J . Phys. Chem. 1989, 93, 736-740
736
the Occurrence of two different rates of heat release under the same average surface temperature (Figure 1) may lead one to conclude erroneously that this reaction exhibits isothermal multiplicity. The existence of isothermal multiplicity can be deduced only from the finding of two uniform temperature states having the same temperature but different heat generation rates. We did not discover any such case in our experiments. The large difference between the observed rate under uniform and nonuniform temperature profiles (Figure 1) indicates that care should be taken in the analysis of kinetic data obtained from heated wires or ribbons. The temperature nonuniformity may lead to pitfalls also in predicting the functional form of the rate expression. Harold et aL9 showed that the direction (counter or clockwise) of the hysteresis loop can be used to infer the functional form of the kinetic expression for lumped (uniform) systems. These rules will lead to pitfalls if applied to nonuniform systems, for which they are not valid. We conclude that it is important to account for any nonuniformities in the surface temperature in the analysis of the mul-
tiplicity features of electrically heated wires and ribbons maintained at a constant resistance. While nonuniform temperatures have been observed also on unheated catalytic surfaces,12J3they are much more likely to appear on electrically heated ribbons kept at a constant resistance. Thus, it is suggested to conduct multiplicity experiments without any electrical heating, such as those described in ref 10. The experiments point out that nonuniformly active wires can lead to rather exotic temperature profiles and bifurcation diagrams (Figures 8 and 9). Failure to account for this nonuniformity may lead one to attribute these erroneously to complex kinetic features. Acknowledgment. We are thankful to the NSF, the Welch Foundation, and the Texas Advanced Research Program for support of this research. Registry No. Pt, 7440-06-4; NH,, 7664-41-7. (12) Pawlicki, P. C.; Schmitz, R. A. Chem. Eng. Prog. 1987,83(2), 40. (13) Cox, M. P.; Ertl, G.; Imbihl, R. Phys. Rev. Lett. 1985, 54, 1725.
Pulse and Contlnuous Radiolysis of Cyano-Bridged Polynuclear Ruthenium Complexes in Aqueous Solution Quinto G. Mulazzani,*.t Margherita and Franco Scandola8
Mila D'Angelantonio,+ Carlo A. Bignozzi,B
Istituto di Fotochimica e Radiazioni d'Alta Energia del C.N.R., Via de' Castagnoli 1 , 40126 Bologna, Italy, Dipartimento di Chimica "G. Ciamician", Universith di Bologna, Via Selmi 2, 401 26 Bologna, Italy, Dipartimento di Chimica, Universith di Ferrara e Centro di Fotochimica del C.N.R., 44100 Ferrara, Italy (Received: April 25, 1988)
The reactions of ea; and C02'- radicals with [(NH3)5RuNCRu(bpy)2CNRu(NH3)5]6+ [3,2,3], [py(NH3),RuNCRu[3',2,3'] in aqueous solution result in (bpy)2CNRu(NH3)5]6+[3',2,3], and [py(NH3)4RuNCRu(bpy)2CNRu(NH3)4py]6+ the formation of the reduced forms [2,2,3], [2',2,3], and [2',2,3'], respectively;exhaustive irradiation results in the formation of the [2,2,2], [2',2,2], and [2',2,2'] species, respectively. The rate constants for the reduction of the three Ru(III)Ru(II)Ru(III) complexes by e,, and by COz'- radicals are - 3 X 1O'O and (1.4 f 0.2) X 1OIo M-' s-l, respectively, independent of the nature of the complex. In acidic solution and in the presence of tert-butyl alcohol, the reaction between H' radicals and [3,2,3], for which k = (4.0 & 0.5) X lo9 M-' s-l, also results in the formation of [2,2,3]; under these conditions, the formation of [2,2,2] was not observed. The intramolecular electron-transfer reaction [3',2,2] [2'2,3], which is believed to follow the formation of [3/,2,2], obtained in parallel with [2',2,3] from the reduction of [3',2,3], was not directly observed, indicating that the associated rate constant has a value of k 2 5 X lo's?.
-
Introduction Polynuclear transition-metal complexes are currently the object of considerable interest from several viewpoints. They are simple molecular analogues of interesting more complex materials, such as inorganic polymers and mixed-valence materials.' They exhibit interesting optical (intervalence transfer, IT) spectra that can be correlated by means of the Hush-Marcus model to the degree of electron delocalization in the molecule and to the thermodynamic and kinetic factors of the correspondent electron-transfer reactions.2 Polynuclear complexes also exhibit interesting types of electrochemical behavior, related to the presence of several more or less independent redox sites in the m o l e c ~ l e . ~For analogous reasons, polynuclear complexes have potential applications as homogeneous multielectron redox catalysts., Recently, polynuclear transition-metal complexes have been used as interesting supramolecular systems for the study of light-induced intramolecular energy and electron transfer proce~ses.~ Istituto di Fotochimica e Radiazioni. * Universiti di Boloena. 'Universiti di Ferrzra.
0022-3654/89/2093-0736$01.50/0
Studies on polynuclear complexes using radiation chemistry methods are, on the other hand, not very frequent6*' These (1) Mixed-Valence Compounds; Brown, D. B., Ed.; Reidel: Dordrecht, The Netherlands, 1980. (2) Creutz, C. In Progress in Inorganic Chemistry; Lippard, S.J., Ed.; Wiley: New York, 1983; Vol 30, p 1 and references therein. (3) See, e.g., Roffia, S.; Paradisi, C.; Bignozzi, C. A. J . Electrwnal. Chem. 1986, 200, 105-1 18. (4) Gilbert, J. A.; Eggleston, D. S.;Murphy, W. R., Jr.; Geselowitz, D. A.; Gerstcn, S.W.; Hcdgson, D. J.; Meyer, T. J. J . Am. Chem. SOC.1985, 107, 3855-3864 and references therein. (5) Scandola, F.; Bignozzi, C. A. In Supramolecular Photochemistry; Balzani, V., Ed.; Reidel: Dordrecht, The Netherlands, 1987; p 121. (6) An outstanding exception is: Isied, S. S.; Vassilian, A.; Magnuson, R. H.; Schwarz, H. A. J . Am. Chem. SOC.1985,107, 7432-7438. (7) Some pulse and/or continuous radiolysis studies of polytungstates and polymolybdatesp" and bimolecular platinum and rhodium comple~es'2'~ have been also reported. (8) Papacostantinou, E. 2.Phys. Chem. (Neue Folge) 1975,97,313-320. (9) Papacostantinou, E. J . Chem. SOC.,Faraday Tram. I 1982, 78, 2769-2772. (10) Papacostantinou, E. Z . Phys. Chem. (Neue Folge) 1983,137,31-35. (1 1) Mulazzani, Q. G.; Venturi, M.; Ballardini, R.; Gandolfi, M. T.; Balzani, V. Isr. J . Chem. 1985, 25, 183-188.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 2, 1989 737
Cyano-Bridged Polynuclear Ru Complexes multisite systems, though, seem likely to exhibit interesting types of behavior in their reactions with the highly reactive reducing and oxidizing species produced under radiolytic conditions. Moreover, pulse radiolysis of such systems can in principle be used to generate transient species that are unstable with respect to intramolecular (intersite) electron transfer. Actually, some of the more important pieces of evidence on the dependence of electron-transfer rates on distance and driving force come from recent studies of pulse radiolysis of organic and inorganic bichromophoric mole~ules.~J~ We would like to report here the results obtained in the study of the pulse and continuous radiolysis of some cyano-bridged trinuclear complexes of ruthenium in aqueous solution. The complexes have the general structure N, N
I
N'
\\R/
Ru
"3
"3
where N-N represents 2,2'-bipyridine (bpy), X and Y are N H 3 and/or pyridine (py), and the overall electric charge is +6. In these complexes, the central ruthenium atom is in the I1 oxidation state, while the terminal ruthenium atoms are in the I11 oxidation state. For this reason, the three complexes will be hereafter indicated with the notations [3,2,3], [3',2,3], and [3',2,3'], where 3 and 3' represent Ru(II1)-pentamino and Ru(II1)-pyridinotetramino moieties, respectively. These complexes, as well as the products of their stepwise reduction, have recently been characterized in some detail by spectroscopy and electrochemical
technique^.^-'^ Experimental Section Materials. T h e preparation and purification of [(NH3)5RuNCRu(bpy)zCNRu(NH3)sI6+[3,2,31, [PY(NH3)4RuNCRu(bpy)2CNR~(NH3)5]6+ [3',2,3], and [py(NH3)4RuNCRu(bpy)zCNRu(NH3)4py]6+ [3',2,3'], as the PF; salts, has been described el~ewhere.'~J~ Sodium formate (Merck) and tert-butyl alcohol (Baker) were used as received. The solutions were prepared by using Millipore (Milli-Q) water and were saturated with purified NzO, purged with argon, or degassed by standard vacuum line techniques. If not otherwise stated, the solutions were at their nominally neutral pH. Procedures. Continuous radiolyses were carried out at room temperature on 10-20-mL samples contained in Pyrex vessels that were provided with silica optical cells on a side arm. The radiation source was a 60Co-Gammacell (A.E.C., Ltd.) with a dose rate of ca. 7 Gy mi&. The absorbed radiation dose was determined with the Fricke chemical dosimeter by taking G(Fe3+) = 15.5, where G(X) = number of molecules of species X formed per 100 eV of energy absorbed by the solution. Absorption spectra were recorded with Perkin-Elmer Model 555 or Lambda 5 spectrophotometers. Pulse radiolyses with optical absorption detection were performed by using the 12-MeV linear accelerator of the FRAE Institute of CNR, Bologna, Italy." The pulse irradiations were
TABLE I: Rate Constants for the Reduction of Ru(III)Ru(II)Ru(III) Species by C02'- Radicals and Electrochemical Data for Related Redox Couples k x 10-lo,' V M-I
redox couples [3,2,31/[2,2,31 [2,2,31/ [2,2,21 [3',2,31/[2',2,31 [2',2,31/[2',2,21 [3',2,3'1/[2',2,3'1 [2',2,3'1/ [2',2,2'1
(vs SCE)b*e
s-I
-0.055 -0.125 +O. 16 -0.11 +0.18 +0.10
1.4 f 0.1
1.5 i 0.1 1.3 f 0.2
'Obtained at pH 7 in N20-saturated solution containing 0.1 M HCOT. Half-wave potentials ( E 1 / * )determined in aqueous solution. Reference 15.
*
performed at room temperature on samples contained in Spectrosil cells of 0.5- or 1- or 2-cm optical path length. The solutions were protected from the analyzing light by means of a shutter and appropriate cutoff filters. The radiation dose per pulse was monitored by means of a charge collector placed behind the irradiation cell and calibrated with a NzO-saturated solution containing 0.1 M HCOT and 0.5 mM methylviologen (1,l'-dimethyl-4,4'-bipyridinium dichloride) using Gc = 9.32 X lo4 at 602 nm.I8 The computer modeling of the reduction of [3',2,3] by COz'radicals and that of [3,2,3] in acidic solutions (vide infra), was performed by numerical integration of the differential equations describing the reaction mechanism, using a program based on that of HindmarshI9 and a Gould Sel Computer 27/32. Generation of Radicals. The radiolysis of an aqueous solution generates eaq-, OH' radicals, H' atoms, and molecular products (H, and H 2 0 2 ) according to reaction 1 where the numbers in parentheses represent the G values for the individual species. In HzO
--
ea; (2.8), OH' ( 2 . Q H'(0.6), H2 (0.45), HZ02 (0.8) (1)
NzO-saturated (25 mM) solution e,, is converted to OH' according to reaction 2. In the presence of HCOz-(O.l M) reactions 3 and 4 take place; in acidic solutions, eaq- is converted to H' according to reaction 5. In the presence of tert-butyl alcohol (0.5 M) reaction 6 takes place, the reaction of H' with tert-butyl Thus, in a alcohol being very slow ( k < 1 X los M-I S-I).~O
- + + + - + + - + + H20
eaq- + N 2 0
k = 8.7
lo9 M-'
X
H'
HCOZ-
k =3 HCOZ-
k =3
X
X
X
CO2'-
HzO
lo9 M-l
s-l
eaq-
(3)
-
C02'-
(4)
(r ef 22)
H'
(5)
1O1OM-I s-l (ref 21)
+ (CH3)3COH k =5
(2)
(r ef 21)
s-I
H2
OH'
H+
OH'
OH-
lo8 M-Is-l (r ef 20)
X
k = 2.2
OH'
N2
HzO
+ 'CHZC(CH3)zOH
(6)
lo8 M-' s-I (r ef 22)
(17) Hutton, A.; Roffi, G.; Martelli, A. Quad. Area Ric. Emilia-Romagna 1974, 5, 67-74.
(12) Che, C.-M.; Atherton, S. J.; Butler, L. G.; Gray, H. B. J. Am. Chem. SOC.1984, 106, 5143-5145. (13) Che, C.-M.; Gray, H. 9.;Atherton, S. J.; Lee, W.-M. J. Phys. Chem. 1986, 90, 6747-6749. (14) Closs, G. L.; Calcaterra, L. T.; Green, N. J.; Penfield, K. W.; Miller, J. R. J. Phys. Chem. 1986, 90, 3673-3683. (15) Bignozzi, C. A.; Paradisi, C.; Roffia, S.; Scandola, F. Inorg. Chem. 1988, 27, 408-414. (16) Bignozzi, C. A,; Roffia, S.; Scandola, F. J. Am. Chem. SOC.1985, 107, 1644-1651.
(18) Mulazzani, Q.G.; D'Angelantonio, M.; Venturi, M.; Hoffman, M. Z.; Rodgers, M. A. J. J. Phys. Chem. 1986, 90, 5347-5352.
(19) Hindmarsh, A. C. "GEAR: Ordinary Differential Equation System Solver"; LLNL Report UCID 30001, Rev. 3, Dec 1974. (20) Anbar, M.; Farhataziz; Ross, A. 9. Natl. Stand. ReJ Data Ser. ( U S . Natl. Bur. Stand.) 1975, 51, 1-56. (21) Anbar, M.; Bambenek, M.; Ross, A. B. Natl. Stand. ReJ Data Ser. ( U S . Natl. Bur. Stand.) 1973, 43, 1-54. (22) Farhataziz; Ross, A. 9. Natl. Stand. ReJ Data Ser. ( U S . Natl. Bur. Stand.) 1977, 59, 1-113.
738 The Journal of Physical Chemistry, Vol. 93, No. 2, 1989
Mulazzani et al.
solution containing HCOF the major reactive species are eaq- and C02'- radicals; if the solution is also saturated with N 2 0 , the reactive species is the C02'- radical only. The latter is a strong reducing agent (E0[CO2/CO2'-] = -2.0 V).23 In acidic Arpurged solutions containing tert-butyl alcohol, the reactive species are H' (Eo[H+/H'] = -2.1 V)24 and 'CH2C(CH3)20H. The latter species (R') is relatively unreactive; it possesses, however, both reducing (Eo[R+/R'] = -0.1 V)24and oxidizing (Eo[R'/R-] = 0.6 V)24 abilities. Results and Discussion [3,2,3], [3',2,3], and [3',2,31 C02'- and eoq-. The rate constants for the reduction of the three Ru(III)Ru(II)Ru(III) complexes by C02'- radicals, reaction 7, were obtained by monitoring at selected wavelengths the spectral changes induced by the pulse irradiation (10-20 Gy per pulse) of N20-saturated solutions containing 0.1 M HC02- and 50-100 p M complex at natural pH; these rate constants are shown in Table I. It can be noted that the value of the rate constant associated with reaction 7 is virtually the same for the three complexes, independent of the redox potential of the couple involved.
+
Ru(III)Ru(II)Ru(III)
+ C02'-
-+
Ru(II)Ru(II)Ru(III)
+ C02
(7)
By pulsing repetitively the same sample, it was shown that the rate constant for the reduction of the second Ru(II1) center was about one half of that obtained for the reduction of the first one. For example, when a N20-saturated solution containing 100 p M [3,2,3] and 0.1 M HCO, was irradiated with pulses delivering a dose of ca. 20 Gy each, kM was seen to decrease from an initial value of 1.4 X lo6 to a value of 0.8 X lo6 s-l after eight pulses; assuming a G value of 6.2, the total dose adsorbed at that stage corresponded to the formation of ca. 1 equivalent of C02'- per equivalent of compound initially present. At the various wavelengths examined, the absorption level reached after completion of reaction 7 did not show any appreciable change over the longer available detection time. From the extent of the spectral changes and from the knowledge of the t values of the species involved, values of G = 7 f 1 were obtained for the first reduction step of the three Ru(III)Ru(II)Ru(III) species. Identical G values were obtained when the solutions were Ar-purged instead of being saturated with N 2 0 ; under these conditions the reduction of the three Ru(III)Ru(II)Ru(III) complexes occurred via reactions 7 and 8. For the latter reaction Ru(III)Ru(II)Ru(III)
+ eaq-
-
Ru(II)Ru(II)Ru(III)
(8)
a value of k8 -3 X 1O'O M-' s-', independent of the nature of the complex, WLS evaluated. In Figure 1 are shown the spectral changes induced by the continuous irradiation of a degassed solution containing 80 p M [3,2,3] and 0.1 M HC02- at pH 7. These changes were fully consistent with the reduction of [3,2,3] to [2,2,3] and that of [2,2,3] to [2,2,2] being the only processes that take place in the system. Similar spectral changes were obtained when [3',2,3] or [3',2,3'] were used in the place of [3,2,3]. In each case the deep blue band25 in the visible region decreased with increasing dose and disappeared when the system had received a dose corresponding to the generation of about 2 equiv of reducing species (eaq-and C02') per equivalent of compound initially present. An experiment performed using a N,O-saturated solution confirmed the pulse radiolytic results, described above, concerning the equivalence of e,; and C02'- radicals, as reducing agents, toward these complexes. As shown in Figure 1, no well-defined isosbestic points were observed during the irradiation. This can be accounted (23) Breitenkamp, M.; Henglein, A,; Lilie, J. Ber. Bunsen-Ges. Phys. Chem. 1916,80, 973-979. (24) Endicott, J. F.In Concepts of Inorganic Photochemistry; Adamson, A. W., Fleishauer, P. D., Eds.; Wile?: New York, 1975;p 88. (25)This band has been assigned to an intervalence-transfer (IT) transition from the Ru(1I) atom of the central bipyridine unit to one of the peripheral Ru(II1) atoms.
0 00 300
400
500
X,nm
600
700
800
,I
Figure 1. Spectral changes (vs air) induced by the continuous irradiation (dose rate = 7.0 Gy min-') of a degassed solution containing 80 p M (3,2,3] and 0.1 M HC02-at pH 7. Optical path = 1.0 cm. (0)Spectrum of unirradiated solution; the other spectra were obtained after 3 min of irradiation, with the arrows indicating the direction of the changes. The bottom line represents the spectrum of H 2 0 vs air.
for by considering that the reduction of the second Ru(II1) center starts before the reduction of the first one is complete; it has to be noted also that the redox potentials in Table I allow the species [3,2,3], [2,2,3], and [2,2,2] to coexist in solution. When the exhaustively irradiated solutions containing the Ru(II)Ru(II)Ru(II) species were equilibrated with air, spectral changes occurred which indicated that the oxidation of one peripheral Ru(I1) center (in the case of [2',2,2]) or both (in the case of [2,2,2]) was taking place. These reactions were completed in less than 20 min. By contrast, the oxidation of the peripheral Ru(I1) centers was not observed over a period of many hours when a solution containing [2',2,2'] was exposed to air. These observations are in agreement with the thermodynamic requirements for the reduction of O2 to H202 ( E O [ 0 2 + 2H+/H202] = 0.28 V at pH 7)26 by the various Ru(II)Ru(II)Ru(II) and Ru(I1)Ru(II)Ru(III) species (Table I). In fact, it can be readily shown that A E > 0 for the reduction of O2by [2,2,2], [2,2,3], and [2',2,2] and AE < 0 for the other cases. [3,2,3] H'. The reaction between [3,2,3] and H' radicals was investigated by pulse irradiating Ar-purged solutions containing 0.5 M tert-butyl alcohol and 50-100 p M compound at pH 2 (adjusted with H2SO4). From the spectral changes observed at 480 nm (formation of an absorption) and 650 nm (bleaching) the value of the rate constant associated with reaction 9 was determined; k9 = (4.0 f 0.5) X lo9 M-I 8. After completion
+
[3,2,3] + H'
-
[2,2,3] + H+
(9)
of this process, some very minor spectral changes were observed on longer time frames. In Figure 2 are shown the spectral changes induced by the continuous irradiation of a 45 pM solution of [3,2,3] containing 0.5 M tert-butyl alcohol at pH 2. Under these conditions, well-defined isosbestic points at 370, 412, 536, and 806 nm were maintained during the irradiation and the exposure of the irradiated solution (147 Gy total dose) to the atmosphere closely reproduced the spectrum prior to irradiation. These two facts indicated that the main process taking place in the system was the reduction of [3,2,3] to [2,2,3]; the reduction of the second Ru(II1) was not taking place under these conditions. The decrease of the visible band at 640 nm and the increase of the absorbance at 480 nm were not linear with the dose; Le., the G value calculated assuming the At values pertaining to the reduction of [3,2,3] to [2,2,3] decreased with increasing dose (Figure 2, inset). These (26)Calculated from E0[02+ 2H+/H202]= 0.695 V." (27) Hoare, J. P. In Standard Potentials in Aqueous Solution; Bard, A. J., Parsons, R., Jordan, J., Eds.; Marcel Dekker: New York, 1985;p 49.
Cyano-Bridged Polynuclear Ru Complexes
0 ’ 4 1 i
0.01
400
300
500
700
600
A ,nm
BOO
The Journal of Physical Chemistry, Vol. 93, No. 2, 1989 739
I
1
Figure 2. Spectral changes (vs air) induced by the continuous irradiation (dose rate = 7.0 Gy min-I) of a degassed solution containing 45 pM [3,2,3] and 0.5 M tert-butyl alcohol at pH 2. Optical path = 1.0 cm. (0) Spectrum of unirradiated solution;the other spectra were obtained after 3 min of irradiation, with the arrows indicating the direction of the changes. The broken line represents the spectrum of the irradiated solution after equilibration with air. Inset: G values, as a function of dose, for the reduction of [3,2,3] to [2,2,3] calculated assuming Ac = 2600 and 1800 M-’cm-l at 640 and 480 nm, respectively. The solid line represents the G values calculated by using the empirical equation described in the text.
G values fitted very nicely empirical eq A, where D represents the dose. From this equation, G values of 5.5 and 0.6 were
G = 0.6 + 1/(0.205
+ 8.3 X 10-3D)
(A)
obtained from the extrapolation of the dose to zero and to m, respectively. The results obtained with this system can be accounted for in terms of a competition between the reduction of [3,2,3] by H’ radicals, reaction 9, and the oxidation of [2,2,3] by the tert-butyl alcohol radical, reaction 10. The zero-dose value [2,2,3]
+ ‘CH2C(CH3)20H
-
[3,2,3]
+ products
(10)
of G ( 5 . 5 ) approaches the sum of the G values for H’ radicals and tert-butyl alcohol radicals (3.4 and 2.8, respectively) indicating that, in the absence of [2,2,3], the tert-butyl alcohol radical acts as a reducing agent toward [3,2,3], reaction 11. This dual [3,2,3]
-
+ ‘CH2C(CH3)20H
[2,2,3]
+ products
(11)
behavior of the tert-butyl alcohol radical is not in contrast with its redox pr0perties.2~ The limiting G value of 0.6 probably reflects the difference between the G values for H’ and tert-butyl alcohol radicals, indicating that under steady-state conditions the system is described solely by reactions 9 and 10. Reaction 9 represents the stoichiometry of the process by which H’ radicals reduce one peripheral Ru(II1) center but does not represent the mechanism of the process. In fact, H’ radicals react rapidly ( k = 9.5 X lo9 M-I s-1)28 with R ~ ( b p y ) ~giving ~ + an H adduct to coordinated bipyridine and slowly ( k = 1.8 X lo6 M-’ s - ’ ) ~with ~ R U ( N H ~ ) giving ~ ~ + RU(”~)~’+. The product of reaction 9 is also likely to be an H adduct to coordinated bipyridine; the dynamics by which this species converts to [2,2,3] is unknown inasmuch as no significant changes in absorbance were detected after completion of process 9. This could indicate that the electron transfer from the coordinated bipyridine radical and one peripheral Ru(II1) center is a fast process. A good fitting of the results shown in the inset of Figure 2 was also obtained by computer simulation of the mechanism represented by reaction 9-1 1 . According to this calculation the ratio between k l o and k l l is ca. 20. An upper limit for k l l of 1 X lo6 M-’ s-l could be evaluated taking into consideration the fact that, (28) Baxendale, J. H.;Fiti, M. J . Chem. SOC.,Dalton Trans. 1972, 1995-1998.. (29) Navon, G.;Meyerstein, D. J . Phys. Chem. 1970, 74, 4067-4070.
Figure 3. Values of kobd for the reduction of [3’,2,3] by C02’- as a function of substrate concentration. The solid lines (a-d) represent the values of koM obtained from the computer modeling of the system, assuming k,,,, = 5 X lo6, 1 X lo7, 5 X lo7, and 1 1 X lo* s-I, respectively. SCHEME I C3’,iS21 + COz C3‘,2.33
+
COz-’
kontr.
kb
C2’,2,33
+
CO2
under the pulse radiolysis conditions employed, reaction 11 was not observed; the latter simulation was performed assuming k = 7 X lo8 M-’ s-I (r ef 30) for the reaction of the tert-butyl alcohol radical with itself. Intramolecular Electron Transfer [3’,2,2] [2’,2,3]. Inasmuch as the rate constants for the reduction of [3,2,3] and [3’,2,3’) are virtually identical (Table I), the reduction of [3’,2,3] by CO2*radicals is expected to generate [2’,2,3] and [3’,2,2] in a ratio close to l.31 In comparison to the value for the [3’,2,3’]/[2’,2,3’] couple, that for the [3,2,3]/[2,2,3] couple is shifted toward more negative potential by 235 mV (from +0.18 to -0.055, Table I). A similar difference presumably exists between Ell2 for the [3’,2,3]/[2’,2,3] couple and that for the [3’,2,3]/[3’,2,2] couple; due to this difference, at a certain stage, the species [3’,2,2] must convert into the [2’,2,3] species, reaction 12. In order to evaluate [3’,2,21 [2’,2,31 (12) k m
-
-
the rate constant associated with reaction 12 (kintra),the spectral changes at 400 nm which followed the pulse irradiation of N20saturated solutions containing [3’,2,3] and 0.1 M HCO, were examined as a function of the initial concentration of [3’,2,3]. At 400 nm the E values for [3’,2,3] and [2’,2,3] are 6200 and 13 500 M-’ cm-I, respectively. The value at 400 nm of [3’,2,2] is unknown but is expected to be similar to that of [3,2,2] (e = 6400 M-’ cm-I). On these assumptions, the electron-transfer process would be a detectable one provided that the formation of [3’,2,2], together with that of [2’,2,3], is fast, Le., does not represent the rate-determining step. The main features of the experimental results obtained were the following: (i) the radiation-induced formation of absorbance at 400 nm could always be fitted by a single exponential; (ii) as (30) Ross, A. B.; Neta, P. Natl. Stand. Ref. Data Ser. ( U S . Natl. Bur. Stand.) 1982, 70, 1-96. (31) By considering that Cot’-radicals react slowly with Ru(bpy)32+(ref 32), their reaction with the Ru(bpy)z2+moiety was ignored. (32) Mulazzani, Q.G.;Venturi, M., unpublished observations.
740
J . Phys. Chem. 1989, 93,140-149
shown in Figure 3, the value of kOw obtained from the formation curves was seen to increase linearly with the initial concentration of [3’,2,3] up to the maximum concentration used (6.4 X lo4 M). The absorption of the parent compound in the 400-nm region prevented the use of more concentrated solutions. In Figure 3 are also shown the values of kobd which were obtained from a computer simulation of the reaction mechanism represented by Scheme I. For these calculations, the value of the rate constant for the reduction of [3’,2,3] was that derived from the experiments shown in Figure 3 ( k = 1.5 X lolo M-I s-l); the curves describing the formation of [2’,2,3], via formation of [3‘,2,2], were always approximated to a single-exponential behavior. The comparison between the values sf kobd obtained experimentally and those obtained from the computer simulation of the
system as described above, indicates that kintramust have a value of ca. 5 X lo7 s-I or higher. This estimate is in agreement with calculations madeI5 using the Hush model and the available spectroscopic information which give a value of ca. lo9 s-l. Acknowledgment. We thank Dr. Nadia Camaioni for solving some software problems and G. Gubellini and A. Monti for technical assistance. This work was partially supported by Minister0 della Pubblica Istruzione of Italy (Quota 40%). Registry NO. [3,2,3](PF6),, 94499-25-9; [3’,2,3](PF&, 1 1 1557-16-5; [3’,2,3’](PF6)6, 1 1 1557-14-3; [2,2,3], 94499-28-2; [2,2,2], 94499-29-3; [2’,2,2], 1 1 1557-22-3; [2’,2,2’], 1 1 1557-25-6; [2’,2,3], 1 1 1557-21-2; [2’,2,3’], 1 1 1557-24-5; H C 0 2 N a , 141-53-7; (CH,)3COH, 75-65-0; H’, 12385-13-6; COz’-, 14485-07-5.
Chemlcal Vortex Dynamics in the Belousov-Zhabotlnsky Reaction and in the Two-Variable Oregonator Model W. Jahnke,+ W. E. Skaggs,t and A. T. Winfree* Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, Arizona 85721 (Received: May 25, 1988)
Like the Belousov-Zhabotinsky reaction, the two-variable Oregonator model supplemented by molecular diffusion supports rotating vortices of chemical activity in two dimensions (“rotors”) and in three dimensions (”scroll rings”). The dynamics of both kinds of organizing center are found computationally to include a regularly patterned “meander” in ways that depend on chemical recipe parameters. A search for comparable regularities in the laboratory was rewarded after detection and removal of a number of experimental artifacts.
Introduction Excitable media tend to organize themselves into periodic wave trains emanating from vortexlike ”rotors”. In three dimensions the vortex is typically a closed ring that moves under its own dynamical rules. Though predicted’ and observed2 many years ago, such vortex rings have not yet been subjected to quantitative laboratory investigation in any chemically excitable medium. Attempts to do so with the Belousov-Zhabotinsky reagent are still frustrated by technical difficulties, but it should at least be possible to preview such experiments c~mputationally~ and perhaps to sidestep some avoidable laboratory difficulties by forethought in the choice of recipe parameters. Accordingly, we undertook computational experiments4 using a qualitatively acceptable caricature of the chemical reactions involved in the BelousovZhabotinsky reaction. There are now several such models, none clearly superior in all respects to the others, and they are hard to compare quantitatively. Because some of the important parameters remain poorly measured, computation from such reaction-diffusion equations still remains to some degree an exercise in curve fitting. For present purposes we chose the earliest, simplest, and most studied model: the two-variable version of the Oregonator with Tyson’s “Lo” parameters (ref 5 ; fitted from diverse “room temperature” data). Any other kinetics could just as easily be plugged into our computational machinery: when the appropriate equations are finally agreed upon and parameters fmed at widely accepted values, these rough trial computations can be rerun to check the models by quantitative comparison with experiment. The two-variable Oregonator has solutions that represent propagation of a steep wave front by interplay of HBrO, diffusion Permanent address: Institut fuer Physikalischeund Theoretische Chemie, Universitaet Tuebingen, 7400 Tucbingen, Federal Republic of Germany. *Permanent address: Department of Psychology, University of Colorado, Boulder, CO 80309.
with an autocatalytic reaction that quickly generates HBrOz (using Br- as an intermediary species that remains always in equilibrium with local instantaneous HBr02). As in an analogous classical problem in evolutionary genetic^,^,^ the square of the propagation speed should be proportional to the product of the HBr02 diffusion coefficient, the rate coefficient of its autocatalysis, the [H+], and the [BrO,-] : this predicted relationship was confirmed experimentally in one version of the reaction, but with a rate coefficient whose value seems uncertain by a factor of 2.5 Behind the front the medium is refractory to immediate reexcitation; the recovery process is mediated by the metal ion catalyst. Our present purpose is a qualitative exploration of rotor behavior, both numerically and in the laboratory. The Oregonator has too many well-known deficiencies to make quantitative results very useful. It predicts a much weaker dependence of rotor period on [H’] than is o b ~ e r v e d . ~It does not predict phase advances observed after a pulse of Br-.8 It does not adequately distinguish the roles of malonic acid and of bromomalonic acid, which seem (1) Winfree, A. T. Sci. Am. 1974, 230(6), 82. (2) (a) Winfree, A. T. Faraday Symp. Chem. Soc. 1975,9,38. (b) Welsh, B.;Gomatam, J.; Burgess, A. Nature (London) 1983, 304, 611. (3) (a) Panfilov, A. V.; Pertsov, A. M. Dokl. Akad. Nauk. USSR 1984, 274, 1500 (in Russian). (b) Panfilov, A. V.; Rudenko, A. N . Physica D (Amsterdam) 1987, 28D, 215. (c) Winfree, A. T.; Winfree, E. M.; Seifert, H. Physica D (Amsterdam) 1985, 170, 109. (4) (a) Winfree, A. T.; Jahnke, W.; Skaggs,W., manuscript in preparation. (b) Jahnke, W.; Henze, C.; Winfree, A. T. Nature, in press. ( c ) Winfree, A. T.; Jahnke, W. J . Chem. Phys., in press. (5) Tyson, J. J. In Oscillations and Traveling Wwes in Chemical System; Field, R. J., Burger, M., Eds.; Wiley: New York, 1985; p 93. (6) (a) Fisher, R. A. Ann. Eugen. 1937, 7, 355. (b) Showalter, K.; Tyson, J. J. J . Chem. Educ. 1987, 64,142. (c) Murray, J. D. J . Theor. Biol. 1976, 56, 329. (7) (a) Smoes, M. L. J . Chem. Phys. 1979,71,4669. (b) Edelson, D. Int. J . Chem. Kinet. 1981, 13, 1175. (8) (a) Ruoff, P. J . Phys. Chem. 1984, 88, 2851. (b) Dolnik, M.; Schreiber, I.; Marek, M. Phys. Lett. A 1984, IOOA, 316. ( c ) Dolnik, M.; Schreiber, I.; Marek, M. Physica D (Amsrerdam) 1986, 2JD, 78.
0022-3654/89/2093-0740$01 .50/0 0 1989 American Chemical Society
