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 6 atm for oxygen and 4 atm for nitrogen, so the answer is (c) 6,4.

Step 1: Determine Moles of Each Gas

First, calculate the number of moles of nitrogen and oxygen. Use the formula \( n = \frac{m}{M} \), where \( m \) is mass and \( M \) is the molar mass. For nitrogen (\( N_2 \)), \( M = 28 \mathrm{~g/mol} \); therefore, \( n_{N_2} = \frac{56}{28} = 2 \mathrm{~mol} \). For oxygen (\( O_2 \)), \( M = 32 \mathrm{~g/mol} \); hence, \( n_{O_2} = \frac{96}{32} = 3 \mathrm{~mol} \).

Step 2: Calculate Total Moles

Add the moles of nitrogen and oxygen to find the total moles of the gas mixture. Total moles \( n_{total} = n_{N_2} + n_{O_2} = 2 + 3 = 5 \mathrm{~moles} \).

Step 3: Calculate Mole Fraction for Each Gas

The mole fraction for each gas is determined using \( x_i = \frac{n_i}{n_{total}} \). For nitrogen, \( x_{N_2} = \frac{2}{5} = 0.4 \), and for oxygen, \( x_{O_2} = \frac{3}{5} = 0.6 \).

Step 4: Calculate Partial Pressure of Each Gas

Using the mole fraction and total pressure \( P_{total} = 10 \mathrm{~atm} \), the partial pressures are calculated by \( P_i = x_i \cdot P_{total} \). For nitrogen, \( P_{N_2} = 0.4 \times 10 = 4 \mathrm{~atm} \), and for oxygen, \( P_{O_2} = 0.6 \times 10 = 6 \mathrm{~atm} \).

Key Concepts

Mole Fraction

When dealing with gas mixtures, the **mole fraction** is a useful component. It offers insight into the contribution of each gas to the entire mixture. The mole fraction, often represented by the symbol \( x_i \), indicates the ratio of the moles of a specific component to the total moles of the mixture.
This is calculated using the formula:

• \( x_i = \frac{n_i}{n_{total}} \)

Here, \( n_i \) is the number of moles of the particular gas and \( n_{total} \) is the total moles of all gases in the mixture.
In our example, the mole fraction of nitrogen was found to be 0.4, while oxygen exhibited a mole fraction of 0.6.
Understanding and calculating mole fractions is crucial because it directly links to the partial pressure of gases, which determines how each gas behaves in a mixture under given conditions.

Molar Mass Calculation

Calculating the **molar mass** is a fundamental step in chemistry, particularly when working with gas mixtures. Molar mass refers to the mass of one mole of a given substance, denominated in grams per mole (g/mol). This is essential for converting mass quantities into the more versatile mole count. To determine the molar mass, it's important to refer to the periodic table, summing up the atomic masses of all atoms present in the molecule.
In our problem, we dealt with nitrogen and oxygen gases:

• Nitrogen (\( N_2 \)): Compound of two nitrogen atoms, each having an approximate atomic mass of 14, giving us a molar mass of 28 g/mol.
• Oxygen (\( O_2 \)): Consists of two oxygen atoms, each with an atomic mass of approximately 16, resulting in a molar mass of 32 g/mol.

Understanding molar masses allows us to transition from a mass-based view to a mole-based one, enhancing our comprehension of how substances interact and transform in chemical reactions.

Gas Mixture

In any **gas mixture**, understanding the behavior and properties of the constituent gases is vital. A mixture's properties are often determined by the proportions and pressures of its components, making calculations of partial pressures and mole fractions essential.Consider a gas mixture under a constant temperature and pressure, often termed isothermal conditions. In such scenarios, individual gases in the mixture behave independently to contribute to the total pressure. This concept follows Dalton's Law of Partial Pressures, which states:

• The total pressure exerted by a gas mixture is the sum of the partial pressures of each individual gas.

Given total pressure and the mole fractions, we can calculate the partial pressures using the formula:

• \( P_i = x_i \cdot P_{total} \)

In our example, the total pressure was \(10 \text{ atm}\). Utilizing mole fractions, we determined partial pressures for nitrogen and oxygen as \(4 \text{ atm}\) and \(6 \text{ atm}\), respectively.
This method highlights how gases interact within mixtures, and predicting their behavior is pivotal in fields as diverse as chemistry, environmental science, and engineering.