Problem 33 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The boiling point of pure benzene is \(353.23 \mathrm{~K}\). When \(1.80 \mathrm{~g}\) of a non-volatile solute was dissolved in \(90 \mathrm{~g}\) of benzene, the boiling point is raised to \(354.11 \mathrm{~K}\). Calculate the molar mass of the solute. \(K_{\mathrm{b}}\) for benzene is \(2.53 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\). (a) \(85 \mathrm{~g} \mathrm{~mol}^{-1}\) (b) \(57.5 \mathrm{~g} \mathrm{~mol}^{-1}\) (c) \(23 \mathrm{~g} \mathrm{~mol}^{-1}\) (d) \(38.4 \mathrm{~g} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

The molar mass of the solute is approximately \(57.5 \mathrm{~g} \mathrm{~mol}^{-1}\).

Step 1: Calculate the Boiling Point Elevation

Determine the change in boiling point by subtracting the boiling point of pure benzene from the boiling point of the solution. \( \Delta T_b = 354.11 \mathrm{~K} - 353.23 \mathrm{~K} \).

Step 2: Apply the Boiling Point Elevation Formula

Use the formula for boiling point elevation, \( \Delta T_b = i \cdot K_b \cdot m \), where \( \Delta T_b \) is the boiling point elevation, \( i \) is the van't Hoff factor (which is 1 for a non-volatile solute), \( K_b \) is the ebullioscopic constant of benzene, and \( m \) is the molality of the solution.

Step 3: Calculate the Molality

Calculate the molality (m) of the solution using the boiling point elevation and ebullioscopic constant: \( m = \frac{\Delta T_b}{K_b} \).

Step 4: Convert Mass of Benzene to Kilograms

Convert the mass of benzene from grams to kilograms to use in the molality equation: \( 90 \mathrm{~g} = 0.090 \mathrm{~kg} \).

Step 5: Calculate the Moles of Solute

Molality is defined as the moles of solute per kilogram of solvent. Calculate the moles of solute as follows: \( moles_{solute} = molality \times mass_{solvent\,in\,kg} \).

Step 6: Calculate Molar Mass of Solute

The molar mass (M) of the solute is the mass of the solute divided by the moles of solute. Use the formula: \( M = \frac{mass_{solute}}{moles_{solute}} \).

Key Concepts

Colligative Properties

Colligative properties are unique characteristics of solutions that depend only on the number of solute particles present, not the identity of these particles. Boiling point elevation is one such property, along with freezing point depression, vapor pressure lowering, and osmotic pressure. Understanding colligative properties is crucial for many scientific applications, such as calculating the molar mass of an unknown solute in a solution, as is demonstrated in the provided textbook exercise.

The principle behind boiling point elevation is that the added solute particles disrupt the solvent’s ability to evaporate, requiring a higher temperature to reach the boiling point. This concept is vital not just in classrooms but also in real-world situations such as adding antifreeze to a car radiator to prevent the water from boiling over in hot weather.

Molality

Molality is a measure of solute concentration in a solution and is defined as the moles of solute per kilogram of solvent. Unlike molarity, which is moles per liter of solution, molality is not affected by changes in temperature or pressure because it is based on mass, not volume.

In the formula for boiling point elevation, molality (symbolized as 'm') plays a key role. Calculating molality involves dividing the number of moles of solute by the mass of the solvent in kilograms. By determining the molality, we can relate the change in boiling point to the quantity of solute present, enabling us to probe deeper into properties of the solution and the solute.

Ebullioscopic Constant

The ebullioscopic constant, represented as \(K_b\), is a specific value for each solvent that reflects how a solute will affect the boiling point of the solvent. It's a measure of the boiling point elevation per mole of solute per kilogram of solvent. Each solvent has its own unique ebullioscopic constant. For instance, benzene has a \(K_b\) of 2.53 K kg/mol.

Understanding and utilizing the ebullioscopic constant is crucial when solving problems related to boiling point elevation. It's important to emphasize that \(K_b\) is derived from the solvent’s properties and is a constant that needs to be accounted for when determining the extent of the boiling point change.

Molar Mass Calculation

Molar mass calculation is the process of determining the mass of one mole of a substance. It plays a significant role in various chemical calculations, including finding the identity of an unknown solute in a solution through colligative properties. From the given exercise, calculating the molar mass of the solute involves dividing the mass of the solute by the number of moles of solute. This is determined by the molality and the mass of the solvent.

The process, as exhibited in the step by step solution, shows how to isolate the molar mass in an equation, utilizing the change in boiling point and the known ebullioscopic constant. This sort of calculation is foundational in stoichiometry and practical applications like formulating pharmaceuticals or designing chemical processes.