Assuming covalent radii to be additive property; calculate the iodine-iodine distances in \(o-, m-, p\) -di-iodobenzene. The benzene ring is regular hexagon and each \(C-\) I bond lies on a line passing through the centre of hexagon. The \(\mathrm{C}-\mathrm{C}\) bond length in \(\mathrm{C}_{6} \mathrm{H}_{6}\) are \(1.40 \AA\) and covalent radius of iodine and carbon atom are \(1.33 \AA\) and \(0.77 \AA\). Also neglect different overlapping effect.

Understanding covalent radii is fundamental to calculating bond lengths and distances between atoms in a molecule. The covalent radius of an atom is approximately half the distance between two identical atoms joined by a covalent bond in the same molecule. It represents the size of an atom that forms part of a single covalent bond.

For instance, in our exercise, the covalent radius of carbon is given as 0.77 Å, and that of iodine is 1.33 Å. When these two atoms form a bond, the sum of their covalent radii gives us the estimated length of that bond - in this case, for a C-I bond, it is 2.10 Å (0.77 Å + 1.33 Å).

It's important to remember that while the additive property of covalent radii simplifies our calculations, in reality, different factors like electronegativity differences and bond multiplicity could slightly alter these distances.

The geometrical structure of benzene is a perfect hexagon, where each side represents a carbon-carbon bond with an equal length. It is described as a planar molecule because all atoms are in the same three-dimensional plane. Because of its symmetry and equal bond lengths, which are 1.40 Å, benzene exhibits a structure where each internal angle is 120 degrees.

In the context of our problem, understanding this hexagonal symmetry is critical for determining the distances between substituents on the ring, such as iodine in di-iodobenzene. The hexagonal shape provides pathways to easily calculate the positions of o-, m-, or p- configurations for substituents.

The Pythagorean theorem is a principle often used in chemistry to determine distances between atoms in molecules, particularly when they form angles with each other. Applied to molecules, it helps us understand the three-dimensional arrangement of atoms.

In our case, for m-di-iodobenzene, which forms an isosceles triangle, the Pythagorean theorem allows us to calculate the I-I distance. By knowing the lengths of two sides (C-I bonds) and the base (C-C bonds), we consider the I-I distance as the hypotenuse of the triangle. The formula tells us that the square of the hypotenuse is equal to the sum of the squares of the other two sides, which gives us the ability to solve for the I-I distance.

In organic chemistry, isomers are compounds with the same molecular formula but different structures. Our exercise deals with such isomers called ortho-, meta-, and para-isomers, distinguished by their unique arrangements of iodine atoms on a benzene ring.

• Ortho- (o-) isomers have substituents adjacent to each other.
• Meta- (m-) isomers have substituents separated by one carbon atom.
• Para- (p-) isomers have substituents opposite to each other on the ring.

Because each isomer has a different geometric arrangement of iodine atoms, the I-I distances vary. Understanding these arrangements is crucial because it impacts the physical and chemical properties of the molecules, making isomerism an important concept in organic chemistry.