The question asks about the mechanism of chlorination of benzene under photochemical conditions. This is a specific reaction condition that leads to a particular reaction pathway.

Understanding Chlorination of Benzene

Chlorination of benzene can occur through different mechanisms depending on the reaction conditions. Under typical conditions with a Lewis acid catalyst like FeCl$_3$ or AlCl$_3$, benzene undergoes electrophilic substitution, where a chlorine atom replaces a hydrogen atom on the benzene ring, producing chlorobenzene.

However, the question specifies photochemical condition. This usually means the presence of UV light (indicated by $\text{h}\nu$). UV light provides the energy needed to break certain chemical bonds and initiate reactions involving free radicals.

Why Photochemical Condition Leads to Free Radical Mechanism

When chlorine gas ($\text{Cl}_2$) is exposed to UV light, the weak chlorine-chlorine bond breaks homolytically, forming two highly reactive chlorine free radicals ($\text{Cl}•$). This is the initiation step of a free radical reaction. The presence of these free radicals dictates the reaction pathway.

Benzene reacts with these chlorine free radicals differently compared to how it reacts with electrophiles. Instead of substitution, the free radicals attack the double bonds of the benzene ring, leading to an addition reaction.

Thus, chlorination of benzene under photochemical condition proceeds via a free radical mechanism.

Steps in the Free Radical Mechanism

The free radical mechanism for the chlorination of benzene under UV light involves three main steps:

1. Initiation: The chlorine molecule absorbs UV light and dissociates into chlorine free radicals. $$ \text{Cl}_2 \xrightarrow{h\nu} 2\text{Cl}• $$
2. Propagation: The chlorine radical adds to a double bond in the benzene ring, forming a new radical. This radical then reacts with another chlorine molecule, adding a chlorine atom and regenerating a chlorine radical. These steps repeat, adding chlorine atoms across the ring's double bonds.
3. Termination: Free radicals combine with each other to form stable molecules, ending the chain reaction. For example, two chlorine radicals can combine to form Cl$_2$.

The Product of Photochemical Chlorination of Benzene

Under photochemical condition, the chlorination of benzene is an addition reaction, not a substitution. Chlorine atoms add to each carbon atom of the benzene ring, breaking the aromaticity.

The final product is Benzene Hexachloride (BHC), which has the formula $\text{C}_6\text{H}_6\text{Cl}_6$. This is formed by the addition of three chlorine molecules to one benzene molecule:

This reaction clearly shows that the mechanism is not substitution but addition reaction, driven by the free radical mechanism initiated by UV light.

In contrast, electrophilic substitution would replace one hydrogen with one chlorine, forming $\text{C}_6\text{H}_5\text{Cl}$ (chlorobenzene).

Therefore, the chlorination of benzene under photochemical conditions takes place by a free radical mechanism.