Problem 126 - Chemistry The Molecular Science

Question

An experiment requires the addition of 0.075 mol gaseous \(\mathrm{NH}_{3}\) to \(1.0 \mathrm{~L}\) of \(0.025-\mathrm{M} \mathrm{Mg}\left(\mathrm{NO}_{3}\right)_{2} .\) Ammonium chloride, \(\mathrm{NH}_{4} \mathrm{Cl}\), is added prior to the addition of the \(\mathrm{NH}_{3}\) to prevent precipitation of \(\mathrm{Mg}(\mathrm{OH})_{2} .\) Calculate the minimum mass in grams of ammonium chloride that must be added. \(K_{\mathrm{sp}}\) of \(\mathrm{Mg}(\mathrm{OH})_{2}=1.8 \times 10^{-11}\)

Step-by-Step Solution

Answer

Add at least 0.255 g of ammonium chloride.

Step 1: Write the dissociation equations

Consider the dissociation of ammonium chloride \( \mathrm{NH}_4\mathrm{Cl} \rightarrow \mathrm{NH}_4^+ + \mathrm{Cl}^- \)and magnesium hydroxide \( \mathrm{Mg(OH)}_2 \rightarrow \mathrm{Mg}^{2+} + 2\mathrm{OH}^- \).\ Ammonium chloride will provide \( \mathrm{NH}_4^+ \) ions to suppress the dissociation of ammonium hydroxide.

Step 2: Calculate hydroxide ion concentration

Calculate the concentration of \(\mathrm{OH}^- \) needed to prevent precipitation of \(\mathrm{Mg(OH)}_2 \) using the solubility product (\(K_{sp}\)). From the equation \[ \mathrm{Mg}^{2+} + 2\mathrm{OH}^- \rightleftharpoons \mathrm{Mg(OH)}_2 \],\(K_{sp} = [\mathrm{Mg}^{2+}][\mathrm{OH}^-]^2 \). Given \([\mathrm{Mg}^{2+}] = 0.025\,\mathrm{M}\), solve for \([\mathrm{OH}^-]^2 \).

Step 3: Solve for [OH^-]

Using \(K_{sp} = 1.8 \times 10^{-11}\) and \([\mathrm{Mg}^{2+}]=0.025\,\mathrm{M}\), we have:\[ 1.8 \times 10^{-11} = 0.025 \times [\mathrm{OH}^-]^2 \] Solve for \([\mathrm{OH}^-]^2 \):\[ [\mathrm{OH}^-]^2 = \frac{1.8 \times 10^{-11}}{0.025} \approx 7.2 \times 10^{-10} \] \([\mathrm{OH}^-] = \sqrt{7.2 \times 10^{-10}} \approx 2.68 \times 10^{-5} \).

Step 4: Calculate ammonia concentration

From the reaction \( \mathrm{NH}_3 + \mathrm{H_2O} \rightleftharpoons \mathrm{NH_4^+} + \mathrm{OH^-} \), use the hydroxide concentration to find ammonia concentration. Ammonia needs to overcome the hydroxide concentration and provide \([\mathrm{NH_4^+}]\) to further suppress \([\mathrm{OH^-}]\) for complexation.

Step 5: Calculate minimum mass of NH4Cl

To suppress \([\mathrm{OH}^-]\), calculate \([\mathrm{NH_4^+}]\) needed using \([\mathrm{NH_4^+}][\mathrm{OH}^-] = K_b = \frac{K_w}{K_a}\) for \(\mathrm{NH_4^+}\), where \(K_w = 1.0 \times 10^{-14}\) and \(K_a = 5.6 \times 10^{-10}\). Given \([\mathrm{OH^-}] = 2.68 \times 10^{-5}\), \([\mathrm{NH_4^+}] = \frac{1.0 \times 10^{-14}}{5.6 \times 10^{-10} \times 2.68 \times 10^{-5}}\). Convert \([\mathrm{NH_4^+}]\) to grams of \(\mathrm{NH_4Cl}\) using molar mass \(53.49 \frac{g}{mol}\).

Key Concepts

Solubility Product Constant

The solubility product constant, abbreviated as \(K_{sp}\), is a vital concept in understanding precipitation reactions. It provides a numerical value that represents the saturated solutions of ionic compounds. At this point, the concentrations of the ions are fixed in a specific ratio, obtaining equilibrium.
The reaction for any precipitation can be written as:
\[ AB(s) \rightleftharpoons A^{+}(aq) + B^{-}(aq) \]
The \(K_{sp}\) for this reaction is expressed as:
\[ K_{sp} = [A^{+}][B^{-}] \]
In the case of magnesium hydroxide (\(\text{Mg(OH)}_2\)), this reaction is:
\[ \text{Mg}^{2+} + 2\text{OH}^- \rightleftharpoons \text{Mg(OH)}_2 \]
The \(K_{sp}\) expression becomes:
\[ K_{sp} = [\text{Mg}^{2+}][\text{OH}^-]^2 \]
A small \(K_{sp}\) value, like \(1.8 \times 10^{-11}\) for magnesium hydroxide, indicates that only a tiny amount of solid dissolves, making it a low solubility substance. Understanding the \(K_{sp}\) is crucial to avoid unwanted precipitates in reactions.

Dissociation Equations

Dissociation equations provide insight into the breakdown of compounds into ions in a solution. Using dissociation equations can help predict whether a precipitation reaction occurs.
For ammonium chloride \((\text{NH}_4\text{Cl})\), the dissociation is straightforward as it splits into ammonium \((\text{NH}_4^+)\) and chloride ions \((\text{Cl}^-)\):
\[ \text{NH}_4\text{Cl} \rightarrow \text{NH}_4^+ + \text{Cl}^- \]
Similarly, when magnesium hydroxide \((\text{Mg(OH)}_2)\) dissociates in solution, it forms magnesium ions \((\text{Mg}^{2+})\) and hydroxide ions \((\text{OH}^-)\):
\[ \text{Mg(OH)}_2 \rightarrow \text{Mg}^{2+} + 2\text{OH}^- \]
These equations are fundamental to understanding how the components interact in a system, affecting factors such as solubility and the potential formation of a precipitate.
This understanding allows for precise control in experimental settings, ensuring reactions proceed as desired.

Hydroxide Ion Concentration

Hydroxide ion concentration \(([\text{OH}^-])\) is a critical factor in predicting and controlling precipitation reactions. It's all about understanding equilibrium and how the presence of certain ions shifts it.
To prevent the precipitation of magnesium hydroxide \((\text{Mg(OH)}_2)\), keeping the \([\text{OH}^-]\) concentration below a certain threshold is essential.
The \(K_{sp}\) equation for magnesium hydroxide demonstrates this:
\[ K_{sp} = [\text{Mg}^{2+}] [\text{OH}^-]^2 \]
Solving for \([\text{OH}^-]^2\), given a fixed \([\text{Mg}^{2+}]\), provides the maximum allowable \([\text{OH}^-]\). In this scenario, calculated to be approximately \(2.68 \times 10^{-5} \text{M}\).
Exceeding this concentration results in the formation of a solid \(\text{Mg(OH)}_2)\) precipitate.
Understanding this balance is crucial for controlling reactions in laboratory and industrial processes.

Ammonia and Ammonium Ion Equilibrium

Ammonia \((\text{NH}_3)\) and ammonium ion \((\text{NH}_4^+)\) equilibrium plays a pivotal role in maintaining the concentration of hydroxide ions (\([\text{OH}^-]\)) in a solution. This equilibrium can be described with:
\[ \text{NH}_3 + \text{H}_2\text{O} \rightleftharpoons \text{NH}_4^+ + \text{OH}^- \]
When ammonium chloride \((\text{NH}_4\text{Cl})\) is added to a solution, it increases the \([\text{NH}_4^+]\) concentration.
This action drives the equilibrium to the left, reducing \([\text{OH}^-]\) and preventing the precipitation of magnesium hydroxide.
This aspect allows the control of \([\text{OH}^-]\) in the solution by adjusting \([\text{NH}_4^+]\) accordingly, which is calculated to be necessary for preventing unwanted reactions.
For students, mastering this concept can open doors to a deeper understanding of chemical equilibria and their practical applications.