Problem 10 - Chemistry and Chemical Reactivity

Question

A saturated solution of milk of magnesia, \(\mathrm{Mg}(\mathrm{OH})_{2},\) has a pH of \(10.52 .\) What is the hydronium ion concentration of the solution? What is the hydroxide ion concentration? Is the solution acidic or basic?

Step-by-Step Solution

Answer

The hydronium ion concentration is approximately \(3.02 \times 10^{-11}\text{ M}\), the hydroxide ion concentration is approximately \(3.31 \times 10^{-4}\text{ M}\), and the solution is basic.

Step 1: Determine Hydrogen Ion Concentration from pH

The pH of a solution is related to the concentration of hydrogen ions \([\text{H}^+]\) in the solution by the formula: \[ \text{pH} = -\log[\text{H}^+] \] Given that the \(\text{pH} = 10.52\), we can rearrange the formula to find \([\text{H}^+]\): \[ [\text{H}^+] = 10^{-\text{pH}} = 10^{-10.52} \] Calculating this gives: \[ [\text{H}^+] \approx 3.02 \times 10^{-11} \text{ M} \]

Step 2: Calculate Hydroxide Ion Concentration from Known Values

We use the water auto-ionization constant (\(K_w\)), which is \(1.0 \times 10^{-14}\) at 25 °C, to find the concentration of hydroxide ions \([\text{OH}^-]\). This relationship is given by \[ K_w = [\text{H}^+][\text{OH}^-] \] Rearranging gives: \[ [\text{OH}^-] = \frac{K_w}{[\text{H}^+]} = \frac{1.0 \times 10^{-14}}{3.02 \times 10^{-11}} \] Which calculates to: \[ [\text{OH}^-] \approx 3.31 \times 10^{-4} \text{ M} \]

Step 3: Determine the Nature of the Solution

A solution is considered acidic if its \(\text{pH}\) is less than 7, neutral if \(\text{pH} = 7\), and basic if the \(\text{pH}\) is greater than 7. For this solution, the \(\text{pH} = 10.52\), which is greater than 7, determining it as basic.

Key Concepts

Hydronium Ion Concentration

Hydronium ion concentration, denoted as \([ ext{H}_3 ext{O}^+]\) or \([ ext{H}^+]\), plays a crucial role in determining the acidity or basicity of a solution. In a solution, the pH value helps us to find the concentration of these ions using a simple formula. To find hydronium ion concentration from a known pH, we use:

• \( [ ext{H}^+] = 10^{- ext{pH}} \).

For example, if the pH is 10.52, plug this value into the formula:

• \( [ ext{H}^+] = 10^{-10.52} \).
• Thus, \([\text{H}^+] \approx 3.02 \times 10^{-11} \text{ M}\).

As observed, the smaller the concentration of \([ ext{H}^+]\), the higher the pH, indicating a more basic solution. Knowing the hydronium ion concentration is essential when assessing chemical reactions and balancing equations in chemistry.

Hydroxide Ion Concentration

Understanding hydroxide ion concentration, denoted as \([ ext{OH}^-]\), is just as important as understanding hydronium ions for interpreting the nature of solutions. The concentration of hydroxide ions in aqueous solutions is inversely related to the hydrogen ion concentration, governed by the water auto-ionization constant \(K_w\).The fundamental relationship between these concentrations is:

• \( K_w = [ ext{H}^+][ ext{OH}^-] \).

Given that at 25 °C, \( K_w = 1.0 \times 10^{-14} \), we can find \([ ext{OH}^-]\) by rearranging this equation:

• \( [ ext{OH}^-] = \frac{K_w}{[ ext{H}^+]} \).

Substituting the hydronium ion concentration obtained earlier, the calculation becomes:

• \( [ ext{OH}^-] = \frac{1.0 \times 10^{-14}}{3.02 \times 10^{-11}} \approx 3.31 \times 10^{-4} \text{ M} \).

High concentrations of hydroxide ions often point to a basic solution, providing insights into equilibrium and reaction tendencies in aqueous environments.

Acidic and Basic Solutions

Solutions can be classified based on their pH levels, which tell us about the balance between hydronium and hydroxide ions in the solution:

• An acidic solution has a pH below 7.
• A neutral solution holds a pH of exactly 7.
• A basic solution has a pH above 7.

In our example, the solution's pH of 10.52 reveals it is basic. The logic behind this classification radiates from the concentration of ions. A higher \([ ext{H}^+]\) means an acidic environment while a smaller \([ ext{H}^+]\) with a higher \([ ext{OH}^-]\) concentration implies a basic one.Understanding the nature of a solution is crucial in fields such as chemistry and environmental science, as it affects reaction rates, corrosion, and biological processes. Thus, determining whether a solution is acidic or basic helps tailor approaches to chemical management and predictions.