The mean free path of molecule of a gas is `10^(-8)` cm. If number density of gas is `10^(9)//cm^(3)`. Calculate the diameter of the molecule.

Explanation

To find the diameter of a molecule given the mean free path and the number density, we can use the formula for the mean free path (\(\lambda\)):

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

Where:
- \(\lambda\) is the mean free path,
- \(n\) is the number density,
- \(d\) is the diameter of the molecule.

### Step 1: Rearranging the formula

We need to rearrange the formula to solve for \(d\):

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

### Step 2: Substituting the values

Now, we can substitute the given values into the equation. We have:
- \(\lambda = 10^{-8} \, \text{cm}\)
- \(n = 10^9 \, \text{cm}^{-3}\)

Substituting these values into the equation:

\[ d^2 = \frac{1}{2 \pi (10^9) (10^{-8})} \]

### Step 3: Calculating the denominator

Calculating the denominator:

\[ 2 \pi (10^9) (10^{-8}) = 2 \pi \times 10^{1} = 2 \times 3.14 \times 10^{1} = 6.28 \times 10^{1} = 62.8 \]

### Step 4: Finding \(d^2\)

Now substituting back into the equation for \(d^2\):

\[ d^2 = \frac{1}{62.8} \]

### Step 5: Calculating \(d\)

Now, we take the square root to find \(d\):

\[ d = \sqrt{\frac{1}{62.8}} \approx \frac{1}{7.93} \approx 0.126 \, \text{cm} \]

### Step 6: Final Answer

Thus, the diameter of the molecule is approximately:

\[ d \approx 0.126 \, \text{cm} \]