$2\mathrm{.}9$ Irreversible consecutive reactions $A{k}_{1}B{k}_{2}C$ occur in a jacketed, stirred$-$tank reactor as shown in Fig. E$2\mathrm{.}9.$ Derive a dynamic model based on the following assumptions:

$($i$)$ The contents of the tank and cooling jacket are well mixed. The volumes of material in the jacket and in the tank do not vary with time.

$($ii$)$ The reaction rates are given by