Snapshot 1: the two concentration ratios shown in the thumbnail matched manually with the corresponding degradation curves, assuming first-order kinetics (

)

Snapshot 2: two concentration ratios at different times matched manually with the corresponding degradation curves, assuming approximate first-order kinetics with

Snapshot 3: predicting isothermal degradation (red curve) at time

and temperature

(red point) from the matched points in Snapshot 2

Consider a degradation reaction of a heat-labile biological compound, for example, a vitamin or pigment, following assumed fixed-order kinetics (

), where the temperature-dependence of the rate constant follows the exponential form

, where

is the rate constant at temperature

. As has already been shown, this simpler model can replace the traditional Arrhenius equation without sacrificing the fit [1].

When the compound is exposed to two constant temperatures

and

for sufficiently long times so that the come-up and cooling times' effects are negligible, the two corresponding degradation curves can be modelled by

and

. The final concentrations

and

are the solutions of these two rate equations for times

and

, respectively. This gives two equations

and

with two unknowns,

and

.

If the final concentration ratios have actually been measured, then one can solve these two equations simultaneously using

FindRoot and extract the values of

and

[2]. Once calculated, these parameters can be used to reconstruct the entire degradation curves of the two temperatures and predict other temperature histories, isothermal or nonisothermal.

In this Demonstration, enter the two temperatures

and

, corresponding concentration ratios

and

at chosen times

and

, the assumed reaction order

(

), and chosen reference temperature

between or equal to one of the two chosen temperatures. Once these are set and displayed, one tries to match the two shown degradation curves with the two chosen points by moving the

and

sliders. The match can be improved by adjusting the numerical values of these two parameters displayed to the right of the sliders. When a satisfactory match is obtained, the corresponding parameters' values, which are displayed on top of the bottom plot, can be used as the initial guesses for solving the equations or considered as estimates of the parameters' values themselves. In that case, by checking the "show prediction for the following conditions" box, you can estimate and display the concentration ratio at any temperature and time entered by the

and

sliders.

Not all time-temperature-concentration ratio entries allowed by the program must have two matching degradation curves. Some entries may reflect experimental errors, a reaction kinetic order outside the permitted range, or nonlinear kinetics.

[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.