Problem 36 - Chemistry The Science in Context

Question

A solution contains 4.5 moles of water, 0.3 moles of sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right),\) and 0.2 moles of glucose. Sucrose and glucose are nonvolatile. What is the mole fraction of water in this solution? What is the vapor pressure of the solution at \(35^{\circ} \mathrm{C},\) given that the vapor pressure of pure water at \(35^{\circ} \mathrm{C}\) is 42.2 torr?

Step-by-Step Solution

Answer

Answer: The mole fraction of water in the solution is 0.9, and the vapor pressure of the solution at 35°C is 37.98 torr.

Step 1: Calculate the total moles in the solution

To find the mole fraction of water, we first need to calculate the total moles of all components in the solution. Add the moles of water, sucrose, and glucose together: Total moles = Moles of water + Moles of sucrose + Moles of glucose

Step 2: Find the mole fraction of water

To find the mole fraction of water, divide the moles of water by the total moles in the solution. The mole fraction of water (X_water) can be calculated as: X_water = (Moles of water) / (Total moles)

Step 3: Use Raoult's Law to calculate the vapor pressure of the solution

Raoult's Law states that the vapor pressure of a solution is equal to the product of the mole fraction of the solvent (in this case, water) and the vapor pressure of the pure solvent. The vapor pressure of the solution (P_solution) can be calculated as: P_solution = X_water * P_pure_water where P_pure_water is the vapor pressure of pure water at 35°C (42.2 torr). Now that we have the steps outlined, let's perform the calculations.

Step 1: Calculate the total moles in the solution

Total moles = 4.5 moles (water) + 0.3 moles (sucrose) + 0.2 moles (glucose) = 5 moles

Step 2: Find the mole fraction of water

X_water = (4.5 moles) / (5 moles) = 0.9

Step 3: Calculate the vapor pressure of the solution

P_solution = 0.9 * 42.2 torr = 37.98 torr So, the mole fraction of water in the solution is 0.9, and the vapor pressure of the solution at 35°C is 37.98 torr.

Key Concepts

Raoult's Law

Raoult's Law is a fundamental principle in solution chemistry, which helps in understanding how the vapor pressure of a solution is influenced by the presence of solutes. In a nutshell, Raoult's Law states that the vapor pressure of a solution is directly proportional to the mole fraction of its solvent. Put simply, the rule applies to ideal solutions where solute particles do not contribute much to changes in the vapor pressure.

The formula used in Raoult's Law is given as:\[ P_{\text{solution}} = X_{\text{solvent}} \times P_{\text{pure solvent}} \]
where:

• \( P_{\text{solution}} \) is the vapor pressure of the solution.
• \( X_{\text{solvent}} \) is the mole fraction of the solvent, which, in most cases, is the major component in the solution.
• \( P_{\text{pure solvent}} \) is the vapor pressure of the pure liquid.

The point is to measure how much the presence of a solute reduces the vapor pressure compared to the pure solvent. When you dissolve nonvolatile substances like sucrose or glucose in water, they don't exert any significant vapor pressure themselves and thus the solution's vapor pressure is primarily a function of the solvent's mole fraction.
This principle is primarily applied to dilute solutions, and it provides key insights into determining various properties of solutions such as boiling points, melting points, and the colligative properties.

Vapor Pressure

Vapor pressure is a key concept in understanding the properties of liquids and solutions. It is defined as the pressure exerted by a vapor in equilibrium with its liquid at a particular temperature. Essentially, it is a measure of the tendency of molecules in the liquid phase to escape into the vapor phase.

At higher temperatures, molecules have more energy and are more likely to evaporate, resulting in higher vapor pressure. Conversely, at lower temperatures, there is less evaporation and therefore lower vapor pressure.

In solution chemistry, the vapor pressure of a solution changes with the addition of solutes. Nonvolatile solutes, such as sucrose and glucose, remain in the liquid phase and reduce the solution's vapor pressure. This is because they occupy space at the liquid's surface and reduce the number of solvent molecules that can escape into the atmosphere.

• Nonvolatile solutes don't vaporize easily, hence they have a minimal or zero vapor pressure.
• Volatile solutes contribute to the overall vapor pressure in a solution.

The change in vapor pressure due to solutes is crucial when calculating phenomena like boiling point elevation and freezing point depression, as they are directly linked to the solution’s vapor pressure characteristics. Understanding vapor pressures helps in the application of Raoult's Law to predict the behavior of solutions.

Solution Chemistry

Solution chemistry explores the interactions between different substances when they are mixed to form a solution. It focuses on understanding what happens when solutes dissolve in solvents, altering various properties of the mixture.

In the context of the given exercise, we are dealing with a binary solution system involving water (solvent) and nonvolatile solutes (sucrose and glucose). It helps in understanding how molecular interactions impact phenomena like dissolution, concentration, and the resultant physical properties.

Key points to note in solution chemistry include:

• Solutions are homogeneous mixtures, wherein the solute particles are uniformly distributed within the solvent.
• Concentration terms such as mole fraction, molarity, and molality describe how much solute is present in a given volume of solvent.
• The presence of solutes can alter physical properties of the solution, including boiling point, freezing point, and vapor pressure.
• Ideal solutions obey Raoult's Law, meaning the partial vapor pressure of each component is directly proportional to its mole fraction.

In our exercise, the mole fraction of water was used to apply Raoult's Law. But solution chemistry also assess product formation, reaction rates, and thermodynamics within solutions. It’s an essential part of chemistry that helps engineer everyday solutions—from food and drink to chemicals and pharmaceuticals.