Question

Calculate the solubility of \(\mathrm{Mg}_{\mathrm{g}}(\mathrm{OH})_{2}\) in \(0.50 \mathrm{M} \mathrm{NH} \mathrm{NCl}_{4}\).

Short answer

The solubility of Mg(OH)₂ in 0.50 M NH₄Cl can be calculated by first finding the Ksp and Kf values, setting up an ICE table, and solving for the concentration of Mg²⁺ ions (s). The Ksp value for Mg(OH)₂ is \(5.61 \times 10^{-12}\) and the Kf value for NH₄OH is \(1.70 \times 10^{7}\). Using these values and simplifying the equilibrium expressions and solubility relations, you can solve for the value of s numerically. Finally, the solubility (M) of Mg(OH)₂ is equal to the concentration of Mg²⁺ ions, which is s.

Step-by-step solution

Write the Equilibrium Equations

First, we need to write the equilibrium equations for the dissolution of Mg(OH)₂ and the formation of the complex ion between NH₄⁺ and OH⁻ ions: Mg(OH)₂(s) ⇌ Mg²⁺(aq) + 2OH⁻(aq) - Dissolution of Mg(OH)₂ NH₄⁺(aq) + OH⁻(aq) ⇌ NH₄OH(aq) - Formation of the complex ion

Find the Values of Ksp and Kf

We need to find the values of Ksp for Mg(OH)₂ and Kf for the complex ion. In general, these values can be found in a textbook or online. For this problem, we will use the following values: \(K_{sp} = 5.61 \times 10^{-12}\) for Mg(OH)₂ \(K_{f} = 1.70 \times 10^{7}\) for NH₄OH

Set Up the ICE Table

Next, we will set up the ICE (Initial, Change, Equilibrium) table for the dissolution of Mg(OH)₂ and the formation of the complex ion. The initial concentration of NH₄⁺ ions is 0.50 M, and the initial concentration of Mg²⁺ and OH⁻ ions is 0. Mg²⁺: 0 OH⁻: 0 NH₄⁺: 0.5 The changes are: Mg²⁺: +s OH⁻: +2s NH₄⁺: -s The equilibrium values are: Mg²⁺: s OH⁻: \(2s+x\) NH₄⁺: \(0.5-s\) We need to add x to the 2s term for the OH⁻ ion because some of the OH⁻ ions will react with the NH₄⁺ ions to form NH₄OH.

Write the Equilibrium Expression and Solve for s

Now, write the equilibrium expressions for Ksp and Kf: \(K_{sp} = [Mg^{2+}][OH^-]^2\) \(K_{f} = \frac{[NH_4OH]}{[NH_4^+][OH^-]}\) We can now use the ICE table values to substitute the concentrations: \(K_{sp} = s(2s+x)^2\) \(K_{f} = \frac{1.70 \times 10^7 * x}{(0.5-s)(2s+x)}\) We can solve for x and s by first assuming that s is very small compared to 0.50. This allows us to simplify our equations: \(K_{sp} = s(2s+x)^2 \approx s(4sx^2)\) - assuming s << 0.5 \(K_{f} = \frac{1.70 \times 10^7 * x}{(0.5)(2s+x)}\) Now solve for x in terms of s using the second equation: \(x = \frac{K_{f}(0.5)(2s+x)}{1.70 \times 10^7}\) Next, substitute this expression into the first equation: \(K_{sp} = s(4s(\frac{K_{f}(0.5)(2s+x)}{1.70 \times 10^7})^2)\) Finally, solve for s numerically using a calculator or a numerical analysis technique.

Calculate the Solubility of Mg(OH)₂

Once you have obtained the value of s, you can calculate the solubility of Mg(OH)₂: Solubility (M) = [Mg²⁺] = s By following these steps and solving for s, you will find the solubility of Mg(OH)₂ in 0.50 M NH₄Cl.

Key concepts

Chemical Equilibrium

In the context of solubility calculations, chemical equilibrium refers to the state in which the rate of the forward reaction (dissolving of the solid into its constituent ions) is equal to the rate of the reverse reaction (recombination of the ions to form the solid). This does not mean the concentrations of the reactants and products are equal, but that their concentrations have stabilized and are not changing over time.

For the dissolution of magnesium hydroxide, Mg(OH)₂, in water, the equilibrium can be disturbed by the presence of other chemical species, such as NH₄⁺ from NH₄Cl. The NH₄⁺ can react with OH^- to form NH₄OH, a new compound, which shifts the equilibrium position. Understanding this concept is vital when predicting the solubility of compounds in different environments.

Solubility Product Constant (Ksp)

The solubility product constant (Ksp) is a crucial concept in solubility calculations. It is specific for every sparingly soluble compound and represents the maximum product of the molar concentrations of the ions that can exist in solution at equilibrium. The Ksp value is constant for a compound at a given temperature.

The formula for Ksp for magnesium hydroxide is given as:\[K_{sp} = [Mg^{2+}][OH^-]^2\]
As the concentration of magnesium and hydroxide ions reaches a specific point, any further dissolution will lead to a supersaturated solution, and excess solid will precipitate out to maintain the Ksp value.

Complex Ion Formation

The formation of complex ions can have a significant impact on the solubility of compounds. A complex ion consists of a central metal ion surrounded by molecules or anions called ligands. These ligands can 'coordinate' to the metal ion, effectively increasing its solubility.

In our exercise, the NH₄⁺ acting as a ligand, reacts with OH^- ions to form the complex ion NH₄OH. This reaction pulls OH^- ions out of equilibrium, influencing the solubility of Mg(OH)₂. The equilibrium constant for complex ion formation, known as the formation constant (Kf), quantifies the extent to which complex ion formation occurs.

ICE Table

An ICE table (Initial, Change, Equilibrium) is an organizational tool used to keep track of the concentrations of species in an aqueous equilibrium system. The 'I' stands for the initial concentrations, 'C' for the change that occurs as the system moves towards equilibrium, and 'E' for the equilibrium concentrations of each species involved.

In the context of our solubility problem, the ICE table helps us determine the changes in concentration of Mg²⁺, OH^-, and NH₄⁺ from their initial values through to equilibrium. It simplifies the complex interactions into a format that is easier to understand and provides a systematic method for calculating the solubility of Mg(OH)₂ in the presence of NH₄Cl.