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PHYSICA ELSEVIER Physica a 237 (1997) 205-219 PHYSICA ELSEVIER Physica A 237 (1997) 205-219 Fluctuating hydrodynamics approach to chemical reactions I. Pagonabarraga*, A. P6rez-Madrid, J.M. Rubi Department de Fisica Fonamental, Faeultat de Fisica, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain Received 27 August 1996 Abstract We have used the thermodynamical description of a chemical reaction as a diffusion process along an internal coordinate to analyze fluctuations in the density of the constituents, which are treated under the framework of fluctuating hydrodynamics. We then obtain a Langevin equation for the density, as a function of the internal coordinate, whose stochastic source statisfies a fluctuation-dissipation theorem. After contraction of the description, by means of integration in the internal coordinate, we derive the Langevin equation for the concentration of reactants and products as well as the statistical properties of the random source which agree with the corresponding results obtained by means of Keizer's theory. Application of the formalism is illustrated by considering particular cases. An extension to coupled chemical reactions is also discussed. 1. Introduction Non-equilibrium thermodynamics has been successfully applied to the study of transport phenomena in different physical systems and for diverse physical situations. The linear laws relating fluxes and forces are derived by the theory in a systematic way, which, when employed in the balance equations permit us to formulate the differential equations describing the evolution of the fields, characterizing the local state of the system [1]. The study of relaxation phenomena reveals, however, the necessity to propose a different scheme from the one used for transport processes. In fact, when consider- ing, for example, chemical reactions, the linear law coupling the reaction rate with the corresponding affinity implicitly restricts its validity in the case when the system is close to the equilibrium state, since the general relationship between both quantities, the law of mass action, is a non-linear function. One then concludes that the * Corresponding author. 0378-4371/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved Pll S0378~43 7 I (96)00377-9 206 L Pagonabarraga et aL /Physica A 237 (1997) 205-219 procedure employed to analyze transport phenomena, which is satisfactorily de- scribed by linear laws, does not completely account for the kinetics of chemical reactions for which the constitutive equations are intrinsically non-linear. To reestablish generality in the application of the theory, one must adopt the proper description of the kinetic process. The reaction is then viewed as a diffusion process through a potential barrier whose minima are related to the initial and final states of the reaction, which correspond to reactants and products, respectively [2]. One then assigns a coordinate (referred to as the internal coordinate or degree of freedom) whose values, ranging from the initial (reactants) to the final (products) states charac- terize the diferent steps of the process. In this description and following Arrhenius, one implicitly assumes the existence of an activated intermediate state between reactants and products [3]. From the entropy production related to the diffusion process, one then derives the law of mass action. This picture has been applied to diverse physical situations for which a diffusion process along an internal coordinate may be defined, and in a general way enables one to obtain kinetic equations of the Fokker-Planck type for the density in the space of the internal coordinate [1,4]. Our purpose in this paper is to show that, as for transport phenomena, fluctuating hydrodynamics provides a solid scheme to describe fluctuation dynamics in systems in which the relaxation phenomena take place. We will show that by using fluctuating hydrodynamics in the space of the internal coordinate, it is possible to arrive at the proper formulation of the properties of the random part of the reaction rate, from which one analyzes the fluctuation dynamics. To this end, we have distributed the paper in the following way. In Section 2 we briefly review the approach of non-equilibrium thermodynamics to study chemical reactions, contrasting the procedure used to analyse transport phenomena with the theory of internal degrees of freedom. Section 3 is devoted to the fluctuation analysis. In Section 4 we extend our previous results to the case of an arbitrary number of coupled chemical reactions. Finally, in the discussion section we emphasize our main results. 2. Non-equilibrium thermodynamics and the law of mass action We consider the chemical reaction in which substance 1 transforms into (1 ~2). This reaction will be characterized by the reaction rate J and the affinity A defined through the expression 2 A = ~ vlPi, (1) i=1 where #i are the chemical potentials of the different species and vl are the stoichiomet- ric coefficients, with sign '-' if they correspond to reactants, and ' +' for products. L Pagonabarraga et al. /Physica A 237 (1997) 205-219 207 Following the scheme of non-equilibrium thermodynamics, one may obtain the entropy production [1] 1 a = - -- JA, (2) T which, as usual, admits an interpretation in terms of flux-force pairs. From this quantity one may derive the phenomenological equation -A J = - 1 T' (3) where i is a phenomenological coefficient, thus establishing a linear relationship between the reaction rate and affinity. This linear law is only valid near equilibrium and one then concludes that non-equilibrium thermodynamics in its standard version is not able to reproduce the law of mass-action, basic to formulate the kinetic equations for the different constituents. As shown in Ref. [5], the form of the entropy production given through Eq. (2) is still valid when the system is far from equilibrium. To avoid this limitation, and maintaining the essentials of the non-equilibrium thermo-dynamics line of reasoning, it is possible to adopt the following scenario. The transformation of 1 into 2 or viceversa may be viewed as a diffusion process along an internal coordinate 7 ranging from an initial state 71 (reactants) to a final state )~2 (products), in which a particle of the activated complex, originating from the reactants [6], crosses the potential barrier between both states. One then assumes local equilibrium in the internal space which enables one to formulate the Gibbs equation [1] ?2 6s- Pt°t1 T f /2(7,t)bn(?, t) d~ , (4) ?l where n(~,, t) is the number density in the internal space,/2(% t) its conjugated chemical potential, T the temperature, and ?2 Pro, = m f n( 7, t) dr, (5) ?1 the total density, with m the mass of a molecule of the activated complex. Additionally, the evolution of the number density is governed by the continuity equation On ~J & QT' (6) 208 L Pagonabarraga et al. /Physica A 237 (1997) 205-219 which introduces the diffusion current in the internal space J(7, t). The entropy production related to the diffusion process then follows from Eq. (4). One obtains a=--~ 1 f J(j-~?0#(7) dT, (7) This expression also has the form of products of currents and thermodynamic forces which now are related to each value of the internal coordinate. The resulting linear laws are then given by L(7) OU(7) J(j - --, (8) T 07 where locality in the internal space has been assumed for which only fluxes and forces corresponding to the same value of 7 are coupled. In this last expression L(J is a phenomenological coefficient. Upon substitution of(8) in (6) and by using the expression for the chemical potential valid for ideal systems I~ = kBT In n(7, t) + ~(7), (9) one then obtains the diffusion equation in the internal space ~1- 07 \ ~7 + nb-~7 " (10) Here 4~ is the potential through which the diffusion process takes place. Although in principle we do not need to give its specific form, it is characterized by two minima at 71 and 72 related to the equilibrium states of reactants and products, respectively, separated by an energy maximum which will be assumed large compared to k~T [6]. Furthermore, we have also introduced the corresponding diffusion coefficient D = kBTb, with b being the mobility in the internal space given by b = L/nT, and kB the Boltzmann's constant. In our subsequent analysis, the mobility and conse- quently the diffusion coefficient will be considered constant as a first approximation. The diffusion current given through Eq. (8) can be alternatively written as J(7) = _ De_•/k. r __c? eu/ker . (11) 07 When the height of the barrier is large compared to thermal energy, the system achieves a quasi-stationary state in which J(7,t) = J(t){O(7 - 71) - 0(7 - 72)}, (12) where J is the quasi-stationary current, assumed uniform, and O(x) is the unit step function. Additionally, due to the height of the barrier we can also consider that equilibrium in each potential well is reached independently; thus, the chemical I. Pagonabarraga et al. / Physica A 237 (1997) 205-219 209 potential is uniform in these domains, and is given by //(7) =//(71)0(7o - 7) +//(72)0(~ - 7o), (13) where 7o is the coordinate at the maximum of the potential. By using Eq. (9) in (13), this last relation yields n(7, t) = na(t)e-r~)-~(~l)l/k"rO(7o -- 7) + n2(t)e-F~')-~el211/k"rO(7 -- 7o), (14) where nl - n(7 = 71) and n2 - n(7 = 72) are the number densities of the activated complex at the initial and final states, respectively. Equating (11) and (12) and integrating over 7 one then obtains the law of mass- action [1] d = kBl(1 -- eA/ksT), (15) where 1 is related to the diffusion coefficient D through the expression D exp (//1/knT) l = k, S~ exp(~(7)/ksV)dT' (16) with//1 =//(71) being the chemical potential of the first species (reactant).
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