Question

Calculate \(\left[\mathrm{OH}^{-}\right]\)given \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\)in each aqueous solution and classify the solution as acidic or basic. (a) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=1.5 \times 10^{-9} \mathrm{M}\) (b) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=9.3 \times 10^{-9} \mathrm{M}\) (c) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=2.2 \times 10^{-6} \mathrm{M}\) (d) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=7.4 \times 10^{-4} \mathrm{M}\)

Short answer

\(\left[\mathrm{OH}^{-}\right]\) for the solutions are (a) 6.67 \times 10^{-6} \mathrm{M} (basic), (b) 1.08 \times 10^{-6} \mathrm{M} (basic), (c) 4.55 \times 10^{-9} \mathrm{M} (acidic), (d) 1.35 \times 10^{-11} \mathrm{M} (acidic).

Step-by-step solution

Review the Concept of pH and pOH

Understand that in any aqueous solution at 25°C, the product of the hydrogen ion concentration \(\left[\mathrm{H}_3\mathrm{O}^{+}\right]\) and the hydroxide ion concentration \(\left[\mathrm{OH}^{-}\right]\) is constant and equals \(1 \times 10^{-14}\). This relationship is expressed by the equation \(\left[\mathrm{H}_3\mathrm{O}^{+}\right] \times \left[\mathrm{OH}^{-}\right] = 1 \times 10^{-14}\).

Calculate \(\left[\mathrm{OH}^{-}\right]\) for (a)

Solve for \(\left[\mathrm{OH}^{-}\right]\) using the given \(\left[\mathrm{H}_3\mathrm{O}^{+}\right]\) of \(1.5 \times 10^{-9} \mathrm{M}\) and the ion product of water. Equation: \(\left[\mathrm{OH}^{-}\right] = \frac{1 \times 10^{-14}}{\left[\mathrm{H}_3\mathrm{O}^{+}\right]}\). Substitute and calculate: \(\left[\mathrm{OH}^{-}\right] = \frac{1 \times 10^{-14}}{1.5 \times 10^{-9}} = 6.67 \times 10^{-6} \mathrm{M}\).

Classify Solution (a) as Acidic or Basic

Because \(\left[\mathrm{OH}^{-}\right] > \left[\mathrm{H}_3\mathrm{O}^{+}\right]\), solution (a) is basic.

Calculate \(\left[\mathrm{OH}^{-}\right]\) for (b)

Using the same method, calculate for (b) where \(\left[\mathrm{H}_3\mathrm{O}^{+}\right] = 9.3 \times 10^{-9} \mathrm{M}\). Equation: \(\left[\mathrm{OH}^{-}\right] = \frac{1 \times 10^{-14}}{9.3 \times 10^{-9}} = 1.08 \times 10^{-6} \mathrm{M}\).

Classify Solution (b) as Acidic or Basic

Since \(\left[\mathrm{OH}^{-}\right] > \left[\mathrm{H}_3\mathrm{O}^{+}\right]\), solution (b) is basic.

Calculate \(\left[\mathrm{OH}^{-}\right]\) for (c)

Proceed with the calculation for (c): \(\left[\mathrm{OH}^{-}\right] = \frac{1 \times 10^{-14}}{2.2 \times 10^{-6}} = 4.55 \times 10^{-9} \mathrm{M}\).

Classify Solution (c) as Acidic or Basic

Given that \(\left[\mathrm{OH}^{-}\right] < \left[\mathrm{H}_3\mathrm{O}^{+}\right]\), solution (c) is acidic.

Calculate \(\left[\mathrm{OH}^{-}\right]\) for (d)

For case (d), calculate: \(\left[\mathrm{OH}^{-}\right] = \frac{1 \times 10^{-14}}{7.4 \times 10^{-4}} = 1.35 \times 10^{-11} \mathrm{M}\).

Classify Solution (d) as Acidic or Basic

Because \(\left[\mathrm{OH}^{-}\right] < \left[\mathrm{H}_3\mathrm{O}^{+}\right]\), solution (d) is acidic.

Key concepts

pH and pOH relationship

The concepts of pH and pOH are fundamental to understanding the acid-base properties of solutions. pH is a measure of the hydrogen ion concentration \(\left[\mathrm{H}_3\mathrm{O}^{+}\right]\), while pOH is a measure of the hydroxide ion concentration \(\left[\mathrm{OH}^{-}\right]\). These two values are inversely related; as the concentration of hydrogen ions increases, the pH decreases, indicating a more acidic solution. Conversely, as hydroxide ion concentration increases, the pOH decreases, indicating a more basic solution.

The relationship between pH and pOH is quantitatively expressed with a simple equation: \(pH + pOH = 14\) at 25°C. This direct relationship allows us to calculate one value if the other is known and is essential for classifying solutions as acidic, basic, or neutral.

Ion product of water

Water has a remarkable characteristic called the ion product, which is fundamental when discussing the pH and pOH of solutions. Pure water partially dissociates into hydrogen ions \(\mathrm{H}_3\mathrm{O}^{+}\) and hydroxide ions \(\mathrm{OH}^{-}\), maintaining an equilibrium. At a temperature of 25°C, the ion product is always \(1 \times 10^{-14}\) M², expressed in the equation \(\left[\mathrm{H}_3\mathrm{O}^{+}\right] \times \left[\mathrm{OH}^{-}\right] = 1 \times 10^{-14}\).

This relationship stays constant for all aqueous solutions, regardless of their acidity or basicity. It allows us to calculate the concentration of hydroxide ions if we know the concentration of hydrogen ions and vice versa, which is crucial for understanding the chemical nature of solutions.

Acidic and basic solutions

When discussing acidity or basicity of solutions, we refer to the concentrations of hydrogen ions \(\left[\mathrm{H}_3\mathrm{O}^{+}\right]\) and hydroxide ions \(\left[\mathrm{OH}^{-}\right]\). A solution is termed acidic if the concentration of hydrogen ions exceeds that of hydroxide ions, which typically results in a pH lower than 7. Conversely, a solution is considered basic or alkaline when the concentration of hydroxide ions is greater than that of hydrogen ions, often leading to a pH above 7.

While pure water is neutral with a pH of approximately 7, the presence of dissolved substances can shift the balance, creating acidic or basic solutions. It is this balance (or imbalance) between hydrogen and hydroxide ions that dictates the pH level and is essential for various chemical and biological processes.

Calculating hydroxide concentration

Calculating the hydroxide ion concentration is a straightforward process when the concentration of hydrogen ions is known. Using the constant ion product of water \(1 \times 10^{-14}\), one can determine the \(\left[\mathrm{OH}^{-}\right]\) by the formula \(\left[\mathrm{OH}^{-}\right] = \frac{1 \times 10^{-14}}{\left[\mathrm{H}_3\mathrm{O}^{+}\right]}\).

For instance, if the hydrogen ion concentration is \(1.5 \times 10^{-9}\) M, the corresponding hydroxide ion concentration is \(\left[\mathrm{OH}^{-}\right] = \frac{1 \times 10^{-14}}{1.5 \times 10^{-9}} = 6.67 \times 10^{-6}\) M, indicating a basic solution. Understanding how to calculate this concentration is key to many scientific fields, including chemistry, biochemistry, and environmental science, where pH plays a crucial role.