# Minimal ORAC Kinetic System: Mathematical Analysis

This Demonstration shows a comparative analysis of the numerical solutions (NS) with analytic solutions (AS) of the first-order ordinary differential equations (ODEs) of the minimal oxygen radical absorbance capacity (ORAC) kinetic system. The NS and AS of the concentration profiles over time of the probe PH, antioxidant XH and peroxyl radical ROO{•} are shown in different panels. The values for the initial concentrations of reactants and the apparent first- and second-order rate constants were determined, based on previous ORAC studies.

### DETAILS

The first-order ODEs that define the minimal ORAC kinetic system can be written [1]:
,
,
,
.
At , the concentrations of the starting and intermediate reactants are:
,
,
,
.
Following this notation, the concentrations of the azo compound, , and at time can be represented as:
,
,
,
,
where , and stand for the reacted concentrations of , and at time , respectively.
The built-in function ParametricNDSolve was used to find numerical solutions to the given differential equations applying the Runge–Kutta method. However, an exact analytical solution of the ODEs of the minimal ORAC kinetic system was not possible applying Lie symmetry theory. The mathematical form of the coefficients of the nonautonomous Chini equation, an ODE that appears after some mathematical transformations and is polynomial in the dependent variable, precluded the reduction of this equation to the autonomous Chini equation. However, an approximate analytical solution was found when . In this former case, the nonautonomous Chini equation was transformed into a nonhomogenous ODE that was linear in the dependent variable. The dependence of the concentration profile of and on time and initial conditions was found to be:
,
,
where , , and stands for the lower incomplete gamma function.
The concentration profile over time of the peroxyl radical was a sum of terms:
.
The goodness of the approximation we introduced to solve the ODEs of the minimal ORAC kinetic system can now be tested in the figure. You can arbitrarily change the input parameters and confirm that the approximate analytical solution for the minimal ORAC kinetic system that we propose is valid when and [PH]0 decreases while [XH]0 remains constant and the initial concentration of the azo compound is kept low enough to ignore the and radical self-reactions and the formation [1].
Reference
[1] J. B. Arellano, E. Mellado-Ortega and K. R. Naqvi, "The ORAC Assay: Mathematical Analysis of the Rate Equations and Some Practical Considerations," International Journal of Chemical Kinetics, forthcoming.

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