The dehydrogenation of propane occurs by the following catalytic gas phase reversible reaction:

C$3$H$8$ C$3$H$6+$ H$2$

The surface reaction exhibits a dual site mechanism in which adsorbed propane reacts on the catalyst surface, the propylene and hydrogen adsorb on the two adjacent active sites, and they desorb finally. Establish expressions for the:

a$)$ rate law

b$)$ fraction of vacant sites

in terms of the rate constant of the reaction, equilibrium constants for the steps in the catalytic reaction, concentration of total active sites on the catalyst surface and partial pressures of the reactant and products, when the surface reaction is the rate limiting step in the catalytic reaction.