Rate Constant Calculation from Four Temperature and Moisture Combinations
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Experimental determination of the dependance of a reaction rate constant on both temperature and moisture is laborious and time consuming by conventional methods. This Demonstration describes a method to calculate the underlying kinetic parameters from a set of four experimental determinations, and thus estimate the rate constant at any temperature-moisture combination within a desired range. The mathematical model used is an exponential temperature-dependence model, a simpler substitute for the Arrhenius equation, in which the two-parameter moisture dependence is described by a similar two-parameter exponential model. The four parameters of the resulting rate constant-temperature-moisture relationship are extracted by numerically solving four simultaneous nonlinear equations with the built-in Mathematica FindRoot function. These extracted parameters are then used to plot a 3D surface depicting the rate constant-temperature-moisture relationship and to calculate and display the rate constant for any chosen temperature-moisture combination.
Contributed by: Mark D. Normand and Micha Peleg (August 2018)
Open content licensed under CC BY-NC-SA
Snapshot 1: same control settings as the Thumbnail after parameter estimation with the FindRoot function
Snapshot 2: rate constant-temperature-moisture control settings that have no FindRoot solution
Snapshot 3: hypothetical control settings for high sensitivity to temperature and to moisture
Snapshot 4: hypothetical control settings for low sensitivity to temperature and to moisture
The rate of many deteriorative reactions in stored dry foods and pharmaceuticals depends on both their temperature and moisture content. However, calculating and mapping the rate constant-temperature-moisture relationship by traditional methods requires the experimental determination of many degradation curves under constant temperature and moisture conditions, which can become a burdensome logistic issue when many different products are to be tested.
A potential shortcut method is shown to estimate the relationship from only four such curves, assuming that it can be characterized and mapped by an empirical four-parameter model in the pertinent temperature and moisture content ranges.
Assume that the relation can be described by the nested model
and are a chosen reference temperature in °C and moisture content on a dry or wet basis, respectively, and
, , and are the adjustable parameters .
having four unknowns, which are the model parameters , , and .
These four simultaneous equations can be solved numerically using the FindRoot function to extract the four desired parameters. Once estimated, these parameter values together with the chosen values of and can be used to calculate and plot the relationship and to estimate the rate constant's value at any chosen temperature-moisture combination within the relevant range.
Use the sliders to set the experimental values of , , and , their corresponding temperatures and moisture content, and the selected and . These are used to calculate and display the four model parameters and plot the relationship as a 3D surface.
When a solution is reached, the slider positions for , , and indicate their calculated values, which can be used as initial guesses in subsequent calculations.
For each parameter setting, the program also calculates and displays the rate constant's magnitude for any chosen pair of and values that are entered with the bottom two sliders. Its position is shown as a movable black dot on the 3D surface.
This Demonstration describes only the concept and mathematical methodology. The applicability of the model and methods to an actual reaction in a particular food or pharmaceutical product should be validated by testing its predictions against experimental observations not used in the kinetic parameters calculation.
 M. D. Normand and M. Peleg. "Reaction Rate Dependence on Temperature and Moisture During Storage" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/ReactionRateDependenceOnTemperatureAndMoistureDuringStorage.