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

)

Snapshot 2: the same two concentration ratios matched manually, used to predict the degradation curve (red) at temperature

and the concentration at time

(red point), assuming first-order kinetics (

)

Snapshot 3: the use of manual matching parameters to predict the degradation curve (red) at temperature

and the concentration at time

(red point), assuming approximate first-order kinetics (

)

The degradation of vitamins or pigments in frozen foods frequently follows fixed-order kinetics of order

or

. The temperature dependence of the rate constant frequently follows the exponential model

, where

is the rate constant at temperature

,

is the rate constant at an arbitrary reference temperature

, all temperatures are in degrees Celsius, and

is a constant. This simpler model can replace the traditional Arrhenius equation without sacrificing the fit [1].

At two constant low temperatures

and

, the two corresponding degradation curves can be described by a pair of rate equations

and

, where the

s are the corresponding concentration ratios. These rate equations' isothermal solutions for times

and

are

and

, and they have to match the experimentally determined concentration ratios, i.e.

and

. This creates two simultaneous equations, where the two unknowns are the exponential model's parameters

and

.

These two simultaneous equations can be solved using

FindRoot to extract the values of

and

[2, 3]. Once calculated, these parameters can be used to reconstruct the entire degradation curves of the two temperatures

and

and predict those of other temperature histories, isothermal or nonisothermal.

To do the calculation, enter the two frozen storage temperatures

and

as negative numbers, corresponding to concentration ratios

and

at the times

and

, the assumed reaction order

(

), and an arbitrary reference temperature

, preferably between the two chosen temperatures. Once these parameters are set and displayed, try to match the two shown generated degradation curves with the two chosen endpoints by moving the

and

sliders. The match can be more finely adjusted by changing the numerical values of these two parameters displayed to the right of the sliders.

When a satisfactory match is obtained, the corresponding

and

parameters' values, which are also displayed above the bottom plot, can be used as the initial guesses for solving the rate 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 recreate the nutrient's degradation curve at any other selected frozen temperature

(red curve) and estimate and display the concentration ratio at any chosen time entered with the

slider.

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, and/or nonlinear kinetics in the particular system.

[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.

[3] M. Peleg, M. D. Normand, and T. R. Goulette, "Calculating the Degradation Kinetic Parameters of Thiamine by the Isothermal Version of the Endpoints Method,"

*Food Research International*,

**79**, 2016 pp. 73–80.

doi:10.1016/j.foodres.2015.12.001.