J. Phys. Chem. 1992,96, 4485-4491
4485
experiments and calculations on this system and on methoxy thermal decomposition at high levels of excitation would help to determine the precise mechanism for CO production.
was provided by the National Science Foundation. P.W.S. acknowledges the award of an SERC/NATO Fellowship during which time this work had been performed.
Acknowledgment. We gratefully acknowledge support of this research by the Department of Energy. Additional equipment
Registry No. CH3, 2229-07-4; CO, 630-08-0; 0, 17778-80-2; SO2, 7446-09-5; CHJ, 74-88-4; acetone, 67-64-1; formaldehyde, 50-00-0.
Investigation of Various Factors Influencing the Effect of Scavengers on the Radiation Chemistry following the High-Energy Electron Radiolysis of Water Simon M. Pimblott Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 (Received: December 11, 1991; In Final Form: February 4, 1992)
Stochastic independent reaction times (IRT) simulations are used to model the fast reaction in spurs similar to those found following the high-energy electron radiolysis of water and aqueous solutions. The time dependence of the rate coefficient for the reaction of the hydrated electron or the hydroxyl radical with a scavenger is shown to be important in modeling scavenging reactions only when the steady-state rate is large, e.g. for the reactions eaq-+ O2or OH + Br-. The calculations are used to test the accuracy of the commonly used Laplace relationship between the time dependence of the radicals and the molecular products resulting from the high-energy electron radiolysis of water and the scavenger concentration dependence of the observed yields following the radiolysis of scavengersolutions. Agreement between modeled yields for the scavenger systems and predictions of the Laplace transforms of the decay and the formation kinetics in the absence of scavenger is excellent once the time dependence of the scavenging rate coefficient has been incorporated in the Laplace relationship.
1 . Introduction The importance of stochastic processes in the period of fast reaction that follows the high-energy electron radiolysis of water has recently received a considerable amount of attention.'v2 The interest has arisen from the recognition that conventional deterministic model^^.^ for the kinetics following electron radiolysis do not take proper account of the separate identities of the reactant^.^.^ This integer nature is of tremendous importance when the reactive system is made up of only a few reactants, as in the high-energy electron radiolysis of water. About 75% of the energy lost by the primary electron results in essentially isolated clusters of reactants which are known as spurs and which contain from 1 to 6 ionization or excitation Two alternative stochastic techniques have been developed to model spur kinetics. The most popular approach involves the Monte Carlo simulation of the kinetics by modeling the encounter times of the reactants.+13 The second method requires the formulation and solution of a (1) Freeman, G. R. Kinetics of Nonhomogeneous Processes. A Practical Introduction for Chemists, Biologists, Physicists and Material Scientists; Wiley-Interscience: New York, 1987. (2) Green, N . J. B.; Pilling, M. J.; Pimblott, S. M. Radiat. Phys. Chem. 1989, 32, 99. (3) Schwarz, H. A. J . Phys. Chem. 1969, 73, 1928. (4) Burns, W. G.; Sims, H. E.; Goodall, J. A. B. Radiat. Phys. Chem. 1984, 23, 143. (5) Clifford, P.; Green, N . J. B.; Pilling, M. J. J . Phys. Chem. 1982, 86, 1318. (6) Clifford, P.; Green, N . J. B.; Oldfield, M. J.; Pilling, M. J.; Pimblott, S. M. J . Chem. Soc., Faraday Trans. 1 1986,82, 2673. (7) Pimblott, S. M.; Laverne, J. A.; Mozumder, A.; Green, N. J. B. J . Phys. Chem. 1990. 94, 488. (8) Pimblott, S. M.; Mozumder, A. J . Phys. Chem. 1991, 95, 7291. (9) Zaider, M.; Brenner, D. J. Radiat. Res. 1984, 100, 245. (10) Turner,J. E.; Ha", R. N.; Wright, H. A.; Magee, J. L.; Chatterjee, A.; Hamm, R. N.; Bolch, W. E. Radiat. Phys. Chem. 1988, 32, 503. (11) Clifford, P.; Green, N. J. B.; Pilling, M. J.; Pimblott, S. M.; Burns, W. G. Radiat. Phvs. Chem. 1987. 30. 125. (12) Green, N.'J. 8.; Pilling, M. J.iPimblott, S. M.; Clifford, P. J. Phys. Chem. 1990, 94, 251. (13) Pimblott, S. M.; Pilling, M. J.; Green, N . J. B. Radiat. Phys. Chem. 1991, 37, 377.
master equation describing the time-dependent probability that a spur has a certain makeup.*." A number of different Monte Carlo simulation techniques have been d e v e l ~ p e d .For ~ the most part these methods follow the trajectories of the diffusing reactants, and then the reaction is modeled by encounter.lOJ1 This type of approach is known as random flights simulation. In a series of ~ a p e r s , ~an . ~alternate J~ and very efficient Monte Carlo simulation method, the independent reaction times (IRT) model, has been developed which relies upon the independent pairs approximation of the Smoluchowski-Noyes treatment of diffusion-controlled kinetics.I5 This approximate simulation model has been thoroughly tested by comparison of its predictions with those of random flights Monte Carlo simulation. The agreement between the results of the two methods under conditions appropriate for high-permittivity solvents such as water is excellent, although some small errors are found for low-permittivitry solvents.'6 The approximate IRT model involves only a small fraction of the computational cost of the random flights method. One of the most frequently used experimental techniques in the radiation chemistry of water involves the study of the effects of an added solute, known as a scavenger, upon the reactions that follow the radiolysis of the liquid.I7J8 This type of experiment provides direct information about the yields of the transients created by the energy transfer from the radiation particle to water. Frequently, the experiments are also used to infer data about the decay kinetics of the reactive species1e22and about the formation (14) Clifford, P.; Green, N. J. B.; Pilling, M. J.; Pimblott, S. M. J . Phys. Chem. 1987, 91, 4417. (15) Noyes, R. M. Prog. React. Kine?. 1961, I , 129. (16) Green, N . J. B.; Pilling, M. J.; Pimblott, S. M.; Clifford, P. J . Phys. Chem. 1989, 93, 8025. (17) Buxton, G. V. In Radiation Chemistry. Principles and Applications; Farhataziz, Rodgers, M. A. J., Eds.; VCH Publishers: Cambridge, U.K., 1987. (18) Spinks, J. W. T.; Woods, R. J. An Introduction to Radiation Chemistry; Wiley-Interscience: New York, 1990. (19) Balkas, T. I.; Fendler, J. H.; Schuler, R. H. J. Phys. Chem. 1970, 71, 4497.
0022-365419212096-4485$03.00/00 1992 American Chemical Society
4486 The Journal of Physical Chemistry, Vol. 96, No. 11, 1992
kinetics of the molecular products H2and H2022923in the absence of the scavenger. The timedependent kinetics are obtained from the scavenger concentration dependence of the yields in scavenger experiments via an inverse Laplace transform relationship. This relationship was fmt applied to the radiation chemistry of ion-pair recombination in hydrocarbons by H ~ m m e and l ~ ~by Warman et aLts The Laplace transform is only strictly valid for a geminate pair (cf. ion recombination in hydrocarbons), but deterministic calculations suggest that it is also accurate for spur kinetics in water In this paper the stochastic IRT technique is used to model the effect of scavengers on the kinetics of spurs similar to those found along the track of a high-energy electron in water. We summarize the IRT methodology and demonstrate the use of the technique to model the kinetics of spurs in water. The extension of the IRT method to include reactions with a homogeneously distributed species is discussed briefly, and the procedure is used to model the reactions of the radiation-induced reactants found in spurs with solutes used as scavengers in experimental systems. The kinetics of a distribution of spurs similar to that expected along the track of a high-energy electron in water is considered: the effect of various scavengers on the reactions occurring is modeled and used to test the accuracy of the Laplace transform relationship between the time dependence of the yields of the radiation-induced species in water and the scavenger concentration dependence of the yields in scavenger studies. The importance of incorporating the time dependence of the scavenging rate coefficient in the Laplace transform relationship is demonstrated using experimental data for the scavenging of the OH radical by Fe(CN):- and for the effect of Br- concentration on the formation of H202. .22323
2. Methodology
The IRT method for modeling nonhomogeneous kinetics has been described in detail e l s e ~ h e r e . ~ -Essentially ~ - ~ ~ J ~ it involves modeling of the reaction times of the reactants in the system of interest, and it makes use of the independent pairs approximation to avoid simulating the trajectories of the reactants. The p r d u r e is as follows: The coordinates of all the reactants are generated from the appropriate spatial distribution which may represent either an isolated spurI3 or a section of a high-energy electron t r a ~ k . ~The J ~ interreactant separations are calculated, and the reaction of overlapping pairs is considered. The interreactant separations of the surviving reactants are used to generate reaction times by sampling from the appropriate reaction-time distribution functions. These distribution functions are taken as those for identical isolated pairs with the same initial separation.26 It is here that the independent pairs approximation is made. The minimum of the resulting ensemble of times is the first t > 0 reaction. The corresponding pair reacts at this time, and any other reaction times for the two reactants are discarded. New reaction times are generated for any reactive products.6 The minimum of the new ensemble of times corresponds to the next reaction. This process is repeated until no reactive pairs remain or a predefined cutoff time is reached. The repeated realization of the procedure gives the kinetics of the reactive system under consideration. Modeling the kinetics of a spur following the procedure described is straightforward as the number of reactants is small
Pimblott TABLE I: Reaction Scheme for the Radhtloa Ckmhlry of WstCP kt-1 .M-1 , x Q-I1010. no. intraspur reaction &n,nm R1 e- + e- HZ+ 2OH0.55 0.16 R2 e- + H30+ H 2.3 0.23 2.5 0.29 R3 e- + H H2 + OHR4 e- + OH OH3.0 0.54 R5 e- + H202 OH + OH1.1 0.22 R6 H,O+ + OH- H20 14.3 1.35 R7 H + H - H , 0.78 0.15 R8 H + OH H 2 0 2.00 0.27 R9 H + H202 OH 0.009 0.001 R10 OH + OH H202 0.55 0.26 R11 O2 + e- 021.90 0.39 R12 02 + H HO2 2.10 0.25 R13 H,O+ + 02- H 0 2 3.80 0.46
------+ - + + -++ + + - + + + - + + -
0.96
0.5W
e,4- Scavenging
CH3C1 CI-
CHI
S1
e-
S2 S3 S4
OH Scavenging OH HC02H H20 HCO2 0.013 H HC02H H2 HCO, 4.4 X l(rs 0 2 HCO2 H0, CO, 1 .o
S5 S6 S7
H2 Precursor Scavenging NO; NOZ20.41 H+NO;-OH-+NO 0.07 1 OH NO; OH- NO2 1 .o
S8
OH
e-
0.1 1
0.024 4 x 10-3
7 X IOd 0.51 0.32 0.01 0.32
H202Precursor Scavenging
Br-
BrOH-
1.10
0.30
Rate coefficients from ref 27. Where no value for R is given, it is the same as Rep cEstimatcdvalue.
(usually between 3 and 20 particles). However, when a scavenger is included in the reaction scheme the calculation becomes more time consuming. As the scavenger is homogeneously distributed, its spatial extent is effectively M i t e , so it is necessary to introduce an outer boundary to the region of space considered. This boundary must be sufficiently distant that the probability of a spur reactant passing through it is negligible, while the number of scavenger reactants enclosed within it has to remain tractable. Even with the imposition of an outer boundary, the number of reactants that are necessary for modeling the kinetics accurately soon becomes excessive. For instance to model the chemistry of a spur in an aerated aqueous solution of 0.1 mol dm-3 formic acid requires about 5 X lo4 scavengers. This problem has led to the development of an alternative method of modeling scavenging reactions within the IRT model. In this alternative approach the scavenger is treated as a continuum and the scavenging times are generated by assuming pseudo-first-order reaction.” The two different techniques are not quite equivalent as the explicit treatment of the scavenger particles takes account of correlations in the initial spatial distribution of the spur and scavenger particles, which is neglected in the continuum treatment. However, no observable discrepancy between calculations using the different techniques has been found.13 The reaction of spur reactants with the scavenger at time t = 0 will be discussed in more detail later in this paper. In the following calculations the chemistry of spurs resulting from the ionization
+
H 2 0 -w- H30+ OH and the excitations
(20) Schuler, R. H.; Hartzell, A. L.; Behar, B. J . Phys. Chem. 1981,85, 192. (21) Schuler, R. H.; Behar, B. In Proceedings ojrhe Fifth Tihany Symposium on Radiation Chemistry; Dobo, J., Hedvig, P., Schiller, R., Eds.; Akademai Kiado: Budapest, 1983. (22) Laverne, J. A.; Pimblott, S. M. J . Phys. Chem. 1991, 95, 3196. (23) Pimblott, S.M.; Laverne, J. A. Radial. Res. 1992, 129, 265. (24) Hummel, A. J . Chem. Phys. 1968, 48, 3268; 1968,49, 4840. (25) Warman, J. M.; Asmus, K.-D.; Schuler, R. H. J . Phys. Chem. 1969, 73, 93 1 . (26) The distribution function used in the generation of the reaction time of a pair depends upon the particular reactants. Functions are available for the diffusion-controlled reaction and for the partially diffusion-controlled reaction of a pair when there is a Coulombic interion potential and a screened interion potential and when there is no electrostatic potential. See ref 28 for details.
R,bnm 0.42 0.5W
HZO H2O
--+
--- + + -
H2
H
+ ea;
OH
“0” H2 + 20H
(D1) (D2) (D3)
of water is considered. The most significant of the reactions of the four reactive species (the hydrated proton, H,O+, the hydrogen atom, H, the hydroxyl radical, OH, and the hydrated electron, e), in deaerated water are R1-R10 of Table I. In aerated water reactions R11-Rl3 also have to be considered. The table includes the steady-state rate anstants (Qn the effective reaction distances (27) Buxton, G. V.; Greenstock, C. L.; Helman, W. P.; Ross,A. B. J . Phys. Chem. Ref. Data 1988, 17, 513.
The Journal of Physical Chemistry, Vol. 96, No. 11, 1992 4481
High-Energy Electron Radiolysis of Water
4.80 4.80 0.60 5.70
1.10 0.35 0.35 0.35
particular size spur. An estimate of the latter is obtained as follows:31 It is apparent from Table I1 that the G-value for the radiation-induced destruction of water at t = 0 is 5.55 molecules/( 100 eV), so the relative frequencies of the dissociations Dl-D3 along an entire track are 0.865, 0.108, and 0.027, respectively. The probability that a dissociation taken at random is a member of an i-event spur is
0.15
0.35
Qi = iPi/CiPi
TABLE II: Parameters for the Spur Species D X 10": GO: reactant m2 s-I molecules/(100 ev) 0.45 0.90 0.70 0.28 0.50 0.50 0.22 0.20 0.20
s,,nm
6
i=l
5.70 a
Taken from ref 4. *Reference 17.
TABLE 111: Relative Freqwncies of Spurs with 1-6 Water DiaPociatioas" pN
1 2 a
0.1635 0.5668
pN
3 4
0.1634 0.0613
pN
5 6
0.0291 0.0159
Now assume that the first (primary) dissociation in a multidissociation spur is always an ionization, D1, and that the probabilities of ionization and of excitation occurring for all the other (secondary) dissociations in such a spur, and in a single dissociation spur, are always pDllpD2, and pD3. The frequency of ionization for the whole track is the sum of the number of single dissociation spurs multiplied by the probability that such an event is an ionization plus the number of multidissociation events plus the the number of secondary dissociations multiplied by the probability that the dissociation is an ionization; i.e.
Obtained from ref 29 using a simple rule of correspondence.
6
0.865 = QlpDl and the reaction distances (R) for the reactions. The rate constant and the effective reaction distance for a reaction, A + B product, are linked by the equation &),
-
k(m) = 4aD'RR,n
(2.1)
where D'is the relative diffusion coefficient of the two reactants. The relationship between the effective reaction distance and the reaction distance depends upon whether or not the reaction is diffusion controlled and upon the interreactant potential. It has been discussed at length by Green and Pimblott.2* The diffusion coefficients of the various radiation-induced species are listed in Table I1 along with experimental estimates of their initial ( t = 0) yields." In modeling the chemistry immediately following the highenergy electron radiolysis of water, it is necessary to consider the kinetics of a distribution of spurs similar to that found along the track of a high-energy electron. The distribution of energy loss events for high-energy electrons in water has been calculated using a simple Monte Carlo simulation method employing accurate cross sections which take into account relativistic and exchange effects and which were calculated using experimentally based dipole oscillator strength distributions.' In liquid water about 75% of the energy is lost in events smaller than 100 eV, i.e. spurs. However, the conversion of this spur distribution which is in terms of energy losses to a distribution in terms of the numbers of the dissociations Dl-D3 is not straightforward because of the existence of ionization and excitation thresholds and is under investigation? In the kinetic calculations presented here, the energy loss distribution has been converted to one in terms of the number of dissociations using a simple rule of comspondence.29 The relative frequencies of events with 1-6 dissociations, Pi with i = 1-6, are given in Table 111. Additionally, the relative widths of spurs were assumed to vary according to a cube root law; i.e.
sN = s,N1/' (2.2) where sN is the width of a spur of N dissociations. This variation has been used in a number of different kinetic calculations3J1J3 and was originally postulated by Samuel and MageeMto keep the spatial energy density within a spur the same irrespetive of the number of dissociations. The nature of the stochastic kinetic simulations is such that the modeling of the nonhomogeneous chemistry of an electron track requires only the distribution given in Table I11 and the relative frequencies of the different dissociations Dl-D3 in a (28) Green,N. J. B.; Pimblott, S. M. J . Phys. Chem. 1989, 93, 5462. (29) Each dissociation is assumed to "cost" 17 eV; 8%: Mozumder, A.; Maget. J. L.Rudiar. Res. 1966, 28, 203; 1966, 28, 215. (30) Samuel, A. H.; Magee, J. L. J . Chem. Phys. 1953, 21, 1080.
(2.3)
6
+ i-2C Q , / i + iC= 2( i - l)QipDl/i
(2.4a)
where Qi is calculated from the distribution given in Table 111 using eq 2.3. This equation can be rearranged to give 6
6
i=2
i=2
p ~ 1 (0.865 - E Q i / i ) / ( l - C ( i - l)Qi/i) = 0.786
(2.4b)
and similar logic gives 6
p ~= 2 0.108/(1
- CQi/i) i=2
(2.5)
6
PO3
= 0.027/(1 - CQi/i) i=2
(2.6)
The kinetics of an electron track are modeled by simulating the reaction in a series of spurs, which is obtained by the repeated random selection of spurs with from 1 to 6 dissociations from the distribution of Table 111. The types of the individual dissociations that make up a particular spur are determined from pDI,pDZ,and pD3, remembering that the first dissociation in a multiple-dissociation spur is always an ionization. The kinetics predicted for the resulting distribution of spurs depends upon the spatial distributions of the different species within each spur. In the following calculations the initial spatial distributions were spherically symmetric G a u s ~ i a n s ,and, ~ , ~ in any particular spur, the spatial distributions of H30+,of H,of OH,and of Hzwere the same and different from that of eaq-.
3. Laplace Relationship The recombination of a geminate pair of reactants has been discussed at length.32 Most conventional treatments make use of the Smoluchowskiequatiod3 and its adjoint,30although recently an alternative kinetic method has been developed.35 Using these methods it is straightforward to show that the probability of scavenging one member of a geminate pair, P, is related to the time-dependent survival probability of the pair if it were in isolation, Q(r), by the equatiod5
~
(31) Green, N. J. B. Private communication. (32) Rice, S.A. In Dif/usion-Limited Reactions, Comprehensive Chemical Kinetics; Bamford, C. H., Tipper, C. F. H., Compton, R. G., Eds.; Elsevier: Amsterdam, 1985; Vol. 25. (33) von Smoluchowski, M. 2.Phys. Chem. 1917, 92, 129. (34) Tachiya, M. J . Chem. Phys. 1978,69, 2375. (35) Green, N. J. B.; Pimblott, S. M. Mol. Phys. 1991, 74, 795; 1991, 74, 881.
4488 The Journal of Physical Chemistry, Vol. 96, No. 11, I992
where k , ( m ) is the steady-state limit of the scavenging rate coefficient and T is a modified time scale which depends upon the time dependence of the scavenging rate coefficient, T = S,lk,(t) dt/ks(m)
Pimblott
5
(3.2) 4
For a diffusion-controlled reaction the time-dependent rate coefficient is given by3’
k,(t) = IC,(-) [ 1 + 2Bt-’/2]
(3.3)
B = R,ff/(4?rD?’I2
(3.4)
where
-
2.
and D’is the relative diffusion coefficient of the scavenger and the scavenged species. Under these conditions T is simply t 4Bt’I2. Equation 3.3 is often an acceptable approximation for partially diffusion-controlled reactions, although more precise forms are a~ailable.~~J’ Generally the scavenging rate coefficient is assumed to be independent of time, and Tis then the real time.36 While convenient, this approximation may be experimentally inappr~priate.~’ The yield of the scavenger reaction is the t limit of eq 3.1, and so is the Laplace transform of the time dependence of the survival probability. This relationship was first used in radiation chemistry by HummelZ4and by Warman et who studied the effect of scavengers on the ion recombination following the high-energy electron radiolysis of hydrocarbons. Both groups fitted their experimental yields to empirical functions with the form
+
I 5\ /
‘ 3
-I
2
-
4,
3
4
5
4
5
log ( t i p s )
-
= G,, + (Go - Ges,)F(ac,)
(3.5)
where G,, and Goare the steady-state yield and the t = 0 yield of the species under consideration and where F(ac,) is a function that describes the scavenger dependence of the observed yield and a is a fitting parameter. The function F(ac,) had a simple inverse Laplace transform, F(Xt)/s in which X = k s ( m ) / a . The time dependence of the ion yield in the absence of the scavenger was then taken to be G ( t ) = G,
+ (Go - G,,)F(Xt)
I
(3.6)
Several different functions have been suggested for F(ac,) (and hence for F(Xt)).22The observed yields in scavenger experiments are fitted to an empirical function (eq 3.5) because the experiments are only possible over a limited range of scavenger concentrations: there are problems associated with the solubility of the scavenger and with the direct radiolysis of the scavenger. Scavenger experiments have also been used to investigate the radiation chemistry of ~ a t e r , ’ and ~ - ~the ~ Laplace relationship has been applied to the time dependences and the scavenger dependences of the radical^'^-^^ and the molecular product^.^^^^^ The only difference in the application of the Laplace relationship to the formation of H2and H202is that X = 2k,(m)/a rather than k s ( ~ ) / aA. detailed discussion has been presented in ref 23. While a single-pair model is appropriate for the kinetics following the radiolysis of hydrocarbons, this is probably not the case for water. Mozumder and Magee have questioned the use of a Laplace transform relationship such as that between eqs 3.5 and 3.6 for describing the radiation chemistry of water where multidissociation spurs are known to occur.3a The accuracy of the Laplace relationship has been tested for idealized multidissociation spurs using (36) Schuler, R. H.; Infelta, P. P. J. Phys. Chem. 1972, 76, 3812. (37) Periasamy, N.; Doraiswamy, S.;Venkataraman, B.; Fleming, G . R. J . Chem. Phys. 1988,89, 4799. (38) Mozumder, A.; Magee, J. L. Radiat. Phys. Chem. 1975, 7 , 8 3 . (39) International Critical Tables of Numerical Data, Physics, Chemistry and Technology; National Research Council: New York, 1933. (40) American Institute of Physics Handbook McGraw-Hill: New York, 1957. (41) Snipes, R. F. S. Statistical Mechanical Theory of Electrolytic Transport of Non-electrolytes;Lecture Notes in Physics, Vol. 24; SpringerVerlag: Berlin, 1973. (42) Anbar, M.; Hart, E. J. Adu. Chem. Ser. 1968, 81, 79.
2
3
log ( t / p s ) Figure 1. Kinetics of a distribution of spurs similar to that produced by the high-energy electron radiolysis of deaerated (a, top) and of aerated (b, bottom) water. The spur distribution was from Table I11 with the spur radii given in Table 11. In both frames curve 1 refers to ea;, curve 2 to OH, curve 3 to H, curve 4 to Hz, and curve 5 to Hz02.
the stochastic master equation technique3s and for the radiation chemistry of water using deterministic methods.22 The agreement between the modeled yields and the predictions of the Laplace transform relationship was remarkable; however, some discripancies between the two sets of results were found in the stochastic calculations. Furthermore, both the stochastic and the deterministic studies employed time-independent rate coefficients for scavenging. Thus a number of questions remain unanswered: do stochastic effects diminish the accuracy of a Laplace relationship between G(t) and G(c,) for the high-energy electron radiolysis of water, and how important is the time dependence of the scavenging rate coefficients in the relationship? 4. Results and Discussion
The spur size distribution listed in Table I11 is appropriate for energy loss events that account for about 75% of the energy lost by a high-energy electron in water. To make the following calculations somewhat realistic, the widths of the spatial distributions of the various reactants have been varied so that the predicted yields at about 1 ws approximate the experimentally observed steady-state yields.22 Figure 1 shows the kinetics predicted for the spur distribution using a spur width, sl, of 0.35 nm for H30+, H,OH,and H2 and of 1.1 nm for ea
