of hypochlorous acid that ionizes, one molecule each of \(\mathrm{H}_{3} \mathrm{O}^{+}\) and \(\mathrm{OCl}^{-}\) are produced.

Understand the Equilibrium Constant Expression

For the reaction, the equilibrium constant expression (often written as \(Ka\), the acid dissociation constant) is given by: \(Ka = \frac{[\mathrm{H}_{3} \mathrm{O}^{+}][\mathrm{OCl}^{-}]}{[HOCl]}\). At equilibrium, the concentration of \(\mathrm{H}_{3} \mathrm{O}^{+}\) and \(\mathrm{OCl}^{-}\) are equal due to the 1:1 stoichiometry of the ionization reaction.

Conclusion

Based on the ionization reaction and the equilibrium constant expression, the equilibrium concentration of \(\mathrm{H}_{3} \mathrm{O}^{+}\) is equal to the equilibrium concentration of \(\mathrm{OCl}^{-}\). This is because the stoichiometry of the ionization dictates that for each hypochlorous acid molecule that ionizes, one hydronium ion and one hypochlorite ion are produced.

Key concepts

Acid Dissociation Constant

Understanding the acid dissociation constant, often symbolized as \(Ka\), is crucial in the study of chemistry, particularly when examining the behavior of acids in solution. The \(Ka\) is a quantitative measure of the strength of an acid in solution, describing the acid's ability to donate protons (or \(H^+\) ions) to the solution. It provides insight into the extent of the dissociation of an acid into its ions. For example, in the ionization of hypochlorous acid (HOCl) in water, which can be represented as \(HOCl \rightarrow \mathrm{H}_{3} \mathrm{O}^{+} + \mathrm{OCl}^{-}\), the \(Ka\) value is expressed as \(Ka = \frac{[\mathrm{H}_{3} \mathrm{O}^{+}][\mathrm{OCl}^{-}]}{[HOCl]}\). A larger \(Ka\) implies a stronger acid, meaning more of the acid will dissociate in solution at equilibrium. It's crucial for students to note that a strong acid will have a larger \(Ka\) and consequently a smaller \(pKa\), which is the negative logarithm of the acid dissociation constant.

Stoichiometry

Stoichiometry is the branch of chemistry that relates to the quantitative relationships and proportions of elements and compounds as they undergo chemical reactions. When we look at the ionization of hypochlorous acid in water, the stoichiometry reveals a 1:1:1 ratio among HOCl, \(\mathrm{H}_{3} \mathrm{O}^{+}\), and \(\mathrm{OCl}^{-}\). This means that for every one molecule of hypochlorous acid that dissociates, one hydronium ion and one hypochlorite ion are produced. Because the ratio is 1:1, the equilibrium concentrations of \(\mathrm{H}_{3} \mathrm{O}^{+}\) and \(\mathrm{OCl}^{-}\) are equal. Critical understanding of stoichiometry is necessary for predicting the outcomes of chemical reactions and for comprehending reaction mechanisms. In classroom settings, balancing chemical equations is one of the foundational skills taught to deploy stoichiometry in practical scenarios.

Equilibrium Constant Expression

The equilibrium constant expression represents the relationship between the concentrations of reactants and products in a reversible chemical reaction at equilibrium. It is derived from the law of mass action, which states that the rate of a chemical reaction is proportional to the product of the concentrations of the reactants. In the case of the dissociation of hypochlorous acid (HOCl), the equilibrium constant expression reflects the concentrations of the products (\(\mathrm{H}_{3} \mathrm{O}^{+}\) and \(\mathrm{OCl}^{-}\)) divided by the concentration of the undissociated acid (HOCl). The expression \(Ka = \frac{[\mathrm{H}_{3} \mathrm{O}^{+}][\mathrm{OCl}^{-}]}{[HOCl]}\) indicates that at equilibrium, the product of the concentrations of \(\mathrm{H}_{3} \mathrm{O}^{+}\) and \(\mathrm{OCl}^{-}\), divided by the concentration of HOCl, remains constant. This concept is a pillar in the understanding of chemical equilibrium and assists in predicting the direction and extent of chemical reactions.