Your company has two liquid streams available containing solutes that are not profitably marketable at the present time. One stream contains an aqueous solution of A$.$ The second stream contains an aqueous solution of species B$.$ Species A can react either with species B or with itself according to the following stoichiometric equations and rate expressions:$A+B,V($ valued product $),r_V=k_1{C}_A{C}_B$

$2A,W($ undesired $by-$product $),r_W,=\frac{k_2{C}_A^2}{1+k_3{C}_V^{\frac{1}{2}}}$

$($a$)$ At a specific temperature $($T$)$: k$1=0\mathrm{.}2($M$-1.$min$-1),$ k$2=0\mathrm{.}1($M$-1.$min$-1),$ k$3=0\mathrm{.}3($M$-0\mathrm{.}5).$ When a $2$L aqueous solution containing $1$M of A mixed quickly with a $5$L aqueous solution containing $1$M of B:

$-$ Plot the concentration of A as a function of reaction time $($up to $95\%$ A reacted$)$

$-$ Plot the concentration of W as a function of reaction time $($up to $95\%$ A reacted$)$

$($b$)$ You have two beakers containing samples of the two streams and desire to carry out a small$-$scale laboratory experiment in which you maximize the formation of species V$.$ In what manner should you carry out this experiment; that is$,$ in what order and at what rate would you add each beaker of the reactants to an empty container?

$($c$)$ If the activation energies for the rate constants k$1,$ k$2,$ and k$3$ are all $40$ kJ$/$mol$,$ what additional statements can you make regarding the operating temperature recommended for increasing production of species V$?$