Treatment of 1.385 g of an unknown metal M with an excess of aqueous HCl evolved a gas that was found to have a volume of 382.6 mL at 20.0 °C and 755 mm Hg pressure. Heating the reaction mixture to evaporate the water and remaining HCl then gave a white crystalline compound, MCl_x. After dis- solving the compound in 25.0 g of water, the melting point of the resulting solution was - 3.53 °C. (c) What is the molality of particles (ions) in the solution of MCl_x?

Verified step by step guidance

1. Identify the freezing point depression formula: \( \Delta T_f = i \cdot K_f \cdot m \), where \( \Delta T_f \) is the change in freezing point, \( i \) is the van't Hoff factor, \( K_f \) is the cryoscopic constant of water (1.86 °C kg/mol), and \( m \) is the molality.
2. Calculate the change in freezing point: \( \Delta T_f = 0.0 \text{ °C} - (-3.53 \text{ °C}) = 3.53 \text{ °C} \).
3. Rearrange the freezing point depression formula to solve for molality: \( m = \frac{\Delta T_f}{i \cdot K_f} \).
4. Assume the van't Hoff factor \( i \) for \( \text{MCl}_x \) is known or can be determined based on the dissociation of the compound in water.
5. Substitute the known values into the rearranged formula to calculate the molality: \( m = \frac{3.53}{i \cdot 1.86} \).

Key Concepts

Gas Laws

Gas laws describe the behavior of gases in relation to pressure, volume, and temperature. The ideal gas law (PV=nRT) is particularly relevant here, as it allows us to calculate the number of moles of gas produced during the reaction. Understanding how to manipulate these variables is essential for determining the amount of gas evolved and its implications for the stoichiometry of the reaction.

Molality

Molality is a measure of the concentration of solute in a solution, defined as the number of moles of solute per kilogram of solvent. It is crucial for understanding colligative properties, such as freezing point depression, which is indicated by the melting point of the solution in this question. Calculating molality requires knowledge of both the mass of the solute and the mass of the solvent.

Dissociation of Ionic Compounds

When ionic compounds like MClx dissolve in water, they dissociate into their constituent ions. This dissociation affects the total number of particles in solution, which is important for calculating colligative properties. Understanding how many ions are produced from the dissolution of MClx is necessary to determine the molality of particles in the solution.