Fit of First-Order Kinetic Model in Degradation Processes

Many chemical, biochemical, and biological degradation processes follow first-order kinetics, which is characterized by a single exponential rate constant. In this Demonstration, we evaluate the possibility that the same experimental data can also be fitted with a general fixed-order kinetic model or with the stretched exponential (Weibull) model with , which implies either a different order or nonlinear kinetics. This is done by generating data using the first-order kinetics model, with and without added random scatter of a chosen amplitude. The data is then fitted to these alternative models, varying the value of . It is shown that there might be numerous situations in which first-order kinetics and the two alternative models with give indistinguishable results.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+