$Q{C}_{in}-Q_{\text{o}\text{u}\text{t}}+vC$

$(40$ points$)$ A $500m^3$ reactor is filled with clean water. At $t=0$; the pump is turned on$,$ pumping in a salt solution having a concentration of $100m\frac{g}{L}$ at a flowrate of $50\frac{m^3}{?}$ day $($approximately $0\mathrm{.}6\frac{L}{s}.$ The volume of solution in the reactor is constant.

a$.$ Draw a picture of the system, showing the control volume, system boundary, flows in and out of the reactor. $(2$ points$)$

b$.$ Write a mass balance equation in words. $(2$ points$)$

c$.$ Write the mass balance equation using mathematic terms. $(6$ points$)$

d$.$ Draw a graph showing how you expect the concentration in the reactor to vary with time.

e$.$ What is the concentration after $20$ days, $60$ days, and $80$ days? $(15$ points$)$

f$.$ Assume the initial concentration is $50m\frac{g}{L}$ and the concentration in the influent is zero. What is the concentration after $20$ days and $80$ days? $(15$ points$)$