Problem 102 - A Complete Resource Book in Chemistry for JEE Main

Question

A closed vessel contains equal number of molecules of \(\mathrm{O}_{2}\) and \(\mathrm{H}_{2} .\) Consider the following statements: 1\. The average speed of hydrogen molecules will be greater. 2\. Weight of hydrogen is \(1 / 8\) th of the weight of oxygen. 3\. Hydrogen molecules strike the walls more often. 4\. The two gases have different average energies. The statements are wrong in (a) 2 and 4 (b) 2 and 3 (c) 1 and 4 (d) 1 and 3

Step-by-Step Solution

Answer

Statements 2 and 4 are wrong.

Step 1: Calculate the Average Speed

The average speed of gas molecules is given by the formula \( v_{avg} = \sqrt{\frac{8RT}{\pi M}} \), where \( R \) is the gas constant, \( T \) is the temperature, and \( M \) is the molar mass. Since the molar mass of hydrogen (H₂) is much lower than that of oxygen (O₂), the average speed \( v_{avg} \) for hydrogen will be greater than that of oxygen. Therefore, statement 1 is correct.

Step 2: Weight Comparison Between Hydrogen and Oxygen

The weight of a gas is proportional to its molar mass. Hydrogen (H₂) has a molar mass of approximately 2 g/mol, while oxygen (O₂) has a molar mass of approximately 32 g/mol. Therefore, given the number of molecules is equal, the weight of hydrogen will be \(\frac{2}{32} = \frac{1}{16} \) times the weight of oxygen, making statement 2 incorrect.

Step 3: Analyze Frequency of Molecular Collisions

The frequency with which gas molecules strike the walls depends on their speed. Since hydrogen molecules move faster, they collide with the walls more frequently compared to heavier oxygen molecules. Thus, statement 3 is correct.

Step 4: Assess Average Energy of Gases

According to the kinetic theory of gases, the average kinetic energy of gas molecules depends only on the temperature and is the same for all gases at the same temperature. Hence, oxygen and hydrogen, having the same temperature in this vessel, have the same average energy. Hence, statement 4 is incorrect.

Key Concepts

Molecular Speed

In gases, molecular speed is an important concept because it influences many properties such as diffusion, effusion, and pressure on container walls. The average speed of gas molecules, also known as the root mean square speed, can be calculated using the formula \[ v_{avg} = \sqrt{\frac{8RT}{\pi M}} \] where:

• \( R \) is the universal gas constant;
• \( T \) is the temperature in Kelvin;
• \( M \) is the molar mass of the gas.

The formula shows that molecular speed is inversely related to molar mass. This means lighter gases, such as hydrogen with a molar mass of about 2 g/mol, have higher average speeds compared to heavier gases, like oxygen, which has a molar mass of about 32 g/mol. Understanding this relationship helps explain why, in our exercise, hydrogen molecules had a greater average speed than oxygen molecules.

Molar Mass Comparison

When comparing gases such as hydrogen and oxygen, understanding molar mass is crucial. Molar mass refers to the mass of one mole of a substance and is typically expressed in grams per mole (g/mol). For hydrogen (H
42 ), the molar mass is 2 g/mol, while for oxygen (O
42 ), it is 32 g/mol.In the exercise, the comparison shows that for equal numbers of molecules, hydrogen is much lighter due to its lower molar mass. To determine the weight ratio of these gases, we can use their molar masses:\[\text{Weight Ratio} = \frac{2}{32} = \frac{1}{16}\]This means that the weight of hydrogen is \( \frac{1}{16} \) times the weight of oxygen. This information is essential for grasping how gas properties change with different molecular masses.

Average Kinetic Energy

A fundamental idea in the kinetic theory of gases is that at a given temperature, all gases have the same average kinetic energy. This is critical because kinetic energy depends not on the type of gas but on the temperature alone.The average kinetic energy of gas molecules can be expressed with the formula:\[ \overline{E_k} = \frac{3}{2}kT \]where:

• \( k \) is the Boltzmann constant;
• \( T \) is the temperature in Kelvin.

In our context, although hydrogen and oxygen have different molecular speeds, their average kinetic energies are the same when they are at the same temperature. Therefore, knowing the temperature allows us to determine that despite their speed differences, the molecular energies of two gases are equivalent. This key tenet helps to correct misconceptions about energy differences in gases at equilibrium.

Collision Frequency

Collision frequency describes how often gas molecules collide with the walls of their container. It is a function of molecular speed and the number of molecules. Higher speeds typically result in more frequent collisions. Therefore, lighter molecules like those in hydrogen gas, which have higher average speeds, will strike the container walls more often than heavier molecules like those in oxygen gas. This can be understood as:

• Faster molecules travel across the container more quickly;
• They cover more distance in less time;
• Leading to more frequent impacts with the walls.

In practical terms, understanding collision frequency helps explain properties like pressure and temperature in gases. It is why in our example, hydrogen molecules collide with the walls more frequently than oxygen molecules, aiding in clarifying dynamics of gas mixtures.