How many kilograms of ethylene glycol (automobile anti- freeze, C2H6O2) dissolved in 3.55 kg of water are needed to lower the freezing point of water in an automobile radiator to - 22.0 °C? The molal freezing point depression constant for water is Kf = 1.86 1°C kg2>mol. (LO 13.13) (a) 0.865 kg (b) 0.0420 kg (c) 9.01 kg (d) 2.61 kg

Verified step by step guidance

1. Identify the formula for freezing point depression: \( \Delta T_f = i \cdot K_f \cdot m \), where \( \Delta T_f \) is the change in freezing point, \( i \) is the van't Hoff factor, \( K_f \) is the freezing point depression constant, and \( m \) is the molality of the solution.
2. Calculate the change in freezing point: \( \Delta T_f = 0 - (-22.0) = 22.0 \) °C.
3. Assume ethylene glycol does not dissociate in solution, so \( i = 1 \). Rearrange the formula to solve for molality: \( m = \frac{\Delta T_f}{i \cdot K_f} = \frac{22.0}{1 \cdot 1.86} \).
4. Calculate the molality \( m \) using the values from the previous step.
5. Use the definition of molality: \( m = \frac{\text{moles of solute}}{\text{kg of solvent}} \). Rearrange to find moles of solute: \( \text{moles of solute} = m \cdot 3.55 \). Convert moles of ethylene glycol to kilograms using its molar mass (\( C_2H_6O_2 \) has a molar mass of approximately 62.07 g/mol).

Key Concepts

Freezing Point Depression

Freezing point depression is a colligative property that describes how the freezing point of a solvent decreases when a solute is added. The extent of this depression is directly proportional to the molality of the solute and is quantified by the equation ΔTf = Kf * m, where ΔTf is the change in freezing point, Kf is the freezing point depression constant, and m is the molality of the solution.

Molality

Molality (m) is a measure of the concentration of a solute in a solution, defined as the number of moles of solute per kilogram of solvent. It is crucial for calculating colligative properties like freezing point depression, as it provides a direct relationship between the amount of solute and the effect on the solvent's properties.

Molar Mass of Ethylene Glycol

The molar mass of ethylene glycol (C2H6O2) is essential for converting between grams and moles when performing calculations related to freezing point depression. Ethylene glycol has a molar mass of approximately 62.07 g/mol, which allows for the determination of how many moles are present in a given mass, facilitating the calculation of molality in the context of the problem.