Analysis of the Reaction Rate of Binary Gaseous Mixtures near from the Chemical Equilibrium

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1 Analysis of the Reaction Rate of Binary Gaseous Mixtures near from the Chemical Equilibrium Adriano W. da Silva May 30, 2013 Abstract A binary gaseous mixture with reversible reaction of type A + A = B + B is studied with Boltzmann equation, assuming hard spheres cross sections for elastic collisions and model of line-of-centers for reactive interactions. The Chapman-Enskog method is used to obtain the solution of the Boltzmann equation in a chemical regime for wich the reactive interactions are of the same order as the elastic one, i.e. in the system is closed to the final stage of a chemical reaction where the affinity is considered to be a small quantity and the sytem tends to the chemical equilibrium. The internal degrees of freedom of the particles of the gas are not taken into account. The aim of this paper is to evaluate the approximation second of the reaction rate coefficient of the mixture. It was verified the reaction heat changes the reaction rate and these changes is bigger for high values of heat reaction. 1 Introduction The influence of chemical reactions on the reaction rate in gas mixtures was first analyzed by Prigogine an co-works [1] using a kinetic theory based on the Boltzmann equation. Since then, several authors have studied this problem as Present [2] and Ross and Mazuer [3]. The teoretical treatment of most papers involves the Chapman-Enskog method [1]-[3], [4]-[8]. For the reactions of the type A+A=B+B, which is the so-called cas of symmetric reactions [9]- [10], the majority of the works is concentrated on the study of the rate of reaction an its relationship with the Arrhenius formula. Concerning the non-equilibrium effects in reactive systems, chemical reactions remove more energetic particles an induce a sensible change of the reaction rate, with respect to the one evaluated with Maxwellian distributions. Conversely, elastic collisions contribute to restoring the equilibrium. The deviations are more relevant for reactions with activation energy in the high altitude atmosphere, where the elastic collisions are less frequent. In the Boltzmann equation the chemical reactions are related to inelastic collisions. The value of the reaction heat distort the Maxwellian distribution function, for large values the effect becomes more importants. In the present work, the influence of the reaction heat is examined for a binary mixture wiht a reversible reaction of type A+A=B+B near chemical equilibrium. This kind of reaction is known as fast reactions, because the reactive processes are of the same order as the elastic ones. The Chapman-Enskog method is used to obtain the solution of the Boltzmann equation in a chemical regime for which the reactive interactions are of the same order the elastic collisions, i.e. for processes close to the beginning of the reaction where the affinity tends to ziro. The resulting integral equation is solved with the expansion of the distribution function in Sonine polynomials up to second-order terms. The calculations have been performed by assuming differential hard-sphere cross sections for elastic 1 429

2 collisions and model with activation energy for reactive interactions namely the line-of-centers model [2]. 2 Basic Definitions and Boltzmann Equation We consider a binary mixture of ideal gases whose constituents denoted by A and B, have binding energies ɛ α = A, B and equal molecular mass m. There exist two kinds of collisions between the molecules of the gas: the elastic one which refers to nonreactive interactions and the one which takes into account the reaction. The gas molecules undergo inelastic collisions with reversible reactions of the type by A + A = B + B. (1) The conservation laws of linear momentum and total energy for a reactive collision are given c A + mc A1 = mc B + mc B1, (2) ɛ A mc2 A + ɛ A mc2 A 1 = ɛ B mc2 B + ɛ B mc2 B 1, (3) where (c A, c A1 ) and (c B, c B1 ) are the velocities of reactants and products, respectively, of the forwad reaction. The subindex 1 is used to distinguish two identical molecules that participate in the collision. The conservation laws for the nonreactive collisions have the same form as the above equations. We denote by g A = c A1 c A and g B = c B1 c B the relative velocities of the reactants and products of the forward reaction, respectively, and write the conservation law of total energy (3) as 1 4 mg2 A = 1 4 mg2 B + E, E = 2(ɛ B ɛ A ), (4) where E, the difference between the binding energies of products and reactants is connected with the reaction heat. In the phase space defined by positions and velocities of molecules, the state of the mixture is defined in terms of the set of one-particle distribution functions f α = f(x, c α, t) (α = A, B), (5) such that f α dxdc α gives at time t the number of molecules of type α in the volume element dxdc α around the position x and the velocity c α. The evolution of te one-particle distribution f α in the phase space is assumed to satisfy the Boltzmann equation which in the absence of external forces is written as where f α t + f α cα i = Q E α + Q R x α, (α = A, B) (6) i Q E α = B β=a Q R α = [f αf β f α f β ] d 2 (g βα.k βα )dk βα dc β, [f γ f γ1 f α f α1 ] σ α(g α.k α )dk α dc α1, (7) 2 430

3 with the indices α and β (α β) representing one of the two constituents A and B, accounts for the source and sink contributions due to inelastic interactions with the chemical reactions. The quantitie σ α denote reactive differencial cross section, k α and dk α denote the unit collision vector and the element of solid angle for elastic collisions, whereas k α and dk α represent the corresponding quantities or reactive interactions. The parameter d denotes the diameter of a molecule and σ α represents the reactive cross section. Due to the principle of microscopic reversibility [11], a definite relationship exists between the forward cross section σ A and that for the reverse reaction σ B. By using this principle, the transformation law bettween the elements for the reactive and forward collisions in the velocity space is given by (g A.k A )σ A dc A dc A1 = (g B.k B )σ B dc B dc B1. (8) The non-equilibrium effect are contained in the one-particle distribution functions, f α, α = A, B and it is characterized after solving the appropriate reactive Boltzmann equation (5). This can be achieved in the framework of the Chapman-Enskog method [1, 2], once a particular model for the reactive cross sections is assumed. The model of the reactive cross section used int this paper is the line-of-centers model, σ α = { 0, εα < ε dr 2 ( 1 ε γ α ), γα > ε, where In equation (8) γ α is the relative translational energy of constituent α and ε is the activation energies of the forward and backward reactions in units of the termal energy of the mixture, kt, T is the temperature of the mixture and k being the Boltzmann constant, i.e., γ α = mg2 α 4kT, ε A = ε A kt, ε B = ε A E, E = E kt. (10) If we multiply the Boltzmann equation (5) by arbitrary function ψ α = 1,ψ α = mc α i and ψ α = mc 2 α/2 + ɛ α, respectively and integrate the resulting equations over all values of c α, we get the equations n α t + t (n αv i + n α u α i ) = (Q E α + Q R α)dc α = τ α, (9) t (mn α)vi α + [p α ij + mn α (u α i v j + u α j v i + v i v j )] = x j mc α i (Q E α + Q R α)dc α, [ 3 t 2 n αkt + n α ɛ α + mn α (u αi v i + 1 )] 2 v2 + { q α x i i + p α ijv j + n α ɛ α u α i [ mn αv 2 u α i + 2 n αkt + n α ɛ α + mn α (u αi v i + 1 )] } 2 v2 v i = ( m 2 c2 α ) (Q E α + Q R α)dc α, (11) where n α is particle densities, vi α is the velocity components, T α is the temperature of each constituent (α = A, B), u α i is the diffusion velocity, p α ij is pressure tensor, qi α is the heat fluxe for each constituent α and τ α denotes the rate of reaction due to the chemical reaction. The quantities above are defined in terms of the distribution function [8]. = 3 431

4 3 Chemical Kinetics and Thermodynamical Equilibrium For mixtures of ideal gases the chemical potential of constituent α reads [ 3 µ α = ɛ α kt 2 lnt lnn α + 3 ( )] 2πmk 2 ln. (12) h 2 In the above equation we have not taken into account the internal degrees of freedom of the molecules and we have considered that all constituents are at the same temperature, which is the temperature of the mixture, T and h is the Planck constant. The affinity is defined by From the definition of the chemical potential, it is also convenient to introduce the affinity for the forward reaction as A = µ A + µ A1 µ B µ B1. (13) This quantity characterizes the deviation of the system from the chemical equilibrium. The chemical equilibrium condition can be defined in terms of the chemical potentials by the equality The equilibrium thermal, mechanical and chemical (A = 0) condition leads to Maxwellian distributions fα M which are f M α = n α ( β π ) 3/2 e βξ2 α, (14) where β = m/2kt. 4 The distribution function The distribution function contains all the information about the non-equilibrium effects induced by the chemical reaction. The solution of the Boltzmann equation (6) in a chemical regime for which the reaction process is close to its final stage (fast process) to this problem is based on the Chapman-Enskog method and Sonine polynomial approximation to the coefficients of the distribution functions [12]. In the case of the fast process, the reactive interactions are same frequent than elastic collisions. Hence, we write the Boltzmann equation (6) as Df α + θξi α f α = 1 B [f ] x i θ αf β f α f β d 2 (g βα.k βα )dk βα dc β A = + 1 [f γ f γ1 f α f α1 ] σ θ α(g α.k α )dk α dc α1, (15) where θ is a parameter of the order of the Knudsen number and the operator D = ( / t) + v i / x i is the so-called material time derivative. The apply the Chapman-Enskog method one should write the distribution functions and the material time derivative as f α = f M α + θf (0) α +..., D = D (0) + θd (1) +... (16) For the solution of the integral equations (20) it is convenient to write f α in terms of Sonine polynomials and retain, at least, the expansion up to the first-order term, f α = f M α [ ( ) a α 0 + a α 1 2 βξ2 α + a α 2 ( βξ2 α + β2 ξ a lpha 4 )], (17)

5 where a α 2 is scalar coefficient to be determined and a α 0 = a 1 α = 0, due to particle number densities conservation and definition of the temperature, respectively. The expansion adoted is capable to reproduce the effects of the reaction heat on the distribution function. The balance equations for particle number density of each constituent, momentum and temperature of the whole mixture, that follow from (19), are used to elimate the material time derivatives in the integral equation through the equalities 5 Reaction Rate The reaction rate of the mixture is defined by τ α = [f γ f γ1 f α f α1 ] σα(g α.k α )dk α dc α1 dc α, (18) When the reaction is forward (A + A B + B), τ A = n 2 B[k (0) r + k (1) r ] n 2 A[k (0) f + k (1) f ] (19) The production terms k (0) f, k(1) f, k(0) r and k r (1) of the particle number density n A can be calculated by inserting the distribution (23) into its definition (25) and by integration over all velocities c α. The terms k (0) f and k r (0) are well known in the literature, see e.g. [?]. In figure 1 is plotted the dimensionless reaction rate coefficient kf defined by k f = k (1) f, (20) dr 2 ε πkt/me th as a function of the ε, considering dr = d. Four cases are analyzed, two for endotermic reaction and two for exothermic reaction. From this figure, can conclude that the effect on the reaction rate is larger for hight values of the reaction heat. Besides, the effect on the reaction rate more evident in regions of low activation energy. As was pointed out by Prigogine and Mahieu [1] the reaction heat modifies the distribution funcitons. On the other hand, in the figure 2, when the activation energy increase the rate of reaction tends to zero, the reactive cross section decreases and therefore it is less suitable to happen a collision that causes a chemical reaction. 6 Conclusion In this paper the analysis of the reaction heat on the reaction rate is held when a bimolecular reversible chemical reaction takes place in mixture of constituents A and B. The chemical kinetic Boltzmann equation with reactive collision terms have been deduced taking into account the presence of product B of the forward reaction. The reactive cross sections with activation energy adopted is the line-of-centers model, whereas the hard-spheres model for elastic cross sections has been considered. The Chapman-Enskog method is used combined with the firstorder polynomials representation of the distribution functions for a flow regime consistent with an final stage of a fast chemical reaction and is shown the reaction to occur only when the relative translational energy is greater than the activation energy. The analytic calculations show that the contribution of the reactions heat on the distribution functions causes a bigger effect on the endothermic reactions than for exothermic reactions. The numerical simulations here employed based on the solution of the Boltzmann equation 5 433

6 via Chapaman-Enskog method, gives a good estimation of the non-equilibrium effects of the reaction heat on the macroscopic properties of the reacting mixture. 7 Acknowlegements AWS acknowledges the support by Instituto Federal do Parana. References [1] I. Prigogine and E. Xhrouet, Physica (Amsterdam)15, 913 (1949). [2] R. D. Present, Proc. Natl. Acad. Sci. U.S.A. 41, 415 (1955). [3] J. Ross and P. Mazur, J. Chem. Phys. 35, 19 (1960). [4] B. D. Shizgal and M. Karplus, J. Chem. Phys. 52, 4262 (1970). [5] B. D. Shizgal and M. Karplus, J. Chem. Phys. 55, 76 (1971). [6] A. S. Cukrowski and J. Popielawski, Chem. Phys. 109, 215 (1986). [7] B. V. Alexeev, A. Chikhaoui and I. T. Grushin, Phys. Rev. E 49, 2809 (1994). [8] A. W. Silva, G. M. Alves and G. M. Kremer. Physica A 374, (2007). [9] B. Shizgal and D. Napier, Phywica A 223, 50 (1996). [10] A. Cukrowski, S. Fritzche and J. Popielawski, Acta Phys. Pol. A 84, 369 (1993). [11] J. C. Light, J. Ross and K. E. Shuler, in Kinetic Processes in Gases and Plasmas, edited by A. R. Hochstim (Academic, New York, 1969), p [12] B. Shizgal and A. Chikhaoui, Physica A 365, 317 (2006). [13] A. S. Cukrowski, S. Fritzche and J. Fort, Chem. Phys. Lett. 341, 585 (2001). [14] A. S. Cukrowski, S. Fritzche and M. J. C. Jr, Chem. Phys. Lett. 379, 193 (2003). [15] G. Munhoz Alves 6 434