Problem 111 - A Complete Resource Book in Chemistry for JEE Main

Question

The solubility of \(\mathrm{AgCl}\) in moles per litre when its solubility product is \(1.56 \times 10^{-10}\) at \(25^{\circ} \mathrm{C}\) is (a) \(0.576 \times 10^{-8} \mathrm{~mol} /\) litre (b) \(1.056 \times 10^{-4} \mathrm{~mol} /\) litre (c) \(1.249 \times 10^{-5} \mathrm{~mol} /\) litre (d) \(1.478 \times 10^{-6} \mathrm{~mol} /\) litre

Step-by-Step Solution

Answer

The solubility of \(\mathrm{AgCl}\) is \(1.249 \times 10^{-5} \, \mathrm{mol/L}\), corresponding to option (c).

Step 1: Understanding solubility product

The solubility product constant, or Ksp, is a measure of how much of a salt can dissolve in water before the solution becomes saturated. For silver chloride, \(\mathrm{AgCl}\), the Ksp is given as \(1.56 \times 10^{-10}\). The characteristic equation for solubility is \[\mathrm{AgCl}(s) \rightleftharpoons \mathrm{Ag}^+(aq) + \mathrm{Cl}^-(aq)\] where the concentration of both ions in a saturated solution is equal to the solubility, \(s\).

Step 2: Setup solubility expression

Since the dissolution of \(\mathrm{AgCl}\) produces one mole of \(\mathrm{Ag^+}\) and one mole of \(\mathrm{Cl^-}\) for every mole of \(\mathrm{AgCl}\) that dissolves, the concentration of each ion is equal to \(s\) (solubility of \(\mathrm{AgCl}\)). Thus, \(\mathrm{Ksp} = [\mathrm{Ag}^+][\mathrm{Cl}^-] = s^2\).

Step 3: Solve for solubility

Substitute the given \(\mathrm{Ksp}\) value into the expression: \[s^2 = 1.56 \times 10^{-10}\] Solve for \(s\) by finding the square root of \(1.56 \times 10^{-10}\).

Step 4: Calculating the solubility

Calculate \(s = \sqrt{1.56 \times 10^{-10}}\) using a calculator. This yields \(s \approx 1.249 \times 10^{-5} \, \mathrm{mol/L}\).

Step 5: Match the calculated solubility with options

Compare the calculated solubility \(1.249 \times 10^{-5} \, \mathrm{mol/L}\) with given multiple-choice options. The matching choice is \(\mathrm{(c)}\).

Key Concepts

Silver Chloride

Silver chloride, chemically represented as \( \text{AgCl} \), is a well-known salt used in various chemical studies and applications. In water, it is nearly white and poorly soluble, meaning it doesn't dissolve well. Even though its solubility is very low, it plays a crucial role in chemistry for understanding dissolution processes.
When silver chloride dissolves in water, it separates into its constituent ions, silver ions (\( \text{Ag}^+ \)) and chloride ions (\( \text{Cl}^- \)). The equation showing this process is:
\[ \text{AgCl}(s) \rightleftharpoons \text{Ag}^+(aq) + \text{Cl}^-(aq) \]
In this equation, "\rightleftharpoons" indicates a dynamic balance, also known as equilibrium, where the rate of dissolution is equal to the rate of precipitation.
The amount that disassociates is dictated by its solubility product constant (Ksp). For silver chloride, Ksp is relatively small, indicating its limited solubility. This low solubility not only makes it a classic example for study but also aids in various applications like photographic processing and the creation of specific chemical solutions.

Dissolution Equilibrium

Understanding dissolution equilibrium is key to predicting how and why a compound dissolves. The term "equilibrium" refers to a state where the concentrations of reactants and products remain constant over time.
For silver chloride's dissolution, we reach a point where the salt continuously dissolves and precipitates at a rate that keeps the concentrations of \( \text{Ag}^+ \) and \( \text{Cl}^- \) ions stable. This phenomenon is described by the solubility product constant, \( \text{Ksp} \), specific to each ionic compound.
The equilibrium expression for silver chloride is:
\[ \text{Ksp} = [\text{Ag}^+][\text{Cl}^-] = s^2 \]
Here, \( s \) stands for solubility, representing the maximum amount of silver chloride that can dissolve to establish equilibrium. In this model, each ion's concentration equals \( s \), meaning that for each molecule of \( \text{AgCl} \) that dissolves, one \( \text{Ag}^+ \) and one \( \text{Cl}^- \) are formed.

• Equilibrium helps to predict solubility under different conditions.
• It also indicates that adding more solid to a saturated solution does not change equilibrium.

By understanding this constant state, chemists can manipulate systems for desired outcomes, like increasing ion concentration or focusing on complete reactions.

Saturated Solution

A saturated solution is one where the maximum amount of solute has been dissolved in a solvent at a given temperature, beyond which no more can dissolve. For silver chloride, saturation occurs at a surprisingly low concentration.
In a saturated solution of \( \text{AgCl} \), any addition of silver chloride simply leads to its accumulation at the bottom of the container without it dissolving further. This is because the dissolution and precipitation processes are in a delicate balance, governed by the solubility product \( \text{Ksp} \).

• Any disturbance in the solution, like a change in temperature, could shift this equilibrium process, affecting solubility.
• The saturated state is crucial for predicting the behavior of solutions in real-world applications, such as chemical manufacturing or environmental chemistry.

Thus, understanding saturation helps to ensure that chemical reactions and formulations proceed smoothly and effectively. Studying saturated solutions also lays a foundation for grasping more complex chemical equilibria in advanced chemistry.