J . Phys. Chem. 1990, 94, 8557-8561 SO2groups. A complex bound by an ion-dipole interaction was found to have a binding energy of 15.6 kcal/mol.
Acknowledgment. We thank the donors of the Petroleum Research Fund, administered by the American Chemical Society,
8557
for financial support. Computer time for this study was made available by the Alabama Supercomputer Network and the NSF-supported Pittsburgh Supercomputer Center. Registry No. SO2,7446-09-5; SO2+,12439-77-9.
Microscopic Solvation Dynamics and Solvent-Controlled Electron Transfer Ilya Rips, Joseph Klafter, and Joshua Jortner* School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978 Tel- Aviv, Israel (Received: October 12, 1989; In Final Form: April 30, 1990)
In this paper we investigate solvent size effects on solvent-controlled electron transfer (ET) in polar liquids. An interrelationship was established between the solvation time correlation function and the ET rate, which was presented within the first passage time approximation. For the case of activationless ET the ET rate, which is equal to the initial solvation rate, was expressed within the framework of the mean spherical approximation. The theory predicts that the activationless ET times are shorter than the average dipole solvation times, in accord with experimental data for intramolecular ET in electronically excited bianthryl in nonassociated polar solvents.
1. Introduction
The traditional treatment of outer-sphere electron transfer (ET) in polar solvents' rests upon the implicit assumption that the dielectric relaxation solvation process of the medium does not affect the ET reaction kinetk2" Marcus in his study of ET assumed that the orientational solvent polarization is in equilibrium with the momentary charge di~tribution.~-~ In the nonadiabatic ET theory of Levich and Dogonadze the microscopic electronic process, rather than the dielectric relaxation, constitutes the rate-determining ~ t e p . The ~ , ~validity conditions of nonadiabatic ET hold for a multitude of ET processes, and this theory has dominated the field for more than two decades.'" Recent progress in the area of ET kinetics7-I9has been mainly due to the realization that the solvation process of the donoracceptor pair can constitute the rate-determining step. This regime will be referred to as the solvent-controlled ET. The criterion for the realization of the solvent-controlled ET is determined by the magnitude of the adiabaticity p a r a ~ n e t e r ~ . ' K~A- '=~ 4 ~ V ' % ~ / f z Ewhere ,, Vis the electronic coupling, E, is the medium reorganization energy, and
( I ) Ulstrap, J. Charge Transfer Processes in Condensed Media; Springer Verlag: Berlin. 1979. (2) Marcus, R. A. J. Chem. Phys. 1956, 24, 966. (3) Marcus, R. A. J. Chem. Phys. 1956, 24, 979. (4) Marcus, R. A. Annu. Reu. Phys. Chem. 1964, 15, 155. (5) Levich, V. G.; Dogonadze, R. R. Dokl. Akud. Nuuk. 1959, 124, 123 [Proc. Acad. Sei. U S S R , Phys. Chem. Sect. 1959, 124, 91. (6) Levich, V. G. Ado. Elecirochem. Eng. 1965, 4, 249. (7) Zusman, L. D. Chem. Phys. 1980, 49, 295. (8) Alexandrov, I. V. Chem. Phys. 1980, 51, 499. (9) Van der Zwan, G.; Hynes,J. T. J . Chem. Phys. 1982,76,2993. Hynes, J. T. J. Phys. Chem. 1986, 90, 3701. (IO) Calef, D. F.;Wolynes, P. G. J. Phys. Chem. 1983,87, 3387; J . Chem. Phys. 1983, 78, 470. ( I I ) Sunk H.; Marcus, R. A. J . Chem. Phys. 1986, 84, 4894. ( I 2) Rips, 1.; Jortner, J. J . Chem. Phys. 1987, 87, 2090. (13) Rips, I.; Jortner, J. J . Chem. Phys. 1987, 87, 6513. (14) Rips, 1.; Jortner, J. J . Chem. Phys. 1988, 88, 818. (15) Rips, 1.; Jortner, J. Chem. Phys. Lett. 1987, 133, 411. (16) Rips, 1.; Jortner, J. hit. J . Quanium. Chem. Symp. 1987, 21, 313. (17) Sparpaglione, M.; Mulkamel, S.J . Chem. Phys. 1988. 88, 3263. (18) Newton, M. D.; Friedman, H.L. J. Chem. Phys. 1988, 88, 4460. (19) McManis, G. E.; Weaver, M. J. J. Chem. Phys. 1989, 90, 1720.
0022-3654/90/2094-8557$02.50/0
7L is the longitudinal dielectric relaxation time of the solvent. For a solvent with a Debye-type dielectric susceptibility 7L = (C,/t?)7D, where cs and t, are the static and the high-frequency (optical) dielectric constants, respectively, and rDis the Debye relaxation time. When K~ >> 1, the solvent-controlled ET regime is realized, with the ET rate, kET, being given by the Arrhenius-type expression k€T = A e x p ( - E a / k B T ) (I. 1) E, is the activation energy'" Ea = (hE - E,)2/4E,,with AE being the (free) energy gap. The frequency factor A in this regime is inversely proportional to the longitudinal relaxation time7J2-I4 A = 6/7L ( K A >> 1) (1.2) with 6 = ( E r / 1~ T ~ ~ T ) ' / ~ [ ~ ( A E , E , ) ] -(1.3) '
h(AE,E,) is a form factor, which depends upon the actual character of the crossing of the parabolic diabatic potential surfaces and varies in the range h(lAE,E,) = 0.7-1.5 over the physically relevant region 0 I AE/E, I 2.0. This result, together with the plausible value of the solvent reorganization energy E, = 1 eV,' implies that 6 is close to unity at T = 300 K (within 50%) so that A 1/ T ~ .Previous studies of solvent-controlled ET'-'9 rest on two central assumptions: ( I ) The dielectric continuum approximation for the polar solvent. The molecular structure of the solvent and, in particular, of the first solvation shell(s) is disregarded. (2) The linear response theory. The effects of dielectric saturation associated with the large electric field in the vicinity of the donor-acceptor pair are neglected. In this paper we extend the theory of solvent-controlled ET to account for the effects of the molecular structure of the solvent on the ET dynamics. This is achieved by replacement of the longitudinal dielectric relaxation time r L by an average solvent relaxation time for ET, which accounts for the finite solvent size effect. We apply the microscopic theory of ion or dipole solvation to describe solvent-controlled ET. From the experimental point of view this extension of the theory is of considerable current interest for the description of activationless intramolecular ET in polar solvents. The remarkable adherence of the intramolecular ET rates in linear alcohol solvents to the relation kET TIL-',^^^^ 0 1990 American Chemical Society
8558 The Journal of Physical Chemistry, Vol. 94, No. 23, 1990
where T I L is the longitudinal relaxation time of the clustered alcohol molecules,24does not provide a conclusive proof for the validity of the continuum result, (1.2), because of two reasons. First, in more recent studies2+28deviations from this simple result have been observed. Second, solvent-controlled ET and solvation kinetics in associated solvents are complicated by specific interactions, whereupon the simple experimental relation in aliphatic alcohols may originate from a cancellation between the effects of structure breaking of the hydrogen-bonded solvent on the one hand and molecular size effects on the other hand. Indeed, a recent experimental study28of intramolecular activationless ET of the bianthryl molecule in Debye-type nonassociated polar solvents reveals that the effective relaxation time, which determines the solvent-controlled ET rate, is longer than the continuum dielectric model predictions. This calls for the extension of the theory, which rests on the calculation of the frequency factor for solvent-controlled ET within the microscopic molecular theory of solvation.29-37 Preliminary results of this work have been presented elsewhere.18 The influence of solvent molecularity on solventcontrolled ET was considered in a recent work by McManis and Weaver,I9 who have utilized the mean spherical approximation (MSA) for solvation dynamics, as developed by W o l y n e ~and ~~ the present author^^^^^^ to calculate the frequency factor and the activation energy for solvent-controlled ET. The characteristics of the free energy of activation emerging from their analysis,I9 which are in accord with previous results,35 will not concern us further, as we shall focus on activationless ET, which is of considerable current experimental i n t e r e ~ t . ~ ~The - ~ *numerical res u l t ~for ’ ~ the frequency factor, A, which rest on the MSA for a single ion, demonstrate the reduction of A relative to the continuum limit, being in accord with our preliminary analysis,18 which is extended in this paper, concerning the effects of dipole solvation dynamics on solvent-controlled ET. It would be useful to gain further insight into solvent-controlled ET by deriving analytical expressions for the ET rate, which allow for the elucidation of the interrelationship between the characteristic lifetimes for solvent-controlled ET and for s ~ l v a t i o n . ~In~ -the ~ ~present paper we explore the relation between the solvent-controlled ET rate and the solvation time correlation function for dipole solvation. For the relevant experimental situation of intramolecular within an electronically excited state of a large polar molecule, we have utilized the MSA for dipole solvation,35rather than for ion s o l ~ a t i o n ,as~ done ~ - ~ ~by McManis and Weaver,I9 to handle the ET dynamics. We have utilized the mean first passage approximation within the framework of the MSA to derive explicit results for activationless ET rates which are confronted with experiment.28 11. The Electron-Transfer Rate
We shall establish the interrelationship between the solvation time correlation function (TCF)33-37and the ET rate. The solvation TCF describes the microscopic solvation dynamics induced by the instantaneous formation of an electronically excited ion (20) Kosower, E. M.; Huppert, D. Annu. Reo. Pfiys. Cfiem. 1986,37. 127 and references therein. (21) Kosower, E. M.; Huppert, D. Cfiem. Pfiys. Left. 1983, 96, 433. (22) Heisel, F.; Miehe, J. A. Cfiem. Pfiys. Leu. 1986, 128, 323. (23) Su.S. G.; Simon, J. D. J. Pfiys. Cfiem. 1986, 90, 6475. (24) Garg, S. K.; Smyth, C. P. J . Pfiys. Cfiem. 1965, 69, 1294. (25) Huppert. D.; Ittah, V.; Kosower, E. M. Cfiem. Pfiys. Lett. 1988,144, 15.
(26) Su,S. G.; Simon, J. D. J . Cfiem. Pfiys. 1988.89, 908. (27) McManis, G. E.; Golovin, M. N.; Weaver, M. J. J . Phys. Cfiem. 1986, 90. 6563. Nielson, R. M.; McManis, G. E.;Golovin, M. N.; Weaver, M. J. J . Pfiys. Cfiem. 1988, 92, 3441. (28) Kang, T. J.; Kahlow, M. A.; Giser, D.; Swallen, S.; Nagarajan, V.; Jarzeba, W.; Barbara, P. F. J . Phys. Cfiem. 1988, 92, 6800. (29) Onsager, L. Can. J. Cfiem. 1977, 55, 1819. (30) Calef, D. F.; Wolynes, P. G. J . Cfiem. Pfiys. 1983, 78, 4145. (31) Friedrich, V.; Kivelson, D. J . Cfiem. Pfiys. 1987, 86, 6425. (32) Loring, R. F.; Mukamel, S. J . Cfiem. Pfiys. 1987, 87, 1272. (33) Wolynes, P. G. J. Cfiem. Pfiys. 1987, 86, 5133. (34) Rips, 1.; Klafter, J.; Jortner, J. J. Cfiem. Pfiys. 1988, 88, 3246. (35) Rips, I.; Klafter, J.; Jortner, J. J. Cfiem. Pfiys. 1988, 89, 4288. (36) Maroncelli, M.; Fleming, G. R. J . Cfiem. Pfiys. 1988, 89, 875. (37) Calef, D. F.; Nichols, A. L. J . Cfiem. Pfiys. 1988. 89, 3783.
Rips et al. (or dipole) in a polar solvent. The solvation TCF is given by33-37
s(t) = - E,(m)l /[E,(O) - E,(m)l (II.1) where E&) is the orientational part of the ion (or dipole) solvation energy at time t . Our aim is to relate the solvation dynamics on a single potential surface to the ET dynamics involving two diabatic surfaces. The current theoretical description of microscopic solvation dynamics rests on Onsager’s “inverted snowball” pict ~ r e , which * ~ asserts that the solvent relaxation time in the immediate vicinity of the ion or dipole is T D , while the relaxation time in the bulk is T L (with TD >> T L ) . Solvent relaxation proceeds from the exterior solvent toward the solute, involving a hierarchy of time scales from T L to T D . A quantification of the “inverted snowball” picture was within the framework of the mean spherical approximation (MSA) being applied for both ion and dipole solvation. This formalism will now be utilized to describe solvent-controlled ET. We shall assume that the ET kinetics is exponential. This was shown to be a reasonable approximation even in the case of activationless solvent-controlled ET, which is a most unfavorable situation from this point of view. An implicit account for the molecular structure of the solvent affects both the Arrhenius factor and the frequency factor. The activation energy depends upon the reorganization energy. The evaluation of the solvent reorganization energy within the MSA is discussed in previous work.35 We shall focus on calculation of the frequency factor within the molecular theory of solvation. The ET rate, kET, can be expressed in terms of the time correlation function (TCF), 4(f), of the reaction coordinate, Q(t) (given in energy units), which is defined by8-I5 (11.2) @J(O= ( Q ( 0 Q(o))/(Q(O) Q(0)) In the case of ET the reaction coordinate is identified with the nonequilibrium solvation energy of the donor-acceptor pair, so that Q(t) = E,W - E(..)
(11.3)
Accordingly, $ ( t ) can be identified with the solvation TCF. The ET rate can now be expressed in the form9J3
with
where Ir/(Q,Q’;t) = (2nA2[1 - S2(t)])-1/2 exp
[Q - Qs(t)12 2A2[1 - S*(t)] (i1.6)
The equilibrium distribution is 4,(Q) = ( 2 ~ A ~ ) -exp(-Q2/2A2) l/~
(11.7)
and A2 = (Q(0)2) = 2ErkBT. Equation 11.6 provides a straightforward way of calculating the ET rate within the microscopic picture of solvation. We shall focus on intramolecular ET where the initial state is neutral and the final state involves a “giant dipole”. One way to proceed is to utilize the numerical result^^^,'^ for the microscopic solvation TCF for the dipole, which replaces the TCF of the ET reaction coordinate. This procedure, however, rests solely on numerical calculations of kET. Alternatively, we have found a simple way to calculate the ET rate based on solvation kinetics, which also provides a physical insight for the dynamics of solvent-controlled ET. 111. The Frequency Factor
We return to the general scheme of the crossing of two diabatic potential surfaces corresponding to the “neutral” and the “ionic“
The Journal of Physical Chemistry, Vol. 94, No. 23, 1990 8559
Solvent-Controlled Electron Transfer states, respectively. For definiteness, we assume normal crossing, Le., AE < E,, between the potential surfaces. We shall also assume that the system is initially prepared with polarization of the solvent being in equilibrium with the charge distribution in the 'neutral" state. The basic connection between solvation and ET dynamics can be established by relating the time-dependent solvation rate k , ( t ) = -d In S(t)/dt (111.1) to the frequency factor, A, for ET. Invoking the first passage time approximation for reaching the crossing point, Qo,of the two potential surfaces, we can define the characteristic time t*, which is required to reach Qo,starting from E,(O) = 0. The frequency factor for the ET is approximately given by the average of k,(t) over the time interval necessary to reach the crossing point A
= (1 / r * ) l "0d t '
k,(t')
which together with ( 1 1 1 . 1 ) results in A = -(l/t*) In S ( t * )
7
3 ' 'I
II
I5
(111.2)
(I I I. 2a)
The characteristic time t* can be expressed in terms of the ratio of Qoand the displacement AQ between the minima of the diabatic potential surfaces. We shall utilize the two parabolic potential surfaces6-'
19 ' 23
27
31
35
39
I 43
ES/COl
Figure 1. Dependence of the activationlesssolvent-controlled ET time normalized by solvent longitudinal relaxation time on the solvent polarity tS/t,. Calculations within the MSA for r , / R = 1.
O
ui(Q)= Q2/W U2(Q) = (Q - 2E,)2/4E, - AE
(111.3)
The relevant parameters of the reaction coordinate are given by Qo = E, - AE and AQ = 2 4 , for the case of normal crossing of the potential surfaces. From the definition of the solvation TCF 1 - S(r*) = Qo/AQ, so that S(t*) = (E, + AE)/2Er (111.4) Equations 111.3 and 111.4 result in A = ( l / f * ) In ( 2 E , / ( E , + A E ) )
(111.5)
with the characteristic time, t*, being determined by (111.4). In general, this can be accomplished only numerically or by using the short-time expansion of the solvation rate k,(t)
= k,(O) + k,(O)t
+ (l/2)ks(0)t2 + ...
(111.6)
The situation is considerably simplified in the practically important case of activationless ET ( E , = AE). In this case the activationless ET rate, kETo = A, is equal to the initial solvation rate, Le., ET' = M O ) (111.7) According to the Onsager picture of solvation, the dominant contribution at short times orginates from distant solvent molecules, for which the point dipole approximation for the solute molecule is justified. Therefore, in evaluation of the frequency factor one can use the initial dipole solvation rate. The latter can be expressed in terms of the Laplace transform, S(p), of the solvation T C F
A derivation of this result together with expressions for the initial solvation rate for an ion and a dipole are presented in the Appendix. We shall focus on the situation for the dipole solvation in the case of r , / R = 1 (where r, and R are the solvent radius and the dipole radius, respectively), which is easy to calculate3s and which is physically relevant. In this case the initial dipole solvation rate is
k,d(O)
+
1
[ 4 . 7 6 ~ , ' / ~41
TL
6 [ 4 . 7 6 ~ , ' /+ ~ 4][(t,/~)'/ ~11
=-
(111.9)
The inverse initial solvation rate is longer than the longitudinal dielectric relaxation time. An obvious implication of this result is that the activationless ET time TET = (kETo)-' = (k,d(O))-l is
3''
7
I1
15
19
23
27
31
35
39
43
Es/Em
Figure 2. Dependence of the activationlesssolvent-controlled ET time normalized by the average dipole solvation time on e,/€,. Calculations within the MSA for r , / R = 1.
longer than the longitudinal dielectric relaxation time. The ratio between the two times can be obtained from (111.9): 6 ( ~ , '+ / ~0.84)[(t,/t,)1/6 TETO N 7L
+
(tm1l6 0.84)
- I]
(111.IO)
This ratio is plotted on Figure 1 vs e,/e,. It is apparent that the rW/TL ratio exhibits a weak dependence on €,/E, and changes from about unity in weakly polar solvents to -3 in higher polar solvents. It should be noted that the result is insensitive to the actual value of the optical dielectric constant within the physically acceptable range of values of e,. On the other hand, the initial dipole solvation time is shorter than the average dipole solvation time T: = J;dt' S(t'),implying that35 TET < 7:. An explicit expression for r,d was previously giver^.'^-^' For the ratio TET/T: in the relevant case r , / R = 1 we obtain from the previous results for T: 3s and from (111.9) TET 3 6 ( ~ . / ~ , [)I ~-/ (~ ~ N 7:
(1 - e m / € $ )
/ t ~ ) ' / ~ ] ~
(111.1 1)
Since the distribution of relaxation times for the dipole solvation is narrow, the difference between TET and T , should be rather small. In Figure 2 the ratio T E T / T , ~ is exhibited as a function of e,/c,. In the physically acceptable range of variation of the parameter e,/:, the average dipole solvation time is longer than the activationless ET time. The difference ranges from a factor of 1.5 for the weakly polar solvents to about 3.0 for the highly polar ones. In the presence of an activation barrier the determination of the frequency factor requires the solution of (111.4). For this purpose one can assume an analytic form of the S ( t ) function.
8560 The Journal of Physical Chemistry, Vol. 94, No. 23, 1990
Rips et al.
TABLE I: Theory and Experiment for Activationless ET and Solvation Dynamics
1.81
36.23
PS 0.19
propionitrile
I .87
28.64
0.31
butyronitrile pentanitrile propylene carbonatd
I .90 1.93 2.63
22.23 19.56 65
0.53 0.74 1.74
t,
6s
lETlTL
T s I ~ L
TL,
solvent acetonitrile
exptUc 4.74" 2.94b 4.82" 2.74b 3.96" 4.86' 3.60" 2.29b
TET/T~
cakd 5.89
cake 2.8
exptc 3.5
cake 0.47
4.92
2.6
4.0
0.53
4.1 1 3.75 6.76
2.4 2.2 3.0
4.0 6.0 2.0
0.57 0.60 0.44
expt0.78'J I . 19b.r 0.80"*c 1.45b.c 0.95a1c 1 .2"J 0.5sa,' 0.86b-c
From experimental solvent relaxation of coumarines (refs 36-40). From recent experimental solvation times of coumarins (ref 41). 'Experimental ET times in bianthryl (ref 28). dCalculated from the theory of ref 35 with r , / R = I . 'Calculated from (1Il.lO) and (111.1 I ) . JThe optical dielectric constant determined from t, = 1 .3n2
As a particular example, one can take the solvation TCF in the form of a stretched e x p ~ n e n t i a l ~ ~ . ~ ~ S(r) = exp(-(t/r)") (111.12) where T and a (aI1) are the parameters. This particular analytic form has been shown to provide a good fit for the experimental solvation kinetics. It also provides a reasonable approximation to the microscopic solvation kinetics predicted by the MSA in the physically relevant time domain. However, it should be borne in mind that the stretched exponential predicts an incorrect behavior both in the short-time limits and in the long-time limits of the TCF. Indeed, stretched exponential is characterized by the vanishing initial relaxation time instead of k,(O) and does not possess the appropriate long-time behavior (which should be exp(-t/rD). For this reason, the stretched exponential cannot be used to study both low- and high-barrier ET reactions. It is only in the intermediate domain that it constitutes a reasonable approximation. With the solvation function in the form of a stretched exponent, one can derive a closed expression for the ET time, TET = 1/ k E T
[A]] Ija-1
TET
=
exp(E,/kBr)
(111.13)
with E, = (E, - AE)2/4E,. The ET time can be compared with the average solvation time, for which a closed analytic expression is obtained: 7 , = 7r(i/a+i) (111.14) where r ( x ) is Euler's gamma function. The ratio between he ET time and the average solvation time is
5) which depends on the Arrhenius factor exp(E,/kBT).
IV. Confrontation with Experiment The rate of the solvent-controlled activationless ET process is given by (111.9) in terms of the initial dipole solvation rate. We now compare the theoretical results with the recent experimental data of Barbara et aL2* for the intramolecular ET rates in the electronically excited state of the bianthryl molecule in polar nonassociated solvents. This ET process is induced by a strong (V > 100 cm-I) intramolecular coupling, so that the adiabaticity parameter, K ~ is, apparently sufficiently large to make the solvent-controlled ET limit applicable. Finally, this ET process is nearly activationless. The experimental data of Barbara et al.28 show that the intramolecular ET in a series of nitriles and acetates TET/TL > I , i.e., the experimental ET time scale, exceeds T~ by a numerical factor of 2-6. This deviation of the time scale for the activationless ET from the prediction of the continuum dielectric model can be attributed to molecular size effects. In Table (38) Castner. E. W . Jr.: Maroncelli, M.; Fleming, G. R.J . Chem. Phys. 1987.86, 1090. (39) Maroncelli. M.: Fleming, G. R. J . Chem. Phys. 1987, 86. 6221.
1 we compare these experimental valuesZ8of TET/TL with theoretical predictions. The calculations were performed for the dipole solvation with r,/R = 1. From these results we infer the following: I . The experimental ET time is shorter than the averaged solvation time, T,, as expected from the truncation of the dipole solvation process at short times (Figure 2). 2. The increase of the TET/TL ratio above the continuum model prediction of unity is due to the molecular size and dielectric screening effects on the short-time solvation process. 3. The experimental data for TET/TL are in reasonable, although not perfect, agreement with theory. The predicted enhancement of TET/TL beyond the continuum model value of unity is borne out by experiment. 4. The ET times are expected to be shorter than the averaged solvation times. The experimental results of Fleming et al.38.39 and of Barbara et aI.@seemed to be in accord with this prediction (Table I). However, the recent experimental results of Kahlow et a1.,4' also presented in Table I, shed doubt on this conclusion, indicating that TET/T, 1 for the nitriles. Obviously, the comparison of T , for coumarins with TET for bianthryl is only tentative and additional experimental data will be desirable. Two limitations of the theory should be noted. The first is its inability to reproduce the experimental trend of increasing TET/TL with the increase of the length of the alkyl chain. This is not surprising as the theoretical estimates in Table I were performed assuming that the solute/solvent radius ratio equals unity. Since the initial solvation rate of a dipole (ion) increases with decreasing r s / R , we can expect that for the proper values of this ratio the calculated values of TET/TL will show an increase with increasing size of the solvent molecule. Second, the comparison between theory and experiment rests on the assumption that the ET process is strictly activationless. The existence of a low-energy barrier (E, C k B T ) for the solvent-controlled ET will enhance the calculated value of TET/TL by the reciprocal Arrhenius factor exp(E,/kBT), washing out the discrepancy between the prediction of the continuum theory and experiment. A small contribution of an activation barrier for ET is expected to increase the ratio TET/T, possibly beyond unity. At present, the available experimental data for 7, (Table I ) cannot distinguish between the possibilities T E T / T ~ < 1 and TET/T, 1. The possibility of a plausible contribution of a small energy barrier renders the interpretation of experimental results somewhat ambivalent. In this context the solvation kinetics on a single potential surface presents a better test for the theory of molecular size effects on solvation dynamics.
-
-
Acknowledgment. This research was supported in part by the BMFT (Contract No. 3 17-4003-0328955A) under the auspices of the University of Munich. Appendix. The Initial Solvation Rate
The initial solvation rate is given by (40) (a) Kahlow, K. A.; Kang, T. A.; Barbara, P. F. J . Phys. Chem. 1987, 91, 6452; (b) J . Chem. Phys. 1988, 88,2372. (41) Kahlow, M. A.; Jarzeba, W.; Kang, T. J.; Barbara, P. F. J . Chem. Phys. 1989, 90, I5 1 .
J . Phys. Chem. 1990, 94. 8561-8567 d k,(O) = -- In dt
~(t)l,=,= ,
-S(O)
(A. 1)
8561
Substitution of (AS) gives for the initial dipole solvation rate
To evaluate the derivative s(O),we use the short-time expansion of the solvation T C F ~ ( t=) I
+ S(O)+ O(t*)
(A.9) (A.2)
The Laplacc transformation of (A.2), $p) = J;dt exp(-pt)S(t), has the form34
For most of the applications, the mated by
fi@)
N
A@)
function can be approxi-
4.7622[?@)]
(A.lO)
leading to the final expression for the initial dipole solvation rate k,'W(o) As a result, the initial solvation rate can be expressed in terms qf the admittance of the system, resulting in (3.8). Expressing
S ( p ) in the form
j.@)= [2@) - 2(O)l/d2(m) - 2(0)l
We now evaluate the initial solvation rate explicitly for the dipole and for the ion. 4. Dipole Solvation. We have shown that in this case jib) .$@),
$(p) =
51Il
7~
6[(eS/e,)1/6
+ 41
- 1][4.76e,'/6
(A.ll)
B. Ion Sohalion. In this case the MSA admittance is given by35
('4.3)
where i ( p ) is the complex admittance of the system, ks(0)assumes the form
a
+
[4.76~,'/~ 41
1 -
-4)
(A.12)
where the dynamic correction factor, Ab),can be approximated by
h@)N
3(r,/Ri)(4.76[Z(p)]'/6
- 21-l
(A.13)
where r, and Ri are the solvent and the ion radius, respectively. Combining (A. 15), (A. 12), and (A. 13), we derive the final result for the initial ion solvation rate in the case of the Debye solvent:
h@P+
C=
2.381 lem'/6(es/em - 1) (4.7622~,'/~ - 2)2
(r,/R)
(A.15)
Temperature Dependence of the Rate Constants for Reactions of the Sulfate Radical, SO4-, with Anions Robert E. Huie* and Carol L. Clifton Chemical Kinetics Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899 (Receiced: February 22, 1990; In Final Form: June I I , 1990)
Rate constants have been measured as a function of temperature over the range 10-60 O C for the reactions of the sulfate radical, SO4-,with azide, chloride, cyanate, cyanide, acetate, and carbonate anions. Room-temperature rate constants ranged from about 5 X IO6 to 5 X IO9 M-' s-I. The variation of the rate constant with substrate depended mainly on the Arrhenius preexponential factor and less on the activation energy. For the reaction of SO4- with CI-, the dependence of the rate constant on ionic strength was determined to be in agreement with the theoretical prediction.
Introduction There have been a large number of rate constant measurements on the reactions of small inorganic free radicals in aqueous solutions,' almost all of which have been carried out only at room temperature. The change in a rate constant with temperature can provide a significant amount of information about the nature of the reaction. For example, abstraction reactions typically exhibit ( I ) Neta. P.:Huie. 1027.
R. E.;Ross, A. B.J . Phys. Chem. Ref. Data 1988,.17,
a systematic variation of the temperature dependence (the activation energy) with the bond strength of the atom being removed. Addition reactions may have a more complicated relationship of the rate constant with temperature, particularly if the reaction involves reversible complex formation. In electron-transfer reactions, the temperature dependence is expected to be related both to the free energy difference between reactants and products and to the solvent reorganization energy. Information on the variation of the rate constant with temperature is needed both for an improved basic understanding of these reactions and because kinetic data on elementary reactions
This article not subject to US. Copyright. Published 1990 by the American Chemical Society
