J. Phys. Chem. 1988, 92, 156-163
156
Salt Effects on Nearty Diffusion Controlled Electron-Transfer Reactions. Bimolecular Rate Constants and Cage Escape Yields in Oxidative Quenching of Tris( 2,2'-bipyridine)ruthenium( I I ) Claudio Chiorboli, Maria Teresa Indelli, Maria Anita Rampi Scandola, and Franco Scandola* Dipartimento di Chimica dell'Universitd and Centro di Fotochimica C.N.R., I-44100 Ferrara, Italy (Received: May 15, 1987)
The rate constants for the quenching of excited Ru(bpy)j+ by methylviologen (MV2+) and Ru(NH&py3+ have been studied in aqueous solution as a function of the concentration (0.01-1 M) and type (NaC1, NaC104, CaCI,) of added electrolyte. With MV2+ as quencher, the yield of products escaping cage recombination and the rate constant of their back electron transfer reaction have also been studied as a function of the concentration of added NaC1. The results have been compared with predictions based on expressions available in the literature for the ionic strength dependence of diffusional parameters kd and k4. With uni-univalent electrolytes, the Debye and Eigen equations appear to be adequate for the calculation of kd and k4, respectively, provided that the appropriate numerical integration over the interreactant distance is performed. Approximations leading to more tractable expressions (such as, e.g., those leading to a Brmsted-Bjerrum ionic strength dependence of kd) give rise to serious disagreement with experiments. Specific counterion effects (C104- faster than C1-) are observed that can be best interpreted in terms of changes in the rate of the unimolecular electron-transfer step within an encounter complex including the counterion. Also, counterion concentration rather than ionic strength better represents (Olson-Simonson effect) the salt effects obtained with the CaC1, electrolyte.
Introduction Three distinct kinetic regimes can be considered for bimolecular reactions in solution. If the overall bimolecular process is split into diffusional and reactive elementary steps'
in the literature is rather confusing as far as the calculation of kd and k4 is concerned, especially for the cases (very frequent with inorganic systems) in which the A and B reactants are charged species and some inert electrolyte is present in the solution. The Debye-Smoluchowski treatment of the diffusion of charged particles gives the following expression7**
the bimolecular rate constant, k, is given by
k = kd[kr/(k4 + k ) ]
(2)
where kd is the bimolecular rate constant for diffusion of the reactants to give the precursor complex (encounter), k4 is the unimolecular rate constant for dissociation of the precursor complex, and k, is the unimolecular rate constant of the reactive step within the precursor complex. The diffusional preequilibrium regime is defined by the condition k , > k-d. In this case, eq 2 reduces to k = kd and the k4, the full eq 2 reaction rate is insensitive to k,. When k, must be used and the reaction rates are slightly lower than diffusion and moderately sensitive to k,. This kinetic regime can be conveniently termed onearly diffusion controlled". Many interesting outer-sphere electron-transfer reactions2" are theoretically expected and experimentally found to belong to the diffusion-controlled or nearly diffusion controlled regimes. In the study of these reactions, the knowledge of the diffusional parameters is of crucial importance. In fact, reliable values of kd are needed in order to decide whether the reaction is in the diffusion-controlled or in the nearly diffusion controlled regime. In the latter case, values of kd and k4 are required in order to extract k, from the experimental k values. Unfortunately, the situation (1) Eigen, M.; Kruse, W.; Maass, G.; De Maeyer, L. Prog. React. Kinet. 1964,2, 287. (2) Cannon, R. D. Electron Transfer Reacfions; Butterworths: London, 1980. (3) Balzani, V.;Scandola, F. In Energy Resources fhrough Photochemistry and Catalysis; GrBetzel, M., Ed.; Academic: New York, 1983; Chapter 1, P 1. (4) Sutin, N.; Creutz, C. J . Chem. Educ. 1983,60, 809. (5) Meyer, T. J. Prog. Inorg. Chem. 1983,30, 389. (6) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985,811, 265.
0022-3654/88/2092-0156$01.50/0
(3) where kB is Boltzmann's constant, N is Avogadro's number, 7) is the solvent viscosity, rAand rBare the radii of the reactants, a = rA rB,r is the distance separating the two reactants, and w(r,p) is given according to the Debye-Huckel theory by9-14
+
2Dr (4)
where =
(
1OOODkBT 8TNe2 >Ii'
and ZAand ZBare the ionic charges of the reactants, D is the static dielectric constant of the solvent, e is the electron charge, OA (or uB) is the radius of reactant A (or €3) plus that of the dominant counterion in the ionic atmosphere, and p is the ionic strength. Because of the double exponential arising from sub(7) Smoluchowski, M. Z . Phys. Chem., Stoechiom. Verwandtschaftsl. 1917,92, 129. For a recent discussion of alternative theoretical models of diffusional effects on bimolecular reactions, see Keizer, J. Chem. Reu. 1987, 87, 167. (8) Debye, P. Trans. Electrochem. SOC.1942,82, 265. (9) Sutin, N. Arc. Chem. Res. 1982,15, 275. (10)Sutin, N. Prog. Inorg. Chem. 1983,30, 441. (1 1) Although different expressions for w(r,fi) are a ~ a i l a b l e , ~ ~the ' ~ ex-'~ pression given in eq 4 is adequate for the present purpose. (12) Logan, S . R. Trans. Faraday SOC.1966,62, 3416. (1 3) Kortiim, G . Treatise on Electrochemistry; Elsevier: Amsterdam, 1965; Chapter V . (14) Friedman, H. L. Pure Appl. Chem. 1981,53, 1277.
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 1, 1988 157
Oxidative Quenching of Ru(bpy),’+ stitution of eq 4 into eq 3, the integration of eq 3 can only be so that the use of these performed by numerical means,12,15-18 theoretical expressions is not very practical. Perhaps for this reason, a number of simplified kd expressions have been used in the literature. For example, it has often been tacitly a ~ s u m e d ~that J ~ -kd~ ~ obeys the same Br~nsted-Bjerrum ionic strength dependence as followed by ordinary bimolecular reactions,24i.e.
It should be remarked that the use of a Brmsted-Bjerrum ionic strength dependence for kd (eq 6) coupled with eq 11 would imply that k-d is independent of ionic strength. It has indeed been proposed” that
On the other hand, the use of the empirical kd expression given in eq 9 coupled with eq 11 yields kBT
k A
In other papers, on the other hand, a different empirical expression (eq 9) has been used25*26 kd =
2kBTN
(2 30007
;:)
+ +
w(a,M)/kBT
exP[w(a&)/kBU
- 1 (9)
where w ( a , p ) is obtained from eq 4 with r = a. A parallel situation is found as far as the calculation of k4 is concerned. Eigen’s treatment of the diffusional separation of encounter pairs gives the following equation:”
If a Debye-Hiickel expression of w(r,p) such as eq 4 is substituted into eq 10, analytical integration is again not feasible and no simple kA expression can be obtained. The ratio of the Debye-Smoluchowski and Eigen equations gives
which is the well-known Fuoss equation for the stability constant of ion (15) Bock, C. R.; Connor, J. A,; Gutierrez, A. R.; Meyer, T. J.; Whitten, D. G.; Sullivan, B. P.; Nagle, J. K. J . Am. Chem. SOC.1979, 101, 4815. (16) Brunschwig, B. S.; DeLaive, P. J.; English, A. M.; Goldberg, M.; Gray, H. B.;Mayo, S. L.; Sutin, N. Inorg. Chem. 1985, 24, 3743. (17) Chiorboli, C.; Scandola, F.; Kisch, H. J . Phys. Chem. 1986,90,2211. (18) Ballardini, R.; Gandolfi, M. T.; Balzani, V.; Scandola, F. Gazz. Chim. Ital., in press. (19) Bolletta, F.; Maestri, M.; Moggi, L.; Balzani, V. J. Am. Chem. SOC. 1973,95,7a64. (20) Demas, J. N.; Addington, J. W. J . Am. Chem. SOC.1976, 98, 5800. (21) Gaines, G. L. J . Phys. Chem. 1979,83, 3088. (22) Ryback, W.; Haim, A.; Netzel, T. L.; Sutin, N . J . Phys. Chem. 1981, 85,2as6. (23) Haim, A. Comments Inorg. Chem. 1985, 4, 119. (24) Frost, A. A,; Pearson, R. G. Kinetics and Mechanism; Wiley: New York, 1961. (25) Balzani, V.; Scandola, F.; Orlandi, G.; Sabbatini, N.; Indelli, M. T. J . Am. Chem. SOC.1981, 103, 3370. (26) Kawanishi, Y.; Kitamura, N.; Tazuke, S. J . Phys. Chem. 1986, 90, 2469. (27) Eigen, M. Z. Phys. Chem. (Munich) 1954, 1, 176. (28) Fuoss, R. M. J . Am. Chem. SOC.1958,80, 5059. (29) Sutin gives a somewhat different expression in which a range of reaction distances is considered instead of a single distance. The two equations give comparable values for many adiabatic reactions.1°
-
1
?.a7 a’(
I‘,
W(W)/kBT
-+:B)
1 - eXp[-W(a,p)/kgT]
(13)
Such an expression has been used by various author^.^^*^^^^' In view of the above-described situation, there seems to be a definite need for critical studies in which the predictions of the various types of kd and kA expressions are compared with experimental data on the ionic strength dependence of diffusioncontrolled or nearly diffusion controlled reactions. Relatively few data of this kind are available in the literature. There are, moreover, two additional important questions that deserve careful attention in studies of this type. First, a number of s t ~ d i e s , ~ -using , ~ inert electrolytes of different stoichiometry have shown that in reactions between ions of the same sign the rates tend to correlate with the concentration of the main counterion rather than with the ionic strength (Olson-Simonson effect). This effect seems to extend to nearly diffusion controlled reactions, and its generality awaits confirmation. Second, several lines of evidence indicate that in many fast electron (and energy) transfer reactions counterions of the “inert” electrolyte may have remarkable specific kinetic e f f e ~ t s . ~ ~Whether s ~ ~ - ~ these ~ specific effects should be incorporated into the diffusional parameters or into the unimolecular reaction rate constant is matter for investigation. Both these effects call for future refinements of the Debye-Huckel model of ionic atmosphere effects. In an attempt to contribute to the clarification of some of the questions addressed in this introduction, we have studied the oxidative quenching of the excited state of tris(2,2’-bipyridine)ruthenium(II), Ru(bpy),’+, by the 4,4’-dimethylbipyridinium dication (methylviologen, MV”), and pentaammine(pyridine)ruthenium(III), R U ( N H ~ ) ~ ~as~a, function +, of the concentration of various added electrolytes. The experimental ionic strength dependences of (i) quenching rate constants, (ii) back electron transfer rate constants, and (iii) yields of products escaping cage recombination are used as a basis for a critical discussion of the various kd and kA expressions available in the literature. Experimental Section Materials. [Ru(bpy),]Clz (Carlo Erba), methylviologen chloride (Ega Chemie), and [ R u ( N H , ) ~ ] C ~(Johnson , Matthey Chemical Ltd.) were commercial products of reagent grade. Methylviologen perchlorate was obtained from the corresponding chloride salt. [Ru(NH,),py] (Clod), was prepared according to the l i t e r a t ~ r e . , ~Triply distilled water was used throughout the work. All other chemicals were of reagent grade. (30) Gore, B. L.; Harriman, A.; Richoux, M. C. J . Photochem. 1982,19, 209. (31) Balzani, V.; Juris, A.; Scandola, F. In Homogeneous and Heterogeneous Photocatalysis; Pelizzetti, E., Serpone, N., Eds.; D. Reidel: Dordrecht, 1986; p 1. (32) Olson, A. R.; Simonson, T. R. J . Phys. Chem. 1949, 17, 1167. (33) Perlmutter-Hayman, B. Prog. React. Kinet. 1971, 6, 239. (34) Zamboni, R.; Giacomelli, A,; Malatesta, F.; Indelli, A. J . Phys. Chem. 1976, 80, 1418. (35) Rampi Scandola, M. A,; Scandola, F.; Indelli, A. J. Chem. SOC., Faraday Trans. 1 1985, 81, 2967. (36) Pethybridge, A. D.; Prue, J. E. Prog. Inorg. Chem. 1972, 17, 327. (37) Bruhm, H.; Nigam, S.;Holzwart, J. F. Faraday Discuss. Chem. SOC. 1982, 74, 129. (38) Braga, T. G.; Wahl, A. C. J. Phys. Chem. 1985, 89, 5822. (39) Gaunder, R.; Taube, H. Inorg. Chem. 1970, 9, 2627.
158
Chiorboli et al.
The Journal of Physical Chemistry, Vol. 92, No. 1, 1988
TABLE I: Electron-Transfer Quenching of *Ru(bpy)?+ by MVZ+;Quenching Rate Constants, k,, and Calculated Diffusional Parameters, k , and kdaSb DE' AIC AII' 0.0 1 0.02 0.04 0.08 0.16 0.32 0.35 0.40 0.60 0.80 1 .oo 1.60
0.41 0.64 0.94 1.2 1.6
1.4
0.40
1.7 2.1 3.6
0.93
I .6 2.0 4.2 2.0 2.4 4.9 2.6
2.1 3.3 3.9 4.7 5.6 6.3 6.4 6.5 6.8 7.0 7.1 7.2
7.4 7.1 6.6 5.9 5.2 4.4 4.3 4.2 3.8 3.6 3.5 3.2
2.0 2.2 2.6 3.0 3.4 3.9 4.0 4.0 4.3 4.5 4.7 4.9
5.6 5.2 4.8 4.4 4.0 3.6 3.6 3.5 3.4 3.3 3.2 3.0
2.4 3.0 3.1 4.7 5.9 7.4 7.6 1.9 8.9 9.6 10.0 11.0
6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9
"Aqueous solutions, T = 293 K. b T h e values reported for the diffusion parameters are calculated for NaCl background e l e c t r ~ l y t e ;for ~ ~details on the methods of calculation and physical parameters used, see Experimental Section. 'The symbols DE, AI, and AI1 refer to different approximations used in the calculation of the diffusional parameters (see Discussion). Background electrolyte used.
Apparatus. Emission lifetime measurements were carried out with a PRA (Photochemical Research Associates, London Ontario) System 3000 nanosecond fluorescence spectrometer equipped with a PRA Model 510B nanosecond pulsed lamp, filled with pure (99.999%) hydrogen and a PRA Model 1551 cooled photomultiplier; data collection was performed on a Tracor-Northern TN- 1750 multichannel analyzer; the data were processed with original PRA software, on a Digital PDP 11/03 computer. Transient absorption measurements were performed with a nanosecond laser flash photolysis apparatus (J&K frequencydoubled ruby laser, Applied Photophysics detection system, Hamamatsu 928 photomultiplier, in a single shot mode, with oscillographic recording). The system works in the collinear mode, with standard 1-cm spectrofluorometric cuvettes. Quenching Experiments. These experiments were carried out in air-equilibrated aqueous solutions, at controlled temperature (20 f 1 "C), by measuring the lifetime of the characteristic (A, = 610 nm). Quenching rate constants R ~ ( b p y ) , ~emission + were obtained from Stern-Volmer plots of the emission lifetime data. At least four different quencher concentrations were used for each quenching rate constant determination. The quencher (MV2+or R U ( N H , ) ~ ~concentrations ~~+) were in the 1 X 104-2 X lo-, M range, and the Ru(bpy)?+concentration was 2.5 X 1O-j M. The ionic strength was adjusted by adding the appropriate amount of NaCl, NaClO,, or CaC12. Transient Absorption Measurements. With MV2+as quencher, the yields of formation of primary redox products escaping cage recombination were obtained as vCe= ( [MV+]/Nhv,,,)( l/vq), where vq = Ksv[Q]/(l Ksv[Q]) is the fraction of excited states quenched (always higher than 30%) and Ksvis the Stern-Volmer constant. The MV+ concentration was calculated from the absorbance changes at the two maximum absorption wavelengths of MV' (A = 393 nm: C M V + = 37500 M-' cm-', t R u ( 1 ~ 1 ) = 1480 M-l cm-', ~ ~ ( =~ 5360 1 ) M-' cm-I; X = 600 nm: t M V + = 1 1 300 M-' cm-I) immediately after the pulse. The concentration of the absorbed laser photons was obtained by using two different actinometric methods that were found to give identical results. The first one is based on the benzophenone triplet a b ~ o r p t i o nwhile ,~~
+
t h e second uses t h e b l e a c h i n g of t h e g r o u n d - s t a t e a b s o r p t i o n in
the quenching of *Ru(bpy)32+by Fe3+.41 The rate constants of the back electron transfer reaction of the primary products of the quenching reaction were straightforwardly measured by monitoring the second-order decay of the absorption of the reduced quencher (MV') at 600 nm. These experiments were performed in deaerated solutions with 347-nm optical density lower than 1.2, since above this limit concentration-dependent rate constants were observed due to inhomogeneous distribution of the primary products within the cell. (40) Bignozzi, C. A.; Scandola, F. Inorg. Chem. 1984, 23, 1540. (41) Mok, C. Y . ;Zanella, A. W.; Creutz, C.; Sutin, N. Inorg. Chem. 1984, 23, 289 1.
Computational Procedures. Values of kd and k4 as a function of the ionic strength (eq 3, 6, 9, 10, 12, 13) were calculated by using the following values for the physical parameters: vH20= 1.002 cP; DH20= 80.2; T = 293 K; rRu(bpy),2+= 0.7 nm,42rMv2+ = 0.33 nm, rRu(N:l,)sPYi+ = 0.38 nm$, rRu(NH3)ajt = 0.35 nm$2 rNa+ = 0.1 16 nm,44rcl- = 0.167 nm,44rCIOa= 0.226 nm.44 The radius ~ 3 , d,, d,, of the MV2+ion was obtained as r = 1 / 2 ( d x d y d z ) 'where and d, are the dimensions measured along three molecular axes on CPK space-filling models. Equations 3 and 10 were numerically integrated (Sympson's method) over a range going from r = a = (ra rb) to the value r = I at which (exp[w(r,p)/kB7'l - 1) I 1 X the remainder of the integral from this value to m being taken equal to 1/1. Comparison with numerical integration over more extended r ranges indicated that this approximation leads to an error of less than 0.1% on the kd and k-d values.
+
Results The bimolecular rate constants, k,, for the quenching of excited R ~ ( b p y ) , ~by+ methylviologen (eq 14) were measured in aqueous
-
* R ~ ( b p y ) , ~++ MV2+
k,
Ru(bpy),,+
+ MV+
(14)
solution as a function of ionic strength, with NaC1, NaC104, or CaClz as background electrolytes. No detectable effects of the background electrolytes on absorption spectrum, emission spectrum, or excited-state lifetime of Ru(bpy)?+ were observed. Good linear Stern-Volmer plots were always obtained by monitoring the excited-state lifetime as a function of quencher concentration. The k, values are reported in Table I, together with the values of the diffusional parameters calculated according to the various expressions available (see D i s c u ~ s i o n ) . ~ ~ Laser photolysis experiments were also carried out using methylviologen as a quencher in order to obtain the yield, vE, of products ( R ~ ( b p y ) ~ and , + MV+) escaping cage recombination and the bimolecular rate constant, kb, of their back electron transfer reaction (eq 15). These experiments were performed in aqueous R ~ ( b p y ) , ~+ + MV'
kb +
Ru(bpy)32+
+ MV2+
(15)
solution, using NaCl as background electrolyte. The results are ~~
(42) Sutin, N. In Tunneling in Biological System; Chance, B., DeVault, D. C., Frauenfelder, H., Marcus, R. A., Schrieffer, J. R., Sutin, N., Eds.; Academic: New York, 1979; p 201. (43) Brown, G. M.; Krentzien, H. J.; Abe, M.; Taube, H. Inorg. Chem. 1919, 18, 3314. (44) Huheey, J. E. Inorganic Chemistry, 3rd ed.; Harper International: Cambridge, 1983. (45) The kd and k4 values reported are calculated for NaCl background electrolyte. Due to the presence of uA and uB in eq 4, the DE and AI expressions (see Discussion) would in principle give different values for the other electrolytes. In practice, however, the differences are negligible (less than 3%) in the whole ionic strength range investigated.
The Journal of Physicat Chemistry, Vol. 92, No. I, 1988 159
Oxidative Quenching of R ~ ( b p y ) , ~ +
TABLE II: Electron-Transfer Quenching of *Ru(hpy)p by MV2+; Yields of Separated Electron-Transfer Products, tee, Rate Constants of the Back Electron Transfer Reaction, kb,and Calculated Diffusional Parameters, kdand kdCC DEd
cct M 0.01 0.02 0.08 0.16 0.35 0.52 0.80 1.60
qce
10-9kb, M-1 s-I
0.38 0.35 0.34 0.32 0.29 0.25 0.24 0.22
10-9kb, M-I s-l 3.6 4.1 5.4 6.0 6.6 6.9 7.1 7.3
2.8 3.4 4.2 4.8 5.3 5.6
AIId
AId
10-9k’4, s-l 6.0 5.7 4.9 4.4 3.9 3.6 3.4 3.1
10-9k’d, M-’ s-I 2.7 3.0 3.6 4.0 4.5 4.7 5.0 5.3
10-9v4, s-I 4.6 4.4 3.8 3.5 3.3 3.1
10-9k’~.M-1 3.3 3.8 5.3 6.4 7.7 8.4 9.2 10.0
3.0 2.8
s-1
10-9k’-,. s-I 5.6 5.6 5.6 5.6 5.6 5.6 5.6 5.6
‘Aqueous solutions, T = 293 K, NaCl background electrolyte. “or details on the methods of the calculation and physical parameters used, see Experimental Section. ‘For the definitions of symbols, see Discussion. dThe symbols DE, AI, and AI1 refer to different approximations used in the calculation of the diffusional parameters (see Discussion). 0.1
TABLE Ill: Electron-Transfer Quenching of *Ru(bpy)?+ by R U ( N H ~ ) ~ P YQuenching ~+; Rate Constant, k,, and Calculated Diffusional Parameters, k d and kdaSb 10-9k,, p,
M
0.02 0.04 0.05 0.08 0.10 0.20 0.50 1.oo
NaCld
M-1
10
P
s-1
10-9kd,cM-I 2.1 2.8 3.0 3.7 4.0 5.1 6.1 6.7
NaC10:
1.4 2.5 2.1 2.3 2.8 3.4 4.1 4.4
1
3 .O 3.5 4.4 5.3 5.4
s-l
10-9k4~SKI 7.9 7.2 7.0 6.3 6.0 4.9 3.7 3.1
‘Aqueous solutions, T = 293 K. bFor details on the methods of calculation and physical parameters used, see Experimental Section. cCalculated according to DE approximation (eq 3 and IO) for NaCI; values for NaC10, are only negligible different.45 Background electrolyte used.
given in Table 11, together with the values of the relevant diffusional parameters calculated according to the various expressions available (see Discussion). Bimolecular rate constants, k,, for the quenching of excited R ~ ( b p y ) ~ ’ by + Ru(NH3),py3+ (eq 16)46 were obtained from
k,
*Ru(bpy)32++ Ru(NH3),py3+ Ru(bPY)33+ + Ru(NH3)spy2+ (16) lifetime quenching experiments in aqueous solution, using NaCl or NaC104 as background electrolytes. The results are given in Table 111, together with calculated values of the relevant diffusional
Figure 1. Typical ionic strength dependence of kd calculated by using DE (-), AI and AI1 expressions (- - -), for two different charge products (+4, -4). For definitions, see text. Aqueous solutions, T = 298 K, r A = 0.7 nm, rB = 0.3 nm, NaCl background electrolyte. (e-),
0.01
log k-dl 10.0
0.1
1
10
P
’
parameter^.^^ Discussion The expressions available for the calculation of the diffusional parameters kd and k4 (eq 3, 6, 9, 10, 12, 13) all lie within the theoretical framework of the Debye-Hiickel theory of electrolyte solutions. It is possible to show that eq 6, 12 and 9, 13 correspond to different degrees of approximation in the procedure required to obtain an analytically integrable expression of eq 3. In fact, the Taylor expansion (ro = a) of the exponential in eq 4 gives
If only the first term is retained in eq 17, then substitution into eq 4 Eollowed by further substitution into eq 3 and integration leads to eq 9. The same approximation applied to eq 10 leads to eq 13. If, on the other hand, both the first and the second terms (46) Although an energy-transfer quenching mechanism cannot be ruled out on thermodynamic grounds for this system, electron transfer seems to be the most plausible mechanism on kinetic grounds. In fact, exergonic energy transfer to Ru(II1) amines is relatively slow (see footnote 49 in Bignozzi, C. A.; Roffia, S.; Scandola, F. J . Am. Chem. SOC.1985, 107, 1644). As far as the comparison with calculated diffusional parameters is concerned, the type of quenching mechanism is obviously immaterial.
I
0.2
0.4
Figure 2. Typical ionic strength dependences of k4 calculated by using D E (-), AI (-), and AI1 expressions (- - -), for two different charge products (+4, -4). For definition, see text. Aqueous solutions, T = 298 K, rA = 0.7 nm, rB = 0.3 nm,NaCl background electrolyte.
are retained in the expansion (eq 17), then substitution into eq 4 (with uA = ug = a ) followed by further substitution into eq 3 and integration lead to eq 6. The same approximation applied to eq 10 leads to eq 12. In the following, we will refer to the k , and k4 values calculated by numerical integration of the “exact” eq 3, 4, and 10 as the Debye-Eigen (DE) values. The values obtained via eq 9 and 13 will be referred to a s approximation I (AI) values, and those obtained via eq 6 and 12 a s approximation I1 (AII) values.
160 The Journal of Physical Chemistry, Vol. 92, No. 1, 1988
Chiorboli et al.
7
SCHEME I
0.01
109K4
0.1
9.5
Before discussing the experimental results, it is worthwhile to examine the general features of ionic strength dependence of kd and k4 predicted by the various sets of equations. Figures 1 and 2 show the results of calculations performed for two different reactant charge products (+4,-4) in aqueous solution at T = 298 K with rA = 0.7 nm, rB = 0.3 nm, and NaCl as background electrolyte (see Experimental Section for details). Inspection of Figures 1 and 2 clearly shows that the AI1 expressions, except for very low ionic strengths, give qualitatively unacceptable predictions. In fact, (i) kd and k4 do not tend asymptotically for high p values, as expected, to the kd and k4 values characteristic of uncharged reactants; (ii) kd is predicted to become higher (positive ZAZB) or lower (negative ZAZB) than the value for uncharged reactants, even at moderate ionic strengths; (iii) at moderate p values, kd becomes lower for oppositely charged reactants than for reactants of the same charge type; (iv) k4 is predicted to be completely insensitive to the ionic strength. The AI expressions, on the other hand, tend to quantitatively underestimate the ionic strength effect on kd and overestimate the same effects on k-d but have the correct asymptotic behavior and qualitatively parallel the predictions of the DE expressions. The above-described qualitative differences in behavior between the various expressions are quite general, as shown by further calculations in which different solvents and reactant charge products were e ~ p l o r e d . ~ ' The actual reliability of the predictions of the various sets of expressions for kd and k4 can be tested, in principle, by using the experimental data on the ionic strength dependence of the quenching rate constants, k,, back electron transfer rate constants, kb, and cage escape yields, qat obtained in this study (Tables 1-111). According to Scheme I, these experimental data are related to diffusional parameters through eq 18-20. If a reaction can be unequivocally assigned to the diffusion-controlled regime (which is not a trivial problem, see below), the experimental rate constant values could be used to directly test the kd expressions. For slower processes, in principle, by using the experimental k,, kb, and qce
9.0
0.1 0.2 0.3
0.4 0.5
4 1I+& Figure 3. Experimental ionic strength dependence of k, values obtained with MVZ+as quencher and NaCl as background electrolyte (Table I). The continuous line is the fit obtained by using eq 18, diffusional parameters calculated according to DE expressions, and k , = l 9 X IO9 s-]. The calculated ionic strength dependence of kd (DE) is also shown for comparison (dotted line).
I 109Kb
9.7'
9 . 5 '
I
/
0.1 0 . 2 0 . 3 0 . 4
0.5
4%-4
Figure 4. Experimental ionic strength dependence of kb obtained with MV2+ as quencher (Table 11). Best fits obtained by using eq 19 and diffusional parameters calculated according to the D E expressions (-) or the AI1 approximation (---). Best-fitting k, value was 1 0 X 1Olo s-' in both cases.
data and calculated values for the diffusional parameters, one could obtain k, or k-, values. Ideally, thus, in these cases the degree of accuracy of a given set of expressions for kd and k-d could be judged from the constancy and ionic strength independence of the values obtained!* In practice, kq and kb values lying in a relatively narrow range (nearly diffusional regime, see Introduction) are (47) It could be noticed that one of the differences between the AI1 expressions and the other ones is the implicit assumption of uA = us = a in eq 4. That this is not the reason for the difference in qualitative behavior between the various types of expressions is shown by the fact that sample calculations which introduce such an assumption in the DE and AI expressions leave the results almost unchanged (f5%). Larger discrepancies due to this approximation are to be expected for higher charge products and less dielectric solvents. (48) In principle, the rate constant of a unimolecular step could be affected by changing the ionic strength since the driving force of the step may change by an amount corresponding to the difference in the work terms between reactants and p r o d ~ c t s .In ~ the systems investigated this difference is either zero ( R U ( N H ~ ) ~ ~as~ quencher) '+ or absolutely negligible (MV2+ as quencher).
only suitable for this type of test. In fact, for reactions in the diffusional preequilibrium regime eq 18 and 19 reduce to k, = (k,/k,)k, and kb = (kb/kl_d)k_,.(Notice that irrespective of the set of expressions used, k d / k 4 = KA is always given by eq 11.) Similarly, the cage escape yields can be used to test the k-d expressions, provided that they are appreciably lower than unity (but sufficiently high to be measured with the required accuracy). The experimental quenching rate constants, k,, obtained with MV2+ as quencher (Table I) are substantially low with respect to the calculated kd values. An example of fit to the experimental data (NaC1) using eq 18, k, = 1.9 X lo9 s-', and kd and k4 values calculated with the DE expressions is shown in Figure 3. The fit is satisfactory. As expected, however, almost equally good fits can be obtained using the AI or AI1 expressions and slightly different k, values. Thus, these data are not suitable for a comparative test of the three types of expressions. The back electron transfer rate constants, kb, and cage escape yields, qa (Table 11), are likely to be more diagnostic in this regard. In Figures 4 and 5 are shown the fits to the experimental points obtained using the DE expressions for k $ and k'4 and eq 19 and
The Journal of Physical Chemistry, Vol. 92, No. I , 1988 161
Oxidative Quenching of Ru(bpy),*+
I
I
9.
0 @
0.2-
0
c
9.
IP b
I I
,
I
I
I
I
0.1 0 . 2 0.3 0 . 4
I
0.5
JP/.+JP Figure 5. Experimental ionic strength dependence of vceobtained with MV2+as quencher (Table 11). Best fits obtained by using eq 20 and k L values calculated according t o the D E expressions (-) (best-fitting k-, value: 1.0 X I O i o s-I) or according to the AI1 approximation ( - - - ) (best-fitting k-, value: 1.4 X IOio s-’).
20. It can be seen that the use of the DE equations gives an acceptable fit to the experimental points for the two types of experiments using a unique, ionic strength independent k-, value (k-, = 1.0 X loLos-l). The figures also show that the AI1 expressions are totally inadequate to fit the experimental results. In particular, Figure 5 clearly shows that, contrary to the ionic strength independence predicted by the AI1 approximation (eq 12 and 20), the cage escape yields of the *Ru(bpy)32+/MV2+ system decrease markedly as the ionic strength is increased. As far as the A I expressions are concerned, in agreement with the previously noted qualitative similarity of its predictions to those of the DE expressions, satisfactory fits (not shown in Figures 4 and 5 ) can be obtained to both kb and qceexperimental data. It is to be noticed, however, that, as expected in view of the opposite deviations predicted for kd and k4 (Figures 1 and 2), such fits require the assumption of two largely different k-, values for the two sets of experiments (k-, I1.0 X 10” s-l for kb and k-, = 8.5 X lo9 s-l for qce). Thus, in these systems, the use of the numerically integrated DE expressions seems to be preferable to that of the AI equations and largely superior to that of the AI1 approximation. It may be worthwhile to point out that the numerically integrated DE expressions are found to give reliable estimates of the diffusional parameters for ionic strength values as high as 1.6 M (NaCl). This might seem surprising, since the Debye-Huckel model is not expected to extend its validity beyond an upper ionic strength limit in the 10-*-10-’ M These results seem to indicate that for systems of the type studied, in spite of the microscopic complications (ionic association, lattice-type effects, etc.) that certainly take place at high electrolyte concentration, the overall salt effects are parametrically reproduced by the Debye-HOckel ionic strength dependent potential function, to a degree of accuracy appropriate for routine estimation of diffusional parameterss2 The quenching rate constants obtained for the * R ~ ( b p y ) , ~ + / MV2+ system with NaCl or NaC104 as supporting electrolytes (49) See, e.g., ref 13, 24, SO, and 51. The upper limit values quoted by various sources depend obviously on differences in the ionic charges of reactants and electrolyte, solvent, and type of experiment. Moreover, in several instances different approximated versions of the Debye-Hiickel expressions are considered by different authors. (50) Clark, D.Wayne, R. P In Comprehensive Chemical Kinelics; Bamford, C . H., Tipper, C. F. H., Eds ; Elsevier: Amsterdam, 1969; Vol. 11, Chapter 4, p 328. (51) Bockris, J . O M . ; Reddy, A. K. M. Modern Electrochemistry; Plenum: New York, 1970; Vol. I, p 236. (52) The same observation has been recently made for reactions of quite different charge (charge product, -4) and solvent (CH,CN or DMF) type.”
Figure 6. Rate constants for the quenching of *Ru(bpy)? by MV2+ in NaCl (0)and in h a C I O 4 (0).The fits are made by using eq 18, the DE expressions for kd and k4, and k, = 1.9 X IO9 s-l for the data in NaCl and k, = 7.6 X lo9 s-’ for the data in NaCIO,. The calculated (eq 3) kd curve is also shown for comparison (dotted line).
(Table I) are compared in Figure 6. The data clearly show a pronounced accelerating effect of C10, relative to Cl-. This effect had already been noticed by Gaines2’ in a single experiment. A similar effect has also been observed recently in the quenching of * R ~ ( b p y ) , ~by+ cobalt(II1) sepulchrate.” Figure 6 shows that the effect is essentially constant throughout the whole ionic strength range investigated. Thus, any explanation attributing the difference in rate with the two anions to changes in kd and/or k4 via different extents of ion-pair f ~ r m a t i o nwould ~ ~ , ~seem ~ unlikely in this case, as any such effect should be ionic strength dependent. Actually, the data in NaCl and in NaC104 can be fitted satisfactorily (Figure 6) by using the same45kd and k4 values for the two sets and different k , values ( k , = 1.9 X lo9 s-l with W a n d kt = 7.6 X lo9 s-I with C104-). Thus, we tend to associate this specific counterion effeGt to changes in the rate of the unimolecular reactive step. This implies that the anion is an important constituent of the precursor complex for the reaction between the two cationic reactants. Possible reasons for the difference in k , could be related to the difference in hydrophilicity between the two anions.55 In principle, the hydration shell surrounding the anion could act in one or both of the following ways: (i) it could provide a more or less “rigid” solvent environment to the precursor complex, requiring a more or less large outer-sphere reorganizational energy;56(ii) it could allow a more or less close approach of the two reactants, reflecting in a more or less adiabatic beh a ~ i o of r ~the ~ reaction. In both cases, the more hydrophilic chloride ion should give a slower unimolecular reactive step than the more hydrophobic perchlorate. In the more extended series of anions used with cobalt(II1) sepulchrate as a quencher (F- < C1- < Br- < ClO,) a hydrophobicity trend was rather evident.35 If the hypothesis that the specific anion effects operate essentially via k, is true, then for an otherwise similar, truly diffusion controlled reaction ( k , >> k4, k, = k d , eq 18) no such specific anion effects should be observable. The problem of identifying a truly diffusion controlled reaction is not a trivial ne.^',^* As (53) Without considering any ion-pair formation, eq 4 does already predict a dependence of kd and k4 on the radii of the counterions. For the system investigated, however, anion effects of this type are practically negligible.45 (54) In electron-transfer reactions between cations in acetonitrile, Wah138 ascribed the accelerating effect of C10,- relative to PF6- to different extents of ion pairing with the two anions and to the larger bimolecular rate constant of the ion-paired reactants (equal for Clod- and PF63 relative to that of the bare reactants. This amounts to consider the charge effect on the k d / k , ratio (preequilibrium regime) for the ion-paired and bare species. In Wal-1’s case, however, the accelerating effect was strongly increasing with ionic strength. ( 5 5 ) Millero, F. J. In Water and Aqueous Solutions; Horne, R. A,, Ed.; Wiley: New York, 1972; Chapter 13, p 519. (56) For general accounts of the electronic and nuclear factors affecting the kinetics of electron-transfer processes, see ref 2-6.
'
162 The Journal of Physical Chemistry, Vol. 92, No. 1 , 1988
IOgkq
Chiorboli et al.
0.01
I
0.1
0
9.2-
0
0 9.0-
0 0
8.8-
0
8.6-
00 0.1
I
1
I
I
0.1 0 . 2 0 . 3 0.4
0.5
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I+&
0.5
fill+
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Figure 7. Rate constants for the quenching of * R ~ ( b p y ) by ~ ~Ru+ (NH3),py3: in NaCl media (0)or in NaC104 media (0). The fits are made by using eq 18, the DE expressions for kd and k4, and k, = 1.1 X 1Olo s-I for NaCl and k, = 4 X 1O'O s-' for NaC104. The calculated (eq 3) kd curve is also shown for comparison (dotted line).
9.0
-
cn
-
al
8.8-
8.6-
far as reactions of * R ~ ( b p y ) ~ ,with + positive quenchers are concerned, Sutin4I has used the rate constant of eq 2146(k, = 2.2
-
0.3 0 . 4
7
I
I
0 . 2
0
0
a 1
8
+
*R~(phen)+ ~ ~R+u ( N H ~ ) ~ ~R+ ~ ( p h e n ) ~ ~R+U ( N H ~ ) ~ * + (21) X lo9 M-' s-l , in 0.2 M LiCI) as an experimental measure of kd for reactants of similar type. In fact, R u ( N H ~ ) is ~ ~considered + to be a substantially adiabatic reactant with small intrinsic barriers, (eq 21) is sufficiently exergonic and its reaction with *R~(phen)~,+ (AG = -0.89 eV) to suggest that it may lie near to or at the diffusional plateau.25 In order to check this point, we have measured the rate constants of an analogous quenching reaction4 involving * R ~ ( b p y ) , ~and + R u ( N H , ) ~ ~as~quencher ~+ (eq 16). This quencher has a more favorable redox potential (for eq 16, AG = -1.12 eV) and a faster self-exchange rate constant43 than Ru(NH3):+. On the other hand, the diffusional parameters are expected to be substantially the same for the two reactions (calculated kd values: kd = 5.1 X lo9 M-' s-] and kd = 5.1 X lo9 M-' s-l at I.L = 0.2 M for reactions 21 and 16, respectively; see Experimental Section for the parameters used). The rate constants for reaction 16 are given in Table 111. In comparable conditions (0.2 M NaCI, k, = 3.4 X lo9 M-' s-l ) this reaction is moderately but definitelys9 faster than that with the hexaammine complex. Thus, reaction 16 seems to be closer to the diffusional limit than reaction 21. The rate constants for reaction 16 in NaCl are compared with those in NaC104 and with calculated (DE) kd values in Table 111 and Figure 7. Here again a specific anion effect is observed, but the difference between the two sets of data is much smaller than in the MV2+ case. This is again consistent with the hypothesis of the anion effect acting on k,, in a reaction which is, however, much closer to diffusion than that of MV2+. Indeed, the data in perchlorate (Figure 7) appear to be very close to, if not coincident with, the calculated diffusion-controlled rate constants. When the background electrolyte is changed from a uni-univalent salt to one involving multivalent ions, further limitations of the simple Debye-Hiickel model show up. The rate constants for the quenching of excited R ~ ( b p y ) , ~by + MV2+ in NaCl or
(57) Wagner, P. J.; Kochevar, I. J . Am. Chem. Soc. 1968, 90, 2232. (58) It should be pointed out that a reaction between charged reactants cannot be assigned to a given kinetic regime as such. In fact, owing to the ionic strength dependence of k.dr reactions may change from one kinetic regime to another as the ionic strength is changed." (59) The difference is real and not due to the difference in the excited reactant or in the cation of the supporting electrolyte. In fact, the same rate constant as given by Sutin4' was reproduced with *Ru(bpy)32+as reactant and
(60) Scatchard, G. Natl. Bur. Stand. Circ. (US.)1953, 524, 185. (61) Friedman, H. L. Ionic Solution Theory; Wiley-Interscience: New York, 1962. (62) Kershaw, M. R.; Prue, J. E. Trans. Faraday SOC.1967, 63, 1198. (63) Hammett, L. P. Physical Organic Chemistry, 2nd ed.; McGraw-Hill: New York, 1970; Chapter 7 . (64) Indelli, A,; De Santis, R. J . Chem. Phys. 1971, 55, 481 I . (65) Indelli, A. Isr. J. Chem. 1971, 9, 301. (66) The authors are grateful to the referees for raising the points that are
NaCl as electrolyte.
discussed
here.
J. Phys. Chem. 1988, 92, 163-166 simple conceptual separation between diffusive and reactive steps becomes somewhat arbitrary. In principle, a statistical model in which the reaction probability is suitably averaged over all the possible interreactant configurations (involving different distance and orientation), each with its own electronic and nuclear kinetic factors, would be better suited to describe such a situation. In practice, however, such a model would be hopelessly complicated for any real system. The approach used in this work maintains the separations between reactive and diffusive steps. In this approach, all the possible intimate aspects of the reaction (e.g., preferential interactions, orientational effects, intercompenetration of reactants), together with their energetic and kinetic consequences, are buried into the behavior of the precursor complex. The nuclear and electronic factors of the unimolecular reactive step are thus taken to represent configurationally averaged properties of the precursor complex. In this framework, the diffusional step is considered as a process leading to an orientationally averaged, nonspecific contact between the reactants. This admittedly simplistic approach has the advantage of being useful as a conceptual tool and does not seem to meet with any serious quantitative problem in the interpretation of the experimental data.
Conclusions The results obtained in this study can be summarized as follows.
163
(1) In the nearly diffusion controlled electron-transfer reactions between cations investigated, two major deviations from ideal salt effects are observed: (i) specific counterion effects and (ii) the Olson-Simonson effect. Both effects are such that substantial errors could be made by comparing rate constants of reactions obtained, at constant ionic strength, with different "inert" electrolytes. (2) The observed salt effects can be parametrically accounted for with good accuracy in terms of the ionic strength dependence of the diffusional rate constants as predicted by the Debye-Huckel model, provided that (i) uni-univalent background electrolytes are considered and (ii) numerical integration over the interreactant distance of the full Debye-Eigen expressions (eq 3 and 10) is used. If multivalent cations are present in the electrolyte, the model can still be used to calculate the diffusional parameters, by using the anion concentration instead of the ionic strength. On the other hand, the use of approximate expressions for the diffusional parameters (eq 6, 12 or 9, 13) may generally lead to substantial errors in the calculated values.
Acknowledgment. We thank Professors L. Moggi and A. Indelli for the benefit of many helpful discussions. Registry No. Ru(bpy),2+, 15158-62-0; MV2+, 4685-14-7; Ru(NH&py3+, 33291-25-7; NaC1, 7647-14-5; NaCIO,, 7601-89-0; CaC12, 10043-52-4.
Measurement of Dispersion Relation of Chemical Waves in an Oscillatory Reacting Mediumt A. Pagola,$J. ROSS,* Department of Chemistry, Stanford University, Stanford, California 94305
and C. Vidal Centre de Recherche Paul Pascal, Universite de Bordeaux I , 33405 Talence cedex, France (Received: May 19, 1987)
We report measurements on the dispersion relation for chemical trigger waves propagating in an oscillatory Belousov-Zhabotinsky reacting medium. The waves are induced by a temperature perturbation (laser heating). The results are in qualitative agreement with a theory of such waves in an excitable medium.
Introduction Nonlinear reactions with a sufficiently complex reaction mechanism, maintained far from equilibrium, can transmit and sustain chemical waves or fronts that are traveling chemical concentration There have been many visual observations of chemical waves, including kinematic waves,5 relaxation oscillation waves,6-8and phase waves.9 The techniques of absorptionlo and transmission, coupled with imaging techniques,"J2 are being used to determine the velocity, profiles of fronts, invariance of structure in relaxation oscillation waves, and the formation of stationary spatial structures. Little is known on the subject of dispersion relations, that is, the dependence of the velocity of a chemical wave on the period of that wave. Keener and TysonI3 have presented analytical results on trigger waves for a simplified Oregonator model. They show that curvature effects, coupled with the dispersion relation, can 'This work has been supported in part by the National Science Foundation, the Air Force Office of Scientific Research, and the French Centre National de la Recherche Scientifique. *Permanent address: Centre de Recherche Paul Pascal, Universite de Bordeaux I, 33405 Talence cedex, France.
0022-3654/88/2092-0163$01.50/0
provide a fairly good description of wave propagation in a twodimensional excitable or oscillatory medium. The only experimental results reported are in ref 13 (unpublished work by A. T. Winfree) and in ref 18. In this article we report experimental ( I ) Eyring, H.; Henderson, D. Theoretical Chemistry; Academic: New York, 1978; Vol. 4. (2) Hanusse, P.; Ortoleva, P.; Ross, J. Ado. Chem. Phys. 1978, 38, 317. (3) Field, R.; Burger, M. Oscillations and Traveling Waves in Chemical Systems; Wiley: New York, 1985. (4) Vidal, C.; Hanusse, P. Inr. Reo. Phys. Chem. 1986, 5 , I . Vidal, C.; Pacault, A. In Evolution of Order and Chaos; Haken, H., Ed.; SpringerVerlag: Heidelberg, 1982. ( 5 ) Kopell, N.; Howard, L. N. Science (Washington, D.C.) 1973, 180, 1172. (6) Field, R. J.; Noyes, R. M. J . A m . Chem. SOC.1974, 96, 2001. (7) Showalter, K.; Noyes, R. M.; Turner, H. J . Am. Chem. SOC.1979, 101, 7463. (8) Sevcikova, H.; Marek, M. Physica D (Amsterdam) 1984, 1 3 0 , 379. (9) Bodet, J. M.; Ross, J.; Vidal, C. J . Chem. Phys. 1987, 86, 4418. (IO) Wood, P. M.; Ross, J. J . Chem. Phys. 1985, 82, 1924. (11) Muller, C.; Plesser, T.; Hess, B. Science (Washington, D.C.)1985, 230, 66 1. (12) Pagola, A.; Vidal, C. J . Phys. Chem. 1987, 91, 503. (13) Keener, P.; Tyson, J. Physica D (Amsterdam) 1986, ZID, 307.
0 1988 American Chemical Society
