Problem 106 - Chemistry: An Atoms-Focused Approach

Question

Some scientists have proposed adding \(\mathrm{Fe}(\mathrm{III})\) compounds to large expanses of the open ocean to promote the growth of phytoplankton that would in turn remove \(\mathrm{CO}_{2}\) from the atmosphere through photosynthesis. Assuming the average pH of open ocean water is \(8.13,\) what is the maximum value of \(\left[\mathrm{Fe}^{3+}\right]\) in seawater if the \(K_{\mathrm{sp}}\) value of \(\mathrm{Fe}(\mathrm{OH})_{3}\) is \(1.1 \times 10^{-36} ?\)

Step-by-Step Solution

Answer

The maximum concentration of Fe³⁺ ions in seawater is approximately 1.03 x 10⁻¹⁸ M.

Step 1: Write the solubility equilibrium equation for \(\mathrm{Fe(OH)_3}\)

The solubility equilibrium is given by the following equation: $$ \mathrm{Fe(OH)_3(s)} \leftrightarrows \mathrm{Fe^{3+}(aq)} + 3\,\mathrm{OH^-(aq)} $$

Step 2: Write the \(K_{sp}\) expression

Write the expression for the solubility product constant for the above equation: $$ K_{sp} = [\mathrm{Fe^{3+}}][\mathrm{OH^{-}}]^3 $$

Step 3: Express the concentration of \(\mathrm{OH^{-}}\) ions in terms of pH

The pH of the seawater is given as \(8.13\). We can use this to calculate the concentration of \(\mathrm{OH^-}\) ions in the solution. Since \(pOH = 14 - pH\): $$ pOH = 14 - 8.13 = 5.87 $$ Now, the concentration of \(\mathrm{OH^{-}}\) ions is given by \([\mathrm{OH^{-}}] = 10^{-pOH}\). So, $$ [\mathrm{OH^{-}}] = 10^{-5.87} $$

Step 4: Substituting values in the \(K_{sp}\) expression and solving for \(\left[\mathrm{Fe^{3+}}\right]\)

Now we can substitute the values for \(K_{sp}\) and \([\mathrm{OH^{-}}]\) in the \(K_{sp}\) expression: $$ 1.1 \times 10^{-36} = [\mathrm{Fe^{3+}}](10^{-5.87})^3 $$ Next, isolate \([\mathrm{Fe^{3+}}]\) by dividing both sides by \((10^{-5.87})^3\): $$ [\mathrm{Fe^{3+}}] = \frac{1.1 \times 10^{-36}}{(10^{-5.87})^3} $$ Calculate the concentration of \(\mathrm{Fe^{3+}}\): $$ [\mathrm{Fe^{3+}}] \approx 1.03 \times 10^{-18}\,\mathrm{M} $$

Step 5: Conclusion

The maximum value of the concentration of \(\mathrm{Fe^{3+}}\) ions in seawater, if the solubility product constant of \(\mathrm{Fe(OH)_3}\) is \(1.1 \times 10^{-36}\) and the average pH of open ocean water is \(8.13\), is approximately \(1.03 \times 10^{-18}\,\mathrm{M}\).

Key Concepts

Solubility Product (Ksp)

The solubility product, denoted as \(K_{sp}\), is a helpful tool in understanding how substances dissolve in water. It's specifically useful for sparingly soluble compounds, like \(\mathrm{Fe(OH)_3}\). In essence, \(K_{sp}\) represents the equilibrium constant for the dissolution of a solid into its constituent ions.
For \(\mathrm{Fe(OH)_3}\), the dissolution process can be depicted as:

• \(\mathrm{Fe(OH)_3(s)} \rightleftharpoons \mathrm{Fe^{3+}(aq)} + 3\,\mathrm{OH^-(aq)}\)

The \(K_{sp}\) expression mathmatically relates to the concentrations of these ions in solution:

• \(K_{sp} = [\mathrm{Fe^{3+}}][\mathrm{OH^{-}}]^3\)

The value of \(K_{sp}\) helps us predict the solubility of the compound in a given solution. A lower \(K_{sp}\) value indicates lower solubility.
In our example, \(\mathrm{Fe(OH)_3}\) has a very low \(K_{sp}\), \(1.1 \times 10^{-36}\), signifying very limited solubility in water.

pH and pOH Calculations

Understanding pH and pOH is crucial for solving problems related to chemical equilibria in aqueous solutions. Here, pH denotes the acidity of a solution, where a pH less than 7 is acidic, and more than 7 is basic.
For our ocean water example, which is basic with a pH of 8.13, calculating pOH is straightforward:

• Use the relation: \(pOH = 14 - pH\)
• Therefore, \(pOH = 14 - 8.13 = 5.87\)

The concentration of hydroxide ions \([\mathrm{OH^{-}}]\) plays a critical role when assessing species solubility, as seen in the \(K_{sp}\) calculation.

• Calculate \([\mathrm{OH^{-}}]\) using: \([\mathrm{OH^{-}}] = 10^{-pOH}\)
• \([\mathrm{OH^{-}}] = 10^{-5.87}\)

This concentration impacts the equilibrium and thus the solubility product of compounds like \(\mathrm{Fe(OH)_3}\).
Knowledge of both pH and pOH allows scientists to manipulate chemical conditions and predict outcomes in natural systems.

Phytoplankton Growth Promoters

Phytoplankton are essential microscopic organisms in ocean ecosystems and play a vital role in carbon dioxide fixation through photosynthesis. When scientists suggest adding \(\mathrm{Fe(III)}\) compounds to oceans, the goal is to stimulate the growth of these tiny plants. This process, known as "ocean fertilization," aims to enhance the phytoplankton population thus increasing carbon uptake from the atmosphere.
Iron is a limiting nutrient in many ocean regions, and its addition can dramatically increase phytoplankton productivity. However, this must be carefully regulated:

• Excess iron can disrupt marine ecosystems. 
• Monitoring is necessary to balance ecosystem needs.

Calculating the maximum permissible concentration of \(\mathrm{Fe^{3+}}\) ions, as discussed using \(K_{sp}\) and pH, ensures that iron levels do not exceed what is soluble in seawater, maintaining a natural balance.
Thus, understanding the chemical equilibrium regarding solubility aids in making informed decisions for environmental interventions.