Problem 34 - A Complete Resource Book in Chemistry for JEE Main

Question

A and B are ideal gases. The molecular weights of \(A\) and \(\mathrm{B}\) are in the ratio of \(1: 4\). The pressure of a gas mixture containing equal weights of \(A\) and \(B\) is \(P\) atm. What is the partial pressure (in atm) of B in the mixture? (a) \(\mathrm{P} / 5\) (b) \(\mathrm{P} / 2\) (c) \(\mathrm{P} / 2.5\) (d) \(3 \mathrm{P} / 4\)

Step-by-Step Solution

Answer

The partial pressure of B is \( \frac{P}{5} \) atm.

Step 1: Establish molecular weight ratio

We know that the molecular weights of gases A and B are in the ratio of 1:4. Let's denote the molecular weight of A as M and that of B as 4M.

Step 2: Calculate moles in terms of weight

For equal weights of A and B, let the mass be W grams each. Using the formula for moles \( n = \frac{\text{mass}}{\text{molecular weight}} \), the moles of A are \( \frac{W}{M} \) and the moles of B are \( \frac{W}{4M} \).

Step 3: Calculate total moles

The total moles in the gas mixture is the sum of moles of A and B: \[\text{Total moles} = \frac{W}{M} + \frac{W}{4M} = \frac{4W}{4M} + \frac{W}{4M} = \frac{5W}{4M}\]

Step 4: Calculate the mole fraction of B

Mole fraction of B is given by the moles of B divided by the total moles:\[\text{Mole fraction of B} = \frac{\frac{W}{4M}}{\frac{5W}{4M}} = \frac{1}{5}\]

Step 5: Determine partial pressure of B

According to Dalton's Law of Partial Pressures, the partial pressure of B is the total pressure multiplied by the mole fraction of B:\[\text{Partial pressure of B} = P \times \frac{1}{5} = \frac{P}{5}.\]

Key Concepts

Molecular Weight

Molecular weight is an important property of a gas, especially when dealing with gas mixtures. It represents the mass of a single molecule of a substance relative to one-twelfth the mass of an atom of carbon-12. For ideal gases, the molecular weight helps us understand how the mass of a gas influences other properties, like moles.

In the given exercise, we're told that gases A and B have molecular weights in the ratio 1:4. If the molecular weight of A is denoted by \( M \), then the molecular weight of B would be \( 4M \). This ratio is crucial as it affects the calculation of the number of moles when equal weights of A and B are considered. Understanding how molecular weight impacts these calculations helps in accurately determining other gas properties.

Partial Pressure

Partial pressure is a concept that explains how pressure contributed by each component of a gas mixture can be determined. In ideal gases, the total pressure of the gas mixture is the sum of the partial pressures of each individual gas.

Partial pressure can be calculated using the formula:

• \[ \text{Partial pressure of a gas} = \text{Total pressure} \times \text{Mole fraction of the gas} \]

In the exercise, the total pressure \( P \) is distributed among gases A and B. The partial pressure of B is calculated by multiplying the total pressure by B's mole fraction, which is \( \frac{1}{5} \). This resulted in the partial pressure of B being \( \frac{P}{5} \). Understanding partial pressure is essential for analyzing gas mixtures in different chemical and physical processes.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is a principle that describes how gases in a mixture contribute to the overall pressure. It states that each gas in a mixture exerts pressure independently of the others, and that the total pressure is the sum of these individual partial pressures.

This can be expressed as:

• \[ P_{\text{total}} = P_{A} + P_{B} + \cdots \]

In the scenario with gases A and B, Dalton's law aids in calculating the individual pressures that A and B contribute to the mixture. As shown in the exercise, it allows us to determine the partial pressure of B by knowing its mole fraction and total pressure. Dalton's Law is fundamental for predicting how gases behave when they mix and for performing calculations in thermodynamics and chemistry.