Problem 78 - A Complete Resource Book in Chemistry for JEE Main

Question

If \(0.44 \mathrm{~g}\) of substance dissolved in \(22.2 \mathrm{~g}\) of benzene lowers the freezing point of benzene by \(0.567^{\circ} \mathrm{C}\), then the molecular mass of substance is, (the molal depression constant \(\left.=5.12^{\circ} \mathrm{C} \mathrm{mol}^{-1} \mathrm{~kg}\right)\) (a) \(128.4\) (b) \(156.6\) (c) \(178.9\) (d) \(232.4\)

Step-by-Step Solution

Answer

The molecular mass of the substance is approximately 178.9, answer (c).

Step 1: Understand the formula

We will use the formula for depression of freezing point: \[ \Delta T_f = K_f \times m \] where \( \Delta T_f \) is the depression of the freezing point, \( K_f \) is the molal depression constant, and \( m \) is the molality of the solution.

Step 2: Calculate the molality (m)

First, calculate the molality \( m \) using the equation: \[ m = \frac{\Delta T_f}{K_f} \] Given: \( \Delta T_f = 0.567^{\circ} \mathrm{C} \) and \( K_f = 5.12^{\circ} \mathrm{C} \cdot \mathrm{mol}^{-1} \cdot \mathrm{kg} \). \[ m = \frac{0.567}{5.12} \approx 0.1107 \] The molality is approximately \(0.1107 \) mol/kg.

Step 3: Use molality to find moles of solute

Using the definition of molality \( m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \), We can rearrange to find moles of solute: \[ \text{moles of solute} = m \times \text{mass of solvent in kg} \] Convert mass of solvent from grams to kilograms: \( 22.2 \mathrm{~g} = 0.0222 \mathrm{~kg} \) \[ \text{moles of solute} = 0.1107 \times 0.0222 \approx 0.0024575 \] So, the moles of solute is approximately 0.0024575 moles.

Step 4: Determine the molecular mass of the substance

The molecular mass (M) can be calculated using the formula: \[ \text{Molecular mass (M)} = \frac{\text{mass of substance in grams}}{\text{moles of substance}} \] Given: mass of substance = \(0.44 \) grams. \[ M = \frac{0.44}{0.0024575} \approx 178.9 \] Therefore, the molecular mass of the substance is approximately 178.9.

Key Concepts

Molal Depression Constant

In the field of chemistry, the molal depression constant (K_f) is a crucial value used in determining how much the freezing point of a solvent will decrease when a solute is added. It essentially tells us how sensitive a solvent is to changes in its freezing point when different substances are dissolved into it. Each solvent has its unique K_f.
To better understand, think of the molal depression constant as a coefficient that links the depression of freezing point (ΔT_f) to the molality of the solution (m). This relationship is expressed through the equation:
\[ \Delta T_f = K_f \times m \]
Here:

• \( \Delta T_f \)is the change in freezing temperature of the solvent
• \( K_f \)is the molal depression constant, specific to each solvent
• \( m \)is the molality, which refers to the amount of solute per kilogram of solvent

For example, if a solvent has a K_f of \(5.12^\circ C \cdot mol^{-1} \cdot kg\), it means that a 1-molal solution of any solute will lower the freezing point of the solvent by \(5.12^\circ C\). Thus, understanding K_f helps us predict how various substances impact the solvent's freezing point.

Molality Calculation

Molality is a way to express the concentration of a solution. Compared to molarity, which depends on volume, molality (m) is focused purely on the mass of the solvent, making it independent of temperature. For students new to chemistry, it's useful to remember that molality measures how many moles of solute are in one kilogram of solvent.
The equation for calculating molality is:
\[ m = \frac{\Delta T_f}{K_f} \]
In this equation:

• \( \Delta T_f \)is the change in the freezing point of the solvent
• \( K_f \)is the molal depression constant
• \( m \)is the molality, which we solve for

Consider a scenario where a solvent's freezing point drops by \(0.567^\circ C\) and the solvent's K_f is \(5.12^\circ C \cdot mol^{-1} \cdot kg\). We can find the molality by dividing \(0.567\) by \(5.12\), which results in a molality of approximately \(0.1107\). This implies there are approximately \(0.1107\) moles of solute in every kilogram of solvent.

Molecular Mass Determination

Determining the molecular mass of a substance is a common exercise in chemistry to understand the molecular composition of a solute in a solution. Molecular mass can be calculated by deducing the number of moles present in a known mass of solute within a given quantity of solvent.
The mathematical formula is:
\[ \text{Molecular mass (M)} = \frac{\text{mass of substance in grams}}{\text{moles of substance}} \]
Here's how it works with numerical values:

• You find moles of solute by multiplying the molality by the mass of solvent in kilograms.
• For instance, if molality equals \(0.1107\) and the mass of solvent is \(0.0222\) kg, the moles of solute will be \(0.0024575\).
• Then, given a substance with a mass of \(0.44\) grams, you use the formula: \[ \text{M} = \frac{0.44}{0.0024575} \]
• This calculation tells us the molecular mass is approximately \(178.9\), showing how many grams one mole of this substance would weigh.

Thus, understanding and calculating the molecular mass reveals essential details about the identity and characteristics of the chemical substance.