Batch experiments were conducted at $50C$ in a laboratory to determine the reaction kinetics data for reactor design. It is predicted that the reaction is elementary and follows the reaction scheme $2AB.$ The initial concentration of reactant A was $3\mathrm{.}0mo\frac{l}{d}{m}^3.$ Samples were taken at various time intervals and the results are reported in TABLE Q$3.$

TABLE Q$3$: Laboratory experimental data

$\backslash$table$[[$Time $($min$),$Conversion of A $(\%)],[5,4\mathrm{.}9],[13,11\mathrm{.}2],[34,25\mathrm{.}6],[45,31\mathrm{.}6],[80,45\mathrm{.}3]]$

a$.$ Prove that the above reaction is elementary. Then develop a complete rate law equation with its specific rate constant.

$[10$ marks$]$

b$.$ Propose the best flow reactor to produce $120mo\frac{l}{m}in$ of product $B$ at $90C($activation energy, $E,$ is $50k\frac{J}{m}ol)$ when liquid reactant $A$ is supplied at $300mo\frac{l}{m}in.$ Justify your answer.

$[15$ marks$]$