Question

At \(27^{\circ} \mathrm{C}\), a closed vessel contains a mixture of equal weights of helium (mol. wt \(=4\) ), methane (mol. wt \(=16\) ) 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) \(\mathrm{P}_{2}>\mathrm{P}_{3}>\mathrm{P}_{1}\)

Short answer

The correct option is (b) \( P_1 > P_2 > P_3 \).

Step-by-step solution

Identify Moles of Each Gas

Since the gases are present in equal weights, let's assume each gas has a mass of 1 gram. The number of moles for each gas is calculated using the formula: \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \). Thus, the moles of helium are \( \frac{1}{4} = 0.25 \), moles of methane are \( \frac{1}{16} = 0.0625 \), and moles of sulfur dioxide are \( \frac{1}{64} = 0.0156 \).

Calculate Total Moles in the Mixture

Add the moles of each gas calculated in Step 1 to find the total number of moles in the mixture.\[ 0.25 + 0.0625 + 0.0156 = 0.3281 \text{ moles} \]

Determine Mole Fraction of Each Gas

To find the mole fraction of each gas, divide the moles of each gas by the total moles: \[\begin{align*}\text{Mole fraction of helium} & = \frac{0.25}{0.3281}, \\text{Mole fraction of methane} & = \frac{0.0625}{0.3281}, \\text{Mole fraction of sulfur dioxide} & = \frac{0.0156}{0.3281}\end{align*}\]

Calculate Partial Pressure Using Mole Fraction

Each gas's partial pressure is found by multiplying its mole fraction by the total pressure (210 mm Hg):\[\begin{align*}P_1 & = 210 \times \frac{0.25}{0.3281}, \P_2 & = 210 \times \frac{0.0625}{0.3281}, \P_3 & = 210 \times \frac{0.0156}{0.3281}\end{align*}\]

Compare and Conclude on Relationships Between Pressures

Calculate and compare each partial pressure. The calculations show:- Helium \( P_1 \approx 160 \text{mm Hg} \)- Methane \( P_2 \approx 40 \text{mm Hg} \)- Sulfur dioxide \( P_3 \approx 10 \text{mm Hg} \)The relationship \( P_1 > P_2 > P_3 \).

Choose Correct Answer Option

Based on the comparison, the correct relationship is \( P_1 > P_2 > P_3 \), which corresponds to option (b).

Key concepts

Mole Fraction

The mole fraction is an important component in calculating gas properties in a mixture. It expresses the ratio of the number of moles of a given gas to the total number of moles in the mixture. This concept is especially handy when applying Dalton's Law of Partial Pressures.

To find the mole fraction of a gas, you use the formula:

• \[\text{Mole fraction} = \frac{\text{Moles of the gas}}{\text{Total moles in the mixture}}\]

In our original exercise, we have a mixture of helium, methane, and sulfur dioxide. By calculating the moles of each using their respective molar masses, we can determine:

• Helium: 0.25 moles
• Methane: 0.0625 moles
• Sulfur dioxide: 0.0156 moles

Adding and calculating these gives us the total moles, 0.3281. Each gas's mole fraction is then computed by dividing the moles of each by this total.

Ideal Gas Law

The Ideal Gas Law is fundamental in understanding the behavior of gases. Given that it relates the pressure, volume, temperature, and number of moles of a gas, it is formulated as:

• \[PV = nRT\]

Where:

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

In our specific problem, while the Ideal Gas Law is not directly applied due to the given conditions and data (a closed vessel with fixed volume and measured pressure), understanding this equation helps clarify why the sum of partial pressures can account for total pressure in accordance with Dalton’s Law. Each component gas behaves much like it would in isolation, thanks to the influence of this law.

Dalton's Law of Partial Pressures

Dalton's Law states that the total pressure exerted by a gaseous mixture is equal to the sum of the partial pressures of each individual component gas. This means if you know the amount of each gas in a mixture and the total pressure, you can determine each gas's partial pressure using its mole fraction.

Essentially, Dalton's Law can be described with the equation:

• \[P_{\text{total}} = P_1 + P_2 + P_3 + \ldots\]

Given the total pressure of 210 mm Hg, we multiply each gas's mole fraction by this total pressure to calculate their partial pressures:

• Helium: \( P_1 = 210 \times \frac{0.25}{0.3281} \)
• Methane: \( P_2 = 210 \times \frac{0.0625}{0.3281} \)
• Sulfur dioxide: \( P_3 = 210 \times \frac{0.0156}{0.3281} \)

By calculating these, we confirm the descending order of pressures: \( P_1 > P_2 > P_3 \), offering clarity on why option (b) was chosen as the answer.