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 \(180 \mathrm{~mm}\).

Step 1: Determine Moles of Each Gas

First, calculate the moles of each gas using the formula \( n = \frac{m}{M} \) where \( n \) is the moles of gas, \( m \) is the mass, and \( M \) is the molar mass. The molar masses are approximately \( 28 \mathrm{~g/mol} \) for \( \mathrm{N}_2 \), \( 44 \mathrm{~g/mol} \) for \( \mathrm{CO}_2 \), and \( 16 \mathrm{~g/mol} \) for \( \mathrm{CH}_4 \). - Moles of \( \mathrm{N}_2: \) \( \frac{56}{28} = 2 \mathrm{~mol} \) - Moles of \( \mathrm{CO}_2: \) \( \frac{44}{44} = 1 \mathrm{~mol} \) - Moles of \( \mathrm{CH}_4: \) \( \frac{16}{16} = 1 \mathrm{~mol} \)

Step 2: Calculate Total Moles of Gas

Add the moles of each gas to find the total moles of gas in the mixture. - Total moles \( = 2 + 1 + 1 = 4 \mathrm{~mol} \)

Step 3: Find Mole Fraction of Methane

The mole fraction of \( \mathrm{CH}_4 \) is given by the formula \( X_{\mathrm{CH}_4} = \frac{\text{moles of } \mathrm{CH}_4}{\text{total moles}} \).- \( X_{\mathrm{CH}_4} = \frac{1}{4} = 0.25 \)

Step 4: Calculate Partial Pressure of Methane

Use the formula for partial pressure given the total pressure and mole fraction: \( P_{\mathrm{CH}_4} = X_{\mathrm{CH}_4} \times P_{\text{total}} \).- \( P_{\mathrm{CH}_4} = 0.25 \times 720 = 180 \mathrm{~mm} \)

Key Concepts

Mole Fraction

In any gaseous mixture, understanding the composition is crucial. One way to express this is through the mole fraction. The mole fraction is essentially a way to describe the ratio of the moles of one component to the total moles of all components in the mixture. It's dimensionless and crucial for calculating properties like partial pressures.
To calculate the mole fraction of a component, use:

• Identify the moles of the given component. For example, if you have 1 mole of methane (\(CH_4\)), that is your numerator.
• Calculate the total moles in the mixture by summing up the moles of all gases included. Given our example of methane, nitrogen, and carbon dioxide, you would add their moles: 2 mol of \(N_2\), 1 mol of \(CO_2\), and 1 mol of \(CH_4\), leading to 4 moles in total.
• The mole fraction for methane is then calculated as \(\frac{1}{4}\).

This ratio tells you that methane constitutes 25% of the total number of moles in the mixture, which is incredibly helpful for further calculations.

Ideal Gas Law

The Ideal Gas Law is a cornerstone of gas behavior in chemistry, relating pressure, volume, temperature, and moles of a gas. The formula can be expressed as:\[PV = nRT\]where:

• \(P\) is the pressure of the gas, often in atmospheres, but sometimes in mmHg as in our problem.
• \(V\) is the volume the gas occupies.
• \(n\) is the number of moles, telling us how much gas is present.
• \(R\) is the ideal gas constant \(0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1}\).
• \(T\) is the absolute temperature in Kelvin, crucial for making sense of the gas's kinetic energy.

While this exercise focuses on partial pressures, the Ideal Gas Law binds everything we calculate here. It shows why moles and pressure relationships matter and how they predict behaviors under different conditions.

Molar Mass Calculation

Calculating molar mass is foundational for determining the amount of substance involved in reactions and gas behaviors. To find the molar mass of a compound, sum the atomic masses of all atoms in its formula:

• Nitrogen's molar mass: Two nitrogen atoms give \(2 \times 14 = 28 \text{ g/mol}\).
• Carbon dioxide's molar mass: Add the atomic masses of one carbon and two oxygen atoms \((12 + 2 \times 16 = 44 \text{ g/mol})\).
• Methane's molar mass: Carbon and four hydrogen atoms yield \((12 + 4 \times 1 = 16 \text{ g/mol})\).

Understanding molar mass allows us to convert between grams and moles, a vital step in chemistry calculations. It provides the bridge between the mass of a substance and the number of moles, crucial in applying the Ideal Gas Law and calculating mole fractions. Accurate molar mass ensures that results, like identifying partial pressures, remain reliable.