Problem 25 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Equal weights of methane and hydrogen are mixed in an empty container at \(25^{\circ} \mathrm{C}\). The fraction of the total pressure exerted by hydrogen is : (a) \(\frac{1}{2}\) (b) \(\frac{8}{9}\) (c) \(\frac{1}{9}\) (d) \(\frac{16}{17}\)

Step-by-Step Solution

Answer

The fraction of the total pressure exerted by hydrogen is \( \frac{8}{9} \).

Step 1: Understand the problem

We need to find out what fraction of the total pressure in a container is due to hydrogen when equal weights of methane (CH₄) and hydrogen (H₂) gases are mixed at a constant temperature of 25°C.

Step 2: Calculate moles of each gas

Given that the weights of methane and hydrogen are equal, let's assume each has weight 1 gram for simplicity. - Molar mass of methane (CH₄) = 16 g/mol. Therefore, moles of CH₄ = \( \frac{1}{16} \) mol.- Molar mass of hydrogen (H₂) = 2 g/mol. Therefore, moles of H₂ = \( \frac{1}{2} \) mol.

Step 3: Find mole fraction of each gas

Calculate the total number of moles in the mixture:- Total moles = \( \frac{1}{16} + \frac{1}{2} = \frac{1}{16} + \frac{8}{16} = \frac{9}{16} \) mol.- Mole fraction of hydrogen (X_H₂) = \( \frac{\text{moles of H}_2}{\text{total moles}} = \frac{\frac{1}{2}}{\frac{9}{16}} = \frac{8}{9} \).The mole fraction of a gas in a mixture directly determines its contribution to the total pressure (assuming ideal behavior), hence the fraction of total pressure exerted by H₂ is \( \frac{8}{9} \).

Key Concepts

Mole Fraction

The concept of mole fraction revolves around understanding how each component in a mixture contributes to the whole. Mole fraction is essentially a way to express the concentration of a component within a mixture. It is a dimensionless quantity and is represented as the ratio of the number of moles of a particular component to the total moles of all components in the mixture.Let's break it down:

• Moles of a component: This is simply the amount of substance, calculated by dividing the mass of the substance by its molar mass.
• Total moles: The total of all moles present in the mixture.
• Mole Fraction Formula: If you have the moles of hydrogen as \( n_{H_2} \) and total moles as \( n_{total} \), then mole fraction \( X_{H_2} = \frac{n_{H_2}}{n_{total}} \).

This concept is crucial when dealing with gas mixtures because each component's contribution to the total pressure of the system is directly proportional to its mole fraction.

Partial Pressure

Partial pressure is a simple extension of the mole fraction concept into the realm of gases. In a mixture of gases, each gas contributes to the total pressure proportional to its mole fraction.This stems from Dalton’s Law of Partial Pressures, which states that the total pressure exerted by a gas mixture is equal to the sum of the partial pressures of each individual gas.When calculating partial pressure:

• Partial Pressure Formula: The partial pressure of a gas \( P_i \) is given by \( P_i = X_i \cdot P_{total} \), where \( X_i \) is the mole fraction of the gas and \( P_{total} \) is the total pressure.
• Understanding Contribution: Each gas in a mixture acts independently and contributes pressure as if it were alone in the container.

By understanding partial pressures, we can predict the behavior of gas mixtures more accurately, observing how the proportional concentration (mole fraction) affects the overall pressure.

Gas Mixtures

Gas mixtures naturally incorporate the concepts of mole fraction and partial pressure due to the ideal behavior of gases. Individual gases within a mixture do not interact strongly with each other, allowing them to be treated separately. Key aspects of gas mixtures:

• Non-reactive Behavior: Gases in a mixture generally do not react with each other unless specific conditions are met or unless they are reactive gases.
• Pressure Contribution: As gases mix, each one contributes to the total pressure, behaving as if it alone occupies the volume, making calculations straightforward with Dalton’s Law.
• Real Gas Considerations: While ideal gas laws work well under many conditions, deviations can occur under high pressure or low temperature.

In practice, understanding and manipulating gas mixtures by adjusting mole fractions and monitoring partial pressures enables industries from pharmaceuticals to aerospace to operate efficiently, ensuring desired outcomes in processes and reactions.