Problem 17 - A Complete Resource Book in Chemistry for JEE Main

Question

A closed container contains equal number of oxygen and hydrogen molecules at a total pressure of \(740 \mathrm{~mm}\). If oxygen is removed from the system then pressure will (a) become double of \(740 \mathrm{~mm}\) (b) become half of \(740 \mathrm{~mm}\) (c) become \(1 / 9\) of \(740 \mathrm{~mm}\) (d) remain unchanged

Step-by-Step Solution

Answer

Pressure becomes half of 740 mm, so the answer is (b).

Step 1: Understand the Composition

The container initially contains equal numbers of oxygen and hydrogen molecules. Hence, the total pressure of 740 mm is due to the combined partial pressures of both gases.

Step 2: Determine Initial Partial Pressures

Since there are equal numbers of oxygen and hydrogen molecules, they contribute equally to the total pressure. Thus, the partial pressure of the oxygen gas is half of the total: \( P_{O_2} = \frac{740}{2} = 370 \mathrm{~mm} \). Similarly, the partial pressure of hydrogen gas is \( P_{H_2} = 370 \mathrm{~mm} \).

Step 3: Consider the Effect of Removing Oxygen

After removing oxygen from the container, only hydrogen remains to exert pressure. Hence, the new total pressure is solely the partial pressure of the hydrogen molecules.

Step 4: Calculate New Pressure

Without the oxygen, the new total pressure in the container, exerted only by hydrogen, will be \( 370 \mathrm{~mm} \).

Step 5: Determine the Correct Option

Given the remaining pressure of 370 mm, which is half of the original total pressure, the pressure will be half of \( 740 \mathrm{~mm} \). This aligns with option (b).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental concept in chemistry and physics that describes the behavior of gases under various conditions. This law is succinctly captured by the equation \( PV = nRT \), where:

• \( P \) is the pressure of the gas,
• \( V \) is the volume it occupies,
• \( n \) is the number of moles of gas,
• \( R \) is the ideal gas constant,
• \( T \) is the temperature in Kelvin.

The Ideal Gas Law is crucial because it provides a good approximation of how real gases behave under many conditions.
However, it is important to note that this law assumes gases are ideal—the interactions between the gas molecules are negligible and the molecules occupy no volume.
In reality, gases deviate from ideal behavior at high pressures and low temperatures.
In the context of the exercise, understanding the Ideal Gas Law helps explain why the pressure changes when oxygen is removed from the container. The law indicates how pressure is directly proportional to the number of gas particles, assuming constant volume and temperature.

Gas Mixtures

Gas mixtures consist of two or more gases that are not chemically combined. They can exist in any proportion and retain their individual properties.
This concept is vital in understanding the behavior of gases in a closed environment, like the container in our example with both oxygen and hydrogen.
When looking at gas mixtures, each type of gas in the mixture will exert its own pressure, known as its partial pressure, irrespective of the other gases present.
A gas mixture behaves according to the combined properties of each individual gas. This means each gas in a mixture will follow the Ideal Gas Law.
In our exercise, both oxygen and hydrogen have equal contributions to the overall pressure initially because the number of molecules for both gases is the same.
This equal contribution directly explains why, after the removal of one, the pressure exerted by the remaining gas is half of the total initial pressure.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is a principle that is exceptionally useful 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 all the gases within it.
Mathematically, this is expressed as:

• \( P_{total} = P_1 + P_2 + P_3 + \, ... \)

Where \( P_1, P_2, \) and \( P_3 \) are the partial pressures of individual gases.
In our scenario, before oxygen is removed, the total pressure of the container is the sum of the pressures exerted by both the oxygen and hydrogen gases.
When Dalton's Law is applied, it predicts the pressure change when one gas is removed.
As illustrated in the exercise, once the oxygen is removed, Dalton's Law indicates that only the partial pressure of hydrogen remains.
Hence, the total pressure drops to half of the original, clearly showing the practical application of Dalton's Law.