Temperature Dependence of Excited-State Electron-Transfer

Jul 3, 1980 - reactions show the expected increase in rate constant as the overall ... free energy, AG*, with the overall free-energy change, AGO, ...
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3012

J. Phys. Chem. 1980, 84, 3012-3019

Temperature Dependence of Excited-State Electron-Transfer Reactions. Quenching of *1RuL3*+ Emission by Copper(11) and Europium(II1) in Aqueous Solution J. E. Baggott and M. J. Pilllng” Physical Chemistry Laboratory, Oxford OX1 3QZ, England (Received: March 5, 1980; In Final Form: July 3, 1980)

The mechanism of electron-transfer reactions of charge-transfer excited states of ruthenium(I1) complexes containing2,2’-bipyridine, 1,lO-phenanthroline,and their Substitutedderivatives with aquo complexes of copper(I1) and europium(II1)has been investigated by using temperature studies in the range 290-350 K. The europium reactions show the expected increase in rate constant as the overall free-energy change for the reaction (-AGO) increases. The copper sequence shows behavior different from that predicted by Marcus’ theory, with rate constants well below the diffusion-controlledlimit yet only weakly dependent on AGO. Indeed, increasing (-AGO) is shown to result in increases in the activation energy of the reactions. Several alternative explanations for this behavior are available, many of which can be dismissed in the light of this work. It is suggested that the rate-determining step in the Cu(1I) sequence involves a configuration change in the encounter pair. A similar step may be present in the Eu(II1) series but it must not be rate determining. One model suggests that this step is associated with the necessary distorted-octahedral/tetrahedralgeometry change in the reduction of Cu(I1). The temperature dependence of the electron-transferquenching of *Ru(4,7-Me2phen)t+by copper(I1) in DzO solvent is also included in the present work in order to test one of the proposed models.

Introduction Several theoretical approaches to the problem of describing electron-transfer reactions are available, but when large, complex inorganic ions are involved the classical statistical-mechanical treatment of Marcus is most widely applied.’ This model predicts a variation of the activation free energy, AG*, with the overall free-energy change, A G O , according to2

Similar studies of the reaction of three of these complexes with europium(II1) allow direct comparison of the activation parameters for apparently Marcusian and nonMarcusian electron-transfer reactions. The experimental activation entropies are compared with the theoretical values obtained by using the treatment of Waisman, Worry, and Marcus.lZ

Experimental Section The synthesis of the R u L ~ complexes ~+ has been outlined earlierlO and is discussed in greater detail by Anderson and Seddon.13 Commercial R ~ ( b p y ) ~ C 1 ~ * 6 H ~ O , neglecting Coulombic work terms, and supplied by Johnson-Matthey, was used in some experiments. In all cases observed luminescence lifetimes at 25 k = 2 exp(-AG*/RT) (2) “C in water agreed well with published values.ll BDH The term X comprises contributions from inner and outer AnalaR grade CuSO4-5H20and Koch-Light Laboratories coordination sphere reorganization and is often termed the EuC13-rH20were used in the quenching experiments. “intrinsic” barrier to electron transfer. 2 is a collision Samples were prepared in doubly distilled water containing frequency in solution, usually given a value of M-’ AnalaR grade H2SO4, HCI, and MgClz.6H20 where aps-1,2 Equation 1 predicts that AG* decreases as A G O bepropriate. In the experiments involving deuterium oxide comes increasingly negative but that, eventually, an as solvent commercial DzO (99.8%), supplied by Fluoro“inverted” region is reached, where AG* begins to increase. chem. Ltd., was used. Some experimental evidence for the “inverted” region The samples were deoxygenated by passing a stream of exists3i4but in many cases this behavior is not o b s e r ~ e d . ~ ~ nitrogen ~ through the solutions in l-cm square Pyrex cells. There are, in addition, many examples where eq 1breaks The technique of time-correlated single-photon counting, down in the “normal” region, i.e., it fails to predict the form used here, has been described previous1y.l4J5 Deconvoluof the dependence of AG* on Among the most tion techniques were not necessary. The temperature was oft-quoted explanations for this are nonadiabaticity and maintained constant between 20 and 80 “C by use of a noncancellation of Coulombic work terms. One such exthermostatically controlled, electrically heated copper ample of non-Marcusian behavior in the normal region is block as the cell housing. Temperature fluctuations are to be found in the electron-transfer quenching of chargeestimated to be f0.2 OC. The luminescence decays were transfer excited states of tris-chelated ruthenium(I1) comanalyzed as single exponentials in every case by using a plexes containing 2,2’-bipyridine (bpy), 1 , l O Research Machines 3802 microcomputer interfaced to the phenanthroline (phen), and their substituted derivatives multichannel analyzer. Plots of In (counts) vs. channel with aqueous copper(I1) in acidic solution?JO At 25 “C the number (time) were linear for at least three half-lives. quenching rate constants show only a weak dependence Results on AGO, in contrast to corresponding reactions involving Figure 1 shows the temperature dependence of the europium(II1) which show “normal” behavior; Le., a despontaneous decay for each of the five ruthenium(I1) pendence on AGO in accordance with eq l and 2.11 complexes studied in this work. Concentrations between Temperature studies of the electron-transfer quenching M in water were used; no variation in the and by aqueous copper(I1) of five related ruthenium(I1) comspontaneous decay rate constants, ko, was found with plexes of general formula RuLt+ have enabled us to disconcentrations in this range.le The temperature range cuss the mechanism in some detail and to examine critcovered does not allow accurate analysis of the dependence ically potential explanations of the observed behavior.

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0022-3654/80/2084-3012$01.00/00 1980 American Chemical Society

The Journal of Physical ChemistIy, Vol. 84, No. 23, 1980 3013

Quenching of +RuL,*+ by Cu(I1) and Eu(II1)

I A'\ :." " " '

- 19.5

--i

-- 190-

v1

'Ill

c

i

--

\

0

\

0 Y

Y

- 194=

I

c

18.5-

- 18.5 18.0-

170

L 29

I

'

3.1

1

t

- 18.0

L2-2d 3 3A 35' ' 2')

31

103~/1

1 x 1

33

3,

103K/T Figure 1. Temperature dependence of the spontaneous decay of the excited state "RuL:' ((3)L = 5-Brphen; (A)L = bpy; (m) L = phen; (0)L = 4,7-Me2phen; (A) L = 3,4,7,8-Me4phen. The decays were studied under identical conditions: neutral, aqueous solution containing 1O-,- 10-5 M complex.

TABLE I: Quadratic Coefficients: Emission Lifetimeqb and Excited-State Redox Potentialsbic for RuL3'+ Complexes in Aqueous Solution 1 0 - 4 ~ 1 io-6c/ *E"/ L A K K a rn/fis V 47.197 -1.6370 1.9197 0.94 -0.76 5-Brphen 53.332 -2.2243 3.1683 0.57 -0.84 bPY phen 51.045 -1.8652 2.2618 1.06 -0.87 4,7-Meaphen 56,404 -2.3612 3.2128 1.58 -1.01 4,7-Me,phend 69.695 -3.0958 4.1925 2.22 3,4,7,8-Me4phen 53.995 -1.9902 2.3244 1.52 -1.11 a Coefficients refer to the equation In k , = A t B/T + C / T 2 . At 25 "C. Taken from ref 11, potential for the reduction o f R u L , ~ +to *RuL$,+. In D,O. RuL3'+ concentration 10-4-10-5M.

-

of the excited state lifetimes in terms of a double exponential expre~si0n.l~For the purposes of calculating quenching rate constants (k,) it is sufficient to fit a quadratic equation in1 T1to the spontaneous decay data allowing interpolation over this temperature range. The quadratic fits are illustrated in Figure 1; coefficients, lifetimes, and reductilon potentials are collected in Table I. In the presence of quencher the observed first-order rate constant of luminesctme decay is given by (3) kobsd = ko +- kJQ1 Interpolated values for ko corresponding to the experimental temperature were used to estimate values of k, from each value of kobd. The quencher concentration was varied to ensure that the system was behaving in a Stern-Volmer fashion. The temperature dependence of the quenching rate constants is given in Figure 2. In each case the excited ruthenium(I1) complexes were quenched by copper(1.I) in 0.25 M H2SO4 Some problems arose in the experiments involving R~(3,4,7,8-Me~phen)~~+ in that this complex is only sparingly soluble under conditions of high sulfate ion concentration.1° A fine suspension of

Figure 2. Temperature dependence of the electron-transfer quenching of *RuL+ : by aqueous Cu(I1). Symbols represent complexes as given in Figure 1. All reactions were studied under identical conditions: 0.25 M HzS04,total ionic strength 0.8-1.0 M.

---I

29

31

33

35

103K/T Figure 3. Temperature dependence of the electron-transfer quenching of *RuL~+by aqueous; Eu(II1). Symbols represent complexes as given in Figure 1. The reactions were studied under identical conditions: 0.025 M HCI wlth added MgCIz to malntain the lonlc strength at 2.8 M.

(presumably) undissolved sulfate complex interfered with the luminescence decays, producing complex functions which could not be analyzed as single exponentials. The data referring to tlhis complex given in Figure 2 were obtained by minimizing the concentration of complex used in the measurements (S105M). The luminescence decays obtained in this way satisfied the criterion mentioned earlier but produced larger standard deviations in the decay plots and, consequently, a more scattered Arrhenius plot. Figure 3 shows the temperature dependence of the rate constants for electron transfer quenching of *Ru(bpy)32+, *Ru(phen)t+, and *Ru(4,7-Me2phen),2+by europium(III), in the form of E U C I ~ + , ~All ~ "three reactions were studied under the same conditions of 0.025 M HCl with added MgC12to maintain the ionic strength constant at -2.8 M. The scatter in the data for the quenching of *Ru(bpy)t+ and *Ru(phen)32+arises because of the very small differences being measured (for example, when L = bpy a total quencher concentration of 0.5 M produces less than a 10%

3014

The Journal of Physical Chemisfty, Vol. 84, No. 23, 1980

Baggott and Piiling

TABLE 11: Rate Constantsa and Activation Parameters* for the Quenching of *RuL12+Emission by C U ( I I ) and ~ Eu(III)~ Ions in Aqueous Solution 10-7kq/

L

Mn+aq

M-1 s-l

AS~/J

AGt/kJ mol-' a

AHt/kJ mol-'

KF1mol-'

3.41 30.1 i 1.0 11.0 f 0.8 -64 f 4 6.24 28.5 f 0.8 11.4 P 0.8 -57 f 4 EU3+aq 0.036 41 f 4 17 i 4 -81 f 18 phen CUZ+BQ 6.37 28.5 i 1.0 13.4 f 0.8 -51 f 4 EU3+aq 0.036 41-t 4 32f 4 -30 f 1 8 4,7 -Me,p hen cu2taq aq 7.93 27.9 k 0.8 17.8 f 0.6 -34 f 4 20.4 f 0.8 -26 f 4 7.77 28.0 f 0.8 cu2+(D,0) E u ~ + ~ ~ 0.125 38.2 i 0.8 25.2 i 0.6 -44 f 4 3,4,7,8-Me4phen cu=aq 6.48 28.5 f 1.4 18.9 f 1.2 -32 f 6 a At 25 "C. Two standard deviations are quoted with -95% confidence limts. All measurements with Cu(I1) taken in 0.25 M H,SO, with ionic strength 0.8-1.0 M. Measurements in 0.025 M HC1 with added MgCl,, ionic strength 2.8 M. In all experiments [ R U L , ~ +=] 10-4-10-5M. 5-Brphen bpy

cu2

cu2+aq

*

N

TABLE 111: Marcus Theory Predictions for the Rate Constants and Activation P a r a m e t e d for the Quenching of *RuL3'+ Emission by Cu(I1) and Eu(II1)

5-Brphen bpy phen

CU'+

CU2+aq

Eu 3 CU2 cas EU3+aq

4,7-Me2phen

cu2+aq Eu 3 +aq 2 +aq aq

2.1 x l o 8 1.5 x 107 1.6 x 107 2.1 x

los

29.3 29.6 45.2 29.1 43.5 23.3 37.4 19.2

22.4 20.1 35.4 20.5 34.6 15.6 29.4 11.8

-23 -32 -33 -29 -30 -26 -27 -25

4.6 x 4.1 x 7.6 X 5.0 X 1.5 x 5.2 X 1.8 x 2.7 x

107 107

lo4 10'

105 10'

lo6

cu 1.3 x 109 109 3,4,7 ,8-Me4phen At 25 "C. Electron exchange rate constant for the excited state: * R U L ~ t~ R ' uL,~+ + RuL13+t *RUL,~+. change in the observed rate constant). Unfortunately, it proved impossible to study the quenching of *Ru(3,4,7,8Me4phen)32fby Eu(II1) under these conditions because of the solubility problems outlined above. The rate and activation data are collected in Table 11. In calculating the activation parameters absolute rate theory has been assumed kET k, = h exp(-AGt/RT) (4)

AIYI = Eact- RT

(5) where Eactis the experimental activation energy. AGt is related to the Marcus free energy AG* (eq 1)by AG? = AG* - RT In

(E)

Comparison of Tables I and I1 indicates that the experimental values of AGt for the Cu(I1) sequence show only a weak dependence on the excited state redox potentials. Earlier work18 has shown that the excited states of Ru(bpy),2+and Ru(phen)32+l7 display strong charge-transfer-to-solvent (CTTS) contributions in addition to the expected CTTL properties, as revealed in solvent isotope studies. The spontaneous decay of *Ru(4,7-Me2phen)?+ has been studied in D20 and displays similar effects to those observed for the other complexes. It is thought that charge transfer via hydrogen-bonded solvent may be responsible for the observed effects.le The electron-transfer quenching of *Ru(4,7-Me2phen):+ by Cu(I1) has been studied in D20 to discover if H-bonding effects are an important feature of the electron-transfermechanism. The data are presented in Figure 4 together with the temperature dependence of the spontaneous decay in DzO. Corresponding measurements in water are included in this figure as the dashed curves. It can be seen from Figure 4 that, within experimental error, the rate constants for quenching of *Ru(4,7-Me2phen):+ by Cu(I1) in H 2 0 and

Figure 4. Solvent Isotope effects. Temperature dependence of the decay of emlssion from "Ru(4,7-Me2phen):+ In D20 (0). Electrontransfer quenching of *Ru(4,7-t~le,phen).,~ by Cu(I1) in D,O (0).Dashed curves refer to corresponding processes In water. Conditions as specified in Figures 1 and 2.

D20 are very similar, AGt being 27.9 f 0.8 and 28.0 f 0.8 kJ mol-l, respectively. Discussion A major aim of this work is to compare the measured rate constants for the Cu(I1) series with those predicted by Marcus theory. Table I11 summarizes the calculated values of AGt, AH+, and AS?, which were obtained from eq 1and 6 and from the equations developed by Marcus et a1.12 The X values were determined as follows: (1)The outer coordination sphere contribution (A,) was calculated by using the expression obtained by Marcus from dielectric continuum theoryS2 (2) The inner coordination sphere contribution (Xi) was estimated from self-exchange rate constants after correc-

Quenching of R ' UL';

The Journal of Physlcal Chemistry, Vol. 84, No. 23, 1980 3015

by Cu(I1) and Eu(II1)

tion for the contribution from A,. This procedure enabled

r

\

a more detailed examination of the rate-determining pa-

rameters to be made than would reliance on the unresolved self-exchange rates. The Xi values calculated in this way were 250 and 220 kJ mol-l for Cu(II)/Cu(I) (kll = lo-' M-* s-l8) and Eu(III)/Eu(II) (kll = 5 X 10"' M-l s-"'), respectively. Since these contributions to the activation free energy are assumed to be constant with changes in the nature of the ligands on Ru(II), any errors in their estimation should not affect the trends found in each electron-transfer series. The contribution to Ai from the excited Ru(I1) complexes themselves were estimated from the data of Hoselton et al.ll on the quenching of *RuL3'+ by Eu(lI1). Thus, these Ai values assume that the Eu(II1) reactions are well behaved and obey the Marcus equations expressed in this form. Consequently, any comparison made between the quenching of the excited ruthenium complexes by Cu(I1) and Marcus theory is essentially a comparison of Cu(I1) and Eu(II1) quenching within the framework of Marcus theory. (3) In all of the calculations the collision frequency, 2, has been estimated by using hard-sphere collision theory,20 rather than assuming a single value of 1011M-l s-'. The relative magnitudes of the contributions to Ai from the excited Ru(I1) complexes may be used to estimate rate constants for the excited-state self-exchange reactions:

-

*RuL,qa++ RuL~'+

*k

R U L ~+ ' *RuL,q2+

(7)

This contribution is found to vary slightly with ligand but remains relatively small with respect to that from Cu(II)/Cu(I) or Eu(III)/Eu(II). The estimated values of *k for each *Ru(II)/Ru(BII)couple are given in Table 111. The low value of *h for the * R u ( b p y ) ~ + / R ~ ( b p yexchange )~~+ is in some disagreement with previously assumed values9J1 but appears to be consistent with the observed Stokes' shift for this complex.21 The values of AGtcdd a H t d c d , and AStcdcd are summarized in Table I11 for the electron-transfer quenching of the *Ru12+complexes by Cu(I1) and Eu(II1). Because the calculations are based on the data of Sutin and co-workersg with regard to the *Ru132+/Eu(III)series, the agreement between AGtcdcd and AGt is good for Eu(III), any differences reflecting the slight disagreement in experimental measurements. The Marcus predictions for A H and A S are, however, less good. Comparison of the activation parameters for the quenching by Cu(I1) given in Tables I1 and I11 shows that, even in cases where relatively good agreement is obtained between calculated and experimental values of AGt, significant discrepancies exist between corresponding values of and ASt. Such behavior is often found when theoretical and experimental activation parameters are compared in this way.22 Experimentally, the invariance of AGT with AGO for the quenching by Cu(II), illustrated in Figure 5, can be seen to be the result of compensation between aHt and A S ; as AHt increases, a concomitant increase in TASf is obtained to the extent that the change in A" is counterbalanced. This "extrathermodynamic" relationship is a feature of many organic and inorganic reaction mechanisms and is generally known as an "isokinetic" effect, discussed in some detail by Leffler and Gr~nwald.'~Essentially, the magnitude of the changes displayed by either AIP or ASt on changing one ligand for another is linked to the constant of proportionality between them, the socalled "isokinetic temperature", p 6fWt = PGASt

(8)

90

100

110

- AGo/kJmol-'

I20

Flgure 5. The dependence of AGt on AGO as given by theory (0) and experlment (O),for the quenchlng of *RuL,*+ emission by Cu(I1): (1) L = 5-Brphen; (2) L = bpy; (3) L = phen; (4) L = 4,7-Me2phen; (5) L = 3,4,7,8-Me4phen.

'Or

-o t

-50 -30 AS*/J k-' mol-' Flgure 6. The isokinetic relationship. The magnitudes of the changes in AHf and ASt are linked In an extrathermodynamic fashion. The plot of AM vs. ASt has a slope 0) of 256 K. A correlation coefficient of 0.98 is obtained from these data.

-70

The magnitude of /3 relative to the experimental temperature determines the extent to which changes in AEF or ASt are communicated to AGt. 6AGt = (1 - ;)&AH+

(9)

The electron-transfer quenching of *RuLZ+ by Cu(I1) displays an isokinetic relationship, as shown in Figure 6, with p = 256 K. This is close to the experimental temperature with the result that only small changes in AGt are observed. Unfortunately, the parameter /3 is purely an experimental one and, as yet, its magnitude cannot be related to any physical property of the reaction mechanism. The establishment of an isokinetic relationship, with /3

3016

The Journal of Physical Chemistry, Vol. 84, No. 23, 1980

close to 300 K, simply recognizes that k, is nearly constant, independent of AGO. It might be argued that attempts to extend the Marcus analysis beyond comparison of experimental and theoretical values for AGt is overoptimistic and fails to recognize our understanding of the effects of changes in solvent structure on the rate of reaction. Thus the trade-off between entropy and enthalpy of activation terms frequently found in reactions of organic compounds can lead to modest and explicable changes in AGt even if the changes in AHt and ASt are large. However, the comparative behavior of the *RuL;+/Eu(III) and *RuL?+/ Cu(I1) series demands some discussion. The major conclusions that may be drawn from this work and that of Sutin et al.9J1are as follows: (1)The rate constants at 298 K show a dependence on the overall free-energy change that is compatible with Marcus theory when Eu(II1) is the electron-transfer oxidant, while those for Cu(I1) do not. Thus, within the framework of Marcu~theory the behavior of the reactions of *RuL;+ with Cu(I1) and with Eu(II1) are incompatible and the experimental evidence suggests that the abnormality must be found in the Cu(I1) series. (2) The comparison between the experimental and theoretical values for aHt and ASt is far from satisfactory for both series of reactions. However, the significant feature is the difference in the behavior of the two reactions. For the Eu(II1) series, (AStl decreases as (AGO( increases, in agreement with theory although (ASt(calcd< lAStlexpt. A much smaller and more erratic change in (AStlexptis found for Cu(I1). The values, on the other hand, behave somewhat erratically for Eu(III), but increase with increasing (AGO1 for Cu(I1) in direct contrast to the theoretical predictions. This behavior supports the conclusion in (1)above that there are significant mechanistic differences in the two series of reactions. Furthermore, given that the Cu(I1) series is the abnormal one, any explanation of its deviation from the Marcus model must be compatible with an approximate invariance in A S and an increase in aHt with increasing IAGOl. Reference can be given here to other experimental work exhibiting a similar form of disagreement with theoretical predictions. An exactly analogous situation exists for the oxidation of aquocobalt(II1)in perchlorate media.% There also rate saturation below the diffusion-controlled limit occurs with similar compensation effects between ivIt and A S being responsible for a levelling-off of AGt well below the values predicted by Marcus theory. It is interesting to compare Figure 5 of this work with Figure 3 of ref 24, where a preequilibrium spin change wm invoked to explain the observed b e h a v i ~ r . ~ ~ ~ ~ ~ The following general discussion examines various potential explanations of the observed behavior. Nonadiabaticity. The consequences of nonadiabatic effects are generally assumed to be a lowering of the transmission coefficient, is, resulting in a large negative contribution to the entropy of activation2esince k = z -kBT exp(-AGt/RT) (10) h This contribution may be estimated from the expression12 ASelect = R In R (11) Nonadiabaticity is often ascribed to spin changes, but such an explanation is not readily invoked in the present context, although electron-transfer reactions depend on orbital overlap and a decrease in the probability of transfer in the activated complex would arise if the distance of closest approach between the reactant, at the preferred

Baggott and Pilling

geometry, were increased. This explanation is particularly attractive since the rate saturation occurs as methyl groups are added to the phenanthroline rings. It is to some extent supported by the charge-transfer valence bond structures which have been suggested to explain some properties of CrL3*+complexes.27

Such forms indicate that spin localization in the 2,9 and 4,7 positions of the phenanthroline ring would be quite marked; some experimental evidence has been obtained for such localization in charge-transfer complexes of Cr(11)27although there is no comparable evidence for such localization in ground-state RuLBp’ complexes.2s However, the existence of such resonance forms in excited state complexes could have the effect of promoting “active sites” for electron transfer in the 2,9 and 4,7 positions which would be sterically blocked in the cases where L = 4,7Me2phen and 3,4,7,8-Me4phen. Although this explanation is attractive it is subject to two fundamental objections. For the two reactions where steric hinderance would be most marked (L = 4,7-Me2phen and 3,4,7,8-Me4phen)the experimental values of A S are not large and negative but are instead in quite close agreement with theoretical predictions. Rate saturation arises, instead, from increases in AH+. Any explanation relying solely on nonadiabaticity must be inadequate because it suggests that the abnormalities should be entropic, in contrast with the experimental behavior. Further evidence against the active site interpretation may be found in the results of pulse radiolysis experiments on reactions of RuL3+with Cu(1I) which also feature rate saturation? The reactions are characterized by even larger negative values of AGO even though the ligands involved are the less basic 5-Clphen and bpy. Steric hindrance in the 4,7 positions, though at first an attractive idea, does not explain why, with L = 5-Clphen, there should be a two-orders-of-magnitudediscrepancy in the calculated and experimental rate constant for its reaction with Cu(I1). Orientation Constraints. The Marcus model assumes that the reactants are spherically symmetric. The presence of active and inactive sites on a large molecule like *Ru(bpy)?’ may lead to anomalous kinetic behavior. A related property of this system is the apparent localization of the electron on e x c i t a t i ~ non ~ ~one l ~ ~of the bipyridyl ligands, although this may only apply at times significantly shorter than the lifetime of an encounter pair. Schmitz and Schurr examined the kinetics of reactions which are diffusion controlled, but in which there is a limited reactive solid angle,31932Under these circumstances rotational diffusion in the encounter pair becomes important and the reaction rate can fall below the diffusion-controlled limit. In the low viscosity limit the diffusion-controlled rate constant may be estimated from32 kdiff

NA = ~~(~o)~o(~TR + ADB)E )(DA

(12)

where Bo is the constraint (half-cone) angle, f(flo) is the fraction of all orientations (or solid angle) allowing reaction, and R A is the radius of the smaller reacting sphere, and DA, DB are the reactant diffusion coefficients. When the relation3’

f@,)= YZ(1

-

cos 60)

(13)

Quenching of *RuL:+

The Journal of Physical Chemistty, Vol. 84, No.

by Cu(I1) and Eu(II1)

23, 1980 3017

Scheme I1

Scheme I

Rut:

[ "Ruc.Cufa',,

v

5

products

products is used the value Bo which would reproduce the observed rate saturation may be calculated. In the case where L = 4,7-Mezphen a constraint angle of -30' is required to account for the observed rate saturation. This seems to be a perfectly realistic value, although it is not clear why an angular constraint should apply to *RuLt+/Cu(II) reactions but not to *RuL?+/Fe(III) reactions. Experiments at variable viscosities are needed to test the angular constraint model; iinfortunately it is difficult, with electron-transfer reactions, to vary the viscosity without, at the same time, affecting other determinants. This model therefore remains feasible but unproved. Kinetic Schemes. There is evidence for rate saturation in other reactions involving Cu(1I) reductionQand it is, therefore, of interest to examine the Cu(II)/Cu(I) system, in which there are changes not only in the equilibrium Cu-OH2 bond distances and angles, but also in the coordination number. It might be argued that the activation barriers that these structural changes represent have already been recognized in the self-exchange rate constant for Cu2+,/Cu+,,, i.e,, in the value of Xi(Cu)that is included in the Marcus analysis, and that this value is unchanged as the ruthenium ligands are varied.33 However, such a critisism does not recognize the complex multistep nature of these electron-transfer reactions and the consequent sensitivity of the overall kinetics to the identity of the rate-determining step, which in turn is dependent on the overall driving force of the reaction. The following discussion examines the effects on the overall kinetics of inserting an additional step in the reaction scheme. This additional step miglht be identified with a change in the Cu(I1) coordination. The first possible reaction scheme, which can be rapidly dismissed, involves reaction of an equilibrium distribution of energized Cu(I1) ions (Scheme 1)* This kinetic scheme proposes that reaction takes place only with an equilibrium distribution of energized Cu(I1) ions represented as tCu(II). The complex enclosed in braces denotes the activated or encounter complex. k{ and k i are diffusion-controlled rate constants, and k3/ is the rate constant for reaction in this encounter pair to form products. The overall bimolecular rate constant for reaction of ground state Cu(I1) and * R u L ~ ~is+ then KdZl'ka)/ ( k i + ka)) which, for a high driving force, reaches a saturation value of Kpkl'. Since kl' is the diffusioncontrol rate constant, such a kinetic scheme can explain rate saturation at low values of the overall rate constant if K p < 1. In the present system, the data require K p 0.03, i.e., the energized form of Cu(I1) must account for -3% of the total Cu(I1) concentration. If tCu(I1) were simply energized, thiri would correspond to an excess energy of -9 kJ molw1at 298 K, hardly enough to explain

-

TABLE IV: Limiting Forms for the Quenching Rate Constant Expression Given in Eq 14 under Various AsymDtotic Conditions primary condition

secondary conditions

k , >> k , k , >> k , k , > k , k 4 > 125, ( k 3 / k s ) (k,/k4)> k , , ( k , / k , ) ( k , / k , ) >> 1 k,,