Problem 128 - Introductory Chemistry Atoms First

Question

True or false? In an aqueous solution at \(25^{\circ} \mathrm{C}\), you will always get the same number when you multiply the equilibrium \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration by the equilibrium OH \(^{-}\) concentration. Explain your answer.

Step-by-Step Solution

Answer

True. In an aqueous solution at \(25^{\circ} \mathrm{C}\), the product of the equilibrium concentrations of \(\mathrm{H}_{3} \mathrm{O}^{+}\) and \(\mathrm{OH}^{-}\) will always be equal to the ion product constant of water, \(K_w\), which is \(1.0 \times 10^{-14} \mathrm{mol}^2\mathrm{L}^{-2}\) at this temperature.

Step 1: Understanding the ion product of water

In any aqueous solution, the following autoionization reaction occurs: \[2\mathrm{H}_{2}\mathrm{O} \rightleftharpoons \mathrm{H}_{3}\mathrm{O}^{+} + \mathrm{OH}^{-}\] The ion product constant of water (\(K_w\)) represents the autoionization equilibrium between water molecules and hydronium and hydroxide ions. At a specific temperature, it is expressed as: \[K_w = [\mathrm{H}_{3}\mathrm{O}^{+}] \times [\mathrm{OH}^{-}]\] where [\(\mathrm{H}_{3} \mathrm{O}^{+}\)] and [\(\mathrm{OH}^{-}\)] represent the equilibrium concentrations of hydronium ions and hydroxide ions, respectively.

Step 2: Ion product of water at \(25^{\circ} \mathrm{C}\)

At \(25^{\circ} \mathrm{C}\), the value of the ion product constant of water, \(K_w\), is constant and equal to \(1.0 \times 10^{-14} \mathrm{mol}^2\mathrm{L}^{-2}\). Therefore, at this temperature, the relationship between the equilibrium concentrations of hydronium and hydroxide ions can be written as follows: \[K_{w25} = [\mathrm{H}_{3}\mathrm{O}^{+}] \times [\mathrm{OH}^{-}] = 1.0 \times 10^{-14} \mathrm{mol}^2\mathrm{L}^{-2}\]

Step 3: Evaluating the statement

Now, let's evaluate if the statement is true or false in the context of our equilibrium relationship at \(25^{\circ} \mathrm{C}\): "True or false? In an aqueous solution at \(25^{\circ} \mathrm{C}\), you will always get the same number when you multiply the equilibrium \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration by the equilibrium \(\mathrm{OH}^{-}\) concentration. Explain your answer." The statement is indeed true. No matter the concentration of either ion, in any aqueous solution at \(25^{\circ} \mathrm{C}\), the product of the equilibrium concentrations of \(\mathrm{H}_{3} \mathrm{O}^{+}\) and \(\mathrm{OH}^{-}\) will always be equal to the constant \(K_w\) value, which is \(1.0 \times 10^{-14} \mathrm{mol}^2\mathrm{L}^{-2}\) at this temperature.

Key Concepts

Autoionization of Water

The phenomenon of water molecules naturally reacting with each other to produce hydronium and hydroxide ions is known as the autoionization of water. This can be represented by the chemical equation:
\[2\mathrm{H}_{2}\mathrm{O} \rightleftharpoons \mathrm{H}_{3}\mathrm{O}^{+} + \mathrm{OH}^{-}\]
Water, although considered neutral, undergoes this process where two water molecules interact, resulting in one molecule donating a proton (H+) to the other, forming a hydronium ion (\(\mathrm{H}_{3}\mathrm{O}^{+}\)) and a hydroxide ion (\(\mathrm{OH}^{-}\)). This spontaneous reaction is crucial as it establishes the baseline for the acidity or basicity of an aqueous solution. Understanding this concept helps clarify why, even in pure water, we have a certain level of ionic activity occurring at all times under stable conditions.

Equilibrium Concentration

In the context of the autoionization of water, equilibrium concentration refers to the steady-state concentration of the hydronium (\(\mathrm{H}_{3}\mathrm{O}^{+}\)) and hydroxide (\(\mathrm{OH}^{-}\)) ions in a solution where the rates of their formation and recombination are equal. This dynamic balance is described quantitatively by the ion product constant of water (\(K_w\)). It is important to note that in any aqueous solution at equilibrium, irrespective of the present amounts of acid or base, the product of the concentrations of these ions is always constant at a given temperature.

Hydronium Ion

The hydronium ion, scientifically referred to as \(\mathrm{H}_{3}\mathrm{O}^{+}\), emerges from the acceptance of a proton (\(\mathrm{H}^{+}\)) by a water molecule. It's essentially a water molecule with an extra hydrogen, resulting in a positive charge. Throughout the autoionization of water, hydronium ions are continuously formed and subsequently revert back to water molecules, contributing to the equilibrium state. This ion plays a fundamental role in determining the pH of the solution because it represents the concentration of acid present.

Hydroxide Ion

The hydroxide ion, simplistically denoted as \(\mathrm{OH}^{-}\), is created when a water molecule loses a proton. Markedly, these ions, together with the hydronium ions, dynamically exist in a perpetual state of balance in water. In the domain of acidity and basicity, the concentration of hydroxide ions dictates the basicity of a solution. The link between hydroxide and hydronium ion concentrations is an excellent illustration of the concept that when one increases, the other must decrease to maintain the equilibrium constant (\(K_w\)).