Solvent dynamics and pressure effects in the kinetics of the tris(bipyridine)cobalt(III/II) electrode reaction in various solvents

Article abstract of DOI:10.1021/ja9923179

The volume of activation ΔV_el? for the Co(bpy)₃^3+/2+ electrode reaction in aqueous NaCl (0.2 mol L^-1) is -8.6 ± 0.4 cm³ mol^-1 at 25.0°C, as expected on theoretical grounds and by analogy wi

Full text of DOI:10.1021/ja9923179

10410
J. Am. Chem. Soc. 1999, 121, 10410-10415
Solvent Dynamics and Pressure Effects in the Kinetics of the
Tris(bipyridine)cobalt(III/II) Electrode Reaction in Various Solvents
Yansong Fu, Amanda S. Cole, and Thomas W. Swaddle*
Contribution from the Department of Chemistry, UniVersity of Calgary,
Calgary, Alberta, Canada T2N 1N4
ReceiVed July 6, 1999
q
3+/2+
Abstract: The volume of activation ∆Vel for the Co(bpy)3
electrode reaction in aqueous NaCl (0.2 mol
L-1) is -8.6 ( 0.4 cm3 mol-1 at 25.0 °C, as expected on theoretical grounds and by analogy with Co(en)3
3+/2+
and Co(phen)33+/2+, and neither the rate constant kel at various pressures nor ∆Vel correlate with the
q
q
corresponding mean diffusion coefficients D for the couple and the diffusional activation volume ∆Vdiff
,
q
respectively. In organic solvents, however, ∆Vel is strongly positiVe (9.1 ( 0.3, 10.2 ( 0.7, and 12.2 ( 0.9
cm3 mol-1 for CH3CN, acetone, and propylene carbonate, respectively, with 0.2 mol L-1 [(C4H9)4N]ClO4 at
25 °C) and correlates with ∆Vdiffq, while kel correlates with D. These results support the proposition of Murray
et al. (J. Am. Chem. Soc. 1996, 118, 1743; 1997, 119, 10249) that solvent dynamics control the rate of the
Co(bpy)33+/2+ electrode reaction in organic solvents. In aqueous solution at near-ambient temperatures, solvent
3+/2+
dynamical influences would not be revealed by pressure effects, but in any event the aqueous Co(bpy)3
electrode reaction appears to be mechanistically different from the nonaqueous cases. For the reduction of
Co(bpy)3 with Co(sep)2+ in homogeneous aqueous solution, the rate constant is lower, and the volume of
3+
activation more negative, than can be accommodated by extended Marcus theory, suggesting nonadiabatic
behavior. These observations are consistent with the view that, although the self-exchange and electrode reactions
are generally adiabatic, cross reactions involving CoIII/II couples (and presumably others) become increasingly
nonadiabatic as the driving potential is increased.
Introduction
η) of the solvent. Such effects have been observed for electrode
reactions of Cr(EDTA)-/2- and Fe(CN)6
in experiments
3-/4-
The question of the possible involvement of solvent dynamics
(solvent friction) in the kinetics of electron transfer processes
has been perceptively reviewed by Weaver1,2 and continues to
attract much theoretical interest.3-5 Experimental evidence for
solvent dynamical influences on the rate constants kel of
electrode reactions is nevertheless still limited and sometimes
ambiguous.1,2,6-15 In essence, if solvent dynamics are rate-
controlling, one expects kel to correlate inversely with the
longitudinal relaxation time τL (and hence with the viscosity
in which η was adjusted by addition of dextrose or sucrose to
the solvent (water or dimethyl sulfoxide).8-10 Solvent friction
has also been inferred in the reduction in aqueous acetone of
(NH3)5Co3+ groups bound to Hg electrodes through thiophene-
carboxylate ligands,11 and in the redox kinetics of metallocene
couples at Hg electrodes in various organic solvents.12,13 Of
particular interest is the recent observation by Murray and co-
3+/2+ 16
workers14,15 that the rate constant kel of the Co(bpy)3
electrode reaction in Debye solvents is proportional to τL-1 and
η-1 and to the diffusion coefficient D of the complexes (which
may be assumed to be the same for both the oxidized and
reduced species); these correlations hold over a spectacular range
of 1011 in kel.
* To whom correspondence should be addressed. Phone: (403) 220-
5358. Fax: (403) 284-1372. E-mail: swaddle@ucalgary.ca.
(1) Weaver, M. J. Chem. ReV. 1992, 92, 463.
(2) Weaver, M. J. J. Mol. Liq. 1994, 60, 57.
(3) Electron Transfer-from Isolated Molecules to Biomolecules; Ad-
vances in Chemical Physics 106 and 107; Jortner, J., Bixon, M., Eds.;
Wiley: New York, 1999.
(4) Khan, S. U. M. In Modern Aspects of Electrochemistry, No. 34;
Bockris, J. O’M., White, R. E., Conway, B. E., Eds.; Plenum Press: New
York, 1997; p 71 and references cited.
(5) Koper, M. T. M. J. Phys. Chem. B 1997, 101, 3168.
(6) Galus, Z. AdV. Electrochem. Sci. Eng. 1995, 4, 217.
(7) Fawcett, W. R.; Opallo, M. Angew. Chem., Int. Ed. Engl. 1994, 33,
2131.
(8) Zhang, X.; Leddy, J.; Bard, A. J. J. Am. Chem. Soc. 1985, 107, 3719.
(9) Zhang, X.; Yang, H.; Bard, A. J. J. Am. Chem. Soc. 1987, 109, 1916.
(10) Khoshtariya, D. E.; Dolidze, T. D.; Krulic, D.; Fatouros, N.;
Devilliers, D. J. Phys. Chem. B 1998, 102, 7800.
(11) Jaworski, J. S.; Kebede, Z. J. Electroanal. Chem. 1994, 370, 259.
(12) Gennett, T.; Milner, D. F.; Weaver, M. J. J. Phys. Chem. 1985, 89,
2787.
(13) McManis, G. E.; Golovin, M. N.; Weaver, M. J. J. Phys. Chem.
1986, 90, 6563.
(14) Pyati, R.; Murray, R. W. J. Am. Chem. Soc. 1996, 118, 1743.
(15) Williams, M. E.; Crooker, J. C.; Pyati, R.; Lyons, L. J.; Murray, R.
W. J. Am. Chem. Soc. 1997, 119, 10249.
These tests for solvent dynamical effects all involve com-
parisons of kel between chemically different solvents or mixtures.
Since η is roughly exponentially dependent on pressure P over
the first few hundred MPa for common solvents other than water
(Figure 1), kinetic studies of electrode reactions at elevated
pressures are potentially capable of revealing solvent dynamical
behavior in nonaqueous systems without changing the solvent
chemically.17 In the case of water, pressure breaks up ice-like
local structures in the liquid phase, so offsetting the normal
increase of viscosity with rising pressure with the result that
the η vs P plot at low temperatures passes through a shallow
minimum around 50-100 MPa and at 25 °C is almost flat to
(16) bpy ) 2,2′-bipyridine; en ) 1,2-diaminoethane; sep ) sepulchrate
) 1,3,6,8,10,13,16,19-octaazabicyclo[6.6.6]eicosane; DMF ) N,N′-di-
methylformamide; TBAP ) tetrabutylammonium perchlorate.
(17) Doine, H.; Swaddle, T. W. Can. J. Chem. 1988, 66, 2763.
10.1021/ja9923179 CCC: $18.00 © 1999 American Chemical Society
Published on Web 10/23/1999

3+/2+
Co(bpy)3
Electrode Reaction in Various SolVents
J. Am. Chem. Soc., Vol. 121, No. 44, 1999 10411
Grade) were dried by storage over freshly baked 4A molecular sieves.
Propylene carbonate (Aldrich, Gold Label, sealed under N2) was used
as received. DMF16 (Burdick and Jackson, distilled in glass) was dried
over molecular sieves and redistilled under reduced pressure. TBAP
(Fluka) was either recrystallized from ethanol (AG grade) or used as
received (Electrochemical grade, puriss.).
Electrochemical Measurements. Cyclic (CV) and alternating
current (ACV) voltammograms were obtained with Pt wire working
electrodes and analyzed as described previously.21,24 Half-wave poten-
tials E1/2 from the CVs were reproducible to within (5 mV. Glassy
carbon working electrodes gave good CVs but distorted ACVs in initial
aqueous studies and were not used further. ACV measurements were
made at 25.2, 45.3, 85.3, and (at 3 °C) 159 Hz for solvent water; 25.2,
45.3, 85.3, and 149 Hz for propylene carbonate; and 45.3, 85.3, 159,
and 258 Hz for acetone and acetonitrile. Measurements at ambient
pressure were made using a glass-jacketed cell, thermostated with
circulating water. Measurements at variable pressure were made in the
apparatus previously described,21,24 with the temperature controlled at
25.0 ( 0.1 °C with water circulating through an external jacket. The
supporting electrolyte was 0.2 mol L-1 NaCl in aqueous solutions (in
which solubility precluded the use of perchlorate media) and 0.2 mol
L-1 TBAP in the organic solvents. The Co complex was insufficiently
soluble in CH3OH, CH2Cl2, and CHCl3, and decomposed slowly in
the course of measurements in the strong donor solvent DMF. The
Ag/AgCl reference electrode used for aqueous solutions was unstable
in organic solvents, in which a Ag/Ag+/TBAP electrode was employed
for the kinetic experiments.
Figure 1. Pressure dependence of viscosity η for typical solvents at
25 °C, recalculated from refs 18 (water), 19 (acetonitrile and DMF),
and 20 (acetone). Data for propylene carbonate are not available.
200 MPa. Consequently, for a given couple, if any pressure
effects (retardations) on kel observed in nonaqueous solutions
are truly due to solvent dynamic effects, they will be conspicu-
ously absent in water near 25 °C. Ordinarily, volumes of
Stopped-Flow Kinetics. Aqueous [Co(sep)](ClO4)3 was reduced to
Co(sep)2+ with amalgamated Zn for 4 h immediately prior to use, and
q
activation ∆Vel ()-RT(∂ ln kel/∂P)T) for cationic couples
3+
this and the Co(bpy)3 solution were handled under Ar at all times.
undergoing electron transfer at an electrode are expected (and
found) to be moderately negative;21 thus, as suggested previously
in the context of self-exchange reactions,17 the observation of
a markedly positiVe ∆Velq value for a nonaqueous solvent would
constitute strong evidence for the incursion of solvent dynamics.
With these considerations and the observations of Murray et
The ionic strength I of all solutions was adjusted to 0.2 mol L-1 with
LiClO4 (pH ∼6). Kinetic measurements were made at 318 nm (relevant
spectra are given in the Supporting Information, Figure S1) in a Hi-
Tech SP-56 pressurizable stopped-flow system, with Co(sep)2+ in >10-
fold excess over Co(bpy)33+; thus, the rate of spectrophotometric change
was first order in [Co(bpy)33+]. A fragment of amalgamated zinc was
placed in the reductant syringe to suppress any reoxidation of
q
al.14,15 in mind, we have measured volumes of activation ∆Vel
for the reduction of Co(bpy)33+ at a Pt electrode in acetonitrile,
acetone, propylene carbonate, and water, the choice of solvents
being limited to these by the solubility and stability of the CoIII
and CoII complexes. The results broadly support the conclusion
of Murray et al.14,15 that solvent dynamics are dominant for this
electrode reaction in the nonaqueous solvents, but also imply
that in water the electrode reaction is mechanistically different.
We have therefore also investigated the effects of pressure on
Co(sep)2+ 25
.
Results
Concentrations specified in this article are temperature and
pressure independent, expressed as if at 22 °C and 0.1 MPa.
Only those variable-pressure experiments in which the low-
pressure measurements at the beginning and end of the pressure
cycle agreed within the experimental uncertainty were accepted.
3+
the chemical reduction of Co(bpy)3 by Co(sep)2+ in homo-
3+/2+
For the Co(bpy)3
electrode reaction in any solvent at
geneous aqueous solution. The results reinforce our interpreta-
tion of the electrochemical results for water, but also suggest
that the cross reaction is not fully adiabatic.
each pressure and temperature, the mean reactant diffusion
coefficient D was obtained from the averaged peak currents of
multiple CV measurements, and the electrode rate constant kel
was calculated from the maximum in-phase and 90° out-of-
phase alternating currents (Ix and Iy, respectively, at potential
Experimental Section
Materials. [Co(bpy)3](ClO4)3‚3H2O was made by the method of
Nyholm and Burstall22 and checked for purity by CHN microanalyses
and its UV-visible spectrum in water (absorbance maxima at 318, 307,
and 222 nm; molar absorbances 3.03 × 104, 3.29 × 104, and 8.76 ×
E
max) of an ACV after correcting for the uncompensated
resistance Ru.24 Although kel can, in principle, be obtained from
the peak separation of CVs, such a procedure will give erroneous
results if Ru is significant, which is generally the case for
nonaqueous solvents.6,26,27 In our ACV experiments, however,
Ru was first obtained directly from the cell impedance, measured
at a high frequency (typically 8 kHz) and a potential g300 mV
away from E1/2 for Co(bpy)33+/2+, and was then specifically
allowed for in the calculation of kel.24 The in-phase and 90°
out-of-phase background currents Ibx and Iby were measured to
obtain the total cell impedance, from which Ru was subtracted
16
104 L mol-1 cm-1, respectively). [Co(sep)](ClO4)3 was made by the
method of Harrowfield et al.23 (Caution! Perchlorate salts of CoIII
complexes containing organic ligands are potentially explosive.)
Distilled water was further purified by passage through a Barnstead
NANOpure train. Acetone and redistilled acetonitrile (BDH, Analytical
(18) Sengers, J. V.; Kamgar-Parsi, B. J. Phys. Chem. Ref. Data 1984,
13, 185.
(19) Salman, O. Ph.D. Thesis, University of Illinois, 1982.
(20) Bridgman, P. W. The Physics of High Pressure; G. Bell: London,
1931; p 342.
(21) Fu, Y.; Swaddle, T. W. J. Am. Chem. Soc. 1997, 119, 7137.
(22) Nyholm, R. S.; Burstall, F. H. J. Chem. Soc. 1952, 3570.
(23) Harrowfield, J. M., personal communication; cf.: Creaser, I. I.;
Harrowfield, J. M.; Herlt, A. J.; Sargeson, A. M.; Springborg, J.; Geue, R.
J.; Snow, M. R. J. Am. Chem. Soc. 1982, 104, 6016.
(24) Fu, Y.; Swaddle, T. W. Chem. Commun. 1996, 1171.
(25) Metelski, P. D.; Fu, Y.; Khan, K.; Swaddle, T. W. Inorg. Chem.
1999, 38, 3103.
(26) Gileadi, E. Electrode Kinetics for Chemists, Chemical Engineers
and Materials Scientists; VCH: New York, 1993; pp 419-420.
(27) Howell, J. O.; Wightman, R. M. Anal. Chem. 1984, 56, 524.

10412 J. Am. Chem. Soc., Vol. 121, No. 44, 1999
Fu et al.
Figure 3. Pressure dependences of ln D (b) and ln kel (2, rising
Figure 2. Arrhenius plots showing the temperature dependences of
the mean diffusion coefficient D (broken line) and the electrode reaction
rate constant kel (solid line) for Co(bpy)33+/2+at a Pt electrode in aqueous
0.2 mol L-1 NaCl.
3+/2+
pressure; 3, descending pressure) for Co(bpy)3
at a Pt electrode
in 0.2 mol L-1 aqueous NaCl at 25.0 °C.
3+/2+
Table 1. Characteristics of Co(bpy)3
at a Pt Wire Electrodea
solvent
to give the double-layer charging impedance Zdl. This in turn
was used to recalculate Ibx and Iby (phase shift φZ). The faradaic
peak currents Ifx and Ify were obtained by subtracting the
recalculated Ibx and Iby from Ix and Iy, respectively, giving a
phase angle φf. Since the expression
propylene
CH3CNc (CH3)2COc carbonatec
H2Ob
(125)d
E1/2/mV
4e,f 0e -52e
∆Vcell
(26.8 ( 0.9)d 15.9 ( 0.4e 12.3 ( 1.0e 14.0 ( 0.4e
kel0/cm s-1
0.172
6.06
0.198
10.6
0.070
9.41
0.0160
1.40
D0/10-6 cm2 s-1
∆Velq/cm3 mol-1
∆Vdiffq/cm3 mol-1
RT(∂ ln η/∂P)T/
Emax ) E1/2 + (RT/F) ln{R/(1 - R)}
-8.6 ( 0.4 9.1 ( 0.3 10.2 ( 0.7 12.2 ( 0.9
0.37 ( 0.03 8.8 ( 0.1 11.1 ( 0.1 15.6 ( 0.4
0.5g
8.2g
8.8g
13.2h
13.1h
showed the transfer coefficient R to be 0.5 in all cases, the
simplified relation
cm3 mol-1
q
∆VSD calc/cm3 mol-1 1.5g
a 1.0 mmol L-1 Co; 25.0 °C; transfer coefficient R ) 0.5 in all cases.
b 0.20 mol L-1 NaCl. c 0.20 mol L-1 Bu4NClO4. d Reference electrode
Ag/AgCl/(4 mol L-1 KCl). e Reference electrode Ag/(0.013 mol L-1
cot φ ) 1 + (2Dω)1/2/kel
AgClO4-0.20 mol L-1 Bu4NClO4-CH3CN). f -85 ( 10 mV vs
where φ is the corrected phase angle ()φZ + φf) and ω is the
angular ac frequency, was used to calculate kel.21,24
Although pressure-dependence parameters such as ∆Vel
+/0
Fe(C5H5)2
.
g Calculated for midrange pressure of 100 MPa. h 30 °C,
0-100 MPa.
q
by other workers.29 At 3.0 ( 0.1 °C, ACV measurements over
the range 0.1-201 MPa again showed an accurately linear
appeared constant within experimental uncertainty over the
pressure range 0-200 MPa, a small secondary dependence on
pressure is expected, so that cited values should be regarded as
valid at 100 MPa, the midpoint of the pressure range.
dependence of ln kel on P,28 giving ∆Vel ) -9.1 ( 0.3 cm3
q
q
mol-1. Thus, the temperature dependence of ∆Vel (cf. Table
The Co(bpy)33+/2+ Electrode Reaction in Water. Figure 2
1) is not significant in view of the standard deviations cited.
The Co(bpy)33+/2+ Electrode Reaction in Organic Solvents.
The choice of solvents was limited to acetonitrile, acetone, and
propylene carbonate by insufficient solubility of Co(bpy)33+ salts
in methanol, dichloromethane, chloroform, etc., and by the
shows the temperature dependences of ln D (from CV data)
3+/2+
and ln kel (from ACV measurements) for Co(bpy)3
(1.0
mmol L-1) in aqueous NaCl (0.2 mol L-1) at a Pt electrode,
3.2-44.7 °C. The data28 yield the respective Arrhenius activa-
tion energies Ea(diff) ) 19.1 ( 0.1 kJ mol-1 and Ea(el) ) 37.0 (
1.7 kJ mol-1 (or, for the electrode reaction rate, the Eyring
parameters ∆Helq ) 34.5 ( 1.7 kJ mol-1, ∆Selq ) -139 ( 6 J
K-1 mol-1), with D298 ) 6.5 × 10-6 cm2 s-1 and kel298 ) 0.31
2+
tendency of the transiently formed Co(bpy)3 to decompose
in strong donor solvents such as DMF. The CV for reduction
of Co(bpy)33+ in acetonitrile (0.3 mol L-1 TBAP) showed redox
waves with E1/2 ) 618 and -655 mV vs Ag/AgCl, correspond-
ing to the Co(bpy)33+/2+ and Co(bpy)32+/+ couples, but decom-
position of the transient CoI species was evident at scan rates
<50 mV s-1. Uncompensated resistance was relatively high and
increased with rising pressure (Table S128), as expected for
organic solvents, but was specifically corrected for in each
experiment and at all temperatures and pressures when calculat-
ing kel.21 This is an important point, because the pressure and
cm s-1
.
Figure 3 shows the linear pressure dependences of ln D and
3+/2+
ln kel for Co(bpy)3
in aqueous 0.2 mol L-1 NaCl at 25.0
°C. The corresponding parameters are given in Table 1. High-
pressure CVs showed that E1/2 was also a linear function of
pressure, 0-200 MPa,28 giving the volume of reaction ∆Vcell
()-F(∂E1/2/∂P)T, where F is the Faraday) relative to the Ag/
AgCl electrode (Table 1) in good agreement with values reported
(29) Swaddle, T. W.; Tregloan, P. A. Coord. Chem. ReV. 1999, 187,
255.
(28) See Supporting Information.

3+/2+
Co(bpy)3
Electrode Reaction in Various SolVents
J. Am. Chem. Soc., Vol. 121, No. 44, 1999 10413
nitrile as solvent, ∆Vcell is in good agreement with the published
value.29
Reduction of Co(bpy)33+ in Homogeneous Aqueous Solu-
tion. For stopped-flow kinetic measurements, the choice of a
suitable reductant for Co(bpy)33+ is severely limited by the need
for the reductant to be substitutionally inert and also for the
reduction to be fast enough (implying a high ∆E1/2 for the cross
reaction) to be substantially complete before the incursion of a
secondary process with rate constant ksec ) 2.4 s-1 at 24.8 °C.
These criteria are met by the cage complex Co(sep)2+ (∆E1/2
) 660 mV, from ref 21 and Table 1). With Co(sep)2+ as the
reductant, ksec was independent of [Co(sep)2+] (0.67-2.0 mmol
L-1). Thus, the secondary process was evidently aquation of
the newly formed Co(bpy)32+, for which Farina et al.30 reported
a rate constant of 0.7 s-1 at 18 °C and pH 5 in the presence of
terpyridine. The secondary process showed a stronger temper-
q
q
ature dependence28 (∆Hsec ) 59.1 ( 1.3 kJ mol-1, ∆Ssec
)
∼
-40 ( 4 J K-1 mol-1) than the initial reduction step (∆H12
q
4 kJ mol-1, not accurately measurable). Thus, by lowering the
temperature to 5.9 ( 0.2 °C, satisfactory separation of the faster
initial electron-transfer step (pseudo-first-order rate constant kobs
) 34 to 145 s-1 for [Co(sep)2+] ) 0.4 to 2.0 mmol L-1) from
the subsequent aquation was obtained. For the cross reaction,
kobs was first order in [Co(sep)2+] as well as [Co(bpy)3+],28
giving the second-order rate constant k12 ) (8.0 ( 0.3) × 104
Figure 4. Dependence of kel on D for Co(bpy)33+/2+ at a Pt electrode
in acetonitrile containing 0.2 mol L-1 TBAP at 25.0 °C.
L mol-1 s-1 at 5.9 °C and I ) 0.2 mol L-1
.
Although the temperature dependence of k12 was not ac-
curately measurable, ln k12 gave satisfactory linear plots against
pressure P up to 150 MPa at 5.3-6.0 °C. Independent runs gave
∆V12q ) -9.9 ( 0.7, -10.9 ( 0.9, -9.7 ( 0.4,28 and -9.5 (
0.428 cm3 mol-1 at 5.3, 5.9, 5.9, and 5.6 °C, respectively; the
weighted mean is -9.7 ( 0.4 cm3 mol-1 at 5.6 ( 0.3 °C and
I ) 0.2 mol L-1
.
Discussion
The rate constant k for an electron-transfer reaction can be
expressed in terms of a free energy of activation ∆G* (compris-
ing internal and solvent reorganizational components, ∆GIR*
and ∆GSR*, respectively), the equilibrium constant Kprec for
assembly of a precursor complex, the electronic frequency factor
(transmission coefficient) κ, and a nuclear frequency factor νn.31
k ) Kprecκνn exp(-∆G*/RT)
(1)
If the electron-transfer step is fully adiabatic, κ ) 1. Solvent
properties can influence k not only energetically through the
contribution ∆GSR* to the height of the activation barrier, but
also dynamically through νn by reducing the barrier crossing
frequency, if the solvent and transition-state motions are
significantly coupled. In the simplest treatment,1,2,7,12-15,32 which
assumes adiabatic electron transfer and a low ∆GIR*, we can
write
Figure 5. Pressure dependences of ln D (b) and ln kel (2, rising
3+/2+
pressure; 3, descending pressure) for Co(bpy)3
at a Pt electrode
in acetonitrile containing 0.2 mol L-1 TBAP at 25.0 °C.
temperature dependences of disregarded Ru could account (at
least qualitatively) for those observed for kel (in other words,
concealed Ru could mimic solvent dynamical effects).1
νn ) τL-1(∆GSR*/4πRT)1/2
(2)
Figure 4 shows that kel is directly proportional to D over the
3+/2+
pressure range 0-201 MPa for Co(bpy)3
in acetonitrile
(slope 1.89 × 104 cm-1, negligible intercept). The pressure
dependences of ln kel and ln D for Co(bpy)33+/2+ in acetonitrile
are shown in Figure 5; these and the corresponding plots for
acetone and propylene carbonate28 are linear within the experi-
mental uncertainty. The derived parameters are summarized in
Table 1, together with cell reaction volumes ∆Vcell relative to
the Ag/Ag+ electrode obtained from the linear pressure de-
pendences of E1/2 over the range 0.1-200 MPa.28 For aceto-
and, if VM is the hard-sphere effective volume and ꢀ0 and ꢀop
are respectively the static and optical dielectric constants of the
solvent,
τL-1 ) (ꢀ0/ꢀop)RT/3VMη
(3)
(30) Farina, R.; Hogg, R.; Wilkins, R. G. Inorg. Chem. 1968, 7, 170.
(31) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265.

10414 J. Am. Chem. Soc., Vol. 121, No. 44, 1999
Fu et al.
For our purposes, ꢀop may be taken to be the square of the
refractive index n of the solvent. The experimental volume of
activation ∆Vobs is given by
experimental ∆Velq values, since ∆VSDq for organic solvents is
predicted to be on the order of +10 cm3 mol-1 (Table 1).
Comparison of Figures 3 and 5 immediately shows that,
qualitatively at least, these predictions are fulfilled for the
Co(bpy)33+/2+ couple. The value of ∆Velq for water (Table 1) is
q
q
∆Vobsq ) ∆VSDq + (∆VIRq + ∆VSRq + ∆VCOULq + ∆VDH
)
3+/2+
3+/2+
closely similar to those for Co(en)3
and Co(phen)3
(4)
(-8.3 ( 0.5 and -9.1 ( 0.4 cm3 mol-1, respectively).21 In
q
where the sum in parentheses is the volume of activation
acetonitrile, acetone, and propylene carbonate, however, ∆Vel
q
is almost as strongly positive as ∆Vdiffq (Table 1), implying rate
control by solvent dynamics. This inference is supported by the
strong linear correlation between kel and D at various pressures
for each organic solvent (shown for acetonitrile in Figure 4).
Murray and co-workers14,15 reported such a correlation over a
wide numerical range using different solvents at atmospheric
pressure; Figure 4 shows that a very tight correlation can be
obtained without changing the chemical identity of the solvent,
using pressure to vary the solvent viscosity. Application of eq
4 to the nonaqueous systems indicates that the barrier-height
associated with barrier height (∆VCOUL representing the
pressure dependence of the Coulombic work terms, i.e., of Kprec
,
q
and ∆VDH the pressure effect on the Debye-Hu¨ckel activity
q
coefficients) and ∆VSD is due to barrier crossing (solvent
dynamics). From eqs 2 and 3, we have
∆VSDq ) -RT(∂ ln νn/∂P)T ) RT[(∂ ln η/∂P)T -
(∂ ln ꢀ0/∂P)T + 2(∂ ln n/∂P)T - ∆VSRq/2∆GSR*] (5)
The last three terms in eq 5 are small for typical solvents
q
part of ∆Vel (that in parentheses) is on the order of 0 to -3
and tend to cancel each other, so that the calculated18-20,33 ∆VSD
q
cm3 mol-1
.
is determined almost entirely by (∂ ln η/∂P)T as shown in Table
1. Values of RT(∂ ln η/∂P)T calculated using literature data18-20
(Table 1) are close to the electrochemically measured activation
For water as solvent, kel increases with increasing pressure,
whereas D remains essentially constant (Figure 3). Qualitatively,
this is as expected, because, in water at near-ambient temper-
atures, solvent friction (if present at all) will not be revealed by
z+
q
volumes for diffusion of Co(bpy)3 (∆Vdiff ) -RT(∂ ln
D/∂P)T), as expected from the Stokes-Einstein relation (D )
kBT/6πaη, where a is the effective radius of the diffusing
particle). We note that the Arrhenius activation energy for
diffusion of Co(bpy)3z+ in water (Ea(diff) ) -RT(∂ ln D/∂T)P )
19.1 kJ mol-1, from Figure 2) is similar to that for viscous flow
of water (16.8 kJ mol-1 at 25 °C, calculated from the data of
ref 18).
q
varying the pressure, as noted above. The magnitude of ∆Vel
in water, however, implies that the barrier-height contribution
is about -9 cm3 mol-1, that is, much more negative than for
the organic solvents. Thus, there may be a mechanistic change
on going from water to nonaqueous solvents. Furthermore, the
3+/2+
Arrhenius activation energy for the Co(bpy)3
electrode
reaction in aqueous NaCl (37 kJ mol-1) is twice as large as
that for diffusion of the complexes or for viscous flow of water
(Figure 2). This is further evidence for a relatively large
activational contribution to the electron-transfer kinetics in water,
regardless of any influence of solvent dynamics.
Equations 1-5 apply to both bimolecular self-exchange
reactions and the corresponding reactions at an electrode. For
the latter, however, the free energy of activation ∆Gel* was
predicted by Marcus34 to be one-half that for homogeneous self-
exchange (∆Gex*). This relationship is difficult to demonstrate
directly through kel and kex because the preexponential factor
in eq 1 for kel is very sensitive to the nature of the electrode
surface. However, because electrode surfaces are not signifi-
cantly affected by pressures in the 0-200 MPa range, a
remarkably precise relationship exists between the corresponding
The inferred mechanistic difference may be due to ion pairing
between the highly charged Co complexes and the anion of the
supporting electrolyte. Such ion association is probably incon-
sequential in 0.2 mol L-1 aqueous NaCl, but will almost
inevitably be important in organic solvents. Its impact on the
q
self-exchange parameters kex and ∆Vex is hard to predict;
q
q
Volumes of activation ∆Vel and ∆Vex for a wide variety of
cation-cation reactions often show mild retardation ascribable
to ion pairing,37 whereas anion-anion electron transfers can
be dramatically catalyzed by cations even in water.38 Counterion
effects are more difficult to interpret for electrode reactions
because of the probable modification of the double layer, and
therefore we have not pursued this question in the present study.
Pyati and Murray,14 however, concluded that ion pairing was
not a major factor in their study of the Co(bpy)33+/2+ electrode
reaction in a wide range of solvents. Thus, the origin of the
inferred mechanistic change between water and organic solvents
remains an open question.
transition-metal complex couples in aqueous solution:21,24,35
q
∆Velq ) (0.50 ( 0.02)∆Vex
(6)
We call eq 6 the “fifty-percent rule”. For aqueous solutions
around 25 °C, ∆VSDq is predicted to be negligible, and so eq 6
can be expected to hold for such systems regardless of whether
q
solvent dynamics are important. In other words, ∆Vel gives
the pressure dependence of ∆G* unambiguously for aqueous
systems at near-ambient temperatures. Furthermore, the theory
that has been developed36 for ∆Vex can be directly applied to
q
q
3+/2+
It might be argued that ∆Vel for Co(bpy)3
in aqueous
q
q
∆Vel for aqueous systems, in particular, ∆Vel for cationic
NaCl is in some way anomalous, although its consistency with
couples can be expected to be negatiVe (-2 to -6 cm3 mol-1
)
q
3+/2+
3+/2+
∆Vel for Co(en)3
and Co(phen)3
suggests otherwise.
and, in the case of tris-bidentate CoIII/II complexes such as
For the latter two couples,21 the measured ∆Vel values were
q
3+/2+
Co(bpy)3
where CoII is high spin, quite strongly so (near
authenticated through ∆Vexq by conformity with the fifty-percent
-9 cm3 mol-1).21 In nonaqueous systems, however, the incur-
sion of solvent dynamics should in general lead to positiVe
rule, but no ∆Vex is available for Co(bpy)33+/2+. As an
q
alternative, we have attempted to estimate ∆Vexq from the cross-
reaction of Co(bpy)33+ with Co(sep)2+ in aqueous LiClO4. The
Marcus cross relation for rate constants of adiabatic, outer-
sphere, bimolecular electron-transfer reactions31
(32) Grampp, G.; Harrer, W.; Jaenicke, W. J. Chem. Soc., Faraday Trans.
1 1987, 83, 161.
(33) Swaddle, T. W. Inorg. Chem. 1990, 29, 5017.
(34) Marcus, R. A. Electrochim. Acta 1968, 13, 995 and references
therein.
(35) Fu, Y.; Swaddle, T. W. Inorg. Chem. 1999, 38, 876.
(36) Swaddle, T. W. Can. J. Chem. 1996, 74, 631.
(37) Wherland, S. Coord. Chem. ReV. 1993, 123, 169.
(38) Metelski, P.; Swaddle, T. W. Inorg. Chem. 1999, 38, 301.

3+/2+
Co(bpy)3
Electrode Reaction in Various SolVents
k12 ) (k11k22K12f)1/2W
J. Am. Chem. Soc., Vol. 121, No. 44, 1999 10415
1) and more negative volumes of activation42 than expected.
Endicott and Ramasami43 have presented data on the kinetics
of reductions of a variety of CoIII complexes by Co(sep)2+ that
indicate that the cross reactions are in general nonadiabatic but
that κ f 1 as ∆E1/2 f 0; in other words, self-exchange (and
electrode) reactions are adiabatic but kinetic parameters for cross
reactions with high ∆E1/2 may deviate from the predictions of
eqs 7 and 8. Our observations25,36,39,42 are certainly consistent
with this view (our early suggestion,42 that the Co(en)33+/2+ and
Co(phen)33+/2+ self-exchange reactions might be nonadiabatic,
has proved to be incorrect36). Significantly, Fu¨rholz and Haim44
have proposed that cross reactions of the Co(bpy)33+/2+ couple
may be “mildly nonadiabatic.”
(7)
[ln K12 + (w12 - w21)/RT]2
ln f )
4[ln(k11k22/Z11/Z22) + (w11 + w22)/RT]
W ) exp[(w11 + w22 - w12 - w21)/2RT]
leads to a corresponding relation in volumes of activation:25,39
∆V12q ) [(∆V11q + ∆V22q + ∆V12)/2] + C
(8)
in which the subscripts 11 and 22 refer to self-exchange
reactions of reactants 1 and 2, and 12 and 21 refer to the cross
reaction of 1 with 2 and its reverse, respectively, kij are rate
constants, K12 is the equilibrium constant ()exp(F∆E1/2/RT)),
and ∆V12 is the reaction volume ()∆Vcell(2) - ∆Vcell(1)). The
symbols w refer to the Coulombic work required to bring the
reactants together. Zij are the frequency factors (preexponential
part of eq 1). For self-exchange reactions and for cross reactions
that are symmetrical with respect to the charges on the reactants,
W ∼ 1 if the reactant radii are comparable, and the term C is
then -(RT/2)(∂ ln f/∂P)T (see Grace et al.39 for further
development). For small ∆E1/2 (<200 mV), f ∼ 1 and C ∼ 0.
Conclusions
Within the limitations and approximations implicit in the
theory underlying eqs 2-5, pressure effects on electrode reaction
rates and reactant diffusion coefficients can provide strong
evidence for the incursion of solvent dynamics in nonaqueous
systems without changing the chemical identity of the solvent.
For the Co(bpy)33+/2+ electrode reaction, solvent dynamics are
found to be important in nonaqueous solvents. This is somewhat
surprising, since ∆GIR* is expected to be substantial for
3+/2+
Co(bpy)3
self-exchange reactions, in which case solvent
3+
dynamics should not be dominant and, indeed, eq 2 should not
be valid.1,2,6,7,32,45 For an electrode reaction, however, the fifty-
percent rule21,24,25 predicts that the activation free energy barrier
height is only one-half that for the corresponding self-exchange
reaction, and besides we have presented here evidence that
For the reduction of Co(bpy)3 (reactant 1), the choice of
reductant (reactant 2) and reaction conditions was limited by
the tendency of most complex-ion reductants as well as the
product Co(bpy)3 to lose ligands, even at neutral pH, on the
2+
time scale of the stopped-flow measurements. Furthermore,
3+/2+
q
suggests a difference in mechanism for the Co(bpy)3
application of eq 8 requires that ∆V22 be known. The only
practical cationic option was the cage complex Co(sep)2+ 40
, for
electrode reaction between organic solvents and the aqueous
systems on which expectations are typically based. Finally,
although the Co(bpy)33+/2+ electrode reaction in water behaves
much as do its analogues Co(en)33+/2+ and Co(phen)33+/2+, for
which adiabaticity in self-exchange reactions has been inferred,36
which the cross relation 7 is unfortunately not closely obeyed
(k12(calc) ) 4.0 × 105 L mol-1 s-1at 5.9 °C and I ∼ 0.2 mol
L-1, cf. k12(obsd) ) 8.0 × 104 L mol-1 s-1). Thus, eq 8 can be
expected to generate a ∆V11q value that is a few cm3 mol-1 too
negative.25 In addition, there are the inevitable mismatches of
temperature, media, etc., with data from different sources. With
the reduction of Co(bpy)3 by Co(sep)2+ and other cross
3+
reactions with a high driving potential appears to be nonadia-
batic, leading to failure of eqs 7 and 8.
∆V12 ) -9.7 cm3 mol-1 from this study, and parameters for
q
the closest matching conditions from the literature,21,29,40,41 eqs
Acknowledgment. We thank Profs. V. I. Birss, R. G.
Compton, A. S. Hinman, and R. Paul for discussions and the
Natural Sciences and Engineering Research Council of Canada
for financial support.
q
3+/2+
6 and 8 predict ∆Vel for Co(bpy)3
) -12.4 ( 1.5 cm3
mol-1 at 5.9 °C, cf. the observed -9.1 ( 0.3 cm3 mol-1 at 3.0
°C (both at I ) 0.2 mol L-1). Although the agreement is not
close, the estimate of ∆Velq from the cross relation does at least
confirm a strongly negative value for the Co(bpy)33+/2+ electrode
reaction in water, in contrast to the positive values found for
organic solvents.
Supporting Information Available: Figures and tables of
spectral changes and pressure and [Co(sep)2+] dependences of
rate constants for the Co(bpy)33+/Co(sep)2+ reaction; temper-
ature dependence of the decomposition of Co(bpy)32+ in water;
pressure dependences of kel, Ru, D, and ∆E1/2 for the Co(bpy)33+/2+
electrode reaction in various solvents (PDF). This material is
available free of charge via the Internet at http://pubs.acs.org.
The apparent failure of eqs 7 and 8 for Co(bpy)33+/Co(sep)2+
reaction in water, like that for the Fe(H2O)63+/Ru(en)32+ reaction
(∆E1/2 ) 534 mV),25 may be taken as evidence for increasing
nonadiabaticity as one goes to higher driving forces. Nonadia-
batic electron transfer would give slower rate constants (κ ,
JA9923179
(39) Grace, M. R.; Takagi, H.; Swaddle, T. W. Inorg. Chem. 1994, 33,
1915.
(42) Jolley, W. H.; Stranks, D. R.; Swaddle, T. W. Inorg. Chem. 1990,
29, 385.
(40) Doine, H.; Swaddle, T. W. Inorg. Chem. 1991, 30, 1858.
(41) Doine, H.; Whitcombe, T. W.; Swaddle, T. W. Can. J. Chem. 1992,
70, 81.
(43) Endicott, J. F.; Ramasami, T. J. Phys. Chem. 1986, 90, 3740.
(44) Fu¨rholz, U.; Haim, A. J. Phys. Chem. 1986, 90, 3686.
(45) Sumi, H.; Marcus, R. A. J. Chem. Phys. 1986, 84, 4894.