A Probabilistic Rate Theory Connecting Kinetics to Thermodynamics Denis Michel

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 Michel‡

‡ 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 equilibrium 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 constants 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; enthalpy; 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- ‡ 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 remains 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

‡ 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 ‡ −(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 ‡ ‡ i ji ji (E −Hi−E +Hj )/kB T = = K = ‡ 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 ‡ E disappears. The equilibrium ratio is independent of ‡ 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 E‡, is kBT

Such a fusion of entropy and enthalpy into free en- E‡ ‡ − hEi ergy will be retained here for whole systems but not P (E > E ) = e (5) for individual reactants, because molecular entropy and Since E‡ 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 difference with current rate equations is the absence of kBT . kBT is generally regarded as the mean single particle energy, acceptable for purely kinetic gases, but not for all heterogeneous systems containing different kinds of energies. An other problem even more critical with the Arrhenius equation of Eq.(1), is that in absence of energetic barrier (Ea = 0), the exponential is equal to 1, suggesting that reactions are no longer energetically re- stricted but only affected by the pre-exponential factor. In other words, the mean particle energy kBT would be fully sufficient for all the transitions. This result is clearly inconsistent with the view of kBT as a mean energy because of a basic principle: the average particle energy in a homogeneous population is no way reached by all the particles. Indeed, even when Ea = 0, that is to say when the barrier corresponds to the mean particle energy, not Figure 2. Randomly moving pens and caps simply illustrate all the particles have a sufficient energy. In the new inter- ‡ the concept of reactional entropy. The set of their equiprob- pretation, Ea = 0 (equivalent to E = H in Fig.1 and to ‡ able relative arrangements (which would be much larger in E = hEi) predicts a probability of 1/e (0.37). Even more 3D) is noted Ω, out of which only one, of probability 1/Ω, strangely when setting H = 0, the new formula gives an allows the assembly of the caps and the pens, as shown at exponential of zero, which means that the transition is the bottom right. But even when conveniently positioned, impossible; but the traditional Arrhenius exponential is the pen and the cap can stably associate only if the energy of non-zero, temperature-dependent and the same for any ‡ their collision is sufficient. This second condition is quantified energyless reactant (Exp[−E /kBT ]), which makes little from 0 to 1 in the energetic term of rate constants defined sense. here. The configurations illustrated in this figure with sepa- rate components, can also be conceived for internal domains 3.2 The rule for particle interconversion of the same molecule. The present approach also fundamentally differs from the Eq.(3b) contrasts with Eq.(4) in that it introduces a classical one in equilibrium. The relative populations of role for activation energies in equilibrium distributions. interconvertible species are determined not only by the

3 difference of their mean energies, but also by the barri- which can be derived from a simple discrete treatment. ers. Interconversion between particles of different nature Mix E white balls and N black balls, E and N being pos- i and j is possible when either a particle i or a particle itive integers, into a large bag and draw them at random. j, has accumulated a minimal number of energy quanta, The number of balls E between two balls N, written E, is ‡ denoted below by Ei,j, which is the same for the con- supposed to correspond to the number of energy quanta versions i → j and j → i. For a particle of type i, in a particle, whose mean number is hEi = E/N. Simple the probability of this accumulation is simply given by probabilities say ‡ ‡ −Ei,j /hEii the exponential distribution P (Xi > Ei,j) = e [3]. As the same holds for the particles of type j, when E‡ E‡ hE i > hE i, the particles of type i would switch more E hEi i j P (E E‡) = = (9a) probably than the reverse. As a consequence, the system > E + N 1 + hEi spontaneously evolves until the reciprocal fluxes equal- ‡ ize. The numbers of particles in the sub-populations i For E and hEi large and of the same order of magnitude, and j necessarily adjust such that this discrete geometric distribution can be approximated as the continuous exponential distribution.

E‡ eq ‡ eq ‡ ‡ − −Ei,j /hEii −Ei,j /hEj i P (E E ) = e hEi (9b) ni Aij e = nj Aji e (6a) > or If we no longer reason in term of threshold but of ! exact value, we have E‡ 1 − 1 ni Aji i,j hE i = e i hEj i (6b) nj eq Aij ‡ E E N P (E = E‡) = Returning to the correspondence between the unitless E + N E + N ‡ (10a) pre-exponential term and entropy, this equation takes the hEi E 1 = form 1 + hEi 1 + hEi

Energetic term Switching once again to the continuous distribution, Entropic term z }| { in the thermodynamic limit ni z }| { ‡ 1 1 ln = Si − Sj + Ei,j − (7) nj eq hEii hEji ‡ − E − 1 P (E = E‡) ∼ e hEi 1 − e hEi Eq.(7) can also be written with the activation energy ‡ −E‡/hEi (10b) E = E − hEii as follows e a(i→j) i,j = P∞ −j/hEi j=0 e ‡ ni hEii Ei,j ln = Si − Sj + 1 − which is, when multiplying the numbers of energy nj hEji hEii eq (8) quanta by the energy unit q and when hEi q = kBT , hEii Ea the Boltzmann probability law where the denominator = Si − Sj + 1 − 1 + hEji hEii can be understood as the sum of all possible microstates, called the partition function. It was considered by Feyn- man as the summit of statistical mechanics [5], but is 4 Reconnection of the energetic recovered in two lines from the exponential distribution. parameter of rate constants to As a consequence, the relative subpopulations of energies E and E are given by the ratio of two Eq.(10b) Boltzmann’s energy distribu- 1 2

E −E tion P(E=E1) 2 1 = e hEi (10c) P(E=E2) 4.1 Starting from the geometric distribution Note that the approximations used above are based on the following property of the exponential function One of the main goals of present proposal is to unify lim (1 + ε x)1/ε = ex kinetic and thermodynamic rules under the same law, ε→0

4 which can be written celebrated Boltzmann’s energy distribution, valid for example for ideal gases. The ideal monoatomic gas is the 1/(1−q) lim (1 + (1 − q) x) = ex perfect example of a system with a maximal entropy but q→1 whose individual components have an energy strictly pro- that is the q-exponential or generalized exponential func- portional to temperature. Inversely, most real systems tion which, when q differs from 1, expands the spectrum have a certain degree of organization (non-maximal en- of statistical mechanics [6] and has applications in a vari- tropy) but are made of molecular components with a ety of fields, thermodynamical and nonthermodynamical certain degree of entropy, which will be examined later. [7]. This function has been proposed as a generalization Let us first deepen the analogies between the exponential of Arrhenius law [8] and is used in rate theories [9, 10, 11]. probability distribution and Maxwell-Boltzmann results. The probability density function (PDF) of the exponential distribution of particle energies is 4.2 Simplified Maxwell-Boltzmann dis-

tribution 1 − E f(E) = e hEi hEi When hEi corresponds to kBT , with energy units, the structure of Eq.(10b) corresponds exactly to that of the represented in Fig.3A.

Figure 3. Fractional distributions of: (A) particle energetic states, (B) particle velocities, (C) energy density, and (D) velocity density. Curves drawn with mass and energy of 1 unit. The fractions of particles with energy higher than E ‡ correspond to the colored surfaces in panel A, quantiﬁed in Eq.(17a). They increase with hEi, itself increasing with temperature. This result meets the traditional interpretation of the role of temperature on rate constants, which supposes that particle energies increase with temperature, contrary to Gibbs free energies.

Given that the nth moment of this distribution is equilibrium. For a monoatomic ideal gas, the energy E[Xn] = hEin n!, its typical ﬂuctuation is simply content of a particle is exclusively translational. With ˜ p 2 2 an energy unit q (in joules) such that hEi = hEi q, a par- σE = E[X ] − E[X] = hEi ˜ ˜ 1 2 ticle of energy Ei has a velocity vi such that Ei = 2 mvi , and a variance of hEi2 is acceptable without breaking giving the distribution of particle velocities, after nor-

5 malization, more elaborate systems with diﬀerent types of particles and energies.

s 2 2m − mv f(v) = e 2hEi˜ πhEi˜ 5 Comparative influences of tem- represented in Fig.3B and which has a mean value q R ∞ ˜ perature in the different treat- v¯ = 0 vf(v)dv = 2hEi/πm. The particles which have an energy content Ei constitute a fraction of the ments system ni/N and bring together in the system a fraction of energy f(Ei) Ei. The most represented particles are, 5.1 Influence of temperature on rate according to the exponential as well as Boltzmann distri- constants butions, the low-energy and low-mobility particles, which poorly contribute to the total energy of the system, thus If the mean reactional energies are proportional to tem- generating a peak in the energy density curve at the level perature (H = CT ), the traditional shape of the Ar- of the largest product Ei f(Ei). The normalized expres- rhenius plot is obtained with the present treatment, but sion of this curve is the density of the Γ2 function (shape through a different way. Increasing the temperature in- shown in Fig.3C). creases the thermal energy stored in the particles and responsible for their reactional capacity, thus reducing E E ‡ − hEi the activation energy Ea = E − H. By contrast using D(E) = γ2(E) = 2 e (11) hEi the conventional formula of Eq.(1), Ea is the constant steering coefficient of the Arrhenius plot [12], in spite of This function can also be understood as the convolu- its frequently noticed dependence on temperature [8, 13], tion of the densities of two independent exponential dis- and the effect of temperature is assumed to come from tributions of identical parameter, corresponding to the the entity k T . bottom and top boundaries of the energetic content. B For exclusively translational particles, when substituting ˜ 1 2 Ei = 2 mvi , 2 mv2 mv − i E f(E ) = i e 2hEi˜ 5.2 Influence of temperature in equilib- i i ˜ 2hEi rium q whose integration between 0 and ∞ gives πhEi˜ /8m. 5.2.1 Where is the temperature in the different Multiplying the latter equation by the inverse of this equations? value directly yields the normalized velocity density function The specificities of the new proposal can be clearly high- lighted by taking up the statistical definition of entropy 3/2 and grouping together the common elements as follows. 1/2 ! 2 2 m − mv The classical relationship can be written D(v) = v2 e 2hEi˜ (12) π hEi˜ eq eq n nj Hj − Hi which gives the bell-shaped curves shown in Fig.3D, with ln i − ln = (13) Ω Ω k T a mode at i j B s 2hEi˜ v = while that proposed here is mp m Therefore, we verify that the most probable values of en- eq eq ni nj ‡ 1 1 ergy and velocity densities (tips of the curves of Fig.3C ln − ln = Ei,j − (14) Ωi Ωj Hi Hj and 3D respectively), naturally coincide through E˜mp = 1 mv2 . The mean velocity density R ∞ vD(v)dv is 2 mp 0 The two major differences between the right sides of s Eqs.(13) and (14) are (i) the absence of the energy barrier 8hEi˜ ‡ hvi = E in the classical form and (ii) the (apparent) absence of πm the temperature in the new one. It is precisely proposed here that the effect of temperature is mediated by en- ˜ When substituting hEi = kBT , Eq.(12) is exactly the thalpies, whereas in the classical approach, temperature celebrated Maxwell velocity distribution, currently intro- is also present explicitly in the component kBT . Using duced in different and longer ways. Let us now consider equilibrium constants for the classical approach, Eq.(13)

6 is equivalent to showing that when the compounds have the same enthalpy, their equilibrium is governed by entropies. Using Gj − Gi the activation energy Ea, ln Kji = (15) kBT Writing ∆G = ∆H − T ∆S and supposing that H Hi Ea(i → j) and S are temperature-independent, has the practical ln Kji = Si − Sj + 1 − 1 + (16b) Hj Hi advantage to yield the expected van’t Hoﬀ plots linearly dependent of 1/T . In the present treatment, using an energy threshold E‡ and a mean enthalpy H for a reactant or a group of reactants according to the Hess’s law, Temperature is not apparent in these equations but hidden in the enthalpies. The relative temperature- ‡ 1 1 dependence of the enthalpies regulate the equilibrium, ln Kji = Si − Sj + E − (16a) Hi Hj as schematized in Fig.4.

Figure 4. Comparative interpretation of the role of temperature in van’t Hoff plots. (A) In the classical interpretation, the heat capacities of the two reactants are always parallel and may vary or not. (B) In the present interpretation, temperature affects in a different manner the reactivity of both reactants because of their different heat capacities. As a consequence, ∆H depends on temperature. The variations of enthalpies are exaggerated in this scheme, but even a faint divergence between the reactant enthalpies would have exponential consequences on the difference of waiting times between the back and forth reactions.

In the classical interpretation (Fig.4A), the enthalpies not correspond to G. The energy landscape of Arrhenius of the reactants are shown to increase in parallel with is governed by enthalpies whereas thermodynamic land- temperature, but note that strangely even if they scapes include entropies. The activated complex theory simply remain constant, the traditional temperature- describes the kinetic barrier in term of difference of Gibbs dependence of van’t Hoff plots would still be obtained. free energies [4, 14]. In this theory, the tip of the barrier is not a temperature-independent threshold but the energy of an activated complex that itself depends on tempera- 5.2.2 The kinetic formalism using Gibbs free en- ture, so that the temperature-dependence of the height of ergies the barrier ∆G‡ results from the relative dependences on temperature of the ground (G) and transition (G‡) states Temperature is intuitively expected to increase the re- [15]. The activated complex is an atypical structure sta- action rates by increasing the energy of the reactants, bly standing on a saddle point of the energetic landscape but using free energies in the kinetic treatments would and with the strange property, when it is reached, to say the opposite. G always decreases with temperature, only evolve in the direction of the complete transition ‡ thereby increasing the difference E − G and raising the but cannot turn back. Moreover, the forbidden transi- height of the barrier. Using the so-called enthalpies of tion is not the same depending of the starting reactant reaction H instead of Gibbs free energies, allows to cir- [3]. In the kinetic theory, temperature is the macroscopic cumvent this contradiction. H is indeed proportional to manifestation of microscopic motions, that is re-scaled at temperature in certain ranges of temperature and does

7 the microscopic level through the Boltzmann constant, so fraction of particles with a energy higher than a given that kBT is an average index of single particle motion. activation threshold. This description implicitly assumes But temperature is also present in Gibbs free energy, not that the Boltzmann distribution of energy is maintained only in its most frequent expression G = H−TS, but also in homogeneous reactants in presence of particle transfor- because H and S are themselves temperature-dependent mations. This rational explanation of the role of temper- [16, 17]. Raising temperature from T1 to T2 (> T1) in- ature mediated by particle energies is formally modeled creases enthalpy by H(T2) − H(T1) = CP (T2 − T1) and using Eq.(17a). On the one hand, it can be understood entropy by S(T2) − S(T1) = CP ln(T2/T1), where CP is for enthalpies but not for Gibbs free energies, but on the the heat capacity supposed constant at constant pres- other hand, even when using enthalpies, the current and sure. present approaches strikingly diﬀer, as explained below.

5.3 Reintroducing temperature in reac- 5.3.2 Temperature increases the reactional en- tional energies ergy of the particles

5.3.1 Increasing particle energy favors reactions The dimensionless exponents of the two competing struc- tures of rate constants compared here, contain ener- Using the energy quanta approach, the dependence on gies in the numerator and denominator, but beyond ‡ temperature is conferred by the ratio E (temperature- this common property, the principles are different. In independent) over hEi (temperature-dependent). This is the usual form, the exponent is H/kBT , whereas in in fact less clear for the traditional Eq.(15) and the ex- the present one, it is E‡/H. In the former ratio, the ponent of the rate equation based on a difference of free numerator and the denominator both depend on tem- energy [15] perature, whereas in the latter one, only the denominator is temperature-dependent. In other words, the ‡ −1 − ∆G k = τ e kB T traditional formulas still work even when H are independent of temperature, in contradiction with the in- where both the numerator and the denominator are terpretation of the role of temperature through parti- temperature-dependent. Reactions are facilitated when cle energies. Thermal energy increases the speed of the energy of the reactants is higher. Possibly because molecules, important for the reactional efficacy of colli- this reasoning does not hold for Gibbs free energies, many sions, but more generally the heat capacity of molecules teachers prefer drawing the Maxwell velocity distribution describes how much energy can be stored by a molecule to show that increasing the temperature increases the in its internal vibrations or rotations, which are often fractional population over the activation energy thresh- the triggering factors of chemical transformations, par- old. This view can in fact be recovered probabilistically, ticularly for biological macromolecules whose heat ca- without returning to the ideal gas, using the partial in- pacity is clearly not negligible. The heat capacity of tegration of the PDF of the exponential distribution. a monoatomic gas is purely translational, but that of more complex particles include the other forms of energy. In the new exponent proposed here, the denom- Z ∞ 1 − E inator represents the temperature-dependent reactional P (E E‡) = hEi dE > e energy in all its forms: translational, rotational and vi- E=E‡ hEi (17a) ‡ brational. Since all these forms define the macroscopic − E = e hEi notion of temperature, the reactional propensity is naturally expected to follow temperature. This property One finds again the energetic part of rate constants pro- precisely characterizes enthalpy, which obeys the ther- posed here, given in Eq.(5) and illustrated by the colored modynamic relation ∂H(T )/∂T = C whose integration surfaces in Fig.3A. Remarkably, Boltzmann probability p with Cp temperature-independent and the postulated ini- can also be obtained by integration of the same function, tial value H = 0 at T = 0, simply gives H ∝ CT . The but in a different manner. variation of enthalpies with temperature is in fact less Z E‡+1 simple in practice. It is calculated using Kirchhoff’s law 1 − E P (E = E‡) = e hEi dE in term of a reference state different from T = 0 [18]. In E=E‡ hEi (17b) addition, the temperature-dependence of CP should be ‡ − 1 − E = 1 − e hEi e hEi introduced using empirical expressions found in tables −2 like CP = α + βT + γT + ... Absolute enthalpies are refractory to experimental measurements contrary to dif- The global shift of the Boltzmann distribution upon ferences of enthalpy calculable by calorimetry. Hence for temperature increase (Fig.3) is commonly used in text- simplicity in the following equations, the proportionality books as a pedagogical explanation of the increase of the between H and T will be retained as a gross approxi-

8 mation for theoretical elementary reactants. Introducing ∂H(T ) = Cp this definition of enthalpy in Eq.(16) gives ∂T The frequency of the preexponential factor is also ‡ Ei,j 1 1 temperature-dependent. Moreover, certain ranges of ln Kji ∼ Si − Sj + − (18) T Ci Cj temperature should not be considered in case of phase transition or of inactivation of certain reactants. For Where Ci and Cj are the average heat capacities of example, proteins are denatured over 320 K and frozen the interconvertible reactants weighted by the stoichio- below 270 K. metric coefficients. Strikingly, Eq.(18) can yield typical van’t Hoff plots. It also shows that at very high tempera- 5.3.3 Deviations from the expected plots ture, the equilibrium constant becomes essentially a matter of entropy, as expected. Considering the exponential Experimental Arrhenius plots are rarely the straight lines term of rate constants as a probability implies profound expected from the Arrhenius equation [ln k = ln A − conceptual changes but moderate apparent changes in Ea 1 ], but can have convex or concave shapes, explained practice, which just impose to reinterpret the experimen- R T by tunneling effects and other hypotheses [8, 19, 20]. The tal plots, as schematized in Fig.5. dependence of heat capacities on temperature is an ad- ditional explanation. Deviations are also the rule for the so-called van’t Hoff plots in equilibrium. Moreover, the two types of enthalpy: (i) calorimetric and (ii) determined from the analysis of experimental plots using the currently admitted functions, are always different [16, 17]. Reintroducing the temperature-dependence of enthalpies and entropies might account for certain deviations [21, 22], but more fundamentally, a difference of enthalpy between the interconvertible reactants is syn- onymous to a heat of reaction, that itself reflects a difference of heat capacity, and heat capacities generally vary with temperature.

5.4 Giving a calorimetric meaning to the Boltzmann constant

Heat capacity is assumed to be closely related to energy fluctuations. We have seen that the typical fluctuation of the exponential distribution is σE = hEi. Considering it Figure 5. Comparative interpretations of the Arrhenius as proportional to temperature through an absolute heat and van’t Hoff plots between the classical and the present capacity C, gives σE = CT , to be√ compared with the treatments of the energetic exponential. The meaning of the result of statistical mechanics σE = CV kB T obtained using hEi = U = − 1 (∂Z/∂β) , β = 1/(k T ) and entropy intersect is unchanged but that of the slopes is com- Z N,V B pletely different. CV = (∂U/∂T )N,V . This comparison suggests a correspondence between the Boltzmann constant and a heat capacity. This correspondence is also supported by the The most striking difference is that in the classical proportionality, in the classical limit and following the interpretation of the van’t Hoff plot, the temperature- Dulong-Petit law, between heat capacity and Boltzmann dependence of enthalpies is denied, whereas the present constant in simple systems like ideal gases. In constant probabilistic model is fundamentally based on calorimet- n volume, the heat capacity of ideal gases is CV = 2 kB, ric enthalpies. The simplistic dependence on tempera- where n is the number of degrees of freedom. For in- ture depicted in Fig.5 should be further refined in these stance, ideal monoatomic gas particles flying in a con- very approximate equations, using Kirchhoff’s law, set- stant volume have a mean kinetic energy ting the standard values H0 at T = 298K, and taking into account the dependence on temperature of heat ca- 3 ˜ 1 2 CV T = hEi = E˜rms = mv pacities, entropies and enthalpies according to the basic 2 2 rms thermodynamic differentials q ˜ where vrms = 3hEi/m is the root mean square velocity, ∂S(T ) C obtained using velocity vectors split over the 3 spatial co- = p ∂T T ordinates. The interest of taking kB as a minimalist heat

9 capacity is twofold: kBT would be a prototypal enthalpy The traditional form of the energetic exponential obvi- for simple systems and the two entropies, thermodynamic ously complies with this rule since the total difference of and statistical, would be related through a heat capacity. energy of a global reaction made of several elementary Volume changes regulate pressure for systems of mutu- steps is simply the sum of energy gaps, with the auto- ally colliding particles like gases, but for macromolecules matic elimination of activation energies, as illustrated in in solution, pressure is essentially due to microscopic col- Eq.(2d). Things are less simple with the new form since lisions with the solvent, water, that is itself incompress- the activation energies should now take part to the de- ible and whose cellular content automatically adjusts to tailed balance relationship. Consider for example a tri- the cellular volume. Heat capacity defines the extent of angular circuit. The compounds a and b can interconvert ‡ heat absorbed or released upon temperature change, ac- either directly, with an activation threshold Ea,b, or in- cording to ∆Q = C∆T . But at constant temperature if directly through an intermediate c. Assuming that the for some reason the heat capacity evolves, then a release reactional enthalpy and entropy of every reactant are the ‡ or uptake of heat is also expected such that ∆Q = T ∆C, same for all its transformations, writing Ea,c the activa- thus making a clear connection between changes of heat ‡ tion threshold between a and c, and Ec,b between c and capacity and of thermodynamic entropy, the latter one b, the detailed balance relation including in addition the configurational information of k k k statistical entropy such that S = CS. Under this postu- ab bc ca = 1 (23a) late, heat capacities would become the main ingredients kba kcb kac of the thermodynamic constants. For the equilibrium implies for the energies constant, E‡ − E‡ E‡ − E‡ E‡ − E‡ a,c a,b + a,b b,c + b,c a,c = 0 (23b) ‡ Ha Hb Hc Si Sj Ei,j 1 1 ln Kji = − + − (19) Ci Cj T Ci Cj generalizable for all the cycles of the system, with any number of components n and for the kinetic constant, n ‡ ‡ X Ei,i−1 − Ei,i+1 ‡ ‡ ‡ ‡ ! E = 0 with E1,0 = En,n+1 = E1,n 1 i,j Hi ln kij = − ln τ − Si + (20) i=1 Ci T (23c) These relationships with circular permutations are mu- Differentiation of Eq.(19) with respect to temperature tually compatible for all the cycles of a reticulated sys- gives tem. In this new treatment, the activation energies are now envisioned as integral components of the system, whereas in the classical theory, they were extra- E‡ d ln Kji i,j 1 parameters whose values can be arbitrary. = (Ci − Cj) 2 (21) dT CiCj T which is interesting since Eq.(21) has the structure 6 Rebuilding discrete thermody- of the original equation of van’t Hoff, based on the heat developed by the reaction [23]. Using conventional Boltz- namics mann entropies S = kBS, differentiation of Eq.(18) gives Matter was discretized in the visionary theory of Boltz- mann and the same was extended to energy by Planck ‡ ! d ln Kji 1 Ei,j [24]. Since handling energy as a number of quanta fa- = (Ci − Cj) + 2 (22) cilitates the formulation of basic equations, let us ap- dT kBT CiCjT ply again this recipe to the rudiments of thermodynamics and then reintroduce the new kinetic rules proposed Once again the theoretical heat capacities of these above. In this section, the notions of free energy and equations are in fact H/T . entropy will apply to the system while the reactants will be supposed purely enthalpic. The concept of system 5.5 Extension of the detailed balance to entropy long proved fruitful to begin statistical physics. activation energies 6.1 Index of uncertainty for a population To the properties of the hypothetical rate constants listed above, is added a complication to satisfy the detailed bal- Two criteria are sufficient for defining this index equiva- ance rule linking the different rates constants of a system. lent to entropy. It should be: (i) a function of the number

10 of possible configurations (microstates) of the system’s interest to reconsider Eq.(26), which could simply mean components and (ii) additive for uncorrelated systems. that every fractional state ni/N is occupied ni times, so (i) The different possible configurations of a macroscopic that Ω is simply the reciprocal of the probability of oc- system (total number written Ω), are invisible and def- cupancy. Based on the same principle, another equation initely not measurable in practice, but the number of can be designed by supposing that the system can evolve. these theoretical ”snapshots” can be deduced statisti- cally. It reflects the uncertainty of the system, a notion closely related to missing information and disorga- 6.2 Evolving system nization. For example, a book is an organized object in In a first-order network, r particle states obey a lin- which the page 189 is located between the pages 187 and ear evolution system, which has a unique stable solution 191 with a good degree of confidence, but if the pages written neq. A new state function allows to describe the are torn and thrown down, this probability strongly de- i neq ni creases because the number of possible arrangements of occupation of the equilibrium proportions i : the pages increases. N (ii) The other requirement for an index of disorganiza- r eq r tion is its additivity for independent systems. This can 1 X n X U = − n ln i = − p ln peq (29) also be easily understood: For each microstate of a sys- N i N i i i=1 i=1 tem 1, all the states of a system 2 are possible, so the total number of states of the global system 1 + 2 is the U is an index of the mean gap between the current product Ω1Ω2. Obviously the function satisfying and equilibrium distributions. This gap can incite the system to evolve, provided it is higher than its degree of

f(Ω1Ω2) = f(Ω1) + f(Ω2) (24) disorganization U > S. The difference F = U − S is r r is the logarithmic function (ln Ω). We will also define a X eq X 1 F = − pi ln pi + pi ln pi single particle average index S = ln Ω. Concretely, N i=1 i=1 (30) r the total number of configurations of a system made of X pi = p ln N noninteracting components falling into r categories is i peq i=1 i N! Ω = (25) F will prove particularly interesting upon derivation. n1!n2! . . . nr! √ Using the Stirling approximation x! ∼ xx e−x 2πx 6.2.1 Introducing time [25] of which a rough approximation is x! ∼ xx, To concretize the notion of system evolution and of inter- changeability between single particle states, one should N n n n −1 N hn1 1 n2 2 nr r i introduce a temporal ingredient, in the form of rate con- Ω ∼ = ... −1 n1 n2 nr stants k examined previously of unit time . As these n1 n2 . . . nr N N N (26) rates depend on the temperature, the system will be considered closed but not isolated and embedded in a ther- so that mal bath (NVT ensemble). As the different states can be interchanged two by two, a simple two-state system can r n X ni i ln Ω = − ln (27) be used to derive the proportions of particles in states i N i=1 and j (or for a single particle, the probability to be at a giving a single particle average index of given moment in state i or j). Let us first derive S.

r r 1 X ni X dS 0 S = − n ln = − p ln p (28) = − (p ln p + p ln p ) N i N i i dt i i j j i=1 i=1 P = (kijpi − kjipj)(1 + ln pi) with pi = ni/N and i ni = N. With respect to these (31a) deﬁnitions, the so-called Boltzmann entropy (S ) and + (kjipj − kijpi)(1 + ln pj) B Gibbs/Shannon entropy (SG) are not equivalent, but re- pi = (kijpi − kjipj) ln lated through SG = SB/N. SG/kB corresponds to S pj whereas S /k corresponds to NS. B B which can be generalized to the r nodes But for deriving a pure theory, using an approxima- dS X pi tion, as acceptable as it can be, is not satisfactory. To by- = (k p − k p ) ln (31b) dt ij i ji j p pass the need for the Stirling approximation, it is of some i,j j

11 Eq.(31) is not particularly illuminating because it can be distinguished, because of the uncertainty principle. be either positive or negative; but things are more in- This value can be obtained in multiple ways including teresting when deriving F. Taking two interconvertible in the context of the Maxwell-Boltzmann system. For a nodes i and j, single particle, the length unit√ is the thermal wavelength of de Broglie λ = h/p = h/ 2πmkBT and the mean par- p ticle velocity determined previously isv ¯ = 2kBT/πm. " !#0 The ratio gives the minimal time step 0 pi pj F (i, j) = pi ln eq + pj ln eq p p λ h i j τ = = ≈ 8 × 10−14 s at T = 300 K (33) " ! # v¯ 2kBT pj pi = (kijpi − kjipj) ln eq − ln eq whose reciprocal is the maximal frequency close to pj pi k T/h, considered by Eyring as the most remarkable eq ! B pi pj achievement of his theory [31], and which was already = − (kijpi − kjipj) ln eq pj pi present in the equation of Herzfeld [32]. In the rate constants, this maximal frequency τ −1 is more or less de- and considering that k peq = k peq, ij i ji j creased by entropic and energetic restrictions. kijpi The arrow of time is a source of endless debates, par- = − (kijpi − kjipj) ln kjipj ticularly interesting when applied to the universe taken (32a) as a whole. The free energy of a closed system should necessarily diminish within the limits of residual fluctu- which can be written in term of flux (J = nk) for the r ations. Then, whence comes the initial free energy of nodes, our universe which does not seem very organized in its youth? Did previously latent forces release free energy dF X Jij = − (J − J ) ln (32b) during the physico-chemical maturation of the universe? dt ij ji J i,j ji or do we live in a local fluctuation in a giant universe in equilibrium? according to an elegant but contested proposal of Boltzmann [33]. Contrary to Eq.(31), Eq.(32) can be only zero or negative. For this reason, it remarkably illustrates the free energy dissipation and can naturally be identified with 6.3 Introducing discrete energy the arrow of time [26, 27, 28]. The discrete statistical functions described above can be reconnected to tradi- The frequencies kij introduced in Section 6.2.1 remain tional thermodynamics by identifying kBT F with the abstract and their nature should be precised. In addi- Gibbs free energy of the system, kBT U with the inter- tion to their temporal parameter, the rate constants in- nal energy and kBS with the entropy. An increase of clude an energy parameter which can be different in the entropy (towards equipartition) logically decreases free different k. Before envisioning reactional systems, the energy and the possibility to extract some work. Further spreading of energy will be first described in simple sys- extensions of this discrete approach to thermodynamics tems without chemical transformations. Instead of dis- are described in [29, 30]. Now let us look at Eq.(32) which tributing particles into ”energy levels”, energy units are has the particularity to bring together the two different distributed over particles. Things appear clearer when aspects of time. energy is handled discretely. Perhaps influenced by what Boltzmann said him in 1891: ”I see no reason why energy shouldn’t also be regarded as divided atomically”, 6.2.2 Time’s arrow and time steps Planck proposed in the founder article which opened the The arrow of time envisioned as the dissipation of free en- way to quantum physics [24], to discretize energy into ergy in Eq.(32) has a clear thermodynamic origin out of quanta of value hν, written more generally q below. equilibrium, but even in full equilibrium conditions, the rates constants keep their time units and the particles in- 6.3.1 Homogeneous system definitely continue to move and transform. This sort of time, completely reversible contrary to the time’s arrow, The mean energy of thermal systems can be recovered in corresponds to the elementary time steps related to an- a discrete manner from the second law. Returning to the other field of physics: quantum physics. Statistical and white balls supposed to correspond to energy quanta in quantum physics are intimately and necessarily related Section.4, the number of ways to distribute E objects in because enumerating configurations would be infinite in N boxes is continuous space and time. Accordingly, the quantum of time can be defined as the time necessary to cross the (N + E − 1)! Ω = (34) length unit below which successive configurations cannot (N − 1)! E!

12 which gives, using the Stirling approximation, a single 6.3.2 Heterogeneous system particle average entropy of When the system is composite and made of r different categories of particles which have different mean energies 1 E E E E P S = ln Ω = 1 + ln 1 + − ln hEii, we have to fill r boxes with N = i ni particles and N N N N N E = P n hE i energy quanta (35a) i i i When the system contains a single category of parti- r cles, the ratio E/N is equivalent to the mean number of Y (ni + Ei − 1)! Ω = (40) energy quanta per particle U/q [24]. (n − 1)! E ! i=1 i i S U U U U S = = 1 + ln 1 + − ln (35b) kB q q q q giving, for large populations and using the Stirling approximation, an entropy of Temperature can now be introduced using the fundamental entropy equation. r X Ei Ei Ei Ei ln Ω = n 1 + ln 1 + − ln dS 1 i n n n n = (36) i=1 i i i i dU T (41) according to which integration of Eq.(35b) yields Keeping N and E constant, this entropy is lower than that of an homogeneous system, showing that the cate- k q 1 B ln 1 + = (37a) gorization of objects is a first step towards organization. q U T As the derivative of a sum is the sum of the derivatives, and U 1 = hEi = (37b) q/k T E 1 q e B − 1 hE i = i = (42) i qi/k T ni e B − 1 The reasoning can be circular in statistical thermodynamics and these results can also be recovered in a reverse way starting from the partition function, as did First-order particle interconversion Einstein [34]. The ratio of Planck (q = hν) over Boltz- mann entities, turns to be the logarithm of the inverse To render this heterogeneous system dynamic, one should proportion of energy quanta in the mixture. introduce the possibility of interconversion between the particles of the different kinds in the form of rate con- hν E 1 stants. The rule for particle interconversion has been = − ln = ln 1 + (38) established in Eq.(8) using the exponential law. For uni- k T N + E hEi B molecular transconversions which do not modify the total Replacing enthalpy by this energy yields the new Ar- number of particles, let us abandon the ratios of pre- rhenius esquation exponential factors. The single particle energy averaged E‡ over the r(r − 1)/2 couples of categories in the system is ln k/A = − eq/kB T − 1 q ‡ E − i,j P P hEii At the thermal scale, q is negligible compared to kBT E i j6=i hEii e hEmi = = ‡ (43) and the expansion of Eq.(37b) gives a mean number E N − i,j (unitless) of energy quanta P P hEii i j6=i e

k T hEi ∼ B (39a) in which the inexisting interchanges are characterized q ‡ by Ei,j = ∞. For a system of enthalpic components, one and a mean energy (joules) of can deﬁne the transition rate constants

hEi q = U ∼ kBT (39b) ‡ −1 −E /hEii As the heat capacity of a substance is its ability to in- kij = τ e i,j (44a) crease its energy content with temperature, it can therefore be deduced from Eq.(37b), ‡ −1 −E /hEj i kji = τ e i,j (44b) d hEi q2 eq/kB T C = q = 2 2 dT kBT eq/kB T − 1

13 where τ is the time step and hEii is the mean energy 6.4 Possible source of confusion for mod- of the particles of type i, both depending on temperature eling the relationships between frac- through Eq.(33) and Eq.(42) respectively. tional populations and energy For two categories n1 and n2, the relationships linking energy and particle numbers are The enthalpic particles of the toy systems described above are sufficient to visualize the important differences in the description of the ratios of particles, depending on −1 n1 E‡ 1 − 1 = 1 + e 1,2 hE1i hE2i (45a) whether they belong to the same category or not. For a N homogeneous system of mean energy hE1i in which energy diffuses without barriers between particles n1, the −1 n2 E‡ 1 − 1 ratio of particles with energies Ea and Eb follows, accord- = 1 + e 1,2 hE2i hE1i (45b) N ing to Eq(10c),

Ea−Eb hE i ‡ n (a)/n (b) = e 1 (48) E1 hE1i E 1 − 1 1 1 = e 1,2 hE1i hE2i (45c) E hE i 2 2 which is a familiar Boltzmann relation when setting and the mean particle energy in the system is hE1i = kBT , whereas interchangeable particles of diﬀer- ent categories with diﬀerent mean energies hE1i and hE2i, ‡ ‡ E E are related through − 1,2 − 1,2 hE1i e hE1i + hE2i e hE2i hEmi = ‡ ‡ (45d) E1,2 E1,2 ‡ 1 1 − hE i − hE i E1,2 hE i − hE i e 1 + e 2 n1/n2 = e 1 2 (49)

Eq.(48) is valid for a single phase in the pioneer statistical theories, applicable for instance to energy spreading Circular equilibria between gas particles, but the object of this manuscript is precisely to propose that this form cannot be extended Three categories of particles are suﬃcient to establish to chemical reactions. The next tasks in this direction the famous detailed balance relationship of Wegscheider will be to implement these rules from ﬁrst order to sec- linking the constants of a cycle [35]. The relation ond order networks, to introduce particle entropies, to extend results to entropic open systems in steady state

n1 n2 n3 like biochemical systems, for modeling the fundamental = 1 (46a) ingredients of life: nonlinearity, retroactions and multi- n2 n3 n1 stability. is obviously always true, in particular at equilibrium, so the constants are necessarily related through 7 Conclusions k k k 12 23 31 = 1 (46b) k k k 21 32 13 The present probabilistic treatment based on the exponential distribution does not invalidate the Maxwell- The activation energies, which were completely arbi- Boltzmann distribution, but encompasses it as a par- trary in previous theories, are now mutually connected ticular case of homogeneous system with temperature- through dependent particle energies. Hence, there is no reason to not extend this approach to the energy-dependence of particle conversions, because rate constants are integral constituents of dynamic systems. The central and E‡ − E‡ E‡ − E‡ E‡ − E‡ 1,3 1,2 + 1,2 2,3 + 2,3 1,3 = 0 (47) original assumption of statistical physics was the model- hE1i hE2i hE3i ing of randomness, but randomness is precisely described in the most fundamental way by the exponential law. Circular equilibria are particularly relevant with re- In fact, the equations of Arrhenius and Boltzmann are spect to catalysts. Contrary to the widespread repre- two facets of the same law of randomness, uniﬁed in the sentation of catalysis as a deformation of the energy present theory. This is perfectly illustrated by Eqs.(17a) landscape along the coordinates of a single transition, and Eq.(17b) respectively: the ﬁrst one is the probability ‡ ‡ it should rather be envisioned as a circuit, at least trian- that E > E and the second one that E = E . To fur- gular and letting unchanged the uncatalysed single step ther enlighten this point, Boltzmann probability is sim- reaction. ply obtained using the exponential distribution only, by

14 subtraction as follows here yield resembling behaviors, so that the comparison between the previous and new treatments is more a ques- P (E = j) = P (E > j) − P (E > j + 1) tion of interpretation of the experimental plots than a = e−j/hEi − e−(j+1)/hEi true discrimination. In the field of rate theories, Laidler already noticed that, surprisingly, the widely different −j/hEi −1/hEi (50) = e 1 − e rate equations proposed in the past can give reasonably e−j/hEi good fit to the same experimental data [2]. Precise ex- = perimental validations in this field are hindered by nu- P∞ e−n/hEi n=0 merous problems, including the knowledge of absolute enthalpies, the complex temperature-dependence of heat The relationships established here are simply derived capacities, the interferences with tunnel effects and the from the single postulate that the exponential term of narrow range of testable temperature. Moreover, exper- rate constants has the status of a probability, in line with imental and calculated values are rarely in good agree- the profoundly probabilistic spirit of Boltzmann’s theory ment [18] and strikingly, calorimetric and van’t Hoff en- and with the intuition of Maxwell: ”The true logic of this thalpies never coincide [16, 17], which is not a surprise world is in the calculus of probabilities”. In this respect, considering that the enthalpy change for a reaction is re- the rate constant defined here is entirely probabilistic sponsible for a heat of reaction, whereas the heat capac- kτ = P (configuration is OK) × P (energy is OK) ity of enthalpies is ignored in the current van’t Hoff plot approach. In the alternative view examined here, the which simply says that the frequency of a transforma- entropies of the reactants are completely identical to the tion follows the probability of favorable conjunctions of previous ones; the calorimetric enthalpies and the Hess’s configurational and energetic conditions, which is law are unchanged; the favored direction of reactions is still driven by entropy increase and/or enthalpy decrease; 1 ‡/hEi the equilibrium constants remain defined by the ratios of kτ = × e−E (51) Ω interconvertible reactants concentrations in equilibrium and still satisfies the van’t Hoff formula. Note in this re- leading to the equilibrium relationship spect that the original formula of van’t Hoff was not the equation currently attributed to van’t Hoff in texbooks, eq eq ni nj ‡ 1 1 since van’t Hoff did not use the Boltzmann’s constant or ln − ln = Ei,j − (52) Ωi Ωj hEii hEji the ideal gas constant, and his reasoning was fundamentally calorimetric [23], in line with the changes of heat ca- The mean reactional energies hEi can logically be pacity on which the present theory is based. In addition, identified with enthalpies, but Gibbs free energies are van’t Hoff indicated that he managed his equation for nu- not appropriate for these definitions because the entropic merical applications and experimental controls only, but and energetic parts are involved in independent probabil- did not enter into the details of thermodynamics neces- ities. If the starting probabilistic postulate is wrong, the sary for its demonstration [23]. The 19th century formula present study would remain a theoretical exercise, but it of Arrhenius also remains an equation of practical conve- is however appealing owing to: (i) its mathematical rele- nience, even if experimental data show that it is a poor vance; (ii) its capacity to recover the Maxwell-Boltzmann quantitative predictor of reaction rates, particularly in distribution and to unify kinetics and thermodynamics; high-dimensional, complex chemical kinetics. Attempts (iii) the recovery of the van’t Hoff relation between equi- to rectify this form include transition state theory [4], librium constants and temperature and (iv) the restora- diffusion from an energy well [36], reactive flux theory tion of the link between the reactional propensity and [37], and transition path theory [38]. The alternative ap- the heat capacity, which is strangely ignored in the tra- proach proposed here is basically simpler since it is just ditional interpretation of the slope of van’t Hoff plots the probability for a particle in a population to have a as ∆H/kb. In the traditional formulas, the enthalpies given level of energy. The energy barrier is conceived are not responsible for the dependence on temperature here as a threshold in the Boltzmann distribution of the that is due only to the division of enthalpies by kBT . reactant itself, but not as the energy of a hypothetical By contrast, it is proposed here that enthalpies directly complex. The present study, however, deals only with mediate this dependence. Singularly, this view is found the basic rate constants of unitary transitions between in textbooks explaining the dependence of reactions on uniform reactants, in which time and energy are clearly temperature using populational energy distributions, but separated and the energy barrier is unique. These ele- without realizing that this explanation contradicts the mentary constants can then be used as ingredients for general formula where the particle energies are at the nu- real reactions which are most often composite and in- merator of the exponent supposed constant, while tem- volve non-uniform reactants. (i) Serial reactions can be perature is at the denominator. But although profoundly treated as Markovian walks with absorbing boundaries restructured, the thermodynamic relationships derived

15 [39] in which the probabilities of jumps are transposed [7] Tsallis C., Introduction to nonextensive statistical me- to reaction chains by introducing elementary transition chanics: Approaching a complex world. Springer Science rates (unidentifiable in practice), each one with its own & Business Media 2009. energy barrier. The resulting mean first passage times [8] Aquilanti V., Mundim K.C., Elango M., et al., Tempera- then correspond to the reciprocal of the global reaction ture dependence of chemical and biophysical rate processes: rates. Such serial reactions could ensure valuable roles in Phenomenological approach to deviations from Arrhenius biochemical interactions [40] and in their accuracy [41]. law, Chem. Phys. Lett. 498 (2010) 209-213. (ii) In biochemistry, real reactants are rarely uniform, [9] Quapp W., Zech A., Transition state theory with Tsallis particularly in the case of large macromolecules like for statistics, J. Comput. Chem. 31 (2010) 573-585. example proteins. A protein is a typical example of a giant macromolecule with many chemical bonds and de- [10] Yin C., Du J., The power-law reaction rate coefficient for grees of freedom, clearly subject to so-called dynamic dis- an elementary bimolecular reaction, Phys. A. Stat. Mech. order, particularly important in intrinsically disordered Appl. 395 (2014) 416-424. domains [40], in which a myriad of microreactions is in- [11] Aquilanti V., Coutinho N.D., Carvalho-Silva V.H., Ki- volved even in simple conformational changes. A protein netics of low-temperature transitions and a reaction rate envisioned as a single species, generally corresponds to theory from non-equilibrium distributions. Philos. Trans. an ensemble of conformers with slightly different ener- A Math. Phys. Eng. Sci. 375 (2017) pii: 20160201. gies. [12] Monk P.M.S., Physical Chemistry: Understanding our The elucidation of the fundamental nature of rate con- Chemical World, Wiley, 2004. stants is essential considering their widespread use from chemistry to systems modeling. Rate constants are the [13] Bao J.L., Meana-PañedaR., Truhlar D.G., Multi-path regulators of systems dynamics and as such allow to in- variational transition state theory for chiral molecules: the troduce the notion of time. If time proves so difficult site-dependent kinetics for abstraction of hydrogen from 2-butanol by hydroperoxyl radical, analysis of hydrogen to conceive, it is probably because it relies on two com- bonding in the transition state, and dramatic temperature pletely different pedestals belonging to different branches dependence of the activation energy, Chem. Sci. 6 (2015) of physics: (i) the step of time, always at work, both 5866-5881. out of and in equilibrium and originating from quantum physics and (ii) time arrow, existing only out of equilib- [14] Evans M.G., Polanyi M., Some applications of the tran- rium and which is a populational phenomenon emerging sition state method to the calculation of reaction velocities, from statistical physics [33]. Dynamic systems clarify especially in solution, Trans. Faraday Soc. 31 (1935) 875- 894. this perception because they combine both aspects: the time step included in the preexponential component of [15] Steinfeld J.I., Francisco J.S., Hase W.L., Chemical ki- the Arrhenius equation, and the time arrow describing netics and dynamics. Englewood Cliffs, Prentice Hall, NJ. the spontaneous evolution of a system of interconvert- 1989. ible particle states. [16] Weber G., van’t Hoff revisited: Enthalpy of association of protein subunits, J. Phys. Chem. 99 (1995) 1052-1059. [17] Liu Y., Sturtevant J.M., Significant discrepancies be- References tween van’t Hoff and calorimetric enthalpies, Prot. Sci. 4 (1995) 2259-2561. [1] Arrhenius S., Ober die reacktionsgeschwindigkeit bei der [18] Atkins P.W., de Paula J., Atkins’ Physical Chemistry. inversion von rohrzucker durch sauren, Z. Physik. Chem. 4 Oxford University Press, 2010. (1889) 226-248. [19] Smith I.W.M., The temperature-dependence of elemen- [2] Laidler K.J., The development of the Arrhenius equation, tary reaction rates: beyond Arrhenius, Chem. Soc. Rev. 37 J. Chem. Educ. 61 (1984) 494-498. (2008) 812-826.

[3] Michel D., Simply conceiving the Arrhenius law and ab- [20] Masgrau L., González-Lafont A., Lluch J.M., The cur- solute kinetic constants using the geometric distribution, vature of the Arrhenius plots predicted by conventional Phys. A Stat. Mech. Appl. 392 (2013) 4258-4264. link canonical transition-state theory in the absence of tunneling, J. Theor. Chem. Acc. 110 (2003) 352-357. [4] Eyring H., The activated complex in chemical reactions, [21] Naghibi H., Tamura A., Sturtevant J.M., Significant dis- J. Chem. Phys. 3 (1935) 107-115. crepancies between van’t Hoff and calorimetric enthalpies, Proc. Natl. Acad. Sci. USA. 92 (1995) 5597-5599. [5] Feynman R., Statistical Mechanics Avalon Publishing, 1998. [22] Sharma G., First E.A., Thermodynamic analysis reveals a temperature-dependent change in the catalytic mecha- [6] Tsallis C., What are the numbers that experiments pro- nism of Bacillus stearothermophilus Tyrosyl-tRNA Syn- vide, Quimica Nova 17 (1994) 468-471. thetase, J. Biol. Chem. 284 (2009) 4179-4190.

16 [23] van’t Hoﬀ J.H., Etudes de Dynamique Chimique (Stud- [33] Lebowitz J.L., Macroscopic laws, microscopic dynam- ies in Chemical Dynamics), F. Muller & Co, Amsterdam, ics, time’s arrow and Boltzmann’s entropy, Physica A. 194 1884. link (1993) 1-27.

[24] Planck M., On the law of distribution of energy in the [34] Einstein A., On the development of our views concern- normal spectrum, Ann. Phys. 4 (1901) 553-563. ing the nature and constitution of radiation, Phys. Z. 10 (1909) 817-826. [25] Michel R., On Stirling’s formula, Am. Math. Month. 109 ¨ (2002) 388-390. [35] Wegscheider R., Uber simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Re- [26] Luo J.L., van den Broeck C., Nicolis G., Stability criteria actionskinetik homogener Systeme, Monatshefte für and fluctuations around nonequilibrium states, Z. Phys. B Chemie/Chemical Monthly 32 (1901) 849-906. Cond. Mat. 52 (1984) 156-170. [36] Kramers H.A., Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (1940) [27] Mackey M.C., Time’s arrow: The origins of thermody- 284-304. namic behavior, Springer-Verlag, New York (1992) [37] Chandler D., Barrier crossings: Classical theory of rare [28] Maes C., NetocnýK., Time-Reversal and Entropy, J. but important events from classical and quantum dynam- Stat. Phys. 110 (2003) 269-309. ics in condensed phase simulations. B.J. Berne, G. Ciccotti, D.F. Coker Editors, 1998. [29] Schnakenberg J., Network theory of microscopic and macroscopic behavior of master equation systems, Rev. [38] Vanden-Eijnden E., Transition path theory, Lect. Notes Mod. Phys. 48 (1976) 571-585. Phys. 703 (2006) 439-478.

[30] Ge H., Qian H., Physical origins of entropy production, [39] van Kampen N.G., Stochastic Processes in Physics and free energy dissipation, and their mathematical represen- Chemistry, 3th Edition, North-Holland Personal Library, tations, Phys. Rev. E 81 (2010) 051133. Amsterdam, 2003;

[31] Glasstone S., Laidler K.J., Eyring H., The theory of rate [40] Uversky V.N., The multifaceted roles of intrinsic disorder processes, McGraw-Hill, New York, 1941. in protein complexes, FEBS Lett. 589 (2015) 2498-2506.

[32] Herzfeld K.F., Zur Theorie der Reaktions- [41] Michel D., Boutin B., Ruelle P., The accuracy of bio- geschwindigkeiten in Gasen, Ann. Phys. 59 (1919) chemical interactions is ensured by endothermic stepwise 635-667. kinetics, Prog. Biophys. Mol. Biol. 121 (2016) 35-44.