$($a$)($i $)$ Explain the difference between the Langmuir $-$ Hinshelwood and Eley $-$ Rideal mechanisms for the catalytic reaction A$($g $)+$B$($g $)->$C$($g $)$

$($ii$)$ The rate of the catalytic reaction A$($g $)+$B$($g $)->$C $($g $)$ is given by the equation r $=$k pa pb $/(1+$Ka pa $)$

where r is the rate of the reaction, pa and pb the pressures of reactants A and B respectively, k is a constant and Ka is the equilibrium constant for the sorption of A$.$ Develop a reaction mechanism and suggest a rate$-$limiting step that satisfies the rate equation.

$($b$)$ The equilibrium constant b $($Langmuir constant$)$ for the dissociative chemisorption of hydrogen $($H$2)$ at a certain temperature on the above Pt catalyst is equal to $0\mathrm{.}8$ kPa$-1.$ Calculate the pressure of hydrogen $($H$2)$ for which $0\mathrm{.}25$ of the active centers of platinum will be occupied by chemisorbed hydrogen atoms $($Ha$).$