Question

Autoionization of Water and pH Calculate \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) in each aqueous solution at \(25^{\circ} \mathrm{C},\) and classify each solution as acidic or basic. $$\begin{array}{l}{\text { a. }[\mathrm{OH}]=1.1 \times 10^{-9} \mathrm{M}} \\\ {\text { b. }\left[\mathrm{OH}^{-}\right]=2.9 \times 10^{-2} \mathrm{M}} \\\ {\text { c. }\left[\mathrm{OH}^{-}\right]=6.9 \times 10^{-12} \mathrm{M}}\end{array}$$

Short answer

The [H3O+] for part a is 9.09 x 10^-6 M (basic), for part b is 3.45 x 10^-13 M (basic), and for part c is 1.45 x 10^-3 M (acidic).

Step-by-step solution

Understanding Autoionization of Water

Water undergoes autoionization where water molecules react with each other to form hydronium (\text{H}_3\text{O}^+) and hydroxide (\text{OH}^-) ions. At 25\(^{\circ}\)C, the ion product for water is Kw = [H3O+][OH-] = 1.0 x 10^-14.

Calculate [H3O+] for Part a

To find [H3O+] for part a, use the equation Kw = [H3O+][OH-]. Since [OH-] is given as 1.1 x 10^-9 M, we can rearrange the equation to [H3O+] = Kw / [OH-] which gives [H3O+] = (1.0 x 10^-14) / (1.1 x 10^-9) = 9.09 x 10^-6 M.

Classify Solution for Part a

Since [H3O+] is less than 1.0 x 10^-7 M, the solution is basic.

Calculate [H3O+] for Part b

For part b, again use the equation Kw = [H3O+][OH-] and input the given [OH-] of 2.9 x 10^-2 M. Then, [H3O+] = (1.0 x 10^-14) / (2.9 x 10^-2) = 3.45 x 10^-13 M.

Classify Solution for Part b

Since [H3O+] is less than 1.0 x 10^-7 M, the solution for part b is also basic.

Calculate [H3O+] for Part c

For part c, use the equation Kw = [H3O+][OH-] with the given [OH-] of 6.9 x 10^-12 M. We find [H3O+] = (1.0 x 10^-14) / (6.9 x 10^-12) = 1.45 x 10^-3 M.

Classify Solution for Part c

Since [H3O+] is greater than 1.0 x 10^-7 M, the solution for part c is acidic.

Key concepts

pH Calculation

The calculation of pH is a critical aspect in chemistry that determines the acidity or basicity of a solution. The pH is defined as the negative logarithm (base 10) of the hydronium ion concentration (\text{H}_3\text{O}^+), represented as \( pH = -\log[H_3O^+] \). In simpler terms, it measures the concentration of hydrogen ions in a solution on a scale from 0 to 14, where 7 is neutral, values lower than 7 are acidic, and values higher than 7 are basic or alkaline.

To calculate the pH of a solution, one must first know the concentration of \text{H}_3\text{O}^+ ions. For instance, if the concentration of \text{H}_3\text{O}^+ ions is found to be 9.09 x 10^-6 M as in part a of the original exercise, the pH is calculated as follows: \( pH = -\log(9.09 \times 10^{-6}) \), which would result in a pH greater than 7, indicating a basic solution.

The knowledge of calculating pH is not only foundational in understanding autoionization of water but also in gauging any substance's acidity or alkalinity, which is a key concept in many industrial, biological, and environmental processes.

Hydronium Ion Concentration

The concentration of hydronium ions (\text{H}_3\text{O}^+) in a solution is a direct measure of its acidity. To find this concentration, one can use the ion-product constant of water (\text{K}_\text{w}) at 25°C, which is always \( 1.0 \times 10^{-14} \).

If the concentration of hydroxide ions (\text{OH}^-) is known, as provided in the exercise, the concentration of hydronium ions can be calculated using the relationship \( \text{K}_\text{w} = [H_3O^+][OH^-] \). By rearranging the formula, one obtains \( [H_3O^+] = \frac{K_w}{[OH^-]} \), which allows for the determination of the hydronium ion concentration when the hydroxide ion concentration is given.

For example, in the original exercise for part c, where \( [OH^-] = 6.9 \times 10^{-12} \text{M} \), the hydronium ion concentration would be calculated as \( [H_3O^+] = \frac{1.0 \times 10^{-14}}{6.9 \times 10^{-12}} \), resulting in \( [H_3O^+] = 1.45 \times 10^{-3} \text{M} \), signifying an acidic solution.

Acidic and Basic Solutions

Understanding the nature of acidic and basic solutions is pivotal in chemistry. Acidic solutions contain higher concentrations of hydronium ions (\text{H}_3\text{O}^+) than hydroxide ions (\text{OH}^-). Conversely, basic solutions contain more hydroxide ions than hydronium ions. The ion-product constant of water (\text{K}_\text{w}) helps to determine the nature of a solution by comparing the concentration of hydronium ions to that of hydroxide ions.

Solutions with a hydronium ion concentration greater than \( 1.0 \times 10^{-7} \text{M} \) are considered acidic. For example, using the given information from part c of the exercise, with \( [H_3O^+] = 1.45 \times 10^{-3} \text{M} \), the solution is acidic as the concentration of \text{H}_3\text{O}^+ exceeds \( 1.0 \times 10^{-7} \text{M} \).

Conversely, if the concentration is less than \( 1.0 \times 10^{-7} \text{M} \) as seen in parts a and b, the solution is basic. This distinction is crucial in many applications, including titration calculations, buffer solutions in biological systems, and the balance of aquatic ecosystems. Therefore, understanding the molecular basis of the acidity or basicity in water via its autoionization plays a critical role in predicting the behavior of substances when mixed with water.