Problem 30 - A Complete Resource Book in Chemistry for JEE Main

Question

A gaseous mixture contains \(56 \mathrm{~g}\) of \(\mathrm{N}_{2}, 44 \mathrm{~g}\) of \(\mathrm{CO}_{2}\) and \(16 \mathrm{~g}\) of \(\mathrm{CH}_{4}\). The total pressure of mixture is \(720 \mathrm{~mm}\) of \(\mathrm{Hg}\). The partial pressure of methane is (a) \(75 \mathrm{~mm}\) (b) \(160 \mathrm{~mm}\) (c) \(180 \mathrm{~mm}\) (d) \(215 \mathrm{~mm}\)

Step-by-Step Solution

Answer

The partial pressure of methane is 180 mm, which corresponds to option (c).

Step 1: Calculate Moles of Each Gas

First, we need to calculate the number of moles of each gas in the mixture. We do that by dividing the mass of each gas by its respective molar mass.- Molar mass of \(\mathrm{N}_{2}\): \(28 \mathrm{~g/mol}\)- Moles of \(\mathrm{N}_{2}\): \( \frac{56}{28} = 2 \mathrm{~mol}\)- Molar mass of \(\mathrm{CO}_{2}\): \(44 \mathrm{~g/mol}\)- Moles of \(\mathrm{CO}_{2}\): \( \frac{44}{44} = 1 \mathrm{~mol}\)- Molar mass of \(\mathrm{CH}_{4}\): \(16 \mathrm{~g/mol}\)- Moles of \(\mathrm{CH}_{4}\): \( \frac{16}{16} = 1 \mathrm{~mol}\)

Step 2: Calculate the Total Moles in the Mixture

Add the moles of each gas to find the total moles in the gaseous mixture:- Total moles \(= 2 \mathrm{~mol} \ (\mathrm{N}_{2}) + 1 \mathrm{~mol} \ (\mathrm{CO}_{2}) + 1 \mathrm{~mol} \ (\mathrm{CH}_{4}) = 4 \mathrm{~mol}\)

Step 3: Calculate Mole Fraction of Methane \((\mathrm{CH}_{4})\)

The mole fraction of a gas is the ratio of the moles of that gas to the total moles in the mixture:- Mole fraction of \(\mathrm{CH}_{4}\): \( \frac{1}{4} = 0.25 \)

Step 4: Calculate Partial Pressure of Methane

Using Dalton's Law, which states that the partial pressure of a gas is the product of its mole fraction and the total pressure, we find:- Partial pressure of \(\mathrm{CH}_{4}\) = Mole fraction of \(\mathrm{CH}_{4}\) \( \times \) Total pressure\(\Rightarrow 0.25 \times 720 \mathrm{~mm} = 180 \mathrm{~mm} \)

Key Concepts

Mole Calculation

Understanding how to calculate moles is crucial when dealing with chemical reactions and gaseous mixtures. A mole is a unit that measures the amount of substance. It helps us quantify tiny entities like atoms, molecules, or ions in a large measurable number. When we want to calculate moles, the formula we use is:
\[ \text{Moles} = \frac{\text{Mass of substance (g)}}{\text{Molar mass (g/mol)}} \]Determining the moles of a gas involves taking the given mass of the gas and dividing it by its molar mass, which is specific to each type of molecule. For instance, in our problem, we have nitrogen (
6N_2), carbon dioxide (
6CO_2), and methane (
6CH_4). Each has its molar mass;

• Nitrogen (
  6N_2): 28 g/mol
• Carbon dioxide (
  6CO_2): 44 g/mol
• Methane (
  6CH_4): 16 g/mol

This straightforward process helps us lay the groundwork for further calculations, like pressure in a mixture.

Partial Pressure

Partial pressure is a critical concept when studying gas mixtures. According to Dalton's Law of Partial Pressures, each gas in a mixture exerts pressure independently as if it were the only gas present, contributing to the total pressure additively. Thus, each gas's partial pressure is dependent on its mole fraction in the mixture and the total pressure. The formula is simple yet powerful:
\[ \text{Partial Pressure of a gas} = \text{Mole fraction} \times \text{Total pressure} \]Let's break it down with our methane (
6CH_4) example: We calculated the mole fraction of methane to be 0.25. Given the total pressure of 720 mmHg, the partial pressure of methane is:

• 0.25
  6times 720 = 180 mmHg.

This formula allows us to identify individual gas pressures in a mix, essential for various scientific and industrial applications.

Gaseous Mixture

A gaseous mixture contains multiple gases, each contributing to the mixture's total properties based on its individual characteristics. Even though gases mix freely and uniformly, they behave independently according to Dalton's Law. This means that each gas in a mixture contributes to the total pressure of the system based on the amount of that gas present in mole terms.
For any mixture:

• Total Pressure = Sum of Partial Pressures of All Gases
• Total Moles = Sum of Moles of Each Gas

In practical terms, understanding gaseous mixtures allows chemists to determine how each component behaves under certain conditions. It explains why we can breathe air, a mixture mainly of nitrogen and oxygen. This mixture's behavior is predictable based on the principles of partial pressures and mole calculations, crucial for various fields like pharmacology, environmental science, and engineering.