Problem 127 - A Complete Resource Book in Chemistry for JEE Main

Question

The solubility product of \(\mathrm{AgI}\) at \(25^{\circ} \mathrm{C}\) is \(1.0 \times 10^{-16}\) \(\mathrm{mol}^{2} \mathrm{~L}^{-2}\). The solubility of AgI in \(10^{-4} \mathrm{~N}\) solution of \(\mathrm{KI}\) at \(25^{\circ} \mathrm{C}\) is (in \(\mathrm{mol} \mathrm{L}^{-1}\) ) (a) \(1.0 \times 10^{-10}\) (b) \(1.0 \times 10^{-8}\) (c) \(1.0 \times 10^{-16}\) (d) \(1.0 \times 10^{-12}\)

Step-by-Step Solution

Answer

The solubility of AgI in \(10^{-4} \ \mathrm{N}\) KI is \\(1.0 \times 10^{-12} \ \mathrm{mol \, L^{-1}}\\). Answer: (d)

Step 1: Set up the Dissolution Equation

The dissolution of silver iodide (AgI) in water can be represented by the equation: \[ \mathrm{AgI(s)} \rightleftharpoons \mathrm{Ag^+(aq)} + \mathrm{I^-(aq)} \] Since we are looking for the solubility of AgI in a potassium iodide (KI) solution, we must consider the common ion effect exerted by the iodide ions from KI.

Step 2: Use Solubility Product Expression

The solubility product constant (\(K_{sp}\)) for AgI is given as \(1.0 \times 10^{-16} \ \mathrm{mol^2 \, L^{-2}}\). It is defined by the expression: \[ K_{sp} = [\mathrm{Ag^+}][\mathrm{I^-}] \] Given the KI solution provides an iodide concentration ([I\(^-\)]) of \(10^{-4} \ \mathrm{mol \, L^{-1}}\), we can incorporate this into the solubility product expression.

Step 3: Substitute Values into Equation

We can substitute the iodine concentration from KI into the solubility expression. Let \(s\) represent the solubility of AgI (which is also \([\mathrm{Ag^+}]\)). The equation becomes: \[ 1.0 \times 10^{-16} = s \times 10^{-4} \] Solving for \(s\) will give us the solubility of AgI.

Step 4: Solve for Solubility

Rearrange the expression obtained in the previous step to solve for \(s\): \[ s = \frac{1.0 \times 10^{-16}}{10^{-4}} = 1.0 \times 10^{-12} \ \mathrm{mol \, L^{-1}} \] This value of \(s\) represents the solubility of AgI in the \(10^{-4} \ \mathrm{N}\) KI solution.

Key Concepts

Common Ion Effect

The common ion effect occurs in a solution where a compound containing an ion, common to both the solute and the solvent, is added. For example, when silver iodide (AgI) is dissolved in a solution of potassium iodide (KI), the iodine ions (I\(^-\)) from KI exert a "common ion effect" on the dissolution of AgI. This happens because both KI and AgI release iodide ions into the solution.

This effect reduces the solubility of the less soluble salt (AgI), as the additional iodide ions from KI shift the equilibrium position of the dissolution of AgI. This shift results in fewer silver ions (Ag\(^+\)) being released into the solution, thereby reducing the overall solubility of AgI.

• The common ion effect is an application of Le Chatelier's principle.
• In practice, this means that adding a common ion reduces the solubility of the solute.
• Understanding this effect helps in predicting how the solubility of a compound will change in the presence of a common ion.

Silver Iodide

Silver iodide (AgI) is a compound that is sparingly soluble in water. It is formed by an ionic bond between silver (Ag\(^+\)) and iodide (I\(^-\)) ions. The solubility product constant (K_{sp}) of AgI at 25 ^\circ C is given as 1.0 \times 10^{-16} \text{ mol}^2 \text{ L}^{-2}. This very small K_{sp} value indicates that AgI only dissolves to a small extent in water.

Solubility product constants are used to quantify the solubility of sparingly soluble ionic compounds. It serves as a measure to predict the extent to which a compound will dissolve in water under specific conditions. Understanding the nature of AgI is crucial for solubility calculations when it's in the presence of common ions or mixed with other substances.

• AgI is typically used in photographic film and cloud seeding for rain enhancement.
• Its low solubility is exploited in various industrial and scientific applications.
• It is important to consider the K_{sp} value when analyzing its solubility behavior in different solutions.

Potassium Iodide

Potassium iodide (KI) is a highly soluble ionic compound and serves as a source of iodide ions in solution. When dissolved in water, KI completely dissociates into potassium ions (K\(^+\)) and iodide ions (I\(^-\)). At a concentration of 10\(^{-4} \text{ mol} \text{ L}^{-1}\), KI provides an excess of iodide ions which can significantly influence the solubility of other iodide-containing compounds, such as silver iodide (AgI).

In the context of solubility calculations and common ion effect, the KI acts as a source of the common ion (I\(^-\)) influencing the equilibrium of AgI. It's important to account for the presence of these ions, as they can shift the equilibrium, thereby altering the calculated solubility of other compounds present in the mixture.

• KI is commonly used in medical treatments and as a nutritional supplement to ensure adequate iodine intake.
• The presence of iodide from KI affects the solubility of AgI, illustrating the impact of added ion concentrations on solubility.
• Understanding the dissociation of KI helps predict the outcome of solubility experiments involving common ions.

Solubility Calculations

Solubility calculations involve using the solubility product constant (K_{sp}) to determine the solubility of ionic compounds like silver iodide (AgI). These calculations are crucial when dealing with solutions that contain additional ions, like potassium iodide (KI), which introduces the common ion (I\(^-\)) that affects solubility.

To perform these calculations, the general formula Ksp = [Ag^+][I^-] is used. Given the iodide concentration from KI, you can solve for the concentration of Ag\(^+\) to find the solubility of AgI in that solution. The formula rearranges into s = \frac{K_{sp}}{[I^-]} to derive solubility s.

• The s value indicates how much of a salt will dissolve in solution.
• Substituting known concentrations into the K_{sp} expression helps predict solubility under given conditions.
• These calculations highlight the impact of additional ions present in the solution, which can either increase or decrease solubility.