Mathematical description of the nonlinear chemical reactions with oscillatory inflow to the reaction field

Abstract

In this paper the arduous attempt to find a mathematical solution for the nonlinear autocatalytic chemical processes with a time-varying and oscillating inflow of reactant to the reaction medium has been taken. Approximate analytical solution is proposed. Numerical solutions and analytical attempts to solve the non-linear differential equation indicates a phase shift between the oscillatory influx of intermediate reaction reagent X to the medium of chemical reaction and the change of its concentration in this medium. Analytical solutions indicate that this shift may be associated with the reaction rate constants k ₁ and k ₂ and the relaxation time τ. The relationship between the phase shift and the oscillatory flow of reactant X seems to be similar to that obtained in the case of linear chemical reactions, as described previously, however, the former is much more complex and different. In this paper, we would like to consider whether the effect of forced phase shift in the case of nonlinear and non-oscillatory chemical processes occurring particularly in the living systems have a practical application in laboratory.

Appendix

Derivation of the approximate solution of the irresolvable nonlinear equation (eqs. (6) and (16)) in this work are based on analytical calculation for the nonlinear autocatalytic reaction with a constant in time inflow of the reactant X to the reaction medium and on the analytical solution for the simple linear reaction with oscillatory, cosine-like inflow of X to the reaction medium described in the ref.[47]

Derivation of Eq. (6) and (16):

The main subject of our consideration is the simplest nonlinear autocatalytic reaction:

$$ A+X \overset{k_{1} k_{2}}{\longleftrightarrow} 2X \quad I_{X} =const. $$

(A1)

where A stands for the initial product, X is the intermediate product, I _X denotes constant in time inflow of the intermediate product X to the reaction field (medium).

We assume that the product X flow to the reaction medium If the inflow I _X of intermediate product X is constant in time, the kinetic equation (1) has the form:

$$ \frac{dX}{dt}=k_{1} AX-k_{2} X^{2}+I_{X} $$

(A2)

The reaction (2) has the following stationary states:

$$ x_{1} =\frac{k_{1} A+\sqrt {\Delta}} {2k_{2}} \quad x_{1} =\frac{k_{1} A-\sqrt {\Delta}} {2k_{2}} $$

(A3)

where:

$$\begin{array}{@{}rcl@{}} \sqrt {\Delta} &\,=\,&\sqrt {\left({k_{1} A} \right)^{2}+4k_{2} I_{X}}~~\text{in case outflow of}~X {\Delta} > 0\\ &&\text{is always positive} \end{array} $$

(A3a)

$$\begin{array}{@{}rcl@{}} \sqrt {\Delta} &=& \sqrt {\left({k_{1} A} \right)^{2} \,-\, 4k_{2} I_{X}}~~\text{in case inflow of} \\ &&~X {\Delta} >0~\text{or}~{\Delta} < 0 \end{array} $$

(A3b)

We assume that Δ is positive.

The analytical solution of the X product change described by equation (2) takes the form:

$$ X=\frac{k_{1} A-\sqrt {\Delta} - [ { ({k_{1} A+\sqrt {\Delta}} )} ]c\exp \left({\sqrt {\Delta} t} \right)}{2k_{2} [ {({1-c\exp (\sqrt {\Delta} t} )} ]} $$

(A4)

where, pre-exponential factor c is,

$$ c=\frac{-X_{0} 2k_{2} +k_{1} A-\sqrt {\Delta}} {-X_{0} 2k_{2} +k_{1} A+\sqrt {\Delta}} $$

(A5)

X ₀– is the initial concentration of the product X. X ₀ is calculated from the following initial conditions:when X = X ₀→t=0 and X ₀ takes the form:

$$ X_{0} =\frac{k_{1} A-\sqrt {\Delta} -({k_{1} A+\sqrt {\Delta}} )c}{2k_{2} -c2k_{2}} $$

(A6)

From the equation (4) follows the relaxation time τ for this reaction which takes the form:

$$ \tau =\frac{1}{\sqrt {\Delta}} =\frac{1}{\sqrt {\left({k_{1} A} \right)^{2}-4k_{2} I_{X}} } $$

(A7)

Analytical solution for the linear reactions with oscillatory, cosine-like inflow of intermediate product X to the reaction medium is described in our earlier paper (see ref.[47]).