$1$ The reaction

AlongrightarrowB

is to be carried out isothermally in a continuous$-$flow reactor that contains a recycle stream $($see below$).$

The following terms are known: recycle ratio $(\frac{Q_R}{Q})$ is $0\mathrm{.}5$ ; reactor volume $(V)$ is $100d{m}^3$; and, the initial concentration $(C_{A,0})$ is $50$mmo$\frac{l}{d}{m}^3.$

$($a$)$ A first order reaction,

$-r_A=k{C}_A$

is occurring in the reactor. Solve the system in general terms, and write an expression for the total molar flow rate of A out of the reactor $($see the star in the figure$)$ as a function of $Q,C_{A,0},k,$ and $V.$ Your expression should not contain any other terms.

$($b$)$ If $k=0\mathrm{.}2s^{-1},$ plot the total molar flow rate $(mol\frac{A}{s})$ out of the reactor as a function of $Q(d\frac{m^3}{s}),$ for $Q=0-10d\frac{m^3}{s}.$

$($c$)$ A second order reaction,

$-r_A=k{C}_A^2$

is occurring in the reactor. Solve the system in general terms, and write an expression for the total molar flow rate of A out of the reactor $($see the star in the figure$)$ as a function of $Q,C_{A,0},k,$ and $V.$ Your expression should not contain any other terms.

$($d$)$ If $k=0\mathrm{.}2d{m}^{3}mmo{l}^{-1}{s}^{-1},$ plot the total molar flow rate $($mol A$/$s$)$ as a function of $Q(d\frac{m^3}{s}),$ for $Q=0-10d\frac{m^3}{s}.$ Add the corresponding curve to the same plot from $($b$).$

$($e$)$ The contractor installed the recycle line backwards $($of course$),$ and $50\%$ of the influent flow is now bypassing the reactor. Resolve parts $($b$)$ and $($d$)$ for this situation and add those curves to the plot. Your plot should contain four curves. Describe and justify the differences between the four curves. Explain how a decrease in the recycle ratio would affect each scenario.