The mean free path of molecules 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 solve the problem of calculating the diameter of a molecule given the mean free path and number density, we can follow these steps:

### Step-by-Step Solution:

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

2. **Rearrange the Formula**:
We need to find the diameter \( d \). Rearranging the formula gives:
\[ d^2 = \frac{1}{\sqrt{2} \pi n \lambda} \]

3. **Substitute the Given Values**:
We are given:
- \( \lambda = 10^{-8} \) cm
- \( n = 10^9 \) cm\(^{-3}\)

Now substituting these values into the rearranged formula:
\[ d^2 = \frac{1}{\sqrt{2} \pi (10^9) (10^{-8})} \]

4. **Calculate the Values**:
First, calculate \( \sqrt{2} \) and \( \pi \):
- \( \sqrt{2} \approx 1.414 \)
- \( \pi \approx 3.14 \)

Now substituting these values:
\[ d^2 = \frac{1}{1.414 \times 3.14 \times 10^9 \times 10^{-8}} \]
\[ d^2 = \frac{1}{1.414 \times 3.14 \times 10^1} \]
\[ d^2 = \frac{1}{4.442 \times 10^1} \]
\[ d^2 \approx 0.0225 \text{ cm}^2 \]

5. **Take the Square Root**:
Now, take the square root to find \( d \):
\[ d \approx \sqrt{0.0225} \approx 0.15 \text{ cm} \]

6. **Final Result**:
Therefore, the diameter of the molecule is:
\[ d \approx 0.15 \text{ cm} \]