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 \(\mathrm{A}\) and \(\mathrm{B}\) is \(\mathrm{P} \mathrm{atm}\). 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; option (a).

Step 1: Understand Given Information

We are given that gases A and B are mixed in such a way that their weights are equal, and the molecular weight ratio between A and B is 1:4. The total pressure of the mixture is P atm.

Step 2: Express Weights and Molecules in Terms of Molecular Weight

Since the weights of A and B are equal, we can let the weight of A be W and B be W as well. If the molecular weights are MW(A) and MW(B) where MW(A) = x and MW(B) = 4x, the number of moles of each gas can be expressed as: - For A: \( n_A = \frac{W}{x} \)- For B: \( n_B = \frac{W}{4x} \)

Step 3: Calculate Total and Partial Pressures

Using the ideal gas law, total pressure P is the sum of partial pressures of A and B. Since the number of moles is related to partial pressure: - Partial pressure of A: \( P_A = \frac{n_A}{n_A+n_B} \times P \)- Partial pressure of B: \( P_B = \frac{n_B}{n_A+n_B} \times P \)

Step 4: Simplify Equations to Find Partial Pressure of B

Calculate the ratio \(\frac{n_B}{n_A+n_B}\): \[\frac{n_B}{n_A+n_B} = \frac{\frac{W}{4x}}{\frac{W}{x} + \frac{W}{4x}} = \frac{1/4}{1 + 1/4} = \frac{1/4}{5/4} = \frac{1}{5}\] Thus, \( P_B = \frac{1}{5} \times P = \frac{P}{5} \).

Step 5: Find Correct Answer Option

Based on the calculation, the partial pressure of B is \(\frac{P}{5}\) atm. So, the correct option is (a) \(\frac{P}{5}\).

Key Concepts

Exploring Molecular Weight Ratio

In understanding the behavior of gas mixtures, molecular weight ratio plays a crucial role. When two gases have a specified molecular weight ratio, it means that their molecular masses have a definitive comparative relation. For instance, if the molecular weight ratio between gas A and gas B is 1:4, it means that the molecular weight of gas B is four times that of gas A. This ratio impacts the way each gas will behave when they are part of a mixture.

Given equal weights in a mixture, the actual number of moles of each gas differs despite their equal mass because of their different molecular weights. A molecule with a higher molecular weight will have fewer moles compared to a lighter molecule when both have the same mass. This concept is fundamental when working with the ideal gas law, which assumes that molecules behave as small, hard spheres that collide elastically with one another."},{

Understanding Partial Pressure Calculation

Partial pressure is an important concept in the study of gas mixtures. It refers to the pressure that each gas in a mixture would exert if it *** were alone in the entire volume occupied by the mixture. Dalton's Law of Partial Pressures states that the total pressure of a gas mixture is equal to the sum of the partial pressures of each individual gas.

To calculate the partial pressure of a gas within a mixture, you need to use the relationship between the number of moles of each gas and the total number of moles present. In simpler terms, the formula to find the partial pressure of a gas B is given by:

• \( P_B = \frac{n_B}{n_A + n_B} \times P \)

This equation implies that you calculate the ratio of the moles of gas B to the total moles of gases A and B in the mixture and then multiply it by the total pressure to find the partial pressure of gas B. By knowing the molecular weight and masses, you can derive the number of moles, which directly influences the individual gas's partial pressure."

Mixing Gases: A Look at Gas Mixtures

Gas mixtures can behave quite differently compared to pure gases due to their mixed molecular characteristics. When dealing with gas mixtures, such as when gases A and B are combined, the properties of the mixture depend significantly on the individual gases' contributions to the total measurable properties like pressure and volume.

Each gas within the mixture contributes to the total pressure in proportion to its mole ratio. The ideal gas law formula,

• \( PV = nRT \)

expresses the relationship between pressure (P), volume (V), the number of moles (n), the ideal gas constant (R), and temperature (T). The contribution of each gas to the total can be calculated independently, given known conditions of temperature, volume, and number of moles, allowing careful calculation of partial pressures and molecular ratios. This behavior showcases the interactions based on their molecular count and individual identities within a mixture."}]} ]} }