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

The statements that are wrong are 2 and 4 (option a).

Step 1: Understand Kinetic Theory

According to the kinetic theory of gases, for gases at the same temperature, the average kinetic energy is the same. Since both hydrogen and oxygen are at the same temperature in the vessel, their average kinetic energies are equal. Thus, statement 4, which claims they have different average energies, is incorrect.

Step 2: Analyze Molecular Speeds

Given that both gases have equal kinetic energy and the formula for kinetic energy is \( KE = \frac{1}{2}mv^2 \), the lightest molecule (in this case, hydrogen) will have a greater speed. Thus, statement 1, asserting that the average speed of hydrogen molecules is greater, is accurate.

Step 3: Examine Molecular Weights

The molecular weight of oxygen \((O_2)\) is 32 g/mol and that of hydrogen \((H_2)\) is 2 g/mol. Therefore, the mass of hydrogen is \(2 / 32\) or \(1 / 16\) of the mass of oxygen, not \(1/8\). So, statement 2 is incorrect.

Step 4: Consider Collision Frequency with Walls

Since hydrogen molecules move faster due to their lower mass, they will strike the container walls more frequently than oxygen molecules. Thus, statement 3, which claims hydrogen molecules strike the walls more often, is true.

Key Concepts

Average Kinetic Energy

In the realm of gases, the concept of average kinetic energy is crucial. According to the kinetic theory of gases:

• The average kinetic energy of gas molecules is directly related to the temperature of the gas.
• Thus, at the same temperature, all gas molecules have the same average kinetic energy.

The formula for kinetic energy is given by \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the molecule, and \( v \) is its velocity. Since this formula holds true for all gases, whether it's hydrogen or oxygen in our exercise, both will possess the same average kinetic energy at the same temperature. Hence, despite differences in individual molecular velocities or masses, the average energy remains constant if the temperature does not change.

Molecular Speed

Molecular speed is an intriguing aspect of gas behavior. Given the kinetic energy formula, \( KE = \frac{1}{2}mv^2 \), it becomes apparent that lighter molecules move faster to maintain the same kinetic energy. This implies:

• For molecules at the same temperature, lighter molecules, such as hydrogen, have higher speeds compared to heavier ones like oxygen.
• The speed of a molecule is influenced by its mass, showing \( v = \sqrt{\frac{2KE}{m}} \).

From the exercise, since hydrogen is significantly lighter than oxygen, its molecules travel at higher speeds despite having equal average kinetic energy with oxygen. This is why statement 1 from the exercise, which states that hydrogen molecules have greater speeds, is true.

Collision Frequency

The rate at which gas molecules collide with the walls of their container is termed collision frequency. This concept is dictated by both the speed of the molecules and their numbers. Important factors include:

• Faster-moving molecules, such as hydrogen, will collide with container walls more frequently than slower ones due to higher molecular speed.
• Despite equal numbers of molecules, the increased velocity of hydrogen results in a higher collision frequency relative to oxygen.

By understanding that collision frequency is intrinsically tied to molecular speed, we see why statement 3 is correct in the exercise context: hydrogen molecules, being lighter and therefore faster, strike the walls more often.