Question

How many grams of sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)\) must be added to \(552 \mathrm{~g}\) of water to give a solution with a vapor pressure \(2.0 \mathrm{mmHg}\) less than that of pure water at \(20^{\circ} \mathrm{C} ?\) (The vapor pressure of water at \(20^{\circ} \mathrm{C}\) is \(17.5 \mathrm{mmHg}\).)

Short answer

Add approximately 134.5 grams of sucrose.

Step-by-step solution

Understand the Concept of Vapor Pressure Lowering

The lowering of vapor pressure due to a solute is a colligative property, which means it depends on the number of solute particles, not their identity. According to Raoult's Law, the change in vapor pressure is proportional to the mole fraction of the solute.

Calculate the Change in Vapor Pressure

We know the vapor pressure of pure water at \(20^{\circ} \mathrm{C}\) is \(17.5\mathrm{mmHg}\). The solution's vapor pressure is \(17.5\mathrm{mmHg} - 2.0\mathrm{mmHg} = 15.5\mathrm{mmHg}\). So, the change in vapor pressure (\(\Delta P\)) is \(2.0\mathrm{mmHg}\).

Apply Raoult's Law

According to Raoult's Law, \(\Delta P = X_{\text{solute}} \cdot P^\circ_{\text{solvent}}\), where \(X_{\text{solute}}\) is the mole fraction of the solute and \(P^\circ_{\text{solvent}}\) is the vapor pressure of pure solvent. Here, \(\Delta P = 2.0\mathrm{mmHg}\) and \(P^\circ_{\text{solvent}} = 17.5\mathrm{mmHg}\).

Solve for the Mole Fraction of Solute

From Raoult's Law, \[X_{\text{solute}} = \frac{\Delta P}{P^\circ_{\text{solvent}}} = \frac{2.0}{17.5}.\] Calculate this to find the mole fraction of the solute.

Calculate Moles of Water

To find the number of moles of water, use the formula \(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\). Water has a molar mass of \(18.015 \mathrm{g/mol}\), so the moles of water is \(\frac{552 \mathrm{g}}{18.015 \mathrm{g/mol}}\).

Use Mole Fraction to Find Moles of Sucrose

Solve for the moles of sucrose using the relationship \[X_{\text{solute}} = \frac{\text{moles of sucrose}}{\text{moles of sucrose} + \text{moles of water}}.\] You already have \(X_{\text{solute}}\) from Step 4 and moles of water from Step 5.

Calculate Mass of Sucrose Needed

Once you have the moles of sucrose, use the molar mass of sucrose \((\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11})\), which is approximately \(342.3 \mathrm{g/mol}\), to find the mass: \(\text{mass of sucrose} = \text{moles of sucrose} \times 342.3 \mathrm{g/mol}\).

Key concepts

Raoult's Law

Raoult's Law is a fundamental principle used to describe how the addition of a solute reduces the vapor pressure of a solvent. It states that the vapor pressure of a solvent above a solution is directly proportional to the mole fraction of the solvent. This implies that adding a non-volatile solute, such as sucrose, decreases the solvent's vapor pressure in proportion to the number of solute particles present.
To understand this concept better, consider that when a solute is added, it occupies space at the liquid's surface that would otherwise be taken up by the solvent molecules. As a result, fewer solvent molecules escape into the vapor phase, thus lowering the vapor pressure.

• The equation representative of Raoult’s Law is: \[\Delta P = X_{\text{solute}} \cdot P^\circ_{\text{solvent}}\]
  where:
  • \(\Delta P\) is the change in vapor pressure.
  • \(X_{\text{solute}}\) is the mole fraction of the solute.
  • \(P^\circ_{\text{solvent}}\) is the vapor pressure of the pure solvent.

This law applies effectively when the solute is non-volatile and the solution behaves ideally, offering a straightforward way to understand the colligative property of vapor pressure lowering.

Colligative Properties

Colligative properties are characteristics of solutions that depend on the number of solute particles and not the type. These properties include vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure.
Vapor pressure lowering, observed in this exercise, occurs because solute particles disrupt the escape of solvent molecules into the vapor phase, as explained by Raoult’s Law. The presence of non-volatile solute particles like sucrose decreases the evaporation tendency, thus lowering the overall vapor pressure.

• Key aspects of colligative properties:
  • Depend only on the solute particle number.
  • Are independent of the solute’s identity.
  • Affect boiling and freezing points, as well as vapor pressure.

Understanding colligative properties is crucial because they directly link the physical change in a solution to the concentration of solute particles. This relationship is especially important in calculating changes in physical properties which are often observed in chemistry and various industrial applications.

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It is the ratio of moles of one component to the total moles of all components in the mixture. For solutions, it is often used to describe the concentration of solute or solvent.
The significance of the mole fraction lies in its role in Raoult’s Law, where it is used to calculate changes in vapor pressure of the solvent upon solute addition.

• The formula for calculating mole fraction is:\[X_{component} = \frac{\text{moles of component}}{\text{total moles in solution}}\]
  where:
  • \(X_{component}\) is the mole fraction of either solute or solvent.
  • "Moles of component" refers to the number of moles of the solute or solvent being considered.

Since the mole fraction is a ratio, it is dimensionless and always lies between 0 and 1. In this exercise, the mole fraction of sucrose is essential for applying Raoult's Law to determine vapor pressure reduction.

Sucrose

Sucrose is a common non-volatile solute used in the study of colligative properties. It is a carbohydrate with the chemical formula \(\mathrm{C}_{12}\mathrm{H}_{22}\mathrm{O}_{11}\). Being non-volatile means it does not easily vaporize, making it ideal for exercises that require the application of Raoult's Law to demonstrate vapor pressure lowering.
In the context of this exercise, the focus is on how sucrose affects the vapor pressure of water. When sucrose is dissolved in water, it inserts itself between water molecules, causing fewer water molecules to escape into the vapor phase, thus reducing the vapor pressure.

• Relevant points about sucrose in this context:
  • Sucrose is non-volatile, suitable for colligative property studies.
  • It has a molar mass of approximately \(342.3 \text{ g/mol}\), which is significant in calculating the mass required for desired vapor pressure changes.

Through understanding sucrose’s role, students can grasp how non-volatile solutes such as sucrose influence solution properties and apply these concepts to practical calculations.