91Ó°ÊÓ

Hydronium ions, \(\text{H}_3\text{O}^+\), play a key role in determining the acidity of a solution. This concentration represents the presence of free hydrogen ions in water that are typically associated with \(\text{H}_2\text{O}\) molecules to form hydronium ions. Measuring \(\text{H}_3\text{O}^+\) concentrations helps us understand how acidic or basic a solution is.

When given the \(\text{OH}^-\) concentration, we can easily find the \(\text{H}_3\text{O}^+\) concentration using the ion-product constant for water. In part (b) of the exercise, you start with \(\left[\text{OH}^-\right] = 8.33 \times 10^{-5} \, \text{M}\). The hydronium ion concentration is calculated by using the formula:

• \(\left[\text{H}_3\text{O}^+\right] = \frac{K_w}{\left[\text{OH}^-\right]}\)

Inserting the known values, \(\left[\text{H}_3\text{O}^+\right] = \frac{1.0 \times 10^{-14}}{8.33 \times 10^{-5}}\), which results in approximately \(1.20 \times 10^{-10} \, \text{M}\). This shows a very low concentration of hydronium ions, indicating that the solution is slightly basic.

Hydroxide ions, \(\text{OH}^-\), are crucial in determining the basicity of a solution. The balance between hydroxide and hydronium ions defines the pH of a solution.

In the exercise, given the hydronium ion concentration \(\left[\text{H}_3\text{O}^+\right] = 4.5 \times 10^{-4} \, \text{M}\) in part (a), you are asked to find \(\left[\text{OH}^-\right]\). The process uses the relationship based on the ion-product constant:

• \(\left[\text{OH}^-\right] = \frac{K_w}{\left[\text{H}_3\text{O}^+\right]}\)

By substituting the given values into the equation, \(\left[\text{OH}^-\right] = \frac{1.0 \times 10^{-14}}{4.5 \times 10^{-4}}\), which results in approximately \(2.22 \times 10^{-11} \, \text{M}\). This low hydroxide concentration suggests that the solution is acidic.

The ion-product constant of water, \(K_w\), is fundamental in understanding the self-ionization of water and its impact on hydronium and hydroxide ion concentrations. \(K_w\) is defined as the product of the molar concentrations of these ions in pure water at 25°C:

• \(K_w = \left[\text{H}_3\text{O}^+\right] \times \left[\text{OH}^-\right] = 1.0 \times 10^{-14}\)

This constant value tells us that at 25°C, any aqueous solution will have the ion product of hydronium and hydroxide ions equal to \(1.0 \times 10^{-14} \, \text{M}^2\).

The constant is vital for solving both parts of the exercise since it allows for the calculation of one ion concentration when the other is known. This interrelationship ensures that when one concentration increases, the other decreases, maintaining the balance dictated by \(K_w\). Thus, water can self-ionize to a very limited extent, but this equilibrium impacts all aqueous acid-base reactions.