$3\mathrm{.}20$ Three stirred$-$tanks in series are used in a reactor train $($see Fig. E$3\mathrm{.}20).$ 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 volumes are constant:

$($a$)$ Derive model expressions for the concentration of tracer leaving each tank, $c_i$ is the concentration of tracer entering the first tank.

$($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 kind of time functions will be involved$)$ if

$V_1=V_2=V_3$

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

$($c$)$ If the amount of tracer injected is unknown, is it possible to back$-$calculate the amount from experimental data? How?