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 find the ratio of the number densities of the two gases, we can use the relationship between mean free path (λ), diameter (D), and number density (n). The mean free path is given by the formula:

\[ \lambda = \frac{1}{\sqrt{2} \pi D^2 n} \]

Where:
- \( \lambda \) is the mean free path,
- \( D \) is the diameter of the gas molecules,
- \( n \) is the number density of the gas.

Given:
- The ratio of diameters of the two gases \( \frac{D_1}{D_2} = \frac{3}{5} \) (where \( D_1 \) is the diameter of gas 1 and \( D_2 \) is the diameter of gas 2).
- The ratio of mean free paths of the two gases \( \frac{\lambda_1}{\lambda_2} = \frac{5}{3} \).

### Step 1: Write the equations for mean free paths

Using the mean free path formula for both gases, we have:

\[ \lambda_1 = \frac{1}{\sqrt{2} \pi D_1^2 n_1} \]
\[ \lambda_2 = \frac{1}{\sqrt{2} \pi D_2^2 n_2} \]

### Step 2: Set up the ratio of mean free paths

From the given ratios, we can express the ratio of mean free paths as:

\[ \frac{\lambda_1}{\lambda_2} = \frac{5}{3} \]

Substituting the expressions for \( \lambda_1 \) and \( \lambda_2 \):

\[ \frac{\frac{1}{\sqrt{2} \pi D_1^2 n_1}}{\frac{1}{\sqrt{2} \pi D_2^2 n_2}} = \frac{5}{3} \]

This simplifies to:

\[ \frac{D_2^2 n_2}{D_1^2 n_1} = \frac{5}{3} \]

### Step 3: Substitute the ratio of diameters

We know \( \frac{D_1}{D_2} = \frac{3}{5} \), so we can express \( D_1^2 \) and \( D_2^2 \):

\[ \frac{D_1^2}{D_2^2} = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \]

Now, substituting this into the equation gives:

\[ \frac{D_2^2 n_2}{D_1^2 n_1} = \frac{5}{3} \]

Rearranging gives:

\[ \frac{n_2}{n_1} = \frac{5}{3} \cdot \frac{D_1^2}{D_2^2} = \frac{5}{3} \cdot \frac{9}{25} \]

### Step 4: Calculate the ratio of number densities

Calculating the right-hand side:

\[ \frac{n_2}{n_1} = \frac{5 \cdot 9}{3 \cdot 25} = \frac{45}{75} = \frac{3}{5} \]

Thus, the ratio of number densities \( \frac{n_1}{n_2} \) is:

\[ \frac{n_1}{n_2} = \frac{5}{3} \]

### Final Answer

The ratio of number densities of the two gases is:

\[ \frac{n_1}{n_2} = \frac{5}{3} \]