Question

The solubility product of iron(III) hydroxide is very small: \(K_{\mathrm{sp}}=4 \times 10^{-38}\) at \(25^{\circ} \mathrm{C} .\) 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 Fe(OH)_3 in water at 25°C is approximately \(6.33 \times 10^{-13}\) mol/L, or \(6.77 \times 10^{-11}\) g/L. This shows that the concentration of iron remaining in solution in a sample after the addition of NaOH or NH3 is indeed very small.

Step-by-step solution

Write the balanced chemical equation

Write the balanced equation for the dissolution of Fe(OH)_3 in water: \[Fe(OH)_3 (s) \rightleftharpoons Fe^{3+} (aq) + 3OH^-(aq)\]

Set up the Ksp expression

Write the Ksp expression for Fe(OH)_3: \[K_{sp} = [Fe^{3+}] [OH^-]^3\] We are given: \[K_{sp} = 4 \times 10^{-38}\]

Solve for solubility

If we let the molar solubility of Fe(OH)_3 in water be represented by "s," then the concentration of Fe^3+ ions in solution will be equal to "s," and the concentration of OH^- ions in solution will be equal to 3s. Substitute these values into the Ksp expression: \[K_{sp} = (s)(3s)^3\] Now, solve for s: \[4 \times 10^{-38} = (s)(27s^3)\] Divide both sides by 27: \[s^3 = \frac{4 \times 10^{-38}}{27}\] Now, take the cube root of both sides to find s: \[s = \sqrt[3]{\frac{4 \times 10^{-38}}{27}} \approx 6.33 \times 10^{-13}\text{ mol/L}\]

Convert to grams per liter

Convert the solubility from moles per liter to grams per liter, using the molar mass of Fe(OH)_3 (which is 106.87 g/mol): \[Solubility = 6.33 \times 10^{-13}\text{ mol/L} \times 106.87\text{ g/mol} \approx 6.77 \times 10^{-11}\text{ g/L}\] Thus, the solubility of Fe(OH)_3 in water at 25°C is approximately \(6.33 \times 10^{-13}\) mol/L, or \(6.77 \times 10^{-11}\) g/L. This shows that the concentration of iron remaining in solution in a sample after the addition of NaOH or NH3 is indeed very small.

Key concepts

Solubility Product Constant (Ksp)

The solubility product constant, often abbreviated as Ksp, is a specialized equilibrium constant that applies to the dissolution process of substances that form sparingly soluble salts. This concept is crucial in understanding how a solute dissolves in a solvent under specific conditions.
In general, Ksp provides insight into the degree of solubility of a particular compound in water. It is calculated based on the concentrations of the ions produced when a compound partially dissolves. The lower the Ksp value, the less soluble the compound is in water. Therefore, a very small Ksp indicates that the substance does not easily dissociate in solution.
For example, in iron(III) hydroxide, a Ksp of approximately \(4 \times 10^{-38}\) at \(25^{\circ} \text{C} \) shows that the solubility is extremely low, meaning that very little \(\text{Fe(OH)}_3\) will dissolve in water. Understanding Ksp is vital for predicting whether a precipitation reaction will occur when solutions are mixed, allowing chemists to control reactions during experiments and industrial processes.

Iron(III) Hydroxide

Iron(III) hydroxide is a solid compound with the formula \(\text{Fe(OH)}_3\). It often appears as a reddish-brown precipitate in chemical reactions and has a very low solubility in water. This limited solubility is largely due to the low Ksp value associated with \(\text{Fe(OH)}_3\).
This compound is typically formed through chemical reactions that involve iron salts and hydroxide ions. For instance, when sodium hydroxide \((\text{NaOH})\) or ammonia \((\text{NH}_3)\) is added to a solution containing iron ions, a precipitation reaction occurs resulting in the formation of \(\text{Fe(OH)}_3\).
Despite its minimal solubility, iron(III) hydroxide is a significant component in various fields, such as water treatment processes. It functions to remove impurities through coagulation and is often found in analytical chemistry procedures for qualitative analysis of iron. Understanding its properties can aid in comprehending other chemical processes where solubility is a key factor.

Precipitation Reaction

Precipitation reactions are a type of chemical reaction where two soluble salts in solution are combined to form an insoluble solid, known as a precipitate. These reactions are used extensively in chemistry to separate substances, identify ions in solution, and even in manufacturing processes.
In the context of iron(III) hydroxide, when a compound like sodium hydroxide \((\text{NaOH})\) or ammonia \((\text{NH}_3)\) is added to an iron(III) solution, a precipitation reaction occurs forming \(\text{Fe(OH)}_3\) as a solid precipitate. This reaction is a hallmark example of how solubility products relate to precipitation reactions. Since the solubility product constant of \(\text{Fe(OH)}_3\) is extremely low \((4 \times 10^{-38})\), it readily precipitates out of solution when the conditions favor this state.
These reactions are not only important for laboratory analysis and industry but also have environmental significance. They play a role in natural processes such as the formation of mineral deposits and the purification of natural water sources. Understanding precipitation reactions helps us manage both manmade and natural chemical processes effectively.