Problem 226 - A Complete Resource Book in Chemistry for JEE Main

Question

The sparingly soluble salt \(\mathrm{M}(\mathrm{OH})_{\mathrm{x}}\) has \(\mathrm{K}_{\mathrm{sp}}=4 \times 10^{-12}\). Its solubility is \(10^{-4} \mathrm{M}\). The value of \(\mathrm{x}\) is

Step-by-Step Solution

Answer

The value of \( x \) is 2.

Step 1: Write the Dissolution Reaction

The dissolution reaction of the salt \( \text{M(OH)}_x \) in water is as follows: \[ \text{M(OH)}_x (s) \rightleftharpoons \text{M}^{x+} (aq) + x \text{OH}^- (aq) \] This equation shows that for every mole of \( \text{M(OH)}_x \) that dissolves, 1 mole of \( \text{M}^{x+} \) and \( x \) moles of hydroxide ions \( \text{OH}^- \) are produced.

Step 2: Write the Expression for Ksp

The solubility product constant, \( K_{sp} \), of \( \text{M(OH)}_x \) is defined as: \[ K_{sp} = [\text{M}^{x+}][\text{OH}^-]^x \] where \([\text{M}^{x+}]\) is the concentration of \( \text{M}^{x+} \) ions, and \([\text{OH}^-]^x\) is the concentration of hydroxide ions raised to the power of \( x \).

Step 3: Substitute Solubility into Ksp Expression

Given that the solubility is \( 10^{-4} \text{ M} \), it means \([\text{M}^{x+}] = 10^{-4}\text{ M}\) and \([\text{OH}^-] = x \times 10^{-4}\text{ M}\). Substitute these into the \( K_{sp} \) expression: \[ 4 \times 10^{-12} = (10^{-4})(x \times 10^{-4})^x \] Simplifying gives: \[ 4 \times 10^{-12} = 10^{-4} \times (x \times 10^{-4})^x \]

Step 4: Simplify the Ksp Equation

Rearrange the equation: \[ 4 \times 10^{-12} = 10^{-4+x} x^x \] Simplify to solve for \( x \): \[ 4 \times 10^{-12} = 10^{-(4+x)} x^x \] Equating powers of 10, we get: \[ -(4+x) \approx -12 \] resulting in \( x \approx 8 \).

Step 5: Apply Logarithms (if necessary)

If necessary, further solve for \( x \) by approximating: \[ 4 = x^x \approx 2^2. \,\, x \text{ must be 2.} \] This falls in line assuming \( \log_{10} x \approx 0 \) for simplicity since actual curves of polymonial comparison in logarithms require full analysis, or graphically approximation shows nearest exact value. \( x = 2 \).

Key Concepts

Ksp expression

The Ksp expression, or solubility product constant expression, is crucial for understanding the solubility of sparingly soluble salts. It provides a mathematical representation of the equilibrium between a solid and its ions in a saturated solution. This expression is specifically formulated for salts that do not dissolve readily in water.
In the case of the salt \( \text{M(OH)}_x \), the Ksp expression is formulated as follows:

• \( K_{sp} = [\text{M}^{x+}][\text{OH}^-]^x \)

This means that the Ksp is calculated by multiplying the concentration of the metal cation \([\text{M}^{x+}]\) by the concentration of hydroxide ions \([\text{OH}^-]\) raised to the power of \( x \). This equation reflects the stoichiometry of the salt's dissolution reaction, allowing us to calculate the solubility and understand the system dynamics at equilibrium.

sparingly soluble salts

Sparingly soluble salts are salts that dissolve very minimally in water, resulting in low concentrations of ions in solution. Their dissolution leads to a state of equilibrium where only a small fraction of the salt is converted into ions. This type of salt is characterized by a very low Ksp value, indicating its limited solubility.
The sparingly soluble salt \( \text{M(OH)}_x \) has a Ksp of \( 4 \times 10^{-12} \), illustrating how little of the salt actually dissolves. Such salts are crucial in many chemical processes, including industrial applications and biological systems.

• They play a significant role in predicting reactions where precipitation might occur.
• Understanding their properties is essential for solving problems involving solubility and chemical equilibrium.

Lower Ksp values imply reduced solubility, an essential consideration when measuring or predicting how substances interact in aqueous solutions.

solubility calculation

Solubility calculations are fundamental for determining how much of a solute can dissolve in a solvent before reaching saturation. To calculate the solubility of a sparingly soluble salt, you often need its Ksp value and the formula of the salt.
For \( \text{M(OH)}_x \), given a Ksp of \( 4 \times 10^{-12} \) and a solubility of \( 10^{-4} \text{ M} \), these values help derive the concentration of ions in solution:

• \([\text{M}^{x+}] = 10^{-4} \text{ M}\)
• \([\text{OH}^-] = x \times 10^{-4} \text{ M}\)

By substituting these concentrations into the Ksp expression, you can solve for \( x \), revealing the stoichiometric coefficients in the dissolution equation. This step-by-step calculation aids in comprehensively understanding the behavior of electrolytes in solution.

dissolution reaction

Dissolution reactions describe the process by which a solid substance dissolves in a solvent, often resulting in the formation of ions. For a sparingly soluble salt like \( \text{M(OH)}_x \), the dissolution can be represented as an equilibrium equation, essential for calculating solubility and interpreting chemical behavior.
Consider the reaction:

• \( \text{M(OH)}_x (s) \rightleftharpoons \text{M}^{x+} (aq) + x \text{OH}^- (aq) \)

This shows how each mole of the salt dissociates into one mole of metal cations and \( x \) moles of hydroxide ions, establishing a dynamic equilibrium in solution.
Understanding this reaction helps:

• Predict the solubility and potential precipitation of salts in various settings.
• Relate changes in ion concentration to shifts in equilibrium according to Le Chatelier's Principle.

Through this approach, you can clearly see how dissolution impacts both the chemical composition and the physical characteristics of the solution.