Problem 21 - General Chemistry: Principles and Modern Applications

Question

Write the equilibrium constant expression for the following reaction, $$\begin{array}{r} \mathrm{Fe}(\mathrm{OH})_{3}+3 \mathrm{H}^{+}(\mathrm{aq}) \rightleftharpoons \mathrm{Fe}^{3+}(\mathrm{aq})+3 \mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \\ K=9.1 \times 10^{3} \end{array}$$ and compute the equilibrium concentration for \(\left[\mathrm{Fe}^{3+}\right]\) at \(\left.\mathrm{pH}=7 \text { (i.e., }\left[\mathrm{H}^{+}\right]=1.0 \times 10^{-7}\right)\)

Step-by-Step Solution

Answer

\([\text{Fe}^{3+}] = 9.1 \times 10^{-18}\) M

Step 1: Write the Equilibrium Constant Expression

Recall that the equilibrium constant expression for a reaction is defined by \(K = \frac{[\text{products}]}{[\text{reactants}]}\), raising concentrations to the power of their stoichiometric coefficients. This gives \(K = \frac{[\text{Fe}^{3+}] [\text{H}_{2}O]^3}{[\text{Fe(OH)}_3][\text{H}^{+}]^3}\). Since H2O is a liquid, its concentration is not included in the equilibrium constant expression, simplifying the expression to \(K = \frac{[\text{Fe}^{3+}]}{[\text{Fe(OH)}_3][\text{H}^{+}]^3}\)

Step 2: Substitute known values into the equation

We are given that \(K = 9.1 \times 10^{3}\) and \([\text{H}^{+}] = 1.0 \times 10^{-7}\). Since Fe(OH)3 is the precipitate (solid), we assign its concentration as 1, leading to: \[9.1 \times 10^{3} = \frac{[\text{Fe}^{3+}]}{(1)(1.0 \times 10^{-7})^3}\]

Step 3: Solve for Fe^{3+ concentration}

Multiply both sides of the equation by \((1.0 \times 10^{-7})^3\) to solve for \([\text{Fe}^{3+}]\): \[ [\text{Fe}^{3+}] = 9.1 \times 10^{3} \times (1.0 \times 10^{-7})^3 \]

Key Concepts

Fe(OH)3

Iron(III) hydroxide, or Fe(OH)₃, is a chemical compound with interesting properties in chemistry. It is an insoluble hydroxide formed when iron ions react with hydroxide ions in an aqueous solution, resulting in a brownish solid precipitate. When Fe(OH)₃ comes into contact with a strong acid like H⁺, it dissolves by releasing iron ions and water, participating in a reversible chemical reaction. This reaction is a type of equilibrium, where the formation of reactants and products happen simultaneously. Consequently, calculating the equilibrium constant expression of such a reaction helps understand the extent to which Fe(OH)₃ can dissolve under given conditions. As you dive deeper into Fe(OH)₃ chemistry, remember: - It is an insoluble compound only dissolving in strong acids. - In chemical reactions, it acts as a source of Fe³⁺ ions. - Understanding its dissolving mechanism is crucial for equilibrium calculations.

pH calculation

The concept of pH is central to understanding chemical reactions in aqueous solutions. pH is a measure of how acidic or basic a solution is, determined by the concentration of hydrogen ions (H⁺) it contains. The pH calculation follows the formula: \[ ext{pH} = - ext{log}_{10} [ ext{H}^+] \]This calculation is not only fundamental in determining the acidity of the solution but also in forecasting the behavior of compounds within it. For example, knowing the pH allows you to determine how much Fe(OH)₃ can dissolve into Fe³⁺ and other ions.Key points about pH calculation:

• Lower pH values indicate higher acidity (higher concentrations of H⁺).
• A neutral pH value is 7, corresponding to a H⁺ concentration of 1.0 × 10⁻⁷ M.
• Understanding pH is essential in calculating equilibrium concentrations and solving chemical equilibrium problems.

chemical equilibrium

Chemical equilibrium is a crucial concept for predicting the composition of a chemical reaction over time. At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, leading to stable concentrations of reactants and products. For a reaction involving Fe(OH)₃, achieving chemical equilibrium means that the dissolved and undissolved forms will maintain a constant ratio, governed by the equilibrium constant (K). Here are some important aspects of chemical equilibrium:

• The equilibrium state can be described by the equilibrium constant expression, K, which allows the calculation of concentrations of all species involved at equilibrium.
• K is derived from the products' concentration divided by the reactants' concentration, each raised to the power of their respective coefficients in the balanced equation.
• In an equilibrium involving solids or liquids, these do not appear in the K expression as their concentrations are constant.

Fe3+ concentration

Determining the concentration of Fe³⁺ ions in solution is an essential part of understanding the dissolution of Fe(OH)₃ at equilibrium. The concentration of Fe³⁺ reveals the extent to which Fe(OH)₃ dissolves under specific conditions, such as pH. By knowing the equilibrium constant (K) and the concentration of hydrogen ions (H⁺), you can calculate the concentration of Fe³⁺ ions. This is achieved by arranging the equilibrium expression to solve for Fe³⁺. Important tips for calculating Fe³⁺ concentration:

• The concentration of Fe³⁺ can be directly calculated via the equilibrium constant expression, once other parameters are known.
• Fe³⁺'s concentration will increase with lower H⁺ concentrations, affecting dissolved Fe(OH)₃.
• Accurate measurement of pH assists in precise calculation of Fe³⁺ levels in your equilibrium analysis.

stoichiometry in chemical reactions

Stoichiometry is fundamental in understanding any chemical reaction, including those at equilibrium. It involves the quantitative relationship between reactants and products in a balanced chemical equation. For the dissolution reaction of Fe(OH)₃, knowing the stoichiometry is essential for setting up the equilibrium constant expression, as it dictates how molecules are counted and used in calculation. Stoichiometric principles to remember:

• Identify the coefficients in the balanced reaction equation, as these dictate the proportional relationships of reactants to products.
• Use stoichiometry to inform the powers each concentration is raised to in the equilibrium constant expression.
• Understanding stoichiometry aids in predicting reaction yields, making it easier to analyze the behavior of chemicals at equilibrium.