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 \(1/3\), corresponding to option (b).

Step 1: Understanding the Problem

We are tasked with finding the fraction of the total pressure exerted by oxygen when equal masses of methane and oxygen are mixed. This involves using their molar masses and the concept of partial pressures.

Step 2: Calculate Moles of Gases

Given equal masses of methane and oxygen, we need to find moles. The molar mass of methane (CH₄) is approximately 16 g/mol, and for oxygen (O₂) it's approximately 32 g/mol. Thus, the ratio of moles is: \[ \text{Moles of CH}_4 : \text{Moles of O}_2 = \frac{m}{16} : \frac{m}{32} = 2:1 \] where \(m\) is the mass.

Step 3: Determine Mole Fraction

Since the total moles are \(2 + 1 = 3\), the mole fraction for oxygen \(O_2\) is: \[ \text{Mole fraction of } O_2 = \frac{\text{Moles of } O_2}{\text{Total Moles}} = \frac{1}{3} \]

Step 4: Relate Mole Fraction to Pressure

In a gas mixture, the pressure exerted by a component is proportional to its mole fraction. Therefore, the fraction of the total pressure exerted by oxygen will be equal to its mole fraction: \(\frac{1}{3}\).

Step 5: Verify Correct Option

Look for the option that matches our calculation. The fraction of the total pressure exerted by oxygen is \(\frac{1}{3}\), which corresponds to option (b).

Key Concepts

Partial Pressure

Partial pressure refers to the pressure that each gas in a mixture would exert if it were alone in the container. It's an essential concept in understanding how gases interact in a mixture. This principle states that the total pressure of a gas mixture is the sum of the partial pressures of each individual gas.

The partial pressure of a gas can be calculated using the Ideal Gas Law if we know the amount, volume, and temperature. In a simple gas mixture like the methane and oxygen example, \[ P_{\text{total}} = P_{\text{CH}_4} + P_{\text{O}_2} \]For our purpose, we can determine the partial pressure of oxygen by multiplying its mole fraction by the total pressure.

Mole Fraction

The mole fraction is how we express the concentration of a component within a mixture. It is defined as the ratio of the number of moles of a component to the total number of moles in the mixture.

For example, in our exercise with methane and oxygen, assuming equal masses, the mole fraction of oxygen (\[ X_{\text{O}_2} \]) was calculated as follows:

• Total moles = 3
• Moles of oxygen = 1
• Mole fraction of oxygen = \[ \frac{1}{3} \]

This ratio helps us determine the proportion of the total pressure that oxygen contributes in the gas mixture.

Molar Mass

Molar mass is the mass of one mole of a given substance, typically expressed in g/mol. It's a crucial factor when dealing with gas mixtures because it helps us convert between mass and moles. For example, knowing the molar mass of oxygen (O₂ is 32 g/mol) and methane (CH₄ is 16 g/mol) was vital in our previous step to calculate the number of moles based on the mass provided.

• Molar mass of methane = 16 g/mol
• Molar mass of oxygen = 32 g/mol

Using molar mass, we converted equal masses of methane and oxygen into moles, which were necessary for determining mole fractions and partial pressures.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that describes the relationships between pressure, volume, temperature, and the number of moles of a gas. The equation is presented as \[ PV = nRT \]where:

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

In our exercise, this law underpins the concept that the pressure a gas exerts is directly related to its number of moles when temperature and volume are held constant. While we did not directly apply the Ideal Gas Law equation in our steps, it conceptually supports why the mole fraction directly translates to the fraction of the total pressure. It is the reason why gases behave predictably under standard conditions.