From statistical thermodynamics to molecular kinetics: the change, the chance and the choice

Abstract

A survey on the principles of chemical kinetics (the “science of change”) is presented here. This discipline plays a key role in molecular sciences, however, the debate on its foundations had been open for the 130 years since the Arrhenius equation was formulated on admittedly purely empirical grounds. The great success that this equation has had in the development of experimental research has motivated the need of clarifying its relationships with the foundations of thermodynamics on the one hand and especially with those of statistical mechanics (the “discipline of chances”) on the other. The advent of quantum mechanics in the Twenties and the scattering experiments by molecular beams in the second half of last century have validated collisional mechanisms for reactive processes, probing images of single microscopic events; molecular dynamics computational techniques have been successfully applied to interpret and predict phenomena occurring in a variety of environments: the focus here is on a key aspect, the effect of temperature on chemical reaction rates, which in cold environments show departure from Arrhenius law, so arguably from Maxwell–Boltzmann statistics. Modern developments use venerable mathematical concepts arising from “criteria for choices” dating back to Jacob Bernoulli and Euler. A formulation of recent progress is detailed in Aquilanti et al. 2017.

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Notes

  1. 1.

    Svante Arrhenius (1859–1927) was a Swedish chemist and physicist, Nobel Prize in chemistry in 1903 for his theory of ion transfer seen as responsible for the passage of electricity. Arrhenius therefore did not get the Nobel Prize for his contribution to chemical kinetics. Actually, we read on the official website of the Nobel Prize that in 1903 he had been nominated for both chemistry and physics, and that the Chemistry Committee had proposed to give him half of each prize, but the Physics Committee did not accept the idea. Arrhenius is also famous for having introduced not only the controversial but currently revisited concept of "panspermia" (possible extraterrestrial origin of life), but above all already a century ago the concept of global warming by the “greenhouse” effect, as generated by human activity: curiously this effect, today documented and feared, appeared to Arrhenius as a hope to ease difficulties of Sweden to face cold winters.

  2. 2.

    Tolman (1881–1948) was an American chemist, physicist and mathematician. The program to apply to chemical kinetics a "new science", statistical mechanics, was developed by Tolman from 1920 (Tolman 1920). The quotation given above concludes his voluminous compendium (Tolman 1927) where he re-elaborates the article Tolman 1920 and other subsequent ones. We will see later that is optimism was later frustrated. He abandoned the field of chemical kinetics, but gained fame in other areas of science. He not only no longer provided any contribution, but even make no mention it in his monumental treatise, which since 1938 is the standard reference on the foundations of statistical mechanics (Tolman 1938). According to Laidler and King 1983, Tolman considered obsolete both the use of the old quantum theory and some hypotheses on the role of radiation in the activation of the chemical process. But Fig. 2 shows the relevance of what is now called "Tolman's Theorem" is well evidenced by a quantum–mechanical perspective provided by Fowler and Guggenheim 1939. As shown in Fig. 8, they dedicated a long chapter on chemical kinetics in their influential statistical thermodynamics treatise.

  3. 3.

    Henry Eyring (1901–1981) was a Mexican chemical from the United States. He dealt with physical chemistry, giving his major contribution to the study of the speed of chemical reactions and reaction intermediates. He received the Wolf Award for Chemistry in 1980 and the National Medal of Science in 1966 for developing the Theory of Absolute Velocities of Chemical Reactions (left panel), one of the most important twentieth-century developments in chemistry. The failure to award the Nobel Prize to Eyring was a surprise for many. Evidently the members of the committee did not promptly understand the importance of the theory as stated in the Nobel Prize website.

  4. 4.

    Enlighted by the hot sun of my lover/I was first shown the road of truth’s grand beauty/approving or disproving, to discover. Dante, Paradiso, incipit of the third canto. The term "provando e riprovando" dates back to Dante, which should be interpreted as "approve and reject", say yes and say no. On this filter Galileo insisted; in fact, the phrase is often considered a Galilean statement and was used by his school and then in the Accademia del Cimento. It seems that we can deduce that Dante is placed at the end of the Middle Ages and at the beginning of humanism: to say that somehow he had anticipated modern science is perhaps excessive, but certainly some signals are recognized. Galileo definitely overcomes the border and proposes the experiment as discriminating between what is true and what is false. Galileo tells his daughter Virginia: “The aim of science is not to open a door to infinite wisdom, but to set a limit to infinite error” Bertolt Brecht, Life of Galileo (ca. 1940), ninth scene. Keeping this attitude may suggest an interesting anticipation of Popper's theory that emphasizes falsifiability, as central to verifiability, thus leading us to the scientific debate of the 20th century.

References

  1. Aquilanti V (1994) Storia e Fondamenti della Chimica, in G. Marino. Rendiconti Accademia Nazionale delle Scienze, Rome

    Google Scholar 

  2. Aquilanti V, Cavalli S, De Fazio D et al (2005) Benchmark rate constants by the hyperquantization algorithm. The F + H2 reaction for various potential energy surfaces: features of the entrance channel and of the transition state, and low temperatur e reactivity. Chem Phys 308:237–253

    CAS  Article  Google Scholar 

  3. Aquilanti V, Mundim KC, Elango M et al (2010) Temperature dependence of chemical and biophysical rate processes: phenomenological approach to deviations from Arrhenius law. Chem Phys Lett 498:209–213

    CAS  Article  Google Scholar 

  4. Aquilanti V, Mundim KC, Cavalli S et al (2012) Exact activation energies and phenomenological description of quantum tunneling for model potential energy surfaces. the F + H2 reaction at low temperature. Chem Phys 398:186–191

    CAS  Article  Google Scholar 

  5. Aquilanti V, Coutinho ND, Carvalho-Silva VH (2017) Kinetics of low-temperature transitions and reaction rate theory from non-equilibrium distributions. Philos Trans R Soc London A 375:20160204

    Article  Google Scholar 

  6. Bell RP (1980) The tunnel effect in chemistry. Champman and Hall, London

    Google Scholar 

  7. Biró T, Ván P, Barnaföldi G, Ürmössy K (2014) Statistical power law due to reservoir fluctuations and the universal thermostat independence principle. Entropy 16:6497–6514

    Article  Google Scholar 

  8. Carvalho-Silva VH, Aquilanti V, de Oliveira HCB, Mundim KC (2017) Deformed transition-state theory: deviation from Arrhenius behavior and application to bimolecular hydrogen transfer reaction rates in the tunneling regime. J Comput Chem 38:178–188

    CAS  Article  Google Scholar 

  9. Cavalli S, Aquilanti V, Mundim KC, De Fazio D (2014) Theoretical reaction kinetics astride the transition between moderate and deep tunneling regimes: the F + HD case. J Phys Chem A 118:6632–6641

    CAS  Article  Google Scholar 

  10. Chapman S, Garrett BC, Miller WH (1975) Semiclassical transition state theory for nonseparable systems: application to the collinear H + H2 reaction. J Chem Phys 63:2710–2716

    CAS  Article  Google Scholar 

  11. Che D-C, Matsuo T, Yano Y et al (2008) Negative collision energy dependence of Br formation in the OH + HBr reaction. Phys Chem Chem Phys 10:1419–1423

    CAS  Article  Google Scholar 

  12. Condon EU (1938) A simple derivation of the Maxwell–Boltzmann law. Phys Rev 54:937–940

    CAS  Article  Google Scholar 

  13. Coutinho ND, Sanches-Neto FO, Carvalho-Silva VH, de Oliveira HCB, Ribeiro LA, Aquilanti V (2018) Kinectics of the OH + HCl →H2O + Cl Reaction: Rate Determining Roles of Stereodynamics and Roaming and of Quantum Tunneling. J Comput Chem. https://doi.org/10.1002/jcc.25597

    Article  Google Scholar 

  14. Coutinho ND, Silva VHC, de Oliveira HCB et al (2015a) Stereodynamical origin of anti-arrhenius kinetics: negative activation energy and roaming for a four-atom reaction. J Phys Chem Lett 6:1553–1558

    CAS  Article  Google Scholar 

  15. Coutinho ND, Silva VHC, Mundim KC, de Oliveira HCB (2015b) Description of the effect of temperature on food systems using the deformed arrhenius rate law: deviations from linearity in logarithmic plots vs. inverse temperature. Rend Lincei 26:141–149

    Article  Google Scholar 

  16. Coutinho ND, Aquilanti V, Silva VHC et al (2016) Stereodirectional origin of anti-arrhenius kinetics for a tetraatomic hydrogen exchange reaction: born-oppenheimer molecular dynamics for OH + HBr. J Phys Chem A 120:5408–5417

    CAS  Article  Google Scholar 

  17. Coutinho ND, Carvalho-Silva VH, de Oliveira HCB, Aquilanti V (2017) The HI + OH → H2O + I reaction by first-principles molecular dynamics: stereodirectional and anti-arrhenius kinetics. In: Lecture notes in computer science. Computational Science and Its Applications – ICCSA. Springer, Trieste

  18. Coutinho ND, Aquilanti V, Sanches-Neto FO et al. (2018) First-principles molecular dynamics and computed rate constants for the series of OH–HX reactions (X = H or the halogens): Non-arrhenius kinetics, stereodynamics and quantum tunnel. In: lecture notes in computer science. Computational Science and Its Applications – ICCSA. Springer, Melbourne

    Google Scholar 

  19. Eyring H (1935) The Activated Complex in Chemical Reactions. J Chem Phys 3:107–115

    CAS  Article  Google Scholar 

  20. Fowler RH, Guggenheim EA (1939) Statistical thermodynamics: a version of statistical mechanics for students of physics and chemistry. Macmillan, London

    Google Scholar 

  21. Fowler R, Guggenheim EA (1949) Statistical Thermodynamics. Cambridge University Press, London

    Google Scholar 

  22. Glasstone S, Laidler KJ, Eyring H (1941) The theory of rate processes: the kinetics of chemical reactions, viscosity, Diffusion and Electrochemical Phenomena. McGraw-Hill, New York City

    Google Scholar 

  23. Hilbert D (1902) Mathematical problems. Bull Am Math Soc 8:437–479

    Article  Google Scholar 

  24. Hinshelwood CN (1940) The kinetics of chemical change. Oxford Clarendon Press. Oxford

    Google Scholar 

  25. Jeans J (1913) The Dynamical Theory of Gases. Dover Publications Incorporated, ‎Mineola

    Google Scholar 

  26. Kasai T, Che D-C, Okada M et al (2014) Directions of chemical change: experimental characterization of the stereodynamics of photodissociation and reactive processes. Phys Chem Chem Phys 16:9776–9790

    CAS  Article  Google Scholar 

  27. Kennard EH (1938) Kinetic theory of gases: with an introduction to statistical mechanics. McGraw-Hill, New York City

    Google Scholar 

  28. Kramers HA (1940) Brownian motion in a field of force and the diffusion model of chemical reactions. Phys 7:284–304

    CAS  Google Scholar 

  29. Laidler KJ (1996) A glossary of terms used in chemical kinetics, including reaction dynamics. Pure Appl Chem 68:149–192

    CAS  Article  Google Scholar 

  30. Laidler KJ, King MC (1983) Development of transition-state theory. J Phys Chem 87:2657–2664

    CAS  Article  Google Scholar 

  31. Lewis WCM (1918) Studies in catalysis. Part IX. The calculation in absolute measure of velocity constants and equilibrium constants in gaseous systems. J Chem Soc 113:471–492

    CAS  Article  Google Scholar 

  32. Lewis GN, Randall M (1923) Thermodynamics and the free energy of chemical substance. McGraw-Hill Book Company, New York

    Google Scholar 

  33. Marcus RA (1993) Electron transfer reactions in chemistry. Theory and experiment. Rev Mod Phys 65:599–610

    CAS  Article  Google Scholar 

  34. Miller H (1993) Beyond transition-state theory: a rigorous quantum theory of chemical reaction rates. Acc Chem Res 26:174–181

    CAS  Article  Google Scholar 

  35. Perlmutter-Hayman B (1976) Progress in inorganic chemistry: On the temperature dependence of Ea. In: Lippard SJ (ed) Wiley, New York, pp 229–297

  36. Polanyi M, Wigner E (1928) Über die Interferenz von Eigenschwingungen als Ursache von Energieschwankungen und chemischer Umsetzungen. Z Phys Chem Abt A 139:439–452

    CAS  Google Scholar 

  37. Ruggeri T (2017) Lecture notes frontiere. Accademia Nazionale dei Lincei, Rome

    Google Scholar 

  38. Sanches-Neto FO, Coutinho ND, Carvalho-Silva VH (2017) A novel assessment of the role of the methyl radical and water formation channel in the CH3OH + H reaction. Physical Chemistry Chemical Physics 19:24467–24477

    CAS  Article  Google Scholar 

  39. Silva VHC, Aquilanti V, De Oliveira HCB, Mundim KC (2013) Uniform description of non-Arrhenius temperature dependence of reaction rates, and a heuristic criterion for quantum tunneling vs classical non-extensive distribution. Chem Phys Lett 590:201–207

    CAS  Article  Google Scholar 

  40. Slater NB (1959) Theory of unimolecular reactions. Cornell University Press, Ithaca

    Google Scholar 

  41. Tolman RC (1920) Statistical mechanics applied to chemical kinetics. J Amer Chem Soc 42:2506–2528

    CAS  Article  Google Scholar 

  42. Tolman RC (1927) Statistical mechanics with applications to physics and chemistry. The Chemical catalog company. New York

    Google Scholar 

  43. Tolman RC (1938) The principles of statistical mechanics. Oxford University, London

    Google Scholar 

  44. Trautz M (1916) Das Gesetz der Reaktionsgeschwindigkeit und der Gleichgewichte in Gasen. Zeitschrift für Anorg und Allg Chemie 96:1–28

    CAS  Article  Google Scholar 

  45. Truhlar DG, Garrett BC (1984) Variational transition state theory. Annu Rev Phys Chem 35:159–189

    CAS  Article  Google Scholar 

  46. Tsallis C (1988) Possible generalization of Boltzman–Gibbs statistics. J Stat Phys 52:479–487

    Article  Google Scholar 

  47. Truhlar DG (2015) Transition state theory for enzyme kinetics. Arch Biochem Biophys 582:10–17

    CAS  Article  Google Scholar 

  48. Uhlenbeck GE, Goudsmit S (1935) Statistical energy distributions for a small number of particles. Zeeman Verhandenlingen Martinus N:201–211

  49. Van Vliet CM (2008) Equilibrium and non-equilibrium statistical mechanics. World Scientific Pub, Singapore

    Google Scholar 

  50. Walter JE, Eyring H, Kimball GE (1944) Quantum Chemistry. Wiley, New York

    Google Scholar 

  51. Warshel A, Bora RP (2016) Perspective: defining and quantifying the role of dynamics in enzyme catalysis. J Chem Phys 144:180901

    Article  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the Brazilian funding agencies Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for financial support. Valter H. Carvalho-Silva thanks PrP/UEG for research funding through the PROBIP and PRO´-PROJETO programs, also thanks CNPq for research funding through the Universal 2016—Faixa A program. Ernesto P. Borges acknowledges National Institute of Science and Technology for Complex Systems (INCT-SC). Vincenzo Aquilanti acknowledgments financial support for the Italian Ministry for Education, University and Research, MIUR. Grant no. SIR2014(RBSI14U3VF) and Elvira Pistoresi for this assistance in the preparation of her manuscript.

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Correspondence to Vincenzo Aquilanti or Valter H. Carvalho-Silva.

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Dedicated to the memory of professor Maria Suely Pedrosa Mundim, physicist (1956–2017).

This contribution is the written peer-reviewed version of a paper presented at the International Conference “Molecules in the World of Quanta: From Spin Network to Orbitals” held at Accademia Nazionale dei Lincei in Rome, April 27 and 28, 2017.

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Aquilanti, V., Borges, E.P., Coutinho, N.D. et al. From statistical thermodynamics to molecular kinetics: the change, the chance and the choice. Rend. Fis. Acc. Lincei 29, 787–802 (2018). https://doi.org/10.1007/s12210-018-0749-9

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Keyword

  • Arrhenius equation
  • Tolman activation energy
  • Super-Arrhenius
  • Sub-Arrhenius