The Arrhenius Equation Revisited

Abstract

The Arrhenius equation has been widely used as a model of the temperature effect on the rate of chemical reactions and biological processes in foods. Since the model requires that the rate increase monotonically with temperature, its applicability to enzymatic reactions and microbial growth, which have optimal temperature, is obviously limited. This is also true for microbial inactivation and chemical reactions that only start at an elevated temperature, and for complex processes and reactions that do not follow fixed order kinetics, that is, where the isothermal rate constant, however defined, is a function of both temperature and time.

The linearity of the Arrhenius plot, that is, Ln[k(T)] vs. 1/T where T is in °K has been traditionally considered evidence of the model's validity. Consequently, the slope of the plot has been used to calculate the reaction or processes’ “energy of activation,” usually without independent verification. Many experimental and simulated rate constant vs. temperature relationships that yield linear Arrhenius plots can also be described by the simpler exponential model Ln[k(T)/k(T_reference)] = c(T-T_reference). The use of the exponential model or similar empirical alternative would eliminate the confusing temperature axis inversion, the unnecessary compression of the temperature scale, and the need for kinetic assumptions that are hard to affirm in food systems. It would also eliminate the reference to the Universal gas constant in systems where a “mole” cannot be clearly identified. Unless proven otherwise by independent experiments, one cannot dismiss the notion that the apparent linearity of the Arrhenius plot in many food systems is due to a mathematical property of the model's equation rather than to the existence of a temperature independent “energy of activation.” If T+273.16°C in the Arrhenius model's equation is replaced by T+b, where the numerical value of the arbitrary constant b is substantially larger than T and T_reference, the plot of Ln k(T) vs. 1/(T+b) will always appear almost perfectly linear. Both the modified Arrhenius model version having the arbitrary constant b, Ln[k(T)/k(T_reference) = a[1/ (T_reference+b)-1/ (T+b)], and the exponential model can faithfully describe temperature dependencies traditionally described by the Arrhenius equation without the assumption of a temperature independent “energy of activation.” This is demonstrated mathematically and with computer simulations, and with reprocessed classical kinetic data and published food results.

Keywords:

• kinetics
• modeling
• temperature effects
• accelerated storage
• shelf life
• time-temperature integrators

ACKNOWLEDGEMENT

Contribution of the Agricultural Experiment Station at Amherst. The opinions expressed in this review are the authors’ own and do not necessarily represent those of the institutions with which they are affiliated.