$(M4\mathrm{.}16)$ A reactor is used in a water reclamation plant. The influent concentration is $100m\frac{g}{L},$ the effluent concentration is $25m\frac{g}{L},$ and the flow rate through the reactor is $500$ gallons$/$min$.$

$($a$)$ What is the first$-$order rate constant for decay of BOD in the reactor? Assume the reactor can be modeled as a CFSTR$.$ Report your answer in units per hour.

$($b$)$ Assume the reactor should be modeled as a PFR with first$-$order decay, not as a CFSTR$.$ In that case, what must be the first$-$order decay rate constant within the PFR reactor?

$($c$)$ It has been determined that the outlet concentration is too high, so the residence time in the reactor must be doubled. Assuming all other variables remain constant what must be the volume of the new CFSTR$?$