$A$ is converted to $B(AB)$ using two CSTR$\text{'}$s connecting in series. The liquid

phase reaction is carried out isothermally with suitable homogeneous catalyst.

During the reaction, the volumes of the reactors remain constant. The reaction

rate expression is given as $-r_A=k{C}_A(mo\frac{l}{m^3}(min)).$ The reaction rate constant is

$0\mathrm{.}5mi{n}^{-1}.$

a$)$ Obtain the expression for transfer function relating outlet concentration of the

second reactor and inlet concentration of $A.$

b$)$ Show that the response is always overdamped or critically damped.

c$)$ If the volumetric flow rate, inlet concentration of $A,$ the volumes of reactors are given as $0\mathrm{.}5\frac{m^3}{m}in,0\mathrm{.}5mo\frac{l}{m^3}$ and $5m^3,$ respectively, find the natural period, damping factor and ultimate values of concentration of $A$ in both reactors for an unit step increase in inlet concentration of $A.$

Please wirte your answer step by step and clearly. I'll remember to evaluate your answer.