$(35$ points$)$ An irreversible, first$-$order reaction takes place in four well$-$mixed reactors as shown in the

figure below according to

$AkB$

Thus, the rate at which $A$ is transformed to $B$ can be represented as

$R_{ab}=kVc$

where $k$ is the $1$ st$-$order reaction rate constant $(\frac{1}{h}),V$ is the volume of the reactor $($L$),$ and $c$ is the

concentration of $A(mo\frac{l}{L}).$ The reactors have different volumes, and because they are operated at

different temperatures, each has a different reaction rate as given in the following table. Q represents the

volumetric flow rate in $\frac{L}{h}.$

a$)$ Make component balances for $A$ on each reactor to derive the system of linear equations which

describe the steady state concentration of $A$ in each reactor.

$\{[$ Rate $of$ accumulation $],[ofA]\}=\{[$ Rate $of],[Ain]\}-\{[$ Rate $of],[$ A out $]\}+-\{[$ Rate $of$ generatio$\frac{n}{c}$onsumption $],[ofA]\}$

b$)$ Determine the concentration of species $A$ in each reactor at steady state using Cramer's rule.

c$)$ Calculate what percentage of the steady state concentration of species A in reactor $2$ is due to the

flow from reactor $1.$