Snapshot 1: the fit of the stretched exponential (Weibull) model with

to slightly scattered simulated data

Snapshot 2: the fit of the fixed-order kinetic model with

to highly scattered data generated with a low

Snapshot 3: the fit of the stretched exponential (Weibull) model with

to highly scattered data generated with a low

A degradation reaction kinetics order

is defined by the differential equation

, where

is the concentration at time

and

is the rate constant with dimensions and units commensurate with

. The boundary condition is

, the initial concentration.

For first-order kinetics with

, the equation's solution is

, where the exponential rate constant

has units of reciprocal time.

For

the rate equation's solution is

. Theoretically, this solution allows for negative concentration values for certain times when

and complex numbers for certain times when

. Replacing these unrealistic values by zero gives a generalized fixed-order kinetics model for

[1], which for

assumes the form

.

A common model for nonlinear kinetics is the stretched exponential (Weibull) model, of which the first-order is a special case. For our purpose it can be written in the form

. This model with

was shown to be useful in predicting nutrient loss under non-isothermal conditions where, traditionally, first-order kinetics has been used [2, 3].

In this Demonstration, you can examine the fit of the three models by generating degradation data with the first-order kinetics model

, with or without added random noise, and fitting them by the two other models

and

using nonlinear regression.

The model is chosen with a setter bar. The seed, number of data points,

value for their generation, the scatter's amplitude, and the

or

in

and

are entered with sliders, as is the maximum of the time axis. By clicking the checkbox, you can select a repeatable seed and choose or decline to plot the generated curve as a dashed curve before the added noise.

The Demonstration plots the generated data points on which the fitted curve is superimposed as a solid curve. It also displays the chosen

or

, calculated

or

, and the fit's

-squared value. It shows that with the values of

and

within the range of 0.8 to 1.2, there can be multiple combinations of the parameters in which the models have a similar fit, as judged by statistical criteria. This suggests that data fit alone might not be sufficient to establish first-order kinetics.

[1] 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.

[2] M. G. Corradini and M. Peleg, "A Model of Non-isothermal Degradation of Nutrients, Pigments and Enzymes,"

*Journal of the Science of Food and Agriculture*,

**84**(3), 2004 pp. 217–226.

doi:10.1002/jsfa.1647.

[3] M. G. Corradini and M. Peleg, "Prediction of Vitamins Loss during Non-isothermal Heat Processes and Storage with Non-linear Kinetic Models,"

*Trends in Food Science & Technology*,

**17**(1), 2006 pp. 24–34.

doi:10.1016/j.tifs.2005.09.004.