Vander Waal's constant 'b' of a gas is 125.57 centilitre `//` mol. How near can be the centres of the two molecules approach each other ?

Answer

To find how near the centers of two molecules can approach each other, we can use the Van der Waals constant 'b'. The constant 'b' represents the volume occupied by one mole of a gas at a certain temperature and pressure, and it is related to the effective volume occupied by the gas molecules themselves.

### Step-by-Step Solution:

1. **Understanding Van der Waals Constant 'b':**
The Van der Waals constant 'b' is defined as the volume occupied by one mole of gas molecules. It accounts for the finite size of the molecules, which means that the volume available for the movement of the gas molecules is less than the total volume of the container.

2. **Convert 'b' to Standard Units:**
The given value of 'b' is 125.57 centiliters per mole. We need to convert this to liters per mole for standardization:
\[ b = 125.57 \, \text{centiliters/mol} = 1.2557 \, \text{liters/mol} \]

3. **Calculate Volume Occupied by One Mole of Gas:**
Since 'b' represents the volume occupied by one mole of gas, we can use this value to find the effective volume occupied by the gas molecules.

4. **Calculate the Volume per Molecule:**
To find the volume occupied by a single molecule, we divide the volume per mole by Avogadro's number (\(N_A\)), which is approximately \(6.022 \times 10^{23} \, \text{molecules/mol}\):
\[ \text{Volume per molecule} = \frac{b}{N_A} = \frac{1.2557 \, \text{liters}}{6.022 \times 10^{23}} = \frac{1.2557 \times 10^{-3} \, \text{liters}}{6.022 \times 10^{23}} = 2.09 \times 10^{-27} \, \text{liters} \]

5. **Calculate the Radius of a Molecule:**
Assuming that the molecules are spherical, we can relate the volume of a sphere to its radius using the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Rearranging for \(r\):
\[ r = \left(\frac{3V}{4\pi}\right)^{1/3} \]
Substituting the volume we calculated:
\[ r = \left(\frac{3 \times 2.09 \times 10^{-27} \, \text{liters}}{4\pi}\right)^{1/3} \]
Converting liters to cubic meters:
\[ r = \left(\frac{3 \times 2.09 \times 10^{-30} \, \text{m}^3}{4\pi}\right)^{1/3} \]

6. **Calculate the Distance Between Centers of Two Molecules:**
The distance between the centers of two molecules is approximately twice the radius:
\[ d = 2r \]

### Final Calculation:
After calculating the radius, we can find the distance \(d\) between the centers of the two molecules.