The mean free path of molecules of a gas (radius r) is inversely proportional to

Options

A. `r^(3)`
B. `r^(2)`
C. r
D. `sqrtr`

Explanation

To solve the question regarding the mean free path of molecules of a gas in relation to their radius, we can follow these steps:

### Step 1: Understand the Mean Free Path Formula
The mean free path (λ) of gas molecules is given by the formula:
\[ \lambda = \frac{1}{\sqrt{2} \pi n d^2} \]
where:
- \( n \) is the number density of the molecules (number of molecules per unit volume),
- \( d \) is the diameter of the molecules.

### Step 2: Relate Diameter to Radius
The diameter \( d \) of a molecule can be expressed in terms of its radius \( r \):
\[ d = 2r \]

### Step 3: Substitute Diameter in the Mean Free Path Formula
Now, substituting \( d = 2r \) into the mean free path formula:
\[ \lambda = \frac{1}{\sqrt{2} \pi n (2r)^2} \]

### Step 4: Simplify the Expression
Simplifying the expression gives:
\[ \lambda = \frac{1}{\sqrt{2} \pi n \cdot 4r^2} \]
\[ \lambda = \frac{1}{4\sqrt{2} \pi n r^2} \]

### Step 5: Analyze the Relationship
From the simplified expression, we can see that the mean free path \( \lambda \) is inversely proportional to \( r^2 \):
\[ \lambda \propto \frac{1}{r^2} \]

### Conclusion
Thus, the mean free path of molecules of a gas is inversely proportional to the square of the radius of the molecules. Therefore, the correct answer is \( r^2 \).

### Final Answer
The mean free path of molecules of a gas (radius \( r \)) is inversely proportional to \( r^2 \).

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