Question

For a given aqueous solution, if \(\left[\mathrm{OH}^{-}\right]=7.11 \times 10^{-10} \mathrm{M},\) what is \(\left[\mathrm{H}^{+}\right] ?\)

Short answer

\([\mathrm{H}^{+}] \approx 1.41 \times 10^{-5} \mathrm{M}\).

Step-by-step solution

Understand the Relationship between Ion Concentrations

The product of the hydrogen ion concentration \([\mathrm{H}^{+}]\) and the hydroxide ion concentration \([\mathrm{OH}^{-}]\) in water at 25°C is a constant known as the ion-product constant for water \(K_w\). Mathematically, this is expressed as \([\mathrm{H}^{+}] \times [\mathrm{OH}^{-}] = K_w\), where \(K_w = 1.0 \times 10^{-14} \) at 25°C.

Insert Known Values into the Equation

Insert the given hydroxide ion concentration \( [\mathrm{OH}^{-}] = 7.11 \times 10^{-10}\) into the equation \([\mathrm{H}^{+}] \times [\mathrm{OH}^{-}] = 1.0 \times 10^{-14}\). This gives \([\mathrm{H}^{+}] \times 7.11 \times 10^{-10} = 1.0 \times 10^{-14}\).

Solve for Hydrogen Ion Concentration \([\mathrm{H}^{+}]\)

Rearrange the equation to solve for \([\mathrm{H}^{+}]\): \([\mathrm{H}^{+}] = \frac{1.0 \times 10^{-14}}{7.11 \times 10^{-10}}\).

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

Perform the division: \([\mathrm{H}^{+}] = \frac{1.0 \times 10^{-14}}{7.11 \times 10^{-10}}\). Calculating gives \([\mathrm{H}^{+}] \approx 1.41 \times 10^{-5}\).

Verify the Calculation

Check the calculation to ensure it is correct, confirming that the product of \([\mathrm{H}^{+}]\) and \([\mathrm{OH}^{-}]\) is indeed \(1.0 \times 10^{-14}\).

Key concepts

Understanding Hydrogen Ion Concentration

In aqueous solutions, the concentration of hydrogen ions (\(\left[\mathrm{H}^+\right]\)) is an essential aspect of defining the acidity of the solution. The hydrogen ion concentration directly relates to the pH of the solution, which is a measure of how acidic or basic the solution is.
The formula to calculate the pH from hydrogen ion concentration is:\[pH = - \log \left( \left[\mathrm{H}^+\right] \right)\] A low pH value means high hydrogen ion concentration, indicating that the solution is acidic. A high pH denotes a low hydrogen ion concentration, showcasing the solution's basicity. Knowing the hydrogen ion concentration helps in various scientific fields, such as biochemistry and environmental science, where maintaining the correct pH is crucial.
Generally, in water at 25°C, \(\left[\mathrm{H}^+\right]\) can be determined from the ion-product constant, \(K_w\), using the relationship with hydroxide ions. This is because the concentration of hydrogen and hydroxide ions in water are interdependent through the equilibrium constant.

Importance of Hydroxide Ion Concentration

Hydroxide ion concentration (\(\left[\mathrm{OH}^-\right]\)) plays a complementary role to hydrogen ions in determining the characteristics of a solution. The balance between hydrogen and hydroxide ions indicates whether a solution is acidic, neutral, or basic.

• High \(\left[\mathrm{OH}^-\right]\): Indicates a basic (alkaline) solution.
• Equal \(\left[\mathrm{H}^+\right]\) and \(\left[\mathrm{OH}^-\right]\): Represents a neutral solution.
• Low \(\left[\mathrm{OH}^-\right]\): Suggests an acidic solution.

Understanding hydroxide ion concentration is necessary when dealing with reactions in water, corrosion processes, and in industries like wastewater treatment. By using the formula \(\left[\mathrm{H}^+\right] \times \left[\mathrm{OH}^-\right] = K_w\), you can calculate one ion concentration if the other and the ion-product constant (\(K_w\)) are known.

Aqueous Solution Equilibrium

Aqueous solution equilibrium involves the constant relationship between hydrogen and hydroxide ion concentrations in water. This equilibrium is defined by the ion-product constant, \(K_w = 1.0 \times 10^{-14}\) at 25°C. Understanding this equilibrium is key when studying solutions in various chemical and biological contexts.
When a solution is at equilibrium, the rate of formation of ions equals the rate of recombination, ensuring the product of \(\left[\mathrm{H}^+\right]\) and \(\left[\mathrm{OH}^-\right]\) remains constant, \(1.0 \times 10^{-14}\). Changes in temperature can shift the \(K_w\), affecting solution equilibrium and requiring careful consideration in temperature-dependent experiments.
Aqueous solution equilibrium serves as the foundation for our understanding of pH, acid and base behavior, and the preparation of various chemical solutions. By mastering these equilibrium concepts, significant advancements in chemistry and other scientific disciplines can flourish.