# Degradation Parameters from Concentration Ratios

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In principle, if a degradation reaction follows known fixed-order kinetics and its rate constant temperature dependence can be described by a two-parameter model, then these parameters can be calculated from measured concentration ratios at two different times along a nonisothermal temperature history. The concept is illustrated in simulated storage under three fluctuating temperature histories, where the assumed kinetic order is between 0.8 and 1.2, and the temperature dependence of the rate constant follows the exponential model whose parameters are and . Once you set , the values of and can be estimated manually by passing a simulated degradation curve using these parameters through the two measured points.

Contributed by: Mark D. Normandand Micha Peleg (March 2015)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: Reconstructed degradation curve for the two points at their default locations and a fluctuating temperature profile having a falling trend. First-order kinetics, .

Snapshot 2: Reconstructed degradation curve of a fluctuating temperature profile having a rising trend. First-order kinetics, .

Snapshot 3: Reconstructed degradation curve of a fluctuating temperature profile having a falling then rising trend where the transition is specified with an If. First-order kinetics, .

Traditionally, the kinetic parameters of a thermal degradation reaction are determined from a set of isothermal concentration decay curves. These are used to establish the kinetic order and the rate constant's temperature dependence, almost always using the Arrhenius equation as a model. It has been shown, though, that data fitted with the Arrhenius equation can also be fitted by the much simpler exponential model , where is the rate constant at temperature , is the rate at reference temperature , and is a constant, without sacrificing the fit [1]; see also Arrhenius versus Exponential Model for Chemical Reactions in Related Links.

In principle, if the reaction's kinetic order is known a priori or can be assumed, and also the temperature profile is nonisothermal, one can retrieve the values of and and reconstruct the entire degradation curve from two points and known to be on the curve. With and , this is done by solving the simultaneous equations and , where in both cases the boundary condition is . In these equations, and are the concentration ratios at times and that you enter with sliders. The right side of the equation is the numerical solution of the corresponding rate equation for the known or assumed kinetic order and user-chosen . The two unknowns are and .

Notice that for there is a time beyond which becomes a complex number. To avoid such situations, the Demonstration restricts time to the range where when .

Because using the built-in *Mathematica* function FindRoot to solve the two equations requires close initial guesses that are hard to come by, especially for a previously untested nutrient or pharmaceutical, the Demonstration solves the equations for the and values set with the sliders and plots the corresponding curve. Manually passing the curve through the two entered points can be used to estimate and , which in turn can serve as the initial guesses in FindRoot. Alternatively, these estimates themselves can be used to predict other concentrations at different times.

This Demonstration illustrates the method using simulated storage data with three hypothetical fluctuating temperature regimes , , and , where an If is included in the temperature profile equation .

Other controls include the two points' coordinates, the reaction's kinetic order , the reference temperature between 15 and 30 °C, and the maximum value of the time axis.

The top plot shows the temperature profile's plot and its equation. The bottom plot shows the two points marked as filled circles and the reconstructed degradation curve created with the displayed values of , , and .

Notice that for the method to work, the points ought to be sufficiently far apart and the difference in the concentration ratios sufficiently large.

Note that not all entered pairs of points can be matched by a curve generated by moving the , , and sliders.

References

[1] M. Peleg, M. D. Normand, and M. G. Corradini, "The Arrhenius Equation Revisited," *Critical Reviews in Foods Science and Nutrition,* 52(9), 2012 pp. 830–851. doi:10.1080/10408398.2012.667460.

[2] M. Peleg, M. D. Normand, and A. D. Kim, "Estimating Thermal Degradation Kinetics Parameters from the Endpoints of Non-isothermal Heat Processes or Storage," *Food Research International*, 66, 2014 pp. 313–324. doi:10.1016/j.foodres.2014.10.003.

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