A drinking water treatment facility operates a JULIA process to destroy concentrated per$-$fluorinated octanoic acid $($PFOA$)$ from a drinking water treatment system. PFOA is concentrated and then sent to the JULIA process for destruction. In the JULIA reactor, PFOA is degraded according to first$-$order reaction kinetics. The system is just starting$-$up$,$ meaning that it is not operating at steady$-$state. At t$=0,$ the reactor is full of tap water $($from testing the equipment$)$ with a PFOA concentration of $0\mathrm{.}000001$ mg$/$L$.$ The PFOA concentration in the influent stream is $0\mathrm{.}25$ mg$/$L and the influent flow rate is $100,000$ gallons per day. The volume of the oxidation reactor is $50,000$ gallons and the first order reaction rate for destruction of PFOA is $3\mathrm{.}2$ d$-1.(1)$ Write a mass balance equation for PFOA. $(2)$ Estimate the steady state concentration of PFOA in the effluent. $(3)$ Integrate the mass balance equation and write an equation that describes the effluent concentration of PFOA over time. $(4)$ What will be the concentration of PFOA after $12$ hours of operation and how does this compare to the steady$-$state concentration?