The ratio of diameters of molecules of the two gases is 3:5 and their mean free path of the molecule is in the ratio of 5:3 then the ratio of number density of the two gases is

Options

A. `5:3`
B. `25:9`
C. `9:25`
D. `1:1`

Answer

To solve the problem, we need to find the ratio of the number densities of two gases given the ratio of their molecular diameters and mean free paths.

### Step-by-Step Solution:

1. **Identify Given Ratios**:
- Let the diameter of the first gas be \( d_1 \) and the second gas be \( d_2 \). We are given:
\[ \frac{d_1}{d_2} = \frac{3}{5} \]
- Let the mean free path of the first gas be \( \lambda_1 \) and the second gas be \( \lambda_2 \). We are given:
\[ \frac{\lambda_1}{\lambda_2} = \frac{5}{3} \]

2. **Mean Free Path Formula**:
- The mean free path \( \lambda \) is given by the formula:
\[ \lambda = \frac{1}{\sqrt{2} \pi d^2 n} \]
- Where \( n \) is the number density.

3. **Setting Up Equations**:
- For the first gas:
\[ \lambda_1 = \frac{1}{\sqrt{2} \pi d_1^2 n_1} \]
- For the second gas:
\[ \lambda_2 = \frac{1}{\sqrt{2} \pi d_2^2 n_2} \]

4. **Forming a Ratio**:
- Taking the ratio of \( \lambda_1 \) to \( \lambda_2 \):
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{1}{\sqrt{2} \pi d_1^2 n_1}}{\frac{1}{\sqrt{2} \pi d_2^2 n_2}} = \frac{d_2^2 n_2}{d_1^2 n_1} \]

5. **Substituting Known Ratios**:
- From the problem, we know:
\[ \frac{\lambda_1}{\lambda_2} = \frac{5}{3} \]
- Therefore, we can write:
\[ \frac{5}{3} = \frac{d_2^2 n_2}{d_1^2 n_1} \]

6. **Substituting Diameter Ratio**:
- We know \( \frac{d_1}{d_2} = \frac{3}{5} \), so:
\[ \frac{d_2}{d_1} = \frac{5}{3} \]
- Squaring this gives:
\[ \frac{d_2^2}{d_1^2} = \left(\frac{5}{3}\right)^2 = \frac{25}{9} \]

7. **Setting Up the Equation**:
- Substitute \( \frac{d_2^2}{d_1^2} \) into the equation:
\[ \frac{5}{3} = \frac{25}{9} \cdot \frac{n_2}{n_1} \]

8. **Solving for the Density Ratio**:
- Rearranging gives:
\[ \frac{n_2}{n_1} = \frac{5}{3} \cdot \frac{9}{25} = \frac{45}{75} = \frac{3}{5} \]
- Therefore:
\[ \frac{n_1}{n_2} = \frac{5}{3} \]

9. **Final Ratio**:
- Thus, the ratio of number densities of the two gases is:
\[ n_1 : n_2 = 5 : 3 \]

### Conclusion:
The ratio of the number density of the two gases is \( 5 : 3 \).