$(3)(50$p$.)$ Autocatalytic reaction in an isothermal batch reactor $($Nobel Prize research$)$

The overall reaction $A+BD+E$ occurs in a well$-$mixed, constant$-$volume, isothermal batch reactor that initially contains only species A and $B$ with concentration $C_{Ao}$ and $C_{Bo},$ respectively. The following mechanism has been proposed for this reaction: $Ax$

with $,r_1=k_1{C}_A$

$B+xY+D$

with $,r_2=k_2{C}_B{C}_x$

$(3)2x+Y3x$ with $r_3=k_3{C}_x^2{C}_Y($a not$-$so unrealistic tri$-$molecular step$)$

$(4),xE$

with $,r_4=k_4{C}_x$

$($a$)$ Write the mass balance equations describing the evolution of the concentrations of the six species present in the system, A$,$ B$,$ D$,$ E$,$ X$,$ and Y$,$ and the initial conditions. $(5$p$)$

$($b$)$ Assume that $x$ and $Y$ are produced and consumed at rates that are much faster than the ones for species A$,$ B$,$ D$,$ and E$.$ Derive expressions for the "steady" concentrations of the intermediates $C_{xs}$ and $C_{Y_s}$ as functions of rate constants, $C_A$ and $C_B.$

$($c$)$ Let's focus now on the time$-$dependent $($temporal$)$ behavior of $C_x$ and $C_Y$ over a time period that is short enough so that $C_A$ and $C_B$ can be assumed to be constant. Linearize the two rate equations that describe the time$-$evolution of $C_x$ and $C_Y$ in the vicinity of the stationary $("$steady state"$)$ solution $(($:$C_{x_s},C_{Y_s}\}$ by using a Taylor expansion. Write the resulting system of two linear ODEs and the two initial conditions with respect to deviation variables $($from the steady state$),C_l=(C_x-C_{x_s})$ and $C_2=(C_Y-C_{Y_s}).$ Show that the resulting system has the form:

$(d\frac{C_l}{d}t)=a{C}_l+b{C}_2$

$((d)\frac{C_2}{d}t)=c{C}_l-b{C}_2$

where $a,b,$ and $c$ are constants and with initial conditions at $t=0$:$(C_l,C_2)=(C_{10},C_{20}),$ and steady state: $.$

$($d$)$ Compute the eigenvalues of the linearized system of ODEs obtained in $($c$)$ and develop conditions that must be obeyed by the four rate constants, $C_A,$ and $C_B$ to guarantee stability of the steady state $(C_{xs},C_{Y_s}).$ What condition that must be obeyed for sustained oscillations? $(20$p$.)$

Note: This problem can produce sustained oscillations that correspond to a limit cycle on the phase plane for a specific set of values of the constants. It was derived from the classical papers on "Symmetry$-$Breaking Instabilities in Dissipative Systems" by I. Prigogine and G$.$ Nicolis, J$.$ Chem. Phys$.46,3542-3550(1967),$ and I. Prigogine and R$.$ Lefever, J$.$ Chem. Phys$.48,1695-1700(1968).$

Ilya Prigogine received the $1977$ Nobel Prize in Chemistry "for his contributions to non$-$equilibrium thermodynamics, particularly the theory of dissipative structures".