Question

If ‘n’ is the number of molecules per unit volume and ‘d’ is the diameter of the molecules, the mean free path ‘\(\lambda\) of molecules is

• A. \(\dfrac{2}{\sqrt{\pi} \, nd}\)
• B. \(\dfrac{1}{2 \, \pi \, nd^2}\)
• C. \(\dfrac{1}{\sqrt{2} \, \pi \, nd^2}\)
• D. \(\dfrac{1}{\sqrt{2} \, \pi \, nd}\)

Hint:

The mean free path is inversely proportional to both the number density of molecules and the square of the diameter of the molecules.

Answer 1

The Correct Option is A

Step 1: Formula for mean free path.
The mean free path (\(\lambda\)) is the average distance a molecule travels before colliding with another molecule. It is given by the formula: \[ \lambda = \dfrac{1}{\sqrt{2} \, \pi \, n \, d^2}. \]
Step 2: Analyzing the options.

- (1) \(\dfrac{2}{\sqrt{\pi} \, nd}\): This is incorrect. The formula for the mean free path includes \(d^2\) and not just \(d\).
- (2) \(\dfrac{1}{2 \, \pi \, nd^2}\): This is incorrect.
- (3) \(\dfrac{1}{\sqrt{2} \, \pi \, nd^2}\): Correct. This is the standard formula for the mean free path.
- (4) \(\dfrac{1}{\sqrt{2} \, \pi \, nd}\): This is incorrect because the formula involves \(d^2\).

Step 3: Conclusion.
The correct formula for the mean free path is \(\dfrac{1}{\sqrt{2} \, \pi \, nd^2}\), corresponding to option (3).