Problem 47 - A Complete Resource Book in Chemistry for JEE Main

Question

By dissolving \(5 \mathrm{~g}\) substance in \(50 \mathrm{~g}\) of water, the decrease in freezing point is \(1.2^{\circ} \mathrm{C}\). The molal depression constant is \(1.85^{\circ} \mathrm{kg} \mathrm{mol}^{-1} .\) The molecular weight of substance is (a) \(105.4\) (b) \(118.2\) (c) \(137.2\) (d) \(154.2\)

Step-by-Step Solution

Answer

The molecular weight of the substance is 154.2, matching option (d).

Step 1: Understand the Problem

The problem requires us to calculate the molecular weight of a substance using the given data on freezing point depression. We have the mass of the substance, the mass of the solvent (water), the decrease in freezing point, and the molal depression constant.

Step 2: Use the Freezing Point Depression Formula

The formula for the freezing point depression is given by \[ \Delta T_f = i \times K_f \times m \]where \( \Delta T_f \) is the decrease in freezing point, \( K_f \) is the molal depression constant, and \( m \) is the molality of the solution.

Step 3: Solve for Molality

Rearrange the freezing point depression formula to solve for molality (\( m \)):\[ m = \frac{\Delta T_f}{K_f} \]Substitute the given values:\[ m = \frac{1.2^{\circ}}{1.85^{\circ} \text{ kg mol}^{-1}} \approx 0.6486 \text{ mol/kg} \]

Step 4: Determine Moles of Solute

Molality (\( m \)) is defined as the number of moles of solute per kilogram of solvent. Therefore,\[ m = \frac{n}{0.05} \](Since 50 g of water is 0.05 kg). Rearrange to find the number of moles \( n \):\[ n = m \times 0.05 = 0.6486 \times 0.05 \approx 0.03243 \text{ moles} \]

Step 5: Calculate Molecular Weight

The molecular weight (M) can be calculated using the formula:\[ M = \frac{\text{mass of substance (g)}}{\text{number of moles}} \]Substitute the known values:\[ M = \frac{5}{0.03243} \approx 154.2 \text{ g/mol} \]

Step 6: Compare with Options

Compare the calculated molecular weight with the given options. The calculated molecular weight is consistent with option (d) \(154.2\).

Key Concepts

Freezing Point Depression

Freezing point depression is a phenomenon that occurs when a solute is added to a solvent, causing the freezing point of the solvent to decrease. This concept is particularly important in chemistry as it helps in determining various properties of solutions, such as the molecular weight of a solute.
When you dissolve a substance in water, for example, the freezing point of the water will be lower than it would be if it were pure water. This is because the solute particles interfere with the formation of the solid phase, requiring the temperature to be lower for the solvent molecules to solidify.
The extent to which the freezing point is lowered is directly proportional to the concentration of the solute in the solution. The formula used to calculate this is:

• \[ \Delta T_f = i \times K_f \times m \] where:
  • \( \Delta T_f \) is the freezing point depression.
  • \( i \) is the van't Hoff factor, typically 1 for non-electrolytes.
  • \( K_f \) is the molal depression constant of the solvent.
  • \( m \) is the molality of the solution.

Molal Depression Constant

The molal depression constant, often symbolized as \( K_f \), is a proportionality constant that relates the freezing point depression of a solvent to the molality of the solution. It is an intrinsic property of the solvent, meaning it is independent of the nature of the solute.
Every solvent has its own unique \( K_f \) value, and it plays a crucial role in calculating the freezing point depression in solutions. For water, a common solvent in many chemistry problems, \( K_f \) is approximately 1.85 \( \degree \text{C kg/mol} \).
In the formula for freezing point depression, \( K_f \) helps determine how much the freezing point of the solvent will decrease per mole of solute particles per kilogram of solvent. This makes it essential for accurate calculations when analyzing solution properties.

Colligative Properties

Colligative properties are properties of a solution that depend on the number of solute particles in a given amount of solvent, rather than the identity of the solute particles themselves. This is a key concept in understanding solutions because it helps explain changes in various physical properties due to solute addition.

Important colligative properties include:

• Freezing point depression
• Boiling point elevation
• Vapor pressure lowering
• Osmotic pressure

These properties are important in fields beyond just chemistry laboratories; they are relevant in everyday applications, such as antifreeze formulations in car engines and the preservation of biological samples at low temperatures.
In the context of freezing point depression, the focus is on how solute particles disrupt the formation of solid structures, thus lowering the temperature at which the solution freezes. This direct relationship between solute quantity and colligative properties is crucial when analyzing molecular weight through experiments like determining the effects on the freezing point.