Problem 26 - A Complete Resource Book in Chemistry for JEE Main

Question

At \(27^{\circ} \mathrm{C}\), a closed vessel contains a mixture of equal weights of helium \((\) mol. \(w t=4)\), methane \((\) mol. \(w t=16\) ) and sulphur dioxide (mol. wt =64). The pressure exerted by the mixture is \(210 \mathrm{~mm}\). If the partial pressures of helium, methane and sulphur dioxide are \(\mathrm{P}_{1}, \mathrm{P}_{2}\) and \(\mathrm{P}_{3}\) respectively, which one of the following is correct? (a) \(\mathrm{P}_{3}>\mathrm{P}_{2}>\mathrm{P}_{1}\) (b) \(\mathrm{P}_{1}>\mathrm{P}_{2}>\mathrm{P}_{3}\) (c) \(\mathrm{P}_{1}>\mathrm{P}_{3}>\mathrm{P}_{2}\) (d) \(\mathrm{P}_{2}>\mathrm{P}_{3}>\mathrm{P}_{1}\)

Step-by-Step Solution

Answer

(b) \( P_1 > P_2 > P_3 \)

Step 1: Determine Moles of Each Gas

Since we have equal weights of each gas, we can find the number of moles of each using the formula: \( n = \frac{\text{weight}}{\text{molar mass}} \). Denoting equal weights as \( w \) for each gas, we have: - Moles of He \( (n_1) = \frac{w}{4} \) - Moles of CH₄ \( (n_2) = \frac{w}{16} \) - Moles of SO₂ \( (n_3) = \frac{w}{64} \) This shows that the number of moles is inversely proportional to the molar mass.

Step 2: Calculate Mole Fraction

The mole fraction of a gas in a mixture is the ratio of the moles of that gas to the total moles of the mixture. Calculate the total moles: \[ n_{\text{total}} = n_1 + n_2 + n_3 = \frac{w}{4} + \frac{w}{16} + \frac{w}{64} \] Simplifying, we find \( n_{\text{total}} = \frac{16w + 4w + w}{64} = \frac{21w}{64} \). Mole fractions are: \[ x_1 = \frac{n_1}{n_{\text{total}}} = \frac{16}{21}, \; x_2 = \frac{n_2}{n_{\text{total}}} = \frac{4}{21}, \; x_3 = \frac{n_3}{n_{\text{total}}} = \frac{1}{21} \].

Step 3: Apply Dalton's Law of Partial Pressures

Dalton's Law states that the total pressure exerted by a gaseous mixture is the sum of the partial pressures of each individual gas. The partial pressure of each gas can be found using \( P_i = x_i \cdot P_{\text{total}} \). Given \( P_{\text{total}} = 210 \text{ mm} \), we have: - \( P_1 = \frac{16}{21} \times 210 \) - \( P_2 = \frac{4}{21} \times 210 \) - \( P_3 = \frac{1}{21} \times 210 \).

Step 4: Compare Partial Pressures

Calculating each partial pressure gives: - \( P_1 = 160 \text{ mm} \) - \( P_2 = 40 \text{ mm} \) - \( P_3 = 10 \text{ mm} \). Arranging these in decreasing order, we get \( P_1 > P_2 > P_3 \). So, option (b) is the correct choice.

Key Concepts

Mole Fraction

In a gaseous mixture, the mole fraction of each gas is an important concept. The mole fraction, denoted as \( x_i \), is defined as the ratio of the number of moles of a particular component to the total number of moles in the mixture. It provides insight into how much of a particular gas is present compared to the entire mixture. The mole fraction is a pure number, meaning it has no units.

To calculate the mole fraction, consider the number of moles of helium, methane, and sulfur dioxide. For each gas, we divide the number of moles of that gas by the total number of moles in the mixture. This gives us:

• Moles of helium: \( n_1 = \frac{w}{4} \)
• Moles of methane: \( n_2 = \frac{w}{16} \)
• Moles of sulfur dioxide: \( n_3 = \frac{w}{64} \)

The total moles in the mixture \( n_{\text{total}} \) is obtained by summing these values: \[ n_{\text{total}} = \frac{w}{4} + \frac{w}{16} + \frac{w}{64} = \frac{21w}{64} \]Using these calculations, we derive the mole fractions:

• Helium: \( x_1 = \frac{16}{21} \)
• Methane: \( x_2 = \frac{4}{21} \)
• Sulfur Dioxide: \( x_3 = \frac{1}{21} \)

Partial Pressure

Partial pressure is the pressure exerted by a single component in a mixture of gases if it alone occupied the entire volume at the same temperature. According to Dalton's Law of Partial Pressures, the total pressure of a gas mixture is the sum of the partial pressures of each individual gas. This means that each gas contributes to the total pressure proportionally based on its mole fraction.

In the given exercise, the total pressure of the gaseous mixture is 210 mm. By multiplying the mole fraction of each gas by the total pressure, we can find the partial pressure:

• Helium: \( P_1 = x_1 \cdot P_{\text{total}} = \frac{16}{21} \times 210 \)
• Methane: \( P_2 = x_2 \cdot P_{\text{total}} = \frac{4}{21} \times 210 \)
• Sulfur Dioxide: \( P_3 = x_3 \cdot P_{\text{total}} = \frac{1}{21} \times 210 \)

This calculation gives us the precise contribution of each gas to the total pressure, showing how the different gases interact within the mixture.

Gaseous Mixtures

A gaseous mixture consists of two or more different gases that share the same physical space while each retains its own chemical properties. In these mixtures, the components do not react chemically but mix physically. Understanding gaseous mixtures is essential in chemistry, especially in contexts where gases differ significantly in their properties such as molar masses and behaviors at varying conditions.

The exercise provided an example of a gaseous mixture consisting of equal weights of helium, methane, and sulfur dioxide. These gases have different molar masses:

• Helium with a molar mass of 4 g/mol
• Methane with a molar mass of 16 g/mol
• Sulfur dioxide with a molar mass of 64 g/mol

By understanding the principles governing gaseous mixtures, such as mole fraction and partial pressure, insights can be gained into the behavior of each gas within the mixture and how they collectively contribute to phenomena such as pressure. Knowing these principles can be crucial in practical applications, from industrial processes to laboratory experiments, and aid in predicting outcomes under various temperature and pressure conditions.