A semi$-$batch reactor initially with $50$ liters is used to process the following set of elementary multiple reactions in an isothermal manner:

$A+BC,k_1=2\mathrm{.}019$

Charr$2E,k_2=0\mathrm{.}850,k_{-2}=0\mathrm{.}57,$ Where $k_2$ is the rate constant of the forward reaction and $k_{-2}$ of the return reaction.

$2$AharrD,$k_3=0\mathrm{.}900,k_{-3}=0\mathrm{.}043$ Where $k_3$ is the rate constant of the forward reaction and $k_{-3}$ of the return reaction.

At the beginning of the reaction $)=(0$min moles of A and $1$ mole of $B$ are available to be added to the system, and the initial concentrations of $C,D$ and $E$ are equal to $0.$ It is known that the solution to this problem involves solving the system of differential equations obtained from the information provided in the stoichiometry of the reaction and the problem formulation data. Knowing this and based on the information given:

$($a$)$ write the mass balance equations necessary to solve the proposed problem, and the respective boundary conditions.

$($b$)$ Establish which of the two available compounds must be continuously added to the other. Justify your answer based on the theory of instantaneous selectivity.

$($c$)$ Obtain graphs for the concentration of the compounds involved in the process as a function of time $($establish the calculation basis that you consider convenient to solve your problem$).$