Question

Calculate \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\)given \(\left[\mathrm{OH}^{-}\right]\)in each aqueous solution and classify each solution as acidic or basic. (a) \(\left[\mathrm{OH}^{-}\right]=2.1 \times 10^{-11} \mathrm{M}\) (b) \(\left[\mathrm{OH}^{-}\right]=7.5 \times 10^{-9} \mathrm{M}\) (c) \(\left[\mathrm{OH}^{-}\right]=2.1 \times 10^{-4} \mathrm{M}\) (d) \(\left[\mathrm{OH}^{-}\right]=1.0 \times 10^{-2} \mathrm{M}\)

Short answer

Solutions (a) and (b) are acidic with \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) equal to \(4.76 \times 10^{-4} \mathrm{M}\) and \(1.33 \times 10^{-6} \mathrm{M}\) respectively. Solutions (c) and (d) are basic with \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) equal to \(4.76 \times 10^{-11} \mathrm{M}\) and \(1.0 \times 10^{-12} \mathrm{M}\) respectively.

Step-by-step solution

Understanding the Relationship between \(\mathrm{H}_{3}\mathrm{O}^{+}\) and \(\mathrm{OH}^{-}\)

The concentrations of hydrogen ions (\(\mathrm{H}_{3}\mathrm{O}^{+}\)) and hydroxide ions (\(\mathrm{OH}^{-}\)) are inversely related by the ion product constant for water, \(K_{w}\), which is \(1.0 \times 10^{-14} \) at 25°C. We use the formula \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right] = \frac{K_{w}}{\left[\mathrm{OH}^{-}\right]}\) to find the concentration of \(\mathrm{H}_{3}\mathrm{O}^{+}\).

Calculate \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) for (a)

For \(\left[\mathrm{OH}^{-}\right]=2.1 \times 10^{-11} \mathrm{M}\), calculate \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) using the formula: \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right] = \frac{1.0 \times 10^{-14}}{2.1 \times 10^{-11}} = \frac{1}{2.1} \times 10^{-3}\).

Classify solution (a)

If \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right] > \left[\mathrm{OH}^{-}\right]\), the solution is acidic. For (a), since \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right] > \left[\mathrm{OH}^{-}\right]\), the solution is acidic.

Calculate \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) for (b)

For \(\left[\mathrm{OH}^{-}\right]=7.5 \times 10^{-9} \mathrm{M}\), calculate \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) using the formula: \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right] = \frac{1.0 \times 10^{-14}}{7.5 \times 10^{-9}} = \frac{1}{7.5} \times 10^{-6}\).

Classify solution (b)

Since \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right] > \left[\mathrm{OH}^{-}\right]\), solution (b) is acidic.

Calculate \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) for (c)

For \(\left[\mathrm{OH}^{-}\right]=2.1 \times 10^{-4} \mathrm{M}\), calculate \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) using the formula: \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right] = \frac{1.0 \times 10^{-14}}{2.1 \times 10^{-4}} = \frac{1}{2.1} \times 10^{-11}\).

Classify solution (c)

Since \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right] < \left[\mathrm{OH}^{-}\right]\), solution (c) is basic.

Calculate \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) for (d)

For \(\left[\mathrm{OH}^{-}\right]=1.0 \times 10^{-2} \mathrm{M}\), calculate \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) using the formula: \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right] = \frac{1.0 \times 10^{-14}}{1.0 \times 10^{-2}} = 1.0 \times 10^{-12}\).

Classify solution (d)

Since \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right] < \left[\mathrm{OH}^{-}\right]\), solution (d) is basic.

Key concepts

Hydroxide Ion Concentration

The hydroxide ion concentration, denoted by \(\left[\mathrm{OH}^{-}\right]\), is a measure of the quantity of hydroxide ions present in a solution. In aqueous solutions, hydroxide ions are an indicator of basicity – the higher the concentration of \(\left[\mathrm{OH}^{-}\right]\), the more basic the solution is. As \(\left[\mathrm{OH}^{-}\right]\) increases, the solution has a greater capacity to accept protons (\(\mathrm{H}^{+}\)) from acids, which is a fundamental characteristic of bases.
To assess the acidity or basicity of solutions, we contrast the concentrations of \(\left[\mathrm{OH}^{-}\right]\) and \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\). A solution with a higher \(\left[\mathrm{OH}^{-}\right]\) compared to \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) is basic, while the opposite indicates an acidic solution. This relation forms the basis for understanding how substances can affect the pH level of a solution.

Hydronium Ion Concentration

Similarly, the hydronium ion concentration, \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\), reflects the number of hydronium ions in a solution and is a direct measure of the solution's acidity. Aqueous solutions containing a high concentration of hydronium ions are acidic because these ions are responsible for donating protons (\(\mathrm{H}^{+}\)) to other molecules or ions – the hallmark of acidic behavior.
To find the \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) when given the \(\left[\mathrm{OH}^{-}\right]\), we use the water ion product constant to calculate the opposing ion concentration. This interrelationship underlines the balance of a solution's pH, determining whether a solution is more inclined towards acidic or basic properties.

pH Calculation

pH is a numeric scale used to specify the acidity or basicity of an aqueous solution. It is defined as the negative logarithm to base 10 of the hydronium ion concentration: \(pH = -\log_{10}\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\). Conveniently, a pH less than 7 indicates an acidic solution, while a pH greater than 7 suggests a basic solution.
Calculating pH provides us with a convenient and unified way to compare the acidity or basicity of various substances. By understanding the principles behind pH and its calculation from the concentration of hydronium ions, students can better grasp how different solutions will react and can predict the outcome of mixing acids and bases.

Water Ion Product Constant

The water ion product constant, \(K_{w}\), is an essential concept in understanding the relationship between \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) and \(\left[\mathrm{OH}^{-}\right]\). At 25°C, \(K_{w}\) is equal to \(1.0 \times 10^{-14}\) and it represents the mathematical product of the concentrations of the hydronium and hydroxide ions in pure water: \(K_{w} = \left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\times\left[\mathrm{OH}^{-}\right]\).
This equilibrium constant remains consistent for all aqueous solutions at this temperature, regardless of the presence of acids or bases. Therefore, if we know the concentration of either the hydronium or hydroxide ions, we can calculate the other one using \(K_{w}\). This invariable property is a cornerstone in the field of acid-base chemistry and forms the foundation for understanding the interrelated behavior of acidic and basic solutions.