Problem 56 - Chemistry: An Atoms-Focused Approach

Question

A solution contains 4.5 mol of water, 0.3 mol of sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right),\) and 0.2 mol of glucose. Sucrose and glucose are nonvolatile. What is the mole fraction of water in the 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 approximately 37.98 torr.

Step 1: Calculate the total number of moles in the solution

Add the moles of each substance (water, sucrose, and glucose) together to find the total moles in the solution: Total moles = Moles of water + Moles of sucrose + Moles of glucose Total moles = 4.5 + 0.3 + 0.2 = 5.0 moles

Step 2: Calculate the mole fraction of water in the solution

Divide the moles of water by the total moles in the solution: Mole fraction of water = (Moles of water) / (Total moles) Mole fraction of water = 4.5 / 5.0 = 0.9

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

Raoult's Law states that the vapor pressure of a component in a solution is equal to the vapor pressure of the pure component multiplied by its mole fraction in the solution. 1. Write the Raoult's Law formula: Vapor pressure (P_solution) = Mole fraction of water × Vapor pressure of pure water (P_pure) 2. Plug in the given values: P_solution = 0.9 × 42.2 torr 3. Calculate the vapor pressure of the solution: P_solution = 37.98 torr The mole fraction of water in the solution is 0.9, and the vapor pressure of the solution at 35°C is approximately 37.98 torr.

Key Concepts

Mole Fraction

The mole fraction is a fundamental concept in solution chemistry. It is a way of expressing the concentration of a component in a mixture.
To calculate it, you divide the number of moles of the component by the total number of moles in the solution.
You can think of mole fraction as a percentage, but instead of being out of 100, it's on a scale from 0 to 1. Let's walk through an example. If you have 4.5 moles of water, 0.3 moles of sucrose, and 0.2 moles of glucose in a solution:

• total moles = 4.5 + 0.3 + 0.2 = 5.0 moles
• mole fraction of water = 4.5 / 5.0 = 0.9

Remember, the sum of all mole fractions in a solution is always equal to 1. This calculation helps in determining other properties of the solution, like vapor pressure.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phase at a given temperature.
It's an important property because it indicates how easily a liquid can evaporate. Every liquid has its own vapor pressure at a particular temperature. For example, pure water at 35°C has a vapor pressure of 42.2 torr.
When you have a solution, the presence of solutes affects the vapor pressure. If a solute is nonvolatile (does not easily evaporate), it lowers the vapor pressure of the solvent. According to Raoult's Law, the vapor pressure of a solution is the product of the mole fraction of the solvent and the vapor pressure of the pure solvent.

• For the above exercise: Vapor pressure of solution = 0.9 (mole fraction of water) × 42.2 torr (vapor pressure of pure water)
• This results in a solution vapor pressure of 37.98 torr

This demonstrates how solutes can lower the vapor pressure of a solvent.

Nonvolatile Solute

A nonvolatile solute is a substance that does not readily evaporate.
This is important in solution chemistry because such solutes can significantly alter the properties of a solvent. In our example, both sucrose and glucose are nonvolatile. When added to water, they reduce the number of water molecules that can escape into the vapor phase, effectively lowering the solution's vapor pressure. The presence of nonvolatile solutes in a solution affects its colligative properties, including:

• lowering the vapor pressure
• elevating the boiling point
• depressing the freezing point
• creating osmotic pressure

Understanding the behavior of nonvolatile solutes helps in applications such as cooking, food preservation, and chemical manufacturing.

Solution Chemistry

Solution chemistry is the study of how substances interact within a solution.
Here, we're particularly interested in mixtures where one or more solutes are dissolved in a solvent. Key features of solution chemistry include understanding:

• how solute and solvent molecules interact
• how properties like boiling point and vapor pressure are altered
• the calculation of concentrations through measures like mole fractions

In a solution, the solute does not settle out but remains uniformly distributed. This makes solutions essential for many chemical reactions and biological processes. Raoult's Law, as used in the exercise, is a crucial concept in solution chemistry. It provides insight into how mixing affects vapor pressure, which is vital in many scientific fields, such as pharmacology and environmental science.