# Determination of products in reactions involving carbocation rearrangement?

How do you determine the "migratory aptitude" of various groups during carbocation rearrangements? Is there a experimentally determined order?
For example, what will be the product in case of the following pinacol-pinacolone type rearrangement:

Will the methyl group be transferred or the cyclopentyl moiety? The same question arises if the cyclopentyl is replaced with phenyl group? In general, which shifts, methyl, hydrogen, phenyl or any other group, will take place? Or equivalently, what determines which group has a greater tendency to undergo shifts during carbocation rearrangements?

## 2 Answers

All else equal, the smaller group migrates preferientially. The energy required to more the group is related to the mass of the group:

$$W=\int_0^\vec{x}\vec{F}\cdot d\vec{x}$$ $$\frac{d\vec{x}}{dt}=\vec{v} \ \ \ \ \ \therefore \ \ \ \ \ d\vec{x}=\vec{v}dt$$ $$W=\int_0^t\left(\vec{F}\cdot \vec{v}\right)dt$$$$\vec{F}=m\vec{a}$$ $$W=m\int_o^t \left(\vec{a}\cdot\vec{v}\right)dt=m\int\left(\vec{v}\cdot\vec{a}dt\right)$$ $$\vec{a}=\frac{d\vec{v}}{dt}$$ $$\vec{a}dt=d\vec{v}$$ $$W=m\int_0^\vec{v}\vec{v}\cdot d\vec{v}=\frac{1}{2}m\vec{v}^2\vert_0^\vec{v}=\frac{1}{2}m\vec{v}^2$$

The group best able to sustain a positive charge migrates according to the following order: $$\ce{H > Me3C > Me2HC > Et > Me}$$