QUESTION $7.$ Deriving and solving a model with Laplace transforms. $(23$ pts$)$

Three stirred$-$tanks in series are used in a reactor train.

The flow rate into the system of some inert species is maintained constant while tracer test are

conducted. Assuming that mixing in each tank is perfect and the volumes are constant:

a$)$ Derive expressions for the concentrations of tracer leaving each tank. Here, $c_i$ is the

concentration of tracer entering the first tank. You can do this by developing an ODE$-$

based model for this system $($do not solve it yet$).$ Note in the image, $q=v$ for what I used

in CHEM E $465($volumetric flow rate$).$

b$)$ If $c_i$ has been constant and equal to zero for a long time and an operator suddenly injects

a large amount of tracer material in the inlet to tank $1,$ what will be the form of $c_3(t)($i$.$e$.,$

what types of time functions will be involved$)$ if:

i$.,V_1=V_2=V_3$

ii$.,V_1{V}_2{V}_3$

c$)$ If the amount of tracer injected is unknown, is it possible to back$-$calculate it from

experimental data? How would you do it$?$ This is a short, conceptual answer $($i$.$e$.,$ imagine

that you have a fully expression for $\{$:$c_3(t)).$