$8\mathrm{.}6$ Consider the following overall reaction:

$\frac{1}{2}{A}_2+BC+D$

The mechanism has the following features:

a$.$ Molecules of $A_2$ dissociatively adsorb on two type $1$ sites $($one A atom on

each site$).$

b$.$ Molecules of B adsorb on type $2$ sites.

c$.$ The rate$-$determining step is the surface reaction between adsorbed A and

adsorbed B$.$

d$.$ Molecules of C adsorb on type $1$ sites and molecules of D adsorb on type $2$

sites.

Derive an LHHW$-$type rate expression that is consistent with this mechanism.