How much would the freezing point of water decrease if 4 mol of sugar were added to 1 kg of water ( [tex]K_{ f }=1.86^{\circ} C /( mol / kg )[/tex] for water and [tex]i=1[/tex] for sugar)?
A. [tex]7.44^{\circ} C[/tex]
B. [tex]2.15^{\circ} C[/tex]
C. [tex]1.86^{\circ} C[/tex]
D. [tex]4.00^{\circ} C[/tex]
Answer (1)
Calculate the molality of the solution: m = 1 kg 4 mol = 4 mol/kg .
Apply the freezing point depression formula: Δ T f = i ⋅ K f ⋅ m .
Substitute the values: Δ T f = 1 ⋅ 1.86 ⋅ 4 .
Calculate the decrease in freezing point: 7.4 4 ∘ C .
Explanation
Problem Analysis and Given Data We are given the following information:
Number of moles of sugar: n s ug a r = 4 mol
Mass of water: m w a t er = 1 kg
Freezing point depression constant for water: K f = 1.8 6 ∘ C / ( m o l / k g )
van't Hoff factor for sugar: i = 1
We need to find the decrease in the freezing point of water, Δ T f .
Freezing Point Depression Formula The freezing point depression is calculated using the formula:
Δ T f = i ⋅ K f ⋅ m
where i is the van't Hoff factor, K f is the freezing point depression constant, and m is the molality of the solution.
Calculating Molality First, we need to calculate the molality ( m ) of the sugar solution. Molality is defined as the number of moles of solute (sugar) per kilogram of solvent (water):
m = m w a t er n s ug a r = 1 kg 4 mol = 4 mol/kg
Calculating Freezing Point Depression Now, we can substitute the given values into the freezing point depression formula:
Δ T f = i ⋅ K f ⋅ m = 1 ⋅ 1.86 mol/kg ∘ C ⋅ 4 kg mol = 7.4 4 ∘ C
Therefore, the freezing point of water decreases by 7.4 4 ∘ C .
Final Answer The decrease in the freezing point of water is 7.4 4 ∘ C . So, the correct answer is A.
Examples
Freezing point depression is a colligative property, meaning it depends on the amount of solute dissolved in a solution, not the solute's identity. This principle is used in real life to de-ice roads during winter. Salt (NaCl) is spread on icy roads to lower the freezing point of water, preventing ice formation even when the temperature drops below 0°C. By adding salt, the freezing point of water is lowered, causing the ice to melt and improving road safety. The amount of salt added needs to be carefully calculated to achieve the desired freezing point depression without causing environmental issues.
