Self-exchange reaction kinetics of metallocenes revisited: Insights from the decamethylferricenium-decamethylferrocene reaction at variable pressure

Article abstract of DOI:10.1021/ic030067y

Rate constants k_ex and volumes of activation ΔV_ex? have been obtained using ¹H NMR for the self-exchange reaction of the [(η⁵-C₅(CH₃)₅)₂ Fe]⁺ hexafluorophosphate and tetrafluoroborate with [(η⁵-C₅(CH₃)₅)₂Fe] in acetone-d₆ (ΔV_ex? = -8.6 ± 0.3 cm³ mol^-1), dichloromethane-d₂, and (semiquantitatively) in acetonitrile-d₃. Under the experimental conditions, ion pairing was significant only in CD₂Cl₂, but even that produced only a minor reduction in k_ex and so had a negligible effect on ΔV_ex? (= -6.4 ± 0.2 cm³ mol^-1 with PF₆-). In all cases, ΔV_ex? is negative and consistent with a simple two-sphere activation model, rather than with that of Weaver et al. (Nielson, R. M.; McManis, G. E.; Safford, L. K.; Weaver, M. J. J. Phys. Chem. 1989, 93, 2152) in which the barrier crossing rate is limited by solvent dynamics. Similarly, the ～5-fold increase in k_ex on going from [(η⁵-C₅H₅)₂Fe]^+/0 to [(η⁵-C₅(CH₃)₅)₂ Fe]^+/0 in acetone can be explained with the two-sphere model on the basis of the effects of reactant size on the solvent reorganization energy, without reference to solvent dynamics.

Full text of DOI:10.1021/ic030067y

Inorg. Chem. 2003, 42, 3718−3722
Self-Exchange Reaction Kinetics of Metallocenes Revisited: Insights
from the Decamethylferricenium−Decamethylferrocene Reaction at
Variable Pressure
,‡
Achim Zahl,† Rudi van Eldik,† Mitsuru Matsumoto,‡ and Thomas W. Swaddle*
Institut fu¨r Anorganische Chemie, UniVersita¨t Erlangen-Nu¨rnberg, Egerlandstrasse 1,
91058 Erlangen, Germany, and Department of Chemistry, UniVersity of Calgary,
Calgary, Alberta, Canada T2N 1N4
Received February 14, 2003
‡
1
Rate constants kex and volumes of activation ∆V have been obtained using H NMR for the self-exchange
ex
reaction of the [(η5-C (CH ) ) Fe]+ hexafluorophosphate and tetrafluoroborate with [(η5-C (CH ) ) Fe] in acetone-d6
(∆V ‡ ) −8.6 ± 0.3 cm mol-1), dichloromethane-d , and (semiquantitatively) in acetonitrile-d . Under the experimental
5
3 5 2
5
3 5 2
3
ex
2
3
conditions, ion pairing was significant only in CD Cl2, but even that produced only a minor reduction in kex and so
2
had a negligible effect on ∆V ‡ ( ) −6.4 ± 0.2 cm mol-1 with PF -). In all cases, ∆V ‡ is negative and consistent
3
ex
6
ex
with a simple two-sphere activation model, rather than with that of Weaver et al. (Nielson, R. M.; McManis, G. E.;
Safford, L. K.; Weaver, M. J. J. Phys. Chem. 1989, 93, 2152) in which the barrier crossing rate is limited by solvent
dynamics. Similarly, the ∼5-fold increase in kex on going from [(η5-C H ) Fe]+/0 to [(η5-C (CH ) ) Fe]+/0 in acetone
5
5 2
5
3 5 2
can be explained with the two-sphere model on the basis of the effects of reactant size on the solvent reorganization
energy, without reference to solvent dynamics.
The kinetics and mechanism of electron transfer in
metallocene couples in nonaqueous solution1-7 continue to
attract attention.8 We report here an investigation of the effect
of pressure on the kinetics of the decamethylferricenium/
decamethylferrocene self-exchange reaction in three organic
solvents, carried out in support of a high-pressure study of
the corresponding electrode reaction in 10 solvents9 but with
the additional purpose of testing the applicability of a
successful treatment of pressure effects on aqueous electron
transfer reactions10 to nonaqueous systems. Whereas recent
theoretical discourse6-8 has focused on the electronic cou-
pling between between the donor and acceptor centers,
pressure dependences of the self-exchange rate constants kex
provide special insights into solvent effects on electron
transfer rates.10 The solvent can influence kex through its
contribution to the activation barrier height11 but also possibly
through solvent dynamical control of the barrier-crossing
frequency (solvent “friction”).
The effect of pressure P on kex for adiabatic outer-sphere
self-exchange electron transfer reactions of metal complexes
in aqueous solution can be accounted for quantitatively in
terms of a simple extension10 of Marcus-Hush theory,11 and
indeed, significant deviations of measured volumes of
activation ∆Vex from the predicted values can serve as
evidence for mechanistic alternatives such as inner-sphere
pathways10 or catalysis by counterions.12
* To whom correspondence should be addressed. E-mail: swaddle@
ucalgary.ca. Phone: (403) 220-5358. Fax: (403) 284-1372.
† Universita¨t Erlangen-Nu¨rnberg.
‡ University of Calgary.
(1) Yang, E. S.; Chan, M.-S.; Wahl, A. C. J. Phys. Chem. 1975, 79, 2049-
2052.
(2) Yang, E. S.; Chan, M.-S.; Wahl, A. C. J. Phys. Chem. 1980, 84, 3094-
3099.
‡
(3) Kirchner, K.; Dang, S.-Q.; Stebler, M.; Dodgen, H.; Wherland, S.;
Hunt, J. P. Inorg. Chem. 1989, 28, 3604-3606.
(4) Nielson, R. M.; McManis, G. E.; Golovin, M. N.; Weaver, M. J. J.
Phys. Chem. 1988, 92, 3441-3450.
(5) Nielson, R. M.; McManis, G. E.; Safford, L. K.; Weaver, M. J. J.
Phys. Chem. 1989, 93, 2152-2157.
∆Vex‡ ) -RT(∂ ln kex/∂P)T
(1)
(6) Gennett, T.; Milner, D. F.; Weaver, M. J. J. Phys. Chem. 1985, 89,
2787-2794.
(7) Newton, M. D.; Ohta, K.; Zhong, E. J. Phys. Chem. 1991, 95, 2317-
(10) Swaddle, T. W. Can. J. Chem. 1996, 74, 631-638.
2326.
(11) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1995, 811, 265-
(8) Baik, M.-H.; Crystal, J. B.; Friesner, R. A. Inorg. Chem. 2002, 41,
5926-5927.
(9) Matsumoto, M.; Swaddle, T. W. Forthcoming publication.
322.
(12) Zahl, A.; van Eldik, R.; Swaddle, T. W. Inorg. Chem. 2002, 41, 757-
764.
3718 Inorganic Chemistry, Vol. 42, No. 12, 2003
10.1021/ic030067y CCC: $25.00 © 2003 American Chemical Society
Published on Web 05/20/2003

Kinetics of Metallocene Self-Exchange Reactions
The success of these calculations is due in part to the
smallness of the contribution ∆VIR‡ for internal reorganization
of the reactants13 (in the absence of major structural effects
due to spin-multiplicity changes10) and the fortuitous near-
cancellation at practical values of the ionic strength I of the
contributions of Coulombic work of precursor complex
formation (∆VCOUL‡) and Debye-Hu¨ckel-type ion activity
terms (∆VDH‡). In water, then, ∆Vex‡ is governed largely by
the negative solvent reorganization term ∆VSR‡, with a very
small positive contribution from the pressure dependence of
the non-Coulombic part of the precursor complex formation
constant (∆VPREC‡):
reactions with impurities leaking from the pressurizing fluid
over the time scale (several hours) of a cycle of measure-
ments at high pressures, and indeed, we were unable to obtain
‡
reliable values of ∆Vel for the Fc+/0 electrode reaction in
acetonitrile.24 Furthermore, for the Fc+/0 self-exchange reac-
tion in acetonitrile, Hunt et al.3 suspected a systematic error
in the kex values of the pioneering study by Wahl et al.,1 in
which context Weaver et al.4,5 noted difficulties in the
1
extraction of reliable kex values from H NMR line width
data in the “fast exchange” regime. Nevertheless, Hunt and
‡
co-workers3 were able to obtain ∆Vex ) -7.0 ( 2.0 cm3
mol-1 for Fc+/0 in CD3CN by working at 0 °C with careful
exclusion of moisture and oxygen but remarked that a more
negative value would be predicted by the model10 based on
Marcus-Hush theory. Moreover, Weaver et al.4-6 have
argued that kex for Fc+/0 in organic solvents is limited by
both nonadiabaticity and solvent dynamical effects. Thus,
the Fc+/0 system is fraught with both experimental and
interpretational difficulties.
‡
∆Vex‡ ) ∆VIR‡ + ∆VSR‡ + ∆VDH‡ + ∆VCOUL‡ + ∆VPREC
(2)
In contrast, for ion-ion electron transfer reactions in
nonaqueous solvents with relatively low dielectric constants
‡
‡
ꢀ, ∆VDH and ∆VCOUL can become numerically very large,
so that even if they roughly cancel the theoretical calculation
of ∆Vex‡ becomes highly susceptible to imperfections in the
model.10,14 Furthermore, the calculations are likely to be
invalidated by extensive ion-pair formation between the
reactants and the counterions, the kinetic consequences of
which are not readily predictable.15 Applications of the
theory to nonaqueous systems are therefore most likely to
succeed when one reactant is electrically neutral, so that
Lay et al.,25 however, have pointed out that the DmFc+/0
couple is much to be preferred to Fc+/0 as a redox-stable
electrochemical standard, particularly in connection with
studies involving solvent effects on electron transfer, as is
the case in our work. Indeed, the self-exchange kinetics have
1
been previously mapped out at variable temperature by H
NMR in CD3CN, (CD3)2CO, CD2Cl2, and C6H5CN by Wahl
et al.2 and Weaver et al.,5 and we have found the electrode
kinetics to be reproducible over lengthy pressure cycles in
10 solvents.9 The high-pressure study of the DmFc+/0 self-
exchange reaction reported here gave excellent reproduc-
ibility but was limited to solutions in (CD3)2CO and CD2Cl2,
with only semiquantitative results in CD3CN, by the low
solubility of DmFc in most polar solvents, because the
sensitivity of static high-pressure NMR probes is relatively
‡
‡
∆VDH ) ∆VCOUL ) 0 and only the ionic reactant is sub-
ject to ion pairing which, for singly charged ions, will be
minimal. With these considerations in mind, the pressure
dependences of the kinetics of the [Ru(hfac)3]0/- 16,17 (hfac-
-
) CF3COCHCOCF3 ) and Fc+/0 3 (Fc ) ferrocene, [(η5-
C5H5)2Fe]) self-exchange reactions have previously been
examined in organic solvents.
1
low and is subject to a H background from impurities in
In this same vein, as part of a wider study9,18-24 that
attempts to relate the kinetic parameters of electrode reactions
the pressurizing fluid.
A special feature of high-pressure kinetic studies is that
‡
(particularly the volumes of activation ∆Vel ) to those of the
reaction rate control by organic solvent dynamics is associ-
corresponding self-exchange reactions, we have measured
‡ 21,23,24
ated with a strongly positive contribution to ∆Vex
,
‡
∆Vex for the obligate-outer-sphere self-exchange reaction
whereas the conventional treatment of outer-sphere electron
of decamethylferrocene (DmFc, [(η5-C5(CH3)5)2Fe]) with its
cation in three nonaqueous solvents. Although the familiar
Fc+/0 couple, traditionally a standard for nonaqueous elec-
trochemistry, might seem a more obvious choice, the Fc+
cation is subject to side reactions and to possible redox
transfer kinetics based on transition-state theory predicts that
‡
∆Vex should be moderately negative.10,14 The “signature”
of solvent dynamical control is an approximate inverse
dependence of rate constant on solvent viscosity η, which,
for organic solvents (unlike water at near-ambient temper-
atures), rises roughly exponentially with increasing pres-
sure.21 Thus, any incursion of solvent dynamics in metal-
locene self-exchange kinetics in organic solvents, as proposed
(13) Stranks, D. R. Pure Appl. Chem. 1974, 38, 303-323.
(14) Swaddle, T. W. J. Mol. Liquids 1995, 65/66, 237-244.
(15) Wherland, S. Coord. Chem. ReV. 1993, 123, 169-199.
(16) Doine, H.; Swaddle, T. W. Inorg. Chem. 1988, 27, 665-670.
(17) Takagi, H. D.; Swaddle, T. W. Chem. Phys. Lett. 1996, 248, 207-
212.
(18) Fu, Y.; Swaddle, T. W. J. Am. Chem. Soc. 1997, 119, 7137-7144.
(19) Fu, Y.; Swaddle, T. W. Inorg. Chem. 1999, 38, 876-880.
(20) Metelski, P. D.; Fu, Y.; Khan, K.; Swaddle, T. W. Inorg. Chem. 1999,
38, 3103-3109.
by Weaver et al.,4-6 would be revealed experimentally by
‡
observation of a substantially positive value of ∆Vex
.
Experimental Section
Materials. Tetrabutylammonium phosphate (Fluka electrochemi-
cal grade, >99%), acetone-d6 (Deutero GmbH, Kastellaun, 99.8%),
(21) Fu, Y.; Cole, A. S.; Swaddle, T. W. J. Am. Chem. Soc. 1999, 121,
10410-10415.
(22) Swaddle, T. W.; Tregloan, P. A. Coord. Chem. ReV. 1999, 187, 255-
289.
(23) Zhou, J.; Swaddle, T. W. Can. J. Chem. 2001, 79, 841-847.
(24) Matsumoto, M.; Lamprecht, D.; North, M. R.; Swaddle, T. W. Can.
J. Chem. 2001, 79, 1864-1869.
(25) Noviandri, I.; Brown, K. N.; Fleming, D. S.; Gulyas, P. T.; Lay, P.
A.; Masters, A. F.; Phillips, L. J. Phys. Chem. B 1999, 103, 6713-
6722.
Inorganic Chemistry, Vol. 42, No. 12, 2003 3719

Zahl et al.
and dichloromethane-d2 (Euriso-top, Saclay, HDO + D2O < 0.05%)
were used as received; acetonitrile-d3 (Euriso-top, 99.6% D, H2O
< 0.01%) was stored over dehydrated 3A molecular sieves.
Decamethylferrocene (Aldrich, 97%) was purified by vacuum
sublimation twice at 140 °C, followed by recrystallization from
acetone; 1H NMR at 400 MHz in acetone-d6, dichloromethane-d2,
and acetonitrile-d3 showed only a single solute peak at ∼650 Hz.
[DmFc]PF6 and [DmFc]BF4 were made by the method of Hen-
drickson et al.26 and checked for purity by CHN microanalysis;
1
they showed only a single, broad H NMR resonance at -14750
to -14790 Hz.
High-Pressure NMR Measurements. The solutions were
prepared by weight, and the concentrations converted to the molar
scale with estimated solution densities. Each sample was sealed in
a cut-down 5 mm o.d. commercial NMR tube with a floating Kel-F
piston with two O-rings of EPDM ethylene-propylene polymer
for use with CD3CN and CD2Cl2, or NBR acrylonitrile polymer
for use with (CD3)2CO, and placed in a home-built high-pressure
static NMR probehead27 resonating at 400.13 MHz for 1H. Sample
temperature was monitored through a Pt-100 resistor and was held
constant ((0.5 °C) with circulating fluid. The chemical shifts
(calibrated against residual proton resonances in the solvents) and
line widths of the single 1H resonances of DmFc, DmFc+, and their
mixtures in solution were measured at selected pressures in the
range 1-150 MPa with a Bruker AVANCE DRX-400 spectrometer
equipped with a wide-bore magnet. Field homogeneities better than
6 × 10-9 were achieved by shimming on the 1H FID with the BOSS
Figure 1. Pressure dependence of ln kex for DmFc+/0 in acetone-d6 at
25.0 °C; anion is BF4-, no added electrolyte. Concentrations as in Table 1.
Measurements taken with upward pressure steps (4) and downward pressure
steps (3) are shown.
Table 1. Volumes of Activation for the Self-Exchange Reaction of
Decamethylferrocene and Decamethylferricenium Ion in Different
Solventsa
[DmFc]/ [DmFc+]/
kex0/107
∆Vex‡/cm3
solvent
anionb mmol L-1 mmol L-1 L mol-1 s-1
mol-1
-
(CD3)2CO PF6- c
14.5
11.5
14.6
25.4
29.1
27.5
9.29
5.63
6.44
11.4
12.5
9.66
0.583
2.24 ( 0.02 -8.5 ( 0.3
2.24 ( 0.02 -8.7 ( 0.3
2.12 ( 0.01 -8.9 ( 0.2
9.58 ( 0.05 -6.5 ( 0.2
8.24 ( 0.08 -6.3 ( 0.3
8.87 ( 0.08 -7.2 ( 0.3
PF6 -
BF4-
1
I shim system. The H 90° pulse width was 16 µs, and the sweep
CD2Cl2
PF6- d
PF6 -
BF4-
PF6
width was varied between 6000 and 20000 Hz. A relaxation delay
of 2 s was applied in a single 30° pulse experiment, with
accumulation of up to 256 scans. Line widths of some very broad
resonances (>1500 Hz) of the chemically exchanging DmFc/
DmFc+ mixtures in the shift region -8 to +5 ppm could not be
determined precisely because of a broad background signal from
the probehead or pressurizing fluid, but in such cases, acceptable
accuracy was achieved by digitally subtracting the background
signal obtained from a pure solvent sample.
CD3CN
3.26
(7 ( 2)e
(-5 ( 2)e
a At 25.0 °C; except as noted, error limits are standard deviations of fit
to eq 4. b Anion concentration ) [DmFc+] except as noted. c Using newly
constructed high-pressure probehead. d In presence of 0.099 mol L-1 [(n-
C4H9)4N]PF6. e Provisional result based on 75-150 MPa data only; error
limits are subjective.
tion. Second-order rate constants kex were calculated from
the formula1
Results
kex ) 4πfDfP(δν)2/(WDP - fDWD - fPWP)c
(3)
1
Measurements were made of the H resonance chemical
shifts νD, νP, and νDP and line widths at half-maximum
WD, WP, and WDP of solutions containing only diamagnetic
DmFc, paramagnetic DmFc+ salts, and the chemically
exchanging mixtures DmFc+/0, respectively, over an ascend-
ing and then descending range of pressures. The mole
fractions fD of DmFc and fP of DmFc+ were calculated from
the solute weights and also from νDP relative to νD and
νP; agreement was excellent except in cases where the
weights were very small, and then, the latter method
was taken as definitive. In general, the chemical shifts
and hence the overall shifts δν () νD - νP) were
almost independent of pressure, but increasing pressure
caused WP to increase (cf. Fc+ in CD3CN3) and WDP to
decrease substantially, while WD was sufficiently small
(7.5-10 Hz) that the barely discernible decrease was often
negligible. Full details are given in the Supporting Informa-
where c is the total metallocene molar concentration; the WP
and WD values used were smoothed on the basis of an
empirical linear dependence on pressure. For solutions in
acetone-d6 and dichloromethane-d2, the pressure dependences
of ln kex were accurately linear within the experimental error
(eq 4 and Figure 1); i.e., ∆Vex‡ was effectively constant over
the pressure range 0-150 MPa. The results are summarized
in Table 1.
ln kex ) ln kex0 - P∆Vex‡/RT
(4)
For acetone-d6 and dichloromethane-d2 as solvents, kex
showed good reproducibility and may be considered accurate
to within (5%. For acetone-d6, kex showed no significant
dependence on the anion and was in excellent agreement
with Wahl’s value,2 which was also adopted by Weaver et
al.5 For CD2Cl2, however, anion effects are evident (Table
1), and Wahl et al.2 gave a value lower than any of ours (kex
(26) Hendrickson, D. N.; Sohn, Y. S.; Gray, H. B. Inorg. Chem. 1971, 10,
1559-1563.
(27) Zahl, A.; Neubrand, A.; Aygen, S.; van Eldik, R. ReV. Sci. Instrum.
1984, 65, 882-886.
-
) 4.4 × 107 L mol-1 s-1 at 25 °C) for DmFc+/0/PF6 , albeit
at higher c. Weaver et al.5 expressed concern that some of
3720 Inorganic Chemistry, Vol. 42, No. 12, 2003

Kinetics of Metallocene Self-Exchange Reactions
the kex values of Wahl et al.1,2 could be too low and, perhaps
kex ) 4000πNσ2κeνnδσ exp(-∆Gex‡/RT)
(5)
significantly, chose to disregard this particular result. We
-
found that addition of a swamping concentration of PF6
where σ is the closest separation of the metal centers (which
may be approximated to the sum of the effective radii r1
and r2 of the reactants), κe is the electronic transmission
coefficient ()1 for a fully adiabatic reaction, otherwise <1),
νn is the nuclear frequency factor, δσ is the range of σ over
(0.1 mol L-1) to force complete ion pairing with DmFc+
produced only a small reduction in kex (Table 1), so the
discrepancy for CD2Cl2 solutions cannot be ascribed entirely
to anion effects. The consistency of the CD2Cl2 data of Table
1 lends credibility to the present results.
‡
which electron transfer rate is substantial, and ∆Gex is the
It is important to note that ln kex values on the upward leg
of each pressure cycle agreed within the experimental
uncertainty with those on the subsequent downward leg at a
particular pressure, i.e., that there was no drift in kex with
free energy of activation. It is difficult, however, to evaluate
νn and δσ precisely,4 so we will use eq 5 to predict relative
‡
rather than absolute values of kex (indeed, ∆Vex is derived
from relative values of kex). The contributions of internal
reorganization to ∆Gex‡ are small and about the same (∼0.6
kJ mol-1)6 for DmFc+/0 and Fc+/0, and the Coulombic and
Debye-Hu¨ckel contributions are effectively nil,29 so the
time, as this could have produced a false slope of the ln kex
‡
versus P plot and hence an erroneous value of ∆Vex
.
The solubility of DmFc in methanol, dimethyl sulfoxide,
and other polar organic solvents was too low for high-
pressure NMR, and that in CD3CN (1.32 mg per g solvent
at 22 °C) was barely sufficient. In practice, WDP in acetonitrile
was too broad to measure except with a limited choice of fP
(which unfortunately placed νDP in a region with substantial
background from the high-pressure probe) and an elevated
pressure range 75-150 MPa where kex was high enough to
give tractable line widths (cf. eq 3). Thus, the provisional
atmospheric pressure kex value for CD3CN in Table 1 is the
result of a long, uncertain extrapolation from high pressures,
and as such, its disagreement with Wahl’s value (3.8 × 107
L mol-1 s-1)2 is not serious; the extent of ion pairing was
<1%, according to calculations based on Fuoss theory,28 and
would not affect this comparison. The essential point is that
‡
difference in ∆Gex for the two couples is the difference in
‡
the solvent reorganization energies ∆GSR
:
‡
∆GSR
)
(Ne2/16πꢀ0)[(1/2r1) + (1/2r2) - (1/σ)](ꢀop-1 - ꢀ-1) (6)
where the optical dielectric constant ꢀop may be approximated
to the square of the refractive index of the solvent.30 Taking
σ for DmFc+/0 to be about 960 pm (the close-contact Fe-
Fe distance in solid DmFc31) and for Fc+/0 to be about 800
31-34
pm,3 and putting r1 ≈ r2 ) σ/2
while assuming δσ to
be proportional to σ, we calculate kex for DmFc+/0 to be about
7kex for Fc+/0 in acetone (cf. 4-5kex, observed), if κe and νn
are the same for the two couples. Since DmFc is bulkier
than Fc, close-contact electronic coupling in DmFc+/0 may
be expected to be weaker than in Fc+/0, and so, any
contribution through κel to the rate difference should be in
the opposite direction. Thus, reactant size differences can
account for this kinetic difference, through solvent reorga-
nization energetics. Weaver et al.4-6 did not consider this
possibility.
‡
∆Vex for CD3CN, though only approximate, is clearly
negative and is at least semiquantitatively consistent with
the values for the other two solvents.
Discussion
The atmospheric pressure value of kex for the DmFc+/0
exchange in acetone-d6 at 25 °C shows no significant
dependence on the nature or concentration of the anion under
the conditions of our experiments and confirms the result of
Wahl et al.2 at roughly comparable concentrations. A Fuoss-
type calculation28 (see Supporting Information) indicated that
A similar calculation predicts that kex for the DmFc+/0
exchange in CD2Cl2 should be 5.4 times faster than in (CD3)2-
CO, in the absence of complications such as anion-cation
pairing. In fact, the kinetic effect of ion pairing is quite small
in this case; Fuoss-type calculations28 indicate 48% DmFc+/
-
only about 2.5% of the DmFc+ would be paired with PF6
at 0.1 MPa, and 1.7% at 150 MPa. Thus, the DmFc+
exchange is 4-5 times faster than that of Fc+/0 in acetone at
25 °C,2 even allowing for a possible 30% error in kex for the
latter as suggested by Hunt et al.3 Weaver et al.5 interpreted
differences in kex for various metallocene couples in several
solvents to barrier-crossing frequency control by donor-
acceptor coupling and solvent dynamics, particularly in the
case of Fc+/0, but for the DmFc-Fc comparison at least the
explanation may be simpler. (The cobalticenium/cobaltocene
couple that Weaver et al.4,5 attempted to include under the
same rubric is a mechanistically different “hole transfer”
case7 which involves significant internal reorganization.8)
Theoretically,11 kex is given in L mol-1 s-1 by the SI
expression
(29) A reviewer has pointed out that quadrupolar interactions in the Fc+/0
and the DmFc+/0 couples may become significant in the encounter
pair involved in electron transfer, leading to enhanced precursor
complex formation constants and thus somewhat higher kex, more so
for the former couple. This would result in a smaller difference in kex
between the two couples than is predicted from the simple two-sphere
model, as is in fact observed. Quadrupolar effects may also lead to
preferred orientations of the reactants within the transition states and
as such may affect the kinetic parameters of the self-exchange process.
However, this is beyond the scope of the two-sphere model presented
‡
here. Any quadrupolar effects on ∆Vex‡ through ∆VCOUL‡ and ∆VPREC
(which are opposite in sign) are expected to be at least an order of
magnitude smaller than for ion-ion interactions and may, therefore,
be neglected.
(30) Swaddle, T. W. Inorg. Chem. 1990, 29, 5017-5025.
(31) Freyberg, D. P.; Robbins, J. L.; Raymond, K. N.; Smart, J. C. J. Am.
Chem. Soc. 1979, 101, 892-897.
(32) Pickardt, J.; Schumann, H.; Mohtachemi, R. Acta Crystallogr. 1990,
C46, 39-41.
(33) Martinez, R.; Tiripicchio, A. Acta Crystallogr. 1990, C46, 202-205.
(34) Takusagawa, F.; Koetzle, T, F. Acta Crystallogr. 1979, B35, 1074-
1080.
(28) Jolley, W. H.; Stranks, D. R.; Swaddle, T. W. Inorg. Chem. 1990, 29,
1948-1951.
Inorganic Chemistry, Vol. 42, No. 12, 2003 3721

Zahl et al.
-
PF6 ion pairs with [anion] ) [DmFc+] ≈ 0.01 mol L-1 at
ion, but because the free DmFc+ ion reacts only some 31%
faster than its ion pair with PF6 , this effect does not emerge
-
0.1 MPa, falling to 34% at 150 MPa, yet addition of 0.1
mol L-1 [Bu4N]PF6 to force virtually complete ion pairing
of DmFc+ caused only a 14% diminution in kex (Table 1).
Presumably, BF4- will pair with DmFc+ more strongly than
above the experimental error limits in Table 1. Thus, it is
acceptable to compare experimental ∆Vex‡ values for DmFc+/0
in CD2Cl2 with the theoretical value ignoring ion pairing,
and indeed, the three experimental values for CD2Cl2 are
effectively the same, but about 2 cm3 mol-1 less negative
than for acetone-d6 solutions, as theory10,30 predicts.
-
-
does the larger PF6 ion, and indeed, kex for DmFc+/0/BF4
without added electrolyte falls between the values for
DmFc+/0/PF6- with and without [Bu4N]PF6 (Table 1). These
results are in accordance with Wherland’s general observa-
tion15 that ion pairing of reactants tends to reduce electron
transfer rates in nonaqueous systems; the effect is small in
this case because, if one reactant is uncharged, it matters
little whether the effective charge on its reaction partner is
reduced by ion-pairing or not, so long as the pairing
counterion has no specific role to play (contrast cation
catalysis of anion-anion exchange in water12). From the data
In summary, the value of kex for DmFc+/0 relative to that
for Fc+/0 in acetone-d6 is consistent with a simple two-sphere
Marcus-Hush-type model without significant influence of
solvent friction; the difference between these rate constants
can be attributed to the fact that the effective radii of the
DmFc species are about 20% larger than those of the Fc
‡
couple. Furthermore, the consistency of ∆Vex for DmFc+/0
in acetone-d6 and in CD2Cl2 with the theoretical predictions
for the simple model is strong evidence against rate control
by solvent friction, which is associated with strongly positive
contributions to ∆V‡ that overwhelm other com-
ponents.9,10,21-24,30 Such a positive contribution to ∆Vex‡ might
conceivably be reduced somewhat23,34 by a negative one due
to nonadiabaticity,10,28,30 but as already noted, nonadiabaticity
-
with PF6 , one can estimate kex ≈ 1.1 × 108 L mol-1 s-1 for
DmFc+/0 couple in CD2Cl2 at 25 °C with no ion pairing,
i.e., 4.9 times larger than in (CD3)2CO and in good agreement
with the foregoing prediction for adiabatic electron transfer
in the absence of solvent friction.
‡
Theoretical values of ∆Vex at selected pressures were
‡
should have a negligible effect on ∆Vex for the Fc+/0 and
obtained by solving eqs 1, 5, and 6 simultaneously over small
(1 MPa) increments at each pressure, allowing σ and δσ to
vary with pressure with linear compressibility â/3, where â
is the isothermal volume compressibility of the solvent (see
Supporting Information for computational procedure). In
practice, whether σ and δσ are taken as pressure-dependent
or not is inconsequential for an adiabatic reaction, as the
effects on ∆VSR‡ and ∆VPREC‡ (eq 5) are small, roughly equal
in magnitude, and opposite in sign.30 For nonadiabatic
electron transfer, although the donor-acceptor coupling
matrix element Hif falls off essentially exponentially with
increasing σ beyond the adiabatic limit, the interaction energy
(which includes the repulsive potential) for the Fc+/0 couple
and presumably also for DmFc+/0 varies only slowly with σ
in the near-approach region;8 consequently, contrary to earlier
expectations that focused on Hif,30 nonadiabatic contributions
presumably also the DmFc+/0 exchanges. In the limiting case
of extreme nonadiabaticity, according to the Sumi-Marcus
model,35 solvent frictional effects would disappear anyway.
The same considerations apply to ∆Vex for the Fc+/0
‡
couple in acetonitrile. Hunt et al.3 considered the measured
‡
∆Vex (-7 ( 2 cm3 mol-1 at 0°C) to be less negative than
that predicted from the simple theory, but it seems that their
prediction was for 0.1 MPa (-17.5 cm3 mol-1, by our
calculation), whereas at the midrange pressure of 100 MPa
the predicted ∆Vex‡ is -9.6 cm3 mol-1, which is a fair match
given the experimental difficulties with the Fc+/0 couple and
pressure dependence of the theoretical value. We were unable
to obtain a reliable experimental ∆Vex‡ value for the DmFc+/0
exchange in acetonitrile because excessive NMR line widths
at low pressures limited useful measurements to the pressure
‡
‡
range 75-150 MPa, over which the average predicted ∆Vex
to the apparent ∆Vex can be small, in this case, negligible
value is about -7.5 cm3 mol-1 at 100 MPa; the approximate
experimental value given in Table 1 is in fair agreement with
this, in view of the large experimental uncertainty, and at
least is unmistakably negative, again militating against any
major influence of solvent dynamics.
(e 0.1 cm3 mol-1). Thus, in the absence of solvent dynamical
rate control, ∆VSR‡ must be by far the dominant component
‡
of ∆Vex for the couples considered here.
Because the pressure dependences of the key variables ꢀ
and ꢀop in eq 6 are closely related to â, which, for organic
solvents, is itself quite strongly pressure dependent,10,14,30 the
predicted ∆Vex‡ values at 25 °C for DmFc+/0 in acetone and
dichloromethane, respectively, vary from -16.8 and -7.9
cm3 mol-1 at atmospheric pressure to -9.5 and -5.9 cm3
mol-1 at 75 MPa, and to -6.7 and -4.6 cm3 mol-1 at 150
MPa. In fact, no significant curvature of the experimental
ln kex versus P plots is discernible for DmFc+/0 in these
solvents (see, e.g., Figure 1), but the measured ∆Vex‡ values
in Table 1 correspond well to the average or midrange (75
MPa) theoretical values. In the case of dichloromethane as
solvent, the partial breaking-up of ion pairs by increasing
pressure should lead to a somewhat more negative apparent
Acknowledgment. T.W.S. thanks the Alexander von
Humboldt Foundation for a Research Award, the Friedrich-
Alexander-Universita¨t Erlangen-Nu¨rnberg for generous hos-
pitality, and the Natural Sciences and Engineering Research
Council of Canada for additional financial assistance. R.v.E.
acknowledges financial support from the Deutsche Fors-
chungsgemeinschaft and the Fonds der Chemischen Industrie.
Supporting Information Available: Tables S1-S6 containing
NMR measurements and rate constants as functions of pressure;
outline of computational procedure. This material is available free
of charge via the Internet at http://pubs.acs.org.
IC030067Y
‡
∆Vex when no swamping concentration of added anion is
present, given that the ion pair is less reactive than the free
(35) Sumi, H. J. Phys. Chem. 1991, 95, 3334-3350.
3722 Inorganic Chemistry, Vol. 42, No. 12, 2003