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 Scenario: For the vapor mixture with $30\%$ A, using Raoult's Law: Liquid Mole Fraction: \[\chi_A = \frac{0.30y}{0.30y + 0.70x}\] and \[\chi_B = 1 - \chi_A\] For the vapor mixture with $50\%$ A: Liquid Mole Fraction: \[\chi_A = \frac{0.50x}{0.50x + 0.50y}\] and \[\chi_B = 1 - \chi_A\] For the vapor mixture with $80\%$ A: Liquid Mole Fraction: \[\chi_A = \frac{0.80x}{0.80x + 0.20y}\] and \[\chi_B = 1 - \chi_A\] Second Scenario: For the liquid mixture with $30\%$ A, using Raoult's Law: Vapor Mole Fraction: \[\chi_A = \frac{0.30x}{0.30x + 0.70y}\] and \[\chi_B = 1 - \chi_A\] For the liquid mixture with $50\%$ A: Vapor Mole Fraction: \[\chi_A = \frac{0.50x}{0.50x + 0.50y}\] and \[\chi_B = 1 - \chi_A\] For the liquid mixture with $80\%$ A: Vapor Mole Fraction: \[\chi_A = \frac{0.80x}{0.80x + 0.20y}\] and \[\chi_B = 1 - \chi_A\]

Step-by-step solution

Determine the initial mole fractions for vapor mixture

For each given mole percentage of A in vapor (30%, 50%, and 80%), we need to determine the mole fraction of both A and B in the vapor phase. Let's start by solving for the 30% A case, then 50% A and finally, 80% A case.

Use Raoult's Law to find mole fraction in the liquid solution for each percentage of A

For each mole percentage of A in vapor, we will use Raoult's Law to set up an equation and find the mole fraction in the liquid solution. Here, we will assume mole fraction of B in liquid is 1 to simplify calculations. We will calculate mole fraction for A and B in the liquid solution. Repeat these calculations for 50% and 80% mole percentage of A in vapor. Second, find the mole fraction of vapor mixture given the liquid percentage:

Determine the initial mole fractions for liquid mixture

For each given mole percentage of A in liquid (30%, 50%, and 80%), we need to determine the mole fraction of both A and B in the liquid phase. Let's start with the 30% A case, then 50% A, and finally, 80% A case.

Use Raoult's Law to find mole fraction in the vapor solution for each percentage of A

For each mole percentage of A in liquid, we will use Raoult's Law to set up an equation and find the mole fraction in the vapor phase. Here, we will assume mole fraction of B in vapor is 1 to simplify calculations. We will calculate mole fraction for A and B in the vapor solution. Repeat these calculations for 50% and 80% mole percentage of A in liquid.

Key concepts

Vapor Pressure

Vapor pressure is a key concept when discussing liquid mixtures and vapor solutions. It refers to the pressure exerted by the vapor in equilibrium with its liquid at a given temperature. Each liquid has its own unique vapor pressure, which depends on factors like temperature and the nature of the liquid itself. For example, liquid A might have a vapor pressure denoted as \( x \), while liquid B could have a vapor pressure of \( y \).
Understanding vapor pressure is crucial because it helps us predict how a liquid will behave when it's mixed with another. In liquid mixtures, the total vapor pressure is a combination of the vapor pressures of the individual components. This total vapor pressure is affected by the mole fractions of the liquids in the mixture, as described by Raoult's Law.
In essence, vapor pressure gives us insight into the volatility of a liquid; the higher the vapor pressure, the more easily a liquid will evaporate into a gas.

Mole Fraction

The mole fraction is a quantity that represents the concentration of a particular component in a mixture. It's defined as the ratio of the moles of one component to the total moles of all components in the mixture. This concept is essential because it allows us to quantify the composition of both liquid mixtures and vapor solutions.
When dealing with Raoult's Law, the mole fraction is used to determine how the vapor pressures of individual liquids contribute to the overall vapor pressure of the mixture. For liquid A in a mixture, its mole fraction can be expressed as \( X_A \), and for liquid B, it would be \( X_B \).
To apply Raoult's Law correctly, you must know these mole fractions. Whether you're calculating the composition of a vapor given a specific liquid mixture or vice versa, the mole fraction is a fundamental part of the calculations.

Liquid Mixture

A liquid mixture consists of two or more liquids mixed together. The composition of this mixture is often expressed in terms of the mole fraction. For instance, if you have a liquid mixture where 30% of it is liquid A by moles, this means 30% of the total number of molecules in the mixture are molecules of A.
When mixing liquids, the interaction between different molecules affects the physical properties of the mixture, such as its vapor pressure. Calculating the vapor pressure of a liquid mixture involves considering the individual vapor pressures and mole fractions of each component, as specified by Raoult's Law.
Knowing the composition of a liquid mixture allows us to predict how it will behave under different conditions and is crucial when calculating the resultant vapor pressure or separating components through processes like distillation.

Vapor Solution

A vapor solution is the gaseous phase that is in equilibrium with a liquid mixture. The composition of this vapor is described in terms of mole fractions, much like the liquid mixture itself. However, these mole fractions often differ from those in the liquid due to the varying volatility of the components.
Raoult's Law plays a pivotal role here. It helps determine the relationship between the mole fractions in the vapor phase and in the liquid phase. For a vapor solution, if liquid A is more volatile, it may have a higher mole fraction in the vapor than in the liquid, resulting in differing compositions.
This concept is important not only in theoretical chemistry but also in practical applications like distillation, where separating components based on differences in vapor pressures is key. Understanding vapor solutions allows us to manipulate and use mixtures effectively.