Problem 126 - A Complete Resource Book in Chemistry for JEE Main

Question

At \(25^{\circ} \mathrm{C}\) the dissociation constant of a base, \(\mathrm{BOH}\) is \(1.0 \times 10^{-12}\), the concentration of hydroxyl ions \(0.01 \mathrm{M}\) aqueous solution of the base would become (a) \(2.0 \times 10^{-6} \mathrm{~mol} \mathrm{~L}^{-1}\) (b) \(1.0 \times 10^{-5} \mathrm{~mol} \mathrm{~L}^{-1}\) (c) \(1.0 \times 10^{-6} \mathrm{~mol} \mathrm{~L}^{-1}\) (d) \(1.0 \times 10^{-7} \mathrm{~mol} \mathrm{~L}^{-1}\)

Step-by-Step Solution

Answer

The concentration of hydroxyl ions is \(1.0 \times 10^{-7} \mathrm{~mol} \mathrm{~L}^{-1}\), which is option (d).

Step 1: Write the Dissociation Equation

The dissociation of a base BOH should be written in the form:\[ \mathrm{BOH} \rightleftharpoons \mathrm{B}^+ + \mathrm{OH}^- \]The dissociation constant \(K_b\) of the base is given as \(1.0 \times 10^{-12}\).

Step 2: Establish Expression for Kb

The expression for the base dissociation constant \(K_b\) is written as:\[ K_b = \frac{[\mathrm{B}^+][\mathrm{OH}^-]}{[\mathrm{BOH}]} \]Assuming the degree of dissociation \(x\), at equilibrium, \([\mathrm{B}^+] = [\mathrm{OH}^-] = x\), and \([\mathrm{BOH}] = 0.01 - x\).

Step 3: Simplifying Assumptions

Since \(K_b\) is very small, we can assume that \(x \ll 0.01\). Thus, \([\mathrm{BOH}] \approx 0.01\).

Step 4: Solve for [OH-]

Substitute into the \(K_b\) expression:\[ 1.0 \times 10^{-12} = \frac{x^2}{0.01} \]So, \(x^2 = 1.0 \times 10^{-14}\), and solving for \(x\):\[ x = \sqrt{1.0 \times 10^{-14}} = 1.0 \times 10^{-7} \]Thus, \([\mathrm{OH}^-] = 1.0 \times 10^{-7}\, \mathrm{mol}\, \mathrm{L}^{-1}\).

Step 5: Verify the Answer

The calculated concentration of hydroxyl ions \([\mathrm{OH}^-]\) is \(1.0 \times 10^{-7}\, \mathrm{mol}\, \mathrm{L}^{-1}\). This matches option \((d)\).

Key Concepts

Dissociation of a Base

When a base dissociates, it separates into its constituent ions when placed in water. Consider an example with a generic base, denoted as BOH. When BOH is dissolved in water, it will dissociate to form one hydroxide ion (OH⁻) and one positive ion (B⁺). Here's what this looks like:

• BOH (aqueous) ⇌ B⁺ (aqueous) + OH⁻ (aqueous)

This reaction is reversible and can reach what is known as equilibrium. At equilibrium, the rate of dissociation of the base is equal to the rate of recombination of the ions. The extent of this dissociation is quantified by the base dissociation constant, known as Kb.
The value of Kb is a measure of how strongly a base, like BOH, dissociates in water. A small Kb value implies that the base is weak, meaning it does not dissociate much, keeping the solution less basic.

Concentration of Hydroxyl Ions

The concentration of hydroxyl ions, [OH⁻], in a solution of a base is crucial for understanding the solution's alkaline nature. When a base like BOH is placed in water and dissociates, it liberates hydroxide ions. These ions increase the basicity of the solution.
To determine the concentration of OH⁻, one must know how much the base has dissociated. In chemical equilibrium scenarios, such as with weak bases, we assume simple dissociation to calculate the hydroxide concentration. This involves solving mathematical equations involving Kb. For our exercise, with a given Kb of 1.0 × 10⁻¹², by establishing a balance between dissociated ions and undissociated base, we found the concentration of OH⁻ to be 1.0 × 10⁻⁷ mol/L. This implies that the base is weakly dissociated.

Degree of Dissociation

The degree of dissociation \( x \) represents the fraction of the base that has dissociated into ions in solution. It helps us understand how much of the original compound remains un-ionized. For weak bases, the degree of dissociation is generally small.
In mathematical terms, the degree of dissociation \( x \) is the change in concentration of the species dissociated divided by the initial concentration of the species. The initial concentration of BOH was set at 0.01 M. Applying the value of Kb to the equation \[ K_b = \frac{x^2}{0.01} \], we calculated \( x \) to be 1.0 × 10⁻⁷.

• This \( x \) value represents the concentration of OH⁻ ions at equilibrium.
• It confirms that the base weakly dissociates.

In our example, x is also approximately equal to [OH⁻], underscoring that only a tiny portion of the base dissociates in water.