Question

What is \(\left[\mathrm{H}^{+}\right]\) in a solution whose \(\left[\mathrm{OH}^{-}\right]\) is \(4.07 \times 10^{-7} \mathrm{M}\) ?

Short answer

The concentration of \(\left[\mathrm{H}^{+}\right]\) in the solution is \(2.46 \times 10^{-8} \, \mathrm{M}\).

Step-by-step solution

Understand the Relationship between \\(\left[\mathrm{H}^{+}\right]\\) and \\(\left[\mathrm{OH}^{-}\right]\\)

In water at 25°C, the ion product constant for water (Kw) is always \(1.0 \times 10^{-14}\). This means that, \[\left[\mathrm{H}^{+}\right] \times \left[\mathrm{OH}^{-}\right] = 1.0 \times 10^{-14}\].

Substitute the Given \\(\left[\mathrm{OH}^{-}\right]\\) into the Formula

We are given that \(\left[\mathrm{OH}^{-}\right] = 4.07 \times 10^{-7} \, \mathrm{M}\). Substitute this value into the formula: \[\left[\mathrm{H}^{+}\right] \times 4.07 \times 10^{-7} = 1.0 \times 10^{-14}\].

Solve for \\(\left[\mathrm{H}^{+}\right]\\)

Divide both sides of the equation by \(4.07 \times 10^{-7}\) to solve for \(\left[\mathrm{H}^{+}\right]\): \[\left[\mathrm{H}^{+}\right] = \frac{1.0 \times 10^{-14}}{4.07 \times 10^{-7}}\].

Perform the Calculation

Calculate \(\left[\mathrm{H}^{+}\right]\) by performing the division: \[\begin{align*}\left[\mathrm{H}^{+}\right] &= \frac{1.0 \times 10^{-14}}{4.07 \times 10^{-7}} \&= 2.46 \times 10^{-8} \, \mathrm{M}.\end{align*}\]

Interpret the Result

The result \(\left[\mathrm{H}^{+}\right] = 2.46 \times 10^{-8} \, \mathrm{M}\), indicates the concentration of hydrogen ions in the solution is relatively low, suggesting it is a basic solution.

Key concepts

Ion Product of Water

The ion product of water, represented as \(K_w\), is a fundamental concept in acid-base chemistry. It defines the product of the concentrations of the hydrogen ions \(\left[\mathrm{H}^{+}\right]\) and hydroxide ions \(\left[\mathrm{OH}^{-}\right]\) in water at a given temperature. At 25°C, this value is always \(1.0 \times 10^{-14}\). This constant value forms the basis for understanding the balance between acids and bases in water.
Essentially, the ion product of water reflects the state of equilibrium in pure water where both ions exist in minute amounts due to the self-ionization of water. This equilibrium can be disturbed by adding acids or bases, which alter the concentrations of \(\left[\mathrm{H}^{+}\right]\) and \(\left[\mathrm{OH}^{-}\right]\) but their product remains constant at \(1.0 \times 10^{-14}\).
Knowing \(K_w\) allows us to calculate either the hydrogen ion or hydroxide ion concentration when the other is known, useful for characterizing the acidity or basicity of the solution.

Hydrogen Ion Concentration

Hydrogen ion concentration, denoted as \(\left[\mathrm{H}^{+}\right]\), is an important measure in determining the acidity of a solution. In essence, the higher the concentration of \(\left[\mathrm{H}^{+}\right]\), the more acidic the solution is considered. This concentration can be directly related to the pH scale, where a lower pH corresponds to a higher \(\left[\mathrm{H}^{+}\right]\) concentration.
To find \(\left[\mathrm{H}^{+}\right]\) in a solution where \(\left[\mathrm{OH}^{-}\right]\) is known, you'd rearrange the water ion product equation: \(\left[\mathrm{H}^{+}\right] = \frac{K_w}{\left[\mathrm{OH}^{-}\right]}\). For the given problem, where \(\left[\mathrm{OH}^{-}\right] = 4.07 \times 10^{-7} \, \mathrm{M}\), substituting into the equation provides \(\left[\mathrm{H}^{+}\right] = 2.46 \times 10^{-8} \, \mathrm{M}\), indicating a basic solution.
Understanding \(\left[\mathrm{H}^{+}\right]\) is crucial for predicting reaction tendencies and the behavior of acids and bases in solution.

Hydroxide Ion Concentration

The hydroxide ion concentration \(\left[\mathrm{OH}^{-}\right]\) is an equally important measure in evaluating the basicity of a solution. As with hydrogen ions, the hydroxide ion concentration is part of the balance described by the water ion product \(K_w\). If the concentration of \(\left[\mathrm{OH}^{-}\right]\) increases, this implies a decrease in \(\left[\mathrm{H}^{+}\right]\), which generally indicates a basic solution.
The importance of hydroxide ion concentration lies in its ability to affect the pH of a solution. For our specific example, with \(\left[\mathrm{OH}^{-}\right]\) known as \(4.07 \times 10^{-7} \, \mathrm{M}\), we observe that the solution is basic because the corresponding \(\left[\mathrm{H}^{+}\right]\) results in a pH greater than 7.
Using \(\left[\mathrm{OH}^{-}\right]\) to ascertain the basic nature of solutions helps in various scientific and industrial applications, like in the understanding of reactions or adjustments necessary to achieve desired pH levels.