The chemical reactions $ABC$ take place in the liquid phase in an adiabatic ideal

CSTR with mean residence time $.$

The reaction $AB$ obeys first$-$order kinetics with rate constant $k_1$ and is endothermic

with enthalpy of reaction $H_1=hkJkmo{l}^1.$

The second reaction $BC$ obeys first$-$order kinetics with rate constant $k_2$ and is

exothermic with enthalpy of reaction $H_2=-2kJkmo{l}^{-1}.$

The feed to the reactor is pure species $A$ at temperature $T_0($in Kelvin$).$ The heat

capacity of all species is $C_p($in $kJmmo{l}^{-1}{K}^{-1}).$

$($a$)$ By performing material balances, derive expressions for the concentration of $A,B$

and $C$ in the reactor in terms of $C_{AO}.$

$($b$)$ Perform a heat balance to show that the temperature $T$ in the reactor obeys:

$T-T_0=\frac{h}{C_p}(\frac{k_1}{1+k_1})(\frac{2k_2}{1+k_2}-1)$

$($c$)$ Identify the temperature change that would take place in the reactor, and give a

physical explanation in each case, if the conditions are such that:

$($i$)k_{1}1$ and $k_{2}1$

$($ii$)k_1>1$ and $k_{2}1$

$($iii$)k_1>>1$ and $k_2>>1.$