$12-8$ A liquid$-$phase reaction $A+BC$ is conducted in a $50-L$ batch reactor. The reaction is first$-$order with respect to each reactant.

$($a$)$ Determine the time required for $90\%$ conversion of $A,if(i)$ the reaction occurs adiabatically; $($ii$)$ the reaction occurs isothermally at $T_o.$

$($b$)$ Determine $Q^{}$ and $T_c($as functions of time$),$ if a cooling coil is placed in the tank to maintain the isothermal conditions required in $($a$)$ part $($ii$).$

$($c$)$ For $($a$)$ part $($i$),$ sketch the conversion$-$versus$-$time and temperature$-$versus$-$time profiles. Data: $c_{Ao}$ and $c_{Bo}$ are $0\mathrm{.}50mol{L}^{-1}$ and $0\mathrm{.}75mol{L}^{-1},$ respectively. The initial temperature $(T_o)$ is $400K,$ and the heat capacity of the reactor contents is $3\mathrm{.}8J{g}^{-1}{K}^{-1}.$ The fluid density is $0\mathrm{.}75gc{m}^{-3},$ and the heat transfer parameter $(U{A}_c)$ for part $($b$)$ is $100W{K}^{-1}.$ The reaction is exothermic $(-145(kJ)((mol)(A){)}^{-1}),$ and $k_A=1\mathrm{.}4{10}^7{e}^{-\frac{7700}{T}}Lmo{l}^{-1}mi{n}^{-1}.$