How to solve

t$-$Butyl alcohol $($TBA$)$ is an important octane enhancer that is used to replace lead additives in gasoline. TBA was produced by the liquid$-$phase hydration $($W$)$ of isobutene $($I$)$ over an Amberlyst$-15$ catalyst. The system is normally a multiphase mixture of hydrocarbon, water, and solid catalysts. However, the use of co$-$solvents or excess TBA can achieve reasonable miscibility.

The reaction mechanism is believed to be

$I+SI*S$

$W+SW*S$

$W*S+I*STBA*S+S$

TBA$*STBA+S$

Derive a rate law assuming:

$($a$)(15$ pts$)$ The surface reaction is rate$-$limiting

$($b$)(15$ pts$)$ The adsorption of isobutene is limiting

$($c$)(15$ pts$)$ The reaction follows Eley$-$Rideal kinetics $,I*S+WTBA*S$

And the surface reaction is limiting$.$

$($d$)(15$ pts$)$ Isobutene $($I$)$ and water $($W$)$ are adsorbed on different sites.

$I+S_{1}I*S_1$

$W+S_{2}W*S_2$

TBA is not on the surface, and the surface reaction is rate$-$limiting$.$

Show all your work after applying each of the assumptions above.