Problem 9 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The vapour pressures of pure liquids \(\mathrm{A}\) and \(\mathrm{B}\) are 400 and \(600 \mathrm{mmHg}\), respectively at \(298 \mathrm{~K}\). On mixing the two liquids, the sum of their initial volumes is equal to the volume of the final mixture. The mole fraction of liquid \(\mathrm{B}\) is \(0.5\) in the mixture. The vapour pressure of the final solution, the mole fractions of components \(\mathrm{A}\) and \(\mathrm{B}\) in vapour phase, respectively are: (a) \(450 \mathrm{mmHg}, 0.4,0.6\) (b) \(500 \mathrm{mmHg}, 0.5,0.5\) (c) \(450 \mathrm{mmHg}, 0.5,0.5\) (d) \(500 \mathrm{mmHg}, 0.4,06\)

Step-by-Step Solution

Answer

(d) 500 mmHg, 0.4, 0.6

Step 1: Understanding Raoult's Law

Raoult's Law states that the partial vapor pressure of each component in a solution is equal to the product of the vapor pressure of the pure component and its mole fraction in the mixture. For a binary solution, the total vapor pressure is given by: \[ P = P_A^0 \cdot \chi_A + P_B^0 \cdot \chi_B \]where \( P_A^0 \) and \( P_B^0 \) are the vapor pressures of pure liquids \( A \) and \( B \) respectively, and \( \chi_A \) and \( \chi_B \) are their mole fractions in the liquid phase.

Step 2: Calculate Total Vapor Pressure

Given the mole fraction of \( B \) is \( 0.5 \), the mole fraction of \( A \) must also be \( 0.5 \) since this is a binary mixture. Using the provided vapor pressures:\( P_A^0 = 400 \text{ mmHg} \) and \( P_B^0 = 600 \text{ mmHg} \), we can substitute into Raoult's Law:\[ P = 400 \cdot 0.5 + 600 \cdot 0.5 = 200 + 300 = 500 \text{ mmHg} \]

Step 3: Calculate Mole Fractions in Vapor Phase

Using Dalton's Law of Partial Pressures, the mole fraction of \( A \) in the vapor phase \( \chi_A' \) is:\[ \chi_A' = \frac{P_A}{P} = \frac{400 \cdot 0.5}{500} = \frac{200}{500} = 0.4 \]Similarly, the mole fraction of \( B \) in the vapor phase \( \chi_B' \) is:\[ \chi_B' = \frac{P_B}{P} = \frac{600 \cdot 0.5}{500} = \frac{300}{500} = 0.6 \]

Key Concepts

Vapor Pressure

Vapor pressure is a key concept in chemistry, particularly in the study of solutions and their properties. It represents the pressure exerted by a vapor in equilibrium with its liquid or solid phase at a given temperature. This is critical for understanding how substances behave when mixed. For instance:

• The vapor pressure of a liquid increases with temperature, which means at higher temperatures, more molecules have sufficient energy to escape into the vapor phase.
• Pure substances have characteristic vapor pressures, such as liquid A having 400 mmHg and liquid B 600 mmHg at 298 K, as mentioned in the exercise.
• In a mixture, each component contributes to the overall vapor pressure in proportion to its mole fraction and its pure vapor pressure. This is a foundational point of Raoult's Law.

This concept helps us determine the conditions under which a liquid will boil or evaporate at a specific temperature and how different mixtures affect these properties. Understanding is key for applications like distillation and refining.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It's calculated by dividing the number of moles of a component by the total number of moles of all components in the mixture. This is a dimensionless quantity and is expressed as a number between 0 and 1.

• In the given problem, the mole fraction of liquid B is provided as 0.5, implying that half of the mixture's moles are B, making it an equally concentrated binary mixture with A.
• The significance of mole fraction extends beyond just concentration; it also directly impacts the vapor pressure in a solution through Raoult's Law, as it's multiplied by the pure component's vapor pressure to determine its partial vapor pressure.

By understanding mole fractions, we can predict how a component behaves in the mixture, such as its likelihood to evaporate, which ties back to vapor pressure and influences the distribution of components between phases.

Dalton's Law

Dalton's Law of Partial Pressures is a principle referring to gas mixtures, where the total pressure exerted by a gaseous mixture is equal to the sum of the partial pressures of each individual gas in the mixture.

• Partial pressure, in this context, is the pressure that a component in the gas mixture would exert if it alone occupied the entire volume of the mixture at the same temperature.
• In the exercise problem, Dalton’s Law helps determine the mole fractions of components A and B in the vapor phase, by relating their partial pressures to the total vapor pressure.
• Using the calculated partial pressures: 200 mmHg for A and 300 mmHg for B, you find the mole fractions in the vapor as 0.4 and 0.6 respectively, confirming the individual contributions to the vapor pressure.

This law is essential for analyzing gas behavior in scientific and engineering applications, ensuring solutions are both understood and predicted accurately, particularly when dealing with mixtures.