Problem 84 - A Complete Resource Book in Chemistry for JEE Main

Question

Equal masses of methane and hydrogen are mixed in an empty container at \(25^{\circ} \mathrm{C}\). The fraction of the total pressure exerted by hydrogen is (a) \(1 / 2\) (b) \(8 / 9\) (c) \(1 / 9\) (d) \(16 / 17\)

Step-by-Step Solution

Answer

The fraction of total pressure exerted by hydrogen is \(8/9\) (option b).

Step 1: Identifying the components and formulae

The gases in the mixture are methane (\(\text{CH}_4\)) and hydrogen (\(\text{H}_2\)). We need to consider their molar masses: Methane has a molar mass of \(16 \, \text{g/mol}\) and hydrogen has a molar mass of \(2 \, \text{g/mol}\). The fraction of total pressure exerted by a gas can be calculated using its mole fraction, which is given by \(\frac{n_2}{n_1+n_2}\) where \(n_1\) and \(n_2\) are the moles of the individual gases.

Step 2: Calculating the moles of gases

Since equal masses of both gases are given, let's assume 1 gram of each. For methane, \(n_1 = \frac{1}{16}\) moles, and for hydrogen, \(n_2 = \frac{1}{2}\) moles. This is because the molar mass of hydrogen is much smaller than that of methane.

Step 3: Calculating the mole fraction and pressure fraction

Calculate the total moles: \( n_{\text{total}} = n_1 + n_2 = \frac{1}{16} + \frac{1}{2} \). This becomes \( \frac{1}{16} + \frac{8}{16} = \frac{9}{16}\) total moles. The mole fraction of hydrogen is \( \frac{n_2}{n_{\text{total}}} = \frac{\frac{1}{2}}{\frac{9}{16}} = \frac{8}{9}\). Thus, the fraction of the total pressure exerted by hydrogen is \(\frac{8}{9}\).

Key Concepts

Mole Fraction

When dealing with gas mixtures, the mole fraction is a crucial concept. It tells us the ratio of the number of moles of one gas to the total moles of all gases in the mixture. In formula form, it's represented as:\(\text{Mole Fraction of Component } i = \frac{n_i}{n_\text{total}}\)Where:

• \(n_i\) is the moles of the individual component.
• \(n_\text{total}\) is the total moles of all components.

The mole fraction is a dimensionless quantity and always less than or equal to 1. It can also be connected to how much each gas contributes to the total pressure of the gas mixture. Understanding this helps in calculating partial pressures.

Ideal Gas Law

The Ideal Gas Law is a key equation in chemistry that relates pressure, volume, temperature, and moles of a gas. It is expressed as:\(PV = nRT\)Where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume occupied by the gas.
• \(n\) is the number of moles.
• \(R\) is the ideal gas constant.
• \(T\) is the temperature in Kelvin.

This law assumes gases behave ideally, which means there are no interactions between the molecules and their volume is negligible. This is a perfect approximation for many gases under typical conditions.

Molar Mass Calculation

Determining the molar mass of a gas is fundamental when solving problems involving gas mixtures. Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). To calculate the moles of any gas, use the formula:\(n = \frac{\text{mass}}{\text{molar mass}}\)Where:

• \(\text{mass}\) is the amount of the gas you're working with.
• \(\text{molar mass}\) is found by summing the atomic masses of the elements in a compound's formula.

For example, for methane (\(\text{CH}_4\)), it is \(16 \, \text{g/mol}\) and for hydrogen (\(\text{H}_2\)), it is \(2 \, \text{g/mol}\). This makes calculations of mole fractions simplified by comparing direct ratios of mass.

Partial Pressure

In a gas mixture, each gas contributes to the total pressure independently. This contribution is known as the partial pressure. To find it, multiply the mole fraction of the gas by the total pressure of the gas mixture:\(P_i = \chi_i P_\text{total}\)Where:

• \(P_i\) is the partial pressure of component \(i\).
• \(\chi_i\) is the mole fraction of component \(i\).
• \(P_\text{total}\) is the total pressure of the gas mixture.

Partial pressures help us understand the behavior of individual gases in a mixture. For instance, knowing partial pressures is vital when determining the total pressure exerted by mixed gases or predicting how a particular gas will behave in a reaction.