Question

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

Short answer

The solubility of Mg(OH)₂ in the presence of 0.50 M NH₄Cl is approximately \(1.31 \times 10^{-4} \mathrm{M}\).

Step-by-step solution

Write down the dissolution reaction

When Mg(OH)₂ dissolves in water, it forms Mg²⁺ and OH⁻ ions. The dissolution reaction can be written as: \[ \mathrm{Mg(OH)}_{2}(s) \rightleftharpoons \mathrm{Mg}^{2+}(aq) + 2\mathrm{OH}^{-}(aq) \]

Determine the solubility product constant (Ksp) of Mg(OH)₂

The solubility product constant, Ksp, is a measure of the solubility of a slightly soluble salt in water. This value is temperature-dependent and can be found in reference tables. For magnesium hydroxide (Mg(OH)₂) at 25°C, Ksp is \(1.5 \times 10^{-11}\).

Find the initial concentration of Cl⁻ ions

Given that NH₄Cl is a strong electrolyte, it will completely dissociate in water. So, the initial concentration of Cl⁻ ions is the same as the concentration of NH₄Cl, which is 0.50 M.

Set up an ICE table and the Ksp expression

Using an Initial-Change-Equilibrium (ICE) table, we can relate the concentrations of the ions involved in the reaction to the solubility (s) of Mg(OH)₂: ``` Mg(OH)₂ (s) ↔ Mg²⁺ (aq) + 2OH⁻ (aq) Initial s 0 0 Change -s +s +2s Equilibrium 0 s 2s ``` Now, we can write the Ksp expression for the dissolution reaction: \[K_{sp} = [\mathrm{Mg^{2+}}][\mathrm{OH^-}]^2\] Plug in the solubility of Mg²⁺ and OH⁻ ions from the ICE table: \[K_{sp} = (s)(2s)^2\]

Use the common ion effect

As NH₄Cl is present in the solution, its Cl⁻ ions will combine with Mg²⁺ ions to form MgCl₂, reducing the concentration of Mg²⁺ ions. On the other hand, some of the OH⁻ ions will combine with NH₄⁺ ions to form NH₄OH, reducing the concentration of OH⁻ ions. Since the concentration of Mg²⁺ ions decreases, the solubility of Mg(OH)₂ will also decrease.

Solve for solubility (s)

We are given the Ksp of Mg(OH)₂ and the ICE table, so we can solve for solubility (s). Substitute the Ksp value we found in "Step 2" into the Ksp expression: \[(1.5 \times 10^{-11}) = (s)(2s)^2\] To solve for s, simplify the above equation: \[(1.5 \times 10^{-11}) = 4s^3\] Now, divide \(1.5 \times 10^{-11}\) by 4: \[s^3 = \frac{1.5 \times 10^{-11}}{4}\] Then, find the cube root of the result to obtain s: \[s = \sqrt[3]{\frac{1.5 \times 10^{-11}}{4}}\] Calculate the final value of s: \[s \approx 1.31 \times 10^{-4} \mathrm{M}\]

Find the solubility of Mg(OH)₂ in the presence of NH₄Cl

The solubility of Mg(OH)₂ in the presence of NH₄Cl is approximated by s, which is \(1.31 \times 10^{-4} \mathrm{M}\).