Problem 153 - A Complete Resource Book in Chemistry for JEE Main

Question

Equal masses of methane and oxygen are mixed in an empty container at \(25^{\circ} \mathrm{C}\). The fraction of the total pressure exerted by oxygen is [2007] (a) \(1 / 3 \times 273 / 298\) (b) \(1 / 3\) (c) \(1 / 2\) (d) \(2 / 3\)

Step-by-Step Solution

Answer

The fraction of the total pressure exerted by oxygen is (b) \(1 / 3\).

Step 1: Understand the problem

We have equal masses of methane (CH extsubscript{4}) and oxygen (O extsubscript{2}) mixed in a container. We need to find the fraction of the total pressure exerted by oxygen.

Step 2: Calculate moles of methane and oxygen

First, calculate the moles of each gas. Use the molar masses: CH extsubscript{4} (approx. 16 g/mol) and O extsubscript{2} (approx. 32 g/mol). Given equal masses of both gases, let's assume the mass is 32 grams for simplicity. Then, moles of CH extsubscript{4} are \( \frac{32}{16} = 2 \) moles, and moles of O extsubscript{2} are \( \frac{32}{32} = 1 \) mole.

Step 3: Calculate the total number of moles

Add the moles of methane to the moles of oxygen to find the total moles in the container. Total moles = 2 moles (methane) + 1 mole (oxygen) = 3 moles.

Step 4: Determine the mole fraction of oxygen

The mole fraction of a gas is the ratio of its moles to the total moles of gas in the mixture. The mole fraction of oxygen is \( \frac{1}{3} \), since it represents 1 out of 3 total moles.

Step 5: Correlate mole fraction to pressure fraction

In an ideal gas mixture, the mole fraction of a component is directly proportional to its pressure fraction. Hence, the fraction of total pressure exerted by oxygen is also \( \frac{1}{3} \).

Key Concepts

Mole Fraction

When you mix different gases in a container, the mole fraction helps you understand the proportion of each gas in the mixture. It is calculated as the ratio of the moles of a specific gas to the total moles of all gases present. In our exercise, the mole fraction of oxygen is \[ \frac{1}{3} \], since its 1 mole out of the 3 total moles in the mixture.

• Moles of Methane: 2 moles
• Moles of Oxygen: 1 mole
• Total Moles: 3 moles

The mole fraction offers insight into the contribution of each gas to the overall number of particles in the mixture.

Ideal Gas Law

The Ideal Gas Law is an equation of state for a hypothetical gas, described by the formula \[ PV = nRT \]where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume.
• \( n \) is the number of moles.
• \( R \) is the ideal gas constant.
• \( T \) is the temperature in Kelvin.

This law allows scientists to predict the behavior of gases under different conditions. It assumes no interactions between molecules. While our exercise doesn't deeply rely on this equation, understanding it is key to grasping how gases behave in mixtures.

Partial Pressure

Partial pressure refers to the pressure that a single gas in a mixture of gases would exert if it occupied the entire volume alone. It's directly related to the mole fraction of the gas in the mixture. According to Dalton's Law of Partial Pressures:\[ P_{\text{total}} = P_1 + P_2 + \ldots \]where each \( P_i \) is the partial pressure of gas \( i \). In our exercise, the partial pressure of oxygen can be determined by:

• Finding its mole fraction: \( \frac{1}{3} \)
• Multiplying it by the total pressure

This means the oxygen exerts \( \frac{1}{3} \) of the total pressure in the container.

Stoichiometry

Stoichiometry is the calculation of reactants and products in chemical reactions. It provides the quantitative "map" for understanding how chemicals combine and react. In our case, stoichiometry isn't directly calculated because we're dealing with gases mixed, not reacting. However, stoichiometry is useful when these gases undergo reactions under specific conditions. Typically, stoichiometry involves:

• Balancing chemical equations
• Using molar ratios to convert between substances
• Predicting yields of products from given reactants

Though not central in this exercise, stoichiometry is a foundational concept for understanding mass and mole relationships in chemistry.