Reaction and reorganization free energies of electron-transfer reactions under restricted geometry conditions

Article abstract of DOI:10.1021/jp1013626

Electron-transfer reactions between iron and cobalt complexes were studied in β-cyclodextrin (βCD), 2-hydroxypropyl-β-cyclodextrin (HβCD), and 18-crown-6 ether (18C6) solutions. The results were rationalized taking as the basis the Marcus-Hush formalism.

Full text of DOI:10.1021/jp1013626

9094
J. Phys. Chem. B 2010, 114, 9094–9100
Reaction and Reorganization Free Energies of Electron-Transfer Reactions under Restricted
Geometry Conditions
P. Perez-Tejeda,† S. Conejero,‡ F. Sanchez Burgos,† and M. Marchena*,†
The Department of Physical Chemistry, Faculty of Chemistry, UniVersity of SeVille, and The Department of
Inorganic, Instituto de InVestigaciones Químicas, CSIC and UniVersity of SeVille
ReceiVed: February 12, 2010; ReVised Manuscript ReceiVed: May 8, 2010
Electron-transfer reactions between iron and cobalt complexes were studied in ꢀ-cyclodextrin (ꢀCD),
2-hydroxypropyl-ꢀ-cyclodextrin (HꢀCD), and 18-crown-6 ether (18C6) solutions. The results were rationalized
taking as the basis the Marcus-Hush formalism. We employed two different approaches, depending on the
kind of receptor and solvent, to obtain the reorganization and reaction free energies that determine the reaction
rate constant. The opposite trends in reactivity observed in ꢀCD and HꢀCD solutions and the behavior in
solutions of 18C6 are explained.
Introduction
precursor complex f successor complex. For electron-transfer
reactions under r.g.c., to obtain λ and ∆G°′ requires developing
some of the strategies that we have employed in micellar
systems.7a,b,h These strategies are extensions of those contained
in papers in which the relation between optical and thermal
processes has been studied.10
At present, there is great interest in the study of changes in
reactivity caused by restricted geometry conditions (r.g.c.), that
is, under conditions in which one or both reactants are bound
to some receptors such as micelles, cyclodextrins (CDs),
polymers, and so forth.1 This interest is due to (i) model studies
on r.g.c. that offer valuable insights into many biological
processes,2 (ii) reactions under r.g.c. which experience changes
in the rate constant of several orders of magnitude that depend
on the concentration of receptors, this allowing control in the
reaction rate,3 and (iii) in some cases, r.g.c. changing the course
of the reaction through a change of its mechanism.4,5 Thus,
different orientations of a given guest in different CDs, as a
consequence of different CD sizes, can lead to conformational
control of reactivity because of the exposition of different
moieties of the guest to a second reactant.5
In this paper, we extend these studies to electron-transfer
reactions in two different kinds of receptors, 18-crown-6 (18C6)
and cyclodextrins ꢀ-cyclodextrin (ꢀCD) and 2-hydroxypropyl-
ꢀ-cyclodextrin (HꢀCD). Two different procedures were em-
ployed depending on the kind of receptor (and solvent) in order
to have the reaction free energy. The analysis permitted
obtaining λ and ∆G°′ in the different cases, as well as explaining
the opposite trend in reactivity observed for ꢀCD and HꢀCD.
Experimental Section
In our studies in this field, we preferred electron-transfer
reactions1,3,6,7 due to the existence of a well-founded theoretical
framework for this kind of reaction, the Marcus-Hush
formalism,8,9 which allows deeper insight into rationalization
of the effects of r.g.c. on electron-transfer reactions. In fact, a
careful analysis of experimental data for electron-transfer
reactions permits the separation of the electron-transfer rate
constant, ket, from the experimentally measured rate constant,
kobs, and then obtaining of the parameters that determine ket
Materials. [Co(NH3)4(H2O)2]3+ was prepared according to a
published procedure11,12 as a perchlorate salt. [Co(NH3)5Cl]2+
was obtained as by Schlessinger.13 [Fe(CN)5(4-phepy)]3- was
prepared in situ from a solution of the commercially available
[Fe(CN)5(NH3)]Na3 (from Aldrich), after adding the stochio-
metric amount of the 4-phenylpyridine ligand (from Fluka). The
rapid hydrolysis of the ammonia complex produced the corre-
sponding aquocomplex14 [Fe(CN)5(H2O)]3-. Complete formation
of the [Fe(CN)5(4-phepy)]3- complex was checked spectropho-
tometrically. Na4[Fe(CN)6] was from Sigma-Aldrich, and
K4[Ru(CN)6] was from the Johnson Matthey Company. Cyclo-
dextrins were commercial products from Fluka and were used
as received. The average degree of substitution per glucose unit
was 0.6-0.8 units of 2-hydroxypropyl (C3H7O) in HꢀCD. 18-
crown-6 (18C6) was from Fluka and was stored in a vacuum
desiccator for several days before being used. Acetonirile from
Merck was used in the experiments. Purified water used in the
preparation of solutions was obtained from a Millipore Milli-Q
water system. Its conductivity was less than 10-6 S m-1 and
was deoxygenated before use.
‡
ket ) Ae-∆G /RT
(1a)
(1b)
(λ + ∆G°′)2
∆G‡ )
4λ
In previous equations, A is the pre-exponential factor, ∆G‡ and
λ are the activation and the reorganization free energies,
respectively, and ∆G°′ is the free energy for the process
Spectra. The spectra corresponding to the metal-to-metal
charge-transfer (MMCT) band within the ion pair [Ru(CN)6]4-/
[Co(NH3)5Cl]2+ were obtained with a Cary 500 (Varian)
spectrophotometer. The final solutions containing this ion pair
were prepared by mixing equal volumes of the ruthenium and
* To whom correspondence should be addressed. Tel.: +34 954 55 71
75. Fax: +34 954 55 71 74. E-mail: marijose@us.es.
† The Department of Physical Chemistry, Faculty of Chemistry, Uni-
versity of Seville.
‡ The Department of Inorganic, Instituto de Investigaciones Químicas,
CSIC and University of Seville.
10.1021/jp1013626  2010 American Chemical Society
Published on Web 06/28/2010

Electron-Transfer under Restricted Geometry Conditions
J. Phys. Chem. B, Vol. 114, No. 28, 2010 9095
cobalt solutions. The concentrations of the final solutions were
always 6 × 10-3 and 6 × 10-4 mol dm-3, respectively.
Electrochemical Measurements. The standard formal redox
potentials of the [Fe(CN)5(4-phepy)]3- in water and in solutions
containing CDs were obtained by differential pulse voltammetry,
after checking the reversibility of the system, using the
apparatus, electrodes, and procedures previously described.15 The
concentration of the iron complex in these experiments was 3
× 10-3 mol dm-3, and the ionic strength of the solutions was
maintained at 0.118 mol dm-3 by addition of NaCl. The
estimated uncertainty in the values of the potentials was (2.5
mV.
SCHEME 1: Schematic Representations of the
Encounter Complexes Formed by Bound Reactantsa
Kinetic Measurements. The kinetics of the reaction between
[Fe(CN)5(4-phepy)]3- and [Co(NH3)4(H2O)2]3+
:
3-
[Fe CN 5(4-phepy)] + [Co(NH3)4(H2O)2]3+
f
(
)
[Fe(CN)5(4-phepy)]2- + [Co(NH3)4(H2O)2]2+ (2)
were followed by measuring the absorbance changes at 405 nm.
This wavelength corresponds to the absorbance maximum of
the iron complex in water (it was also checked that changing
the observation wavelength does not change the value of the
rate constants).
a The charges are omitted.
TABLE 1: Values of ket, ∆G‡, ∆G°′, and λ for the Reaction
[Fe(CN)54phepy]3- + [Co(NH3)4(H2O)]3+ in Aqueous
Solutions of ꢀCD
Kinetic runs were performed in a stopped-flow apparatus from
Applied Photophysics. Concentrations of the iron and cobalt
complexes were 10-4 and 6 × 10-3 mol dm-3, respectively.
That is, the work was done under pseudo-first-order conditions.
The estimated uncertainty in the values of the rate constants
[ꢀCD] × 103/
∆G‡b/
∆G°′b/
λc/
mol dm-3
keta/s-1
kJ mol-1
kJ mol-1
kJ mol-1
0
1
2
4
6
8
10
12
14
16.2
14.7
14.9
13.6
13.1
12.0
13.3
12.1
12.5
63.9
64.1
64.2
64.4
64.5
64.5
64.6
64.6
64.6
-36.6
-36.5
-36.4
-36.0
-35.8
-35.6
-35.4
-35.2
-35.1
325
325
325
325
325
325
325
325
325
was (0.6 s-1
.
The temperature in all of the experiments in this work was
maintained at 298.2 ( 0.1 K.
Results and Discussion
General. As mentioned in the Introduction, the first step in
the determination of the reorganization free energies was to
separate ket (eq 1) from the experimentally measured rate
constant (kobs). There are different approaches to accomplish
this. Here, irreversible electron-transfer reactions were studied.
In this case, this irreversible character comes from the decom-
position of Co(II) complexes. Under this circumstance, and
working in the excess of one of the reactants, as here, it can be
shown that the experimental rate constant is given by
a The values of the parameters obtained from the fit to eq 4 are K
) 291 mol-1 dm3 and (ket)CD ) 11 s-1 b The uncertainties cal-
culated using the usual procedure on the propagation errors are
(0.1 kJ mol-1 c The uncertainties calculated using the usual
procedure on the propagation errors are (2 kJ mol-1
.
.
.
CD Solutions. Tables 1 and 2 contain the values of ket for
the reaction of [Co(NH3)4(H2O)2]3+/[Fe(CN)5(4-phepy)]3- in
solutions which contained ꢀCD and HꢀCD, respectively. The
influence of the CD concentrations can be described by a two-
state model18 (see Figure 1)
ketKIP[A]
kobs
)
(3)
1 + KIP[A]
where KIP is the equilibrium constant for the formation of the
precursor (encounter) complex from the separate reactants (see
Scheme 1) and A is the acceptor complex (the cobalt one), which
is in high excess over the donor. Reactants highly charged and
bearing opposite charge signs were selected. Therefore, in our
experiments, KIP[A] . 1. Thus, kobs ) ket.
All of the kinetic data in this paper are ket rate constants.
This was checked experimentally by changing the concentration
of the reactant in excess. For different concentrations of this
reactant in the range of 4 × 10-3-8 × 10-3 mol dm-3, no
changes in ket were observed. Therefore, we are measuring, in
fact, rate constants for the electron transfer within the precursor
complex. Schematic representations of encounter complexes,
corresponding to the case in which one of the reactants is bound
to the receptor, are given in Scheme 1. These representations
are inspired by the structures given in refs 16 and 17.
(ket)w + (ket)CDK[CD]
ket )
(4)
1 + K[CD]
In this equation (ket)w is the rate constant for the precursor
complex formed from the free reactants, (ket)CD is the rate
constant corresponding to the precursor complex formed from
the reactants bound to the CD (see Scheme 1), and K is the
equilibrium constant of the binding process. It is worth pointing
out that, for reactions in which two reactants participate, the
fact that kinetic data fit eq 4 implies that only one of them is
present in the two states. In the present case, this reactant is the
iron complex, as established in a previous paper.19 Despite the
fact that the same model holds in the presence of the two CDs
studied here, there is a striking difference between the binding
effect; in ꢀCD solutions, the rate constant decreases when the

9096 J. Phys. Chem. B, Vol. 114, No. 28, 2010
Perez-Tejeda et al.
TABLE 2: Values of ket, ∆G‡, ∆G°′, and λ for the Reaction
[Fe(CN)54phepy]3- + [Co(NH3)4(H2O)]3+ in Aqueous
Solutions of HꢀCD
[HꢀCD] × 103/
∆G‡b/
∆G°′b/
λc/
mol dm-3
keta/s-1
kJ mol-1
kJ mol-1
kJ mol-1
0
1
2.5
5
10
15
20
25
30
16.2
18.2
19.7
21.8
24.9
25.3
25.8
26.9
26.9
63.9
63.7
63.4
63.2
62.9
62.8
62.7
62.7
62.7
-36.6
-37.5
-38.3
-38.9
-39.4
-39.7
-39.8
-39.9
-39.9
325
325
326
326
326
326
326
326
326
a Data obtained from ref 19. The values of the parameters
obtained from the fit to eq 4 are K ) 162 mol-1 dm3 and (ket)CD
29 s-1 b The uncertainties calculated using the usual procedure on
the propagation errors are (0.1 kJ mol-1 c The uncertainties
calculated using the usual procedure on the propagation errors are
(2 kJ mol-1
Figure 2. Plot of the electron-transfer rate constant, ket/s-1, for the
reaction [Fe(CN)54phepy]3- + [Co(NH3)4(H2O)2]3+ in aqueous solutions
at different temperatures, T/K.
)
.
.
.
6b can be fitted directly using a nonlinear fitting procedure.
Using this method, the following results were found (see Figure
2): a ) 8.6 × 109 ( 1 × 109 s-1 K-1, ∆H‡ ) 101 ( 5 kJ
mol-1, ∆S‡ ) 124 ( 17 J mol-1 K-1. This would imply that A
) aT ) 2.6 × 1012 ( 0.3 × 1012 s-1 at 298 K.
In order to check the values of the activation parameters
obtained following this fitting procedure, we have also obtained
the value of ∆H‡ following the conventional procedure, that is,
from the slope of the Eyring plot ln(keth/kBT) versus 1/T. The
value obtained was 104 ( 5 kJ mol-1, which is close to the
value obtained from the nonlinear fitting. This gives support to
the values of the activation parameters obtained by us.
Notice that this value of A is obtained directly from
experimental data. Therefore, its value is independent of using
κelkBT/h as a frequency factor. However, if one accept this value
for the pre-exponential term, one obtains κel ) Ah/kBT ) 0.42
( 0.05. Therefore, the reaction would be close to the adiabaticity
limit (κel ) 1). Notice that our procedure for obtaining the
preexponential term permits the separation of κel and ∆S‡,
whereas that based on the Eyring plot does not permit this
separation.
Figure 1. Plot of the electron-transfer rate constant, ket/s-1, for the
reaction [Fe(CN)54phepy]3- + [Co(NH3)4(H2O)2]3+ in aqueous solutions
at different concentrations of ꢀCD (b) and HꢀCD (O).
concentration of ꢀCD increases; the opposite effect is found in
the presence of HꢀCD.
In order to shed light on the cause of this different behavior,
it is necessary to determine λ and ∆G°′. The first step is to
calculate ∆G‡ from ket. This can be done using eq 1a. In this
equation, the pre-exponential term, A, is given by20,21
With the value of the pre-exponential term obtained as
previously indicated, ∆G‡ can be calculated from ket
ket
∆G‡ ) -RT ln
(7)
2.6 × 1012
kBT
The values of ∆G‡ are given in Tables 1 and 2 for ꢀCD and
HꢀCD, respectively.
A ) κel
(5)
h
Once ∆G‡ is known, one can have λ, provided that the
reaction free energy, ∆G°′, is known. This free energy can be
calculated from the redox potentials of the iron and cobalt redox
couples. The redox potentials of the iron couple were measured
in the presence of CDs and appear in Table 3. In order to
combine these values with those of the activation free energy,
the potential values were corrected from the ionic strength, and
the potential data for the same CD concentrations used in the
kinetic measurements were interpolated. However, the redox
potential for the cobalt couple cannot be obtained by conven-
tional electrochemical techniques due to the electrochemical
irreversibility of this couple. (Notice that this irreversible
character guarantees that eq 3 holds.)
where κel is the electronic transmission coefficient (∼1 for an
adiabatic process), kB is the Boltzmann constant, h is the Planck
constant, and T is the work temperature.
As the purpose was to calculate ∆G‡, it is necessary to know
κel. There are several procedures to estimate this factor. We have
used a procedure that consists of taking as a starting point
‡
ket ) aTe-∆G /RT
(6a)
(6b)
‡
‡
ket ) aTe-(∆H -T∆S )/RT
For this reason, a different procedure was employed. This
procedure is based on the relation between the optical and
thermal electron-transfer processes.22 According to Hush, the
Obviously, a ) κelkB/h if one accepts the frequency factor given
by the Transition-State Theory, as in refs 20 and 21. Equation

Electron-Transfer under Restricted Geometry Conditions
J. Phys. Chem. B, Vol. 114, No. 28, 2010 9097
TABLE 3: Values of the Redox Potentials for the Complex
[Fe(CN)54phepy]3- (E° ) in Solutions of ꢀCD and HꢀCD
Fe
[ꢀCD] × 103/
[HꢀCD] × 103/
mol dm-3
E°Fe/V
mol dm-3
E°Fe/V
0
1
2.5
3.5
5
8
12
16
0.442
0.439
0.444
0.445
0.452
0.452
0.458
0.458
0
1
3
6
10
16
0.442
0.431
0.422
0.416
0.413
0.408
energy of the maximum of the MMCT band, Eop, is related to
the parameters of the corresponding thermal electron transfer
by
Figure 3. Plot of the redox potentials of the cobalt couples,
E°(CoIII/II(NH3)5L)/V, determined by the procedures in refs 23 and 24
versus the redox potentials of the corresponding ruthenium couples,
E°(RuIII/II(NH3)5L)/V, obtained from conventional electrochemical
measurements. L ) Cl (chloride), Pz (pyrazine), Py (pyridine), and
DMSO (dimethyl sulfoxide).
Eop ) λ + ∆G°′
(8)
Starting from eq 8, some of us developed a procedure for the
determination of redox potentials of chemically and electro-
chemically irreversible inorganic redox couples, such as the
cobalt couple here.23,24 Following this procedure, the redox
potentials of a series of cobalt complexes were determined. The
reliability of this method is illustrated in Figure 3. In this figure,
the redox potentials of the cobalt couples determined by the
procedures in refs 23 and 24 (see Appendix) are plotted versus
the redox potentials of the corresponding ruthenium couples
(obtained from conventional electrochemical measurements).
The linear relationship of the two sets of potentials is in line
with the ideas of Lever.25 This gives support to our determina-
tions of the potentials of the redox couples of the cobalt
complexes. On the other hand, Figure 4 gives the plot of ln ket
for the oxidation of [Fe(CN)6]4- by the cobalt complexes versus
the redox potentials of these complexes. There is a linear
relationship between ln ket and E°′, as expected according to
the Marcus theory. This gives further support to the procedures
in refs 23 and 24. In fact, we have used the relation illus-
trated in Figure 4 to determine the redox potential of
[Co(NH3)4(H2O)2]3+. Thus, ket ) 32 s-1 for the oxidation of
[Fe(CN)6]4- with the cobalt complex. From this value of ket, a
value of E° ) 0.82 V is obtained (the estimated uncertainty is
(30 mV, which is obtained from standard errors on the slope
and the intercept, appearing in Figure 4, and the usual
propagation errors).
Figure 4. Plot of ln ket for the oxidation of [Fe(CN)6]4- by
[CoIII(NH3)5L] or [CoIII(NH3)4L′] complexes versus the redox potentials
(V) of the corresponding CoII/III couples. L ) Phepy (4-phenylpyridine),
Py (pyridine), Pz (pyrazine), and DMSO (dimethyl sulfoxide); L′ )
PzCO2 (pyrazine-2-carboxylate) and (H2O)2 (diaquo).
However, the free energy in eq 10 is not the value of ∆G°′ to
be used in the calculation. The latter corresponds to the process
precursor complex ∆G°'8 successor complex
(11)
The linearity of the plot in Figure 4 deserves some comments
because, according to eq 1
λ
4RT
∆G°'
2RT
(∆G°')2
4λRT
and the free energy in eq 10 corresponds to the process
ln k ) ln A -
+
+
(9)
(
)
et
separate reactants ∆G°8 separate products
(12)
(13)
(14)
Therefore, a quadratic dependence of ln ket on ∆G°′ should be
expected. However, as Marcus and Sutin established,26 this
quadratic term is small for reactions with low values of ∆G°′
and/or high values of λ. In fact, for data in Figure 4, the change
in the quadratic term is about 2 kJ mol-1, whereas the linear
However, ∆G° and ∆G°′ are related by
∆G°' ) ∆G + (wp - wr)
term changes by about 25 kJ mol-1
.
Once the value of the potential for he [Co(NH3)4-
(H2O)2]3+/2+ couple has been estimated and the values of E°
for the iron complex have been measured, ∆G° for the reaction
can be calculated
The difference of these work terms can be expressed as
Kp
wp - wr ) -RT ln
(
)
Kr
∆G° ) -F(E° - E° )
(10)
Co
Fe

9098 J. Phys. Chem. B, Vol. 114, No. 28, 2010
Perez-Tejeda et al.
TABLE 4: Values of ket, ∆G‡, (Eop)exp, (Eop)corr, ∆G°′, and λ
for the Reaction [Fe(CN)6]4- + [Co(NH3)5Cl]2+ in a Mixture
of Acetonitrile and Water (xACN ) 0.4) at Different
Concentrations of 18C6
where Kp and Kr represent the equilibrium constants corre-
sponding to formation of the successor complex from the
separate products and the precursor complex from separate
reactants, respectively, which were calculated by
c
c
[18C6]/
∆G‡b/
(Eop)exp
/
(Eop)corr
/
∆G°′d/
λe/
mol dm-3 keta/s-1 kJ mol-1 kJ mol-1 kJ mol-1 kJ mol-1 kJ mol-1
4
3
Ki ) 103πNAr3 exp(-ꢀVAEFD(r, I))
(15)
0
1.06
1.22
1.34
1.46
1.61
1.72
1.77
1.87
1.9
70.7
70.3
70.1
69.9
69.7
69.4
69.4
69.3
69.3
324
321
320
319
317
316
316
316
316
275
272
271
270
268
267
267
267
267
7
9
9
10
10
10
10
10
10
268
263
261
260
258
257
257
257
256
0.02
0.035
0.05
0.1
0.2
0.3
VADEF(r,I) being the mean force potential between the reactants
or products (electronic acceptor and donor) at the contact
distance (r), ꢀ ) 1/kBT, and NA being Avogadro’s constant. The
interaction potential was calculated by the Eigen-Fuoss
approach27
0.4
0.5
a Data obtained from ref 31. The values of the parameter
obtained from the fit to eq 4 are K ) 13 mol-1 dm3 and (ket)18C6
2 s-1 b The uncertainties from the experimental errors are (0.1 kJ
mol-1 c The uncertainties calculated using the usual procedure on
the propagation errors are (1 kJ mol-1 d The uncertainties
calculated using the usual procedure on the propagation errors are
(3 kJ mol-1 e The uncertainties calculated using the usual
procedure on the propagation errors are (2 kJ mol-1
)
ZAZDe2
.
VEAFD(r, I) )
(16)
.
4πεoεsr(1 + κr)
.
.
ZA and ZD are the charge type of the electronic acceptor and
donor, respectively, e is the proton charge, εo is the vacuum
permittivity, εs is the relative dielectric constant, r is the
encounter distance between the ions (taken as the sum of ionic
radii)28 (see also ref 29), and κ is the Debye screening parameter
that is a function of the ionic strength, I. Taking into account
an ionic strength of 0.037 mol dm-3, the value for wp - wr was
7.9 kJ mol-1, in agreement with the estimation performed in
refs 23 and 24 for similar reactants. In this way, the values of
∆G°′ appearing in Tables 1 and 2 were obtained. Finally, from
∆G‡ and ∆G°′, the values of λ were calculated.
As can be seen in the tables, the reorganization free energy
does not change appreciably in the presence of CDs. Therefore,
the different behavior is due to the differences in reaction free
energies, ∆G°′, which, in turn, are due to the differences in the
behavior of the redox potentials of the iron complex in the two
CDs; in the case of ꢀCD, ∆G°′ becomes less favorable when
the iron complex is bound, whereas in the case of HꢀCD, the
opposite effect is observed. Notice that the influence of the
receptor on the redox potential of an included guest depends
on the binding of the oxidized and reduced forms of this couple
to the host30
.
the fact that kinetic results, which appear in Table 4, were taken
from a previous paper31 in which a mixture of acetonitrile and
water (xACN ) 0.4) was used as the solvent. For this solvent,
redox potentials of Co3+/2+ couples were not available. Conse-
quently, a different approach was used. This approach follows
from the consideration of eqs 1b and 8, which permits one to
write
E2op
∆G‡ )
(19)
4λ
This equation implies that by combining kinetic (∆G‡) and
optical (Eop) data, one can obtain λ, and then, once λ is known,
it is possible to use eq 8 to obtain ∆G°′.
However, having Eop for a reactive system presents difficul-
ties. For this reason, instead of measuring Eop for the MMCT
band in the [Fe(CN)6]4-/[Co(NH3)5Cl]2+ system, we have
measured the values of Eop for the [Ru(CN)6]4-/[Co(NH3)5Cl]2+
system. Of course, this implies correction of the measured values
of Eop. Briefly, the corrections are the following:23,24 (i)
correction for the difference in the spin-orbit coupling of the
iron and ruthenium complexes, (ii) corrections for the difference
for the redox potentials in these complexes, and (iii) corrections
for the differences of spin of Co(II) complexes resulting in
optical (low-spin) and thermal (high-spin) electron transfer. After
applying these corrections (see Appendix), the values of Eop to
be used in eq 19, (Eop)corr appearing in Table 4, were obtained.
Notice that (Eop)corr corresponds to the same optical transition
γox
1 + Kred[CD]
1 + Kox[CD]
E° ) E° + RT ln
) E° + RT ln
o
o
γred
(17)
Thus, one must conclude that the presence of the 2-hydrox-
ypropyl moiety in HꢀCD implies that the binding of the oxidized
form is stronger than the binding of the reduced form of the
iron complex couple. The opposite is true for the ꢀCD.
Crown Ether Solutions. In the case of solutions containing
18-crown-6, our task was the same as that in the case of CD
solutions, to find out if the influence of the binding in the kinetics
of the reaction
as that in the thermal process. That is, [Fe(CN)6]4-
/
[CoIII(NH3)5Cl]2+ (low spin) f [Fe(CN)6]3-/[CoII(NH3)5Cl]1+
(high spin).
From the values of (Eop)corr in Table 4 and those of ∆G‡, λ
was determined using eq 19 for the different concentrations of
18C6. Finally, from the values of λ and employing eq 1b (or
8), the values of ∆G°′ were obtained (see Table 4).
It is important to realize that by using eq 19, one can obtain
λ without employing the bandwidth22 because the latter proce-
dure overestimates λ values due to the problem of inhomoge-
neous broadening.32
[Fe(CN)6]4- + [Co(NH3)5Cl]2+ f [Fe(CN)6]3-
+
[Co(NH3)5Cl]+ (18)
comes from the effects of this binding on λ or ∆G°′ (or both).
However, in this case, the redox potential of the cobalt complex
couple cannot be determined through the same procedure as
that employed in the case of CDs. This difficulty arises from
The data in Table 4 show that the behavior of the primary
kinetic data, ket, is similar to the behavior of this parameter in
CD solutions in the sense that it can be described employing a

Electron-Transfer under Restricted Geometry Conditions
J. Phys. Chem. B, Vol. 114, No. 28, 2010 9099
two state model (eq 4). However, in the case of 18C6, the
increase in ket is mainly controlled by λ, whereas ∆G°′ exerts
only a minor influence (in the opposite sense to λ but in
agreement with eq 1717). In the case of CDs, on the contrary,
changes in ∆G°′ cause the changes in ket, λ being practically
constant.
(c) the differences between free energies of thermally
equalibrated high-spin and low-spin Co(II) complexes, ∆Ei, and
the inner-shell reorganization free-energy difference between
the thermal and optical processes, owing to the differences
in the spin of Co(II), δλin
In conclusion, two electron-transfer processes, in which one
of the reactants participates as a bound species, were studied.
Working under conditions in which the measured rate constant
is directly ket and following two different approaches, λ and
∆G°′ were calculated for these electron-transfer processes. These
data show that similar behavior in ket (in 18C6 and HꢀCD) can
arise from different causes. It also made it possible to explain
the different behavior in the two CD-type receptors. We think
that this kind of analysis may be useful in gaining a deeper
knowledge of the electron-transfer reactions under restricted
geometry conditions.
Co/Fe
op
(Eop)corr ) E
+ ∆Ei + δλin
(A-3)
A value of -79.5 kJ mol-1 was taken for ∆Ei according to the
data of Brunschwig,28b and a value of 127.8 kJ mol-1 was
calculated for δλin using eq A-4 and the corresponding force
constants and bond lengths (see ref 23)
1
2
λ )
fk(∆qk)2
(A-4a)
(A-4b)
∑
in
k
Acknowledgment. This work was financed by D.G.I.C.Y.T.
(CTQ2008-00008/BQU) and the Conserjer´ıa de Educacio´n y
Ciencia de la Junta de Andaluc´ıa.
2frkfpk
frk + fpk
fk )
Appendix
Scheme 2 summarizes the steps of the correction procedure.
Once one has (Eop)corr, from experimental ket and the corre-
sponding values of ∆G‡ by using eqs 19 and 8, the values of
∆G°′ and λ for a given reaction can be calculated. In this way,
the values of these parameters appearing in Table 4 were
obtained.
It is clear that one cannot obtain the energy of the maximum
optical transfer band directly within the ion pairs [Co(NH3)5Cl]2+/
[Fe(CN)6]4-, (Eop)corr, because the reactions are too fast at room
temperature. Instead, the energy of the maximum optical transfer
band within the ion pairs [Co(NH3)5Cl]2+/[Ru(CN)6]4-, (Eop)exp
was measured. Thus, (Eop)exp must be corrected for
,
On the other hand, once one has ∆G°′, to have the values of
E° for the cobalt couples is a straightforward matter using eq
13 and the redox potential of the [Fe(CN)6]3-/4- couple.
(a)the spin-orbit coupling contribution of the hexacyanoru-
thenate (III) complex by using
Co/Ru
op
E
) (Eop)exp - λso
(A-1)
References and Notes
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where λso is the spin-orbit coupling parameter. A value of 14.9
kJ mol-1 was used for this parameter.28b
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(b) the difference in the redox potential of [Fe(CN)6]3-/4-
(E° ) and [Ru(CN)6]3-/4- (E° ) couples in order to obtain the
Fe
Ru
maximum of the MMCT band corresponding to the ion pair
[Co(NH3)5Cl]2+/[Fe(CN)6]4-
(6) Prado-Gotor, R.; Jimenez, R.; Perez-Tejeda, P.; Lopez-Cornejo, P.;
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Co/Fe
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Co/Ru
op
E
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+ F(EFe - ERu)
(A-2)
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