From the Kinetic Theory of Gases to the Kinetics of Rate Processes: On the Verge of the Thermodynamic and Kinetic Limits

Abstract

A variety of current experiments and molecular dynamics computations are expanding our understanding of rate processes occurring in extreme environments, especially at low temperatures, where deviations from linearity of Arrhenius plots are revealed. The thermodynamic behavior of molecular systems is determined at a specific temperature within conditions on large volume and number of particles at a given density (the thermodynamic limit): on the other side, kinetic features are intuitively perceived as defined in a range between the extreme temperatures, which limit the existence of each specific phase. In this paper, extending the statistical mechanics approach due to Fowler and collaborators, ensembles and partition functions are defined to evaluate initial state averages and activation energies involved in the kinetics of rate processes. A key step is delayed access to the thermodynamic limit when conditions on a large volume and number of particles are not fulfilled: the involved mathematical analysis requires consideration of the role of the succession for the exponential function due to Euler, precursor to the Poisson and Boltzmann classical distributions, recently discussed. Arguments are presented to demonstrate that a universal feature emerges: Convex Arrhenius plots (super-Arrhenius behavior) as temperature decreases are amply documented in progressively wider contexts, such as viscosity and glass transitions, biological processes, enzymatic catalysis, plasma catalysis, geochemical fluidity, and chemical reactions involving collective phenomena. The treatment expands the classical Tolman’s theorem formulated quantally by Fowler and Guggenheim: the activation energy of processes is related to the averages of microscopic energies. We previously introduced the concept of “transitivity”, a function that compactly accounts for the development of heuristic formulas and suggests the search for universal behavior. The velocity distribution function far from the thermodynamic limit is illustrated; the fraction of molecules with energy in excess of a certain threshold for the description of the kinetics of low-temperature transitions and of non-equilibrium reaction rates is derived. Uniform extension beyond the classical case to include quantum tunneling (leading to the concavity of plots, sub-Arrhenius behavior) and to Fermi and Bose statistics has been considered elsewhere. A companion paper presents a computational code permitting applications to a variety of phenomena and provides further examples.

The Basic Theorem of Temperature-Dependent Processes

“…In this review, we discussed the universal applicability of Arrhenius' law, as the key equation of the theory of the temperature dependence of the flow of various chemical and physical processes. We understand that the linearization of thermodynamic parameters can be carried out in a very restricted range, deviations being well studied and quantitatively accounted for once the temperature range confining the system is known [112]. Still, the regularities of the relationships of thermodynamic functions, even in narrow ranges, provide insight into various phenomena.…”

Temperature effect on water dynamics in tetramer phosphofructokinase matrix and the super-arrhenius respiration rate

“…( 11 ) similar to Eq. ( 7 ): the fact is here emphasized that the transmissibility may not be constant, and the temperature-dependent diffusion coefficient D(T) can be described according to a deformed Arrhenius type of formula, so that low order derivatives of with respect to 1 /RT can be extracted by molecular dynamics simulation and compare to experiments when available 12 , 17 – 19 . Thus the parameters measure deviations between equilibrium and non-equilibrium situations leading to the continuous generalization of the diffusion states as series of replicas of microscopic ‘entities’, the specific configurations depending on temperature, The coefficient B and C account for transport behavior attributed to the occurrence of long series of events.…”

“…Thus the parameters measure deviations between equilibrium and non-equilibrium situations leading to the continuous generalization of the diffusion states as series of replicas of microscopic ‘entities’, the specific configurations depending on temperature, The coefficient B and C account for transport behavior attributed to the occurrence of long series of events. Collective phenomena such as the rate of the enzymatic catalysis process, food preservation process, and the basic features of the dynamics of complex or glass-forming liquids and solids have exhibited Super Arrhenius behavior 12 , 17 – 19 : a significant number of studies on the temperature dependence of rates of biophysical reactions have been analyzed and found to manifest the super-Arrhenius kinetics, where transport phenomena accelerate as the temperature increases faster than according to the Arrhenius law. The role of transitivity 12 , 17 – 19 is a promising target of further studies.…”

“…Therefore, it is worthwhile to investigate how this diffusion kinetics term works at the molecular level [14][15][16] . There has been an increasing number of studies measuring the temperature dependence of enzyme reactions, showing super-Arrhenius behaviors 12,[17][18][19][20][21][22][23] . Although theoretical model studies have been advanced for the interpretation of such super-Arrhenius behaviors 12,17,18,20 , space remains for more specific interpretations, such as enzyme complex dynamics in conjunction with a temperature-dependent structural reorganizations and reaction processes 12 .…”

The Fuzziness in Molecular, Supramolecular, and Systems Chemistry

“…It is not fair to define just one kinetic constant and one activation energy for all the coexistent conformers. It is necessary to add that both the original transition-state theory and the Arrhenius law have been already questioned by the most recent theoretical and experimental developments, as evidenced by Carvalho-Silva, Coutinho, and Aquilanti [ 35 ]. Quantum mechanical effects, such as tunneling and resonance, stochastic motions of particles in condensed environments, and non-equilibrium effects in classical and quantum formulations, are responsible for deviations from the traditional Arrhenius equation.…”