The chlorination of methane proceeds via a free radical chain mechanism involving initiation, propagation, and termination steps. During the propagation step, chlorine free radicals attack methane to generate methyl free radicals ($\text{H}_3 \dot{\text{C}}$).

In the chain terminating step, these highly reactive methyl free radicals can combine with each other. When two methyl free radicals react, they form a C-C bond, resulting in the formation of ethane ($\text{H}_3\text{C} - \text{CH}_3$).

The reaction is: $\text{H}_3 \dot{\text{C}} + \dot{\text{CH}}_3 \rightarrow \text{H}_3\text{C} - \text{CH}_3$. This accounts for the formation of ethane as a byproduct during the chlorination of methane.