Question

What is the freezing point of an aqueous solution that boils at \(105.0{ }^{\circ} \mathrm{C} ?\)

Short answer

The freezing point of the aqueous solution is approximately -18.18°C.

Step-by-step solution

Calculate the boiling point elevation

First, we have to determine the boiling point elevation (ΔT_b) using the given boiling point and the normal boiling point of water (100°C): ΔT_b = boiling point of the solution - normal boiling point of water ΔT_b = 105.0°C - 100°C = 5.0°C

Find the ratio of ΔT_b / K_b

The boiling point elevation constant for water (K_b) is known to be 0.512°C/m. Now, we can find the ratio of ΔT_b / K_b: ΔT_b / K_b = 5.0°C / 0.512°C/m = 9.7656 m^-1

Calculate the freezing point depression (ΔT_f)

Since ΔT_b / K_b = ΔT_f / K_f, we can now find the freezing point depression ΔT_f using the known value of the freezing point depression constant for water (K_f = 1.86°C/m). ΔT_f = K_f * (ΔT_b / K_b) ΔT_f = 1.86°C/m * 9.7656 m^-1 ΔT_f ≈ 18.18°C

Determine the freezing point of the solution

Finally, we can find the freezing point of the solution by subtracting the freezing point depression (ΔT_f) from the normal freezing point of water (0°C): Freezing point of the solution = normal freezing point of water - ΔT_f Freezing point of the solution = 0°C - 18.18°C ≈ -18.18°C The freezing point of the given aqueous solution is approximately -18.18°C.

Key concepts

Colligative Properties

Colligative properties are fascinating aspects of chemistry that involve the change in physical properties of solvents when a solute is dissolved in them. These alterations are merely dependent on the quantity of solute particles and not on their identity. This is a crucial concept for students to understand, as it unravels the mystery behind several phenomena in solutions.

There are four main colligative properties we consider: vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure. Each of these relies strictly on the number of solute molecules in a solution, resulting in predictable changes. For example, when salt is dissolved in water, it disrupts the orderly structure of the water molecules, leading to a significant shift in freezing and boiling points—a critical concept for applications ranging from culinary arts to antifreeze in cars.

Boiling Point Elevation

Boiling point elevation occurs when a non-volatile solute is added to a solvent, such as water, causing the boiling point of the solution to be higher than that of the pure solvent. Imagine you're cooking pasta, and you add salt to the water; this causes the boiling point to increase slightly, though often imperceptibly in the kitchen. The scientific explanation lies in the solute’s interference with the evaporation of the solvent.

Understanding Elevation

A solution won’t boil until its vapor pressure equals atmospheric pressure. By dissolving a solute in it, the solution’s vapor pressure is reduced, and it needs to be heated more to reach the same vapor pressure as the pure solvent's boiling point—that's why the boiling point goes up. This phenomenon is guided by Raoult's Law, which explains how vapor pressure depends on the concentration of the solute and is quantified using the boiling point elevation constant (\( K_b \)).

Solution Concentration

Solution concentration is integral in predicting and calculating colligative property changes. It signifies the amount of solute present per amount of solution or solvent. Concentration is commonly expressed in molarity (moles of solute per liter of solution), molality (moles of solute per kilogram of solvent), and mole fraction (the ratio of the number of moles of solute to the total number of moles in the solution).

Calculating Impact

Each concentration unit has its own applications. For instance, molality is the preferred unit for boiling point elevation and freezing point depression because it doesn't change with temperature. By accurately determining the concentration, we can use constants like the boiling point elevation constant (\( K_b \) ) and the freezing point depression constant (\( K_f \) ) to calculate how much the boiling or freezing point of a solution will be altered from its pure state. This knowledge is not only academically significant but also essential for real-world applications such as designing antifreeze or manufacturing medications.