Problem 16 - A Complete Resource Book in Chemistry for JEE Main

Question

Pressure of a mixture of \(4 \mathrm{~g}\) of \(\mathrm{O}_{2}\) and \(2 \mathrm{~g}\) of \(\mathrm{H}_{2}\) confined in a bulb of \(1.0 \mathrm{~L}\) capacity at \(0^{\circ} \mathrm{C}\) is (a) \(25.18 \mathrm{~atm}\) (b) \(31.205 \mathrm{~atm}\) (c) \(40.215 \mathrm{~atm}\) (d) \(15.210 \mathrm{~atm}\)

Step-by-Step Solution

Answer

The pressure is 25.18 atm, which matches option (a).

Step 1: Convert Mass to Moles

Start by converting the mass of each gas to moles using their molar masses. The molar mass of \(\mathrm{O}_2\) is \(32 \mathrm{~g/mol}\), and the molar mass of \(\mathrm{H}_2\) is \(2 \mathrm{~g/mol}\). The number of moles of \(\mathrm{O}_2\) is \( \frac{4\, \mathrm{g}}{32\, \mathrm{g/mol}} = 0.125\, \text{mol} \).The number of moles of \(\mathrm{H}_2\) is \( \frac{2\, \mathrm{g}}{2\, \mathrm{g/mol}} = 1\, \text{mol} \).

Step 2: Apply the Ideal Gas Law

Use the ideal gas law \( PV = nRT \) to find the total pressure exerted by the gases. The temperature \( T \) is \( 0^{\circ}\mathrm{C} = 273 \mathrm{~K} \). The gas constant \( R \) is \(0.0821 \mathrm{~L~ atm~ K^{-1}~ mol^{-1}} \). First, find the total number of moles: \( n_{\text{total}} = 0.125 \mathrm{~mol} + 1 \mathrm{~mol} = 1.125 \mathrm{~mol} \).Now use the equation: \[ P = \frac{nRT}{V} = \frac{1.125 \times 0.0821 \times 273}{1.0} \].

Step 3: Calculate Pressure

Perform the calculation:\[ P = \frac{1.125 \times 0.0821 \times 273}{1.0} = 25.18 \mathrm{~atm} \].

Step 4: Select the Closest Answer

The calculated pressure is \( 25.18 \mathrm{~atm} \), which matches option (a). Therefore, the correct answer is (a) \( 25.18 \mathrm{~atm} \).

Key Concepts

Understanding Partial Pressure

Partial pressure is a fundamental concept in chemistry, particularly when dealing with gas mixtures. It refers to the pressure that a single type of gas in a mixture would exert if it were alone in the container. The total pressure of a gas mixture is the sum of the partial pressures of all individual gases. To find the partial pressure of a specific gas, you can use Dalton's Law of Partial Pressures.\[\P_{\text{total}} = P_{1} + P_{2} + \ldots + P_{n} \\] where \(P_{\text{total}}\) is the total pressure, and \(P_{1}, P_{2}, ..., P_{n}\) are the partial pressures of each gas in the mixture. This makes it easier to predict how each component of the gas mixture behaves. In our exercise, even though individual calculations of partial pressures were not shown, understanding this concept helps in understanding how each gas contributes to the total pressure measured by the formula \(PV = nRT\).

When adding gases together, the overall behavior of the mix is governed by these individual pressures, and it helps to describe real-world applications like breathing air under various conditions, where different gases like oxygen and nitrogen together create a total pressure.

Calculating Molar Mass

Molar mass is the mass of one mole of a substance and is expressed in g/mol. For gases, knowing the molar mass allows us to convert between the mass of a gas and its amount in moles, which is crucial for using the ideal gas law. In calculations involving molar mass, we employ the formula:

• \(\text{Moles} = \frac{\text{Mass (g)}}{\text{Molar Mass (g/mol)}}\).

This calculation is necessary to determine how much of a gas you have in mole terms. For example, in our exercise, we found the moles of \(\text{O}_2\) as \(0.125\) and \(\text{H}_2\) as \(1\) by using the molar masses from the periodic table: \(32\, \text{g/mol}\) for oxygen and \(2\, \text{g/mol}\) for hydrogen. Using these values in the ideal gas law requires precise molar mass calculations, as they directly impact the outcome of any pressure or volume change scenarios in a gas law application.

Understanding molar mass helps you connect the mass of different substances to the number of particles present, bridging micro- and macroscopic descriptions of substances.

Behavior of Gas Mixtures

Gas mixtures consist of different gases occupying the same volume. Understanding how gas mixtures behave requires using concepts like the ideal gas law and Dalton's Law of Partial Pressures. The ideal gas law \(PV = nRT\) tells us about the relationship between the gas's pressure, volume, and temperature, but it applies to the sum of all gases present.

In our exercise, we had a mix of oxygen and hydrogen gases confined in a bulb. To find the total pressure, we considered their collective effect, leveraging the total moles of the gases. What makes gas mixtures interesting is the interaction between different gases, which do not chemically combine but physically occupy the same phase in the container.

Knowledge of gas mixtures is critical in various real-world applications. For instance, breathing air, cooking gas, or atmospheric studies all involve understanding gas mixtures and their corresponding properties to make necessary calculations or to anticipate their behaviors under various atmospheric or environmental conditions.