Question

A solution is prepared from 4.5701 g of magnesium chloride and \(43.238 \mathrm{~g}\) of water. The vapor pressure of water above this solution is 0.3624 atm at 348.0 K. The vapor pressure of pure water at this temperature is 0.3804 atm. Find the value of the van't Hoff factor ( \(i\) ) for magnesium chloride in this solution.

Short answer

To find the van't Hoff factor, calculate the molality of the solution, use Raoult's Law to find the mole fraction of water, and then use the mole fraction of the solute to calculate the van't Hoff factor based on the expected dissociation of magnesium chloride in solution.

Step-by-step solution

Calculate the Molality of the Solution

First determine the molality of the magnesium chloride solution by using its mass and the mass of the water. Calculate the number of moles of MgCl2 using its molar mass (MgCl2 = 95.211 g/mol). Then, calculate molality (mol/kg) by dividing the moles of solute by the mass of solvent in kilograms.

Use Raoult's Law to Find the Mole Fraction of Water

Apply Raoult's Law, which states that the vapor pressure of the solvent over the solution (P_solution) is equal to the vapor pressure of the pure solvent (P_pure) multiplied by the mole fraction of the solvent (X_solvent). Solve for the mole fraction of water: X_solvent = P_solution / P_pure.

Calculate the Moles of Water

Assuming the density of water is 1 g/mL, calculate the number of moles of water using its mass and molar mass (18.015 g/mol).

Calculate Total Moles in the Solution

Add the moles of MgCl2 to the moles of water to find the total moles in the solution.

Determine the Mole Fraction of MgCl2

Now calculate the mole fraction of MgCl2 by dividing the moles of MgCl2 by the total moles in the solution.

Calculate the van't Hoff Factor (i)

Knowing that magnesium chloride dissociates into three ions in solution (Mg2+ and 2 Cl-), calculate the van't Hoff factor (i) by taking the ratio of the mole fraction of MgCl2 calculated using Raoult's Law to the mole fraction of MgCl2 assuming complete dissociation into ions. The equation is i = (X_solvent / (1 - X_solvent)) / (moles of MgCl2 / total moles in solution).

Key concepts

Molality

When we talk about molality, we're referring to how concentrated a solution is, specifically in terms of the amount of solute in a certain mass of solvent. This is different from molarity, which is concerned with the volume of the solution. To calculate molality (m), you use the number of moles of solute and the mass of the solvent in kilograms (kg). The formula for molality is:
\[ m = \frac{\text{number of moles of solute}}{\text{mass of solvent in kg}} \]
This measure is particularly useful because unlike molarity, molality doesn’t change with temperature since it's based on mass, not volume.

Raoult's Law

Coming to Raoult's Law, it's a fundamental principle in physical chemistry that helps us understand how the vapor pressure of a solvent is affected by the presence of a solute. It states that the vapor pressure of a solvent over a solution (P_solution) is directly proportional to the vapor pressure of the pure solvent (P_pure) and the mole fraction of the solvent (X_solvent) in the mixture. It's given by the formula:
\[ P_{\text{solution}} = P_{\text{pure}} \times X_{\text{solvent}} \]
The mole fraction, which is another key term here, represents the ratio of the moles of solvent to the total moles of all components in the solution. If a non-volatile solute is added to the solvent, the vapor pressure will lower. This relationship is crucial for calculating the van’t Hoff factor in colligative property exercises.

Mole Fraction

The mole fraction is an expression of the concentration of a component in a mixture. For a given component, it is calculated as the number of moles of that particular component divided by the total number of moles of all components in the mixture. The mole fraction (X) is a dimensionless quantity and is especially useful in the application of Raoult's Law and when considering the colligative properties of solutions. The mole fraction is typically used when dealing with mixture's vapor pressures and boiling points. Here is the formula for calculating the mole fraction of a component:
\[ X_{\text{component}} = \frac{\text{moles of component}}{\text{total moles in solution}} \]
In a binary solution, such as our magnesium chloride and water example, you would calculate the mole fraction of water and magnesium chloride separately.

Colligative Properties

Colligative properties are fascinating because they depend only on the number of solute particles in a solvent, not the identity of the solute particles. These include boiling point elevation, freezing point depression, vapor pressure lowering, and osmotic pressure.

These properties are essential when considering solutions in various scientific and industrial processes. The van't Hoff factor (i) plays a critical role in colligative properties because it provides a correction for the number of particles a solute forms in a solution. For ionic compounds like magnesium chloride, which dissociates into multiple ions, the van’t Hoff factor allows us to calculate the actual effect of the solute on the colligative property in question. If a compound completely dissociates into three particles, for instance, the van't Hoff factor would be 3, indicating that each formula unit of solute increases the colligative effect by a factor of three compared to a non-dissociating solute of equal concentration.