Question

Calculate the \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration in an aqueous solution at \(25^{\circ} \mathrm{C}\) with each of the following \(\mathrm{OH}^{-}\) concentrations: (a) \(2.50 \times 10^{-2} M\) (b) \(1.67 \times 10^{-5} M\) (c) \(8.62 \times 10^{-3} \mathrm{M}\) (d) \(1.75 \times 10^{-12} \mathrm{M}\).

Short answer

(a) \(4.0 \times 10^{-13} \) M; (b) \(5.99 \times 10^{-10} \) M; (c) \(1.16 \times 10^{-12} \) M; (d) \(5.71 \times 10^{-3} \) M.

Step-by-step solution

Understanding the Problem

To find the concentration of \( \mathrm{H}_3\mathrm{O}^+ \), we need to use the relationship between \( \mathrm{H}_3\mathrm{O}^+ \) and \( \mathrm{OH}^- \) in water. The ion product of water \( K_w \) at \( 25^{\circ} \mathrm{C} \) is \( 1.0 \times 10^{-14} \). So, \( [\mathrm{H}_3\mathrm{O}^+][\mathrm{OH}^-] = K_w \).

Calculate \( [\mathrm{H}_3 \mathrm{O}^+] \) for Case (a)

Given \([\mathrm{OH}^-] = 2.50 \times 10^{-2} \) M, use the equation:\[[\mathrm{H}_3\mathrm{O}^+] = \frac{K_w}{[\mathrm{OH}^-]} = \frac{1.0 \times 10^{-14}}{2.50 \times 10^{-2}}\]Calculate and find:\[[\mathrm{H}_3\mathrm{O}^+] = 4.0 \times 10^{-13} \, \mathrm{M}\]

Calculate \( [\mathrm{H}_3\mathrm{O}^+] \) for Case (b)

Given \([\mathrm{OH}^-] = 1.67 \times 10^{-5} \) M, use the equation:\[[\mathrm{H}_3\mathrm{O}^+] = \frac{K_w}{[\mathrm{OH}^-]} = \frac{1.0 \times 10^{-14}}{1.67 \times 10^{-5}}\]Calculate and find:\[[\mathrm{H}_3\mathrm{O}^+] = 5.99 \times 10^{-10} \, \mathrm{M}\]

Calculate \( [\mathrm{H}_3\mathrm{O}^+] \) for Case (c)

Given \([\mathrm{OH}^-] = 8.62 \times 10^{-3} \) M, use the equation:\[[\mathrm{H}_3\mathrm{O}^+] = \frac{K_w}{[\mathrm{OH}^-]} = \frac{1.0 \times 10^{-14}}{8.62 \times 10^{-3}}\]Calculate and find:\[[\mathrm{H}_3\mathrm{O}^+] = 1.16 \times 10^{-12} \, \mathrm{M}\]

Calculate \( [\mathrm{H}_3\mathrm{O}^+] \) for Case (d)

Given \([\mathrm{OH}^-] = 1.75 \times 10^{-12} \) M, use the equation:\[[\mathrm{H}_3\mathrm{O}^+] = \frac{K_w}{[\mathrm{OH}^-]} = \frac{1.0 \times 10^{-14}}{1.75 \times 10^{-12}}\]Calculate and find:\[[\mathrm{H}_3\mathrm{O}^+] = 5.71 \times 10^{-3} \, \mathrm{M}\]

Key concepts

Ion Product of Water

The ion product of water, often denoted as \(K_w\), is a fundamental concept in chemistry. It refers to the balance between the concentrations of hydrogen ions \( \left[ \mathrm{H}_3 \mathrm{O}^+ \right] \) and hydroxide ions \( \left[ \mathrm{OH}^- \right] \) in water. At 25 degrees Celsius, this equilibrium constant is always equal to \(1.0 \times 10^{-14}\). This relationship is expressed mathematically as:\[ [\mathrm{H}_3\mathrm{O}^+] \cdot [\mathrm{OH}^-] = K_w \]This equation illustrates that, even in pure water, there is a tiny amount of ionization that creates these ions. Understanding the ion product of water is essential for calculating concentrations in other solutions and predicting how they behave in reactions.

Kw Value

The \(K_w\) value is crucial in acid-base chemistry. It indicates the extent to which water self-ionizes to form \(\mathrm{H}_3\mathrm{O}^+\) and \(\mathrm{OH}^-\). This constant helps determine pH and pOH, which are the measures of acidity and basicity in solutions.

Why is \(K_w\) important?

• Predicts equilibrium in acid-base reactions.
• Helps to understand solution acidity or basicity.
• Vital for calculations involving acidic and basic solutions.

Since the \(K_w\) value is always \(1.0 \times 10^{-14}\) at \(25^{\circ}C\), this provides a benchmark value that can be used effectively to calculate unknown ion concentrations by relating them to known values.

Acid-Base Chemistry

Acid-base chemistry revolves around the transfer of hydrogen ions between substances. The hydronium ion \( \left( \mathrm{H}_3 \mathrm{O}^+ \right) \) represents the acidic properties, while the hydroxide ion \( \left( \mathrm{OH}^- \right) \) is indicative of basic properties. This area of chemistry helps us understand many natural processes, like digestion and even battery function.

Key Principles

• Acids increase \(\mathrm{H}_3\mathrm{O}^+\) in solution.
• Bases increase \(\mathrm{OH}^-\) in solution.
• The product of \(\mathrm{H}_3\mathrm{O}^+\) and \(\mathrm{OH}^-\) concentrations equals \(K_w\).

Understanding these fundamentals helps in predicting since acids and bases will neutralize each other when mixed, eventually affecting the pH of solutions.

Calculating Concentrations

Calculating concentrations of ions within solutions is a core skill in chemistry, particularly when dealing with acidic or basic solutions. Given the concentration of one ion and knowing \(K_w\), you can easily calculate the concentration of the other ion.For example, to find \( \left[ \mathrm{H}_3 \mathrm{O}^+ \right] \) given \( \left[ \mathrm{OH}^- \right] \), use the formula:\[ [\mathrm{H}_3\mathrm{O}^+] = \frac{K_w}{[\mathrm{OH}^-]} \]This approach allows for determining solution properties and predicting how changes in concentration will affect the overall chemical environment. Remember, practice with different examples solidifies one's understanding of this scientific principle.