746
J. Phys. Chem. 1992, 96, 746-752
Scavenger Concentration Dependences of Yields in Radiation Chemlstry Simon M. Pimbiott* and Jay A. LaVerne Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 (Received: July 18, 1991)
Analytic models are used to show that at low concentrations the effect of scavengers on radiation chemical yields should be expressed as a power series in the square root of the scavenger concentration. The coefficients of the different terms in the series are determined by the initial distribution of separations of the reactants and by the interparticleforces. Conventional empirical functions used to describe the scavenger concentration dependence of radiation chemical yields are shown to be equivalent to a power series in which all the coefficients are defined by a single fitting parameter. The importance of the terms of higher order than the square root of the scavenger concentration is emphasized, and the need for a second fitting parameter to account for the difference between the coefficients of these terms and the square root term is demonstrated. A new function to describe the scavenger dependence of radiation chemical yields is discussed, and it is shown to reproduce other existing empirical fitting functions, even those which do not have the correct square root of scavenger concentration asymptote, over the limited concentration range that is experimentally accessible.
1. Introduction
One of the primary goals in radiation chemistry is the understanding of the nonhomogeneous kinetics that follow the passage of ionizing radiation through matter. A significant fraction of the total chemistry is contained in the short period of fast reaction that occurs as the initially nonhomogeneous spatial distribution of reactants evolves to a homogeneous distribution.1,2 The reactions of the radiation-induced species can be studied by using a number of experimental methods. Perhaps the most common of these techniques involves the analysis of the effects of added solutes, known as scavengers, on the yields of stable products. These experiments have provided information about the radiation chemical yields, about the decay and formation kinetics, and about the reaction mechanisms in a variety of different However, there are several limitations to the use of scavengers in the study fast kinetic processes: the effective time scale which can be probed is restricted by the solubility and by the reactivity of the solute, there are experimental limitations on the detection of the observed product, and one must consider solute radiolysis. Consequently, the available data are usually fitted to an empirical function that can be used to make extrapolations to concentration regimes that are not experimentally accessible. There is a considerable amount of uncertainty as to the applicability of this type of treatment and as to the appropriate functional form for the equation used to fit the data obtained in the scavenger experiments. Sworski was the first to report the effect of scavenger concentration on radiation chemical yield^.^,^ He found, empirically, that the yields of hydrogen peroxide produced in the radiolysis of aqueous halide solutions depended on the cube root of the halide concentrati~n.~,~ Williams later observed that the scavenging of protons in cyclohexane depended on the square root of the scavenger concentration.' This square root dependence has been found in numerous other studies which involved the scavenging of ions in hydrocarbon^.^^^ (1) Freeman, G. R. Kinetics of Nonhomogeneous Processes. A Practical Introduction for Chemists, Biologists, Physicists and Material Scientists; Wiley-Interscience: New York, 1987. (2) Farhataziz; Rodgers, M. A. J. Radiation Chemistry. Principles and Applications; VCH Publishers: New York, 1987. (3) Allen, A. 0. The Radiation Chemistry of Water and Aqueous Solutions; Van Nostrand: New York, 1961. (4) Draganic, I. G.; Draganic, Z . D. The Radiation Chemistry of Water; Academic: New York, 1971. ( 5 ) Sworski, T. J. J . Am. Chem. SOC.1954, 76,4687. (6) Sworski, T. J. Radiat. Res. 1955, 2, 26. (7) Williams, F. J. Am. Chem. SOC.1964, 86,3954. (8) Scala, A. A.; Lias, S. G.; Ausloos, P. J. Am. Chem. SOC.1966, 88, 5701. (9) Warman, J. M.; Asmus, K.-D.; Schuler, R. H. J . Phys. Chem. 1969, 73, 931.
A number of empirical equations have been suggested for fitting experimental data, and these equations have been applied to scavenging both in hydrocarbons and in aqueous solutions. Since the amount of scavenging cannot increase without limit as the scavenger concentration increases, typically the dependence of the observed radiation chemical yield on the scavenger concentration, csris assumed to vary as f/(l +A, where f is a function of the square root of the scavenger concentration. According to this type of equation, the power of the scavenger concentration dependence changes monotonically from 0.5 at the low-concentration asymptote to zero in the high-concentration limit. We have examined the scavenger concentration dependences of the yields of the radicals and the products produced by the radiolysis of water, and we found that although a fitting equation in which f depended only on the square root of the scavenger concentration was acceptable, a fitting function in which f contained higher order terms in scavenger concentration fitted the data more acc~rate1y.l~A number of doubts have been raised about the indiscriminate use of a square root concentration dependence inf. In a series of fluorescence quenching experiments, Lipsky and co-workers' l-I3 obtained results that support a dependence off on the scavenger concentration to the power 0.7. Meanwhile, studies at Delft have suggested a power dependence of 0.6 for the scavenging of ions in hydro~arbons.'~J~ The theoretical description of the competition between scavenging and recombination has received a great deal of attention.IG2* Much of this interest has centered on the reaction and
(10) Laverne, J. A.; Pimblott, S. M. J. Phys. Chem. 1991, 95, 3196. (11) Choi, H. T.; Wu, K. C.; Lipsky, S. Radiat. Phys. Chem. 1983, 21, 95.
(12) Choi, H. T.; Haglund, J. A,; Lipsky, S . J. Phys. Chem. 1983, 87, 1583. (13) Tweeten, D. W.; Lee, K.; Lipsky, S . Radiat. Phys. Chem. 1989,34, 771.
(14) van den Ende, C. A. M.; Luthjens, L. H.; Warman, J. M.; Hummel, A. Radiat. Phvs. Chem. 1982. 19. 455. (15) van den Ende, C. A. M.;Warman, J. M.; Hummel, A. Radiat. Phys. Chem. 1984, 23, 5 5 . (16) Noyes, R. M. J. Am. Chem. SOC.1955, 77,2042. (17) Danckwerts, P. V. Trans. Faraday SOC.1951, 47, 1014. (18) Hummel, A. J . Chem. Phys. 1968, 48, 3268. (19) Mozumder, A. J. Chem. Phys. 1971, 55, 3026. (20) Magee, J. L.; Taylor, A. B. J. Chem. Phys. 1972, 56, 3061. (21) Tachiya, M. J . Chem. Phys. 1979, 70, 238. (22) Raaen, S.; Hemmer, P. C. J . Chem. Phys. 1982, 76,2569. (23) Petersen, J. B.; Larsen, J. E. J. Chem. Phys. 1982, 77, 1061. (24) Hong, K. M.;Noolandi, J. J . Chem. Phys. 1978,68, 5163. (25) Friauf, R. J.; Noolandi, J.; Hong, K. M. J . Chem. Phys. 1979, 71, 143. (26) Mozumder, A. High Energy Chem. (Engl. Transl.) 1977, 11, 150.
0022-365419212096-746$03.00/0 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 741
Effect of Scavengers on Radiation Chemical Yields scavenging of a geminate pair of reactants. The effect of a scavenger on the recombination of a pair of radicals is easily s0lved;l7 however, the effect on the reaction of a pair of ions is more complex. In the early treatments of the ion pair problem, a number of simplifying approximations were used to derive the scavenger dependence in the low scavenger concentration limit.IsJ9 More complete and more accurate approximations have been developed since then,m23and recently Hong and Noolandi derived the exact solution for the scavenging of a geminate ion pair.24JsAll of these solutions suggest the asymptotic square root dependence on scavenger concentration that is observed experimentally. The scavenging of multiple ion pairs has seen less attention. Mozumder has described the scavenging of multiple ion pairs in hydrocarbons using the prescribed diffusion appr~ximation.’~ Although analytic solutions to his equations were not found, Mozumder was able to show a square root dependence at low scavenger concentrations. Mozumder has also considered scavenging in multiparticlesystems, like those produced by the radiolysis of aqueous solutions, using the prescribed diffusion model and found a square root concentration dependence?6 However, Byakov used similar methods and found a cubic root dependen~e.~’The disagreement between the analyses is a consequence of the different approximations used in the analysis, and the two results are probably appropriate over different concentration ranges.28 In this paper we examine the most common empirical equations used to describe the scavenger concentration dependence of radiation chemical yields. We use expansions of the solutions given by kinetic models for single and for multiple pair systems to show why the different empirical fitting functions successfully match radical chemistry both in hydrocarbons and in aqueous solutions. We also present arguments emphasizing the importance of including terms of higher order in scavenger concentration than the square root in these functions. We show that these terms, and in particular their accompanying coefficients, are vital for describing ionic processes. Finally, we attempt to resolve some of the argument over the appropriate function to use for the fitting of experimental data. 2. Empirical Scavenging Functions A number of empirical functions have been postulated as describing the scavenger dependences of radiation chemical yields. Early experiments suggested that at low concentrations G(c,) = G,,
+ Act
(2.1)
where G(c,) is the observed yield of the species at a scavenger concentration c,, , G is the yield measured in the absence of the solute, A is a constant which is proportional to the difference between the initial (time zero) yield Goand G,, and 1.1 is the power dependence of the scavenger concentration. Experiments involving the scavenging of cations or electrons produced by radiolysis of hydrocarbons gave a value of 1.1 equal to 1/2.7,8 However, earlier experiments on the radiolysis of halide ions in water originally for p.5’6 The apparent disagreement over suggested a value of the correct value of 1.1 is resolved once it is realized that G(c,) cannot increase without limit, and so 1.1 also depends upon the scavenger concentration. Warman et al.9 suggested that the scavenger dependence should be given by
where a is obtained by empirical fitting and will depend on the scavenged species and the scavenging rate constant. The observed square root dependence is equal to the asymptotic limit of this equation at low scavenger concentrations, and as the scavenger concentration increases the apparent power of the dependence of the yield on scavenger concentration decreases to zero. Clearly, the power dependence observed in a series of scavenging experiments is determined by the particular system under study. ~
(27) Byakov, V. M. Dokl. Akad. Nauk S S S R 1963, 153, 1356. (28) Byakov, V. M. High Energy Chem. (Engl. Transl.) 1977, 11, 153.
A second empirical fitting equation with the same low-concentration asymptotic limit as eq 2.2 was suggested by H u m e l l s based on his analysis of scavenging in hydrocarbons. He postulated that
Both this equation and that of Warman et al. are members of the family of equationslO t(ac,y/Z/j!
j= I
G(c,) = Ge,
+ (Go - G,)
”
(2.4)
where n = 1 for the version of Warman et al. and n = m for the version of Hummel. The two equations have been shown to give good fits to a variety of sets of experimental data describing the electron radiolysis of hydrocarbons, and the equation of Warman et al. has also been used to fit data for the hydrated electron yieldB and the O H yield30*31 in aqueous solution. Neither the Hummel equation nor that of Warman et al. can accommodate a power dependence of the yield on scavenger concentration,p , greater than Lipsky and co-workers1*12have analyzed a number of experimental studies on the luminescence quenching of hydrocarbons and found a better fit to equations with the following form The value of v is usually found to be about 0.7. While this equation for the dependence of G(c,) on scavenger concentration has a similar structure to the earlier equations, it has a very different lowancentration asymptote, and this fact is of considerable kinetic significance. It implies a very different asymptotic time dependence for the decay of the reactive species to that suggested by the majority of the theoretical studies. In a recent paper lo we thoroughly discussed the scavenger concentration dependences of the radiation-inducedtransients and their products in the electron radiolysis of water. We used deterministic kinetic calculations to show that both eq 2.2 and eq 2.3 could be used to describe the scavenging of the hydrated electron and the OH radical but that a much more acceptable fit was obtained using eq 2.4 with n = 2. This equation has the same square root limit at low c, but givcs a different dependence at higher concentrations. The equation was then used to fit a compilation of experimental data for the high-energy electron radiolysis of water. Similar fits, and calculations, were also made for the molecular products H2 and H 2 0 2for the first time. Although that equation worked reasonably well, no theoretical justification for its use was given. The random searching for the appropriate functional dependence of scavenger concentration is clearly unacceptable, and so in this paper we consider the expansions of the analytic solutions resulting from a number of simple kinetic models to gain insight.
3. Methodology for Geminate Pair The recombination of a geminate pair of reactants can be described by the adjoint to the Smoluchowski equation,32that is (3.1)
where Q(r) is the time-dependent survival probability of the pair, D’is its relative diffusion coefficient, U is the interreactant po(29) Balkas, T. I.; Fendler, J. H.; Schuler, R. H. J. Phys. Chem. 1970, 74, 4497. (30) Schuler, R. H.; Hartzell, A. L.; Behar, B. J . Phys. Chem. 1981,85, 192. (31) Schuler, R. H.; Behar, B. In Proceedings of the Fifth Tihany Symposium on Radiation Chemistry; D o h , J., Hedvig, P., Schiller, R., Eds.; Akademai Kiado: Budapest, 1983; Vol. 1 , p 183. (32) Sano, H.; Tachiya, M. J. Chem. Phys. 1979, 71, 1276.
748 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992
tential, k is Boltzmann’s constant, Tis the absolute temperature, and the operator V operates in the original coordinate space. In the presence of a homogeneously distributed scavenger, an extra term is added to the right-hand side of this equation
Here s is the pseudo-first-order rate constant that is the product of the scavenger concentration,c,, and the second-order scavenging rate constant, k,. This new equation is easily returned to its original form by making the transformation w = Q exp(-st). It is easy to show that the time dependence of the scavenging probability is given by P(s) = s A ‘ Q ( u ) exp(-su) du
(3.3)
Pimblott and LaVerne to that suggested by Hummel.‘* For an exponential or a Gaussian distribution, the resulting expression for G(c,) is not so simple, but both expressions expand to give a series expansion in terms of s1l2. An exponential distribution with a width parameter of I gives
(3.1 1) and G, is obtained by setting s to 0. I’(x,a) is the incomplete Gamma f~nction.)~A Gaussian distribution of separation distances with a standard deviation u gives
and the total probability of scavenging is the corresponding Laplace transform, i.e., eq 3.3 in the limit t = m. Now the survival probability of a geminate pair in the absence of a scavenger is Q(t)
= I gr ( r ) ( l - W(r,R;r))dr
(3.4)
where g(r) is the distribution function describing the interreactant separation and Wis the time-dependent reaction probability for a pair initially at a distance r. Substitution of eq 3.4 into eq 3.3 gives P(s) = s S - S g ( r ) ( l - W(r,R;t))exp(-st) dr dt ( 3 . 5 ) O
r
which expresses the amount of scavenging in terms of the timedependent reaction probability.)) First, consider the geminate recombination of a pair of radicals. If the reaction of the two radicals is diffusion controlled and occurs when the reactants are separated by a distance R , then34 W(r,R;t) = ( R / r ) erfc [ ( r - R ) / ( 4 D ’ t ) 1 / 2 ] ( 3 . 6 ) When the Laplace transform of this P(r,R;s) = ( R / r s ) exp[-(r - R ) ( s / D ’ ) ’ / ~ ] ( 3 . 7 )
is substituted into eq 3.5 and the exponential in s1I2is expanded, one obtains P(s) = 4 r L - 4 r - R ) g(r) dr + ( ~ / D ’ ) ~ 1 ~ 4 ? r R L - r ( rg(r) - R )dr -
+
( S / D ’ ) ~ ~R R J - ~ (g(r) ~ - dr R ) ~ ... (3.8)
This equation shows that, no matter what the initial distribution of separations of the reactants, the effect of a scavenger on the radiation chemical yield is described by an equation of form
+
C(C,) = Gesc ac,I/2
+ j3cs + yc,3/2 + ...
(3.9) It is the coefficients (a,B, etc.) that are defined by the spatial distribution. Three different spatial distributions are commonly used in modeling radiation chemistry: delta, exponential, and Gaussian. A delta distribution implies that the radical pair is initially at a fixed distance ro and leads to the following expression for scavenging G(G) = G,,,
+ (Go - Gesc)[1- exp(-(ro - R)(k,c,/D?’”)I (3.10)
with G, = Go(1 - R/ro). This equation, of course, corresponds (33) Since nonhomogeneous recombination is inherently transient, it can be argued that a time-dependent rate coefficient, ks(t),should be used for modeling scavenging. The effect of using a time-dependent rate coefficient is merely a change of time scale in eq 3.3 (a substitution of T = Jzk,(,t) dt/k,(m) for u), which speeds up the kinetics. The Laplace relationship between P(s) and Cl can still be used if the time scale change is made (see ref 55). Additionally, the effect of the time dependence of k, is only important when k s ( - ) and c, are both large. (34) Chandrasekar, S. Rev. Mod. Phys. 1943, 15, 59.
and 2%
exp(-R2/2u2)/I’
(?2’
”)] 2u2
(3.13)
where eerc (x) = exp(x2) erfc (x). Figure 1 shows the variation of G(cJ with solute concentration for a radical pair with an initially Gaussian distribution of separations. This example is included to show the precision of the various empirical functions when they are applied to a simple system which can be modeled analytically. The reaction distance and the relative diffusion coefficient of the geminate pair were assumed to be 0.5 nm and 5.0 X m2 s-’, respectively, and the distribution of separations had a standard deviation of 1 .O nm. The best nonlinear least-squares fits to the empirical functions of Warman et al.9 (eq 2.2), Hummell* (eq 2.3),and Laverne and PimblottIo (eq 2.4 with n = 2 ) are included. The best fits were obtained by allowing the yields G, and Goand the fitting parameter a to vary independently. The values obtained by the fitting for the three curves are given in the figure legend. All three functions accurately match the scavenger yields at low scavenger concentrations,but at higher concentrationssome errors are apparent: Warman et al.’s function crosses the data three times, initially overestimating the yield, then a slight underestimate, and finally a small overestimate. Hummel’s function shows the largest deviations from the data including a 5% underestimate of the high-concentration limit; our function matches the data accurately over the whole range of concentrations. The discussion thus far has considered the kinetics of a geminate radical pair, and we now turn our attention to a pair of ions. The recombination of an ion pair initially at a fixed distance has been discussed in detail.36 An exact analytic solution for a Coulomb interion potential was presented by Hong and N ~ l a n dalthough i~~ numerous different analytic approximations have been developed.2w23137-40 The solution of Hong and Noolandi is in “Laplace space” and not the conventional “time space” that is usually considered. However, it is exactly this result which is of interest to us as the probability of scavenging is merely s times the Laplace transform of the survival probability. Unfortunately, Hong and Noolandi‘s solution is a complicated sum of Bessel functions, and numerical evaluation is necessary to obtain @(r,R;s). Friauf et al.25have shown that for the simple limiting case of low-permittivity solvents (i.e., for hydrocarbons), where the Onsager (35) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1970. (36) See: Rice., S. E.Difjusion-limited Kinetics. ComprehensiueChemical Kinetics; Bamford, C. E., Tipper, C. F. H., Compton, R. G., Eds.; Elsevier: Amsterdam, 1985; Vol. 25. (37) Mozumder, A. J . Chem. Phys. 1968, 48, 1659. (38) Flannery, M. R. Phys. Rev. A 1982, 25, 3403. (39) Clifford, P.; Green, N. J. B.; Pilling, M. J. J . Phys. Chem. 1984, 88, 4171. (40) Traytak, S. D. Chem. Phys. 1990, 140, 281.
The Journal of Physical Chemistry, Vol. 96, No. 2. 1992 149
Effect of Scavengers on Radiation Chemical Yields
1 3
-(x2
- x + Ei(l/x)
- y)) (3.16)
y is Euler's constant, E,(x) is the exponential integral, and Ei(x)
I
0.4
4
-. .
4
1
I
10
12
I
6
6
8
10
8
14
12
14
log ( k , c , / d ) I .o
I
I
I
I
~
..............
(C)
0.4 4
1
6
I
I
I
8
10
12
14
log (k,c,/s-')
Figure 1. Effect of scavenger concentration on the scavenging yield for a geminate radical pair with a Gaussian initial distribution of interradical separations of standard deviation, u = 1 .O nm, a reaction distance, R = 0.5 nm, and a relative diffusion coefficient, D'= 0.5 X m2 s-l. In all three. panels the dotted line is the analytic solution to the problem, eq 3.12. The solid lines are the nonlinear least-squares best fits of the empirical functions: (a) Warman et a1.: eq 2.2,G, = 0.61,Go= 0.97, a / k , = 1.08 X 1O-Ios; (b) LaVerne and Pimblott,Io eq 2.4 with n = 2, ,G = 0.62,Go= 0.96,a / & , = 6.09 X 10-" s; (c) Hurnmel,l8 eq 2.3,,G = 0.62,Go = 0.95,a / k , = 5.17 X lo-".
-
distance, r,, is large and where R/rc 0, the scavenged yield from an ion pair initially at a separation ro is 112 (3.14) C(cJ = , G + Gea( O(s)
$)
= -El(-x). Equation 3.15 is only an approximation, but it demonstrates that the coefficients of the terms in and s depend upon the interparticle potential as well as the interparticle distance. F w e 2 shows how x vanes with the reduced distance ro/rcwhere ro is the interion separation. The coefficient x is considerably larger than one when ro/rcis small and decreases as ro/rcincreases. Eventually x becomes negative when ro is of similar magnitude to r,, but by this stage the approximation is no longer valid. For water r, is small, and so under reasonable conditions x < 1; however, for low-permittivity solvents r, is large, and frequently x will be significantly larger than one. 4. Methodology for Multiparticle Spurs A number of different techniques have been developed to model the nonhomogeneous kinetics following high-energy electron radi~lySis!l-~ The essential aim of all of these models is to describe the reaction in a small cluster of reactants, which is commonly known as a spur.51 Two different approaches are commonly used: the more conventional method relies upon the solution of a set of coupled diffusion-reaction equations in which reaction is modeled using a deterministic rate law,1034143while the other approach uses a stochastic formalism and involves either the simulation of the reaction times of the radiation-induced speci e or the ~ construction ~ ~ and ~ solution ~ of a stochastic master equation describing the probability that the spur has certain contents."-% The deterministic models have a number of faults which have been discussed at length else~here.51~~ The principal criticism is that they do not incorporate statistical effects correctly in modeling reaction, and so they do not describe the relative amounts of the different products accurately. However, we have shown that deterministic models do give a fairly adequate description of the decay kinetics of the reactive and we feel that this type of model can be used to obtain a qualitative description of the effects of a scavenger on a radiation-induced reactant. According to the conventional deterministic model for spur kinetics, the reaction in a idealized singlespecies spur is described by the following diffusion-reaction equatiod4 dc/dt = DV2c - k 2 - kcc,
(4.1)
where c and D are the concentrationand diffusion coefficient of the reactant, respectively, and the concentration profile of the reactant is initially a Gaussian of standard deviation u
+
with , G = exp(-r,/ro). They do not give the coefficient of the term in s, but this is understandable. The evaluation of the coefficient is difficult, and it is too complicated to be of use. The exact Hong-Noolandi theory will always give an asymptotic cs1/2 dependence whatever the initial distribution of anion-ation separations. At intermediate concentrations, apparent concentration dependences of greater than are possible, and preliminary results (to be published elsewhere) suggest that in many experimental systems the cS1l2asymptote will be practically inaccessible. Many of the approximations for the geminate recombination kinetics also deal with the Laplace transform. Following the approach outlined by Tachiya?l it is easy to show that for small s
~
~
1318.
where
~~
(41) Schwarz, H. A. J . Phys. Chem. 1969, 73, 1928. (42) Burns, W. G.; Sims, H.E.; Goodall, J. A. B. Radiat. Phys. Chem. 1984,23, 143. (43) Pimblott, S . M.; Laverne, J. A. Radiat. Res. 1990, 122, 12. (44) Pimblott, S.M.; Pilling, M. J.; Green, N. J. B. Radiat. Phys. Chem. 1991, 37, 317. (45) Green, N. J. B.; Pilling, M. J.; Pimblott, S.M.; Clifford, P. J. Phys. Chem. 1990, 94, 251. (46) Turner, J. E.; Ha", R. N.; Wright, H. A.; Ritchie, R. H.;Magee, J. L.; Chatterjee, A.; Bolch, W. E. Radiat. Phys. Chem. 1988, 32, 503. (47) Bartczak, W. M.; Hummel, A. J . Chem. Phys. 1987,87, 5222. (48) Clifford, P.; Green, N. J. B.; Pilling. M. J.: Pimblott. S.M.: Burns. W. G. Radiar. Phys. Chem. 1987, 30, 125. (49) Green, N. J. B.; Pilling, M. J.; Pimblott, S . M. Radiat. Phys. Chem. 1989, 34, 105. (50) Green, N. J. B.; F'imblott, S . M. J . Phys. Chem. 1990, 94, 2922. (51) Mozumder, A,; Magee, J. L. J. Chem. Phys. 1966, 45, 3332. (52) Clifford, P.; Green, N. J. B.; Pilling, M. J. J. Phys. Chem. 1982,86, (53) Pimblott, S. M. D.Phil. Thesis, Oxford University, 1988. (54) Ganguly, A. K.; Magee, J. L. J . Chem. Phys. 1956, 25, 129.
Pimblott and Laverne
750 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992
are more difficult to determine. By expanding the error function in X(t+to) in the second term, we find that integration of this term
tl
J
gives
x 1+-
-0
0.2
0.4
0.6
I .o
0.8
rO’rC
Figure 2. Dependence on the initial interion separation of the function x in eq 3.15 for a geminate ion pair. The regimes appropriate for ionic recombination in liquid water and in most hydrocarbons are shown.
Here n(0) is the average number of reactants initially in a spur and is equal to eGo/lOO with t being the mean spur energy. Equation 4.1 can be solved either n ~ m e r i c a l l yor ~ ~by, ~analytic ~ approximation using the well-known prescribed diffusion technique.4l.” The latter technique gives more insight into the problem since it leads to an analytic result. In the simplest form of the prescribed diffusion model the concentration profde of the reactant in the spur is constrained to be always a Gaussian, whose width is determined by the parameter bz = (4Dt + 2d). In other words, the concentration profile is determined only by diffusion and not by reaction. The prescribed diffusion model has received considerable attention, and it is easy to show that the decay kinetics of a single-species system in the presence of a scavenger are given by c
(s~)l/~ erfc ((s(t
+
I1
- erfc ((sto)’/2)
where s is kscsand to is a2/2D. Now
(4.3)
(4.4) The terms inside the square brackets of eq 4.3 can be expressed as 1 X(to) - X(t+to) where
1
erfc ((sx)”~)
(4.5)
Since 1 + .%‘(to) is always greater than X ( t + f o ) ,the terms in the square brackets of eq 4.3 can be expanded using a Maclaurin series. When the expansion is inserted into eq 4.4, it gives G(cs) = 1
x +cGX(t0) Jmexp(-st) 0 1 + X(t+to) + X(t+to) )2 + 1 + X(t0) 1 + X(t0)
[
(
. .I
dt (4.6)
The integration of the first term in the expansion 1 +‘19 X(t0) J-exp(-st) 0
dr
gives GO/(1 + X(to));however, unfortunately the following terms
1
A similar treatment of the higher order terms is possible, but tedious, and the result is an expansion of the same form as eq 3.9. It is evident that this type of expansion in the square root of s is qualitatively general. The effect of scavenger concentration on G(cs) for a singlespecies spur initially containing eight radicals is shown in Figure 3 along with the best nonlinear least-squares fits of the empirical scavenger functions of Warman et al? (eq 2.2), of Hummell* (eq 2.3), and of Laverne and PimblottIo (eq 2.4 with n = 2). As was the case for the scavenging of a geminate pair, all three functions match the data in the limit of low scavenger concentrations. Of the three functions, that of Warman et al. shows the greatest deviations from the data. Small errors appear in the predicted scavenger yield when the pseudo-first-order scavenging rate constant is only about lo7 s-*, and the fit overestimates the value of Go by about 10%. Hummel’s function is fairly accurate although it does underestimate Go. Our function performs very well over the entire range of concentrations. The use of a time-dependent rather than a time-independent scavenging rate coefficient in these calculations does not affect the results in the low-concentration regime; however, there is a slight increase in scavenging at high concentration^.^^ Good fits of the empirical functions to the modeled yield are still found. The implications of using a time-dependent scavenging rate coefficient are discussed in detail in ref 55. Calculations similar to those presented in Figure 3 were also made for spurs with initially fewer and initially greater numbers of particles, and very much the same standard of agreement was found. A similar treatment to that presented for the deterministic prescribed diffusion model is also possible for the stochastic master equation model, and it gives the same qualitative conclusion^.^^ The scavenging of multiple ion pairs in hydrocarbons has been examined by Mozumder using the prescribed diffusion approximation.lg He found a limiting c1/2dependence at low scavenger concentrations, but he did not present any higher order terms. More recently, stochastic master equation techniques have been used to show that the number of ion pairs surviving in a multiple ion pair spur in the absence of scavenger, (N), is given byS0
+
X(X) =
-
exp(sto) r(3/2+n,2sto) 21/2 F 1 . 3...(2n + 1)
NO
( N ) = CA,n(t)f12 n= I
(4.7)
where the A, are constants. The amount of scavenging occurring and the time dependence of (N) are frequently related using a Laplace transform.I0 This relationship and the link between (N) and the geminate pair of survival probability, n, suggests that the amount of scavenging in the presence of a scavenger might be described using a power series in cl/’. 5. Discussion
The empirical functions and the analytic models for one pair and for multiple pair spurs that we have discussed all suggest a dependence of yields on scavenger concentration that is given by
= Ges, + (Go - Ge,)F(cs) (5.1) and the form of F(cs)depends upon the particular set of experWCS)
( 5 5 ) Green, N. J . B.; Pimblott, S . M. Mol. Phys., in press.
The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 751
Effcct of Scavengers on Radiation Chemical Yields 1.21
I
I
I
and led to the postulation that F(c,) was given by
I
F(c,) = f(c,1'2)/( 1 +f(csl'2))
(5.2)
with f being some function of the square root of the scavenger concentration, c,. When the scavenger concentration is low, the denominator of F(c,) can be expanded using a Maclaurin series56 to give
0
(3
\
-
F(c,) =A1 - f + f 2 -
w 0
(3
f3
+ ...I
(5.3)
The resulting final expansion depends upon the form off. If we take the empirical fitting equation of Warman et al. (eq 2.2), then m
4
1
I
I
I
I
6
8
10
12
14
log (k,c,/s")
F(c,) = C(-ly+yac,y'2 1
(5.4)
Alternatively, the fitting equation suggested by Hummel, eq 2.3, gives m
F(c,) = C(-ly+l(ac,y/2/j! 1
(5.5)
Both these expansions (5.4) and (5.5) are crude approximations in terms of a single parameter to the true description of F(c,), and ultimately of C(cs),and their validity depends upon how well the empirical coefficients,(-lY+'dZ and (-l)'+id/z/j?, respectively, match the true coefficients a,j3, y, etc. Recently, other experimental data have been used to justify different empirical scavenger functions from those postulated by Warman et al.9 and by Hummel.18 In these expressions, the scavenger dependence of the yield is assumed to vary according to
0
W
\
w
0
u (3
F(c,) = (ac2)'/(1 0
4
6
8
10
12
14
I
I
log (k,c,/C') 1.2r
I
I
I
0 \ w 0
u (3
log (k,c,/s-')
Figure 3. Variation of the scavenging yield as a function of scavenger concentrationfor a spur initially containing eight identical radicals with a Gaussian concentration profile of width u = 1.26 nm, diffusion coefficient of radicals, D = 0.5 X lo-* m2 s-I, and a recombination rate, 2k = 3.8 X 1O1O M-I s-l. In all three panels the points ( 0 )are the predictions of the prescribed diffusion model obtained by the numerical integration of eq 4.3 according to eq 4.4. The solid lines are the nonlinear least-squaresbest fits of the empirical functions: (a) Warman et a].: eq 2.2, C,/8 = 0.35, Go/8 = 1.09, a/k, = 2.66 X 1O-Io s; (b) Laverne and Pimblott,Ioeq 2.4 with n = 2, G,/8 = 0.35, Co/8 = 1.01,a/k, = 2.1 1 1O-Io s; (c) Hummel,18eq 2.3, G,/8 = 0.36, Go/8 = 0.99, o/ks = 2.07 x 10-10 s. X
iments fitted or the assumptions of the model used. Most of the early experimental studies of the effects of an added solute on radiation chemical yields in the electron radiolysis of hydrocarbons suggested a square root dependence on the solute concentration
(5.6)
over the range of scavenger concentrations considered. The low-concentration asymptote of this type of dependence is clearly a t odds with the expressions developed in sections 3 and 4. However, the fitting equations do accurately match the experimental data. This apparent discrepancy requires explanation. Consider the different forms suggested for the scavenger dependence of the radiation chemical yields. Each can be expressed in the following form F(cJ =
(3
+ (ac,)")
r#JCS#
(5.7)
where p is a parameter that varies with the scavenger concentration. Figure 4 shows how 1.1 varies with c, for the fitting functions postulated by Warman et ala9(eq 5.4 or alternatively eq 5.6 with u = OS), by HummelI8 (eq 5.3,and by Lipsky"-I3 (eq 5.6 with u = 0.7). For the first two the scavenger dependence of p varies from 0.5 to 0, and for the last it changes from 0.7 to 0. Now the range of 1.1given by the analytic solutions discussed in the preceding two sections depends on the particular system considered, but in all cases the high-concentration asymptotic is 0 and .the lowconcentration value is 0.5. The important point, which must be emphasized, is that the low-concentration asymptotic can be approached from the larger values of p; Le., p can be greater than 0.5, and the change from 0.5 to 0 is not necessarily monotonic. This point is clearly demonstrated if we consider the expansions in powers of presented in the preceding two sections. The value of p in the low-concentration regime is determined by the relative values of the coefficients of the various terms. The experimental data of Lipsky only supports the use of the scavenger dependence described by eq 5.6 over the particular concentration range studied, and any fitting equation that matches the corresponding curve for 1.1 as a function of c, over that concentration range is equally valid. Outside this range other dependences are possible, and there is no reason to believe that a limiting square root dependence should not be found at very low concentrations. (56) Stephenson, G. Mathematical Methods for Science Students; Longman: London, 1961.
Pimblott and Laverne
752 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992
* 1
I*o 0.8
Ex erimentol gepime
0.8
t--------. 0.6
0.6
3. 0.4
0.2
-
n -6
-4
-2
0
2
4
log (c,/ M-') Figure 4. Effect of scavenger concentration on the apparent scavenger concentration dependence of scavenging yields. Each curve refers to the appropriate equation with u = 50 M-I. Curve a refers to the empirical function of Warman et a1.9 (eq 5.6 with v = OS), curve b to that of Laverne and Pimblottio(eq 5.8 with b = a/2), curve c to that of Hum(eq 5.6 with v = 0.7). me]'* (eq 5.5), and curve d to that of Lipsky11*12 It is straightforward to develop forms for F(c,) that have the correct limiting asymptote by introducing a second fitting parameter, and in fact this approach is warranted as we have shown that only for the most trivial conditions (U= 0, g(r) = &(pro)) is a single parameter appropriate. The function that we introduced to fit the scavenger experiments for the high-energy electron radiolysis of waterlo (eq 2.4 with n = 2) only has one fitting parameter, like the functions of Warman et al. and of Hummel. The function is also shown in Figure 4. This empirical scavenger function does not offer a general alternative to the equations of Warman et al. and of Hummel as the power parameter p is always less than or equal to 0.5. Some experimental data do seem to require values of p greater than 0.5. The extended function does, however, suggest a suitable form for an empirical equation to describe scavenger dependences that has two fitting parameters57 (ac,)1/2+ bc,
F(cs) = 1
+
(UC,)'/*
+ bc,
where a and b are the two parameters. For low scavenger concentrations expansion of this equation gives F(c,) = ( U C , ) ' / ~
+ ( b - U)C, + ( u ~-/ 2~ ~ ' / ~ 6+) ...~ ~ / (5.9) ~
The effect of scavenger concentration on the power parameter for this function is illustrated in Figure 5 . The ratio of b to a determines the effect of scavenger concentration on p, while the value of a determines position of the curve: increasing (I shifts the curve to higher concentrations, and vice versa. A comparison of our new equation for F(c,) with one of the experimental Y = 0.7 curves of Lipsky12 is included in Figure 5. The two sets of curves have been arranged to agree over an experimentally accessible range of concentrations. The Y = 0.7fitting requires only one parameter which is rather different from the a parameter obtained in the two-parameter fit. In ref 12 Lipsky considered the radiolysis of decalin with perfluorodeelin as a scavenger. Figure 5 shows that for this system the concentration range over which the c,lj2 dependence is valid is c, < lo4 M. A general estimate of the concentration range is not possible as it will depend on the particular systems under examination. Both the equation of Lipsky (eq 5.6 with Y = 0.7)and that presented here (eq 5.8) are modificationsto the original functions of Warman et al. and of Hummel. The aim of the corrections p
( 5 7 ) A scavenger function with form F(c,) = ( u c , ) ~ / * / + [ ~(bc,)'/']does not give values of p greater than 0.5.
0 -6
I
-4
1
-2
0
2
1 4
log (c*/M-') Figure 5. Influence of the ratio h a in eq 5.8 on the apparent scavenger concentration dependence of scavenging yields. The solid lines refer to the empirical function with the ratio b:a changing from 0 to 8 (0,2,4, 6, 8) and u always being 15 M-l. The dotted line refers to the function of Lipsky with u = 70 M-l.
is to improve the fit of the function to the data. Equation 5.6 makes the correction by altering the concentration dependence to give a function with an inappropriate low-concentration asymptote. However, our alternative attempts to take into account the different coefficients of the terms of higher order in c, and so has the correct asymptote. As empirical fits to data both functions are acceptable, but predictions made using eq 5.6 outside the range of the data must be viewed with extreme caution. The scavenger function we have suggested has the advantage that it correctly suggests the limiting square root dependenceon scavenger concentration and so has a firmer foundation. The only question is whether the experimental data available are sufficiently precise to allow the determination of two fitting parameters in addition to C, and Go. 6. Conclusions
In this paper we have examined the influence of scavenger concentration on yields in radiation chemistry. We used a single pair model to show that for low concentrations the scavenger dependence can be expressed as a power series in c,I/~ in which the coefficients depend upon the initial interparticle separations and upon the interparticle potential. A prescribed diffusion treatment was used to show that the same type of power series dependence is also appropriate for multiple particle spurs. We have looked at the different empirical equations that are used to describe the scavenger dependences of yields. These expressions reduce to a power series in c,I/~ with all the coefficients determined by a single fitting parameter. This series is appropriate in the asymptotic limit where only the first cS1I2term is important or under trivial conditions, for example when the potential U = 0 and the distribution of separations g(r) = &(pro).We have shown that it is necwary to introduce a second fitting parameter to match the true expansion in the experimentally studied range of concentrations. The failure to recognize the importance of higher order terms in cS1I2even at low scavenger concentrations has been the cause of considerable controversy and has led to the development of several empirical scavenger functions which do not have the correct asymptotic dependence on c,I/~. The introduction of a second fitting parameter allows the fitting function to match experiment and to have the correct asymptote. Acknowledgment. We thank Prof. R. H. Schuler for a number of interesting discussions. The research described herein was supported by the Office of Basic Energy Sciences of the US. Department of Energy. This is Contribution No. NDRL-3394 of the Notre Dame Radiation Laboratory.
