Problem 169 - A Complete Resource Book in Chemistry for JEE Main

Question

The degree of dissociation \((\alpha)\) of a weak electrolyte, \(\mathrm{A}_{x} \mathrm{~B}_{y}\) is related to van't Hoff factor (i) by the expression: (a) \(\alpha=\frac{x+y-1}{i-1}\) (b) \(\alpha=\frac{x+y+1}{i-1}\) (c) \(\alpha=\frac{i-1}{(x+y-1)}\) (d) \(\alpha=\frac{i-1}{x+y+1}\)

Step-by-Step Solution

Answer

The correct option is (c) \( \alpha = \frac{i-1}{(x+y-1)} \).

Step 1: Understanding Degree of Dissociation

The degree of dissociation, \( \alpha \), represents the fraction of the original solute molecules that dissociate into ions. For the weak electrolyte \( \text{A}_x \text{B}_y \), the dissociation process can be represented as: \[ \text{A}_x\text{B}_y \rightleftharpoons x\text{A}^{y+} + y\text{B}^{x-} \].

Step 2: Define Van't Hoff Factor

The van't Hoff factor, \( i \), measures the effect of solute particles on colligative properties. For a dissociating substance, it is calculated as: \[ i = 1 + \alpha (n - 1) \] where \( n \) is the number of resultant ions, which is \( x + y \) for this dissociation.

Step 3: Expressing \( \alpha \) in Terms of \( i \)

Rearrange the van't Hoff factor equation to solve for \( \alpha \): \[ i = 1 + \alpha ((x+y) - 1) \]. Thus, \[ \alpha = \frac{i - 1}{x + y - 1} \].

Step 4: Matching with Given Options

The derived expression for \( \alpha \) is \( \alpha = \frac{i-1}{x+y-1} \). Compare this with the options provided. The correct choice is (c) \( \alpha = \frac{i-1}{(x+y-1)} \).

Key Concepts

Van't Hoff Factor

The van't Hoff factor, denoted as \( i \), is essential when studying solutions, especially those involving electrolytes. It quantifies the effect of solute particles on colligative properties such as boiling point elevation and freezing point depression. This factor accounts for the number of particles a solute dissociates into when dissolved.
For non-electrolytes that do not dissociate, the van't Hoff factor is generally 1, as a single molecule of solute corresponds to a single molecule in solution. However, for electrolytes like the weak electrolyte \( \text{A}_x \text{B}_y \), the picture changes. The solute ionizes into several particles, hence increasing the number of entities in the solution. This makes the van't Hoff factor greater than one.
Simply put, the van't Hoff factor is expressed for a dissociating solute like this:- \( i = 1 + \alpha(n - 1) \)where \( n \) is the number of particles formed after dissociation. Knowing \( i \) helps in understanding how much a solute can affect the solution's colligative properties.

Weak Electrolyte

Weak electrolytes only partially dissociate into ions in solution. Compared to strong electrolytes, which dissociate completely, weak electrolytes such as \( \text{A}_x \text{B}_y \) produce a equilibrium consisting of the undissociated molecules along with the dissociated ions in the solution.
The degree of dissociation \( \alpha \) quantifies how much of the solute dissociates into ions. A weak electrolyte has a small \( \alpha \), indicating limited dissociation due to its inherent properties.
For the equilibrium reaction:- \( \text{A}_x \text{B}_y \rightleftharpoons x\text{A}^{y+} + y\text{B}^{x-} \) This equilibrium showcases the balance between the ionized and unionized forms of the electrolyte.Understanding weak electrolytes is crucial in calculating the right van't Hoff factor and evaluating the colligative properties of the solution. It highlights one reason why not all solutions behave identically.

Colligative Properties

Colligative properties are the characteristics of solutions that depend on the number of solute particles rather than their identity or chemical nature. The primary colligative properties include:

• Boiling Point Elevation
• Freezing Point Depression
• Osmotic Pressure
• Vapor Pressure Lowering

Each of these properties is directly influenced by the van't Hoff factor, \( i \), or the degree of dissociation, \( \alpha \), in the case of electrolytic solutions.
For example, more solute particles result in a greater impact on boiling point elevation and freezing point depression. This means that for a weak electrolyte, where dissociation into ions is incomplete, these effects will not be as pronounced compared to a strong electrolyte.
These properties are crucial in practical and industrial applications for predicting the behaviors of solutions under various conditions. Understanding them helps students and professionals alike to manipulate and use solutions to their advantage in different scientific and engineering contexts.