$P$ is an important industrial product that can be produced from the gas$-$phase reaction $A+BP$ over a platinum catalyst. The limiting step in the reaction is known to be an irreversible surface reaction. The following data giving the reaction rate for $P$ were obtained in a differential reactor at $200C.$

$\backslash$table$[[P_A(atm),0\mathrm{.}15,0\mathrm{.}15,1\mathrm{.}5,0\mathrm{.}15,0\mathrm{.}15,3\mathrm{.}0,0\mathrm{.}015],[,,,,,,,],[P_B(atm),0\mathrm{.}15,1\mathrm{.}5,0\mathrm{.}15,3\mathrm{.}0,3\mathrm{.}0,0\mathrm{.}15,0\mathrm{.}15],[,,,,,,,],[P_P(atm),0\mathrm{.}3,0\mathrm{.}3,0\mathrm{.}3,0\mathrm{.}3,1\mathrm{.}5,0\mathrm{.}3,0\mathrm{.}3],[\backslash$table$[[$Reaction Rate$],[($mol$/$gcat$/$s$)]],0\mathrm{.}017,0\mathrm{.}170,0\mathrm{.}027,0\mathrm{.}340,0\mathrm{.}139,0\mathrm{.}028,0\mathrm{.}004]]$

$($a$)$ What is the rate law consistent with the experimental data?

$($b$)$ What is the reaction mechanism consistent with the rate law?

$($c$)$ The entering partial pressures of A and $B$ are $1\mathrm{.}5$atm in a packed$-$bed reactor. If the adsorption equilibrium constant for A and $P$ are: $K_A=20\mathrm{.}4at{m}^{-1}$ and $K_P=6\mathrm{.}0at{m}^{-1},$ what is the ratio of active sites occupied by $A$ to that occupied by $P$ at $50\%$ conversion of $A?$ What is the conversion if the number of sites occupied by A and $P$ are equal?