Question

A solution is prepared by dissolving \(396 \mathrm{~g}\) of sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)\) in \(624 \mathrm{~g}\) of water. What is the vapor pressure of this solution at \(30^{\circ} \mathrm{C}\) ? (The vapor pressure of water is \(31.8 \mathrm{mmHg}\) at \(\left.30^{\circ} \mathrm{C} .\right)\)

Short answer

The vapor pressure of the solution is approximately 30.74 mmHg.

Step-by-step solution

Calculate the moles of sucrose

Find the molar mass of sucrose, \( \mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11} \), by summing the atomic masses: \( (12 \times 12.01) + (22 \times 1.01) + (11 \times 16.00) = 342.30 \; \mathrm{g/mol} \). Then, calculate the moles of sucrose using the formula: \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \approx \frac{396 \; \mathrm{g}}{342.30 \; \mathrm{g/mol}} \approx 1.16 \; \mathrm{mol} \).

Calculate moles of water

Determine the molar mass of water, \( \mathrm{H}_2\mathrm{O} \), which is \( (2 \times 1.01) + 16.00 = 18.02 \; \mathrm{g/mol} \). Calculate the moles of water as follows: \( \text{moles} = \frac{624 \; \mathrm{g}}{18.02 \; \mathrm{g/mol}} \approx 34.62 \; \mathrm{mol} \).

Determine mole fraction of water

The mole fraction of water is given by the formula: \( X_{\mathrm{H}_2\mathrm{O}} = \frac{\text{moles of water}}{\text{moles of sucrose} + \text{moles of water}} \). Substitute the values: \( X_{\mathrm{H}_2\mathrm{O}} = \frac{34.62}{1.16 + 34.62} \approx 0.967 \).

Apply Raoult's Law

According to Raoult's Law, the vapor pressure of the solution is given by: \( P_{\text{solution}} = X_{\mathrm{H}_2\mathrm{O}} \times P^0_{\mathrm{H}_2\mathrm{O}} \), where \( P^0_{\mathrm{H}_2\mathrm{O}} \) is the vapor pressure of pure water. Using the values provided, \( P_{\text{solution}} = 0.967 \times 31.8 \; \mathrm{mmHg} \approx 30.74 \; \mathrm{mmHg} \).

Key concepts

Raoult's Law

Raoult's Law is a principle used to determine the vapor pressure of a solution. It states that the vapor pressure of a solvent in a solution is directly proportional to the mole fraction of the solvent present in the solution.
This means the more solvent molecules present, the higher the vapor pressure. In mathematical terms, the law is expressed as:

• \( P_{\text{solution}} = X_{\text{solvent}} \times P^0_{\text{solvent}} \)

Where:

• \( P_{\text{solution}} \) = vapor pressure of the solution
• \( X_{\text{solvent}} \) = mole fraction of the solvent
• \( P^0_{\text{solvent}} \) = vapor pressure of the pure solvent

This relationship helps us to understand how solutions affect the volatility of the solvent. When a non-volatile solute (like sucrose) is added, it lowers the vapor pressure by reducing the number of solvent molecules that can escape into the vapor phase.

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. Specifically, in a solution, it represents the ratio of the number of moles of a component to the total number of moles of all components present. To calculate the mole fraction of water in the sucrose solution, you use the formula:

• \( X_{\mathrm{H}_2\mathrm{O}} = \frac{\text{moles of water}}{\text{moles of sucrose} + \text{moles of water}} \)

Mole fraction is a unitless number that provides insight into the composition of a solution.
It is crucial for determining properties such as vapor pressure using Raoult's Law. In this example, the mole fraction of water was calculated to be approximately 0.967.

Sucrose Solution

A sucrose solution is created by dissolving sucrose, a common sugar, into water. Sucrose is a large molecule, chemically represented as \( \mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11} \).
When sucrose is added to water, it completely dissolves, meaning it doesn't contribute to the vapor pressure since it is non-volatile.This kind of solution is an example of a non-ideal solution because the presence of sucrose affects the physical properties like boiling point and vapor pressure of the water.
Specifically, the vapor pressure of the solution is less than that of pure water due to the sucrose molecules taking up space where water molecules would otherwise evaporate.

Vapor Pressure of Water

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid phase. For water, this indicates how readily water molecules escape from the liquid into the vapor phase.
At any given temperature, pure water has a specific vapor pressure, which increases as temperature rises.In this exercise, the vapor pressure of pure water at \( 30^\circ \mathrm{C} \) is provided as \( 31.8 \mathrm{mmHg} \). This serves as a benchmark. The presence of a solute like sucrose reduces the vapor pressure by making it slightly more difficult for water molecules to escape, as indicated by the decreased vapor pressure of the solution (30.74 mmHg) calculated using Raoult's Law.
Understanding vapor pressure is crucial in fields like meteorology, cooking, and chemistry as it influences phenomena like evaporation and boiling.