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

Option (b) \(\mathrm{P}_{1} > \mathrm{P}_{2} > \mathrm{P}_{3}\) is correct.

Step-by-step solution

Convert weight to moles

First, since the mixture has equal weights of helium, methane, and sulphur dioxide. Let's assume each gas weighs 16 grams for simplicity. The number of moles of any substance is calculated by dividing its weight by its molar mass. Thus, moles of helium = \( \frac{16}{4} = 4 \), moles of methane = \( \frac{16}{16} = 1 \), and moles of sulphur dioxide = \( \frac{16}{64} = 0.25 \).

Determine mole fraction

Next, calculate the total number of moles of the mixture: \(4 + 1 + 0.25 = 5.25\). Then, find the mole fraction of each gas: - Helium: \( \frac{4}{5.25} \)- Methane: \( \frac{1}{5.25} \)- Sulphur dioxide: \( \frac{0.25}{5.25} \)

Calculate partial pressures

Since the total pressure exerted by the mixture is \(210\) mm Hg, apply Dalton's Law of Partial Pressures. Calculate each partial pressure: - \( P_1 = 210 \times \frac{4}{5.25} \approx 160\) mm Hg (for helium)- \( P_2 = 210 \times \frac{1}{5.25} \approx 40\) mm Hg (for methane)- \( P_3 = 210 \times \frac{0.25}{5.25} \approx 10\) mm Hg (for sulphur dioxide)

Compare the partial pressures

The partial pressures are therefore: \(P_1 \approx 160\) mm Hg, \(P_2 \approx 40\) mm Hg, \(P_3 \approx 10\) mm Hg. So, \(P_1 > P_2 > P_3\). Thus, the statement (b) \(\mathrm{P}_{1} > \mathrm{P}_{2} > \mathrm{P}_{3}\) is correct.

Key concepts

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is an essential concept in chemistry, particularly when dealing with gas mixtures. This law states that the total pressure of a gas mixture is equal to the sum of the partial pressures of each individual gas in the mixture. Simply put, if you have a container filled with different gases, each gas contributes to the overall pressure. This contribution is known as its partial pressure.

Here's how to apply Dalton's Law:

• Identify all the gases in the mixture.
• Calculate the partial pressure of each gas separately.
• Add all the partial pressures together to find the total pressure.

In our exercise, you calculated partial pressures for helium, methane, and sulfur dioxide individually based on their mole fractions. Since each partial pressure adds up to the total pressure of 210 mm Hg, understanding these relationships helps in solving related chemistry problems effectively.

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It's especially useful in gas mixtures, where it helps determine each gas's contribution to the total pressure. To find the mole fraction of a gas, you need to know the number of moles of that individual gas, as well as the total moles in the mixture.

Calculation of mole fraction is straightforward:

• Determine the total number of moles in the mixture.
• Divide the moles of the gas you are interested in by the total number of moles.

For example, if a mixture contains 4 moles of helium, 1 mole of methane, and 0.25 moles of sulfur dioxide, it's crucial to find their individual mole fractions to calculate the partial pressures accurately. In this task, you computed:

• Helium: \( \frac{4}{5.25} \)
• Methane: \( \frac{1}{5.25} \)
• Sulfur Dioxide: \( \frac{0.25}{5.25} \)

By using mole fractions, you can cross-check how much a specific component affects the total pressure in a gaseous mixture.

Gas Mixture Calculations

Understanding gas mixture calculations is crucial for solving many real-world chemistry problems and projects. These calculations involve converting weights to moles, determining mole fractions, and employing Dalton’s Law to find partial pressures. This comprehensive approach ensures all elements in the mixture are accounted for, which leads to accurate solutions.

To solve gas mixture problems effectively:

• Convert the given weights into moles by using the molar mass of each gas.
• Calculate the total number of moles in the system.
• Find mole fractions of each gas to allow the calculation of partial pressures.
• Apply Dalton's Law to determine which gas contributes most to the overall pressure.

In scenarios like our exercise, converting equal weights of gases into moles shows how one gas may dominate the total pressure due to having more moles, as was the case with helium here. Recognizing these relationships boosts your ability to tackle complex problems related to gas mixtures.