Problem 64 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The molal boiling point elevation constant of water is \(0.513^{\circ} \mathrm{C} \mathrm{kg} \mathrm{mol}^{-1}\). When \(0.1\) mole of sugar is dissolved \(200 \mathrm{~g}\) of water, the solution boils under a pressure of 1 atm at (a) \(100.513^{\circ} \mathrm{C}\) (b) \(102.565^{\circ} \mathrm{C}\) (c) \(100.256^{\circ} \mathrm{C}\) (d) \(101.025^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

The solution boils at \(100.256^\circ C\).

Step 1: Understanding the concept of boiling point elevation

The boiling point elevation is a colligative property, which means it depends on the number of solute particles in a solvent and not on their identity. The formula to calculate the change in boiling point is given by \( \Delta T_b = i \cdot K_b \cdot m \), where \( \Delta T_b \) is the boiling point elevation, \( i \) is the van't Hoff factor (number of particles the solute breaks into), \( K_b \) is the molal boiling point elevation constant and \( m \) is the molality of the solution.

Step 2: Calculate the molality

Molality (\( m \)) is defined as the moles of solute per kilogram of solvent. Since we have 0.1 mole of sugar dissolved in 200 g of water, we first convert 200 grams to kilograms which is 0.2 kg. Then we calculate the molality: \( m = \frac{0.1\, mole}{0.2\, kg} = 0.5\, mol/kg \).

Step 3: Identify the value of the van't Hoff factor

Sugar (sucrose, C12H22O11) does not dissociate into ions in water, so its van't Hoff factor (\( i \)) is 1.

Step 4: Calculate the change in boiling point

Use the formula \( \Delta T_b = i \cdot K_b \cdot m \) with the given values. Plug in the values \( i = 1 \), \( K_b = 0.513^\circ C \cdot kg/mol \) and \( m = 0.5 \, mol/kg \), to get: \( \Delta T_b = 1 \cdot 0.513^\circ C \cdot kg/mol \cdot 0.5 \, mol/kg = 0.2565^\circ C \).

Step 5: Determine the new boiling point

The normal boiling point of water is \(100^\circ C\). Add the change in boiling point to find the new boiling point: \(100^\circ C + 0.2565^\circ C = 100.2565^\circ C\). However, for the answer choices we have discrete values, therefore, the closest value to our calculated result is \(100.256^\circ C\).

Key Concepts

Colligative Properties

Colligative properties are those properties of solutions that depend on the number of solute particles in a given amount of solvent and not on the nature of the chemical species present. These properties arise because the solute particles disrupt the solvent's molecular interactions, which affects various physical aspects of the solution. Boiling point elevation is one such colligative property, alongside others like freezing point depression, osmotic pressure, and vapor pressure lowering.

Understanding how the addition of solute to a solvent can change the boiling point is critical in many industrial and laboratory processes, such as antifreeze in car radiators and the cooking of food at high elevations. Boiling point elevation occurs because the solute particles interfere with the escape tendency of solvent molecules from the liquid to the gas phase, thus requiring a higher temperature to boil.

Molality

Molality is a measure of the concentration of a solute in a solution. It is expressed as the number of moles of solute per kilogram of solvent, not the volume of the solution, making it temperature-independent. This is an important factor to consider, as the volume of liquids can change with temperature.

To calculate the molality, you divide the moles of solute by the mass of the solvent in kilograms. For example, dissolving 0.1 moles of sugar in 200 grams (0.2 kilograms) of water results in a molality of 0.5 moles per kilogram. In the context of boiling point elevation, molality provides a direct measure that, when multiplied by the molal boiling point elevation constant, gives us the increase in boiling point due to the presence of the solute.

van't Hoff Factor

The van't Hoff factor, symbolized by 'i', represents the number of particles into which a compound dissociates in solution. For example, common table salt (NaCl) dissociates into two particles (Na+ and Cl-), giving it a van't Hoff factor of 2. On the other hand, a molecule like sugar, which does not dissociate into ions when dissolved, has a van't Hoff factor of 1.

The van't Hoff factor is crucial in calculating the extent of boiling point elevation or freezing point depression in solutions. It's the multiplying factor in the equation that connects the number of particles resulting from a solute to the magnitude of the colligative effect observed. However, it is also essential to note that real solutions can have van't Hoff factors that deviate from their ideal values due to solute-solute and solute-solvent interactions.