Question

A closed vessel contains equal number of oxygen and hydrogen molecules at a total pressure of \(740 \mathrm{~mm}\). If oxygen is removed from the system, the pressure (1) becomes half of \(740 \mathrm{~mm}\) (2) remains unchanged (3) becomes \(1 / 9\) th of \(740 \mathrm{~mm}\) (4) becomes double of \(740 \mathrm{~mm}\)

Short answer

The pressure becomes half of 740 mm Hg.

Step-by-step solution

Understanding the problem

A closed vessel contains an equal number of oxygen and hydrogen molecules. The total pressure is given as 740 mm Hg. We need to find the pressure after oxygen is removed.

Using Dalton's Law of Partial Pressures

According to Dalton's Law, the total pressure of a mixture of gases is the sum of the partial pressures of each individual gas. Let the partial pressure of oxygen be \(P_O\) and that of hydrogen be \(P_H\). Since there are equal numbers of oxygen and hydrogen molecules, \(P_O = P_H\).

Calculate the partial pressures

Given that the total pressure is 740 mm Hg and the partial pressures are equal, we can set up the equation \(P_O + P_H = 740\). Because \(P_O = P_H\), we have \(2P_O = 740\). Therefore, \(P_O = P_H = 370\) \mathrm{mm} \mathrm{Hg}\.

Determine the new pressure after oxygen removal

If oxygen is removed, only hydrogen is left in the vessel. So the total pressure will be the partial pressure of hydrogen, which is 370 mm Hg.

Compare the new pressure to the given choices

The new pressure is 370 mm Hg. Comparing this to the multiple choice options, the correct answer is (1) becomes half of 740 mm Hg.

Key concepts

Understanding Gas Laws

Gas laws explain the behavior of gases in different conditions. One of the fundamental concepts includes Dalton's Law of Partial Pressures, which helps us understand how individual gases contribute to the total pressure of a gas mixture.

• Gas laws combine principles such as temperature, volume, and pressure.
• They are crucial for solving problems related to gases in chemistry and physics.

Dalton’s Law specifically states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of each individual gas.

The formula for Dalton's Law is:
\[ P_{total} = P_1 + P_2 + P_3 + ... + P_n \] Where:

• \(P_{total}\) is the total pressure of the mixture.
• \(P_1, P_2, P_3, ..., P_n\) are the partial pressures of the individual gases.

This principle allows us to isolate and measure the pressure of each gas within a mixture.

The Concept of Partial Pressure

Partial pressure is the pressure that a single gas in a mixture of gases would exert if it occupied the entire volume by itself.

• Partial pressure depends on the amount of gas, its volume, and temperature.
• In a scenario with equal moles of gases, each gas exerts an equal portion of the total pressure.

Consider a sealed container with oxygen and hydrogen molecules at equal amounts:

• Given: total pressure is 740 mm Hg.
• Oxygen and hydrogen contribute equally, so each partial pressure is half the total pressure.

Mathematically:
\[ P_{O_2} + P_{H_2} = 740 \text{ mm Hg} \] Since they contribute equally:
\[ 2P_{O_2} = 740 \text{ mm Hg} \] Therefore:
\[ P_{O_2} = P_{H_2} = 370 \text{ mm Hg} \] Removing one gas (oxygen) leaves us with the partial pressure of the remaining gas (hydrogen), which is now the total pressure: 370 mm Hg.

Understanding Total Pressure in Gas Mixtures

Total pressure is the sum of all partial pressures in a gas mixture. This concept is essential in understanding how mixed gases behave.
With Dalton’s Law, we can break down the total pressure into contributions from each gas:

In our example:

• Total pressure = 740 mm Hg.
• Each gas (oxygen and hydrogen) has a partial pressure of 370 mm Hg.

By removing oxygen, the total pressure changes as follows:

• The total pressure equals the remaining hydrogen's partial pressure.

Resulting in:

• New total pressure = 370 mm Hg.

Understanding how these pressures add up and change in different scenarios helps in numerous practical applications, from chemical reactions to atmospheric studies.