The mean free path l for a gas molecule depends upon diameter d of molecule as

Options

A. `l proportional to 1/d^2`
B. `l proportional to d`
C. `l proportional to d^2`
D. `l proportional to 1/d`

Answer

To determine how the mean free path \( l \) for a gas molecule depends on the diameter \( d \) of the molecule, we can follow these steps:

### Step 1: Understand the Concept of Mean Free Path
The mean free path \( l \) is defined as the average distance a molecule travels between collisions with other molecules. It is an important concept in kinetic theory of gases.

### Step 2: Identify the Factors Affecting Mean Free Path
The mean free path depends on several factors:
- The number density \( n \) of the gas molecules (the number of molecules per unit volume).
- The diameter \( d \) of the gas molecules.

### Step 3: Use the Formula for Mean Free Path
The mean free path \( l \) can be expressed mathematically as:
\[ l = \frac{1}{\sqrt{2} \cdot n \cdot \sigma} \]
where \( \sigma \) is the collision cross-section, which can be approximated as:
\[ \sigma \approx \pi d^2 \]
Thus, substituting \( \sigma \) into the mean free path formula gives:
\[ l = \frac{1}{\sqrt{2} \cdot n \cdot \pi d^2} \]

### Step 4: Analyze the Relationship
From the above equation, we can see that:
- The mean free path \( l \) 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, and vice versa.

### Final Expression
The final relationship can be summarized as:
\[ l \propto \frac{1}{d^2} \]

### Summary
The mean free path \( l \) for a gas molecule is inversely proportional to the square of the diameter \( d \) of the molecule.

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