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Our teacher told us that $\ce{OCH3}$ is a less bulky group when compared to other bulky groups like $\ce{CH3}$, $\ce{NO2}$ etc. Is this true? If yes, why?

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In organic chemistry bulkiness isn't just a matter of geometry; one can find the volume of the group using vdW radii, but it's a crude approximation. Quantitatively steric bulk relies on energy measurements and can be described with a cyclohexane A-value — energy required for a group to switch between axial and equatorial positions in cyclohexane. From the table of A-values methoxy group is indeed less bulk in comparison to nitro and methyl groups:

$$ \begin{array}{lc} \hline \text{Substituent} & \text{A-Value}/\pu{kcal mol-1} \\ \hline \ce{-OCH3} & 0.6 \\ \ce{-NO2} & 1.1 \\ \ce{-CH3} & 1.7 \\ \hline \end{array} $$

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