Question

What is \(\left[\mathrm{H}^{+}\right]\) in a solution whose \(\left[\mathrm{OH}^{-}\right]\) is \(2.09 \times 10^{-2} \mathrm{M} ?\)

Short answer

The \([\mathrm{H}^+]\) in the solution is \(4.78 \times 10^{-13} \, \mathrm{M}\).

Step-by-step solution

Understanding the Ion Product of Water

Water self-ionizes to form a balance of hydrogen ions (H⁺) and hydroxide ions (OH⁻). At 25°C, the ion product of water is constant: \[ [\mathrm{H}^+] \cdot [\mathrm{OH}^-] = 1.0 \times 10^{-14} \, \mathrm{M}^2 \].This relationship allows us to find one ion concentration if we know the other.

Rearranging the Formula

We need to find the concentration of hydrogen ions \([\mathrm{H}^+]\) given the concentration of hydroxide ions \([\mathrm{OH}^-] = 2.09 \times 10^{-2} \, \mathrm{M} \).Using the ion product of water, rearrange the equation to solve for \([\mathrm{H}^+]\):\[ [\mathrm{H}^+] = \frac{1.0 \times 10^{-14}}{[\mathrm{OH}^-]} \].

Substituting Known Values

Substitute the given value of \([\mathrm{OH}^-]\) into the rearranged formula:\[ [\mathrm{H}^+] = \frac{1.0 \times 10^{-14}}{2.09 \times 10^{-2}} \].

Calculating the Result

Perform the division to calculate the concentration of hydrogen ions. Using a calculator, divide\(1.0 \times 10^{-14}\) by \(2.09 \times 10^{-2}\):\[ [\mathrm{H}^+] \approx 4.78 \times 10^{-13} \, \mathrm{M} \].

Interpreting the Result

The calculated hydrogen ion concentration \([\mathrm{H}^+]\) is \(4.78 \times 10^{-13} \, \mathrm{M} \), indicating a basic solution since it is less than \(1.0 \times 10^{-7} \, \mathrm{M}\).

Key concepts

Hydrogen Ion Concentration

When studying aqueous solutions, especially those involving water, it's crucial to understand the balance between hydrogen ions (\( [\mathrm{H}^+] \)) and hydroxide ions (\( [\mathrm{OH}^-] \)). These ions are the result of water self-ionizing, a process where water molecules convert into ions.

The concentration of \( [\mathrm{H}^+] \) in pure water at 25°C is typically \( 1.0 \times 10^{-7} \mathrm{M} \). However, when we introduce substances that alter water’s pH, this concentration changes.

• The ion product for water is constant, which means \( [\mathrm{H}^+] \times [\mathrm{OH}^-] = 1.0 \times 10^{-14} \, \mathrm{M}^2 \).
• This relationship allows calculation of one ion’s concentration if the other is known.

Given a hydroxide concentration, you can easily find the hydrogen ion concentration using: \[ [\mathrm{H}^+] = \frac{1.0 \times 10^{-14}}{[\mathrm{OH}^-]} \]
For the problem at hand with \( [\mathrm{OH}^-] = 2.09 \times 10^{-2} \, \mathrm{M} \), substituting into the formula will reveal \( [\mathrm{H}^+] \approx 4.78 \times 10^{-13} \, \mathrm{M} \).

Hydroxide Ion Concentration

Hydroxide ions, denoted as \( [\mathrm{OH}^-] \), play a critical role in determining the nature of a solution, whether acidic, neutral, or basic.

In pure water, at 25°C, the concentration of hydroxide ions is \( 1.0 \times 10^{-7} \mathrm{M} \), due to water undergoing self-ionization. This process generates equal concentrations of both \( [\mathrm{H}^+] \) and \( [\mathrm{OH}^-] \) at equilibrium.

• An increase in hydroxide ions typically makes a solution more basic, as seen in solutions with \( [\mathrm{OH}^-] \) greater than \( 1.0 \times 10^{-7} \mathrm{M} \).
• Conversely, solutions where \( [\mathrm{OH}^-] \) is less than \( 1.0 \times 10^{-7} \mathrm{M} \) are generally acidic.

In the given exercise, \( [\mathrm{OH}^-] = 2.09 \times 10^{-2} \, \mathrm{M} \), a clear indication of a solution trending towards basic. This concentration is part of the calculation for determining the corresponding hydrogen ion concentration.

pH Calculation

The pH of a solution is a measure of its acidity or alkalinity, where a lower pH indicates an acidic solution and a higher pH indicates a basic one. Understanding how to calculate pH is essential in chemistry.

The relationship between pH and hydrogen ion concentration is represented by the formula:\[ \text{pH} = -\log_{10}([\mathrm{H}^+]) \]This logarithmic scale implies that every unit change in pH reflects a tenfold change in hydrogen ion concentration.

• Neutral solutions have a pH of 7, corresponding to \( [\mathrm{H}^+] = 1.0 \times 10^{-7} \mathrm{M} \).
• Acidic solutions have \([\mathrm{H}^+]\) higher than \(1.0 \times 10^{-7} \mathrm{M} \), hence a pH less than 7.
• Basic solutions have \([\mathrm{H}^+]\) lower than \(1.0 \times 10^{-7} \mathrm{M} \), resulting in a pH greater than 7.

Using the calculated \( [\mathrm{H}^+] \approx 4.78 \times 10^{-13} \, \mathrm{M} \) from the exercise, the pH can be directly computed, showing a distinctly basic solution with a high pH value.