Problem 37 - Chemistry An Atoms First Approach

Question

Calculate the solubility (in moles per liter) of \(\mathrm{Fe}(\mathrm{OH})_{3}\) \(\left(K_{\mathrm{sp}}=4 \times 10^{-38}\right)\) in each of the following. a. water b. a solution buffered at \(\mathrm{pH}=5.0\) c. a solution buffered at \(\mathrm{pH}=11.0\)

Step-by-Step Solution

Answer

The solubility of $\mathrm{Fe}(\mathrm{OH})_{3}$ is as follows: a. In water, the solubility is \(2.11 \times 10^{-10}\) mol/L. b. In a pH 5.0 buffered solution, the solubility is \(4 \times 10^{-11}\) mol/L. c. In a pH 11.0 buffered solution, the solubility is \(4 \times 10^{-29}\) mol/L.

Step 1: Case (a) - Water

When Fe(OH)₃ is dissolved in water, it dissociates into its ions: Fe(OH)₃(s) → Fe³⁺(aq) + 3OH⁻(aq) Let the solubility of Fe(OH)₃ be s mol/L. Thus, [Fe³⁺] = s and [OH⁻] = 3s. The Ksp expression is Ksp = [Fe³⁺][OH⁻]³. Substituting the concentrations, we get: Ksp = s × (3s)³ = 4 × 10⁻³⁸ Now we need to solve for s.

Step 2: Solving for s in Case (a)

Ksp = s × (3s)³ 4 × 10⁻³⁸ = s × 27s^3 s^4 = 4 × 10⁻³⁸ / 27 s = \(\sqrt[4]{\frac{4 \times 10^{-38}}{27}}\) s = 2.11 × 10⁻¹⁰ mol/L The solubility of Fe(OH)₃ in water is 2.11 × 10⁻¹⁰ mol/L.

Step 3: Case (b) - pH 5.0 Buffer Solution

In a buffer solution with pH 5.0, we can calculate the concentration of OH⁻ ions using the following formula: pOH = 14 - pH pOH = 14 - 5 = 9 Now, use the formula for pOH to find [OH⁻]: [OH⁻] = 10^(-pOH) = 10⁻⁹ M Now we can use the Ksp expression to solve for [Fe³⁺]: Ksp = [Fe³⁺][OH⁻]³ 4 × 10⁻³⁸ = [Fe³⁺](10⁻⁹)³

Step 4: Solving for [Fe³⁺] in Case (b)

4 × 10⁻³⁸ = [Fe³⁺](10⁻²⁷) [Fe³⁺] = \(\frac{4 \times 10^{-38}}{10^{-27}}\) = 4 × 10⁻¹¹ mol/L The solubility of Fe(OH)₃ in a pH 5.0 buffered solution is 4 × 10⁻¹¹ mol/L.

Step 5: Case (c) - pH 11.0 Buffer Solution

For a pH 11.0 buffer solution, we can follow the same steps as in case (b): pOH = 14 - 11 = 3 [OH⁻] = 10^(-pOH) = 10⁻³ M Now, use the Ksp expression with the [OH⁻] value: Ksp = [Fe³⁺][OH⁻]³ 4 × 10⁻³⁸ = [Fe³⁺](10⁻³)³

Step 6: Solving for [Fe³⁺] in Case (c)

4 × 10⁻³⁸ = [Fe³⁺](10⁻⁹) [Fe³⁺] = \(\frac{4 \times 10^{-38}}{10^{-9}}\) = 4 × 10⁻²⁹ mol/L The solubility of Fe(OH)₃ in a pH 11.0 buffered solution is 4 × 10⁻²⁹ mol/L. To summarize: a. In water, the solubility of Fe(OH)₃ is 2.11 × 10⁻¹⁰ mol/L. b. In a pH 5.0 buffered solution, the solubility of Fe(OH)₃ is 4 × 10⁻¹¹ mol/L. c. In a pH 11.0 buffered solution, the solubility of Fe(OH)₃ is 4 × 10⁻²⁹ mol/L.

Key Concepts

Chemical Equilibrium

Chemical equilibrium is a state in which the rate of the forward chemical reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of reactants and products over time. In the context of solubility equilibrium, the solubility product constant (\( K_{sp} \)) is a key aspect. It represents the level at which a solute's ions are saturated in solution, preventing further dissolution or prompting precipitation if exceeded.

When dealing with exercises like the dissolution of \( \mathrm{Fe}(\mathrm{OH})_{3} \), understanding \( K_{sp} \) is crucial because it allows us to calculate the point at which the solution is saturated with Fe(OH)₃ ions. Moreover, manipulations of this constant can provide insights into how the solubility will change as the environment—such as pH—is altered.

Buffer Solution

A buffer solution is a special chemical system that resists changes in pH when small amounts of acid or base are added. Buffers usually consist of a weak acid and its conjugate base or a weak base and its conjugate acid. They function by reacting with any added acid or base to minimize changes in pH.

In our exercise example, the impact of a buffer's pH on the solubility of \( \mathrm{Fe}(\mathrm{OH})_{3} \) is explored. In a buffered solution at pH 5.0, the concentration of hydroxide ions (OH⁻) is much lower compared to a buffer with pH 11.0. This difference in [OH⁻] concentration directly influences the solubility of Fe(OH)₃, as demonstrated by the stark contrast in solubility between the two buffered solutions. Understanding buffers is hence vital in predicting how a compound's solubility can differ in various environments.

pH Calculation

The pH scale is a measure of the acidity or basicity of a solution. It is defined as the negative logarithm (base 10) of the hydrogen ion concentration (\( pH = -\log[H^+] \)). Conversely, pOH measures the concentration of hydroxide ions and similarly follows that \( pOH = -\log[OH^-] \). pH and pOH are related by the equation \( pH + pOH = 14 \) for aqueous solutions at 25°C.

In the context of our solubility problem, calculating the pH helps determine the [OH⁻] present in the solution, which is then used to find the solubility of Fe(OH)₃ in a buffered environment. For instance, knowing the pH is 5.0 or 11.0 allows us to compute pOH and subsequently the [OH⁻], which is essential in solving for the solubility using the \( K_{sp} \) expression. pH calculation is an indispensable tool in many chemistry scenarios, from understanding solubility to gauging the properties of a chemical solution.