Develop the complete batch$-$reactor model for a exothermic elementary reaction:

$3$A$->5$B

Derive the overall material balance, component balances and energy balance equations starting from the basic laws of conservations. The reactor is jacketed and the temperature is controlled by a coolant flow $F_{}$ through the jacket. If the reactor temperature is controlled by a proportional controller with gain Kc$,$ explain the modification to be done in the final energy balance equation to accommodate the control law.