Question

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: If the average kinetic energy of $H_2$ and $O_2$ molecules, kept in two different sized containers are same, then their temperatures will be same.
Reason R: The r.m.s. speed of $H_2$ and $O_2$ molecules are same at same temperature.
Choose the correct answer from the options given below

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is NOT the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
According to the kinetic theory of gases, the average kinetic energy of a gas molecule depends exclusively on the absolute temperature. However, the root mean square (r.m.s.) speed of a gas molecule depends on both the temperature and the molar mass of the gas.

Step 2: Key Formula or Approach:
Average Translational Kinetic Energy: $K_{avg} = \frac{3}{2} k_B T$
Root Mean Square Speed: $v_{rms} = \sqrt{\frac{3RT}{M}}$

Step 3: Detailed Explanation:
Let's analyze the Assertion (A):
The average translational kinetic energy of any gas molecule is given by $\frac{3}{2}k_BT$, where $k_B$ is the Boltzmann constant and $T$ is the temperature in Kelvin. This implies that kinetic energy is solely determined by temperature and is entirely independent of the nature of the gas, its mass, or the container size. If the kinetic energies are the same, their temperatures must be the same. Thus, Assertion A is absolutely true.
Let's analyze the Reason (R):
The r.m.s. speed is given by $v_{rms} = \sqrt{\frac{3RT}{M}}$. At the same temperature $T$, the speed is inversely proportional to the square root of the molar mass $M$.
Since Hydrogen ($H_2$, $M \approx 2$ g/mol) is much lighter than Oxygen ($O_2$, $M \approx 32$ g/mol), the $H_2$ molecules will have a significantly higher r.m.s. speed than $O_2$ molecules at the same temperature.
Therefore, the r.m.s. speeds are NOT the same. Reason R is false.

Step 4: Final Answer:
Assertion A is true, but Reason R is false.