Aquilanti–Mundim deformed Arrhenius model
Arrhenius plots, which are used to represent the effects of temperature on the rates of chemical and biophysical processes and on various transport phenomena in materials science, may exhibit deviations from linearity.
Account of curvature is provided here by a formula, which involves a deformation of the exponential function, of the kind recently encountered in treatments of non-extensivity in statistical mechanics.
Svante Arrhenius (1889) equation is often used to characterize the effect of temperature on the rates of chemical reactions.
[1] The Arrhenius formula gave a simple and powerful law, which in a vast generality of cases describes the dependence on absolute temperature
is as the minimum energy, which molecules need have to overcome the threshold to reaction.
(1) was motivated by the 1884 discovery by van't Hoff [2] of the exponential dependence from the temperature of the equilibrium constants for most reactions: Eq.
(1), when used for both a reaction and its inverse, agrees with van't Hoff's equation interpreting chemical equilibrium as dynamical at the microscopic level.
In case of a single rate-limited thermally activated process, an Arrhenius plot gives a straight line, from which the activation energy and the pre-exponential factor can both be determined.
However, advances in experimental and theoretical methods have revealed the existence of deviation from Arrhenius behavior (Fig.1).
To overcome this problem, Aquilanti and Mundim[3] proposed (2010) a generalized Arrhenius law based on algebraic deformation of the usual exponential function.
, coincides with the usual exponential according to the well-known limit due to Euler, that is, This definition was first used in thermodynamics and statistical mechanics by Landau.
[5] In the most recent scientific literature, there is a variety of deformed algebras with applications in different areas of science.
[6][7] Considering the d-exponential function, we introduce the deformed reaction rate coefficient,
(5), we obtain the following expression for the non-Arrhenius plot, The logarithm of the reaction rate coefficient against reciprocal temperature shows a curvature, rather than the straight-line behavior described by the usual Arrhenius law (Figs.1 and 1a).
In Tolman’s[8] definition the barrier or activation energy is a phenomenological quantity defined in terms of the slope of an Arrhenius law; it is usually assumed to be independent of absolute temperature (T), requires only local equilibrium and in general is given by where
To generalize Tolman´s definition, in the case chemical reactions, we assume that the barrier or activation energy is a function of the temperature given by the following differential equation, where
and the usual activation energy law is recovered as a constant.
Noticeably, on the contrary of the usual Arrhenius case, the barrier or activation energy is temperature dependent and
This general result is explained by a new Tolman-like interpretation of the activation energy through Eq.(8).
It was postulated as the basic expansion the reciprocal-activation reciprocal-temperature relationship, for which can provide a formal mathematical justification by Tolman Theorem.
function when written as the logarithmic derivative of the rate constants with respect to
(7), the concept to an activation energy represents an energetic obstacle to the progress of the reaction: therefore its reciprocal can be interpreted as a measure of the propensity for the reaction to proceed and defined as the specific transitivity (
) of the process: This notation emphasizes the fact that in general the transitivity can take a gamma of values, but not including abrupt changes e.g. in the mechanism or in the phases of reactants.
If it is admit a Laurent expansion in a neighbourhood around a reference value, it is possible recover the Eqs.
[19] What it is call the sub-Arrhenius behaviour would be accounted for traditionally by introducing a tunnelling parameter (
in the TST rate constant by the deformed exponential function, Eq.
is the (translational, vibrational and rotational) partition functions of the reactants, and
and an explicit procedure for its calculation were proposed, which it is inversely proportional to the square of the barrier height (
)and directly proportional to the square of the frequency for crossing the barrier (
) at a saddle point in the potential energy surface: This theory was initially developed for applications in chemical kinetics problems as above discussed, but has since been applied to a wide range of phenomena:
