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

Question

An aqueous solution of a solute which neither associates nor dissociates has a freezing point depression of \(X^{\circ} \mathrm{C}\). An equimolar solution of a second has a freezing point depression of \(4 \mathrm{X}^{\circ} \mathrm{C}\). The second solution could be a salt of formula (assuming \(100 \%\) dissociation ) (a) \(\mathrm{AB}_{2}\) (b) \(\mathrm{A}_{2} \mathrm{~B}_{2}\) (c) \(\mathrm{A}_{3} \mathrm{~B}\) (d) \(\mathrm{AB}_{3}\)

Step-by-Step Solution

Answer

The correct salt formula is either \( \text{A}_2\text{B}_2 \) or \( \text{A}_3\text{B} \).

Step 1: Understanding Freezing Point Depression Formula

The formula for freezing point depression is given by \( \Delta T_f = i \cdot K_f \cdot m \), where \( \Delta T_f \) is the freezing point depression, \( i \) is the van't Hoff factor, \( K_f \) is the freezing point depression constant, and \( m \) is the molality of the solution. Here, it is mentioned the first solute does not associate or dissociate, so \( i=1 \).

Step 2: Analyzing the Equimolar Solution

For the second solution, the freezing point depression is \( 4X^{\circ} \mathrm{C} \). Since both solutions are equimolar and use the same solvent, their molality \( m \) is the same. The difference in freezing point depression arises due to different values of \( i \). Thus, the van't Hoff factor for this solution is \( i = 4 \).

Step 3: Relating Van't Hoff Factor to Dissociation

The van't Hoff factor \( i \) reflects the total number of particles in solution after dissociation. For a fully dissociated salt \( \text{A}_x\text{B}_y \), the total number of ions produced is \( x + y \). Since the second solution has \( i = 4 \), \( x + y = 4 \).

Step 4: Determining the Salt Formula

Given \( x + y = 4 \), the possible formulas for the salt that satisfy this are (a) \( \text{AB}_3 \) with \(1+3=4\), (b) \( \text{A}_2\text{B}_2 \) with \(2+2=4\), or (d) \( \text{AB}_3 \) with \( 1 + 3 = 4 \). Double-check the original options: (c) \( \text{A}_3\text{B} \) with \(3+1=4\). Only \( \text{A}_2\text{B}_2 \) (option b) and \( \text{A}_3\text{B} \) (option c) fit the criteria.

Key Concepts

van't Hoff factor

The van't Hoff factor, denoted as \( i \), is a crucial component when discussing freezing point depression. It represents the number of particles that a solute splits into when dissolved. In essence, the van't Hoff factor quantifies the effect of solute particles on the colligative property of a solution.

For non-electrolyte solutions, like sugar in water, the van't Hoff factor is \( i = 1 \) because the molecules do not dissociate. However, for electrolytes, like salts, the factor increases as the compound dissociates into ions when dissolved in water. For example, when sodium chloride (NaCl) dissolves, it dissociates into two ions (\( ext{Na}^+ \) and \( ext{Cl}^- \)), making \( i \) approximately equal to 2. Similarly, the factor will be higher for solutes that form more ions upon dissociation, which directly influences properties like freezing point depression.

Understanding the van't Hoff factor helps in analyzing how solutions behave under various conditions, making it essential for solving problems related to colligative properties.

aqueous solutions

Aqueous solutions are mixtures where water is the solvent, commonly encountered in chemical reactions and everyday phenomena. When a solute is added to water, it may dissolve, producing a homogeneous solution. The dissolving process is influenced by several factors including the nature of the solute and solvent interaction.

Since water is a polar molecule, it interacts effectively with other polar substances, such as salts and sugars, enhancing their dissolving capacity. When an ionic compound, like table salt, is added to water, it dissociates into its constituent ions due to water's polarity. This ability to dissolve diverse substances makes aqueous solutions versatile in chemical processes and contributes to various colligative properties like boiling point elevation and freezing point depression.

Understanding aqueous solutions is fundamental in chemistry as it forms the foundation for studying reactions in which water acts as a medium, offering insights into how solute and solvent interactions govern solution behaviors.

dissociation

Dissociation is a process where molecules or ionic compounds split into smaller particles like atoms, ions, or radicals, typically when dissolved in a solvent. It is a crucial phenomenon, particularly in electrolytes, substances that form ions when dissolved in water.

In the context of electrolytes, dissociation is the reason these compounds conduct electricity. For instance, when sodium chloride (NaCl) is added to water, it dissociates into sodium (\( ext{Na}^+ \)) and chloride (\( ext{Cl}^- \)) ions. The extent of dissociation can affect several solution properties, including the van't Hoff factor. Typically, more dissociation means more particles in the solution and therefore a higher impact on colligative properties like freezing point depression.

Understanding dissociation helps elucidate why some substances significantly alter the physical properties of solutions while others do not. This provides a clearer picture of how and why particular chemical reactions proceed within aqueous solutions.

molality

Molality is a measure of solute concentration in a solution, defined as the moles of solute per kilogram of solvent. This unit is distinct from molarity, which is based on the volume of the solution. Molality is particularly useful when dealing with colligative properties as it remains unaffected by temperature changes.

The formula for calculating molality \( m \) is: \[ m = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \] The consistent nature of mass (as opposed to volume) in response to temperature fluctuations makes molality a preferred unit when discussing properties like boiling point elevation and freezing point depression.

In cases where precise calculations relating to temperature-dependent properties are required, using molality ensures accuracy. This precision is particularly important in experiments where temperature varies significantly, affecting solute concentration and thereby impacting the outcome of reactions and properties being measured.