8904
J . Phys. Chem. 1992, 96, 8904-8909
8, is defined as the time necessary to wait before observing the state of n particles in any v after n has been observed previously in any u; 7 is the observation interval. For all calculations in this paper, 7 is taken as 1 s and the total volume, V,is 6 mL. The recurrence times for fluctuations of [HBrO,] with varying size of perturbation area are calculated with the above formula. Their magnitudes determine the likehood of observing a trigger wave which has been induced by an internal fluctuation in the time the experiment is carried out. Tables VI and VI1 summarize the results of the calculations for the perturbations in [HBrO,]. In both tables, Y is the average number of HBr02 moleculescontained in a sphere of a given radius in the initial profile; n is the number of HBr0, molecules which must be contained in that volume in order for a trigger wave to propagate, as determined by the calculations of the reaction-diffusion equations presented in section 11. The size of the fluctuation is the number of HBr02molecules (the difference between n and v) necessary for trigger wave generations, and 6 is the mean fluctuation for the given average number of HBr02 molecules within the considered volume. 8, is the recurrence time for the critical fluctuation. By comparing the magnitude of critical fluctuations and the average thermal fluctuations in HBr02, we see that thermal fluctuations are orders of magnitude too small for initiation of a trigger wave; further, calculated recurrence times are many orders of magnitude larger than the observation times of traveling wave experiments. The critical perturbations predicted by the deterministic equations necessary for the initiation of a trigger wave in an oscillatory chemical system have vanishingly small probabilities of occurring spontaneously in solution. Calculations have also been carried out by solving three variable reaction-diffusion equations from eqs 1-3 with perturbations in [Br-] in the initial profiles. The results are very similar to the results we discussed above for perturbations in HBrO,. The minimum critical radius to initiate a trigger wave is even larger than the values in Tables IV and V. V. Conclusions We have used numerical solutions of the deterministic reaction-diffusion equations and equilibrium fluctuation theory to investigate the probability of trigger wave generation in an oscillatory Belousov-Zhabotinskii system. Critical perturbations
in the phase of the oscillations, in [HBrO,] and in [Br-1, as well as critical radii of perturbation area are obtained by solving the reaction-diffusion equations in one dimension of a modified Oregonator model. Phase-diffusion waves are generated when weak perturbations are applied to the system in a given area of space. Estimates of the critical radii may change with the change of the dimension of the system but not significantly. Equilibrium stochastic calculations show that the probability of a thermal fluctuation initiating a trigger wave through internal thermal fluctuations is vanishingly small. Therefore, we conclude that the spontaneous traveling waves observed experimentally in oscillatory BZ systems are most likely initiated by a heterogeneous mechanism. Acknowledgment. We thank Igor Schreiber and Eugenia Mori for helpful discussions. This work was supported in part by the National Science Foundation and the Air Force Office of Scientific Research. References and Notes Mori, E.; Ross, J. J . Phys. Chem. 1992, 96, 8053. Bodet, J. M.; Ross,J. J . Chem. Phys. 1987, 86, 4418. Ross, J.; Muller, S. C.; Vidal, C. Science 1988, 240, 460. Vidal, C.; Pacault, A. In Evolution of Order and Chaos; Haken, H., Ed.; Springer-Verlag: Heidelberg, 1982. (5) Sadoun-Goupil, M.; De Kepper, P.; Pacault, A,; Vidal, C. Acta Chim. (1) (2) (3) (4)
1982, 7, 37. (6) Kopell, N.; Howard, L. N. Science 1973, 180, 1171. (7) Winfree, A. Faraday Symp. Chem. SOC.1974, No. 9, 38. (8) Ortoleva, P.; Ross, J. J . Chem. Phys. 1974, 60, 5090. (9) Beck, M. T.; Varadi, 2.B. Nature (Phys. Sci.) 1972, 235, 15. (10) Tyson, J. J. J. Chim. Phys. 1987, 84, 1359. (1 1) Walgraef, D.; Dewel, G.; Borckmans, P. J . Chem. Phys. 1983, 78, 3043. (12) Orteleva, P.; Ross, J. J. Chem. Phys. 1973, 58, 5673. (13) Vidal, C.; Pagola, A. J. Phys. Chem. 1989, 93, 271 1. (14) Nicolis, G.; Prigigine, I. Self-organization in Nonequilibrium Systems; Wiley: New York, 1977. (15) Showalter, K.; Noyes, R. M.; Turner, H. J . Am. Chem. SOC.1979, 101, 7463. (16) Foester, P.; Muller, S. C.; Hess, B. Proc. Natl. Acad. Sci. U.S.A. 1989,86, 683 1. (17) Vidal, C.; Pagola, A.; Bodet, J. M.; Hanusse, P.; Bastardie, E. J . Phys. 1986, 47, 1999. (18) Mori, E.; Schreiber, I.; Ross, J. J . Phys. Chem. 1991, 95, 9359. (19) NAG. Fortran Library, Mark 13. Vol. 2, 1988. (20) Reusser, E. J.; Field, R. J. J. Am. Chem. SOC.1979, 101, 1063. (21) Wood, P. M.; Ross, J. J. Chem. Phys. 1985, 82, 1924. (22) Tyson, J. J.; Keener, J. P. Phys. D 1988, 32, 327.
Cooperative Effects of Scavengers on the Scavenged Yield of the Hydrated Electron Simon M. Pimblott* and Jay A. LaVerne Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 (Received: April 2, 1992; In Final Form: July 23, 1992)
An analytic description of the cooperative effects of scavengers on the radiation chemistry of the hydrated electron in aqueous solution is developed by using the available experimental data and the results of deterministic diffusion-kinetic calculations. The formulation has two components: the first corresponds to the effect of the primary (electron) scavenger in isolation and the second represents the suppression of the eaq- OH spur reaction by the secondary (hydroxyl radical) scavenger. The two aqueous systems that have been considered in detail are nitrate/formate and nitrous oxide/Zpropanol. The agreement between the analytic treatment, the experimental data, and the kinetic modeling is good.
+
1. Introduction
A variety of different experimental techniques have been used to probe the factors influencing the radiation chemistry of aqueous so1utions.I One area of extensive effort has been the examination of the effect of solute concentration on the yield of an observed product.2d These scavenger experiments can give the yields of the different radiation-induced species,’,8 and in addition they can 0022-36S4/92/2096-8904$03.00/0
provide direct information that helps to elucidate the details of the radiation-hemical reaction scheme? It is frequently desirable to be able to estimate the yield in a particular scavenger experiment. To this end the concentration dependence of measured yields are usually fitted to one of several simple empirical equations so as to allow for facile e x t r a p o l a t i ~ n . ~ Thus . ~ ~ ~far, ~ ~ lthese ~ experimental formulas have only been developed to describe the 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 8905
Effects of Scavengers on the Hydrated Electron
effects of scavengers on the chemistry of a particular reactant and on the yield of the corresponding molecular p r o d u ~ t . ~In? ~this paper the cooperative effect of the scavenging of one radiationinduced species on the chemistry of its siblings is examined. We focus on the chemistry of the hydrated electron, although the methodology can be applied equally well to the hydroxyl radical.12 The cooperative effect of scavengers is of considerable importance in experimental radiation chemistry and radiation biology, as frequently several scavengers are present in c o n j ~ n c t i o n . ' ~ J ~ Clearly, a simple predictive treatment will be of considerable use. In a series of papers7*8*15v16 we have discussed the radiation chemistry following the high-energy electron radiolysis of water using both deterministic and stochastic models. We have scrutinized the influence of an added scavenger on the chemistry of a particular radiation-induced specie^.^*^-^^ In a chemical system with two different types of scavengers, the kinetics of one of the radiation-induced reactants may be altered when a sibling reactant, which might have reacted with the first, is scavenged before an intraspur reaction between the two can occur. Obviously, it is important to determine the extent to which this effect influences the overall chemistry of irradiated aqueous solutions. Our goal is to develop a simple predictive formulation similar to that presented in refs 7 and 8 that can be used to estimate the observed experimental yields. To quantify the effects of cooperative scavenging and to investigate the origins of the observed changes in the radiation chemistry, a series of predictive calculations were performed by using a diffusion-kinetic m ~ d e l . ~This , ~ analysis was then used to develop an empirical formulation of the modeled problem. Finally, the formulation was further modified to account for the documented, and well understood, discrepancies between the predictions of experiment and the kinetic modeL7J7 The diffusion-kinetic model utilized is discussed briefly in the following section. Section 3 describes the arguments used in developing an expression for predicting the effects of cooperative scavengers. The chemistry of the hydrated electron in several aqueous systems is considered in depth. The final section summarizes our comments and conclusions.
2. Kinetic Model The calculations presented in the following section were performed by using a deterministic diffusion-kinetic model based upon the numerical technique outlined by Burns et a1.18 The methodology was described in detail in refs 7 and 17. In this treatment the nonhomogeneous radiation chemistry of an aqueous solution is modeled in terms of the chemistry of a single, typical s p ~ r . ' ~ * ~ O The diffusion and reaction of each species is described by a differential equation: &,/at
= DiV2Ci - Ck,C,Cj
+ Ck,,c,c,
(2.1)
where D, and ci are the diffusion coefficient and concentration of species i, respectively. The terms on the right-hand side of the differential equation refer to the diffusion, the reaction and the production of species i. The concentration of the radiation-induced reactants is spatially nonhomogeneous and it is assumed to be initially Gaussian.21g22 Each spur is divided into concentric shells that are sufficiently narrow that the concentration of any species within it can be regarded as homogeneous. The set of coupled equations describing diffusion between the shells and the reaction within each shell is solved numerically by using the FACSIMILE algorithm.23 In our modeling of the effects of cooperative scavenging on the reactions of the hydrated electron in aqueous solutions, we have used the same reaction scheme for the chemistry of water that was presented in ref 7 as well as the same spur parameters obtained in that work. The agreement between the experimental data and the predicted kinetics from that scheme and those parameters was shown to be excellent. Alternative stochastic treatments for modeling nonhomogeneous radiation chemical kinetics in aqueous solutions have been developed.24 It is not expected that the conclusions obtained by using stochastic techniques would be significantly different from those presented.I6 The reactions, and the corresponding rate coefficients, of the reactants produced by the radiolysis of water with the various
TABLE I: Scavenger Reactions _____
reaction nitrate scheme2?
--- +++ - +
+ NO3-
NO2 20HNO2 OHH + HC02H2 C02OH + HCOT H ~ O COT
s1
eaq-
H + NO3-
s2 s3 s4
---
nitrous oxide scheme s5 eaq-+ N 2 0 N2 00-+ H20 OH OHS6 Sl 0-+ e,; 20HS8 0-+ OH H02s9 O-+H+-OH OH C3H70H H 2 0 c3H.10 s10 H C3H70H H2 c3H.10 s11 0- C 3 H 7 0 H OH- + C3H70 s12
+ + +
+ +
--- + +
k,25
M-l
s-I
9.7 x 1.4 X 2.1 x 3.2 x
109 lo6 108 109
9.1 x 1.8 x 2.2 x 2.0 x 2.3 X 1.9 X 7.4 X 1.2 X
109 106 1010 1010 1O'O lo9 lo7 lo9
scavengers considered in this paper are listed in Table I.25 All of the scavenging reactions are assumed to be pseudo-first-order and not to result in solute depletion in the vicinity of the spur.26 3. Discussion 3.1. Effects of a Single Scavenger. We have previously considered the radiation chemistry following the high-energy electron radiolysis of aqueous solutions in some depth.7-8J5J6In particular we have focused on the effect of a single scavenger on the chemistry of the radiation-induced reactant scavenged. A typical set of scavenging experiments might measure the effect of nitrate concentration on the yield of the hydrated electron scavenged and on the yield of molecular hydrogen. The hydrated electron is the principal precursor of molecular hydrogen. Henceforth a scavenger, in this case NO3-,of the radiation-inducedspecies under investigation, e,; is referred to as a primary scavenger. The dependence of the amount of scavenging reaction or of the yield of molecular product on the concentration of a primary scavenger can be described by using a relationship such as7*8J0J3 G ~ ( s =) GXCSc+ (GxO - G x C S C ) F ( a x ~ ) (3.1) where X is the radiation induced species of interest (e.g., ea< or H2), GXW is its steady-state or escape yield, and Gxo is its initial yield. The scavenger efficiency, s, is the pseudo-first-order rate coefficient that is the product of the scavenging rate coefficient, k,, and the scavenger concentration, [SI. The parameter a is uniquely defined for a particular species X irrespective of the scavenger as it is equivalent to a phenomenological decay or formation time in the absence of the scavenger.28 Experimental measurements of G ( s ) are available only over a limited concentration range and the parameters (a,P,and Go) in relationship (3.1) are obtained by nonlinear least-squares fitting using a simple empirical function for F. A number of different forms have been suggested for the function F and for aqueous solutions we have found the most accurate to be798 F ( x ) = ( x ' / ~+ t / z x ) / ( l + x 1 / 2+ x x ) (3.2) By convention, experimental data are usually presented as a function of scavenger concentration, [SI,rather than scavenger efficiency, The resulting fitting parameter, a = aks, is not unique; however, the simple scaling [SI = [S']k,'/k, allows experimental data obtain by using different scavengers to be presented t ~ g e t h e r . ~ ~ ~ The dependence on primary scavenger concentration of the amount of scavenging of the hydrated electron is shown Figure 1. The data are presented with the effective nitrous oxide concentration, [N20],as the concentration scale. The figure includes the predictions of two sets of diffusion-kinetic calculations, one using nitrous oxide as the hydrated electron scavenger and the other using nitrate.29 The best fit of eq 3.1 with the function (3.2) to this modeled data is shown. The parameters obtained are a = 0.552 ns, Gc" = 2.60,and = 4.7K30 Also included in the diagram are the available experimental data obtained with nitrous oxide:' with and with methyl chloride5as the scavenger. Regardless of the nature of the primary scavenger the results are s.576
Pimblott and Laverne
8906 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992
*L
l-7
.' I
I
-
6
:i
4
I
I
i
-
I01
0
z
Y
c3
-3
2l
e
10-6
I
I
I
I
10-2
I
102
io4
I
0 IO-'
10-2
I
IO'
10-6
io4
[NO;] Figure 1. Scavenged yield of e, - as a function of effective N20concentration. The results of the difhon-kinetic calculations are denoted by the following: (0)N20, (0) NO1 M). Even here the error in the empirical equation is less than 5%. In ref 7 the parameters of our diffusion-kinetic model were optimized to match the body of the available experimental scavenger data. The final agreement was generally very good; however, there were a number of minor discrepancies. To use eq 3.4 to predict yields in experiments, it is clearly appropriate to have the parameters for the experimental system. While the values of a,P,and Go for Gk-(sI) are available in ref 7, estimates of the parameters for Gc,,+OH(~1,~2) are still required. From the observed chemical products and model calculations, it can be inferred that the majority of the nonhomogeneous chemistry occurring within spurs involves the four reactions" eaq-+ eaq- H2 20H(R1)
1
I
,2.0
-
+
-
eaq-+ H,O+ eaq- + OH
H
(R2)
OH-
W4)
and
OH + OH I .6
1
I
+
The relatively low yields of the molecular products compared to the total decay of the hydrated electron and the hydroxyl radical suggest that the "recombination* reactions R2 and R4 dominate the chemistry of the hydrated electron and the hydroxyl radical. This fact implies the time dependence of the ea( OH reaction a should complement the electron decay and that %H. In fact the phenomenological decay times for the hydra& electron and for the hydroxyl radical and the production times for molecular hydrogen and for hydrogen peroxide are all within a factor of 5 . 7 ~These ~ results suggest that the parameter %-+OH must be. in the range 0.2 ns C < 0.9 ns and that a good estimate is probably aCq-+oH = 0.55 ns, in good agreement with model calculations. An appropriate experimental value for G=-+oH is obtained by considering the oxidizing species rather t h a n x e reducing species, as the hydrated electron decay is influenced by reaction e,; + H30+(eq R2), the yield of which is not known. The maximum possible yield of the reaction eq- + OH (eq R4) is given by PoH - - 2 q 0 , . The available experimental measurements tell us that this maximum is about 1.4 and our diffusion-kinetic calculations predict that it is 1.42. This maximum yield overestimates the amount of reaction, as it does not include the effects of the reaction H + OH H20 (R7)
+
-
i 0 10-6
10-2
I
102
io4
[SI ks /kN*O (M) Figure 7. Yield of e a i + OH reaction as a function of scavenger efficiency normalized to the effective nitrous oxide concentration. The results of diffusion-kinetic using nitrous oxide as a scavenger of eaq-are given by 0 and using 2-propanol as an OH scavenger arc denoted by 0. The solid line is the best fit to eq 3.1 with function (3.2).
by using the two scavengers in conjunction also lie on the fitted line. Following the logic that we have espoused thus far, to a first order of approximation, the yield of nitrogen following the fast electron radiolysis of an aqueous solution of both nitrous oxide and 2-propanol is made up of two components: GNJSI~FZ) = Ge,;(s1,~2) GCq-(si) + G%
