Given a batch reactor with a series reaction in which species A reacts reversibly to form the desired specie B$.$ In this reaction, it is observed that specie $B$ react to form an undesired specie $C$

where $k_{1f}$ and $k_{1r}$ represent the rate constants for the forward and reverse reactions for the conversion of species A and $B,$ while $k_2$ is the rate constant for the conversion of species $B$ to species $C.$

If it is assumed that each of the reactions is of first$-$order then the modeling equations are given by

$\frac{d{C}_A}{dt}=-k_{1f}{C}_A+k_{1r}{C}_B$

$\frac{d{C}_B}{dt}=k_{1f}{C}_A-k_{1r}{C}_B-k_2{C}_B$

$\frac{d{C}_C}{dt}=k_2{C}_B$

where $C_A,C_B$ and $C_C$ are the concentrations $($mol$/$volume$)$ of components A$,B,$ and $C.$

a$.$ Given the following dimensionless quantities:

The dimensionless time: $=k_{1}t$

Conversion of species A: $x_1=\frac{C_{A0}-C_A}{C_{A0}}$

Dimensionless concentration of species B: $x_2=\frac{C_B}{C_{A0}}$

Ratio of rate coefficients: $=\frac{k_2}{k_{1f}}$

Ratio of forward and reverse rate coefficients: $=\frac{k_{1r}}{k_{1f}}$

Show that the equation for the dimensionless concentration of species $B$ is

$\frac{d^2{x}_2}{d{}^2}+(++1)\frac{d{x}_2}{d}+x_2=0$

and that the roots of the characteristic equation can never be complex or unstable for positive rate coefficients.

b$.$ Determine $x_2(,,).$

c$.$ Given $k_{1f}=2,k_{1_r}=1,$ and $k_2=1\mathrm{.}25h{r}^{-1}$ use a computer algebraic system $($e$.$g$.,$ Mathematica ${?}^{})$ to determine the maximum conversion of species A to species B and the reaction time required for this conversion.

d$.$ Suppose the real value of $k_2$ is $1\mathrm{.}5h{r}^{-1}$ instead of the previous $(1\mathrm{.}25h{r}^{-1})$ and if the reaction is run for the time found in part c$),$ what will be the actual conversion of $A$ to $B?$