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

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding Van't Hoff Factor

The van't Hoff factor, denoted as \( i \), is a measure of the effect of solute on colligative properties. It represents the number of particles into which a solute dissociates in solution. For a weak electrolyte, complete dissociation into ions is not achieved, leading to a van't Hoff factor less than the ideal value.

Step 2: Weak Electrolyte Dissociation

Consider a weak electrolyte \( \mathrm{A}_{\mathrm{x}} \mathrm{B}_{\mathrm{y}} \) dissociating as \( \mathrm{A}_{\mathrm{x}} \mathrm{B}_{\mathrm{y}} \rightleftharpoons x\mathrm{A}^{y+} + y\mathrm{B}^{x-} \). If \( \alpha \) is the degree of dissociation, initially, before dissociation, there is 1 mole of \( \mathrm{A}_{\mathrm{x}} \mathrm{B}_{\mathrm{y}} \) which dissociates to give \( x\alpha \) moles of \( \mathrm{A}^{y+} \) and \( y\alpha \) moles of \( \mathrm{B}^{x-} \).

Step 3: Determine Van't Hoff Factor Expression

The total number of moles after dissociation becomes \( 1-\alpha + x\alpha + y\alpha = 1 + (x+y-1)\alpha \). Hence, the van't Hoff factor \( i \) which is the ratio of total particles after dissociation to initial particles is given by: \[ i = 1 + (x+y-1)\alpha \]

Step 4: Solving for Degree of Dissociation

Rearrange the equation for \( \alpha \): \[ i = 1 + (x+y-1)\alpha \] Subtract 1 from both sides: \[ i - 1 = (x+y-1)\alpha \] Divide by \((x+y-1)\) to solve for \( \alpha \): \[ \alpha = \frac{i-1}{x+y-1} \]

Step 5: Matching with given Options

Review the given options and match our derived expression: \( \alpha = \frac{i-1}{x+y-1} \) matches option (a) when rearranged.

Key Concepts

Van't Hoff Factor

The van't Hoff factor is a crucial concept in understanding how solutes affect the properties of a solution, specifically colligative properties like boiling point elevation and freezing point depression. It is represented by the symbol \( i \). Essentially, the van't Hoff factor conveys the number of particles that a solute produces when it dissolves in a solvent.

When a solute dissolves, it may dissociate into multiple ions or remain as individual molecules. For instance:

• If a solute completely dissociates, \( i \) would be equal to the number of particles formed.
• If a solute doesn’t dissociate at all, \( i \) stays at 1.
• For weak electrolytes, \( i \) tends to be more than 1 but less than the maximum potential dissociation because these electrolytes don't dissociate completely.

By understanding \( i \), we can predict how a solute will behave in a solution and calculate essential properties that are dependent on the number of solute particles present.

Weak Electrolyte

Weak electrolytes are substances that only partially dissociate into ions when dissolved in water. This partial dissociation is because the ions reform the original compound, resulting in a dynamic equilibrium between the undissociated compound and the dissociated ions.

A common representation of a weak electrolyte could be \( \mathrm{A}_{\mathrm{x}} \mathrm{B}_{\mathrm{y}} \rightleftharpoons x\mathrm{A}^{y+} + y\mathrm{B}^{x-} \). Initially, you start with 1 mole of the undissociated compound. Upon dissociation:

• \( x\alpha \) moles of \( \mathrm{A}^{y+} \) are formed.
• \( y\alpha \) moles of \( \mathrm{B}^{x-} \) are formed.

Here, \( \alpha \) represents the degree of dissociation. Since weak electrolytes don't dissociate completely, solutions of weak electrolytes exhibit properties different from those expected in strong electrolytes, as complete conversion doesn’t occur.

Degree of Dissociation

The degree of dissociation, symbolized by \( \alpha \), communicates the fraction of solute molecules that dentatively dissociate into ions in solution. This concept is particularly pertinent for weak electrolytes, where dissociation is incomplete.

For a weak electrolyte expressed as \( \mathrm{A}_{\mathrm{x}} \mathrm{B}_{\mathrm{y}} \), the degree of dissociation can influence the solution's behavior profoundly. Through the dissociation process:

• \( \alpha \) values range from 0 (no dissociation) to 1 (complete dissociation).
• We can quantify \( \alpha \) by the equation \( \alpha = \frac{i-1}{x+y-1} \), derived from the balance of immediate particles and those formed after dissociation.

By calculating \( \alpha \), we ascertain how much of the weak electrolyte actually forms ions in a solution, thereby impacting properties such as the van't Hoff factor and colligative properties.