Problem 68 - Chemistry The Central Science In Si Units

Question

At \(20^{\circ} \mathrm{C}\), the vapor pressure of benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) is \(10 \mathrm{kPa}\), and that of toluene \(\left(\mathrm{C}_{7} \mathrm{H}_{8}\right)\) is \(2.9 \mathrm{kPa}\). Assume that benzene and toluene form an ideal solution. (a) What is the composition in mole fraction of a solution that has a vapor pressure of \(4.7 \mathrm{kPa}\) at \(20^{\circ} \mathrm{C} ?(\mathbf{b})\) What is the mole fraction of benzene in the vapor above the solution described in part (a)?

Step-by-Step Solution

Answer

To find the mole fraction of benzene and toluene, use Raoult's Law with given vapor pressures.

Step 1: Understanding the Problem

We have an ideal solution composed of benzene and toluene with given vapor pressures at a certain temperature. We need to determine the composition of the solution required to achieve a specific total vapor pressure and the composition of the vapor above this solution.

Step 2: Raoult's Law for Ideal Solution

According to Raoult's Law, the total vapor pressure of an ideal solution is the sum of the partial pressures of its components. Each component's partial pressure is its vapor pressure multiplied by its mole fraction in the liquid phase. Mathematically, this is represented as:\[ P_{ ext{total}} = P_{ ext{benzene}} + P_{ ext{toluene}} \]And, \[ P_{ ext{benzene}} = x_{ ext{benzene}} P^{ ext{benzene}} \]\[ P_{ ext{toluene}} = x_{ ext{toluene}} P^{ ext{toluene}} \]where \( P^{ ext{benzene}} = 10 \text{ kPa} \) and \( P^{ ext{toluene}} = 2.9 \text{ kPa} \).

Key Concepts

Ideal Solution

An ideal solution is a type of solution where the intermolecular interactions between components are similar to those within each pure component. This means that when forming a solution, a perfect mixture occurs without any change in volume or heat. Ideal solutions follow Raoult's Law precisely, meaning their total vapor pressure can be directly calculated based on the individual components' vapor pressures and mole fractions.

In an ideal solution, each molecule is surrounded by others of both types, and the forces between them are the same, whether the surroundings are benzene, toluene, or a mixture. This concept assumes no deviations from ideal behavior, as seen in real-world solutions where interactions can cause positive or negative deviations.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid phase at a given temperature. Every liquid has a specific vapor pressure that depends on the temperature and the nature of the liquid itself. In the context of Raoult's Law and ideal solutions, the vapor pressure of a component contributes to the total vapor pressure of the solution, which can be expressed as a sum of partial pressures.

For this exercise, the vapor pressure of benzene is 10 kPa, and for toluene, it is 2.9 kPa at 20°C. When forming a mixture of benzene and toluene into an ideal solution, Raoult's Law dictates that we can determine the total vapor pressure by using the individual vapor pressures and their respective mole fractions.

• The partial pressure of benzene in the vapor is given by: \( P_{\text{benzene}} = x_{\text{benzene}} P^{\text{benzene}} \)
• The partial pressure of toluene is expressed as: \( P_{\text{toluene}} = x_{\text{toluene}} P^{\text{toluene}} \)

These equations help us calculate how the components contribute to the mixture's total vapor pressure.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a solution. It is defined as the ratio of the number of moles of one component to the total number of moles in the solution. Mole fractions are dimensionless and are used in Raoult's Law to determine the contributions of different components to the total vapor pressure.

To calculate the mole fraction of benzene and toluene in the solution, you'll sum the partial pressures, which rely on these mole fractions:

• For benzene, the mole fraction \( x_{\text{benzene}} \) can be found using its partial pressure equation \( P_{\text{benzene}} = x_{\text{benzene}} P^{\text{benzene}} \).
• Similarly, \( x_{\text{toluene}} \) relates to the partial pressure of toluene.
• Since the solution is composed of only benzene and toluene, the sum of their mole fractions equals 1: \( x_{\text{benzene}} + x_{\text{toluene}} = 1 \).

Using these relationships allows us to find the composition of the solution by considering both the mole fractions and the resulting vapor pressures.