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.
Theoretical model
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 of the rate constant as following,
(1)
where is the absolute temperature, is the gas constant and the factor varies only slightly with temperature. The meaning attached to the energy of activation is as the minimum energy, which molecules need have to overcome the threshold to reaction. Therefore, the year 1889 can be considered as the birth date of reactive dynamics as the study of the motion of atoms and molecules in a reactive event. Eq. (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).
Fig.1 Arrhenius plot as a function of parameter. The Arrhenius plot concavity is depending on the value of the parameter.
To overcome this problem, Aquilanti and Mundim[3] proposed (2010) a generalized Arrhenius law based on algebraic deformation of the usual exponential function. Starting from the Euler[4] exponential definition given by,
(2)
defining the deformed exponential function as,
(3)
Identifying the deformation parameter as a continuous generalization of . At the limit the d-exponential function, , coincides with the usual exponential according to the well-known limit due to Euler, that is,
(4)
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, , in the following way,
(5)
Fig.1a Aquilanti-Mundim plot as a function of parameter. At the limit the usual Arrhenius Plot is recovered. At is Usual Arrhenius, is concave and at convex plot.
and at the limit the usual Arrhenius reaction law is recovered (Figs.1 and 1a). is pre-exponential factor. Taking the logarithm of , Eq.(5), we obtain the following expression for the non-Arrhenius plot,
(6)
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
(7)
where is constant and is the ideal gas constant.
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,
or → (8)
where (constant) at limit 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 has different concavities depending on the value of the d parameter (see Figs.1 and 1a). Thus, a positive convexity means that decreases with increasing temperature. This general result is explained by a new Tolman-like interpretation of the activation energy through Eq.(8).
In the recent literature, it is possible to find different applications to verify the applicability of this new chemical reaction formalism[9][10][11][12][13][14][15][16][17][18]
Fig.2 – Reaction rate coefficient and activation energy equations, in both theories.
Apparent Reciprocal Activation Energy or Transitivity
can be considered as temperature dependent. It was postulated as the basic expansion the reciprocal-activation reciprocal-temperature relationship, for which can provide a formal mathematical justification by Tolman Theorem. The function when written as the logarithmic derivative of the rate constants with respect to , Eq. (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:
(9)
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. (6) and (8).[19]
What it is call the sub-Arrhenius behaviour would be accounted for traditionally by introducing a
tunnelling parameter () in the conventional
Transition-State-Theory. In the -TST formulation, it is replace the factor in the TST rate constant by the deformed exponential function, Eq. (3), yielding:[18]
(10)
where is Planck constant, is Boltzmann constant and is the (translational, vibrational and rotational) partition functions of the reactants, and is the partition function of the activated complex. In Ref.,[11] the significance of the parameter 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:
(11)
Fields of Applications and Related Subjects
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:
the characterization of reaction rates in Chemistry,[20]
^Tolman, Richard C. (1920). "Statistical Merchanics Applied to Chemical Kinetics". Journal of the American Chemical Society. 42 (12). American Chemical Society (ACS): 2506–2528.
doi:
10.1021/ja01457a008.
ISSN0002-7863.
^
abLuiggi Agreda, Ney J. (2016-07-06). "Aquilanti–Mundim deformed Arrhenius model in solid-state reactions". Journal of Thermal Analysis and Calorimetry. 126 (3). Springer Nature: 1175–1184.
doi:
10.1007/s10973-016-5566-8.
ISSN1388-6150.
S2CID195333962.
^Cavalli, S.; Aquilanti, V.; Mundim, K. C.; De Fazio, D. (2014-06-13). "Theoretical Reaction Kinetics Astride the Transition between Moderate and Deep Tunneling Regimes: The F + HD Case". The Journal of Physical Chemistry A. 118 (33). American Chemical Society (ACS): 6632–6641.
Bibcode:
2014JPCA..118.6632C.
doi:
10.1021/jp503463w.
ISSN1089-5639.
PMID24893210.
^Coutinho, Nayara D.; Silva, Valter H. C.; de Oliveira, Heibbe C. B.; Camargo, Ademir J.; Mundim, Kleber C.; Aquilanti, Vincenzo (2015-04-13). "Stereodynamical Origin of Anti-Arrhenius Kinetics: Negative Activation Energy and Roaming for a Four-Atom Reaction". The Journal of Physical Chemistry Letters. 6 (9). American Chemical Society (ACS): 1553–1558.
doi:
10.1021/acs.jpclett.5b00384.
ISSN1948-7185.
PMID26263312.
^Coutinho, Nayara D.; Aquilanti, Vincenzo; Silva, Valter H. C.; Camargo, Ademir J.; Mundim, Kleber C.; de Oliveira, Heibbe C. B. (2016-05-26). "Stereodirectional Origin of anti-Arrhenius Kinetics for a Tetraatomic Hydrogen Exchange Reaction: Born–Oppenheimer Molecular Dynamics for OH + HBr". The Journal of Physical Chemistry A. 120 (27). American Chemical Society (ACS): 5408–5417.
Bibcode:
2016JPCA..120.5408C.
doi:
10.1021/acs.jpca.6b03958.
ISSN1089-5639.
PMID27205872.
^Aquilanti, V.; Mundim, K.C.; Cavalli, S.; De Fazio, D.; Aguilar, A.; Lucas, J.M. (2012). "Exact activation energies and phenomenological description of quantum tunneling for model potential energy surfaces. The F+H2 reaction at low temperature". Chemical Physics. 398. Elsevier BV: 186–191.
Bibcode:
2012CP....398..186A.
doi:
10.1016/j.chemphys.2011.05.016.
ISSN0301-0104.
^K. C. Mundim and M. S. P. Mundim, Rev. Proc. Químicos. 2013, 14, 21.
^V. H. C. Silva, H. C. B. Oliveira and K. C. Mundim, Rev. Proc. Químicos. 2013, 14, 9.
^
abCarvalho-Silva, Valter H.; Aquilanti, Vincenzo; de Oliveira, Heibbe C. B.; Mundim, Kleber C. (2016-11-17). "Deformed transition-state theory: Deviation from Arrhenius behavior and application to bimolecular hydrogen transfer reaction rates in the tunneling regime". Journal of Computational Chemistry. 38 (3). Wiley: 178–188.
doi:
10.1002/jcc.24529.
ISSN0192-8651.
PMID27859380.
S2CID32714721.
^Carvalho-Silva, Valter H.; Aquilanti, Vincenzo; de Oliveira, Heibbe C. B.; Mundim, Kleber C. (2016-11-17). "Deformed transition-state theory: Deviation from Arrhenius behavior and application to bimolecular hydrogen transfer reaction rates in the tunneling regime". Journal of Computational Chemistry. 38 (3): 178–188.
doi:
10.1002/jcc.24529.
ISSN0192-8651.
PMID27859380.
S2CID32714721.
^Alvarado-Contreras, José A.; Olevsky, Eugene A.; Maximenko, Andrey L.; German, Randall M. (February 2014). "A continuum approach for modeling gravitational effects on grain settling and shape distortion during liquid phase sintering of tungsten heavy alloys". Acta Materialia. 65: 176–184.
Bibcode:
2014AcMat..65..176A.
doi:
10.1016/j.actamat.2013.10.059.
ISSN1359-6454.
^de Moura Pedroso, Diogo; Vieira, Gustavo Soares; Passaro, Angelo (February 2017). "Modelling of high-temperature dark current in multi-quantum well structures from MWIR to VLWIR". Physica E: Low-dimensional Systems and Nanostructures. 86: 190–197.
Bibcode:
2017PhyE...86..190D.
doi:
10.1016/j.physe.2016.10.021.
ISSN1386-9477.
^Tabernero, A.; Vieira de Melo, S.A.B.; Mammucari, R.; Martín del Valle, E.M.; Foster, N.R. (September 2014). "Modelling solubility of solid active principle ingredients in sc-CO2 with and without cosolvents: A comparative assessment of semiempirical models based on Chrastil's equation and its modifications". The Journal of Supercritical Fluids. 93: 91–102.
doi:
10.1016/j.supflu.2013.11.017.
ISSN0896-8446.
^Coutinho, Nayara D.; Silva, Valter H. C.; Mundim, Kleber C.; de Oliveira, Heibbe C. B. (2015-04-04). "Description of the effect of temperature on food systems using the deformed Arrhenius rate law: deviations from linearity in logarithmic plots vs. inverse temperature". Rendiconti Lincei. 26 (2): 141–149.
doi:
10.1007/s12210-015-0407-4.
ISSN2037-4631.
S2CID95276538.
^Lin, Xiao; Negenborn, Rudy R.; Lodewijks, Gabriel (2015), Survey on Operational Perishables Quality Control and Logistics, Lecture Notes in Computer Science, Springer International Publishing, pp. 398–421,
doi:
10.1007/978-3-319-24264-4_28,
ISBN978-3-319-24263-7
^Conner, J.M.; Piggott, J.R.; Paterson, A.; Withers, S. (1996), "Interactions Between Wood and Distillate Components in Matured Scotch Whisky", Flavour Science, Elsevier, pp. 419–424,
doi:
10.1533/9781845698232.7.419,
ISBN978-1-85573-779-2
^Chendynski, Letícia Thaís; Signori Romagnoli, Érica; da Silva, Paulo Rogério Catarini; Borsato, Dionisio (2017-08-30). "Activation Energy's Deviations of the Oxidation Reaction of Biodiesel in Mixtures with Ferric Ions and Natural Antioxidant". Energy & Fuels. 31 (9): 9613–9618.
doi:
10.1021/acs.energyfuels.7b01911.
ISSN0887-0624.
^Gregório, Ana Paula H.; Borsato, Dionísio; Moreira, Ivanira; Silva, Elisângela T.; Romagnoli, Érica S.; Spacino, Kelly R. (2017-06-02). "Apparent activation energy and relative protection factor of natural antioxidants in mixture with biodiesel". Biofuels. 10 (5): 607–614.
doi:
10.1080/17597269.2017.1332297.
ISSN1759-7269.
S2CID98960194.
^Romagnoli, Érica Signori; Borsato, Dionísio; Silva, Lívia Ramazzoti Chanan; Chendynski, Letícia Thais; Angilelli, Karina Gomes; Canesin, Edmilson Antônio (December 2018). "Kinetic parameters of the oxidation reaction of commercial biodiesel with natural antioxidant additives". Industrial Crops and Products. 125: 59–64.
doi:
10.1016/j.indcrop.2018.08.077.
ISSN0926-6690.
S2CID105380721.
^Rangel, Fernando C.; Mamiya, Arthur A.; de Oliveira, Heibbe C. B.; Vieira, Fernando M. C.; Mundim, Kleber C. (2013-07-17). "Alternative Approach to Calculate Two-Center Overlap Matrix through Deformed Exponential Function". The Journal of Physical Chemistry A. 117 (30): 6622–6628.
Bibcode:
2013JPCA..117.6622R.
doi:
10.1021/jp401631a.
ISSN1089-5639.
PMID23815463.
^de Oliveira, H. C. B.; Rangel, F. C.; Esteves, C. S.; Vieira, F. M. C.; Mundim, K. C. (2009-12-31). "Calculation of MP2and Coupled-Cluster Molecular Properties Using the q-Integral Method†". The Journal of Physical Chemistry A. 113 (52): 14691–14698.
Bibcode:
2009JPCA..11314691D.
doi:
10.1021/jp904807b.
ISSN1089-5639.
PMID20028167.