To determine the rate of addition of a 1-hexyl radical to the given molecules, we must consider the stability of the resulting radical intermediates. The more stable the radical, the faster the reaction is likely to proceed.
Molecule P is an ester with a vinyl group adjacent to the ester functional group. The addition of a 1-hexyl radical to the double bond will generate a primary radical stabilized only through hyper-conjugation. This is relatively less stable compared to radicals that can benefit from other forms of stabilization.
Molecule Q features an additional methyl group adjacent to the double bond. This offers some hyper-conjugation and inductive effects that can stabilize the radical, but not significantly better than P.
Molecule R contains two ester groups on its double bond. When a radical adds to this molecule, significant resonance stabilization is possible because of the involvement of these electron-withdrawing groups. This makes the resulting radical highly stable, favoring a faster rate of addition.
Based on the above analysis, the order of the rate of radical addition is determined by the stability of the intermediate radicals. Thus, the correct order is:
R > P > Q
This is because the resonance and electron-withdrawing groups in R make it the most favorable for radical stability and therefore the fastest for the addition reaction.