\(\mathrm{C}_{6} \mathrm{H}_{5}-\mathrm{CH}_{3}+\mathrm{Cl}_{2} \stackrel{800 \mathrm{~K}}{\longrightarrow} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2} \mathrm{Cl}+\mathrm{HCl}\). This reaction takes place mainly by (a) \(\mathrm{S}_{\mathrm{N}} 1\) mechanism (b) \(\mathrm{S}_{\mathrm{N}} 2\) mechanism (c) free radical mechanism (d) E1 mechanism

In this reaction, toluene (\(\mathrm{C}_{6} \mathrm{H}_{5}-\mathrm{CH}_{3}\)) reacts with chlorine (\(\mathrm{Cl}_{2}\)) at 800 K to form benzyl chloride (\(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2}\mathrm{Cl}\)) and hydrogen chloride (\(\mathrm{HCl}\)).

We need to examine the characteristics of each mechanism and see which one is most suitable for this reaction. (a) \(\mathrm{S_N}1\) mechanism: This mechanism involves the formation of a carbocation intermediate where the nucleophile attacks the positively charged carbon atom. However, in this reaction, the intermediate formed would be a benzyl carbocation, which is not very stable. Thus, it is less likely for the reaction to proceed via this mechanism. (b) \(\mathrm{S_N}2\) mechanism: This mechanism involves a direct, concerted attack of the nucleophile on the substrate and does not involve the formation of intermediates. However, the reaction conditions (800 K) are very unfavorable for the \(\mathrm{S_N}2\) mechanism, and the benzyl group is not a good leaving group in this case. Thus, it is less likely for the reaction to proceed via this mechanism. (c) Free radical mechanism: This mechanism involves the formation of free radicals as intermediates. At high temperatures (such as 800 K), the homolytic cleavage of \(\mathrm{Cl}_{2}\) forms two highly reactive chlorine atoms, which can initiate the free radical mechanism. (d) E1 mechanism: This mechanism involves the formation of a carbocation intermediate and subsequent elimination of a leaving group in two separate steps. Since the formation of a benzyl carbocation is not favorable, it is less likely for the reaction to proceed via this mechanism.

Based on the analysis of each mechanism, the free radical mechanism seems to be the most plausible mechanism for this reaction. This is because of the high reaction temperature (800 K), which results in the formation of reactive chlorine atoms, capable of initiating the free radical pathway. Therefore, the correct answer is (c) free radical mechanism.