$(30$ points$)$ Mass Balance: A completely mixed sewage lagoon receives $15,000\frac{m^3}{d}$ of sewage containing $200m\frac{g}{L}$ of biodegradable organic matter. The lagoon has a surface area of $10$ hectare $($ha$)$ and a depth of $1\mathrm{.}0$ m $.$ The organic matter $($OM$)$ in the sewage degrades biologically $($decays$)$ according to first$-$order kinetics with a reaction coefficient of $0\mathrm{.}65d^{-1}.$ Assume no additional inputs and outputs $($evaporation$,$ rainfall, etc.$),$ steadystate conditions, and that the volume of the lagoon remains constant. Find the following:

a$.(10)$ The OM concentration in the effluent $($outflow$)$ from the lagoon in $m\frac{g}{L}($Do a mass balance to solve this part$).$

b$.(5)$ The OM concentration in the lagoon $($in $m\frac{g}{L})$ after mixing and degradation

c$.(5)$ The retention time of the lagoon in days

d$.(10)$ If you were to model the same reactor as a PFR$,$ what would the OM concentration in the effluent be using same kinetics and steady$-$state conditions? Compare this result to the value found in $($a$)$ and comment.