There are two stereoisomers of 1-tert-butyl-4-chloro-cyclohexane. One of these isomers reacts with sodium ethoxide in an E2 reaction that is 500 times faster than the reaction of the other isomer. Identify the isomer that reacts faster, and explain the difference in rate for these two isomers.

The cis-isomer quickly follows E2 reaction due to the fact that it is locked in a chair conformation whereby chlorine dominates an axial position, as displayed in drawing C as well as the leaving group Cl which is antiperiplanar to a beta proton.

Due to the bulkiness of tert-butyl, the trans-isomer is Basically trapped in a chair configuration, with chlorine occupying an equatorial position as displayed in drawing A below, thus leaving the group isn't antiperiplanar to a beta proton.

Tert butyl favors an equatorial position due to its bulkiness. This makes the Cl to assume an axial position in an isomer

The difference:

From the second image attached below, in the first diagram(1):

Suppose H is antiperiplanar, C-H bond electrons reach antibonding orbitals of C-Cl, assisting in the breakdown of the C-Cl bond and therefore speeding up the process.

From the second diagram(2) in the second image:

Unless Cl is in the axial position would it have β-hydrogen in the antiperiplanar position. Because tert-butyl prefers equatorial location, the Cl occupies equatorial position in the trans isomer.

As a result, the reaction is slower and there is no β-hydrogen antiperiplanar to Cl.