Question

The solubility product of iron(III) hydroxide is very small: \(\mathrm{K}_{\mathrm{sp}}=4 \times 10^{-38}\) at 25 C. A classical method of analysis for unknown samples containing iron is to add \(\mathrm{NaOH}\) or \(\mathrm{NH}_{3}\). This precipitates \(\mathrm{Fe}(\mathrm{OH})_{3},\) which can then be filtered and weighed. To demonstrate that the concentration of iron remaining in solution in such a sample is very small, calculate the solubility of \(\mathrm{Fe}(\mathrm{OH})_{3}\) in moles per liter and in grams per liter.

Short answer

The solubility of iron(III) hydroxide (Fe(OH)₃) at 25°C is approximately 7.09 × 10⁻¹⁰ moles per liter or 7.58 × 10⁻⁸ grams per liter.

Step-by-step solution

1. Write the balanced chemical equation for the dissolution of Fe(OH)₃

The balanced chemical equation for the dissolution of iron(III) hydroxide is: Fe(OH)₃ (s) ⇌ Fe³⁺ (aq) + 3OH⁻ (aq)

2. Write the Kₛₚ expression

For the reaction above, the solubility product constant (Kₛₚ) expression can be written as: Kₛₚ = [Fe³⁺][OH⁻]³ We are given Kₛₚ = 4 × 10⁻³⁸.

3. Set up an ICE table to find the solubility of Fe(OH)₃

An ICE table (Initial, Change, Equilibrium) helps to find the solubility (x) of Fe(OH)₃ using the Kₛₚ expression. The equilibrium concentrations are [Fe³⁺] = x and [OH⁻] = 3x. | Fe(OH)₃ (s) | Fe³⁺ (aq) | 3OH⁻ (aq) ----------------------------------------- I | - | 0 | 0 C | - | +x | +3x E | - | x | 3x Now substitute the equilibrium concentrations into the Kₛₚ expression: 4 × 10⁻³⁸ = (x)(3x)³

4. Solve for x (solubility in moles per liter)

Solve the equation for x (the solubility of Fe(OH)₃ in moles per liter): 4 × 10⁻³⁸ = (x)(27x³) x⁴ = (4 × 10⁻³⁸) / 27 x⁴ = 1.481 × 10⁻³⁹ x = ∛√(1.481 × 10⁻³⁹) x ≈ 7.09 × 10⁻¹⁰ moles per liter The solubility of Fe(OH)₃ in moles per liter is approximately 7.09 × 10⁻¹⁰.

5. Convert solubility from moles per liter to grams per liter

To convert the solubility from moles per liter to grams per liter, we'll use the molar mass of Fe(OH)₃, which is: 1 Fe = 55.845 g/mol 3 O = 3 × 16.00 g/mol = 48.00 g/mol 3 H = 3 × 1.008 g/mol = 3.024 g/mol Molar mass of Fe(OH)₃ = 55.845 + 48.00 + 3.024 = 106.869 g/mol Solubility in grams per liter = (solubility in moles per liter) × (molar mass of Fe(OH)₃) = (7.09 × 10⁻¹⁰ mol/L)(106.869 g/mol) ≈ 7.58 × 10⁻⁸ g/L The solubility of Fe(OH)₃ in grams per liter is approximately 7.58 × 10⁻⁸.

Key concepts

Chemical Equilibrium

In chemistry, the concept of chemical equilibrium is fundamental when discussing the solubility product of a substance like iron(III) hydroxide. Chemical equilibrium occurs when the rates of the forward and reverse reactions in a chemical process become equal, leading to no net change in the concentration of reactants and products over time. When considering the solubility of a compound, equilibrium is reached between the solid and its ions in solution.

For the solubility product (\(K_{sp}\)), it is the equilibrium constant for the solubility process of a sparingly soluble ionic compound. The smaller the value of the solubility product, the less soluble the compound is. In our case, the iron(III) hydroxide has a very small solubility product, indicating extremely low solubility, which means in equilibrium, the concentration of dissolved ions in solution is very tiny compared to the undissolved solid.

Precipitation Reaction

A precipitation reaction is another key concept at play when discussing the solubility product. In such reactions, soluble reactants combine to form an insoluble product—a precipitate—that settles out of the solution. It occurs when the product of the concentration of the ions in the solution exceeds the solubility product.In our exercise, when NaOH or NH₃ is added to a solution containing iron, iron(III) hydroxide forms as a precipitate. This demonstrates that the dissolved iron ions can interact with hydroxide ions to form a solid, which can then be removed from the solution. This kind of reaction is highly useful in analytical chemistry for separating and assessing the amount of iron in various samples. The low solubility of iron(III) hydroxide ensures that most of the iron can be efficiently removed from the mixture upon the addition of hydroxide ions.

Molar Mass

The molar mass is the mass of one mole of a substance and is usually expressed in grams per mole (g/mol). It plays a crucial role in converting the solubility from moles per liter to grams per liter, allowing us to understand the amount of solute in a common unit of measure. The molar mass is calculated by summing the atomic masses of all the atoms in the molecule.

In our example with iron(III) hydroxide (\(Fe(OH)_3\)), the molar mass is determined by adding the atomic masses of one iron (Fe) atom and three hydroxide (OH) groups. As given in the solution, the molar mass of \(Fe(OH)_3\) is 106.869 g/mol. Knowing the molar mass is essential for translating the theoretical solubility calculated in moles per liter to a practical quantity in grams per liter, which can be easily measured in a laboratory setting.