Q$5$ Hydrogen adsorption on copper catalyst can be fairly modeled considering $H_2$ dissociates upon adsorption, with each atom occupying one site on the catalyst.

$($a$)$ Derive the relation between the concentrations of hydrogen in the bulk and adsorbed phases.

$($b$)$ If the adsorption needs to occur on a pair of adjacent vacant sites, how do the equations change?

$($c$)$ If $O_2($which also dissociates upon adsorption$)$ is also present in the gas phase, how are the concentrations of adsorbed and bulk phases related?