Question

For a given aqueous solution, if \(\left[\mathrm{OH}^{-}\right]=1.0 \times 10^{-5} \mathrm{M},\) what is \(\left[\mathrm{H}^{+}\right] ?\)

Short answer

\([\text{H}^+] = 1.0 \times 10^{-9} \, \text{M}\).

Step-by-step solution

Understand the Relationship

The ion product constant of water, \[K_w = [ ext{H}^+][ ext{OH}^-] = 1.0 \times 10^{-14} \, \text{M}^2 \]relates the concentrations of hydrogen ions and hydroxide ions in water.

Insert Given Values

Given that \([\text{OH}^-] = 1.0 \times 10^{-5} \, \text{M}\), we can substitute this value into the equation.\[K_w = [\text{H}^+](1.0 \times 10^{-5} \, \text{M}) = 1.0 \times 10^{-14} \, \text{M}^2\]

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

To find \([\text{H}^+]\), divide both sides of the equation by \([\text{OH}^-]\):\[[\text{H}^+] = \frac{1.0 \times 10^{-14} \, \text{M}^2}{1.0 \times 10^{-5} \, \text{M}}\]

Calculate \\(\\left[\mathrm{H}^{+}\\right]\\) Value

Perform the division:\[[\text{H}^+] = 1.0 \times 10^{-9} \, \text{M}\]

Key concepts

Ion Product Constant of Water

The ion product constant of water, denoted by \(K_w\), is a crucial concept in understanding the chemistry of aqueous solutions. It represents the product of the concentrations of hydrogen ions \([\text{H}^+]\) and hydroxide ions \([\text{OH}^-]\) in water. For water at 25°C, \(K_w\) is a constant value of \(1.0 \times 10^{-14} \, \text{M}^2\). This means that regardless of the state of the solution—whether it is acidic, neutral, or basic—the product of the concentrations of hydrogen ions and hydroxide ions will always equal this constant.
This relationship is fundamental for calculating either the hydroxide ion concentration or the hydrogen ion concentration in a solution when the other is known. Knowing this can help us predict the acidity or basicity of a given solution.

• If a solution is neutral, \([\text{H}^+]=[\text{OH}^-]\).
• If it is acidic, \([\text{H}^+]>[\text{OH}^-]\).
• If it is basic, \([\text{H}^+]<[\text{OH}^-]\).

This constant provides a benchmark against which you can measure different solutions.

Hydroxide Ion Concentration

The hydroxide ion concentration, \([\text{OH}^-]\), is a key factor in determining the basicity of a solution. In basic solutions, there is a higher concentration of hydroxide ions compared to hydrogen ions. The concentration of \([\text{OH}^-]\) helps in computing \([\text{H}^+]\) using the ion product constant of water, \(K_w\).
For example, when you know the \([\text{OH}^-]\) is \(1.0 \times 10^{-5} \, \text{M}\), you use the relationship
\[ K_w = [\text{H}^+][\text{OH}^-] = 1.0 \times 10^{-14} \, \text{M}^2 \] to find the \([\text{H}^+]\) concentration. By dividing \(K_w\) by \([\text{OH}^-]\), you can solve for \([\text{H}^+]\).
This process showcases the inverse relationship between \([\text{OH}^-]\) and \([\text{H}^+]\):

• A high \([\text{OH}^-]\) indicates a low \([\text{H}^+]\), meaning the solution is basic.
• A low \([\text{OH}^-]\) suggests a higher \([\text{H}^+]\), pointing to an acidic solution.

Understanding \([\text{OH}^-]\) not only helps in calculating pH and hydroxide but also in assessing the overall chemical nature of the solution.

Hydrogen Ion Concentration

The hydrogen ion concentration, \([\text{H}^+]\), serves as an indicator of a solution's acidity. In an aqueous solution, \([\text{H}^+]\) is inversely related to \([\text{OH}^-]\) as governed by the ion product constant of water, \(K_w\).
To find \([\text{H}^+]\), especially when dealing with an alkaline solution where \([\text{OH}^-]\) is known, you apply:
\[ [\text{H}^+] = \frac{K_w}{[\text{OH}^-]}\]
In the exercise provided, the \([\text{OH}^-]\) concentration is \(1.0 \times 10^{-5} \, \text{M}\). To find \([\text{H}^+]\), you divide the \(K_w\) value of \(1.0 \times 10^{-14}\) by \(1.0 \times 10^{-5}\), giving \([\text{H}^+]\) as \(1.0 \times 10^{-9} \, \text{M}\).

• This indicates the solution's nature based on how \([\text{H}^+]\) compares to neutral conditions (\([\text{H}^+]=1.0 \times 10^{-7} \, \text{M}\)).
• A \([\text{H}^+]\) below \(1.0 \times 10^{-7} \, \text{M}\) confirms a basic solution.

This concentration is essential for calculating the pH of a solution, where \( pH = -\log([\text{H}^+]) \). A lower \([\text{H}^+]\) indicates a higher pH, reflecting a more basic environment.