Problem 24 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Liquids A and B form ideal solution for all compositions of \(\mathbf{A}\) and \(\mathbf{B}\) at \(25^{\circ} \mathrm{C}\). Two such solutions with \(0.25\) and \(0.50\) mole fractions of \(\mathbf{A}\) have the total vapor pressures of \(0.3\) and \(0.4\) bar, respectively. What is the vapor pressure of pure liquid \(\mathbf{B}\) in bar?

Step-by-Step Solution

Answer

The vapor pressure of pure liquid B is 0.2 bar.

Step 1: Apply Raoult's Law

For ideal solutions, Raoult's Law states that the vapor pressure of a component in a solution is equal to the mole fraction of that component times the vapor pressure of the pure component. The total vapor pressure is the sum of the partial pressures for all components. If \( P_A \) and \( P_B \) represent the vapor pressures of pure liquids A and B, then for a solution with mole fraction \( x_A = 0.25 \): \[ P_{total,1} = x_A P_A + x_B P_B = 0.3 \text{ bar} \] and for \( x_A = 0.50 \): \[ P_{total,2} = x_A P_A + x_B P_B = 0.4 \text{ bar} \] where \( x_B = 1 - x_A \).

Step 2: Write the Equations for Each Solution

Using Raoult's Law, write the equations for the total vapor pressures: For the first solution (\(x_A = 0.25\)): \[ 0.25 P_A + 0.75 P_B = 0.3 \] For the second solution (\(x_A = 0.50\)): \[ 0.50 P_A + 0.50 P_B = 0.4 \]

Step 3: Solve the System of Equations

You have the following simultaneous equations: Equation 1: \( 0.25 P_A + 0.75 P_B = 0.3 \) Equation 2: \( 0.50 P_A + 0.50 P_B = 0.4 \) First, solve Equation 2 for \( P_B \): \[ P_B = \frac{0.4 - 0.50 P_A}{0.50} = 0.8 - P_A \] Substitute \( P_B = 0.8 - P_A \) into Equation 1: \[ 0.25 P_A + 0.75(0.8 - P_A) = 0.3 \] \[ 0.25 P_A + 0.6 - 0.75 P_A = 0.3 \] \[ -0.5 P_A + 0.6 = 0.3 \] \[ -0.5 P_A = -0.3 \] \[ P_A = 0.6 \] Now, substitute back to find \( P_B \): \[ P_B = 0.8 - 0.6 = 0.2 \text{ bar} \]

Step 4: Verify Solution

Check that the found values satisfy the initial conditions. For \( P_A = 0.6 \text{ bar} \) and \( P_B = 0.2 \text{ bar} \): First solution: \[ 0.25 \times 0.6 + 0.75 \times 0.2 = 0.15 + 0.15 = 0.3 \text{ bar}\] Second solution: \[ 0.50 \times 0.6 + 0.50 \times 0.2 = 0.3 + 0.1 = 0.4 \text{ bar}\] Since both conditions hold true, the calculations are verified.

Key Concepts

Ideal Solutions

In chemistry, an ideal solution refers to a mixture of two or more components where the interactions between unlike molecules are similar to those between like molecules. This means that the forces between molecules of different components (e.g., A and B) are akin to those in the pure components themselves. Because of this similarity, the properties of the solution are often predictable using Raoult's Law.

An ideal solution is characterized by the absence of volume change or heat change upon mixing. For instance, if you mix liquid A and liquid B, the total volume and enthalpy do not change in an ideal solution.

Real solutions, however, can deviate from ideal behavior due to differences in interactions, which is measured by deviations in the vapor pressure or under thermodynamic analysis. Deviations occur when interactions between dissimilar molecules are either stronger or weaker than those between similar ones. Yet, for many dilute solutions, Raoult’s Law holds as a good approximation.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid phase. When a substance is in a closed system, some molecules escape from the liquid to form a gas above the liquid. The vapor pressure measures the tendency of particles to escape from the liquid phase to the gas phase.

In the context of an ideal solution, the presence of different components can affect the vapor pressure of the solution. The vapor pressure of each component is influenced by its mole fraction within the solution, according to Raoult's Law. This law states that the partial vapor pressure of each component is equal to the vapor pressure of the pure component times its mole fraction in the solution.

• The vapor pressure of a solution is the sum of the partial pressures of each component.
• This means if you know the vapor pressures of the pure components and the mole fractions, you can determine the total vapor pressure of the solution.

The concept of vapor pressure is key to understanding boiling point elevation and how substances behave when mixed, as liquids with higher vapor pressures tend to evaporate more easily.

Mole Fraction

The mole fraction is a way of expressing the composition of 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. For a solution with components A and B, the mole fraction of A, denoted as \( x_A \), is given by:\[ x_A = \frac{n_A}{n_A + n_B} \]where \( n_A \) and \( n_B \) are the number of moles of components A and B, respectively.

Mole fraction is a dimensionless quantity and always lies between 0 and 1. It provides a simple way to quantify the concentration of each component in a solution, and it is particularly useful in the application of Raoult's Law.

• In a binary mixture, the sum of mole fractions is always equal to 1: \( x_A + x_B = 1 \).
• Since mole fraction does not depend on temperature or pressure, it is a preferred way to express solution compositions compared to other concentration measures like molarity or molality.

The mole fraction has a direct effect on the vapor pressure of the components in a solution. Understanding and calculating mole fractions are essential to predict how different substances will behave when mixed.