Problem 22 - Chemistry in 30 days

Question

The solubility of mercurous chloride in water will be given as (a) \(S=K_{s p}\) (b) \(S=K_{\mathrm{m}} / 4\) (c) \(S=\left(K_{s p} / 4\right)^{1 / 2}\) (d) \(S=\left(K_{\mathrm{sp}} / 4\right)^{1 / 3}\)

Step-by-Step Solution

Answer

The correct answer is (d) \( S = \left(\frac{K_{sp}}{4}\right)^{1/3} \).

Step 1: Understand the Compound

Mercurous chloride, also known as calomel, has the chemical formula \( \text{Hg}_2\text{Cl}_2 \). It dissociates in water as \( \text{Hg}_2\text{Cl}_2 \rightleftharpoons \text{Hg}_2^{2+} + 2\text{Cl}^- \). The solubility product \( K_{sp} \) is expressed in terms of its ionic concentrations at equilibrium.

Step 2: Set up the Solubility Product Expression

For the dissociation of \( \text{Hg}_2\text{Cl}_2 \), the solubility product expression is given by \( K_{sp} = [\text{Hg}_2^{2+}][\text{Cl}^-]^2 \). If \( S \) is the solubility of \( \text{Hg}_2\text{Cl}_2 \), then at equilibrium \([\text{Hg}_2^{2+}] = S\) and \([\text{Cl}^-] = 2S\).

Step 3: Substitute in Solubility Product Expression

Substitute the equilibrium concentrations in the solubility product expression: \( K_{sp} = (S)(2S)^2 = 4S^3 \). Hence, we have the equation \( K_{sp} = 4S^3 \).

Step 4: Solve for the Solubility \( S \)

From \( 4S^3 = K_{sp} \), solve for \( S \): \( S^3 = \frac{K_{sp}}{4} \). Therefore, \( S = \left(\frac{K_{sp}}{4}\right)^{1/3} \).

Step 5: Compare with Given Options

Compare the expression \( S = \left(\frac{K_{sp}}{4}\right)^{1/3} \) to the options provided. The correct expression for \( S \) matches option (d).

Key Concepts

Understanding Solubility Product

Solubility product, often denoted as \( K_{sp} \), is a crucial concept in understanding how salts dissolve in water. It refers to the equilibrium constant of a solid substance dissolving into its constituent ions in a solution.
For any given sparingly soluble salt, like mercurous chloride (Hg_2Cl_2), the solubility product is the product of the concentrations of the ions, each raised to the power of its stoichiometric coefficient in the balanced dissolution equation.
What this means, in simple terms, is that \( K_{sp} \) helps us predict how much of the salt will dissolve in water and reach equilibrium.

• It is temperature-dependent.
• Each salt has a unique \( K_{sp} \) value.
• It gives insight into solubility without performing experiments.

Understanding \( K_{sp} \) allows us to make educated guesses about solubility and helps in various applications ranging from industrial processes to pharmaceutical formulations.

Chemical Equilibrium Basics

Chemical equilibrium refers to the state in a chemical reaction where the rates of the forward and reverse reactions are equal, leading to no net change in the concentrations of reactants and products. It is a dynamic process. In the case of a dissociation reaction like that of mercurous chloride ( Hg_2Cl_2 ightleftharpoons Hg_2^{2+} + 2Cl^- ), equilibrium is reached when the ions and solid remain constant over time.
The dissolved ions are in a state where their formation and recombination into the solid form occur at the same rate.

• Equilibrium is a balance point, not a cessation of reactions.
• Even while at equilibrium, bonds break and form constantly.
• This balance dictates the concentration of ions in solution at any given temperature.

By understanding chemical equilibrium, students can better predict concentrations of different species in solution and solve solubility problems effectively.

The Dissociation Reaction Process

A dissociation reaction is when a compound separates into its constituent ions when in solution. For salts like mercurous chloride, this reaction is particularly insightful for predicting solubility.Mercurous chloride (Hg_2Cl_2) dissociates in water into one mercury (I) ion and two chloride ions, represented as:\[Hg_2Cl_2 ightleftharpoons Hg_2^{2+} + 2Cl^-\]Knowing the stoichiometry of this reaction is fundamental to calculating how much of a salt can dissolve. The amount of substance that dissolves defines the solubility (\( S \)).
The expression for the solubility product \( K_{sp} \) relies on these dissociated ion concentrations at equilibrium.

• Stoichiometry determines the coefficients for ion concentrations in \( K_{sp} \).
• In this case, every mole of \( Hg_2Cl_2 \) yields one mole of \( Hg_2^{2+} \) and two moles of \( Cl^- \).
• This helps set up the calculation for solubility based on known \( K_{sp} \) values.

Understanding dissociation reactions ensures accurate approaches to solving solubility exercises.

Having a Closer Look at Mercurous Chloride

Mercurous chloride, chemically known as \( Hg_2Cl_2 \), is a unique compound with multiple applications, especially in historical medical use and electrochemistry. Understanding its chemical properties helps illustrate solubility principles effectively.Here are some key points:

• It is often referred to as calomel in the chemistry world.
• This compound is poorly soluble in water, a common trait exploited in analytical chemistry for precipitation reactions.
• It dissociates under equilibrium conditions into its constituent ions, which is critical in various solubility calculations and reactions.

In textbooks, mercurous chloride is frequently used as an example to teach the fundamental concepts of solubility and solubility product, demonstrating how we predict and calculate how much of a substance can go into solution from the solid state.