What mass of ethanol, C2H6O, (molar mass = 46.1 g/mol), must be dissolved in 125 g of water to raise the boiling point of the solution to 102.1°C? The normal boiling point of water is 100.°C and Kb for water is 0.512°C/m.
The Correct Answer and Explanation is :
To solve this problem, we use the boiling point elevation formula:
[
\Delta T_b = K_b \cdot m
]
where:
- (\Delta T_b) is the boiling point elevation ((102.1^\circ\text{C} – 100.0^\circ\text{C} = 2.1^\circ\text{C})),
- (K_b) is the boiling point elevation constant for water ((0.512^\circ\text{C/m})),
- (m) is the molality of the solution (( \text{moles of solute} / \text{kilograms of solvent} )).
Step 1: Rearrange the formula to find (m):
[
m = \frac{\Delta T_b}{K_b} = \frac{2.1}{0.512} \approx 4.10 \, \text{m}
]
Step 2: Calculate the moles of ethanol required:
[
m = \frac{\text{moles of solute}}{\text{mass of solvent (kg)}}
]
[
\text{moles of solute} = m \cdot \text{mass of solvent (kg)} = 4.10 \cdot 0.125 = 0.5125 \, \text{moles}
]
Step 3: Convert moles of ethanol to mass:
[
\text{mass of ethanol} = \text{moles} \cdot \text{molar mass} = 0.5125 \cdot 46.1 \approx 23.6 \, \text{g}
]
Final Answer:
The mass of ethanol required is 23.6 g.
Explanation (300 words):
Boiling point elevation occurs when a solute is dissolved in a solvent, raising the solvent’s boiling point. The change in boiling point ((\Delta T_b)) is directly proportional to the molality ((m)) of the solution and the boiling point elevation constant ((K_b)) of the solvent. For water, (K_b = 0.512^\circ\text{C/m}).
The problem specifies a desired boiling point of (102.1^\circ\text{C}), so (\Delta T_b = 2.1^\circ\text{C}). Using the formula (m = \Delta T_b / K_b), the molality is calculated to be (4.10 \, \text{m}). This means 4.10 moles of ethanol must be dissolved per kilogram of water.
Since the solvent mass is 125 g (0.125 kg), the required moles of ethanol are (0.5125). To determine the mass of ethanol, we multiply its moles by its molar mass ((46.1 \, \text{g/mol})), yielding (23.6 \, \text{g}).
This calculation assumes ideal behavior (no significant solute-solvent interactions affecting boiling point) and that ethanol fully dissolves in water. Such problems are common in colligative property studies, demonstrating the impact of solute concentration rather than its chemical identity.