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

(c) \(AX>AX_3>A_2X\)

Step 1: Understanding Solubility Products

The solubility product, denoted as \(K_{sp}\), is a constant for a given salt at a specific temperature. It represents the product of the molar concentrations of the ions, each raised to the power of their coefficients in the balanced equation.

Step 2: Expressions for Solubility Products

For the three salts, the expressions for their solubility products are: 1. For \(A_2X\): \(K_{sp} = [A^+]^2[X^{2-}] = (2s)^2(s) = 4s^3\) where \(s\) is the solubility of the salt.2. For \(AX\): \(K_{sp} = [A^+][X^-] = (s)(s) = s^2\).3. For \(AX_3\): \(K_{sp} = [A^{3+}][X^-]^3 = (s)(3s)^3 = 27s^4\).

Step 3: Equating Solubility Products

Since the solubility products are the same, we equate them:1. \(4s_1^3 = s_2^2 = 27s_3^4\)Where \(s_1\), \(s_2\), and \(s_3\) are the solubilities of \(A_2X\), \(AX\), and \(AX_3\) respectively.

Step 4: Comparing Solubilities

From the equations, solve for \(s\):- Using \(s^2 = 4s_1^3\), we find \(s_1 = \left(\frac{s^2}{4}\right)^{1/3}\)- Using \(s^2 = 27s_3^4\), we find \(s_3 = \left(\frac{s^2}{27}\right)^{1/4}\)Compare these:- For \(s_1\), \(s_2\), and \(s_3\), note that cubes of smaller numbers yield smaller results than fourth roots, hence \(s_2 > s_3 > s_1\).

Step 5: Determining Solubility Order

Based on the calculations above, the solubility order is \(s_2 > s_3 > s_1\) which corresponds to \(AX > AX_3 > A_2X\).

Key Concepts

Sparingly Soluble Salts

Sparingly soluble salts are salts that do not dissolve well in water. This means only a small amount of the salt is able to form ions in a solution before reaching its equilibrium. This level of solubility is always tied to specific conditions, such as temperature, and its measurement can be quantified using the solubility product constant, or \( K_{sp} \).

The characteristics of sparingly soluble salts are as follows:

• They produce very little ionic concentrations in solutions compared to more soluble salts.
• They often precipitate in solution more readily.
• The solubility product \( K_{sp} \) is used to help predict their solubility under certain conditions.

Understanding these salts and their behavior in solutions is key to solving problems related to their solubility and the reactions they undergo.

Chemical Equilibrium

Chemical equilibrium in the context of sparingly soluble salts refers to the balance between the solid salt and its dissolved ions in a solution. When equilibrium is reached, the rate of dissolution and the rate of precipitation become equal, resulting in no net change of concentration.

For each sparingly soluble salt, the chemical equilibrium can be described using an equation representing this balance. For instance:

• \( A_2X \) in equilibrium with \([A^+]^2[X^{2-}]\)
• \( AX \) in equilibrium with \([A^+][X^-]\)
• \( AX_3 \) in equilibrium with \([A^{3+}][X^-]^3\)

Each equation reflects the ionic concentration of the salts at equilibrium and is used to form the expression for \( K_{sp} \), which helps solve for solubility in varied chemical environments.

Solubility Calculation

Solubility calculations involve determining how much of a sparingly soluble salt dissolves in a particular solvent to form a saturated solution. For each type of sparingly soluble salt, the solubility can be calculated using its specific solubility product (\( K_{sp} \)) expression.

In the problem, we were given the solubility expressions for three salts:

• For \( A_2X \), \( K_{sp} = 4s^3 \)
• For \( AX \), \( K_{sp} = s^2 \)
• For \( AX_3 \), \( K_{sp} = 27s^4 \)

By setting these \( K_{sp} \) expressions to be equal, we can solve for specific solubilities \( s_1 \), \( s_2 \), and \( s_3 \) using algebraic methods. Comparisons of these values allow us to discern the order of solubility among different salts, crucial for predicting and explaining their behavior in chemical solutions.