Calculate molecular diameter for a gas if its molar exclued volume is `3.2 pi ml`. (in nenometer) (Take `N_(A) = 6.0 xx 10^(23))`

Explanation

To calculate the molecular diameter of a gas given its molar excluded volume, we can follow these steps:

### Step 1: Understand the relationship between molar excluded volume and molecular diameter
The molar excluded volume (B) is related to the molecular diameter (d) of a gas by the formula:
\[ B = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the gas molecule. The diameter \( d \) is related to the radius by:
\[ d = 2r \]

### Step 2: Substitute the given value
We are given that the molar excluded volume \( B = 3.2 \pi \, \text{ml} \). We can set up the equation:
\[ \frac{4}{3} \pi r^3 = 3.2 \pi \]

### Step 3: Simplify the equation
We can cancel \( \pi \) from both sides:
\[ \frac{4}{3} r^3 = 3.2 \]

### Step 4: Solve for \( r^3 \)
Multiply both sides by \( \frac{3}{4} \):
\[ r^3 = 3.2 \times \frac{3}{4} \]
\[ r^3 = 2.4 \]

### Step 5: Calculate \( r \)
Now, take the cube root of both sides to find \( r \):
\[ r = \sqrt[3]{2.4} \]

### Step 6: Calculate the diameter \( d \)
Now that we have \( r \), we can find the diameter \( d \):
\[ d = 2r = 2 \times \sqrt[3]{2.4} \]

### Step 7: Convert to nanometers
To convert the diameter from milliliters to nanometers, we need to remember that:
1 ml = \( 10^{-3} \, \text{L} \) and \( 1 \, \text{L} = 10^{-3} \, \text{m}^3 \).
Thus, \( 1 \, \text{ml} = 10^{-6} \, \text{m}^3 \).

Since we are calculating in nanometers, we need to convert meters to nanometers:
\[ 1 \, \text{m} = 10^{9} \, \text{nm} \]

### Final Calculation
Putting it all together, we get:
\[ d \text{ (in nm)} = 2 \times \sqrt[3]{2.4} \times 10^{9} \]

### Step 8: Numerical Calculation
Calculating \( \sqrt[3]{2.4} \) gives approximately \( 1.34 \):
\[ d \approx 2 \times 1.34 \times 10^{9} \]
\[ d \approx 2.68 \times 10^{9} \, \text{nm} \]

Thus, the molecular diameter is approximately **20 nanometers**.