Which of the following represents the mean free path for a gas ? (where n is number of gas molecules per unit volume, d is diameter of gas molecules)

Options

A. `(1)/(sqrt(2)pi m)`
B. `(1)/(sqrt(2)pi nd^(2))`
C. `(1)/(sqrt(2)pi nd^(3))`
D. `(1)/(sqrt(2)pi nd^(3//2))`

Explanation

To determine the mean free path (λ) for a gas, we can use the formula that relates it to the number density of gas molecules (n) and the diameter of the gas molecules (d). The mean free path is the average distance a molecule travels between collisions with other molecules.

### Step-by-Step Solution:

1. **Understand the Mean Free Path**: The mean free path (λ) is defined as the average distance traveled by a gas molecule between successive collisions.

2. **Identify the Variables**: In the formula for mean free path, we have:
- \( n \): Number of gas molecules per unit volume (density of gas molecules).
- \( d \): Diameter of the gas molecules.

3. **Use the Formula**: The formula for the mean free path is given by:
\[ \lambda = \frac{1}{\sqrt{2} \pi n d^2} \]
This formula indicates that the mean free path is inversely proportional to both the number density of molecules and the square of the diameter of the molecules.

4. **Analyze the Formula**: From the formula, we can see that:
- As the number density (n) increases, the mean free path (λ) decreases, meaning molecules collide more frequently.
- As the diameter (d) of the molecules increases, the mean free path (λ) also decreases for the same reason.

5. **Select the Correct Option**: Based on the formula provided, the correct representation of the mean free path for a gas in terms of n and d is:
\[ \lambda = \frac{1}{\sqrt{2} \pi n d^2} \]
Therefore, the option that matches this formula is the correct answer.

### Final Answer:
The mean free path for a gas is represented by:
\[ \lambda = \frac{1}{\sqrt{2} \pi n d^2} \]