USE EXCEL: The irreversible first$-$order reaction: $AB$ takes place in

two identical well$-$mixed reactors in series $($as shown in

the schematic$).$ The unsteady$-$state mass balances for

each chemical species in both reactors are:

$\frac{d{C}_{A1}}{dt}=\frac{1}{r}(C_{A0}-C_{A1})-k{C}_{A1}$

$\frac{d{C}_{B1}}{dt}=-\frac{1}{r}{C}_{B1}+k{C}_{A1}$

$\frac{d{C}_{A2}}{dt}=\frac{1}{r}(C_{A1}-C_{A2})-k{C}_{A2}$

$\frac{d{C}_{B2}}{dt}=\frac{1}{r}(C_{B1}-C_{B2})+k{C}_{A2}$

where $C_{A0}$ is the feed concentration to the first reactor,

$C_{A1}$ and $C_{B1}$ are the exit concentrations from the first

reactor and feed to the second reactor.

$C_{A2}$ and $C_{B2}$ are the exit concentrations from the second

reactor, $$ is the residence time in a reactor, and $k$ is the

reaction rate constant.

$($a$)$ Using Excel and a $4^{th}$ Order Runge$-$Kutta approach,

find the concentrations of A and B in each of the reactors

during the first $10$ minutes of operation if $C_{A0}$ is $1M,=4$

min, and $k=\frac{0\mathrm{.}5}{m}in.$ The initial concentrations of all

chemical species are $0$ in both reactors. For consistency,

use a time spacing $(h)$ of $15$ seconds $(0\mathrm{.}25$min $).$ Plot your

results for all four changing concentrations $,$

and $C_{32}).$

$($b$)$ Report the concentrations $($M$)$ of species A and B in

the final reactor after $5$ minutes have elapsed.