Problem 112 - A Complete Resource Book in Chemistry for JEE Main

Question

If the solubility of sodium hexafluoroaluminate is 'a' mol/litre, its solubility product is (a) \(a^{8}\) (b) \(27 \mathrm{a}^{4}\) (c) \(180 \mathrm{a}^{3}\) (d) \(2916 \mathrm{a}^{8}\)

Step-by-Step Solution

Answer

The correct solubility product is \(27a^4\), option (b).

Step 1: Write the Dissociation Reaction

Sodium hexafluoroaluminate (Na₃AlF₆) dissociates in water according to the chemical equation: \[ \text{Na}_3\text{AlF}_6 \rightleftharpoons 3\text{Na}^+ + \text{AlF}_6^{3-} \]This reaction shows the dissociation of one formula unit of sodium hexafluoroaluminate into 3 sodium ions (Na⁺) and one hexafluoroaluminate ion (AlF₆³⁻).

Step 2: Express Solubility in Terms of Ion Concentrations

The solubility 'a' gives the molar concentrations of sodium hexafluoroaluminate and its ions at equilibrium. Thus, \[ \text{[Na}]^+ = 3a, \quad \text{[AlF}_6^{3-}] = a \]

Step 3: Determine Solubility Product Expression

The solubility product (Ksp) is given by the product of the ions raised to the power of their stoichiometric coefficients. Thus, \[ K_{sp} = [\text{Na}^+]^3 [\text{AlF}_6^{3-}] \]Substituting the concentrations from step 2, we have \[ K_{sp} = (3a)^3 (a) = 27a^3 \times a = 27a^4 \]

Step 4: Identify the Correct Option

We calculated the solubility product as \(27a^4\). Therefore, the correct answer is option (b), which matches our calculation.

Key Concepts

Dissociation Reaction

When sodium hexafluoroaluminate (Na₃AlF₆) is dissolved in water, it undergoes a dissociation reaction. This is when the compound breaks down into its individual ions. For Na₃AlF₆, the dissociation can be represented by the equation: \[ \text{Na}_3\text{AlF}_6 \rightleftharpoons 3\text{Na}^+ + \text{AlF}_6^{3-} \] This means that one formula unit of sodium hexafluoroaluminate dissociates into three sodium ions (Na⁺) and one hexafluoroaluminate ion (AlF₆³⁻).

• The 'right-facing' arrow in the equation symbolizes that the reaction can proceed in both directions, indicating an equilibrium state.
• Understanding dissociation reactions helps predict how compounds will behave in a solution.

A compound reaching its dissociation means achieving a balance where the rate of dissociation equals the rate of re-association.

Chemical Equilibrium

Chemical equilibrium is a state within a reaction where the concentrations of reactants and products remain constant over time, even though the reactions continue to occur. For our sodium hexafluoroaluminate scenario, once equilibrium is reached, the reaction between Na₃AlF₆ and its ions stabilizes. At equilibrium:

• The rate of dissociation of Na₃AlF₆ into its ions is equal to the rate at which the ions re-form into the original compound.
• The concentrations of \[ \text{Na}^+ \] and \[ \text{AlF}_6^{3-} \] stay constant.

The equilibrium is not static, but rather a dynamic state where reactants and products are formed at an equal rate. This implies there are no visible changes happening at a macroscopic level, but at a molecular level, the exchange is dynamic.

Ionic Concentrations

Ionic concentrations refer to the amount of ions generated from the dissolved compound in a given volume of solution. In the exercise, sodium hexafluoroaluminate has a solubility of 'a' mol/L. This solubility translates into its ionic concentrations, as follows:

• The concentration of sodium ions \([\text{Na}^+]\) generated is three times the solubility of the compound, or \(3a\).
• The concentration of hexafluoroaluminate ions \([\text{AlF}_6^{3-}]\) equals the solubility 'a'.

Ionic concentrations are crucial in calculating the solubility product \(K_{sp}\). In the given exercise, we determine \(K_{sp}\) by multiplying the ionic concentrations raised to the power of their respective stoichiometric coefficients, yielding \(27a^4\). Hence, a deep understanding of ionic concentrations is vital for predicting solubility product and the behavior of ions in solution.