A probabilistic rate theory connecting kinetics to thermodynamics Denis Michel To cite this version: Denis Michel. A probabilistic rate theory connecting kinetics to thermodynamics. 2017. hal- 01569569v7 HAL Id: hal-01569569 https://hal.archives-ouvertes.fr/hal-01569569v7 Preprint submitted on 15 Mar 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A probabilistic rate theory connecting kinetics to thermodynamics∗ Denis Michelz z Universite de Rennes1-IRSET. Campus de Villejean. 35000 Rennes France. Email: [email protected]. Abstract. Kinetics and thermodynamics are largely classical version (without tunneling effects), the empiri- disconnected in current theories because Arrhenius acti- cal Arrhenius equation vation energies (Ea) have strictly no influence on equilib- rium distributions. A first step towards the incorporation − Ea of rate theories in thermodynamics is the identification k = A e kB T (1) of the pre-exponential term of the Arrhenius equation as an entropic quantity. A second step examined here where kB is the Boltzmann constant and T is the tem- is the possible contribution of Ea in equilibrium land- perature, includes (i) a so-called preexponential factor A, scapes. Interestingly, this possibility exists if envisioning containing the elementary frequency and configurational the energetic exponential term of Arrhenius rate con- restrictions, and (ii) a unitless exponential function con- stants as the probability that the energy of the reactant taining an energy barrier called activation energy Ea. is sufficient for the transition. This radically new ap- In the present paper, the exponential factor of Arrhe- proach encompasses Maxwell-Boltzmann distributions nius will be reformulated as the probability to reach a and solves inconsistencies in previous theories, in partic- threshold energy, as initially proposed but incompletely ular on the role of temperature in kinetics and thermo- appraised in [3]. This simple hypothesis will lead to ma- dynamics. These probabilistic rate constants are then jor changes in thermodynamic relationships. reintroduced in dynamic systems to provide them with the two distinct facets of time: the time step and the time arrow. Keywords: Kinetics; equilibrium; heat capacity; en- thalpy; entropy; Gibbs free energy. Notes to readers: The unitless statistical entropy usually written H will be renamed here S to avoid confusion with enthalpy. The letter C will be used for heat capacity but not for concentration. 1 Introduction Rate constants are the drivers of all dynamic systems and are widely used in physical chemistry and modeling stud- ies. By the restrictions they impose on chemical transi- Figure 1. Representation of the energy barrier according to tions, rate constants prevent our organized world from the Arrhenius principle. The barrier is supposed to restrict falling directly into its state of weaker free energy and reaction kinetics but to have no influence on equilibrium ra- z maximal entropy. Yet the profound nature of these con- tios. E is a fixed energy threshold necessary for crossing the stants remains obscure and the main focus of this study barrier in both directions and H are the mean enthalpies of is to clarify their energetic component. After multiple reaction of the particles. attempts of description, its most common expression re- mains the exponential factor of Arrhenius [1, 2]. In its Using the notations of Fig.1, the Arrhenius approach ∗Reference: Michel, D. 2018. A probabilistic rate theory connecting kinetics to thermodynamics. Physica A 503, 26-44 1 reads 2.3 A non-probabilistic exponential law z The exponential term of the Arrhenius law has cer- k = A e−(E −Hi)=kB T (2a) ij ij tain appearences of a probability, but as explained in and for the reverse reaction the following section, as it stands it cannot correspond z −(E −Hj )=kB T to the exponential distribution. The purpose of the kji = Aji e (2b) present study is precisely to examine the theoretical con- In equilibrium the forward and reverse fluxes equalize sequences of the hypothesis that it is a probability of such that sufficient energy for the transition. This simple hypoth- eq eq ni kij = nj kji (2c) esis justified later leads to the novel formulas giving n k A z z i ji ji (E −Hi−E +Hj )=kB T = = K = z E ji e − E − 1+ a nj kij Aij sufficient energy H ( H ) eq (2d) P ( ) = e = e (3a) Aji Aji = e−(Hi−Hj )=kB T = e−∆H=kB T Aij Aij which gives the equilibrium relationship z E disappears. The equilibrium ratio is independent of z ni Aji Hi E the energetic barrier and depends only on the difference ln = ln + 1 − nj Aij Hj Hi of energy between the reactants. These currently admit- eq (3b) ted relationships can be naively questioned. Aji Hi Ea = ln + 1 − 1 + Aij Hj Hi 2 Some concerns and alternative whose structure is completely different from the usual proposal result ni Aji Hj − Hi Gj − Gi 2.1 On the universality of kBT as a mean ln = ln + = (4) n A k T k T energy j eq ij B B 1=kBT is introduced in statistical mechanics as a con- In Eq.(4), the pre-exponential terms A disappear stant called β corresponding to a Lagrangian multiplier from the latter form containing Gibbs free energies G. or a distribution parameter, and finally connected to This modification results from the identification of the temperature by analogy with classical thermodynamics. pre-exponential factor as an entropic parameter, which This introduction of kBT is elegant and perfectly rational can then be incorporated in the Gibbs free energies. This in this context but could be too restrictive for an exten- pertinent operation will also be used here and applied to sion to more complex systems. kBT is concretely useful Eq.(3b), but let us first explain the entropic meaning of here to adimension the exponent of the Arrhenius expo- the pre-exponential factor. This factor can be defined nential and is usually interpreted as the mean particle as A = τ −1Ω−1 where τ −1 is the universal thermal fre- energy. This interpretation holds in the kinetic theory quency (Section 6.2.2), giving their time−1 unit to the of ideal gases but is much less clear for complex chem- rate constants, and Ω−1 is the reciprocal of the number ical systems in which the mean reactional energies are of possible configurations of the reactant, one of which precisely different for each type of particle. allowing the reaction to proceed, as simply illustrated in Fig.2. As a consequence, ln(Aji=Aij) can be identi- 2.2 Insensitivity of dynamic equilibria to fied, after elimination of the identical frequencies, as the difference of statistical entropies activation energies The accepted conclusion that the presence and the height Aji Ωij ln = ln = Sij − Sji of energy barriers do not participate to equilibrium ra- Aij Ωji tios between the reactants, can appear not completely intuitive because one could expect that increasing the Note the inversion of the suffixes between the ratios height of the barrier exponentially amplifies the differ- of A and Ω. In addition, if a given reactant is involved ential capacity of the reactants to jump over, including in several reactions, one can assume that it is not nec- in equilibrium. By contrast, in the potential surface of essary to precise the nature of the product in reactional the transition state theory [4], the activated complexes entropy, because it is expected to be the same for all reside on mountain crests but their energy levels do not the reactions starting from this reactant. Indeed, even if influence the populational distributions. each reaction requires a specific reactional configuration, 2 this configuration belongs the same set of total configura- Large Ea do not invert but accentuate the differences de- tions Ω of this reactant. Converting configurational into termined in both cases by the relative energy levels. This thermodynamic entropy through the Boltzmann formula point results from a mathematical interpretation of the S = S=kB, allows to detail Eq.(4) step-by-step exponential factor. n A H − H ln i = ln j + j i 3 A new probabilistic interpreta- n A k T j eq i B tion of the exponential factor Ω H − H = ln i + j i Ω k T j B 3.1 The exponential distribution S − S H − H = i j + j i kB kBT The exponential component of the Arrhenius law and of Hj − Hi − T (Sj − Si) the Boltzmann distribution, can be simultaneously de- = kBT rived from the exponential distribution, which is basi- (Hj − TSj) − (Hi − TSi) cally mathematical. According to this law, the proba- = bility that a particle from an homogeneous population kBT with an average number of energy quanta per particle Gj − Gi = hEi, reaches a fixed threshold number Ez, is kBT Such a fusion of entropy and enthalpy into free en- Ez z − hEi ergy will be retained here for whole systems but not P (E > E ) = e (5) for individual reactants, because molecular entropy and Since Ez is a temperature-independent threshold and enthalpy play different roles in reactions, which are de- hEi is temperature dependent, temperature remains at scribed by different probabilities (Fig.2). the denominator of the exponent, but an important dif- ference with current rate equations is the absence of kBT .