A well$-$mixed vessel of volume, $V=50m^3,$ is half$-$filled with acetic acid solution at a concentration of $C_0(20\frac{kg}{m^3}).$ Pure water is fed at a steady flow rate of $Q_0(5\mathrm{.}0\frac{m^3}{h})$ into the vessel and the well$-$mixed solution is pumped from the vessel at the same rate. The concentration of salt in the exit fluid $C(t)k\frac{g}{m^3},$ is monitored.

a$.$ Derive the unsteady state differential material balance for the concentration of salt in the exit stream flowing from the vessel and show that it follows the following exponential relationship:

$C(t)=C_0$exp$(\frac{-Q_{0}t}{\frac{V}{2}})$

b$.$ Determine the concentration of acetic acid present in the fluid in the vessel after a period of $10$ hours.

c$.$ If the inlet flow rate had been $7\mathrm{.}5\frac{m^3}{h}$ and the exit flow was

$[10\%]$ maintained at $5\frac{m^3}{h},$ derive the unsteady state mass balance relationship for this case.

d$.$ Determine the volume of solution in the vessel after $10$ hours $30\%$ and the concentration of the acetic acid in the stream leaving the vessel.

e$.$ What would you need to do after the $10$ hour mark has been reached in d$?$

$[30\%]$

$5\%$

Please explain with steps, especially part d