Consider the reversible, isothermal, gas$-$phase reaction:

$A+1\mathrm{.}5B2C$

$k_1$ is the rate constant for the forward reaction and $k_2$ the rate constant for the backward reaction. The forward and backward reactions are elementary.

$($a$)$ Write the reaction rate expressions below in terms of the concentration of species $A,B$ and $C.$

$($i$)-r_A$

$($ii$)r_B$

$($iii$)rc$

$[8$ marks$]$

$($b$)$ If the species A and $B$ are fed to a flow reactor at a molar ratio $\frac{F_{A0}}{F_{B0}}=1\mathrm{.}25,$ further express the rate law for this reversible reaction exclusively in terms of the conversion $(x)$ of the limiting reactant, $k_1$ and $k_2,$ with all other terms of the rate law reported as numerical values. The concentration of the limiting reactant in the feed stream to the reactor is $1mold{m}^{-3}.$

$[12$ marks$]$