Final answer:
The trans-1-bromo-3-tert-butylcyclohexane isomer will react faster in an E2 elimination reaction because of its favorable anti-periplanar geometry, which is not as readily available in the cis isomer due to steric hinderance.
Step-by-step explanation:
E2 Elimination in Stereoisomers of Cyclohexane
In an E2 elimination reaction, the reaction rate is dependent on the ability of a base to abstract a proton from a carbon adjacent to the one bearing the leaving group, in this case, a bromide ion. For a cycloalkane like bromocyclohexane, the trans isomer will react faster in an E2 reaction compared to the cis isomer. This difference in reactivity is due to the fact that in the trans isomer, the hydrogen atom to be abstracted and the leaving group bromide are positioned in an anti-periplanar arrangement which is geometrically ideal for E2 reactions.
The cis isomer, on the other hand, does not readily allow for this arrangement due to steric hinderance between the bulky tert-butyl group and the base that hinders the extraction of the adjacent hydrogen and the leaving of the bromide. Therefore, the trans-1-bromo-3-tert-butylcyclohexane will undergo E2 elimination more readily than the cis counterpart.