Problem 40 - A Complete Resource Book in Chemistry for JEE Main

Question

\(56 \mathrm{~g}\) of nitrogen and \(96 \mathrm{~g}\) of oxygen are mixed isothermally and at a total pressure of \(10 \mathrm{~atm}\). The partial pressures of oxygen and nitrogen (in \(\mathrm{atm}\) ) are respectively (a) 4,6 (b) 5,5 (c) 6,4 (d) 8,2

Step-by-Step Solution

Answer

The partial pressures are 4 atm for nitrogen and 6 atm for oxygen, option (a).

Step 1: Determine Molar Masses

First, we need to find the molar masses of nitrogen (N₂) and oxygen (O₂). The molar mass of N₂ is approximately 28 g/mol and for O₂ it is approximately 32 g/mol.

Step 2: Calculate Moles of Each Gas

Using the formula for moles, \( n = \frac{m}{M} \), where \( m \) is the mass and \( M \) is the molar mass, calculate the moles of nitrogen and oxygen. For nitrogen: \[ n_{N_2} = \frac{56 \, \text{g}}{28 \, \text{g/mol}} = 2 \, \text{mol} \]For oxygen: \[ n_{O_2} = \frac{96 \, \text{g}}{32 \, \text{g/mol}} = 3 \, \text{mol} \]

Step 3: Total Moles and Mole Fraction

Find the total moles of the gas mixture. \[ n_{\text{total}} = n_{N_2} + n_{O_2} = 2 + 3 = 5 \, \text{mol} \]Calculate the mole fractions for each gas:\[ \text{Mole fraction of } N_2, \chi_{N_2} = \frac{2}{5} = 0.4 \]\[ \text{Mole fraction of } O_2, \chi_{O_2} = \frac{3}{5} = 0.6 \]

Step 4: Calculate Partial Pressures

Use the formula \( P_i = \chi_i \times P_{\text{total}} \) to find the partial pressures.For nitrogen:\[ P_{N_2} = 0.4 \times 10 \, \text{atm} = 4 \, \text{atm} \]For oxygen:\[ P_{O_2} = 0.6 \times 10 \, \text{atm} = 6 \, \text{atm} \]

Step 5: Review and Conclusion

The calculated partial pressures are 4 atm for nitrogen and 6 atm for oxygen. Therefore, the correct option is (a) 4,6.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is an essential equation in chemistry that simplifies the study of gases. It relates four fundamental properties of a gaseous system: pressure (P), volume (V), temperature (T), and the number of moles (n). The formula is expressed as:\[ PV = nRT \]where \( R \) is the universal gas constant.This equation is incredibly useful because it allows us to predict how a gas will respond when its state conditions change, given that we know three out of the four variables. It assumes that the gas behaves ideally, meaning there are no interactions between gas molecules, and they occupy no volume. In the context of the exercise, although the Ideal Gas Law directly doesn't get used to calculate partial pressures, understanding it aids in comprehending the behavior of gases in mixtures. The exercise simplifies the scenario by providing constant pressure, hence focusing on mole calculations in a consistent temperature environment.

Mole Fraction

A mole fraction is a way to express the concentration of a component in a mixture. It's defined as the ratio of the number of moles of a component to the total number of moles in the mixture. It is denoted by \( \chi \) (chi).For any component in a mixture, the mole fraction \( \chi_i \) can be calculated using the formula:\[ \chi_i = \frac{n_i}{n_{\text{total}}} \]where \( n_i \) is the number of moles of the component, and \( n_{\text{total}} \) is the total moles of the mixture.In our problem, we determined the mole fractions of nitrogen and oxygen as:

• Mole fraction of nitrogen, \( \chi_{N_2} = 0.4 \)
• Mole fraction of oxygen, \( \chi_{O_2} = 0.6 \)

These fractions give a clear indication of how each gas contributes to the mixture. This is crucial because these mole fractions are directly used to compute the partial pressures of each gas using Dalton's Law. As such, understanding both concepts connect well to solve this problem.

Dalton's Law of Partial Pressures

Dalton's Law is a fundamental principle in the study of gas mixtures. It states that in a mixture of non-reacting gases, the total pressure exerted is the sum of the partial pressures of individual gases. This implies:\[ P_{\text{total}} = P_1 + P_2 + P_3 + ... + P_n \]Each \( P_i \) represents the partial pressure of a gas, which is the pressure the gas would exert if it alone occupied the entire volume.The formula used to calculate each gas's partial pressure in terms of the mole fraction \( \chi_i \), as seen in the exercise, is:\[ P_i = \chi_i \times P_{\text{total}} \]For the described exercise:

• The partial pressure of nitrogen was calculated as 4 atm, using \( \chi_{N_2} \times 10 \) atm.
• The partial pressure of oxygen was calculated as 6 atm, using \( \chi_{O_2} \times 10 \) atm.

This law is particularly significant because it simplifies how we calculate and predict the behavior of gases in mixtures without complex interaction considerations. This makes Dalton's Law an indispensable tool for chemists and students alike.