At home, she dissolves 40.0g of Epsom salt, MgSO4, in 5.0 kg of water. What is the freezing point of this solution? (Kf water = 1.86 oC/m)
The correct answer and explanation is:
To calculate the freezing point of the Epsom salt solution, we use the freezing point depression formula: ΔTf=i⋅Kf⋅m\Delta T_f = i \cdot K_f \cdot m
Where:
- ΔTf\Delta T_f = freezing point depression
- ii = van ‘t Hoff factor (number of particles the solute splits into)
- KfK_f = cryoscopic constant (1.86 °C·kg/mol for water)
- mm = molality of the solution (mol/kg)
Step 1: Calculate moles of MgSO₄
Molar mass of MgSO₄: 24.3 (Mg)+32.1 (S)+4×16.0 (O)=120.4 g/mol24.3\ (\text{Mg}) + 32.1\ (\text{S}) + 4 \times 16.0\ (\text{O}) = 120.4\ \text{g/mol} Moles of MgSO₄=40.0 g120.4 g/mol≈0.3323 mol\text{Moles of MgSO₄} = \frac{40.0\ \text{g}}{120.4\ \text{g/mol}} \approx 0.3323\ \text{mol}
Step 2: Calculate molality (m)
Molality=0.3323 mol5.0 kg water=0.0665 mol/kg\text{Molality} = \frac{0.3323\ \text{mol}}{5.0\ \text{kg water}} = 0.0665\ \text{mol/kg}
Step 3: Determine van ’t Hoff factor (i)
Magnesium sulfate (MgSO₄) dissociates in water: MgSO₄→Mg2++SO₄2−\text{MgSO₄} \rightarrow \text{Mg}^{2+} + \text{SO₄}^{2-}
So, i=2i = 2
Step 4: Calculate freezing point depression
ΔTf=2⋅1.86 °C\cdotpkg/mol⋅0.0665 mol/kg=0.2477 °C\Delta T_f = 2 \cdot 1.86\ \text{°C·kg/mol} \cdot 0.0665\ \text{mol/kg} = 0.2477\ \text{°C}
Step 5: Calculate new freezing point
Freezing point of solution=0.0 °C (pure water)−0.2477 °C≈−0.25 °C\text{Freezing point of solution} = 0.0\ \text{°C (pure water)} – 0.2477\ \text{°C} \approx \boxed{-0.25\ \text{°C}}
Explanation (300 words)
Freezing point depression is a colligative property, meaning it depends on the number of solute particles in a solvent rather than the type of particles. In this case, we’re adding 40.0 grams of Epsom salt (magnesium sulfate, MgSO₄) to 5.0 kg of water. When MgSO₄ dissolves, it dissociates into two ions: Mg²⁺ and SO₄²⁻. This dissociation doubles the effective number of solute particles, increasing the freezing point depression.
First, we calculated the number of moles of MgSO₄ by dividing its mass (40.0 g) by its molar mass (120.4 g/mol), giving 0.3323 mol. Then we determined the molality of the solution by dividing the moles of solute by the mass of water in kilograms, resulting in 0.0665 mol/kg.
Next, we applied the formula for freezing point depression. The van ’t Hoff factor (i) for MgSO₄ is 2 because it splits into two particles. The freezing point depression constant (Kf) for water is 1.86 °C·kg/mol. Multiplying these together with the molality gives a freezing point depression of 0.2477 °C.
Finally, we subtract this value from the normal freezing point of water (0.0 °C) to get the new freezing point of the solution, which is approximately –0.25 °C. This means the water will now freeze at a slightly lower temperature due to the presence of dissolved ions, which disrupt the formation of the solid ice lattice.
This principle is also used in real life—for example, in salting icy roads or using antifreeze in car engines.