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Modeling of Interactions Between Iodine Interphase Transport and Oxygen Production in the Modified Briggs-Rauscher Reaction with Acetone Erik Szabo and Peter Sˇevcˇ´ık* Department of Physical and Theoretical Chemistry, Faculty of Natural Sciences, Comenius UniVersity in BratislaVa, 842 15, BratislaVa, SloVakia ReceiVed: May 11, 2010; ReVised Manuscript ReceiVed: June 7, 2010
Novel aspects of influence of gas evolution on oscillating chemical reaction are considered on a model of the modified Briggs-Rauscher reaction with acetone. The mechanism proposed by Noyes with Furrow and De Kepper with Epstein, adapted for acetone and expanded with oxygen evolution and first-order escape of iodine by interphase transport, was subjected to further extension, and calculations assuming direct proportionality of the rate of iodine escape on the surface area of the oxygen gas trapped in the solution as well as calculations considering first-order back-flow of iodine from the gas phase into the solution were performed. Furthermore, the concentrations of iodine vapors were considered to be variable within the bubble population, as all the bubbles are not in contact with the reaction mixture for the same length of time. Distribution functions of the surface size over the range of concentrations of iodine in the contacting gas phase were introduced and derived, and whereas they were proved not to play any role in the total intensity of iodine return when it is first-order, the article also demonstrates their explicit consideration in calculations to be essential if the dependence of the rate of iodine return into the solution on its concentration is not linear, for example, in future extensions of the model with bubble radius also considered variable. 1. Introduction For many years now, chemical oscillators have no longer been mere laboratory obscurities but an important building block in our understanding of complexity and a valuable class of examples in the field of nonlinear dynamics, also allowing practical rather than just theoretical explorations. The first recorded oscillator with predominantly chemical control mechanism is most probably the Bray-Liebhafsky reaction,1,2 prepared from acidic solutions of iodate and hydrogen peroxide at elevated temperatures, whereas the most extensively examined oscillating chemical reaction is without a doubt the BelousovZhabotinsky reaction,3,4 employing bromate with various organic substrates and typically also a redox catalyst. It is, however, a more recent hybrid of the previous two, the Briggs-Rauscher (BR) reaction,5 consisting of iodate, an organic substrate, hydrogen peroxide, and a redox catalyst in acidic medium, that not only found a common application in determination of antioxodants6,7 but also is perhaps the most popular for elementary demonstrations because of its straightforward preparation as well as for its rather spectacular behavior. The core mechanism of the BR reaction has been considered to be established for the most part since the works of Furrow with Noyes8 and De Kepper with Epstein,9 who simultaneously proposed almost identical representations of the reaction, consisting of 11 pseudoelementary steps, as designed in eqs R1-R11 of Table 1, although its detailed examination remains a fruitful field even today.10,11 Growing attention has recently also been dedicated to the production of gas,12–14 to some extent accompanying a vast majority of oscillating reactions, however, with particularly dramatic examples in the family of the BR reaction and its substitutions or additions.8,15–17 The BR reaction with acetone as the organic substrate has already been considered for its exceptionally well-defined oscillations in the rate of gas * Corresponding author. E-mail:
[email protected].
production and their practical measurement18 and effects of the interphase transport of oxygen and iodine on the reaction have been modeled,19 suggesting a possible interplay of the physical process of gas production and the chemistry of the reaction, which is normally neglected. This article aims to pursue the possibility within a theoretical study of the reaction, considering a portion of the gas produced to remain temporarily trapped in the reaction mixture and the rate of iodine transport to be directly proportional to the total size of the interphase surface, including the interphase with the trapped gas. Moreover, the reverse process is also considered, and circumstances under which the individual history of each part of the surface (i.e., each part of the total bubble population) must be taken in account instead of using only the average concentration of the iodine vapors in the hold-up. 2. Formulation of the Model The core of the model implements the mechanism proposed simultaneously by Furrow with Noyes8 and De Kepper with Epstein,9 as previously adapted for the acetone modification and extended with interphase transport steps.19 The governing equations were assembled assuming pseudoelementary character of all steps, mass balance corresponding to the reaction stoichiometry, and constant acidity. As summarized in Table 1, the rate constants were employed as defined by Furrow and Noyes,8 but with k2, k10, and k-10 fixed19 to yield results in better agreement with experimental data obtained for the reaction with acetone. The initial conditions were set to [IO3-]0 ) 0.035 M, [H2O2]0 ) 0.4 M, [acetone]0 ) 1.3 M, [H3O+]0 ) 0.2 M, [Mn2+]0 ) 5.0 × 10-3 M, and [I-]0 ) 10-8 M and all other species were assumed to be absent in the initial mixture. Integration was performed on a PC using Fortran subroutine LSODE20 with absolute error tolerance set to 10-30 and relative error tolerance to 10-6, visualizing the data thus obtained with gnuplot.21
10.1021/jp104283j 2010 American Chemical Society Published on Web 07/14/2010
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TABLE 1: Model of the BR Reaction Employeda (R1) (R2) (R3) (R4) (R-4) (R5) (R6) (R7) (R8) (R9) (R-9) (R10) (R-10) (R11) (R12) (R13)
+ I- + IO3 + 2H f HIO + HIO2 HIO2 + I- + H+ f 2HIO HIO + H2O2 f I- + O2 + H+ + H2O + · IO3 + HIO2 + H f 2IO2 + H2O + + HIO 2IO2· + H2O f IO3 2 + H 2+ · IO2 + Mn + H2O f HIO2 + Mn(OH)2+ Mn(OH)2+ + H2O2 f Mn2+ + H2O + HO2· 2HO2· + f H2O2+O2 + 2HIO2 f IO3 + HIO + H + HIO + I + H f I2 + H2O I2 + H2O f HIO + I- + H+ Org(oxo) f Org(enol) Org(enol) f Org(oxo) Org(enol) + I2 f I-Org + I- + H+ I2 f escape O2 f escape
+ 2 V1 ) k1[IO3 ][I ][H ] V2 ) k2[HIO2][I-][H+] V3 ) k3[HIO][H2O2] V4 ) k4[H+][HIO2][IO3] V-4 ) k-4[IO2· ]2 V5 ) k5[IO2· ][Mn2+] V6 ) k6[Mn(OH)2+][H2O2]
[k1] ) 1.4 × 103 [k2]* ) 4.0 × 109 [k3] ) 37.0 [k4] ) 1.516 × 104 [k-4] ) 1.607 × 109 [k5] ) 1.0 × 104 [k6] ) 3.2 × 104
V7 ) k7[HO2· ]2 V8 ) k8[HIO2]2 V9 ) k9[H+][HIO][I-] V-9 ) k-9[I2] V10 ) k10[Org(oxo)] V-10 ) k-10[Org(enol)] V11 ) k11[Org(enol)][I2] V12 ) k12[I2] V13 ) k13[O2]
[k7] ) 7.5 × 105 [k8] ) 45.3 [k9] ) 3.1 × 1012 [k-9] ) 2.2 [k10]* ) 4.0 × 10-6 [k-10]* ) 14.0 [k11] ) 9.1 × 105 [k12] ) parameter [k13] ) paramater
a
Source model of the BR reaction with steps R1-R11 as suggested by Noyes with Furrow8 and De Kepper with Epstein,9 adapted for acetone with fixed rate constants (marked with asterisks) and supplemented with first-order iodine and oxygen interphase transport R12, R13.19 The rate constants are expressed in the units of s-1, s-1 M-1, or s-1 M-2 with respect to the form of rate law.
Oxygen Hold-Up. Having successfully reproduced the original model,19 we started its further extension by considering the hold-up of produced oxygen gas by the reaction mixture. Its evolution was therefore divided into two steps
O2(aq) f O2(ghld)
(R13.1)
O2(ghld) f O2(gfree)
(R13.2)
Whereas R13.1 differs from the original process (R13) only in the product and so does not alter the formulation of the original model as long as V13 and k13 are maintained, a new variable has to be introduced to follow the amount of oxygen hold-up O2(ghld). This was best expressed in terms of its absolute volume Vhld captured in Vrxn ) 30 cm3 of the reaction mixture, assuming ideal gas molar volume Vm at 25 °C and 1 atm, and the dynamics of the original model was thus supplied with
dVhld ) VmVrxnV13 - V13.2 ) VmVrxnV13 - k13.2Vhld dt
(1)
where VmVrxn was approximated to be 734 cm3 M-1. The results were then observed with respect to the changing values of k13.2, and a suitable value representing a realistic course of gas evolution was chosen. Subsequently, the volume of oxygen held by the mixed reaction matrix was assumed to be a monodispersion of perfectly spherical cavities of radius r, thus bringing the reacting solution to contact with surface Shld given by
Shld ) 3Vhld /r
(2)
Iodine Transport into Gas. The rate of iodine evaporation V12, which has already been proven to be capable of affecting the overall chemistry of the oscillator,19,22 was now assumed to be proportional to the size of the interphase surface, consisting of the top surface S∞ and the surface of oxygen trapped in the solution Shld
V12 ) k12[I2(aq)] ) k12(Shld + S∞)[I2(aq)]
(3)
Approximated from previous results,18,19,22 k12 ) 2.74 × 10-4 s was considered for circular surface with 2 cm radius representing S∞ ) 12.57 cm2, affording the surface-independent rate constant k′12 ) 2.18 × 10-5 cm-2 s-1. With this rate constant fixed, the course of reaction was observed with respect to varying radius r of the particles of dispersed oxygen, at the same time calculating the total amount of iodine trapped with the oxygen -1
dn(I2(ghld)) Shld V V - k13.2n(I2(ghld)) ) dt Shld + S∞ 12 rxn
(4)
from which its average concentration [I2(ghld)] in the total volume Vhld was also determined. Iodine Back-Flow into the Reaction Mixture. In the next step, iodine evaporated into the gas phase was assumed to flow in the reverse direction as well, although only regarding the surface Shld, as if the top surface S∞ was continuously stripped off all iodine vapors by an auxiliary gas flow above the surface. In the first approximation, the rate of the back-flow V14 was formulated similarly to eq 3 as directly proportional to the average concentration of iodine in the gas phase [I2(ghld)] affording
V14 ) k14[I2(ghld)]Shld
(5)
and after the product [I2(ghld)]Shld was evaluated with respect to variable radius r of gas particles at k14 ) 0, the parameter k14 was varied for fixed r ) 0.25 mm, and its effects on the course of the reaction were observed. Substituting the Average [I2(ghld)] with Concentration Distributed over the Surface Unevenly: Introducing the Surface Distribution Function. Whereas the bubble size was still maintained constant, the population of bubbles was now considered to contain various concentrations of iodine vapors because all of the bubbles are not in contact with the reaction mixture for the same length of time. According to the individual history of each part (i.e., each “generation” of bubbles), the total surface is modeled to be distributed over the whole range of iodine concentrations cI for the ease of computation employing
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Figure 1. Effects of extending the original model of the BR reaction stated in Table 1 (a) with oxygen hold-up in the reaction mixture and the rate constant k12 directly proportional to the surface of the trapped gas phase, assumed to be monodispersed into particles with radii of (b) 2, (c) 1, (d) 0.5, and (e) 0.25 mm demonstrated on the traces of total volume of O2 produced (I) and the average concentrations of I2 in the hold-up (II).
a continuous representation of the surface distribution function s(cI,t) defined as
dShld(t) ) s(c1, t) dc1
(6)
The calculations of s(cI,t) were performed within an extra module appended to the LSODE subroutine, recursively building the functions from a single starting point of s(cI ) 0, t ) 0) ) 0 by consecutive alternations of the first order decay, analogous to the decay terms of eqs 1 or 4
∂s(cI, t) ) -k13.2s(cI, t) ∂t
Figure 2. Details of the effects as described in Figure 1, here demonstrated on the traces of average concentrations of I2 in the trapped gas phase (I), concentrations of I2 in the reaction solution (II), and the volume of hold-up (III).
(7.1)
with shifting of the whole concentration grid according to the definition of the rate of iodine transport into the gas phase, common over all values of cI
dcI Vrxn Shld V ) dt Vhld Shld + S∞ 12
(7.2)
and with the new values of s(0,t) for freshly evolved gas (with cI ) 0) determined as
∂Shld(0, t) dShld dcI ) ∂cI dt dt
/
(8)
The values of the iodine back-flow intensity I obtained as [I2(ghld)]Shld from average iodine concentrations in the trapped gas phase for k14 ) 0 and r varied were then compared with the values of the corresponding integral of the product cI s(cI,t) over all values of cI.
Figure 3. Effects of further modification of the model of the BR reaction evaluated in Figure 1, assuming the trapped gas phase to be monodisperse into particles with radii of 0.25 mm and considering the iodine interphase transport to be a reversible process, with the rate constant of the back-flow k14 equal to (a) 0, (b) 0.5k12 ′ , (c) k12 ′ , (d) 2k12 ′ , (e) 4k12 ′ , and (f) 8k12 ′ demonstrated on the traces of total volume of O2 produced (I) and average concentrations of I2 in the hold-up (II).
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Figure 4. Distribution functions s(cI,t) of the hold-up surface according to the concentration of iodine cI in the contacting gas phase for the model of the BR reaction as stated in Table 1 with the rate constant k12 directly proportional to the surface of the trapped gas phase assumed to be monodispersed into particles with radii of (a) 2 and (b) 0.25 mm, rendered as parametric plots depicting the first oscilation periods (I) and parts of 3D plots (II) and 2D grayscale maps (III) depicting the first 1500 s.
Finally, nonlinear dependence of the rate of iodine back-flow into the solution, which may arise, for example, in further extensions of the current model after setting the bubble radius r individually variable within the bubble population, was considered. For simplicity, we modeled the nonlinearity by considering the order of the iodine back-flow with respect to iodine concentration to be equal to 2, and thus we established the effect on correlation between rates calculated from average iodine concentrations and from its corresponding distributions over the surface by using the data for k14 ) 0 and r varied to evaluate the ratio
f)
[I2(ghld)]2Shld
∫0∞ c2I s(cI, t) dcI
(9)
3. Results and Discussion With the rate constants k12 and k13 fixed as in the previous works at 2.74 × 10-4 and 8.33 × 10-3 s-1, respectively, and with other parameters as in Table 1, yielding ∼10 cm3 of oxygen in the first couple of periods, ∼3.3 cm3 of gas seemed to be a realistic maximum volume of oxygen trapped in 30 cm3 of reaction mixture, and the value of k13.2 ) 0.01 s-1 was therefore selected for all subsequent calculations. According to eq 2, when the bubble diameters of 4, 2, 1, and 0.5 mm are considered, this volume of trapped gas phase affords approximately 50, 100, 200, and 400 cm2 of maximum extra surface for iodine interphase transport, compared with the original surface of 12.57 cm2 roughly representing an increase by factors from 4 to 32. Effects of this increase on the overall dynamics of the model are summarized in Figures 1 and 2. As clearly shown in Figure 1, on the traces of total volume of oxygen evolved (Figure 1-I), enhancement of iodine interphase transport clearly speeds up the course of the reaction, even though maintaining the total volumes of oxygen evolved in individual periods practically unchanged. Whereas these results reproduce the previous findings in models very closely, the traces of iodine concentration in the reaction solution depicted in Figure 2-II, increasingly diverging from a perfect saw-wave record with decreasing
bubble diameters, also reveal the effects of variations in the rate of iodine interphase transport throughout the individual reaction periods. Moreover, if we compare the concentrations of iodine in the reaction solution with the average concentrations of iodine calculated inside the gas phase trapped within the solution, we find that under the circumstances specified, the latter not only compares to the former but also even tends to exceed it considerably. We can therefore propose a conclusion that besides the quite likely possibility of having overestimated the value ′ , the back-flow of iodine from its vapors in the trapped of k12 gas phase should also be considered, especially for finer gas dispersions, where both the concentration of iodine in the gas phase and the size of the interphase surface are greater, causing their combined impact to grow dramatically with decreasing bubble radii. Nevertheless, the results shown in Figures 1 and 2 clearly illustrate the theoretical possibility of the influence of gas evolution on oscillating reaction by supplying extra interphase surface for transport of volatile intermediates, leading to a direct dependence of the course of reaction on the extent of dispersion of the evolving gas. Demonstrating the dynamics of the model for r ) 0.25 mm and the iodine back-flow rate constant k14 varied from 0 to 8k′12, the results in Figure 3 support the idea that this interphase transport should be considered as a two-way process. As seen from the trace of total volume of oxygen evolved (Figure 3-I), at least in the case of a finer dispersion of the gas, the return of iodine to the solution can affect the dynamics of the system just as its evaporation. However, the corresponding traces of the average concentrations of iodine trapped with the gas phase (Figure 3-II), clearly showing new minima, ascertain that the effect of the back-flow of iodine into the solution is not mere compensation of the evaporation and that it also influences the dynamics of the system in a more complex fashion. Finally, the results of calculating the surface distributions according to the concentration of iodine in the contacting gas phase and their evolution in time are presented in Figure 4, showing how a significant portion of gas comes from a short phase of intensive gas evolution yet negligible iodine evaporation and how the peak and the more uniform signal that follows
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of oxygen evolution and iodine transport confirmed the theoretical possibility of the influence of gas production on the chemistry of an oscillating system by enhancement of the transport of volatiles through extra interphase surface supplied by the gas transiently trapped in the solution, here assumed monodispersed. Whereas the magnitude of the effect has most probably been overrated in our calculations, both the evaporation of iodine into the gas phase as well as its return into the solution have been shown to be capable of affecting the overall kinetics of the reaction. In addition, distributions of the surface with respect to the iodine concentration in the contacting gas phase have been derived, evaluated, and visualized, and whereas they were proven to be dispensable for first-order return of iodine into the solution, they were found to be essential for correct treatment of more complex dynamics of the back-flow, as may be anticipated, for example, in further extensions of the model for nonuniform populations of the trapped gas bubbles. Acknowledgment. This work was supported by the Grant of the Ministry of Education of Slovakia VEGA 1/0039/09 and the Grant of Comenius University in Bratislava UK/124/2010. Figure 5. Comparison of the total intensity of iodine back-flow from the trapped gas phase assumed in monodispersion with radii of 2 mm (I) for a hypothetical second-order kinetics calculated from the average iodine concentration (dark line) and by integrating with the surface distribution function (pale line) and the values of the ratio f of the two (II) for radii of the gas-phase particles of (a) 2, (b) 1, (c) 0.5, and (d) 0.25 mm.
decay and simultaneously grow in concentration of trapped iodine with time. Subsequent comparisons of the intensities of iodine return into the solution based on the distributions and on the average iodine concentration in the trapped gas not only confirmed that in the case of a first-order iodine back-flow these intensities are equal in agreement with
∫0∞ cI
( )
∂Shld 3 dcI ) ∂cI r
∫0∞ cI
( )
∂Vhld dcI ) ∂cI
Shld 3 n(I (g )) ) Shld[I2(ghld)] (10) n(I (g )) ) r 2 hld Vhld 2 hld
but also verified the consistency of our calculations. As can be seen in Figure 5, depicting the situation for a hypothetical second-order iodine return into the solution, it is also clear that in such a case, the discrepancy introduced by working only with average iodine concentration in the gas phase can be considerable, and the actual distribution of iodine within the gas phase therefore must not be disregarded in accurate calculations of models that are expected to be somewhat more complex. 4. Conclusions Calculations based on a classic mechanism of Briggs-Rauscher reaction adapted for acetone and extended with the dynamics
References and Notes (1) Bray, W. C.; Caulkins, A. L. J. Am. Chem. Soc. 1921, 43, 1262– 1267. (2) Bray, W. C.; Liebhafsky, H. A. J. Am. Chem. Soc. 1931, 53, 38– 48. (3) Zhabotinsky, A. M. Biophysics 1964, 9, 329–335. (4) Degn, H. Nature 1967, 213, 589–590. (5) Briggs, T.; Rauscher, W. J. Chem. Ed. 1973, 50, 496. (6) Cervellati, R.; Honer, K.; Furrow, S. D.; Neddens, C.; Costa, S. HelV. Chim. Acta 2001, 84, 3533–3547. (7) Honer, K.; Cervellati, R. Eur. Food Res. Technol. 2002, 215, 437– 442. (8) Furrow, S. D.; Noyes, R. M. J. Am. Chem. Soc. 1982, 104, 3842, 42-45, 45-49. (9) De Kepper, P.; Epstein, I. R. J. Am. Chem. Soc. 1982, 104, 49–55. (10) Furrow, S. D.; Aurentz, D. J. J. Phys. Chem. A 2010, 114, 2526– 2533. (11) Lawson, T.; Fu¨lo¨p, J.; Wittmann, M.; Noszticzius, Z.; Muntean, N.; Szabo´, G.; Onel, L. J. Phys. Chem. A 2009, 113, 14095–14098. (12) Szabo´, G.; Csavda´ri, A.; Onel, L.; Bourceanu, G.; Noszticzius, Z.; Wittmann, M. J. Phys. Chem. A 2007, 111, 610–612. (13) Onel, L.; Bourceanu, G.; Wittmann, M.; Noszticzius, Z.; Szabo´, G. J. Phys. Chem. A 2008, 112, 11649–11655. (14) Muntean, N.; Szabo´, G.; Wittmann, M.; Lawson, T.; Fu¨lo¨p, J.; Noszticzius, Z.; Onel, L. J. Phys. Chem. A 2009, 113, 9102–9108. (15) Furrow, S. D. J. Phys. Chem. 1995, 99, 11131–11140. (16) Furrow, S. D.; Cervellati, R.; Amadori, G. J. Phys. Chem. A 2002, 106, 5841–5850. (17) Cervellati, R.; Furrow, S. D.; De Pompeis, S. Int. J. Chem. Kinet. 2002, 34, 357–365. (18) Szabo, E.; Sˇevcˇ´ık, P. J. Phys. Chem. A 2009, 113, 3127–3132. (19) Sˇevcˇ´ık, P.; Kissimonova´, K.; Adamcˇ´ıkova´, L′. J. Phys. Chem A 2003, 107, 1290–1295. (20) Hindmarsh, A. C. ODEPACK, Double Precision, version 2005; Center for Applied Scientific Computing: Livermore, CA, 2005. (21) Williams, T.; Kelley, C. gnuplot, version 4.4.0, 2010. (22) Sˇevcˇ´ık, P.; Adamcˇ´ıkova´, J. Phys. Chem. A 1998, 102, 1288–1291.
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