Computational Simulation of Bray-Liebhafsky (BL) Oscillating Chemical Reaction

Transcription

1 Portugaliae Electrochimica Acta 26/4 (2008) PORTUGALIAE ELECTROCHIMICA ACTA Computational Simulation of Bra-Liebhafsk (BL) Oscillating Chemical Reaction Jie Ren, Jin Zhang Gao *, Wu Yang Chemistr & Chemical Engineering College, Northwest Normal Universit, Lanzhon, , PR China Received 19 th Januar 2008; accepted 3 rd April 2008 Abstract The computational simulation of the Bra-Liebhafsk (BL) oscillating chemical reaction b differential kinetic methodolog is carried out in this work. According to the mechanism of Treindl and Noes involving 10 reaction steps, the changes of the concentrations of I 2 and O 2 in solution are simulated. When the control parameters are = 0.55, β = and <, the differential equations present periodic solution, and the oscillations can be observed in 150 min. If, β and are taken as the control parameters, respectivel, the bifurcation points would be observed in the processes of control parameters, changing successivel with the critical values of = 0.55,β = ,and =. The acidit of solution on the nonlinear phenomena is also investigated in detail. Kewords: computational simulation, Bra-Liebhafsk (BL) oscillating chemical reaction, differential kinetic methodolog, bifurcation. Introduction A large number of oscillating chemical reactions have been investigated both eperimentall and theoreticall [1-4]. It is generall accepted that at least two different conditions are necessar for the occurrence of chemical oscillations in a homogeneous chemical sstem: (1) the sstem should be far from thermodnamic equilibrium, and (2) the corresponding kinetic equations must be nonlinear. The first eample of homogeneous oscillating chemical reaction was reported b Bra [5] after studing iodate-catalzed decomposition of hdrogen peroide in acidic medium over 80 ears. The oscillating phenomena and the various component reactions were eamined etensivel b Bra and * Corresponding author. address: jzgao@nwnu.edu.cn

2 Liebhafsk, and the Bra-Liebhafsk (BL) oscillating chemical reaction was proposed. According to references [6,7], microwaves and light have obvious effect on the BL reaction. The overall chemical change in the BL sstem is described as progress (A): 2 H 2 O 2 (aq) 2 H 2 O (aq) + O 2 (g) (A) The concentration of iodine and the rate of ogen gas evolution change periodicall in this sstem. The main reason of this phenomena results from the dual effects of hdrogen peroide in the whole process. That is to sa, hdrogen peroide acts not onl as the oidant but also as the reductant in the sstem. The process (A) is assumed to be the net result of processes (B) and (C): 2 IO H 2 O H + I O H 2 O (B) I H 2 O 2 2 IO H H 2 O (C) Processes (B) and (C) are not the reverse of each other and both proceed with negative Gibbs free energ ( G < 0) [8]. If iodate was indeed acting onl as a catalst, the total concentration of iodate would be constant and the total chemical change of the processes (B) and (C) should be equal during an etended period of time. However, one process might be faster than the other during a short interval. Processes (B) and (C) alternatel dominate over the whole sstem and result in the concentration of iodine and the rate of ogen gas evolution change periodicall [9-12]. Although the overall stoichiometr of the BL reaction seems to be simple, it is not eas to describe the mechanism. The difficult lies on two aspects: (1) because of the sstem has few variables which could var independentl, it is difficult to observe the effects of each variable on the sstem; (2) it is ver difficult to make clear the kinetic effect of ogen on the BL reaction. In the BL reaction sstem ogen ields continuousl and its concentration in the solution is higher than that under normal conditions because of the eistence of supersaturation phenomenon [13]. On one side, the ogen is gaseous and it is too difficult to control it. On the other hand, the concentration of ogen becomes supersaturated and the degree of super-saturation depends on the eperimental conditions, such as the concentrations of reactants and temperature. There are several papers of the analsis of mechanism of BL reaction [8,11,14,15]. Matsuzaki [12] proposed a mechanism involving I 3 - ; the question is how to verif it b eperiment. Schmitz and Rooze [17] assumed that the HOI should be oidized at fictitious equilibrium state; meanwhile, Sharma and Noes [18] considered further that the free radical might be formed in the redo processes. Recentl, Schmitz found that in an open reactor, where the flow rate was the variable parameter, different simple and comple oscillations and different routes to chaos were observed through numerical calculations [19]. So, up to now, there is no agreement about the BL reaction mechanism. 350

3 This work simulates the nonlinear behavior of the BL reaction based on a model involving two variables. The bifurcations are observed in the process of changing parameters. The acidit of the solution on the nonlinear phenomena is investigated in detail. Mathematical model Treindl and Noes [16] suggested the mechanism involving 10 individual steps in which the transport of ogen from the supersaturated solution to atmosphere was a significant component of the overall mechanism of the batch BL sstem and radicals I and IO 2 were involved in the whole process. The 10 individual steps are listed below: IO I H + HIO 2 + HOI (R1) HIO 2 + I - + H + 2 HIO (R2) HOI + I - + H + I 2 (aq) + 2 H 2 O (R3) HOI + H 2 O 2 I - + H + + O 2 (aq) + H 2 O I - + H + + H 2 O 2 HOI + H 2 O I 2 (aq) 2 I (R4) (R5) (R6) I + O 2 (aq) IOO (R7) IOO IO 2 (R8) 2 IO 2 + H 2 O IO H + + HIO 2 (R9) O 2 (aq) O 2 (g) (R10) In fact, process (B) is the sum of R1, R2, R3, R4 and R10. The species IO 3 - and H 2 O 2 are the major reactants in the process (B), but the do not react with each other at a finite rate even though both are consumed in proportionate amounts in process (B), which takes place at a significant rate. The rate-determining step that initiates process (B) is step R1. I - is the principal reactant in step R1 and it is formed b rapid hdrolsis of the I 2 product. The eistence of autocataltic process is one of the necessar conditions for the occurrence of oscillations. In the BL sstem the process (B) is autocataltic. Process (C), in which the iodine is oidated to iodate, is even more complicated than autocataltic process (B). The rate-determining step in process (C) is step R1. Zimmerman [20] found that in degassed iodine atoms recombined with each other in one tenth of a second. However, in the presence of air, iodine atoms would persist for almost a minute. So iodine might be stabilized b ogen in water. 351

4 According to the mechanism proposed b Treindl and Noes, this work simulates the nonlinear behavior of the BL reaction involving two variables. The variables indicate the concentrations of the components. Table 1. Identification of smbols for variables and parameter. X I 2 (aq) Y O 2 (aq) - A IO 3 B O 2 (eq) C H 2 O 2 The acidit of the solution is an important factor for the nonlinear behavior, but the more of variables in the model, the more difficult of calculation. Thus, in this paper, we neglect the acidit of solution. According to steps R6 and R7 the following equations can be obtained: d[ I2( aq) ] 1 = vr3 kr7[ I ][ O2( aq) ] dt 2 d[ O2( aq) ] = kr4[ H 2O2 ][ HOI ] kr 10{[ O2( aq) ] [ O2( eq) ]} dt (1) (2) where k is the rate constant of each reaction, respectivel, and v R3 is the general rate constant of reversible step R3. And step R1 is the rate-determining step in process (B). 1 vr3 = kr 1[ IO3 ][ I ] 2 (3) If K d and K h are the dissociation equilibrium constants and hdrolsis equilibrium constants of I 2, respectivel, then, the following equations can be gained: K d = 2 [ I ] [ I2( aq) ] (4) K h + = [ H ][ I ][ HOI] [ I ] 2( aq) (5) It can be seen from equation (5) that, if the concentration of [I - ] was proportional to X 1, then the concentration of HOI would be proportional to X. If k 1, k 2, k 3 and k 4 were empirical rate constants for a particular sstem, then the above equations could be combined to generate equation (6) as the fundamental equations of the model: 352

5 dx 0.5 = k1 AX k2 X Y dt dy = k CX k Y B dt 1 ( ) 3 4 (6) The above differential equations show the relationship of concentrations with time among the components in the sstem, and concentration and time are dimension variables and parameters. So the above differential equations must be transformed into dimensionless equations before resolving, the concentrations X and Y, the time t and the rate constants also should be transformed into dimensionless variables and parameters. Here, these transformations involve replacing X, Y and t in the above equation b,,, β and, which are defined as ( k2 A) X k k C = 2 3, k 2 ( k1a) Y k A k k C = k1a, = t( k2k3c) k2k3c 2 1 2(5 6 ), 2 1 2(5 6 ) 1 k1 A = k4 0.5 ( k2k3c) k2k3c k 2 ( k1a), β = B k 1 A k 2 k 3 C After the dimensionless transformation, the differential equations become as follows: d = d d d = ( β ) (7) Double-precision computations are performed on a personal computer and the mathematic 5.0 is used in all the simulations. The temporal concentrations of reaction sstem are calculated b Gear method. The precision of calculation is more than 20 bit validit numerals and the simulation time is 150 minutes. The differential equations are resolved and the graphs of - and - are obtained. Results and discussion Results Fig. 1 shows the simulation results of the BL oscillating chemical sstem. In the batch sstem, the concentration of iodine and the product rate of ogen change periodicall, and the amplitude damped but the period hold the line. The oscillations can maintain 150 minutes. Parameters, β and affect the nonlinear behaviors obviousl, and it will be discussed later. 353

6 0 = riables = Figure 1. Simulation of the BL oscillations obtained b numerical integration with the differential equations ( =0.55, β =0.03, = ). Discussion Effect of on BL nonlinear behavior Equation (7) indicates that the value of should be in the range of 0-1. The simulation results showed that the number of period increased with increasing the value in the range of That is to sa, the larger in this range, the slower decaed of oscillation in the sstem. In addition, the maimal concentrations of both I 2(aq) and O 2(aq) are decreased with the increase of (see Fig. 2), where the value of indicates the hdrolsis degree of I 2(aq) in the solution. As mentioned above, the change of BL oscillating chemical could be divided into two processes (B) and (C); the H 2 O 2 is oidated b iodate to I 2 in - process (B) and I 2 goes back to IO 3 in process (C). If the reaction rates of process (B) and process (C) were equal in an time, the sstem must achieve to chemical equilibrium state and no oscillation appeared. Just because of the reaction rate of process (B) and process (C), one is faster than the other at a certain time, so process (B) and process (C) are dominant on the oidation and reduction processes alternativel, resulting in the oscillations of both I 2 and I - concentrations and the production rate of O 2. If was too small, then - substantive I 2 produced in process (B) translated into IO 3 and process (C) dominated the sstem for a long time. It is too difficult to shift from process (C) to process (B), so the oscillations deca quickl and the sstem achieves to the chemical equilibrium state. On the contrar, if was too large, then substantive I 2 produced in process (B) hdrolzes and the reaction rate of process (B) dominates the sstem and can not shift to the process (C) and lead the sstem to reach the chemical equilibrium state, so no oscillation appears. In order to investigate the effect of on the nonlinear behavior of the BL oscillating chemical reaction, the value of changes with step 0.01 in the range of was researched in detail (see Fig. 3). Results show that the slight change of brings to obviousl change of the oscillation curves. More interesting is that when =0.55, the sstem oscillates in 150 minutes; but when =0.56, the differential equations described b the sstem have no solution in =7. From a mathematician view, if was chosen as the control parameter and in 354

7 the process of changing the qualitative character (there is stabilit) changes suddenl when =0.55. So it could be thought that the sstem could appear bifurcation when =0.55 and the critical value was =0.1 = =0.2 = =0.3 = = =0.4 =0.5 = Figure 2. Effect of in the range of on the oscillation (β = 0.03, = ). 355

8 =0.48 =0.48 =0.49 =0.49 =0.50 =0.50 =0.51 = =0.53 =0.53 =0.54 =0.54 =0.55 =0.55 =0.52 =0.52 Figure 3. Effect of in the range of on the oscillation (β=0.03, =0.2882). 356

9 =0.55 =0.55 =0.56 = Figure 4. Comparison of =0.55 and =0.56 (β = 0.03, = ). Effect of on BL nonlinear behavior 2(5 6 ) 1 k1 A From the definition of = k4 0.5, epresses the composition ( k2k3c) k2k3c effects of the initial concentration of reactant, reaction rate constants and. Simulation results show that when , the differential equations described b the sstem have no solution; when , the differential equations have solution, and onl for the value of in the range of the differential equations have periodic (or oscillating) solution, and the number of periodic solutions decreases with increasing in this range (see Fig. 5). When 882, the differential equations have solution but no periodic one. The same as, if was chosen as the control parameter and in the process of changing the qualitative character changes suddenl when = So the sstem appears bifurcation when = and the critical value is Effect of β on BL nonlinear behavior According to the simulation results, the effect of parameter β on the oscillating curve is as not sensitive as and. When β <, and the other parameters are consistent with the above discussion, the oscillation can be observed. The change of β has little effect on the oscillation curve. There is no change between the curves of β = 0.03 and β = In the same wa, when β = the sstem appears bifurcation and the critical value is. 357

10 = = = = = = = = = = = = Figure 5. Effect of on the oscillation ( = 0.55,β = 0.03). 358

11 Effect of solution acidit on BL nonlinear behavior As pointed out in the introduction, the BL oscillating chemical reaction consists of processes (B) and (C). Not accounting for the reaction rate of them, the total iodine elementar in the sstem is equal to the initial concentration of iodate and keeps constant. The relative concentrations of I 2 and IO - 3 depend on the acidit + of the solution. According to equation (5), H I [ HOI ] = K h I 2( aq), if [H + ] is small ([H + ]=0.01M), then [I - ][HOI] is larger [17] and the concentration of I 2 increases even to saturation [18]. Noes [8] reported a much larger formation of I 2 at low than at high acidit. At lower acidit, formation of I 2 cannot be oidized to IO - 3 and no oscillation is observed. In addition, lower acidit corresponds to high value of. If the acidit of the solution is high ([H + ]=0. 1 M), the concentration of I 2 would be much smaller than that in lower acidit, so it corresponds to higher value of in the model. For < 0.5, the mathematical simulation showed that equation (7) wouldn t be able of generating long-lasting oscillations. It is consistent with lower and higher acidit. However, when the concentration of [H + ] = M, the sstem was alternatel dominated, firstl b process (B) and then b process (C), resulting long-lasting oscillation. This situation corresponds to slightl greater than 0.5. This is the range of values where oscillations appear in the simulation results reported in the above section. =882 = Figure 6. Curves of =882 ( = 0.55,β = 0.03) Conclusion According to the mechanism proposed b Treindl and Noes involving 10 reaction steps, the simulation of the BL oscillating chemical reaction b differential kinetic methodolog was carried out in this work. In addition, the parameters of the differential equations were investigated in detail. The simulation results show that (1) when the variable parameters are = 0.55, β = and <, the differential equations present periodic solution, and the oscillation can be observed in 150 min; (2) if, β and were taken as the control parameters respectivel, the bifurcations would be observed in the 359

12 process with the critical values of = 0.55,β = ,and = ; (3) the acidit of the solution is an important factor on the nonlinear phenomena. When the concentration of H + is about M, that equivalent to = 0.5, the oscillation can be observed. Acknowledgment The authors are grateful to the financial support in part b the National Natural Science Foundation ( ), the Project of Ecellent Teachers of Education Ministr (3070), and the project of KJCXGC-01 of Northwest Normal Universit, as well as the Gansu Ke Lab of Polmer Materials, China. References 1. V. Vukojević, S. Anić, Lj. Kolar-Anić, Phs. Chem. Chem. Phs. 4 (2002) J. Ren, J.Z. Gao, W. Yang, Comput. Visual. Sci. (2008) DOI /s (online first). 3. S. Anić, Lj. Kolar-Anić, E. Korös, React. Kinet. Catal. Lett. 61 (1997) Lj. Kolar-Anić, Ž. Čupič, S. Anić, G. Schmitz, J. Chem. Soc. Farada Trans. 93 (1997) W.C. Bra, J. Am. Chem. Soc. 43 (1921) D.R. Stanisavljev, A.R. Djordjević, V.D. Likar-Smiljanić, Chem. Phs. Lett. 423 (2006) S. Keki, G. Szekel, M.T. Beck, J. Phs. Chem. A 107 (2003) R.M. Noes, L.V. Kalachev, R.J. Field, J. Phs. Chem. 99 (1995) P. Ševčík, K. Kissimonová, L. Adamčíková, J. Phs. Chem. A 104 (2000) F.G. Buchholtz, S. Broecker, J. Phs. Chem. A 102 (1998) I. Valent, L. Adamčíková, P. Ševčík, J. Phs. Chem. A 102 (1998) I. Matsuzaki, T. Nakajima, H.A. Liebhafsk, Farada Smp. Chem. Soc. 9 (1974) P.G. Bowers, Ch. Hofstetter, C.R. Letter, R.T. Toome, J. Phs. Chem. 99 (1995) E. David, R.M. Noes, J. Phs. Chem. 83 (1979) K. Kissimonová, I. Valent, L. Adamčíková, P. Ševčík, Chem. Phs. Lett. 341 (2001) L. Treindl, R.M. Noes, J. Phs. Chem. 97 (1993) G. Schmitz, H. Rooze, Far from Equilibrum Snergetics, Springer-Verll, Berlin, 1979, p V. Sharma, R.M. Noes, J. Am. Chem. Soc. 98 (1976) G. Schmitz, Lj. Kolar-Anić, S. Anić, J. Phs. Chem. A 110 (2006) J. Zimmerman, R.M. Noes, J. Phs. Chem. 18 (1950)