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

Question

The degree of dissociation \((\alpha)\) of a weak electrolyte, \(\mathrm{A}_{\mathrm{x}} \mathrm{B}_{y}\) 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 expression is (c) \(\alpha=\frac{i-1}{(x+y-1)}\).

Step 1: Understanding the Problem

The problem asks us to find the correct expression that relates the degree of dissociation \((\alpha)\) of a weak electrolyte \(\mathrm{A}_{\mathrm{x}} \mathrm{B}_{y}\) with the van't Hoff factor \(i\). We have four options to determine which one is correct.

Step 2: Conceptual Background

The van't Hoff factor \(i\) accounts for the number of particles in solution after dissociation compared to the number initially. For a weak electrolyte \(\mathrm{A}_{\mathrm{x}} \mathrm{B}_{y}\), the dissociation can be represented as: \[ \mathrm{A}_{\mathrm{x}} \mathrm{B}_{y} \rightarrow x \mathrm{A} + y \mathrm{B} \] The total number of particles after dissociation is \(1 + \alpha(x + y - 1)\). Thus, the van't Hoff factor is \(i = 1 + \alpha(x + y - 1)\).

Step 3: Relating \(\alpha\) and \(i\)

We have derived the relationship \(i = 1 + \alpha(x + y - 1)\). To find \(\alpha\), rearrange this equation: \[ \alpha(x + y - 1) = i - 1 \] Thus, \[ \alpha = \frac{i - 1}{x + y - 1} \]

Step 4: Comparing with Options

Now, compare the derived expression \(\alpha = \frac{i - 1}{x + y - 1}\) with the options provided in the question: (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) \(\alpha=\frac{i-1}{x+y+1}\). The correct option is (c) as it matches our derived expression.

Key Concepts

Van't Hoff Factor

The van't Hoff factor \(i\) is a crucial concept in understanding how solutes behave in a solution, especially in the case of electrolytes. It signifies the number of particles present in the solution following the dissociation of the solute.

For non-electrolytes, which do not dissociate in solution, \(i\) is typically 1, as there are no additional particles formed. However, for electrolytes, it becomes important to account for the breaking down into ions.

• In the context of weak electrolytes like \(A_xB_y\), the dissociation is partial, hence the van't Hoff factor may not be an integer.
• The van't Hoff factor can be expressed as \(i = 1 + \alpha(x + y - 1)\), where \(\alpha\) is the degree of dissociation.

This equation helps to quantify the difference between the expected and observed number of particles in solution, providing insight into the nature of solute interactions.

Weak Electrolyte Dissociation

Weak electrolytes are compounds that only partially dissociate into ions when dissolved in water. Unlike strong electrolytes, which fully dissociate, weak electrolytes establish a dynamic equilibrium between the dissolved ions and undissociated molecules.

When a weak electrolyte such as \(A_xB_y\) dissociates, it results in:

• \(A_xB_y \) \(\rightarrow\) \(xA^+ + yB^- \).
• Instead of completely turning into ions, only a fraction represented by the degree of dissociation \(\alpha\) converts, affecting the total number of particles in solution.

This partial dissociation is central to calculating both the van't Hoff factor and the extent of ion formation in solution. Understanding weak electrolytes thus involves assessing the balance between molecular and ionic forms in equilibrium.

Chemical Equilibrium

Chemical equilibrium is a key concept in chemistry, pertinent to reactions involving weak electrolytes. It describes a state where the rate of the forward reaction (dissociation into ions) equals the rate of the reverse reaction (recombination into molecules).

At equilibrium in a solution of a weak electrolyte like \(A_xB_y\), the concentration of reactants and products remains constant over time. This balance allows for:

• Predicting ion concentrations using equilibrium constants.
• Deriving relationships between measurable properties, such as in the case of the van't Hoff factor and degree of dissociation.

Understanding chemical equilibrium helps clarify why the dissociation of weak electrolytes is partial and how this influences observable properties like ion concentration and colligative properties in a solution. It is essential to grasp these dynamics to predict and manipulate the behavior of weak electrolytes in various contexts.