A feed stream with concentration $c_{f,1}$ enters Tank $1.$ The outlet of Tank $1$ is fed to Tank $2$ along with an additional feed stream with concentration $c_{f,2}.$ Mole balances for this system give the following differential equations for the solute concentrations in Tank $1$ and Tank $2,$ respectively:

$\frac{d{c}_1}{(d)t}={}_1^{-1}(c_{f,1}-c_1)$

$\frac{d{c}_2}{(d)t}={}_2^{-1}(\frac{c_{f,2}+c_1}{2}-c_2),$

where ${}_1$ and ${}_2$ are the residence times for each tank. Both tanks initially contain no solute.

$($a$)$ What are the steady$-$state concentrations $c_{,1}$ and $c_{,2}$ in each tank?

$($b$)$ Transform this system of equations to use the differences $c_1$ and $c_2$ between the concentrations $c_1$ and $c_2$ and their steady$-$state values $($i$.$e$.,c_i=c_i-c_{,i}$ for Tank $i)$ as the dependent variables, rather than $c_1$ and $c_2$ themselves.

$($c$)$ Solve for $c_1(t)$ and $c_2(t)$ by first finding $c_1(t)$ then finding $c_2(t).$

$($d$)$ If $c_{f,1}=1M,c_{f,2}=0\mathrm{.}5M,{}_1=2$min, and ${}_2=4$min, plot $c_1(t)$ and $c_2(t)$ along with their steady$-$state values.

$($e$)$ Rewrite the system of equations from $($b$)$ using matrices and vectors. Determine the eigenvalues and eigenvectors of the matrix for the parameters given in $($d$).$ Show that these eigenvalues and eigenvectors lead to the same solution as $($c$).$

$($f$)$ Use the result from $($c$)$ to derive expressions for $c_1(t)$ and $c_2(t)$ if ${}_1={}_2=.$