7333
J. Phys. Chem. 1992,96,1333-7331 thank Kenneth McMurtrey for performing the mass spectral analysis and Emory Howell for a critical reading of the manuscript.
References and Notes (1) Farage, V. J.; Janjic, D. Chimia 1980, 34, 342. (2) Mpez-Tomb, L.; SaguC, F. J . Phys. Chem. 1991, 95, 701-705.
(3) Noszticzius, Z.; Horsthemke, W.; McCormick, W. D.; Swinney, H. L. Stirring Effects in the BZ Reaction with Oxalic Acid-Acetone Mixed Substrate in a Batch Reactor and in a CSTR. In Spatial Inhomogeneities and Transient Behavior in Chemical Kinetics; Gray, P., Baras, G. N. F.,Borckmans, P.,Scott, S.K.,Eds.; Manchester University Press: Manchester, U.K., 1990; pp 647-652. (4) Noszticzius, Z.; Bodnir, Z.; Garamszegi, L.; Wittmann, M. J. Phys.
Chem. 1991,9, 6575-6580. (5) Menzinger, M.; Jankowski, P.J. Phys. Chem. 1986,90, 1217-1219. (6) Menzinger, M.; Jankowski, P. J . Phys. Chem. 1986, 90, 6865.
(7) Menzinger, M.; Jankowski, P.J. Phys. Chem. 1990, 94,4123-4126. (8) Dutt, A. K.; Menzinger. M. J . Phys. Chem. 1990, 94, 4867-4870. (9) Farage, V. J.; Janjic, D. Chimia 1981, 35, 289. (10) Field, R. J.; Burger, M. Oscillations and Traueling Waves in Chemical Systems; Wiley: New York, 1985. (11) Sevcik, P.;Adamkcikovi, I. Chem. Phys. Lert. 1988,146,419-421. (12) Sevcik, P.; Adamkcikovi, I. J . Chem. Phys. 1989, 91, 1012-1014. (13) Noszticzius, Z.; Bodiss, J. J . Am. Chem. SOC.1979, 101, 3177. (14) Noszticzius, Z.; Stirling, P.;Wittmann, M. J . Phys. Chem. 1985, 89, 49 14. (15) Noszticzius, Z. M a w . Kem. Foly. 1979, 85, 330. (16) Ouyang, Q.; Tam, W. Y.; DeKepper, P.; McCormick, W. D.; Noszticzius. Z.; Swinney, H. L. J. Phys. Chem. 1987, 91, 2181-2184. (17) Field, R. J.; Koros, E.; Noyes, R. J . Am. Chem. Soc. 1972, 94, 8649-8664. (18) Eigen, M.; Kustin, K. J. Am. Chem. SOC.1962,84, 1355. (19) Nagy, I. P.;Baua, G. React. Kinet. Catal. Lert. 1991, 45, 15-25.
Analysis of the Transient Effect for a Bimolecular Fluorescence Quenching Reaction between Ions in Aqueous Solution A. D. Scully, S. Hirayama, Laboratory of Chemistry, Kyoto Institute of Technology, Sakyo-ku, Kyoto 606, Japan
D. Hachisu, and T. Tominaga* Department of Applied Chemistry, Okayama University of Science, 1 - 1 Ridai-cho, Okayama, 700. Japan (Received: February 12, 1992; In Final Form: May 13, 1992)
The influence of diffusion on the quenching of fluorescence from electronically excited 5,10,15,20-tetrakis(4-sulfonatopheny1)porphine by methylviologen in aqueous solution was investigated as a function of ionic strength using time-resolved fluorescence decay measurements. The resulting nonexponential fluorescence decay curves were analyzed using the expression for the timedependent rate coefficient derived by Hong and Noolandi for bimolecular diffusion-influenced reactions between ions in solution. The results of this analysis indicate that this expression provides a satisfactory description of the kinetics for this reaction under the experimental conditions used in this work. Values for the reaction distance and the intrinsic reaction rate constant of 1.4 f 0.1 nm and (2.7 f 0.5) X 1O'O M-' s-I, respectively, were calculated on the basis of the results of this analysis.
Introduction The verification experimentally of equations derived from theory to describe the time dependence of the rate coefficient for diffusioncontrolled bimolecular reactions in solution, k(t), has been the subject of a number of recent publications.'-'* The measurement of fluorescence decay curves has been found to be an effective method for probing the influence of diffusion on chemical reactions. In particular, fluorescence decay data obtained using the technique of time-correlated single photon counting (TCSPC)'-5 for fluorophore solutions containing a known concentration of quencher, where both the fluorophore and quenching species are ionic, have been analyzed according to the equations for k ( t ) derived by either Flannery13 or Hong and Noolandi14 which are based on the DebyeSmoluchowski model with the C~llins-Kimball'~boundary condition (DSCK model), Analysis using the expression for k ( t ) derived by Flannery of data obtained by using TCSPC is not straightforward due to correlations between the fitted parameters3s4and it has been found necessary to fu at least one of the fitted parameters in order to recover physically realistic values for the other parameters. However, despite this severe limitation in the analysis of data, the results of recent e ~ p e r i m e n t s indicate ~-~ that this equation provides a satisfactory description of the kinetics of bimolecular reactions between ions in solution. The Hong-Noolandi expression, which is effectively a long-time approximation of the Author to whom correspondence should be addressed.
Flannery equation for k(t), has a form identical with that of the long-time approximation to the equation for k ( t ) derived for diffusion-controlled bimolecular reactions in the absence of Coulombic interactions between the reactants. Periasamy et al.' have demonstrated the general validity of the Hong-Noolandi expression for k(t) by analyzing fluorescence decay curves measured using TCSPC for a number of fluorescence quenching reactions between ions in aqueous solution. The breakdown of the DSCK model in the subpicosecond time regime has been proposed recently4s5based on the results of analysis using the Flannery equation for k(t) of fluorescence decay curves measured using the technique of fluorescence upconversion. Measurements using the technique of TCSPC of fluorescence decay curves for aqueous solutions of a tetraanionic porphyrin in the presence ofthe methylviologen dication over a range of ionic strengths are described in this report. The results of analysis of these decay curves are discussed in terms of the DSCK model for the kinetics of diffusion-controlledbimolecular reactions between ions in solution. Theory
The expression for the time-dependent rate coefficient for the quenching of an electronically excited ionic fluorophore, A*, by an ionic quencher, Q,that was derived by FlanneryI3 from the DSCK model is k(t) = a b exp(6t) erfc(ctl/*) (1) where
+
0022-3654/92/2096-7333$03.00/00 1992 American Chemical Society
7334 The Journal of Physical Chemistry, Vol. 96, No. 18, 1992
Scully et al. MCA CHANNEL NUMBER
In the above equations, R is the distance required for reaction between species A* and Q in the absence of ionic interactions, DAQis the sum of the diffusion coefficients of these entities, kaCt is the intrinsic rate constant for the reaction, N is Avogadro's number, and rc is the Onsager distance which is defined as
I
I
In this equation zAe and z e are the effective charges on the fluorophore and quencher, respectively, Q and c are the dielectric constants of a vacuum and the reaction medium, respectively, ke is Boltzmann's constant, and Tis the absolute temperature of the system. At sufficiently long times eq 1 can be approximated by the following equation which was derived by Hong and N001andi.l~
where RHN=
C r
[1
+ (4rr$Afl/kact)l
exp(rc/R)
-1
(4)
The parameter RHNmay be interpreted as an effective encounter distance at which the reaction proceeds with certainty.14 The time dependence of the concentration of A* in the presence of quencher, Q, is given by d[A*It/dt = -[A*lt(70-' + k(t)[Qlo)
(5)
where T~ is the unquenched fluorescence lifetime of A*, [A*], is the concentration of A* at time 1 and [Ql0 is the concentration of quencher. The function that is obtained for the timedependent decay of fluorescence intensity, F(t), upon substitution of eq 3 for k(r) in eq 5 and then integration is F ( t ) = F(0) exp(-Ar - Bt1l2)
(6)
where A = T0-l
+ ~TDAQ&.JN[Q]o
(7)
B = ~(XDAQ)'/~RHN~N[Q]O (8) The nonexponentiality of the fluorescence decay curves at early times in the decay that is implied in eq 6 is the swalleds "transient effect". Experimental Section The tetrasodium salt of 5,10,15,20-tetrakis(4-sulfonatopheny1)porphine (H2TPPS4-)was prepared by neutralization of H6TPPS.2HSO4-4H2O(Dojin Laboratories) using NaOH. The chloride salt of methylviologen (MV2+) (Nacalai Tcsque) was recrystallized from a methanol-acetone mixture. All solutions were prepared using doubly-distilled water. Dissolved oxygen was removed by at least four freeze-pump-thaw cycles. The concentration of H2TPPSe used for all measurements reported in this work was 1.0 pM, for which no evidence of reabsorption or self-quenching effects could be detected. The temperature of
0 0
IO 0
20 0
30 0
40 0
50 0
TIME I NANOSECONDS Figure 1. Fluorescence decay curves (dots) measured for H2TPPS' in 0.03 M NaCl solution at MV2+concentrations of (a) 2 mM, (b) 3 mM, (c) 4 mM, (d) 5 mM, (e) 7 mM, and (f) 8 mM. The solid lines are the best-fit curves calculated according to the function given in cq 6. Curve g is the instrument response function.
sample solutions was controlled at 25.0 f 0.1 OC by a thermoregulated water bath. Fluorescence decay curves were measured using a commercially available time-correlated single photon counting apparatus (Horiba, NAES-SSO), the excitation source for which is a hydrogen-filled flash lamp. The samples were excited at a wavelength of 370 nm by using a combination of filters (Hoya Glass, U-330and B-390). Fluorescence was detected at a wavelength of 640 nm by using a 10-nm bandpass filter (Corion, P10-640-F). The decay of fluorescence from aqueous solutions of H2TPPSC in the absence of MV2+ was measured using a channel width of 0.40 ns. All other fluorescence decay curves were measurtd using a channel width of 0.20 ns. These decay curves were analyzed using the method of iterative nonlinear least squares where the optimization of the fitted function with respect to the reduced x2 parameter was performed using an algorithm based on the Levenberg-Marquardt The time zero shift was incorporated as an adjustable parameter in these analyses. The analysis of all fluorescence decay curves was performed using a DEC 5000/200 minicomputer.
Results and Discussion The decay of fluorescence from H2TPPSe in aqueous solution in the absence of MV2+was found to be exponential for the entire range of ionic strengths used in this work, where the ionic strength was varied by the addition of NaCl. A slight decrease in the unquenched exponential decay time, T ~ from , 10.4 f 0.2 to 10.0 f 0.2 ns occurred as the concentration of added NaCl increased from zero to 0.5 M. The fluorescence decay curves measured for aqueous solutions of H2TPPSe in the presence of 0.03 M of added NaCl, for various concentrations of MVz+,are shown in Figure 1. The distributions of reduced residuals that correspond to the fitting of these decay curves using either a single-exponential function or the function given in eq 6 are presented in Figures 2 and 3, respectively. The decrease in the fluorescence decay time and the corresponding deviation from exponentiality that occurs upon the addition of MV2+ is illustrated clearly in these figures. The extent of quenching decreases significantly with increasing concentrations of added NaCl with the consequence that at higher ionic strengths higher concentrations of MV2+were required in order to detect deviations from exponentiality in the fluorescence decay curves. Analysis according to eq 6 of the nonexponential
The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 7335
Fluorescence Quenching Reaction between Ions
5 00124
0
4 48602
0
5 41117
6 51167
0.091,. 0
0 66037
0
.. ,...,... ... 4
;2
8
I
16
... . I
20
[MV2'] (mM) 1
I
0 Ill76
Figure 2. Distributions of reduced residuals corresponding to the analysis of the fluorescence decay curves shown in Figure 1 using a single-exponential function. The distributions associated with the decay curves a-f in Figure 1 are shown from top to bottom. The corresponding values for ~ the reduced x2 parameter are (a) xR2= 2.26, (b) XR' = 2.51, (c) X R = 4.16, (d) x R 2 = 3.95, (e) x R 2 = 5.81, and (f) X R = ~ 6.83.
Figure 4. Plots of A versus MV2+ concentration and the corraponding lines of best fit calculated using eq 7 for concentrations of added NaCl of (a) 0.00 M, (b) 0.01 M, (c) 0.03 M, (d) 0.1 M, (e) 0.2 M, and (f) 0.5
M.
0
2 1960
0
2 88089
0
2 41241
0
IMV2'l (mM)
3.00348
0
I
I
3 13223
F'igure 3. Distributions of reduced residuals corresponding to the analysis of the fluorescence decay curves shown in Figure 1 using the function given in eq 6. The distributions associated with the decay curves a-f in Figure 1 are shown from top to bottom. The corresponding values for the reduced xz parameter are (a) XR' = 0.97, (b) XR' = 1.06, (c) X R ~ 1.19, (d) XR' = 1.11, (e) XR' = 1.02 and (f) XR* = 1.27.
fluorescence decay curves enables both A and B to be plotted as a function of MV2+ concentration (see eqs 7 and 8). These plots are shown in Figures 4 and 5 , respectively, for a range of concentrations of added NaCl and in all cases a good linear relationship for both A and B as a function of MV2+concentration is observed. This result supports the finding by Periasamy et al.' that, for sufficiently long times, the expression for k(t) proposed by Hong and Noolandi provides a satisfactory description of the reaction kinetics for bimolecular fluorescence quenching reactions between ions in solution. The slight increase with increasing concentrations of added NaCl in the ordinate value of the intercept for the straight lines shown in Figure 4 is consistent with the decrease observed for the values of r0for H2TPPS4 with increasing concentrations of added NaCI. Values for RHNand DAQat each concentration of added NaCl were calculated from the slopes of the straight lines shown in Figures 4 and 5 by using the relationships given in eqs I and 8 and the results of these calculations are summarized in Table I. The values for RHN become progressively smaller with increasing
Figure 5. Plots of E versus MV2+ concentration and the corresponding lines of beat fit calculated using eq 8 for concentrations of added NaCl of (a) 0.00 M, (b) 0.01 M, (c) 0.03 M, (d) 0.1 M, (e) 0.2 M, and (0 0.5 M.
TABLE I: R d b of Analysis According to Eq 6 of the Queaebed Fluomceoce Decay Cunes Meunwd for H2TPPSc in the Reaeoce of w+ [NaCI]? M RHN?nm DAnX m2 s-* 0.00 0.01 0.03 0.1 0.2 0.5
4.8 i 0.4 3.4 0.1 2.7 0.2 1.4 0.1 1.16 0.07 1.07 i 0.06
** *
'
5.5 5.4 4.7 8.7 7.7 6.0
1.3
i 0.4 1.0 & 0.6
i 1.2 i 0.5
"Concentration of added NaC1. bCalculated from the slopes of the lines of best fit shown in Figures 4 and 5.
concentrationsof added electrolyte until a limiting value of 1.12 0.05 nm is attained when the concentration of added NaCl reaches 0.2 M. The observation of a limiting value for RHN at sufficiently high ionic strengths is attributed to complete screening of the charges on the reactants, resulting in the nullification of the Coulombic interactions which exist between H2TPPSC and MVz+in solutions of lower ionic strength. The lack of a theoretical description of the dependence of r,, and therefore R H N(see . eq 4), on ionic strength prevents a more quantitative analysls of the ionic-strength dependence observed in this work for the values Of RHN.
*
7336 The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 There is no clear trend in the values recovered for DAQ over the range of concentrationsof added NaCl used in this work. The values for DAQ are expected to decrease to some extent with increasing ionic strength due to the reduced mobility of the reactants. However, this effect appears to be obscured by the rather large uncertainties in the recovered values for DAQ. Nevertheless, the value obtained in this work for DAQ in the absence of added salt is in reasonably good agreement with the value of 9.9 X 1O-Io m2 s-I that is estimated for the sum of the bulk diffusion coefficients of H2TPPS4-(2.8 X m2 s-l)l9 and MV2+ (7.1 X m2s-1)20in aqueous solution. In the limit of complete screening of the effective charges on the reactants, lrcl = 0 (see eq 2) and eq 1 reduces to the equation for k(t) derived for noninteracting species derived by Smoluchowski with the Collins-Kimball boundary conditi~n.*~~*I~ The mean value obtained for RHN in this limit is 1.12 f 0.05 nm and this value can be treated as being equivalent to the apparent reaction distance for noninteracting reactants, R,which is defined in the following equation.
Scully et al.
Q
HF
(9) The value for the intrinsic bimolecular reaction rate constant, k,,, at the reaction distance, R, can be calculated by the rearrangement of eq 4 to give exp(rc/ R, kact = ( ~ J R H N )+ 1 - exp(rc/R) 4*DAflrc
exp(x) - x - 1 = 0
Figure 6. Hard-sphere radii for H2TPPSe and MV2+ calculated from
(11)
where CY=
( ~ ~ / R H+ N 1) ; x= / R')
HF
(10)
followed by the substitution of eq 9 into eq 10 and subsequent rearrangement which results in the following equation: CY
0 T+
4*DAflrc
kact
The only unknown parameter in eq 11 is k,,,. The value of r, for the present system was calculated using eq 2 to be -5.7 nm. If the quenching reaction involves contact between the reactants then a value for R can be estimated from the hard-sphere radii of HzTPPS" and MVz+calculated from space-filling molecular models (see Figure 6). The hard-sphere contact distance varies depending on the direction of approach of the reactants and has a maximum value of approximately 1.8 nm. From these estimates it is reasonable to assume that the value for lrcl is significantly larger than that for R. In this case it can be seen from eq 4 that if the fluorophore and quencher are mutually attractive (r, < 0) then, for an ionic strength of zero, RHN = Ircl. It is this value for RHN which was used for the calculation of k,,, in this work. The value obtained for RHN in the absence of added NaCl of 4.8 f 0.4 nm is somewhat smaller than that calculated for Ir,l, and this is attributed to the effects of partial screening of the charges on the reactants by the respective counterions. The value that is calculated for k,, upon substitution of values for R'and RHN of 1.12 f 0.05 and 5.7 nm, respectively, into eq 11 is (2.7 f 0.5) X 1Olo M-I s-I, where the value for DAQused in this calculation was (6.3 f 1.2) X m2 s-l, which is the mean of those values listed in Table I. Insertion of this value for k,,, into eq 9 gives a value for R of 1.4 f 0.1 nm, which suggests that this reaction requires contact of the reactants. The quenching of emission from electronically excited derivatives of H2TPPS4-by alkylviologen cations in solution has been shownz1to be the result of electron transfer. The change in free energy, AGO, associated with electron transfer from HzTPPS4-*, to MV2+can be calculated using22 where E ( S l ) is the singlet excitation energy of H2TPPS4-,
Eo12(D/D') is the oxidation potential of the ground-state donor, HzhPS4-,E01/2(A/A-)is the reduction potential of the acceptor,
space-filling molecular models. MVZ+and wp and w, are the product and reactant work terms, respectively. The value for E @ , ) that is calculated from the wavelength of the 0-0 transition in the Q-band regions of the absorption and fluorescence emission spectra of H2TPPSe in water is 1.95 eV. The values for E',I~(D/D+)and EO1/z(A/A-)in aqueous solution are reported to be 1.10 eV21 and -0.686 eV,23 respectively. The values for the work terms for the present system are calculated to be -0.039 and -0.10 eV for wp and w,,respectively. The value for AGO calculated using these values for the parameters in eq 12 is -0.10 eV, indicating that the electrontransfer reaction between H2TPPS4-* and MVZ+is exergonic. The expression derived24for the rate constant for adiabatic electron-transfer reactions, k,,, is 6 k,, = v exp
[
-(A
+ AGO)' 4AkBT
]
where v is an effective frequency factor which is related to the vibrational frequencies of the reactants and solvent and A is the sum of the inner- and outer-sphere reorganization energies. On the basis of eq 13, values for k,,, which is equivalent,to kad in the present context, are predicted to increase with decreasing values of AGO in the so-called "normal regime", where (A AGO) > 0, and to reach a maximum value when (A AGO) = 0. Values for k,, are predicted to decrease with decreasing values of AGO in the "inverted regime" for which (A AGO) < 0. Values for A are typically in the range of approximately 0.5-1.0 eV. The variation in the values of k,, with (A AGO) for bimolecular electron-transfer reactions in solution is described satisfactorily by eq 13 for systems located in the normal regime. However, instead of the predicted inversion, the values of k,, for systems in which the value of (A + AGO) is significantly less than zero have been shown recently1IJ2to reach a plateau region for which the limiting value for k,, is approximately 4 X 10" M-I s-l. The value of AGO for the bimolecular electron-transfer reaction between H2TPPS4-*and MV2+indicates that this system lies in the normal regime. This is consistent with the recovered value for k,,, of (2.7 f 0.5) X 1 O l o M-I s-I being significantly smaller than the reported limiting value. According to eq 12, the value of AGO for the reaction examined in this work is expected to be influenced to a small extent by changes in the ionic strength of
+
+ +
+
Fluorescence Quenching Reaction between Ions
The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 7337
is required for electron transfer from H2TPPSe* to MV2+. The the reaction medium, resulting in a value for k,,, which may be value calculated for the intrinsic rate constant for this reaction slightly dependent upon ionic strength. However, the magnitude is consistent with this system being located in the so-called " n o d of this effect has been assumed to be negligible in this work. region" for bimolecular electron-transfer reactions. Eads et al.4 have reported recently values for kaaand R of 1.7 x 1O1O M-' s-*and 2.3 f 0.3 nm, respectively, for the quenching Acknowledgment. A.D.S. expresses appreciation for financial by electron transfer of the fluorescence from the rhodamine B support from a postdoctoral fellowship from the Japan Society cation by ferrocyanide (Fe(CN)b") in aqueous solution. These for the Promotion of Science. This work was also supported by values were obtained from the analysis according to eq 1 of Grants-in-Aid for Scientific Research from the Japanese Ministry fluorescence decay curves measured using TCSPC. The inapof Education, Science and Culture Nos.02245102,02245101, and plicability of eq 1 to the description of the kinetics for this same 03231102. system on the subpicosecond time scale was also demonstrated Registry No. H2TPPS4-, 39174-47-5; MV2+, 4685-14-7; NaC1, by these author^^.^ for fluorescence decay curves measured using 7647-14-5. the technique of fluorescence upconversion. However, as also noted by these authors? the contribution by diffusional processes to the References and Nom overall bimolecular quenching reaction on these extremely short (1) Periasamy, N.; Doraiswamy, S.;Maiya, G. B.; Venkataraman, B. J. time scales will be minimal and the rate of reaction is dominated Chem. Phys. 1988,88, 1638. by the intrinsic reaction rate. This phenomenon is generally (2) Periasamy, N.; Doraiswamy, S.;Venkataraman, B.; Fleming, G. R. J. Chem. Phys. 1988,89,479. referred to as "static" quenching. It is, therefore, perhaps not (3) Das, R.; Periasamy, N. Chem. Phys. 1989, 136, 361. surprising that the description by the DSCK model for diffu(4) Ea&, D. D.; Periasamy, N.; Fleming, G. R. J . Chem. Phys. 1989,W. sionoontrolled bimolecular reactions on these ultrashort time scales 3876. is inadequate. A further contribution to the apparent inappli( 5 ) Eads, D. D.; Dismer, B. G.; Fleming, G. R. J. Chem. Phys. 1990,93, 1136. cability of the DSCK model on these time scales would be the (6) Joshi, G. C.; Bhatnagar, R.; Doraiswamy, S.; Periasamy, N . J. Phys. effects of the finte relaxation rate of the ionic atmosphere around Chem. 1990, 94, 2908. the fluorophore upon excitation. It has been shown r e c e n t l ~ * ~ a ~ ~ (7) Wijnaedts Van Resandt. R. W. Chem. Phvs. Leu. 1983. 95..~205. that these effects will be significant for reactions probed using (8) Nemzek, T. L.; Ware, W. R. J. Chem. PhG. 1975,62, 477. (9) Scully, A. D.; Yasuda, H.; Okamoto, M.; Hirayama, S.Chem. Phys. a time resolution on the order of picoseconds or less.
Conclusions It has been demonstrated in this work that at sufficiently long times the time dependence of the rate coefficient for the diffusion-influenced bimolecular fluorescence quenching reaction between electronically excited H2TPPSk and MV2+ in aqueous solution is described well by the expression derived by Hong and Noolandi. The effective encounter distance, RHN,for this reaction decreases with increasing ionic strength due to the reduction in the strength of the Coulombic interaction between the reactants. The value for RHNin the absence of added electrolyte is somewhat smaller than the Onsager distance and this is attributed to screening of the charges of the reactants by the respective counterions. The true encounter distance for the reaction between electronically excited H2TPPS4-and MV2+ in the absence of ionic interactions, R, is comparable with that estimated for the hardsphere contact between these reactants. This suggests that contact
1991, 157, 271. (10) Lakowicz, J. R.; Johnson, M. L.; Gryczynski, I.; Joshi, N.; Laczko, G. J. Phys. Chem. 1987,91, 3277. (11) Angel, S.A.; Peters, K. S. J. Phys. Chem. 1991, 95, 3606. (12) Nishikawa, S.;Asahi, T.; Okada, T.; Mataga, N.; KakitaN, T. Chem. Phys. Lerr. 1991, 185, 237. (13) Flannery, M. R. Phys. Reu. A 1982, 25, 3403. (14) Hong, K. M.; Noolandi, J. J. Chem. Phys. 1978,68, 5172. (15) Collins, F. C.; Kimball, G. J. Colloid Sci. 1949, 4, 425. (16) OConnor, D. V.; Phillip, D. Time-correlated Single Phoron Counting; Academic Press: New York, 1985. (17) Marquardt, D. W. 1 . Soc. Ind. Appl. Marh. 1963, 11, 431. ( 18) Bevington, P. R. Dara Reduction and Error Analysis for rhe Physical Sciences; McGraw-Hill: New York, 1969. (19) Tominaga, T.; Endoh, S.;Ishimaru, H. Bull. Chem. Soc. Jpn. 1991, 64, 942. (20) Harriman, A.; Millward, G. R.; Neta, P.; Richoux, M. C.; Thomas, J. M. J. Phys. Chem. 1988, 92, 1286. (21) Schmehl, R. H.; Whitten, D. G. J. Phys. Chem. 1981, 85, 3473. (22) Rhem, D.; Weller, A. Isr. J. Chem. 1970, 8, 259. (23) Hunig, S. Jusrus Liebigs Ann. Chem. 1973, 324. (24) Marcus, R. A. J. Chem. Phys. 1956, 24, 966, 979. (25) Ibuki, K.; Nakahara, M. J. Chem. Phys. 1990, 92,7323. (26) Chapman, C. F.; Maroncelli, M. J. Phys. Chem. 1991. 95, 9095.
