The first order exothermic irreversible reaction a- b is carried out in a perfectly mixed batch reactor with no change in volume. the rate constant is given by an arrhenius expression with pre-exponential factor k and activation energy ea. in addition, the molar density of the fluid is p, the molar heat capacity is cp, and the heat of reaction is δηr, where δhrs 0. you may assume that heat is lost from the reactor to the environment with constant overall heat transfer coefficient u and a heat exchange area a, so that the rate of heat removal from the reactor is proportional to t - to, where to is the ambient temperature. if the initial concentration of a is cao and the initial reactor temperature is the ambient temperature to, derive mass and energy balances for the concentration ca and temperature t as a function of time. next, linearize these equations for perturbations in ca and t around the initial conditions. what two methods are available to solve the resulting set of equations?