The mean free path of collision of gas melecules varies with its diameter (d) of the molecules as

Options

A. `d^(-1) `
B. `d^(-2)`
C. `d^(-3)`
D. `d^(-4)`

Answer

To solve the question regarding the relationship between the mean free path of gas molecules and their diameter, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Mean Free Path**:
The mean free path (λ) is defined as the average distance traveled by a molecule between collisions. It is an important concept in kinetic theory of gases.

2. **Mean Free Path Formula**:
The formula for the mean free path is given by:
\[ \lambda = \frac{k \cdot T}{\sqrt{2} \cdot n \cdot \pi d^2} \]
where:
- \( \lambda \) = mean free path
- \( k \) = Boltzmann constant
- \( T \) = absolute temperature
- \( n \) = number density of molecules (number of molecules per unit volume)
- \( d \) = diameter of the molecules

3. **Analyzing the Formula**:
From the formula, we can see that the mean free path \( \lambda \) is inversely proportional to the square of the diameter \( d \) of the molecules. This means that as the diameter increases, the mean free path decreases.

4. **Expressing the Relationship**:
We can express the relationship mathematically as:
\[ \lambda \propto \frac{1}{d^2} \]
This indicates that if the diameter \( d \) of the molecules increases, the mean free path \( \lambda \) decreases proportionally to the square of the diameter.

5. **Conclusion**:
Therefore, we conclude that the mean free path of gas molecules varies inversely with the square of the diameter of the molecules. The correct option is:
\[ \lambda \propto d^{-2} \]

### Final Answer:
The mean free path of collision of gas molecules varies with its diameter (d) as \( d^{-2} \).

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