Problem 167 - A Complete Resource Book in Chemistry for JEE Main

Question

Solubility of \(\mathrm{AgCl}\) in water, \(0.01 \mathrm{M} \mathrm{CaCl}_{2}, 0.01 \mathrm{M}\) \(\mathrm{NaCl}\) and \(0.05 \mathrm{M} \mathrm{AgNO}_{3}\) are \(\mathrm{S}_{1}, \mathrm{~S}_{2}, \mathrm{~S}_{3}\) and \(\mathrm{S}_{4}\) respec- tively then (a) \(\mathrm{S}_{1}>\mathrm{S}_{2}=\mathrm{S}_{3}>\mathrm{S}_{4}\) (b) \(\mathrm{S}_{1}>\mathrm{S}_{3}>\mathrm{S}_{4}>\mathrm{S}_{2}\) (c) \(\mathrm{S}_{1}<\mathrm{S}_{2}<\mathrm{S}_{3}<\mathrm{S}_{4}\) (d) \(\mathrm{S}_{1}>\mathrm{S}_{3}>\mathrm{S}_{2}>\mathrm{S}_{4}\)

Step-by-Step Solution

Answer

(a) \( S_1 > S_2 = S_3 > S_4 \).

Step 1: Understand AgCl Solubility Product

The solubility product (Ksp) of AgCl is the product of the concentrations of its constituent ions in a saturated solution. Given: \[ \text{AgCl (s) } \rightleftharpoons \text{ Ag}^+ \text{(aq) } + \text{ Cl}^- \text{(aq) } \]The Ksp expression is \( K_{sp} = [\text{Ag}^+][\text{Cl}^-] \). In pure water, AgCl dissociates only into \( \text{Ag}^+ \) and \( \text{Cl}^- \). The solubility \( S_1 \) will be the highest due to the absence of any additional ions.

Step 2: Solubility in CaCl2 Solution

In a \(0.01\ \mathrm{M}\ \mathrm{CaCl}_{2}\) solution, the concentration of \( \text{Cl}^- \) is already high. This common ion effect decreases the solubility of AgCl. Higher \( \text{Cl}^- \) from \( \text{CaCl}_2 \) means AgCl solubility \( S_2 \) is reduced, possibly equal to or less than in \(0.01\ \mathrm{M}\ \mathrm{NaCl}\).

Step 3: Solubility in NaCl Solution

In a \(0.01\ \mathrm{M}\ \mathrm{NaCl}\) solution, the presence of \( \text{Cl}^- \) also reduces solubility of AgCl due to the common ion effect as in \( \text{CaCl}_2 \). Hence, \( S_3 \approx S_2 \) since both solutions provide the same \( \text{Cl}^- \) concentration of \(0.01\ \mathrm{M}\).

Step 4: Solubility in AgNO3 Solution

In the \(0.05\ \mathrm{M}\ \mathrm{AgNO}_3\) solution, the concentration of \( \text{Ag}^+ \) is already very high. The additional \( \text{Ag}^+ \) severely reduces the solubility of AgCl through the common ion effect. Therefore, \( S_4 \) is significantly smaller than \( S_1 \), \( S_2 \), and \( S_3 \).

Step 5: Compare the Solubility

Based on the effects above, \( S_1 > S_2 = S_3 > S_4 \). The solubility order reflects agitating the highest in pure water, equating in \( \text{CaCl}_2 \) and \( \text{NaCl} \), and the least in the presence of AgNO3 due to the high \( \text{Ag}^+ \) concentration.

Key Concepts

Common Ion Effect

When a solution contains an additional source of an ion that is also a product of a salt's dissociation, this can influence the salt's solubility. This phenomenon is known as the common ion effect. For example, when silver chloride (AgCl) is dissolved in a solution that already contains chloride ions (Cl^-), like in CaCl2 or NaCl solutions, this additional chloride from the common ion source shifts the dissolution equilibria of AgCl, reducing its solubility. This happens because adding more Cl^- ions drives the reaction \[\text{AgCl (s)} \rightleftharpoons \text{Ag}^+ \text{(aq) } + \text{Cl}^- \text{(aq)}\]towards the left, forming more solid AgCl rather than dissolving it.

• The common ion effect is crucial in the context of solubility, as it helps predict how the presence of additional ions can shift equilibria.
• It is defined by Le Chatelier's principle, which predicts a system's response to stress or change in concentration, pressure, or temperature.

By understanding and utilizing the common ion effect, chemists can manipulate solubility outcomes in various chemical applications.

Dissolution Equilibria

Dissolution equilibria refer to the balance achieved between dissolved ions and the undissolved solid in a saturated solution. For salts like AgCl, this concept is rooted in equilibrium chemistry where the rate at which the ions go into solution equals the rate at which they precipitate out. In mathematical terms, for AgCl dissolving into Ag⁺ and Cl⁻ ions, the equilibrium condition is expressed by the solubility product constant (Ksp):\[K_{sp} = [\text{Ag}^+][\text{Cl}^-]\]This equilibrium constant is a fixed value at a given temperature and indicates the maximum concentration of ions that a solution can hold before the salt begins to precipitate.

• Understanding dissolution equilibria is crucial for predicting how changes in conditions, such as concentration of reactants or products, affect solubility.
• The dynamic equilibrium concept suggests that although the concentrations remain constant, ions continually exchange between the dissolved and solid states.

This concept is foundational for not only chemistry students but also for professionals managing processes such as water treatment and pharmaceuticals.

AgCl Solubility

The solubility of silver chloride (AgCl) depends significantly on the presence of other ions in the solution. In pure water, AgCl has a certain inherent solubility due to the absence of competing ions, allowing it to dissociate sufficiently into Ag⁺ and Cl⁻ ions.

• Under standard conditions, AgCl's solubility is quite low, which is typical for many silver halides.
• In solutions containing common ions, such as in CaCl2 or NaCl, the solubility decreases due to the common ion effect, where the availability of more chloride ions suppresses further dissociation of AgCl.
• In an AgNO3 solution, the high concentration of Ag⁺ further reduces AgCl solubility, again highlighting the role of equilibrium and Le Chatelier's principle in solubility processes.

Understanding these factors allows chemists to predict and control the extent of AgCl solubility under various conditions.

Solubility Comparison

Comparing the solubility of AgCl in different solutions involves considering the influences of common ions and the solubility product principle. Pure water provides the baseline solubility where only Ag⁺ and Cl⁻ are present from the dissociation of AgCl. In contrast, in a 0.01 M CaCl2 or NaCl solution, the added Cl⁻ ions significantly lower the solubility due to the common ion effect.

• For both CaCl2 and NaCl at 0.01 M, the solubility of AgCl is similar as each introduces the same concentration of Cl⁻ ions.
• In a 0.05 M AgNO3 solution, the decreased solubility of AgCl is even more pronounced because the additional Ag⁺ ions further impede the dissociation of AgCl.
• The solubility ordering becomes \( S_1 > S_2 = S_3 > S_4 \), with \( S_1 \) being in pure water, \( S_2 \) and \( S_3 \) in CaCl2 and NaCl respectively, and \( S_4 \) in the AgNO3 solution.

These comparisons reflect the intricate balance between chemical equilibria and the presence of common ions, reinforcing the importance of considering all variables affecting solubility in analysis and application.