Question

Given below are two statements :
Statement (I) : The mean free path of gas molecules is inversely proportional to square of molecular diameter.
Statement (II) : Average kinetic energy of gas molecules is directly proportional to absolute temperature of gas.
In the light of the above statements, choose the correct answer from the option given below:

• A. Statement I is false but Statement II is true
• B. Statement I is true but Statement II is false
• C. Both Statement I and Statement II are false
• D. Both Statement I and Statement II are true

Answer 1

The Correct Option is D

To determine the correctness of the given statements, let's analyze each one regarding kinetic theory and properties of gases:

1. Statement I: The mean free path of gas molecules is inversely proportional to the square of the molecular diameter. 
   The mean free path (\(\lambda\)) of a gas molecule is given by the formula: \(\lambda = \frac{kT}{\sqrt{2} \pi d^{2} P}\) where:
   • \(k\) is the Boltzmann constant,
   • \(T\) is the absolute temperature,
   • \(d\) is the diameter of the molecule,
   • \(P\) is the pressure.
2. Statement II: Average kinetic energy of gas molecules is directly proportional to absolute temperature of gas. 
   According to the kinetic theory of gases, the average kinetic energy (\(E\)) of a gas molecule is: \(E = \frac{3}{2}kT\) Here, \(E\) is directly proportional to the absolute temperature (\(T\)). Therefore, Statement II is true.

After analyzing both statements with theoretical backing:

• Statement I is true because the mean free path is inversely proportional to the square of the diameter.
• Statement II is true because the average kinetic energy is directly proportional to the absolute temperature.

Thus, the correct answer is: Both Statement I and Statement II are true.

Answer 2

The mean free path (\(\lambda\)) of gas molecules is given by:
\[\lambda = \frac{RT}{\sqrt{2} \pi d^2 N_A P}.\]
Here, \(\lambda \propto \frac{1}{d^2}\), verifying Statement (I).
The average kinetic energy of gas molecules is:
\[KE = \frac{f}{2} nRT,\]
where \(KE \propto T\), confirming Statement (II).
Thus, both Statement I and Statement II are correct.