Question

Liquid A has vapor pressure \(x\), and liquid B has vapor pressure \(y\). What is the mole fraction of the liquid mixture if the vapor above the solution is \(30 . \%\) A by moles? \(50 . \%\) A? \(80 . \%\) A? (Calculate in terms of \(x\) and \(y .\) ) Liquid A has vapor pressure \(x\), liquid B has vapor pressure \(y\). What is the mole fraction of the vapor above the solution if the liquid mixture is \(30 . \%\) A by moles? \(50 . \%\) A? \(80 . \%\) A? (Calculate in terms of \(x\) and \(y .\) )

Short answer

First problem: Using the relationship between mole fractions in vapor and liquid phases: \(\frac{x_A^l}{x_B^l} = \frac{x_A^v \times y}{x_B^v \times x}\), and the given vapor mole fractions, we find the mole fractions of A in the liquid mixture as follows: Case 1: \(x_A^l = \frac{0.3y}{0.7x}\) Case 2: \(x_A^l = \frac{0.5y}{0.5x} = \frac{y}{x}\) Case 3: \(x_A^l = \frac{0.8y}{0.2x} = 4y\) Second problem: Using the relationship between mole fractions in liquid and vapor phases: \(\frac{x_A^v}{x_B^v} = \frac{x_A \times x}{x_B \times y}\), and the given liquid mole fractions, we find the mole fractions of A in the vapor phase as follows: Case 1: \(x_A^v = \frac{0.3x}{0.7y}\) Case 2: \(x_A^v = \frac{0.5x}{0.5y} = \frac{x}{y}\) Case 3: \(x_A^v = \frac{0.8x}{0.2y} = 4x\)

Step-by-step solution

Write Raoult's law equations for A and B

Write the expressions for partial pressures of A and B using Raoult's law given the mole fraction vapor phase: \(P_A = x_A \times x\) \(P_B = x_B \times y\) Here, \(x_A\) and \(x_B\) are the mole fractions of A and B in the liquid mixture.

Find the relationship between xA and xB in the vapor phase

The vapor phase has given mole fractions 30%, 50%, and 80% of A. We will use these values to find the relationship between xA and xB in the liquid mixture: \(x_A^v = 0.3, 0.5, 0.8\) Because mole fractions must add up to 1, we have \(x_B^v = 0.7, 0.5, 0.2\)

Use mole fraction ratios to find xA and xB in the liquid phase

The given mole fractions in the vapor phase (v) relate to the mole fractions in the liquid phase (l) as follows: \(\frac{x_A^l}{x_B^l} = \frac{x_A^v \times y}{x_B^v \times x}\) Use the mole fractions given in the vapor phase and the relationship above to find the mole fraction of A in the liquid mixture for each case. Second problem: To solve the second problem, we will follow these steps:

Write Raoult's law equations for A and B

Similarly to the first problem, we can write the Raoult's law equations: \(P_A = x_A \times x\) \(P_B = x_B \times y\) Here, \(x_A\) and \(x_B\) are the mole fractions of A and B in the liquid mixture.

Find the relationship between xA and xB in the liquid phase

The liquid phase has given mole fractions 30%, 50%, and 80% of A. We will use these values to find the relationship between xA and xB in the vapor phase: \(x_A = 0.3, 0.5, 0.8\) Because mole fractions must add up to 1, we have \(x_B = 0.7, 0.5, 0.2\)

Use mole fraction ratios to find xA and xB in the vapor phase

The given mole fractions in liquid phase (l) relate to the mole fractions in the vapor phase (v) as follows: \(\frac{x_A^v}{x_B^v} = \frac{x_A \times x}{x_B \times y}\) Use the mole fractions given in the liquid phase and the relationship above to find the mole fraction of A in the vapor phase for each case.

Key concepts

Vapor Pressure

Understanding vapor pressure is crucial when studying the behavior of liquids in different conditions. Vapor pressure refers to the pressure exerted by a vapor in equilibrium with its liquid or solid form at a given temperature. It's a measure of a liquid's tendency to evaporate, and it depends on both the substance and the temperature.

When a liquid is placed in a closed container, some of the molecules will evaporate and form vapor above the liquid. As the vapor accumulates, it exerts pressure on the walls of the container. Eventually, the rate of evaporation equals the rate of condensation, creating a dynamic equilibrium. At this point, the pressure exerted by the vapor remains constant and is specific to the temperature and the liquid. That constant pressure is the vapor pressure.

For any two volatile liquids, like the Liquid A and B in the textbook exercise, each has its own inherent vapor pressure denoted as 'x' for Liquid A and 'y' for Liquid B. These pressures can help us predict the composition of the vapor when these liquids are mixed.

Mole Fraction

A mole fraction is a way of expressing the concentration of a component in a mixture. It's defined as the ratio of the number of moles of a particular substance to the total number of moles in the mixture. To illustrate, if we have a solution made up of substances A and B, their mole fractions (\( x_A \text{ and } x_B \text{ respectively} \) ) would be calculated as follows:

\( x_A = \frac{\text{moles of A}}{\text{moles of A + moles of B}} \text{ and } x_B = \frac{\text{moles of B}}{\text{moles of A + moles of B}} \)

Since mole fractions are proportions, they must always add up to 1 (\( x_A + x_B = 1 \) ). This simple yet robust ratio can provide deep insights into the behavior of mixtures. In the exercise, we examine how the mole fraction in the liquid phase influences the vapor composition and vice versa.

Partial Pressure

In a mixture of gases, each component exerts a pressure as if it were alone in the container. This is known as the partial pressure. The total pressure of the mixture is the sum of the partial pressures of each gas. According to Raoult's law, the partial pressure (\( P_A \text{ or } P_B \) ) of a component in a solution is directly proportional to its mole fraction in the solution and to its pure vapor pressure.

\( P_A = x_A \times \text{vapor pressure of A} \text{ and } P_B = x_B \times \text{vapor pressure of B} \)

This relationship allows one to predict the behavior of a solution in terms of the pressures that the constituents would contribute to the overall pressure of the vapor. In the context of the textbook exercise, the mole fractions of A and B in the vapor (\( x_A^v \text{ and } x_B^v \) ) can be determined by using the partial pressures resulting from the mole fraction of A and B in the liquid state combined with the pure vapor pressures 'x' and 'y' of the respective liquids.