Question

40 g of glucose (Molar mass = 180) is mixed with 200 mL of water. The freezing point of solution is _________ K. (Nearest integer)
[Given: K\(_f\)= 1.86 K kg mol\(^{-1}\); Density of water=1.00 g cm\(^{-3}\); Freezing point of water = 273.15 K]

Hint:

Remember the difference between molarity (moles/L of solution) and molality (moles/kg of solvent). Colligative property calculations like freezing point depression and boiling point elevation always use molality.

Answer 1

Correct Answer: 271

Step 1: Understanding the Question:
We are asked to calculate the freezing point of an aqueous solution of glucose. This requires using the formula for freezing point depression, a colligative property.
Step 2: Key Formula:
The depression in freezing point (\(\Delta T_f\)) is given by:
\[ \Delta T_f = i \cdot K_f \cdot m \] where \(i\) is the van't Hoff factor, \(K_f\) is the cryoscopic constant, and \(m\) is the molality of the solution.
Step 3: Calculate Molality (m):
Molality is defined as moles of solute per kilogram of solvent.
- Solute: Glucose (C\(_6\)H\(_{12}\)O\(_6\)). It is a non-electrolyte, so its van't Hoff factor \(i = 1\).
- Moles of glucose = \( \frac{\text{mass}}{\text{molar mass}} = \frac{40 \, \text{g}}{180 \, \text{g/mol}} = \frac{2}{9} \) mol.
- Solvent: Water.
- Mass of water = Volume \(\times\) Density = \(200 \, \text{mL} \times 1.00 \, \text{g/mL} = 200\) g.
- Mass of water in kg = \( \frac{200 \, \text{g}}{1000 \, \text{g/kg}} = 0.2 \) kg.
- Molality \(m = \frac{\text{moles of solute}}{\text{kg of solvent}} = \frac{2/9 \, \text{mol}}{0.2 \, \text{kg}} = \frac{2}{9 \times 0.2} = \frac{10}{9} \) mol/kg.
Step 4: Calculate the Freezing Point Depression (\(\Delta T_f\)):
\[ \Delta T_f = 1 \times (1.86 \, \text{K kg mol}^{-1}) \times \left(\frac{10}{9} \, \text{mol/kg}\right) = \frac{18.6}{9} \approx 2.067 \, \text{K} \] Step 5: Calculate the New Freezing Point:
The freezing point of the solution is the freezing point of the pure solvent minus the depression.
\[ T_{f, \text{solution}} = T_{f, \text{water}} - \Delta T_f \] \[ T_{f, \text{solution}} = 273.15 \, \text{K} - 2.067 \, \text{K} = 271.083 \, \text{K} \] Rounding to the nearest integer, the freezing point is 271 K.