A CSTR must be designed for a liquid phase reaction $A$ products. The reaction is pseudo$-$first order

with respect to $A,$ with $A=4\mathrm{.}0{10}^{13}{s}^{-1},$ and $E_A=90,000\frac{J}{m}ol.$ The feed is an aqueous solution of

$A(2\mathrm{.}0M),$ with a density of $1\mathrm{.}02k\frac{g}{L}$ and a temperature of $320K.$ The heat capacity of the mixture is

$3\mathrm{.}9kJk{g}^{-1}{K}^{-1},$ and the reaction is strongly exothermic, with an enthalpy of $-50kJ$ per mol of A reacted.

a$)$ The mathematical relationship between $f_A$ and $T$ based upon the steady state material

balance is shown in graphical form on the next page. Show how this relationship is

developed, from first principles, based upon the information provided above.

b$)$ Determine the maximum and minimum possible steady state conversion and operating

temperature that could be achieved in this system.

c$)$ The failure of an upstream heat exchanger means that the inlet temperature of the feed is

reduced by $15C.$ What effect, if any, will this have upon system operation and steady state

operating conditions?