A 50.0-mL solution is initially 1.55% MgCl₂ by mass and has a density of 1.05 g/mL. What is the freezing point of the solution after you add an additional 1.35 g MgCl₂? (Use i = 2.5 for MgCl₂.)

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Calculate the initial mass of the solution using its volume and density: \( \text{mass} = \text{volume} \times \text{density} \).

Determine the initial mass of MgCl_2 in the solution using the percentage by mass: \( \text{mass of MgCl}_2 = \text{total mass} \times \frac{1.55}{100} \).

Add the additional 1.35 g of MgCl_2 to the initial mass of MgCl_2 to find the total mass of MgCl_2 in the solution.

Calculate the molality of the solution: \( \text{molality} = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \). First, find moles of MgCl_2 using its molar mass, then find the mass of the solvent by subtracting the mass of MgCl_2 from the total mass of the solution.

Use the freezing point depression formula: \( \Delta T_f = i \cdot K_f \cdot m \), where \( i \) is the van't Hoff factor, \( K_f \) is the freezing point depression constant for water, and \( m \) is the molality, to find the change in freezing point. Subtract \( \Delta T_f \) from the normal freezing point of water to find the new freezing point.

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Colligative Properties

Colligative properties are physical properties of solutions that depend on the number of solute particles in a given amount of solvent, rather than the identity of the solute. These properties include boiling point elevation, freezing point depression, vapor pressure lowering, and osmotic pressure. In this question, the freezing point depression is relevant as it quantifies how the addition of solute affects the freezing point of the solution.

Freezing Point Depression

Freezing point depression is a colligative property that describes the decrease in the freezing point of a solvent when a solute is added. The extent of freezing point depression can be calculated using the formula ΔTf = i * Kf * m, where ΔTf is the change in freezing point, i is the van 't Hoff factor (number of particles the solute dissociates into), Kf is the freezing point depression constant of the solvent, and m is the molality of the solution.

Van 't Hoff Factor (i)

The van 't Hoff factor (i) is a dimensionless number that indicates the number of particles into which a solute dissociates in solution. For ionic compounds like MgCl2, which dissociates into one Mg²⁺ ion and two Cl⁻ ions, the van 't Hoff factor is 2.5 in this case, accounting for the degree of ionization and any association that may occur in solution, which is crucial for accurately calculating colligative properties.

A 50.0-mL solution is initially 1.55% MgCl2 by mass and has a density of 1.05 g/mL. What is the freezing point of the solution after you add an additional 1.35 g MgCl2? (Use i = 2.5 for MgCl2.)