T H E
J O U R N A L
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PHYSICAL CHEMISTRY Registered in U . S . Patent Office
0 Copyright, 1977, by the Amsrican Chemical Society
VOLUME 81, N U M B E R 2 J A N U A R Y 27, 1977
Decay Kinetics of the Photochemical Hydrated Electron‘ L. I. Grossweher” and J. F. Baugher Biophysics Laboratoty, Department of Physics, Illinois Institute of Technology, Chicago, Illinois 606 16 (Received June 23, 1976) Publication costs assisted by the National Institutes of Health and the US. Energy Research and Development Administration
A model is proposed for the decay of hydrated electrons generated by photoionization of inorganic anions and aromatic solutes, in which the electron diffuses through a considerable volume of the medium before recombining with its radical coproduct, during which time it may react with scavengers or electrons and radicals generated in other geminate pairs. The analysis based on diffusional recombination theory leads to a new decay function in good agreement with electron decays observed by 265-nm laser flash photolysis of aqueous I-, Fe(CN):-, tryptophan, and tyrosine. The dependence of the electron lifetimes on scavenger concentration and the initial electron concentration are in quantitative agreement with the theory, where the latter is developed in terms of a time-dependent rate constant to include the bimolecular electron-electron and electron-radical reactions. The recombination lifetimes are -lo4 times longer than predicted by the Noyes recombination theory and comparable to those deduced in the earlier photochemical scavenging experiments of Stein and co-workers. It is proposed that the initial separation of the hydrated electron and radical supresses fast “Noyes type” recombination and permits the electron to enter a regime in which diffusion-limited back reactions with the original radical remain probable in the absence of high scavenger concentrations.
Introduction the “long time” photochemical electron are identical with the hydrated electron generated by water radiolysis was The generation of hydrated electrons by ultraviolet flash questioned in recent work of Bryant et al.7 where it was photolysis of inorganic anions and aromatic molecules in aqueous solution was demonstrated many years a g ~ . ~ - ~observed that electrons generated by 265-nm laser photolysis of inorganic anions and aromatic amino acids decay The hydrated electron was identified by its characteristic faster than can be explained by the (ea; + eaq-) and red absorption band, the quenching action of electrophilic agents such as O2and N20, and the observation of apscavenger reactions in the bulk. It was proposed that the hydrated electron and its radical coproduct form a loosely propriate counterradical spectra, e.g., 12-from I-5 and the bound complex of 1 ps duration in which the back rephenoxy1 radical from phenol.6 The detailed analysis of the electron decay kinetics was not feasible in this early action competes with separation into the free species. The work because the flash lamp durations were comparable “loose complex” model is consistent with the significant to the electron lifetimes. However, it was assumed that loss of electrons during the time period from about 50 ns the reactions of the photochemical hydrated electron are to 1p s , the approximately exponential time dependence similar to the radiolysis hydrated electron, i.e., competition during the initial decay stage, the comparable activation energy for the observed decay rate with that for “inverse between the bimolecular (eaq-+ eaq-) reaction and the viscosity” of water, and the similarity of the initial and pseudo-first-order reactions with available scavengers “long time” electron absorption spectra. including the original photolyte. In addition, bimolecular back reactions are possible between the electron and its Prior to these flash photolysis studies, Stein and his radical coproduct which would not be easily separated from colleagues proposed that hydrated electrons are generated (ea; + ea;). The tacit assumption that the reactions of from aqueous I-8-10and phenolate” ions based on steady
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a3
94
L. I. Grosswelner and J. F. Baugher
irradiation quantum yields in the presence of electron scavengers. The quantum yield of electron scavenging was found to obey the square-root concentration dependence predicted by the Noyes theory of diffusive recombinat i ~ n , ' ~ -in' ~ which the geminate coproducts escaping primary recombination undergo secondary diffusive recombination in competition with scavenging in the "cage" and diffusive separation into the bulk. According to Noyes, the efficiency of scavenger action is given by
p(t) = e - k s ( s ) t [ l- Joth(t')dt']
J , " h ( t ' ) [ l - e-ks(S)t'] dt'
p ( t ) N e - k s ( s ) t [ l- p'+ 2a/t1/2]
where h(t) is the probability (per s) that a radical pair generated at t = 0 (or interacting at t = 0 without reaction) recombines at time t, k, is the bimolecular rate constant for the scavenging reaction, and (S) is the bulk scavenger concentration. The choice of h(t) based on the random walk of a particle in three dimensions:
These equations predict that the electron decay function is the product of the exponential rate as determined by reactions with scavenger S (which may be the photolyte) modulated by a scavenger-independentterm related to the diffusive recombination of the geminate pair. Although the decay predicted by (6) is nonexponential, the mean electron lifetime can be defined as
(5) We proceed formally by substituting h(t) of (1)leading to the decay function
~ ( t= )e - h s ( S ) t [-l p'erfc (a/p')(n/t)1/21 (6) where erfc x = (2/#) Sx"e-r2dx. The expansion: erfc x = 1 - (2x/7#)[1 - x 2 / 3 + x4/10 - ...I leads to an approximate form of (6) valid for t
-
t
leads to the approximate solution
Y = Yr + 2ar [ ~ k ~ ( s ) ] ' / ~
(2) where y is the total quantum yield for electron scavenging, yr is the "residual yield" for the scavenging of electrons that escape recombination at low scavenger concentrations, and r is the quantum yield for generation of the geminate radicals. The parameter p' is the total probability for recombination of the geminate pair
$ = Jo"h(t') dt'
(3)
and is related to the photochemical quantum yields by: 0' = 1- yr/I', Jortner et obtained the exact solution of (1): y = r[1- p'e-(2a/P')Cnks(S)1'/2] (4) which reduces to (2) at low scavenger concentrations. The functional dependence predicted by (2) or (4)was found for various photolytes (I-, Br-, Fe(CN),4-,phenolate) in the presence of electron scavengers (H', NzO, 02, H2P04-, acetone) with approximately correct relative ~ ~wever, ~ - ~ ' the values of k, from system to ~ y s t e m . " ~ ~Ho parameter 2 a ( ~ k , )ranges ~ / ~ from 2 to 60 M-lj', which is much too large to be consistent with the Noyes theory of diffusive displacements.ls These results imply values of a N lo4 s1i2 and "cage" lifetimes -lo4 s, while the Noyes theory leads to u < lo* and lifetimes s. Nevertheless, there has been no apparent explanation for the discrepancy excluding the total inapplicability of the diffusive recombination theory. Dainton and Loganlg proposed an alternative mechanism in which the rapid reaction of the photoelectron with a scavenger may not permit adequate time for the formation of the ionic atmosphere of the electron, leading to a decreased rate constant for scavenging by charged solutes at high solute concentrations. However, this model does not explain the concentration dependence of scavenging quantum yields observed by Stein and co-workers. Electron Decay Theory The photochemical scavenging studies and laser flash photolysis measurements provide different probes of the same photochemical process; Le., competition between the fast back reaction of the electron with its radical coproduct and the bimolecular reactions of the electron with an initially randomly dispersed, scavenging solute. Accordingly, the probability that an electron survives both decay process from t = 0 to t = t is given by The Journal of Physlcal Chemistw, Vol. 81, No. 2. 1977
>> TU^//^'^:
(7)
= JoO't
dP/So"dP (8) The integrations in (8) are easily carried out using (l),( 5 ) , and the approximation of (7) for the [ l - S;h(t? d t l term in dp leading to
-
t=
[Formally, each of the 6 integrals in (8) is a Laplace transform L, with p = k,(S).] In comparing the predicted dependence of t on (S) with experimental data it is convenient to use the reciprocal form:
l/t= k , ( S )
The probability that a geminate pair eventually recombines is also of interest. Since (y - yr)/I' is the probability for scavenging and y,/r is the probability that the electron escapes both recombination and scavenging, the recombination p r follows from (4)as
pr = p'e-2(a/P')[sks(S)]'/2 (10) Alternatively, the same result can be calculated directly from pr = Jowh(t)e-hs(s)f dt
The attempt to calculate t in the absence of scavengers by substituting (1)in (8)leads to a divergent integral. This result is not surprising because h(t) is based on random walk in three dimensions, in which there is a finite probability that two particles executing independent steps will never meet.2O Nevertheless, the mean electron lifetime in the absence of scavengers can be estimated as the time required for h(t) to fall from the maximum value h(t*) to h(t*)/e h(t 3. Direct differentiation of (1)leads to t* = ( 2 ~ / 3 ) ( a / p ' ) ~ t' = 9 . 1 8 ( ~ / p ' ) ~ For example, if alp' equals or s1/2, the corresponding electron lifetimes are and s, respectively. Alternatively, the mean lifetime may be defined as the time required for the integrated probability of recombination to attain half the maximum value. Taking S t h(t) dt = p'/2 gives
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Decay Kinetics of the Hydrated Electron
t” = 1 3 . 8 ( ~ / / 3 ’ ) ~
(12)
The electron decay function (5) can be related to a time-dependent scavenging rate constant kt by differentiation:
i.e.
For the case where t
k , = k,(S) +
1
>> . l r ~ ~ / p ’ ~ : 1 I
2t
+ (t3/2/U)(I- 0‘)
t (,used
3
4
Figure 1. Comparison of eq 6 with electron decay after 265-nm laser photolysis of aqueous 330 pM tryptophan. The lines are calculated for = 1, k, = 3.6 X lo8, and for following values of B (a) 4 X (b) 3.5 X (c) 3.0 X (d) 2.5 X (e) 2.0 X (f) 1.5 x 10-4.
@‘
The integrated solution (6) is preferable to (14) because it can be compared directly to experimental data over the full decay period. However, the neglect of reactions between electrons generated in different pairs cannot be strictly correct. The bimolecular contribution to electron decay may be estimated for the case of high recombination probability by substituting the binomial expansion of (14) into the differential rate equation:
-d(eaq-)/dt = kt(ea,)
2
+ k’(ea,)2
I ah
(15)
leading to the integrated solution for the case where y (1- p’)/2a
