Problem 133 - A Complete Resource Book in Chemistry for JEE Main

Question

There sparingly soluble salts \(\mathrm{A}_{2} \mathrm{X}, \mathrm{AX}\) and \(\mathrm{AX}_{3}\) have the same solubility product. Their solubilities will be in the order (a) \(\mathrm{AX}_{3}>\mathrm{AX}>\mathrm{A}_{2} \mathrm{X}\) (b) \(\mathrm{AX}_{3}>\mathrm{A}_{2} \mathrm{X}>\mathrm{AX}\) (c) \(\mathrm{AX}>\mathrm{AX}_{3}>\mathrm{A}_{2} \mathrm{X}\) (d) \(\mathrm{AX}>\mathrm{A}_{2} \mathrm{X}>\mathrm{AX}_{3}\)

Step-by-Step Solution

Answer

Given the same solubility product, \( \mathrm{AX} > \mathrm{AX}_{3} > \mathrm{A}_{2} \mathrm{X} \). Option (c) is correct.

Step 1: Understand the solubility product

The solubility product, denoted as \( K_{sp} \), is an equilibrium constant for the dissolution of a sparingly soluble salt. It is specific for each solid at a given temperature and represents the product of its ions' concentrations raised to the powers of their stoichiometric coefficients in the balanced chemical equation.

Step 2: Write dissolution equations for each salt

The equations for dissociation are:- \( \mathrm{A}_{2} \mathrm{X} \rightarrow 2\mathrm{A}^{+} + \mathrm{X}^{2-} \)- \( \mathrm{AX} \rightarrow \mathrm{A}^{+} + \mathrm{X}^{-} \)- \( \mathrm{AX}_{3} \rightarrow \mathrm{A}^{3+} + 3\mathrm{X}^{-} \)

Step 3: Write expressions for \( K_{sp} \)

From the dissolution equations, the expressions for \( K_{sp} \) are obtained:- For \( \mathrm{A}_{2} \mathrm{X} \): \[ K_{sp} = [\mathrm{A}^{+}]^2 [\mathrm{X}^{2-}] \]- For \( \mathrm{AX} \): \[ K_{sp} = [\mathrm{A}^{+}] [\mathrm{X}^{-}] \]- For \( \mathrm{AX}_{3} \): \[ K_{sp} = [\mathrm{A}^{3+}] [\mathrm{X}^{-}]^3 \]

Step 4: Assume solubility and relate to \( K_{sp} \)

Assuming solubility \( s \) for each compound:- \( \mathrm{A}_{2} \mathrm{X} \): \( \mathrm{A}^{+} = 2s \) and \( [\mathrm{X}^{2-}] = s \) so \( K_{sp} = (2s)^2 (s) = 4s^3 \)- \( \mathrm{AX} \): \( [\mathrm{A}^{+}] = s \) and \( [\mathrm{X}^{-}] = s \) so \( K_{sp} = s^2 \)- \( \mathrm{AX}_{3} \): \( [\mathrm{A}^{3+}] = s \) and \( [\mathrm{X}^{-}] = 3s \) so \( K_{sp} = s(3s)^3 = 27s^4 \)

Step 5: Compare solubilities

Given \( 4s_1^3 = s_2^2 = 27s_3^4 \), solve for \( s \). It follows the inequality:- For \( \mathrm{AX} \), solubility \( s_2 \) is highest: \( s_2 = (K_{sp})^{1/2} \).- For \( \mathrm{AX}_{3} \), \( s_3 = (K_{sp}/27)^{1/4} \).- For \( \mathrm{A}_{2} \mathrm{X} \), \( s_1 = (K_{sp}/4)^{1/3} \).Thus, \( s_2 > s_3 > s_1 \), hence \( AX > AX_3 > A_2X \).

Step 6: Conclusion

Based on calculations, the solubility order is found to be \( \mathrm{AX} > \mathrm{AX}_{3} > \mathrm{A}_{2} \mathrm{X} \). The correct answer is option (c).

Key Concepts

Sparingly Soluble Salts

Sparingly soluble salts are fascinating compounds that only dissolve slightly in a solvent, usually water. Unlike salts that completely dissolve, sparingly soluble salts only dissociate to a small extent. This means that only a small amount of ions are present in the solution. Despite their low solubility, these salts still reach a saturation point where no more solute can dissolve in solution.
When discussing sparingly soluble salts, we often focus on the balanced equations that describe their dissolution. This helps us understand how much of each ion is released into the solution.
These salts are crucial in many natural and industrial processes. They can form scales in pipes, influence the hardness of water, and are involved in numerous geochemical cycles. Essentially, knowing about their behaviors helps us manage and predict these phenomena.

Ksp Expressions

The solubility product constant, or Ksp, is a key concept used to describe the equilibrium between a sparingly soluble salt and its ions in solution. This constant is essentially a measure of the solubility of a compound in solution.
For a specific sparingly soluble salt, each distinct equation represents how the salt dissociates into its ions. The Ksp expression is then formulated by raising the concentration of each ion to the power of its coefficient in the balanced equation.

• For the salt \(AX\), we have \[K_{sp} = [A^+][X^-]\]\
• For \(A_2X\), the expression is \[K_{sp} = [A^+]^2[X^{2-}]\]\

Dissolution Equations

To delve deeper into the solubility of sparingly soluble salts, you start with dissolution equations. These equations show how salts dissociate into their respective ions.
Take for example the dissolution of \(A_2X\)\; it dissociates into two \(A^+\) ions and one \(X^{2-}\) ion: \[A_2X \rightarrow 2A^+ + X^{2-}\]. This helps us understand the composition of the solution when the salt dissolves.
Similarly, for \(AX_3\), the dissociation equation would be: \[AX_3 \rightarrow A^{3+} + 3X^-\]. This equation gives the ratio and amount of ions released when \(AX_3\) dissolves.

• Each salt dissociates differently based on its composition.
• This affects the number of ions available and the resulting Ksp expression.

Solute Solubility Calculations

Calculating the solubility of a solute from its Ksp expression involves some algebra. We generally assume solubility 's' as the concentration of ions produced in the dissolution process.
For example, to calculate solubility for \(AX\), where \[K_{sp} = s^2\], we simply solve for 's' given the Ksp value. This shows us how much of the salt can dissolve under equilibrium conditions.
With more complex salts like \(AX_3\), where \[K_{sp} = 27s^4\], the calculation involves a greater coefficient for the ions, making the process slightly more challenging. But knowing \(s\) for each salt helps us determine their relative solubility in comparison to each other.