The van der Waals' constant 'b' of a gas is `4pixx 10^(-4)L//mol.` How near can the centeres of the two molecules approach each other? [Use :`N_(A)=6xx10^(23)`]

Options

A. `10^(-7)` m
B. `10^(-10)` m
C. `5xx10^(-11)` m
D. `5xx10^(-9)` m

Explanation

To solve the problem, we need to determine how near the centers of two gas molecules can approach each other, given the van der Waals constant 'b' and Avogadro's number.

### Step-by-Step Solution:

1. **Understand the van der Waals constant 'b':**
The van der Waals constant 'b' is related to the volume occupied by one mole of gas molecules. It is given by the equation:
\[ b = N_A \cdot \frac{4}{3} \pi R^3 \]
where \( R \) is the radius of a gas molecule, and \( N_A \) is Avogadro's number.

2. **Substitute the given values:**
We have:
\[ b = 4 \pi \times 10^{-4} \, \text{L/mol} \]
and
\[ N_A = 6 \times 10^{23} \, \text{molecules/mol} \]

3. **Rearranging the equation for R:**
Rearranging the equation for \( R \):
\[ R^3 = \frac{3b}{4\pi N_A} \]
Now substitute the values of \( b \) and \( N_A \):
\[ R^3 = \frac{3 \times (4 \pi \times 10^{-4})}{4 \pi \times (6 \times 10^{23})} \]

4. **Simplifying the equation:**
The \( 4\pi \) terms cancel out:
\[ R^3 = \frac{3 \times 10^{-4}}{6 \times 10^{23}} = \frac{3}{6} \times 10^{-4} \times 10^{-23} = \frac{1}{2} \times 10^{-27} \]
\[ R^3 = 0.5 \times 10^{-27} \]

5. **Calculating R:**
Now, take the cube root:
\[ R = \left(0.5 \times 10^{-27}\right)^{1/3} \]
This can be calculated as:
\[ R = \frac{1}{\sqrt[3]{2}} \times 10^{-9} \, \text{m} \approx 0.7937 \times 10^{-9} \, \text{m} \approx 7.937 \times 10^{-10} \, \text{m} \]

6. **Finding the distance between two molecules:**
The distance between the centers of two molecules when they are just touching each other is:
\[ d = 2R \approx 2 \times 7.937 \times 10^{-10} \, \text{m} \approx 1.5874 \times 10^{-9} \, \text{m} \approx 1.59 \times 10^{-9} \, \text{m} \]

### Final Answer:
The centers of the two molecules can approach each other to a distance of approximately \( 1.59 \times 10^{-9} \, \text{m} \).