Problem 108 - A Complete Resource Book in Chemistry for JEE Main

Question

Equal weights 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 (a) \(1 / 2\) (b) \(2 / 3\) (c) \(1 / 3 \times 273 / 298\) (d) \(1 / 3\)

Step-by-Step Solution

Answer

The fraction of total pressure exerted by oxygen is \(\frac{1}{3}\) (option d).

Step 1: Understanding the Problem

We are given equal weights of methane (5) and oxygen (9) mixed in a container, and we need to find the fraction of the total pressure exerted by oxygen.

Step 2: Calculate Moles of Methane and Oxygen

The molar mass of methane is 16 g/mol, and the molar mass of oxygen is 32 g/mol. Given the same mass of each, the number of moles of methane is double the number of moles of oxygen since methane is half as heavy.

Step 3: Determine Mole Fractions

If we have 1 mole of methane, we must have 0.5 moles of oxygen by weight equivalency. So, the mole fraction of oxygen, \(X_{O_2} = \frac{\text{moles of } O_2}{\text{moles of } CH_4 + \text{moles of } O_2} = \frac{0.5}{1 + 0.5} = \frac{1}{3}.\)

Step 4: Conclusion on Partial Pressure Fraction

The partial pressure is directly proportional to the mole fraction, thus the fraction of the total pressure exerted by oxygen is also \(\frac{1}{3}\).

Key Concepts

Mole Fraction

The concept of mole fraction is essential when dealing with mixtures of gases. It tells us how much of a particular component is present in the gas mixture compared to the total amount.
The mole fraction, represented by the symbol \( X \), is calculated using this equation:
\[ X_i = \frac{n_i}{n_{total}} \]where \( n_i \) is the number of moles of the component of interest, and \( n_{total} \) is the total number of moles of all components in the mixture.
In our case with methane and oxygen, you calculate the mole fraction of oxygen as \( X_{O_2} = \frac{0.5}{1 + 0.5} \), resulting in \( \frac{1}{3} \).
This means one-third of the moles in the container are oxygen. The mole fraction is useful for understanding other properties, such as partial pressure.

Partial Pressure

Partial pressure is the pressure that a gas in a mixture would exert if it were alone in the container. This concept helps us understand how individual gases contribute to the total pressure of a gas mixture.
According to Dalton's Law, the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of individual gases. The partial pressure of a gas \( P_i \) is given by:
\[ P_i = X_i \times P_{total} \]where \( X_i \) is the mole fraction, and \( P_{total} \) is the total pressure.
In this exercise, since the mole fraction of oxygen is \( \frac{1}{3} \), the partial pressure it exerts is also \( \frac{1}{3} \) of the total pressure. This demonstrates that the partial pressure is directly proportional to the gas's mole fraction.

Ideal Gas Law

The ideal gas law is a fundamental equation in describing the behavior of gases. It relates the pressure, volume, temperature, and number of moles of a gas. The ideal gas law is expressed as:
\[ PV = nRT \]where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume of the container.
• \( n \) is the number of moles of the gas.
• \( R \) is the ideal gas constant (approximately 0.0821 L·atm/mol·K).
• \( T \) is the temperature in Kelvin.

The ideal gas law assumes that gases behave perfectly, meaning their molecules have no volume, and no forces act between them. This law is instrumental in deriving the relationships for partial pressures and mole fractions in gas mixtures, as it implies a constant relationship among volume, pressure, temperature, and amount of gas.
Although real gases may deviate from this behavior, especially under high pressure and low temperature, the ideal gas law serves as an excellent approximation for many practical situations. In the context of our exercise, while we didn't directly use the ideal gas law, it underpins the principles that allow us to understand the behavior of the gas mixture in terms of mole fractions and partial pressures.