Most Likely Migrating Group in Carbocation Rearrangement

To determine which group is most likely to migrate to the positively charged carbon ($\text{C-3}$), we evaluate the relative stability of the resulting carbocations formed via 1,2-shifts from the adjacent carbons ($\text{C-2}$ and $\text{C-4}$):

1. Shift from $\text{C-4}$:

   • Hydride ($\text{H}^-$) shift from $\text{C-4}$:
     Moving the hydride from $\text{C-4}$ to $\text{C-3}$ produces a tertiary ($3^\circ$) carbocation at $\text{C-4}$, which is stabilized primarily by hyperconjugation and inductive effects:
     $\text{CH}_3-\text{CH(OH)}-\text{CH}_2-\overset{+}{\text{C}}(\text{CH}_3)_2$
   • Methyl ($\text{CH}_3^-$) shift from $\text{C-4}$:
     Hydride shifts have a much higher migratory aptitude than methyl shifts; furthermore, a methyl shift from $\text{C-4}$ yields a less stable secondary carbocation.

2. Shift from $\text{C-2}$:

   • Hydride ($\text{H}^-$) shift from $\text{C-2}$:
     Moving the hydride from $\text{C-2}$ to $\text{C-3}$ creates a carbocation directly on $\text{C-2}$:
     $\text{CH}_3-\overset{+}{\text{C}}(\text{OH})-\text{CH}_2-\text{CH}(\text{CH}_3)_2 \longleftrightarrow \text{CH}_3-\text{C}(=\overset{+}{\text{O}}\text{H})-\text{CH}_2-\text{CH}(\text{CH}_3)_2$
     This carbocation is extraordinarily stable due to the strong resonance donation ($+M$ effect) of the lone pair of electrons on the oxygen atom of the $-\text{OH}$ group, which allows every atom to attain a complete octet.
   • Methyl ($\text{CH}_3^-$) shift from $\text{C-2}$:
     There is no methyl group attached directly to $\text{C-2}$ (only $-\text{H}$, $-\text{OH}$, $\text{C-1}$ methyl group, and $\text{C-3}$). Even considering migratory aptitude, hydride shifts are significantly faster than alkyl shifts ($\text{H}^- > \text{CH}_3^-$).

Since the resonance-stabilized (octet-complete) oxocarbenium ion formed by the 1,2-hydride shift from $\text{C-2}$ is far more stable than the $3^\circ$ carbocation formed by migration from $\text{C-4}$, the migration of $\text{H}$ at $\text{C-2}$ provides the greatest thermodynamic driving force and proceeds with the lowest activation energy.

Therefore, the group that is most likely to migrate is $\text{H}$ at $\text{C-2}$.

Correct Answer: D