J. Phys. Chem. 1988, 92, 163-166 simple conceptual separation between diffusive and reactive steps becomes somewhat arbitrary. In principle, a statistical model in which the reaction probability is suitably averaged over all the possible interreactant configurations (involving different distance and orientation), each with its own electronic and nuclear kinetic factors, would be better suited to describe such a situation. In practice, however, such a model would be hopelessly complicated for any real system. The approach used in this work maintains the separations between reactive and diffusive steps. In this approach, all the possible intimate aspects of the reaction (e.g., preferential interactions, orientational effects, intercompenetration of reactants), together with their energetic and kinetic consequences, are buried into the behavior of the precursor complex. The nuclear and electronic factors of the unimolecular reactive step are thus taken to represent configurationally averaged properties of the precursor complex. In this framework, the diffusional step is considered as a process leading to an orientationally averaged, nonspecific contact between the reactants. This admittedly simplistic approach has the advantage of being useful as a conceptual tool and does not seem to meet with any serious quantitative problem in the interpretation of the experimental data.
Conclusions The results obtained in this study can be summarized as follows.
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(1) In the nearly diffusion controlled electron-transfer reactions between cations investigated, two major deviations from ideal salt effects are observed: (i) specific counterion effects and (ii) the Olson-Simonson effect. Both effects are such that substantial errors could be made by comparing rate constants of reactions obtained, at constant ionic strength, with different "inert" electrolytes. (2) The observed salt effects can be parametrically accounted for with good accuracy in terms of the ionic strength dependence of the diffusional rate constants as predicted by the Debye-Huckel model, provided that (i) uni-univalent background electrolytes are considered and (ii) numerical integration over the interreactant distance of the full Debye-Eigen expressions (eq 3 and 10) is used. If multivalent cations are present in the electrolyte, the model can still be used to calculate the diffusional parameters, by using the anion concentration instead of the ionic strength. On the other hand, the use of approximate expressions for the diffusional parameters (eq 6, 12 or 9, 13) may generally lead to substantial errors in the calculated values.
Acknowledgment. We thank Professors L. Moggi and A. Indelli for the benefit of many helpful discussions. Registry No. Ru(bpy),2+, 15158-62-0; MV2+, 4685-14-7; Ru(NH&py3+, 33291-25-7; NaC1, 7647-14-5; NaCIO,, 7601-89-0; CaC12, 10043-52-4.
Measurement of Dispersion Relation of Chemical Waves in an Oscillatory Reacting Mediumt A. Pagola,$J. ROSS,* Department of Chemistry, Stanford University, Stanford, California 94305
and C. Vidal Centre de Recherche Paul Pascal, Universite de Bordeaux I , 33405 Talence cedex, France (Received: May 19, 1987)
We report measurements on the dispersion relation for chemical trigger waves propagating in an oscillatory Belousov-Zhabotinsky reacting medium. The waves are induced by a temperature perturbation (laser heating). The results are in qualitative agreement with a theory of such waves in an excitable medium.
Introduction Nonlinear reactions with a sufficiently complex reaction mechanism, maintained far from equilibrium, can transmit and sustain chemical waves or fronts that are traveling chemical concentration There have been many visual observations of chemical waves, including kinematic waves,5 relaxation oscillation waves,6-8 and phase waves.9 The techniques of absorptionlo and transmission, coupled with imaging techniques,"J2 are being used to determine the velocity, profiles of fronts, invariance of structure in relaxation oscillation waves, and the formation of stationary spatial structures. Little is known on the subject of dispersion relations, that is, the dependence of the velocity of a chemical wave on the period of that wave. Keener and TysonI3 have presented analytical results on trigger waves for a simplified Oregonator model. They show that curvature effects, coupled with the dispersion relation, can 'This work has been supported in part by the National Science Foundation, the Air Force Office of Scientific Research, and the French Centre National de la Recherche Scientifique. *Permanent address: Centre de Recherche Paul Pascal, Universite de Bordeaux I, 33405 Talence cedex, France.
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provide a fairly good description of wave propagation in a twodimensional excitable or oscillatory medium. The only experimental results reported are in ref 13 (unpublished work by A. T. Winfree) and in ref 18. In this article we report experimental ( I ) Eyring, H.; Henderson, D. Theoretical Chemistry; Academic: New York, 1978; Vol. 4. (2) Hanusse, P.; Ortoleva, P.; Ross, J. Ado. Chem. Phys. 1978, 38, 317. (3) Field, R.; Burger, M. Oscillations and Traveling Waves in Chemical Systems; Wiley: New York, 1985. (4) Vidal, C.; Hanusse, P. Inr. Reo. Phys. Chem. 1986, 5 , I . Vidal, C.; Pacault, A. In Evolution of Order and Chaos; Haken, H., Ed.; SpringerVerlag: Heidelberg, 1982. ( 5 ) Kopell, N.; Howard, L. N . Science (Washington, D.C.) 1973, 180, 1172. (6) Field, R. J.; Noyes, R. M. J . A m . Chem. SOC.1974, 96, 2001. (7) Showalter, K.; Noyes, R. M.; Turner, H. J . Am. Chem. SOC.1979, 101, 7463. (8) Sevcikova, H.; Marek, M. Physica D (Amsterdam) 1984, 1 3 0 , 379. (9) Bodet, J. M.; Ross, J.; Vidal, C. J . Chem. Phys. 1987, 86, 4418. (IO) Wood, P. M.; Ross, J. J . Chem. Phys. 1985, 82, 1924. (11) Muller, C.; Plesser, T.; Hess, B. Science (Washington, D.C.)1985, 230, 66 1. (12) Pagola, A.; Vidal, C. J . Phys. Chem. 1987, 91, 503. (13) Keener, P.; Tyson, J. Physica D (Amsterdam) 1986, ZID, 307.
0 1988 American Chemical Society
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The Journal of Physical Cheniistrj', Voi. 92, N o . I , 1988
Pagola et al.
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1.50 Figure 1. Schematic drawing of the experimental configuration: ( 1 ) perturbation laser beam, wavelength = 488 nm; (2) convergent lens, focal length = 50 mm; (3) Petri dish cover which touches the reacting solution; (4) reacting solution of depth 1 mm; ( 5 ) Petri dish, diameter = 93.8 mm: (6) irradiation point (diode 512); (7) area of measurement, beginning at 2.5 mm (diode 41 2 ) and ending at 12.8 m m from the irradiation point (diode 1). The expanded laser beam used for absorption meawrements is not shown.
data on the dispersion relation in the Belousov-Zhabotinsky reaction, under conditions where the reacting solution has relaxation oscillations.
Experimental Section W e prepared the Belousov-Zhabotinsky reagent with initial concentrations of [H2SO4], = 0.2 M, [CH2(COOH)2],= 0.08 M, [NaBr03], = 0.31 M, and [ferroin], = 2 X M; with these conditions the reaction is oscillatory (after an induction period) with a period To = 110 s. The period remains fairly constant for a t least 15 min. The apparatus is the same as that used in ref 9 and is very similar to that used earlier in ref 10. It allows us to perform spatially resolved absorption measurements on the ferroin wave profile of the Belousov-Zhabotinsky reaction. W e used an argon ion laser with a wavelength of 488 nm. The beam is expanded and then sent through the sample Petri dish, normal to the surface of the thin layer of the reacting solution: then the transmitted light is measured in time and space with a one-dimensional reticon system with 1024 photosensitive diodes and spatial resolution of 25 microns. Each measurement, that is. the recording of a complete scan of 1024 photodiodes, takes 0.14 s, and for technical reasons we chose a time interval of 1.53 s between two successive scans. The solution is covered (see Figure 1) to prevent evaporation and possible consequent hydrodynamic effects. In order to change locally the period of oscillation, a second laser beam is focused on the oscillatory reaction in the Petri dish. The irradiated region is approximately 4 mm2. A part of the incoming light is absorbed and turned into heat. This raises the temperature of the solution a t the point of irradiation: that temperature rise constitutes our imposed perturbation. In this experiment the solution is irradiated for the entire interval of the experiment, which is normally 3 min. In that interval the temperature rise for a power input of 100 pW is about 4 O C (estimated value). Measurements and Results Measurements of the Effect o f t h e Perturbation. In Figure 1 we show a schematic drawing of the laser perturbation and also of the region of measurement. The light impinges on the reticon a t a location centered on the photodiode numbered 5 12, and the diameter of the laser beam a t the reticon is 1.125 mm. The period of the unperturbed solution, To, is measured away from the region of perturbation, a t photodiode 1, and as a check a t other photo-
0.75 0
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Time (min)
Figure 2. (a) Plot of transmitted light received a t photodiode 1 (Figure 1) as a function of time in minutes. The difference in time between the peaks is the period of oscillation of the unperturbed oscillatory reaction ( T o= 1 IO s ) . (b) Measurement of transmitted light a t photodiode 412 (Figure 1 ) as a function of time. The first, second. third, and fourth fronts emitted from the laser irradiation location are marked by A, B, C, and D, respectively. The difference in time between two peaks is the period of the emitted aave.
diodes similarly placed far from the origin (see Figure 2a). As the light is absorbed by the solution the temperature rises and, due to that perturbation,'' one or more trigger waves are generatedi5at the point of the perturbation and propagate outward. W e measure the front of the propagating wave in the solution at the ambient temperature (0 = 22 f 0.5 "C) sufficiently far from the perturbed region, such that, at the time of measurement, heat conducted away from the perturbed region has not yet affected the temperature at this location of measurement (as will be shown later on). We show such measurements at laser diode 412 (Figure 1) in Figure 2b. The peak at approximately 2 min is the first front emitted from the perturbed region and detected at photodiode 41 2; the second peak arrived at 2.75 min, the third one at 3.51 min, and the fourth one a t 4.32 min. W e have made measurements as shown in Figure 2b for laser powers ranging from 30 to 130 p W . As the power is increased, so is the temperature difference between the perturbed and unperturbed regions, and as a consequence, the period of wave emission is decreased. '4 plot of the variation of the period of the emitted wave (the time interval between maxima in the waves observed a t a given location) and its dependence on the power of the laser irradiation is shown in Figure 3. Measurements of the Velocity of Fronts and Dispersion Relation. The velocity of each front emitted from the center is determined by measuring the position of the maximum in amplitude of the transmitted signal as a function of time. A plot of the time of measurement of a given maximum of amplitude against its location is shown in Figure 4. The slope of the curves between 2.5 and 7.8 mm gives the reciprocal of the velocity. .4t approximately 7.8 mm the oscillation of the homogeneous (un(14) Thoenes, D. LVarure (London),P h j s . Sci. 1973, 243, 18. ( 1 5) For a discussion of relaxation oscillation waves, kinematic waves, and phase waves, see ref 8 and Ortoleva, P.; Ross, J. J Chem. Phys. 1975, 63, 3398.
Chemical Waves in an Oscillatory Reacting Medium
The Journal of Physical Chemistry, Vol. 92, No. I, 1988 165
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Figure 6. Dispersion relation. Plot of velocity of a given front (maximum in amplitude) versus period of successive emitted fronts: A, first front; 0, second front; +, third front; X , fourth front; 0,fifth front. The horizontal dashed line represents the mean value of the propagating velocity of the first front = 0.0903 mm s-l); it corresponds to the maximum value to be expected in the unperturbed medium.
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Figure 7. Comparison with calculated dispersion relation (curve) and Winfree's data (bars) taken from ref 13. Two sets of scaling factors have been used to plot our measurements on this dimensionless graph. +: K , A = 5 s-l; K 5 B = 0.05 s-l; D = 1.5 X cm2K1. 0: K 3 A = 2 s d ; K 5 B = 0.09 sd; D = 1.5 X cm2d.
(16) Kuhnert, L.; Krug, M. J.; Pohlmann, L. J . Phys. Chem. 1985, 89, 2022.
constant. The velocity varies with the number of the emitted front; t h e first front, which by definition has no other front preceding it, has nearly constant velocity as t h e period of the emitted wave is varied, since the first front always proceeds through t h e unperturbed medium. The velocity of the second front is smaller than that of the first, and the difference of velocity increases with decreasing period of the wave. The velocities of subsequent fronts decrease still further. We ascribe these variations in velocity with front number and period to incomplete relaxation of the reacting
J. Phys. Chem. 1988, 92, 166-171
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solution behind each front. The extent of incompleteness of relaxation increases with increasing difference between the period of the wave and the period of the unperturbed oscillation of the reacting solution. Hence, the decrease in velocity is predicted to become larger with decreasing period of the wave, as is in fact observed.
Comparison of Theory and Experiments Keener and TysonI3 calculated a dispersion curve for a simplified Oregonator model of the Belousov-Zhabotinsky reaction and values of parameters for which the solution is an excitable medium rather than an oscillatory one, as in our experiments."-I9 Their results and the experimental values of A. T. Winfree (unpublished by Winfree) adjusted by Keener and Tyson are shown in Figure 7 taken from ref 13. Furthermore, we have plotted on (17) Let us also refer to two other articles. Tatterson and Hudson'* filled a long capillary tube with an oscillating Belousov-Zhabotinsky reaction. The tube was connected to a stirred tank reactor which served as a pacemaker. With pairs of electrodes, they measured wavelength, period of emission, and wave velocity of oxidation fronts propagating through the tube. By changing the stirring rate, they induced a change of the period of the oscillating reaction in the reactor and initiated a front which, then, propagated through the tube. In this one-dimensional geometry, they suprisingly found a small rise of the velocity of the front with higher excitation frequencies. This effect is exactly opposite to our observation. Using a fairly similar one-dimensional device, Sevcikova and MarekI9 have also reported data which allow drawing at least a qualitative comparison with our measurements. In their article, Figure 4b shows the dependence of the velocity on the pulse number, at various excitation frequencies (notedfl. For successive fronts beyond the third one, this velocity decreases respectively to 87% (,f= 0.6 min-') and to 75% cf= 0.75 min-') of its value at an excitation frequencyf = 0.1 1 min-l. Therefore, the trend is this time the same as we report here. (18) Tatterson, D. F.; Hudson, J. L. Chem. Eng. Commun. 1973, 1 , 3. (19) Sevcikova, H.; Marek, M. Physica D (Amsferdam)1983, 9D, 140.
that figure our measurements of the lowest observed speed at each frequency of emission of waves. Indeed, this lowest value is our best estimate of the asymptotic value of the speed of a wave coming after a large number of other waves. Thus the relaxation of the medium has approached its asymptotic value (see the end of the previous section), while the time elapsed since the start of the experiment (3 min a t most) is not yet enough to alter the oscillatory characteristics of the medium. In order to allow for comparison, our measurements must be scaled to the results of the theory. Using the same scaling factors (K3A = S s-I: K5B = 0.05 s-'; D = 1 .S X lo-, cm2,s-') as Keener and Tyson, we obtain the signs appearing in Figure 7 . Though there is no obvious discrepancy between experimental values and the calculated curve, no real fit is observed in that case. However, the values of the rate constants chosen by Keener and Tyson are solely to be considered as orders of magnitude. It thus seems much more appropriate to scale our results so that the speed of the waves in the unperturbed medium (dashed line in Figure 6) corresponds to the comparable calculated velocity (plateau of the upper branch of the curve). If we take, for instance, K,A = 2 s-' and K,B = 0.09 s-'-that is, K3 26[H+] M-*.s-' and K, 1.1 M-'.s-', values which are plausible as well-we are led to the open circles shown in Figure 7. With these scaling factors, the agreement between experiment and theory becomes excellent. Nevertheless, as long as a deeper justification of the values of the scaling parameters is not provided, the agreement must still be considered as mainly qualitative.
"+"
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Acknowledgment. We thank Dr. John J. Tyson for a clarifying discussion. Registry No. CH2(COOH)2,141-82-2; Br03-, 15541-45-4; ferroin, 14708-99-7.
Electronic-State Dependence of Intramolecular Proton Transfer of o -Hydroxybenzaldehyde Shin-ichi Nagaoka,* Umpei Nagashima, Institute for Molecular Science, Okazaki 444, Japan
Nobuhiro Ohta, Masahisa Fujita, and Takeshi Takemura Division of Chemistry, Research Institute of Applied Electricity, Hokkaido University, Sapporo 060, Japan (Received: May 19, 1987)
The electronic-state dependence of the intramolecular proton transfer of o-hydroxybenzaldehydein the vapor phase has been studied by means of emission spectroscopy. High-resolution fluorescence s tra were obtained by excitation at several bands of the absorption from the ground state to the second excited ' ( x , n * )( S E state in the vapor phase at room temperature. The decay rates at several vibronic levels of Sp) were evaluated from the analysis of line width of the individual vibronic bands in the fluorescence-excitation spectrum in a supersonic free jet. It is considered that the intramolecular proton transfer does not play an important role in the decay process from the Sp' state contrary to the case of the first excited '(?r,a*) (Sp)) state. The observed electronic-state dependence of the intramolecular proton transfer of o-hydroxybenzaldehyde is consistent with the explanation that in contrast to the case of the So and Sp) states the enol tautomer is stabilized in the SF) state owing to the character of the wave function.
Introduction The proton transfer in the excited states of intramolecularly hydrogen-bonded molecules is a topic of current interest.'-*O It ( 1 ) Nagaoka, S.;Hirota, N.; Sumitani, M.; Yoshihara, K. J . Am. Chem. 1983. .~ _ _ , 105. 4220. (2) Nagaoka, S.; Hirota, N.; Sumitani, M.; Yoshihara, K.; LipcznskaKochany, E.; Iwamura, H. J . Am. Chem. SOC.1984, 106, 6913. (3) Nagaoka, S.; Fujita, M.; Takemura, T.; Baba, H. Chem. Phys. Lett. 1986, 123, 489.
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is a chemically very simple process and would be of great interest in view of the possibilities of making a proton-transfer laser2' and (4) References cited in ref 1-3. (5) Ernsting, N. P. J. Am. Chem. SOC.1985, 107,4564. (6) Bouman, T. D.; Knobeloch, M. A,; Bohan, S. J. Phys. Chem. 1985,89, 4461-1 .
(7) Flom, S. R.; Barbara, P. F. J. Phys. Chem. 1985, 89, 4489. (8) Ernsting, N. P. J . Phys. Chem. 1985, 89, 4932. (9) Sinha, H. K.; Dogra, S. K. Chem. Phys. 1986, 102, 337. (10) Chou, P.; Aartsma, T . J. J . Phys. Chem. 1986, 90, 721.
0 1988 American Chemical Society
