Ratio of excluded volume to molecular volume:
In the gas phase, the ratio of excluded volume to molecular volume for a spherical molecule is 8. This means that the excluded volume, which refers to the space that is unavailable for other molecules due to the presence of a molecule, is eight times larger than the molecular volume of the molecule itself.

Explanation:
When considering the gas phase, molecules are assumed to be in constant motion and have negligible interactions with each other, except for brief collisions. In this scenario, the size and shape of the molecules become important factors in determining their behavior.

Molecular volume:
The molecular volume of a spherical molecule refers to the actual physical space occupied by the molecule. It is calculated based on the molecular dimensions and can be visualized as a sphere with a certain radius.

Excluded volume:
The excluded volume, on the other hand, takes into account the repulsive forces between molecules that prevent them from occupying the same space simultaneously. It represents the additional space required around a molecule to prevent overlap with other molecules.

Ratio of excluded volume to molecular volume:
The ratio of excluded volume to molecular volume is a measure of the relative size of the space that a molecule effectively occupies compared to the space it physically occupies. In the case of a spherical molecule, this ratio is 8.

Derivation:
To understand why the ratio is 8, we can consider a simplified model where molecules are treated as hard spheres with no attractive forces. In this model, the excluded volume can be approximated using the Van der Waals equation of state.

The Van der Waals equation takes into account the size of the molecules by introducing a correction term, called the excluded volume term. This term represents the space that is excluded or unavailable for other molecules due to the presence of a molecule.

For a spherical molecule, the excluded volume term can be calculated using the formula: V_ex = 4/3 * π * (r + b)^3, where r is the radius of the molecule and b is the excluded volume correction parameter.

Comparing this with the molecular volume formula V_mol = 4/3 * π * r^3, we can calculate the ratio of excluded volume to molecular volume:

Ratio = V_ex / V_mol = (4/3 * π * (r + b)^3) / (4/3 * π * r^3)
= (r + b)^3 / r^3

Assuming that the molecule is spherical and has a radius of r, we can simplify the equation further:

Ratio = (r + b)^3 / r^3
= (r/r)^3 + 3 * (r/r)^2 * (b/r) + 3 * (r/r) * (b/r)^2 + (b/r)^3
= 1 + 3 * (b/r) + 3 * (b/r)^2 + (b/r)^3

In the limit of small values of b/r, where the excluded volume is much larger than the molecular volume, the higher-order terms become negligible, and the ratio simplifies to:

Ratio ≈ 1 + 3 * (b/r)

Since the ratio is given as 8, we can solve the equation:

8 = 1 + 3 * (b/r)
3 * (b/r)