Calculate the ratio of the mean free paths of the molecules of two gases having molecular diameters `1 Å` and `2 Å`. The gases may be considered under identical conditions of temperature, pressure and volume.

Options

A. `2:1`
B. `3:1`
C. `4:3`
D. `4:1`

Explanation

To calculate the ratio of the mean free paths of molecules of two gases with molecular diameters of `1 Å` and `2 Å`, we can use the formula for mean free path \( l \):

\[ l = \frac{1}{\sqrt{2} \pi d^2 n} \]

Where:
- \( l \) is the mean free path,
- \( d \) is the molecular diameter,
- \( n \) is the number of molecules per unit volume.

Since the gases are under identical conditions of temperature, pressure, and volume, we can simplify our calculations. The number density \( n \) will be the same for both gases under these conditions.

### Step 1: Write the mean free path for both gases

Let:
- \( l_1 \) be the mean free path for the gas with diameter \( d_1 = 1 \, \text{Å} \),
- \( l_2 \) be the mean free path for the gas with diameter \( d_2 = 2 \, \text{Å} \).

The mean free paths can be expressed as:

\[ l_1 = \frac{1}{\sqrt{2} \pi d_1^2 n} \]

\[ l_2 = \frac{1}{\sqrt{2} \pi d_2^2 n} \]

### Step 2: Calculate the ratio of the mean free paths

Now, we can find the ratio \( \frac{l_1}{l_2} \):

\[ \frac{l_1}{l_2} = \frac{\frac{1}{\sqrt{2} \pi d_1^2 n}}{\frac{1}{\sqrt{2} \pi d_2^2 n}} = \frac{d_2^2}{d_1^2} \]

### Step 3: Substitute the values of diameters

Substituting the given values:

- \( d_1 = 1 \, \text{Å} \)
- \( d_2 = 2 \, \text{Å} \)

\[ \frac{l_1}{l_2} = \frac{(2 \, \text{Å})^2}{(1 \, \text{Å})^2} = \frac{4 \, \text{Å}^2}{1 \, \text{Å}^2} = 4 \]

### Conclusion

Thus, the ratio of the mean free paths of the two gases is:

\[ \frac{l_1}{l_2} = 4 \]