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The Missing Link in the Kinetic Theory of Chemical Reactions and Epidemics L

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The Missing Link in the Kinetic Theory of Chemical Reactions and Epidemics L The missing link in the kinetic theory of chemical reactions and epidemics L. Vanel1* 1Institut Lumière Matière, Univ Claude Bernard Lyon 1, Univ Lyon, CNRS; F-69622, Villeurbanne, France. *Corresponding author. Email: [email protected] In thermodynamics, affinity is the driving force of chemical reactions. At equilibrium, affinity is zero, and the reaction quotient equals the constant of equilibrium. Thermodynamics predicts the timescale of chemical reactions but not rate laws that are purely phenomenological relations. Here, we derive an affinity relaxation theory (ART) that predicts chemical rate laws. Strikingly, the simplest chemical reaction in the ART model has a close resemblance with epidemic models instead of the mass-action rate law, revealing that epidemic models are actually more accurate chemical rate laws than the mass-action law. However, epidemic models appear in turn as an approximation of the ART model, where conceptual incoherencies of epidemic models are exposed and resolved. Furthermore, the mathematical Gompertz solution of the model links chemical rate laws with empirical biological and cancer growth laws. The ART model provides an entirely new framework to describe dynamic phenomena in chemical reactions, epidemics and biology. Introduction The mass-action law introduced in 1864 by Guldberg and Waage (1) was the first equation linking the rate of a chemical reaction to the concentration of reactants. A main idea that quickly followed is that a reaction can occur in two opposite directions and that equilibrium corresponds to the point where the speeds of the forward and backward reaction cancel each other (2). Chemical thermodynamics established a long time ago that mass-action rate laws are compatible with the thermodynamic constant of equilibrium of a chemical reaction (3) and that rate constants depend on the reaction activation energy (4). However, the kinetic form of the rate laws does not follow first principles in thermodynamics (5). These rate laws remain phenomenological relations requiring experimental validation (6). Prigogine et al. have shown that close to equilibrium, the speed of a chemical reaction should be proportional to affinity (7), a quantity defined as the negative derivative of the Gibbs energy with respect to the extent of the reaction (3). Affinity represents the thermodynamic driving force of a chemical reaction. Prigogine and Defay derived an equation of evolution for affinity, which predicts an exponential relaxation towards equilibrium when the rate of variation of the other state parameters (pressure, temperature) is small enough (3). However, they used a linear dependence of the reaction rate on affinity, an assumption expected to be valid only close to equilibrium. Indeed, far from equilibrium, compatibility with mass-action rate laws implies a nonlinear dependence of the reaction rate on affinity (7). Predicting the reaction rate from affinity was attempted, but a purely empirical decay rate law was used (8). In this article, we postulate that exponential relaxation of affinity provides a general mechanism of convergence towards equilibrium and derive a new theory for the rate of chemical reactions as a replacement for phenomenological rate laws. This convergence mechanism is analogous to stress relaxation in a Maxwell fluid suddenly put out-of-equilibrium (9,10). In the affinity relaxation theory (ART) of chemical reactions, the reaction rate depends linearly on affinity, even far away from equilibrium. The theory removes the need to consider separate forward and backward reactions and predicts a significant time delay in the peak reaction rate that is nonexistent in conventional rate laws. We did not find chemical reaction data that could directly confirm the existence of this peak. Instead, we find an unforeseen application to the peak rate of an epidemic wave, reinstating the historical link between the rate laws of chemistry and mathematical epidemiology (11). We also find that the mathematical solution for the evolution of the reaction rate has the same generic form as the growth rate laws used in biology (12,13). Results and discussion To illustrate the consequences of this new theory, we consider here the simplest chemical reaction AB. We note xA and xB as the number fractions of molecules A and B and assume they form an ideal solution. The equilibrium reaction constant is Ke=xB,e/xA,e. We note K=xB/xA the reaction quotient corresponding to the out-of-equilibrium value and the value at an initial reference time Ko=xB,o/xA,o. The affinity of the chemical reaction at any time is then A=kBT ln(Ke/K) and at the initial reference time Ao=kBT ln(Ke/Ko). Assuming an exponential relaxation of affinity towards -t/ equilibrium A=Aoe (3), we find that the reaction quotient K verifies: et/ (1et/ ) K t Koe K (1) This prediction has a mathematical form that falls into the family of Gompertz functions (14). Eq. (1) is a generic result that applies to any chemical reaction provided the ideal solution assumption is valid (15). To find the corresponding reaction rate, we need to express the link between the extent of reaction and the reaction quotient K. In the simple chemical reaction considered here, the number of molecules are NA=NA,o and NB=NB,o+(No=NA,o+NB,o), allowing us to express number fractions xA=NA/No and xB=NB/No as a function of the extent of reaction . We find that the reaction rate verifies (13): dξ No xxABA (2) dt kB T This relation is valid for all values of Ao, however small or large. Thus, the speed of reaction varies linearly with affinity even very far from equilibrium. Second, there are no forward or backward reactions in this expression. The direction in which the reaction will take place depends only on the sign of affinity. If Ko <K initially, affinity is positive, and the reaction will proceed in the forward direction. The backward reaction will occur for a negative affinity obtained when Ko >K. In this approach, there is no longer a need for the concept of forward and backward reactions that cancel each other at equilibrium (2). Third, the mass-action rate law appears to be an approximation of this theoretical prediction. The usual mass-action rate law states that d/dt=kNoxAkNoxB, where k and k are the kinetic constants of the forward and backward reactions. The solution of this rate law is then an exponential variation of the extent of reaction ξξe 1 exp( (k k ) t ) and of the reaction rate d/dt. In the case of a complete forward reaction (k+≫ k_), Eq. (2) shows that k+ is not a constant but evolves as the product of affinity and fraction of B molecules. Since affinity decreases exponentially and fraction xB increases in the course of the reaction, the product xBA has a bell-shaped variation, and the mass-action rate law appears as an approximate expression of the reaction rate. Here, we consider the time evolution of the rate law Eq. (2) (Fig. 1A, 1B, 1C) for various equilibrium fractions xB,e and initial fractions xB,o<xB,e. When the initial fraction is small enough, a peak in the reaction rate (15) occurs at a fraction xB,p that depends only on the equilibrium fraction xB,e (Fig. 2A). The peak reaction rate is then (Fig. 2b): xx1 dξ No B,, p B p (3) dt p 12 xBp, and occurs at time (Fig. 2C): Ao t p ln (4) A p where Ap is the value of affinity at fraction xB,p. The overall shape of the reaction rate depends on the equilibrium fraction (Fig. 2D). Eq. (4) shows that a peak time exists only if Ao > Ap (here, A>0), which means xB,o < xB,p. This is a necessary condition for the appearance of a delayed peak rate. The time delay increases with smaller initial fractions of B molecules (Fig. 2C). In the opposite case where xB,o > xB,p, the maximum reaction rate occurs at the initial time and decreases almost purely exponentially with time. In that case, the mass-action rate law remains a very good approximation. Fig. 1. Reaction rate of a simple chemical reaction in the ART model. (A), Molecule B number fraction when the initial fraction is smaller than xp. (B), Reaction rate showing a peak at time tp. (C) Reaction rate in logarithmic scale. Fig. 2. Parameters controlling the reaction rate shape. (A) The fraction xp below which a peak will appear only depends on the equilibrium fraction xe. (B) When the initial fraction xo is smaller than xp, the peak rate value depends only on the equilibrium fraction xe, while (C) the time at which the peak appears depends crucially on the initial fraction xo. (D) The corresponding overall shape of the reaction rate for a given initial fraction xo depends on the equilibrium fraction xe (curves are shifted vertically for better visualization). We are not aware of any chemical reaction data to compare with the prediction of a delayed peak in the reaction rate of a simple reaction. One may then wonder to what extent Eq. (2) gives a valid prediction for the reaction rate. One problem is that chemical reactions are usually fast, so it would be difficult to measure with enough precision the variation of reaction rate with time. In addition, the initial instant of a chemical reaction is often ill defined experimentally, especially when it involves a mixing step of reactants or catalysts. From a theoretical perspective, rate laws proportional to both the amount of reactant and product exist in the distant but historically related (11) field of mathematical epidemiology (16). A central assumption in most epidemic models is indeed that the rate of infection grows with the product of the number of susceptible persons by the number of infected persons, which is analogous to the product of the A and B molecule fraction that appears in Eq.
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