Why is the excluded volume v four times the actual volume of molecules?

Why is the excluded volume $v$ four times the actual volume of molecules?

To understand why the excluded volume, represented as $v$, is four times the actual volume of molecules, we need to delve into the concept of excluded volume and its calculation.

Explanation:

When we talk about excluded volume, we're referring to the space around a molecule that is inaccessible to other molecules. This concept plays a crucial role in the Van der Waals equation for real gases, which considers the finite volume of molecules.

Step-by-Step Derivation:

1. Van der Waals Equation: The Van der Waals equation for real gases is given by: $$ \left( P + \frac{an^2}{V^2} \right) (V - nb) = nRT $$ Here, $b$ represents the excluded volume per mole of gas.

2. Considering Two Molecules: Let’s consider two spherical molecules, A and B. Assume that each molecule has a radius $r$.

3. Distance Between Molecules: The centers of the two molecules cannot come closer than a distance of $2r$. This is because the molecules occupy space and, thus, cannot overlap.

4. Volume Calculation: The excluded volume for a pair of molecules is equivalent to the volume of a sphere with radius $2r$ (since the center-to-center distance is $2r$):

   $$ \text{Volume} = \frac{4}{3} \pi (2r)^3 = \frac{4}{3} \pi (8r^3) = 8 \left( \frac{4}{3} \pi r^3 \right) $$

   This equation means the excluded volume for a pair of molecules is 8 times the volume of a single molecule.

5. Excluded Volume Per Molecule: Since the aforementioned calculation is for a pair of molecules, the excluded volume per molecule is:

   $$ \frac{1}{2} \times 8 \left( \frac{4}{3} \pi r^3 \right) = 4 \left( \frac{4}{3} \pi r^3 \right) $$

6. Actual Volume of a Molecule: The actual volume of a single molecule (a sphere with radius $r$) is:

   $$ \frac{4}{3} \pi r^3 $$

7. Conclusion: Comparing the excluded volume per molecule with the actual volume of the molecule, we get:

   $$ b = 4 \left( \frac{4}{3} \pi r^3 \right) = 4 \left( \text{actual volume of the molecule} \right) $$

Final Answer:

Hence, the excluded volume $v$ is four times the actual volume of the molecules. This is because the excluded volume for a pair of molecules was found to be 8 times the volume of a single molecule, and thus, for an individual molecule, the excluded volume ratio is reduced to 4 times its actual volume.