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258

CHAPTER V
INFLUENCES 0F SPECIFIC REACTANT-SOLVENT

INTERACTIONS ON ELECTRON-TRANSFER KINETICS AND THERMODYNAMICS

A. The Influence 9; Specific Reactant-Solvent Interactions 93

Intrinsic Activation Entrogies for Outer-Sphere Electron Transfer
m

(Accepted for publication in J. Phys. Chem.)

1- W
In recent years increasingly detailed and sophisticated theories

of outer-sphere electron-transfer kinetics have been formulated.56

These enable rates and activation parameters to be calculated from
reaction thermodynamics together with reactant and solvent structural
information. Although treatments of inner-shell (intramolecular
reactant) reorganization have reached a high degree of sophisti-

23

cation, the important contribution to the free energy barrier arising

from outer-shell (noncoordinated solvent) reorganization is usually
treated in terms of the classical dielectric continuum model as

13

originally formulated by Marcus. While comparisons between theory

and experiment for bimolecular outer-sphere processes show reasonable

agreement in a number of cases,10 significant and often large

259

discrepancies still remain (cf. Chapter VII). Among other things, such
discrepancies call into question the quantitative validity of the
dielectric continuum model, especially in view of the well-known
failure of similar treatments to describe the thermodynamics of ion
salvation.

In principle, a useful way of monitoring the influence of outer-
shell solvation upon electron-transfer energetics is to evaluate en-
tropic parameters since these are expected to arise chiefly from the
changes in the degree of solvent polarization associated with electron
transfer. The activation entropy,AS*, as for other reorganization
parameters, can usefully be divided into "intrinsic" and

"thermodynam ic" factors: 245 ’2“

AS 'AS. + aAS

m (5.1)

where the coefficient a is predicted usually to be close to 0.5.247
The intrinsic activation entropy, ASE“, is that component of [38* that
remains in the absence of the entropic driving force A80. When esti-
mating values of 138* from Equation 5.1, it is usual to employ experi-
mental values of A80 and yet also employ values of ASL"; calculated
from dielectric continuum theory. Although these calculated values of
A82“ are often small, given that the values of [33° are often much
larger and more variable than those calculated from the dielectric

continuum model, it is reasonable to inquire if a more trustworthy

:-
method for estimating A8 int could be formulated.

260

A useful and often enlightening approach for understanding
electron transfer processes both on a conceptual and an experimental
basis is to examine the thermodynamics and kinetics of electrochemical

reactions:53-55’248

Ox + e'(¢m) Red (1.12)

where o!“ is the (Galvani) electrode-solution potential difference.
Although absolute values of on cannot be evaluated with useful
accuracy. the temperature dependence of ¢m can be obtained using a

nonisothermal cell arrangement?“55

This enables the entropic change
induced by reduction of a single redox center. the so-called “reaction
entropy" Asgc, to be determined from the temperature dependence of the

55

standard (or formal) potential ¢: under these conditions. Activation

parameters for such electrochemical "half reactions" can be obtained

using an analogous procedure.53‘54

These quantities provide insights
into the structural changes accompanying electron transfer at each
redox center that remain hidden for homogeneous bimolecular reactions.
The aim of this section is first to provide a simple physical
picture, based on electrochemical half-reactions, of the origin of the
intrinsic activation entropy in homogeneous and electrochemical redox
reactions. With this background a new approach for estimating As:nt
will be outlined based on reaction entropy data whereby the effects of
specific reactant-solvent interactions can be taken into account. De-

spite their potential importance, such interactions have yet to be con-

sidered even in the more sophisticated theories of electron transfer.

261

2- mgmmmm

The actual entropic barriers, A3: and As:, to electron transfer
for the forward (reduction) and reverse (oxidation) electrochemical
reactions at a given electrode potential have been termed "ideal"

53 ,54,248

activation entropies. These can be formulated as

*

* o
A a A A
8f a 3re + sint,e (5'23)
* s:
I — A o A .
ASr (a 1) Sn + Sint,e (5.2b)
*
where a is the electrochemical transfer coefficient and Asint e is the
D

so-called ”real" (or intrinsic) electrochemical activation entropy,
i.e. that which remains after accounting for the entropic driving
force.53 For convenience, we shall assme that the interactions
between the reactant and electrode, and between the reactant pair in
homogeneous solution, are weak and nonspecific (i.e. the "weak
249

*
Under these circumstances AS. is related
int,e

to the intrinsic activation entropy for the corresponding self-exchange

interact ion" 1 imit) .

reaction by

(5.3)

Relationships such as Equation 5.3 reflect the fact that homogeneous
outer-sphere reactions can be regarded as coupled reductive and

oxidative electrochemical reactions.

262

Equations 5.2a and 5.2b point to a key difference between homo-
geneous self-exchange and electrochemical exchange reactions: the
latter are characterized by a net entropy driving force A8:c even when
the free energy driving force is zero. This results from the inherent
chemical asymmetry of the electrochemical half reactions. This entropy
driving force Contributes to the forward or reverse entropic barrier
for each redox center to an extent determined by the difference in
(hypothetical) charge between the oxidized or reduced reactant and the
transition state, namely aor (o-l). The transition state of course
never acquires a fractional charge since electron transfer occurs
approximately independently of nuclear motion, but nonetheless is char-
acterized by a polarized solvent environment appropriate to a molecule
possessing such a charge.13’56

According to the theoretical approach of Marcus,13 solvent reor-
ganization to form the transition state can be viewed as occurring by a
hypothetical two-step process. First the charge of the reactant is
slowly adjusted to a fractional charge approximately midway between the
reactant and product charges, with attendant reorientation of the sur-

rounding solvent.250‘251

Then in a rapid step (much faster than
solvent motion) the transitionrstate charge is reset to that of the
reactant. Taken together, the energies of the two steps are equivalent
to the nonequilibrium.solvent polarization energy. 0n the basis of the
conventional dielectric continuum approach, the energetics of the first
step are determined by the static solvent dielectric constant es,
while the optical (i.e. infinite frequency) dielectric constant e

0?
determines the energy of the fast second step. We shall term these two

263

steps the “static” and "optical" components, respectively. Generally
the optical component is anticipated to provide the dominant contri-
bution to the free energy of solvent reorganization due to the relative
magnitudes of 80p and 5'. However, the temperature coefficients of the
two dielectric constants are such that in many solvents the optical and
static components are calculated to contribute roughly equally to the
entropic component of the solvent barrier.
The conventional calculation of the solvent reorganization

energetics involves an application of the Born ionrsolvation model to

250.251

transition state theory. The Born model predicts that entropies

252 It is

of ions will vary with the square of the charge number.
reasonable to suppose that the static component of the electrochemical
transition-state entropy will also depend on the square of the
effective charge. The differences in static entropy between the
transition and ground states should be appropriately weighted fractions
of the total entropy difference Asgc between the two ground redox

states. We can therefore express the static components of the forward

and reverse electrochemical activation entropies as
Aspmm) - {[(n + 1)2_(n + 1 -a)2]/[(n + nz-uznAsgc (5.4.)
A8:(static) ' {In2 - (n + l -o)2]/[(n + l)2 - n21}As:c (5.4b)

where n and n+1 are the charge numbers of the two forms of the redox

couple, and (n+l-a) is the effective transitionestate charge.

264

It can be seen from Equations 5.4a and 5.4b that even for a
transition state that is symmetrical with respect to charge, i.e.cz-
0.5. that A8;(static) will differ from -ZB:(static). In other words,
the transition state will not lie midway in terms of entropy between
the reduced and oxidized states even though it may be equally
accessible in terms of free energy from either oxidation state. This
mismatch of the energetics of the forward and reverse half reactions
follows from the liggg; variation of driving force contributions with
charge (Equation 5.2), coupled with the quadratic dependence of static
entropy on charge.

Equations 5.2a and 5.2b can be combined to yield

*
int,e

AS - (1-0)AS; +aA S: (5.5)

The intrinsic activation entropy therefore is a measure of the extent
of the mismatch between forward and reverse half-reaction entropic

barriers after normalizing for driving force contributions. This is

*
f

a . .
and A8r exactly cancel. The connections between the various entropic

seen most clearly whena.- 0.5 and the driving force components of AS

quantities are illustrated schematically in Figure 5.1. The magnitude
*

of Asint is given by the vertical displacement of the curve AB from the

chord (dashed line) to this curve. The curve AB describes the depen-

*
dence of ASin upon the effective ionic charge.

1:

Equations 5.4a and 5.4b can be combined with Equation 5.5 to

yield an expression for the static component of As:nt e:
5

265

158*

int,e(static) - [o(l—o)/(2n+1)]AS:c (5.6)

Taking a . 0.5 and inserting the Bornian expression for the reaction

entropy:253

As:c -(Ne2/2r€§)(d€sldT) [(n+l)2-n2] (5.7)
into Equation 5.6 yields:

*
A3

int,e(static) - -(Ne2/8rۤ) (dealdT) (5.8)

where N is the Avogadro number, e is the electronic charge, and r is
the reactant radius. Note that the apparent dependence of Asint,e
(static) on reactant charge in Equation 5.6 has now been eliminated.
Equation 5.7 can be compared with the relation obtained from the
temperature derivative of the usual dielectric continuum.expression for

the reorganization free energy:13’54

*
AS

int,e ' [(Ne2/8)(1/r-l/m] [(l/eip)(deop/dT)-(l/c§) (desldTH (5.9)

Equations 5.8 and 5.9 differ in that the latter takes account of image
stabilization of the ion in the vicinity of the electrode by including
the ionrimage separation distance R; furthermore, the optical portion
of the activation entropy is included. This term is similar in form to
the static term since it is assumed, based on a linear response of

solvent polarization to the field of the ion, that the optical portion

266

 

 

 

 

 

 

 

 

 

 

 

 

 

g, A
\
\\ A53... , AS‘,‘
3 g. (a-MS}. \\
o
.3 A55, \
LIJ \\ .
.53 \ 1531
c \
.9. 0A5}. \
\\
\\
$3, 5 s
l l l
n m l-a ml

Effective Ionic Charge

Figure 5.1. Schematic representation of the ionic entropy
of an individual redox center as a function of its effective
ionic charge during the electron transfer step. See text

for details.

267

Of As:nt e also varies with the square of the effective charge of the
9

transition state.56b

Similarly. from Equations 5.3 and 5.6 the static portion of the

intrinsic entropy for homogeneous self-exchange reactions can be

expressed as

ASInthtatic) ' [205(1-o)/(2n+1)] As:c (5.10)

Again, for <1 - 0.5 and on the basis of the Bornian model (Equation

5.7), this leads to

* . 2 2
Asint(static) - (Ne l4r88) (dEBIdT)] (5.11)

This is identical in form to the dielectric continuum expression for

A * ( f ' .
8int c . Equation 5.9).

As;t - (Ne2/4) ($.- é) ((1/efip) (deep/dT)-(1/€:)(d€8/dT)] (5.12)

allowing again for the addition of the optical term and the presence of

the nearby coreactant through the internuclear distance term R.

3. 3&5; Chemical Environments. Incorpogatigg Specific
Egactant-Solvent Interactions ig.Activation Entropy Calculations,
In general, the experimental values of Asgc differ widely from
the continuum predictions of Equation 5.7. In water, for example as:

for the Cr(1120)2+/2+ couple is seven times greater than predicted,

268

while the experimental value of AS:c for F'e(bpy)g+/2+ is less than a

third of the theoretical value. Furthermore, the expected variation of
reaction entropies with solvent dielectric properties is not
observed.253-255 The discrepancies between theory and experiment have
variously been attributed to dielectric saturation, hydrogen bonding

between reactants and solvent,55 long range solvent structuring,253-256

and hydrophobic interactions.257

Consequently, in view of Equation
5.10 dielectric continuum theories of solvent reorganization are not
expected to provide accurate estimates of intrinsic activation
entropies.

Nevertheless Equation 5.10 suggests a means of incorporating the

numerous factors neglected in the dielectric continuum treatments.
Rather than employing estimates of As:c based on Equation 5.8, experi-

mental values of AS:c can be used to determine the static component of

*

A81“. Therefore instead of Equation 5.12 the intrinsic activation

entropy can be expressed as

* _, 2 1 1 2
As,1m (Re /4) ‘F" E) [(1/eop) (deop/dTH + Asgc/(an + 2) (5.13)

*
int’

Equation 5.13, is unchanged from Equation 5.14; however the static

The optical component of AS the first term on the r.h.s. of
component embodied in the second term is taken instead from Equation
5.10 with a - 0.5 (As expected, a is commonly observed to be close to
0.5 for outer-sphere electrochemical reactions). Equation 5.13 is
therefore anticipated to yield more reliable values of Asint’ at least

in the weak-interaction limit, since it circumvents the known severe

269

limitations of the Born model for calculating static entropies.
The latter model is retained for estimating the optical component

.1; lieu of any direct experimental information to the contrary. The

 

justification for this approach is that the Born model is likely to be
much more reliable for estimating the optical rather than the static
component in view of the relative insensitivity of 80p to solvent
structure. Thus the extensive local perturbations in solvent structure
induced around an ionic solute that are responsible for the failure of
the dielectric continuum model for predicting ionic solvation thermo-
dynamics should have a much smaller influence on the intramolecular
electronic perturbations which constitute the optical component of the
reorganization barrier.

A comparison between values of As:at calculated from Equations

5.12 and 5.13 for some representative redox couples in aqueous media is
presented in Table 5.1. Whereas the dielectric continuum model

*
(Equation 5.12) predicts that Asia will be small and nearly inde-

t

pendent of the chemical nature of the redox couple, somewhat larger and

a
more varying values of 13in are predicted by Equation 5.13 since this

t

takes into account specific reactant-solvent interactions via inclusion
of the experimental values of Aszc. Further, the latter relation

*
predicts markedly larger variations in ASint with solvent than obtained

with the former relation, resulting from the much greater sensitivity

0f Asgc to the solvent than predicted by the Born model.253-255

Although the differences between Equations 5.12 and 5.13 have
been emphasized here, it should be noted that the As:nt values obtained
by the latter are still relatively small. An interesting result is

270

that for multicharged reactants very large thermodynamic solvation

effects translate to much smaller intrinsic entropic barriers. For

example, the Fe(HZO)Z+/2+ self-exchange reaction involves thermodynamic

l l l

entropy changes amounting to 360 J deg- mol-1 (180 J deg- mol- for

each half reaction) which yields an entropic contribution of just 13 J

-l 1

deg mol- to the Franck-Condon barrier (Table 5.1). Still, the

effects are large enough to warrant consideration. For example, for

3+/2+

*
the “(320)6 reaction ASin should contribute a factor of five to

t
the self-exchange rate constant at room temperature. This effect is

therefore comparable in magnitude to the nuclear tunneling corrections
and nonadiabatic electron tunneling factors which have been emphasized

in the recent literature.23’258

4. Copparisons with Experipgnt

 

In addition to the calculated values of Asint’

some
“experimental" values for these homogeneous self-exchange reactions,
*
Asint(°xp)’ are given in Table 5.1. The latter were extracted from the
measured activation enthalpies, Ant, and the rate constants, k, using
*
I - *

k lipl‘nvn exp(ASint/R) exp( AH IRT) (5.14)
where KP is the equilibrium constant for forming the precursor complex
immediately prior to the electron-transfer step, rn is the nuclear
tunneling factor. and x”: is the nuclear frequency factor.9’23’258

[Note that the activation entropy in Equation 5.14 can be directly

. . . . *
identified with A31“ since A30 - 0 for self-exchange reactions

271

(Equation 5.1)]. The values of RP and Tn were calculated as described
in Chapter VII and reference 10. The values of As:nt(exp) were
corrected for the variation of Pa with temperature by calculating this
quantity using the relationships given in reference 23.

It is seen in Table 5.1 that the values of Asznt(exp) are uni-
formly smaller, i.e. more negative, than the estimates of ASEfit from
both Equation 5.12 and Equation 5.13. Such negative values of
As;nt (exp) are commonly observed for homogeneous outer-sphere reac-
tions. They have been variously attributed to an unfavorable contri-
bution to the precursor work term arising from reactant-solvent inter-
actions, to the occurrence of nonadiabatic pathways, and to steric

factors.“’207

In any case, in view of the present discussion it ap-
pears likely that these negative values of Asint reflect properties of
the bimolecular precursor complex rather than those of the individual
redox couples; i.e. reflect the modification of the solvation environ-
ment around each redox center brought about by its proximity to the co-
reactant necessary for electron transfer. In fact, the inclusion of
specific reactant-solvent interactions in the calculation of As:nt for
the weak interaction limit by employing Equation 5.13 rather than Equa-
tion 5.12 leads in most cases to more positive values of As:nt (Table
5.1).

Before accepting this conclusion, however, it is worth examining
further the various assumptions embedded in Equation 5.13. Given the
breakdowns observed thus far in the Born solvation model, it is

possible that the assumed quadratic variation of entropy with charge is

also incorrect. The magnitude of the intrinsic activation entropy

272

Table 5.1. Intrinsic Activation Entropiem for Selected Homogeneous Self-Exchange Reactions, ASLc

(J deg-1 mol-1). calculated without (Equation 5.12) and with (Equation 5.13) Consideration
of Specific Esactant-Solvent Interactions. and Comparison with Experiment.

 

 

a b a d
Eedox Cou 1e Solvent ' r R A 3* A s' 35*
P ' ' int int int
Equation 5.12 Equation 5.13 (experiment)
3+/2+
Fe(0!lz)6 a 820 3.3 - -1.5 ‘ 13 -62
3+/2+
Hafiz)6 320 3.3 -l.5 12.5 -61
Eu(NE3):+/2+ s20 3.3 -1.5 2.5 -24‘
Co(ed§+’z*f 320 «.2 -1.5 9.5 -as
9
many)?”2+ 320 6.7 -1.0 -4.5 --
ferricinium- 820 3.8 -l.5 ~15 -—
ferrocene
ferricinium- methanol 3.8 -6.5 -9.5 -64
ferrocene
ferricinium- nitromethane 3.8 -6.0 16 -29
ferrocene '

akeactant radius, used to calculate ASInt (Equation 5.12). Values taken from references 10 and 255.
bIntrinsic activation entropy, calculated from Equation 5.12 using the listed values of r and assuming

that 'Zr (Reference “.82)- Literature values of t ,(dc Id'I'). c . (dc /d‘l‘): water -e -78.3.

(deg/d )--0.365, :0 -1.7s, (dc /dT)--0.00024 (valuessfrom retereeEB 165,°Bp. E61, 3224); Serhenei-

r -32.6, (dc /dT)--B.20 (reterSBcee 263); c -1.76. (dt ldT)--0.0011 (reference 260, p. 145);

n tromethanegc.-35.5, (dc./dT)--O.l6 (rererSBee 265); cop-1.90, (drop/dr)--o.ooio (reference 266, p. 391).

c
Intrinsic activation entropy, calculated similarly to footnote b above, using the experimental values
of 68:: taken from references 55 and 255.

dValues extracted from published rate dattzby_!sipg Equation 5.16 and correcting for nuclear tunneling
effects. Values of K v are circa. 3 x 10 H 5 (see reference 10 and Chapter VII). Literature
sources for rate datg:n Fe(l!20)63+/2+ - reference 266, V(820)g+/2+ - reference 267, Ru(NHg)a+/2+ -
reference 268,Co(en) 3442* - reference 269, ferricinium/ferrocene-reference 270.

‘Brown and Sutin (reference 28) have questioned the accuracy of this result. based on the more negative
AS‘ value for the au(qu+ - Eu(Nfl3)§ cross reaction.

fen - ethylenediammine

gbpy-2,2'-hipyridine

273

obtained from Equations 5.13 is closely connected to the functional
dependence of entropy on charge. For example a linear dependence leads
to a value of zero for Asint' Other functions might lead to large
imbalances of forward and reverse entropic barriers and therefore
substantial intrinsic activation entropies. Since most couples exhibit
positive values of As:c a fractional dependence of entropy on charge

a
would normally be required to deduce negative values of ASin The

t'
entropy-charge relation was the subject of a number of detailed
examinations and some controversy in previous years, and apparently was
never unambiguously resolved.259"261 One reason for this was the
difficulty of varying the ionic charge while holding constant the other
relevant parameters such as ionic size, ligand composition,
coordination number. etc.

In order to determine the relation between entropy and charge for
a prototype system we examined the reaction entropies of ruthenium tris
bipyridine, for which oxidation states 0, I, II, and III are accessible
in acetonitrile. The experimental details are given in reference 253.
By employing the same compound in various oxidation states the numerous
complications and ambiguities inevitably involved in previous studies
are avoided. The reaction. entropies thus obtained for the
Eu(bpy)g+/2+. Ru(bpy)§+/+. and Eu(bpy);/o couples, respectively, in
acetonitrile (containing 0.l‘l_l_KPF6 supporting electrolyte) are 117, 71

1 mol-l. If these data are recast as relative single ion

and 23 J deg-
entropies, 8° + K, where K is an unknown constant quantity, a straight-
forward variation of entropy with the square of ionic charge is evident

(Figure 5.2), supporting the validity of Equation 5.13.

274

It is possible of course that the ruthenium trisbipyridine
reactions represent an atypical case. (However, additional evidence is
assembled in Section V. C). Another way of exploring the possibility
that the negative experimental values of A8;t might arise in part from
mismatches in the thermodynamic entropic changes occurring in each half
reaction is to examine if the magnitude of A8:nt depends on the sum of
the constituent As:c values. The larger these entropy changes, the

a a
larger should be the mismatch in ASf and ASr for each half reaction,

*
yielding larger (or more negative) values of ASin However. such an

t.
examination for about thirty self-exchange and cross reactions shows no

signs of such a systematic trend. (Details are given in Chapter VII).

*
In addition, the experimental values of ASin also show no discernable

t

dependence on the magnitude of the reorganization barrier. comparable

. * . . .
negative values of AS. being obtained even for extremely rapid

int
207

reactions. This provides evidence that these negative values are

associated either with an entropically unfavorable work term and/or
nonadiabaticity, rather than residing in the elementary reorganization
barrier to electron transfer of which the estimates of As:gt obtained
from Equation 5.13 form a part.

Nevertheless, the method of calculating Asint embodied in

Equation 5.13 is considered to be useful since it provides a reliable

* . .
estimate of ASin for the limiting "weak interaction" case where the

t
solvating environments of the two reactants do not modify each other,
while accounting properly for the influence of the actual reactant-

solvent interactions upon the entropic reorganization barrier for these

isolated redox environments. It therefore provides a more trustworthy

275

 

 

 

p

l l l l

o 4 s
(Ionic Charge)2

Figure 5.2. Plot of relative ionic entropy of Ru(bpy); (bpy-
2,2'-bipyridine) in acetonitrile as a function of the square
of the ionic charge n. The solid line is the best fit line

through the experimental points; the dashed line is the slope

of this plot predicted by the Born model.

 

276

means of gauging the extent of influence of reactant-reactant inter-
actions upon the activation entropy than is obtained by employing the
conventional relationship (Equation 5.12), as well as supplying useful
insight into the physical and chemical factors that determine this
quantity.

A related approach to that described here can also be employed to
estimate the effects of isolated reactant-solvent hydrogen bonding on
the intrinsic enthalpic component of the Franck-Condon barrier. This

involves examining the solvent dependence of the half-cell redox

262

thermodynamics. Preliminary results indicate that such enthalpic

effects are markedly larger than the corresponding entropic factors
examined here, contributing several kJ mol-1 to the intrinsic free

energy barriers for a number of reactions.262

These findings suggest
that such specific reactant-solvent interactions may indeed account in
part for the common observation that the experimental rate constants
for homogeneous outer-sphere reactions are significantly smaller than
the theoretical predictions where the outer-shell reorganization energy
is calculated using the conventional dielectric continuum model.10
B. Utility pf Surface Reaction Entropies £2; Examining
Eeactant-Solvent Interactions pp Electrochemical Interfaces,

FerriciniumrFerrocene Attached to Platinum Electrodes

(Accepted for publication in J. Electrochem. Soc.)

1.111531111122122
Reactant-solvent interactions are of prime importance to both the

kinetics and thermodynamics of electrode processes. Since electro-

277

chemical reactions inevitably occur within the interfacial region, it
is desirable to gain information on the nature of reactant salvation at
the electrode surface as well as in bulk solution. Weaver. et al. have

demonstrated that useful information on the latter for simple redox

couples can be obtained from the so-called "reaction entropy", A836,
determined from the temperature dependence of the formal potential, Ef,
using a nanisathermal cell arrangement:55
As° - Nerf/er) (1 21)
re ni °

Since the temperature dependence of the thermal liquid junction poten-
tial in such a cell can be arranged to be negligibly small, A8:c

essentially equals the difference, (3°

0 . .
red 80:), between the ionic

entropies of the reduced and oxidized forms of the redox couple in the
bulk solution. The reaction entropies of simple transitionrmetal redox
couples have been found to be extremely sensitive to the chemical
nature of the coordinated ligands and the surrounding solvent, illus-
trating the importance of specific ligand-solvent interactions to the
overall redox thermodynamics.””84’2”-255

It would clearly be desirable additionally to determine reaction

entropies for redox couples residing in the interfacial region. Such

0

"surface reaction entropies", Asrc s
9

would yield insight into the
salvation changes accompanying the elementary electronrtransfer step
for the redox couple in a particular interfacial environment. For
redox couples present at sufficiently high concentrations at the inter-

face to enable the formal potential for the interfacial (adsorbed)

278

redox couple, E2, to be measured, values of A8:c s can be obtained
8

directly from (cf. Equation 1.21):

o f
asrc’s F<dEaldT)ni (5.15)

Whereas As:c corresponds to the overall entropy driving force for

transforming the bulk-solution reactant to product, As:c s equals the
8

thermodynamic entropy change for the heterogeneous electranrtransfer

186

step itself. Thus Asa and Asgc are related by

rc,s

° -As° + 13° - 33° (5.16)

ASrc,s re p s

where As; and A8: are the entropic work terms associated with farming

the "precursor“ state for electron transfer from the bulk reactant, and

186

the “successor“ state from the bulk product, respectively. We
report here values of Asgc s for a surface-attached ferricinium-
8

ferrocene couple in several solvents in order to illustrate the virtues
of such measurements for elucidating the nature of reactant-solvent
interactions at electrochemical interfaces.

For surface redox couples where the redox center lies within the
inner layer, A8; and A8: are expected to be both nonzero and different.
so that As:c s #Asgc. Indeed we have recently obtained such a result

8
for a specifically adsorbd Co(III)/(II) sepulchrate couple versus the
corresponding bulk solution couple.271 For electrode reactions where
the redox center is present in the diffuse layer or at the outer

Helmholtz plane, it is conventional to assume that the work terms are

279

272 a

so that A8; As: O and hence A8: A3 .

purely coulombic in nature c,s rc

This assumption is required in order to extract true frequency factors
for electron transfer from the temperature dependence of electro-

53,54,186

chemical kinetics. However, in actuality even As:c s for an

a
outer-sphere reaction might be expected to differ significantly from

Aszc in a given solvent medium, bearing in mind the structure sensi-
tivity of ionic entropies and the possibility that the solvating

environment in the vicinity of the surface may differ significantly
from that in the bulk solution. Indeed, one reason for pursuing the
present study was to discover whether differences which recently have
been observed between the energetics of structurally similar electro-
chemical reactions involving surface-bound and solution-phase redox

couples204 could be rationalized in terms of differences between the

bulk and interfacial salvation environments.

2. Measurement 2; Surface Reaction Entropies

Although it is not feasible to evaluate.AS§c'8 for outer-sphere
(i.e. unadsorbed) redox couples, a suitably high interfacial concen-
tration of normally unadsorbed, and presumably fully solvated, reactant
can be achieved by attaching the redox center to the electrode surface
via an inert covalent linkage. As a model system, we studied the
ferricinium-ferrocene redox couple attached to a platinum electrode as
shown in Figure 5.3. The surface-bound ferrocene was prepared by using
the chemical modification procedure described in reference 70. This

system was selected since both the bond to the platinum surface and the

electroactive center itself are exceptionally stable, placing the redox

280

70 In addition it was

center about 6-8 8 from the electrode surface.
anticipated that the surface-attached couple would exhibit reversible
behavior in a variety of solvents.

Efforts to prepare the same ferrocene derivative in solution were
unsuccessful. Nevertheless, nrferrocenemethylene-p-toluidine (Alfred
Bader Chemicals), shown in Figure 5.4, was selected as a reasonably
close analog of the surface-attached complex since in the vicinity of
the redox center the structures of the two substituents are closely
similar. Formal potentials for either the surface-attached or bulk-
solution redox couples were obtained from the average peak potentials
of the cyclic voltammograms. (Quasi-reversible, rather than reversible,
behavior was typically observed for the surface-attached, as well as
bulk-solution, couples, with anodic-cathodic peak separations up to ca
50 mV even in the presence of in compensation; cf. reference 70).

Values of E: for the reduction of the surface-bound ferricinium

derivative could be measured with sufficient accuracy to enable A336
-1

,s
f

mol-l. Representative E8

to be determined to within about 6 J. deg
data obtained in aqueous solution are plotted against temperature in
Figure 5.5. Data also were obtained in methanol, acetonitrile,
dimethylsulfoxide and sulfolane. Attempts were made to measure Asgc,s
in formamide, nitromethane and acetone. but were unsuccessful with the
first solvent due to instability of the surface complex and with the
other two because of irreproducible behavior. Either 0.1 31

tetraethylammoium perchlorate or 0.2 11 LiClO4 was used as the

supporting electrolyte.

281

Fe

55
.3:

Figure 5.3. Made of attachment of ferrocene to platinum surface.

Figure 5.4. Ferrocene derivative (n-ferrocenemethylene-p-toluidine)

used as a solution analog of surface-attached ferrocene in Figure 5.3.

282

 

340 I I I l

320

.mV

 

300

 

 

 

Figure 5.5. Representative plot of formal potential for surface-
attached ferricinium-ferrocene couple, Bi, versus temperature in
aqueous 0.1 §;TEAP. Potentials versus saturated calomel electrode

at 24 C, using nonisothermal cell arrangement.

283

The resulting values of Asgc and E: are summarized in Table

1
5.2. together with As:c data for the ferrocene and n-ferrocene-
methylene-p-toluidine couples in bulk solution. Contrary to our
initial expectations, the reaction entropies for the surface couple and
its solution analog were found to be in reasonable agreement in each
solvent. Evidently the solvent interactions experienced by the
surface-attached couple are not noticeably different from the reactant-
solvent interactions occurring in bulk solution. At least for this
couple, therefore, it appears that differences between. solvent
structure in bulk solution and in the double layer where the surface

redox site is located do not greatly influence the electron-transfer

energetics.

3. Interpretapipn pf Surface Reaction Entropy xplppp

Although the two derivatized ferrocene couples (Figures 5.3 and
5.4) yield similar reaction entropies, these tend to be less positive
than the Asia values for the unsubstituted ferriciniumrferrocene couple
(Table 5.2). Furthermore. the formal potentials for the surface-
attached couple in various solvents are positive of those for ferrocene
itself, while the Ef values for the nrferracenemethylene-p-toluidine
couple are still more positive (Table 5.2; note that the Ef values
quoted are versus that for ferrocene itself in the same solvent). The
differences in formal potentials between ferrocene and its derivatives
are probably manifestations of the greater electron-withdrawing
273

capabilities of carbon-nitrogen double bonds compared to hydrogen.

Thus such an electron-withdrawing substituent would tend to stabilize

284

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285

the relatively electronerich ferrocene redox center to a greater extent
than for ferricinium, leaving the former more difficult to oxidize and
thereby yielding a positive shift in the formal potential. The syste-
matic differences in reaction entropies seen between ferrocene and the
derivatized couples can also be rationalized on this basis. Whether
the differences in formal potentials between the adsorbed couple and
its solution analog result from surface attachment or from differences
in substitutent properties is not entirely clear.

A curious aspect of the results is the marked solvent dependence

of both the Asgc s and Asgc values. The magnitude of these quantities

expected from purely continuum electrostatic considerations is given by

Equation 5.17:255

Mano/133:) - -(e2N/2r€T)(dln€/dlnT) (5.17)

where e is the electronic charge, N is Avogadra’s number, s is the
dielectric constant of the solvent, and r is the radius of the

ferricinium cation. The A8:c a values listed in the last column of
8

Table 5.2 are obtained from Equation 5.17. using literature values of e

274

and assuming that r - 3.88. There is clearly no general pattern of

agreement between the experimental and these calculated quantities, the

Born treatment predicting a much milder solvent dependence of asgc s
9

than is observed. Similar breakdowns of the dielectric continuum model

in predicting reaction entropies have been found for several bulk

solution couples in a number of solvents.55’84’253'255

286

A probable reason for the failure of Equation 5.17 is that the
major property determining the entropy Of charge-induced solvent
reorientation is the degree of "internal order" of the solvent (i.e.,
self-association and long range structuring induced by hydrogen
bonding), rather than the macroscopic dielectric properties.275’276
Thus, a solvent having a high degree of internal order would be
relatively unperturbed by a charged molecule, whereas considerable
solvent ordering around the ion would occur in a medium having little
intermolecular structure. Since such charge-dipole interactions will
be absent for neutral ferrocene, a positive contribution to the
reaction entrepy (S:ed - 83“) would be anticipated for the present
redox couples, especially in relatively nonassaciated solvents. Criss
and co-workers have suggested estimating the degree of internal order
of a solvent from the difference in boiling point, Apr, compared to
that for a structurally analogous hydrocarbon. These values of Apr
are also listed in Table 5.2. Indeed, the Asgc,s values for the
surface-attached ferrocene couple do for the most part vary as expected
with the corresponding values of AT p'

An unusual result which merits comment is the large negative

1

value of A8:c s (-50 J deg- mol-l) found in water (Table 5.2). A

8
small negative value of A8:c has previously been oberved for the bulk-

solution ferrocene couple, also in water.255

This indicates that the
net solvent ordering in the vicinity of the surface-attached redox
center is less extensive in the cationic than in the neutral state, in

qualitative disagreement with the expectations from an electrostatic

treatment. These negative entropy values possibly result from donor-

287

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"accumuo ._ "mu:u>H0m cu aux .ochOHuou :oHumHomanmc "moHomHo mono "Ac.n omamumv acouomuou momma
Im>Huom coHusHomuxHam "moHucmHuu coao “an.n ouomHmv ocooonuou cozomuumuoumenzm "moncmHuu mmHHHm
.HHN oucmnouon aouu cmxmu .uocaaz mauaooo< uco>Hom momuo> muco>Hom maoqnm> cu monaaoo unoccuuom

(ancHoHunou mmocaucoHusHom mam mocomuumuoomuusm Haw mouaouuco :oHuommu mo uon .o.m shaman

as}; Shnwu u< hzmgm

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288

acceptor interactions between the cyclopentadienyl rings and the acidic

water hydrogens.255

(An alternate and additional explanation in terms
of an entropy change associated with "solvent disruption" is offered in
Section V. C). Since the electron density on the cyclopentadienyl
rings will be greater in the reduced state, such specific solvent
interactions should be enhanced leading to increased solvent ordering
and a decrease in entropy compared with that for the oxidized state.
If such an explanation were correct a correlation between A8:C,B and
the acidity of the solvent might be expected. Figure 5.6 shows a plot
of the reaction entropies for the adsorbed couple and its solution
analog as well as for unsubstituted ferrocene versus the solvent
"acceptor number“ which is an empirical measure of the electron-

accepting capabilities of the solvent.277

A reasonable correlation is
indeed observed. Apparently, both noanornian ianrdipole interactions
and specific donor-acceptor interactions might be important in
determining the redox properties of the surface-attached as well as
solution ferrocene couples.

The present work demonstrates the feasibility of determining
surface reaction entropies and illustrates the utility of these
measurements for elucidating the various elements of interfacial
reactant-solvent interactions. Given the sensitivity of Asgc,8
measurements to the solvent structure it is suggested that this
approach might also usefully be employed to gain insight into reactant

salvation in polymer film electrodes for which the question of solvent

penetration within the film is of current interest.

289

C. Size, Charge, Solvent and Ligand Effects pp Reaction Entropies

1. Introduction
Relative entropies of redox reactions were widely measured in the

1950’s and 19602s in order to examine basic notions concerning ionic

281-287

salvation. Interest in this topic was revived in 1979 with the

report by Weaver and co-workers that absolute measures of the entropy

0

0 A 0 - - 0 m a
difference 31“: ( sred sax) between the two ions forming a redox

couple could readily be obtained from nonisothermal electrochemical

experiments.55 Numerous papers on reaction entropies have appeared

since then.84,253-255,271.28l-304

These have tended to emphasize
either the value of such measurements in unraveling the details of

solvent reorganization in connection with electron transfer
55.84,253-255.293,294,297,304

dynamics or their usefulness for gaining
. . . . . 300-302 .
information concerning the salvation of metalloproteins, peptide
complexeszgs’299 and other biological model compounds.303 Although

many insights have been gained into both problems, a number of puzzles
remain.

This section describes some empirical relations which have been
uncovered concerning the dependence of As:c values for simple tran-
sition metal couples on reactant size and charge and an the nature of
the solvent, and seeks to provide interpretations at the molecular
level. Besides offering predictive power. it is suggested that these
correlations and interpretations can rationalize some of the more

curious findings of earlier studies.

290

Z-EmLLta

The data examined here have been taken largely from previous

reportsss’84’89’253-255

although a fair number of new results are
included. (The new results are summarized in Appendix I). In each
case the redox couple is substitutionally inert in both oxidation
states.

Values of Asgc in. water, dimethylsulfoxide (DNSO) and
acetonitrile for several "3+/2+" couples are plotted against reactant
radius in Figure 5.7. For spherically nonsymmetrical complexes (e.g.

3+/2+)

Ru(NH3)5py , the effective radius is taken as equal to half of the

cube root of the product of the diameters along the three ligand-metal-
ligand axes.28 Excluded from the comparisons at this point are redox
couples where a difference in spin state occurs between the oxidized
and reduced forms. It is evident that there is a good linear fit of
the data in water. with the exception of the results for three couples
containing aqua ligands. (These three could not be examined in the
other solvents). In acetonitrile and DHSO the reaction entropy also
varies with -r, but apparently not in a strictly linear fashion.
Somewhat better linear correlations, at least in nonaqueous solvents,
are found, with l/r (Figure 5.8). Similar correlations were obtained
in solvents other than the three for which data are shown, but these
are omitted from the plots for clarity.

The dependence of As:c on charge was examined by monitoring
consecutive reduction reactions of ruthenium- and chromium tris

bipyridine complexes. For both of these, at least four redox states

are accessible in acetonitrile and acetone. Figure 5.9 shows that the

291

 

 

 

 

i
zoo~ ‘ ‘ _
12
(3
1225
7
"r fig C:
3 9A 4
E!
E! ‘A 1:
—- s
1 14 (a
8’ 8 4‘ 2
-:: 1()()r- ‘3 -
'-5 12
b 8 ‘8
‘2‘;_ 11 g;
(a
(D ()7
4
‘°<>5(5 1
<3
223
()- 3
L 1 1 1'7
13 '7

ES
r,A

Figure 5.7. Reaction entropy versus effective radius of reactant.

Key to solvents:

Key to reactants:

(0) water; (A) dimethylsulfoxide; (D) acetonitrile.

+

(1) c:(bpy)33+; (2) Fe(bpy)33 ; (3) Ru(bPY)33+;

(4) c-Ru(NH3)2(bPY)23+; (5) c-Ru(H20)2(bPY)23+; (6) c-Ru(n20)2(bpy),3+;

3+
(7) Ru(NH3)4bpy

3+ 3+
(11) RU(NH3)6 ; (12) 05(NH3)6 _: (13) Ru(NH3)5H20

3+

; (8) Ru(NH3)aphen : <9) Ru<en>33+i (1°) Ru(“3’5”

3+.

3+ 3+
(14) Ru(NH3)4(H20)2 ; (15) R“(H20)6 .

3+

292

 

200

7 /
/9///
T 4/’/ 95/]
'5 ,0 A/
s- /D // 8
(u /’
'5 100 b /
fl
. a 8 /9/
(1) 7° /
C)
< /// 9
4 l’,/’

 

 

Figure 5.8. Reaction entropy versus l/r. Keys to solvents and

reactants as in Figure 5.7.

 

J (:leg“I mol‘I

o
I'C'

AS

293

 

100 "

0|
(3
l

 

 

 

2 2
zox' 2 red

Figure 5.9. Reaction entropy for ruthenium tris bipyridine couples in

acetonitrile versus the difference in the squares of the charges on the

oxidized and reduced states.

294

reaction entr0py for Ru(b1>y)§"'un in acetonitrile varies with (lied
-z§x), where 2°x and zred are the charges of the oxidized and reduced
states of each couple. Unfortunately. there are only a few complexes
and solvents for which consecutive electron transfer reactions can be
examined.

The connection between reaction entropies and the nature of the
solvent is illustrated in Figures 5.10 and 5.11. Linear correlations
are found between A8:c and the so-called solvent acceptor number.277
regardless of the reactant charge, size, electronic state or ligand

composition. The As:c values for Ru(NH3)SN032+/+

are taken from the
dissertation of Dr. Saeed Sahami,89 while those for
Co(EFNE-axosar-H)2+/+ (structure in Figure 5.12) were measured by Dr.

Peter Lay and this author.

3. Discussion

The simplest theoretical treatment of reaction entropies is based

on the Born electrostatic model in which the solvent is treated as a

continuum.252 According to this model:
a _ 2 2 _ 2
Asre (e Nl2e'l'r)(dlr1e/d'lf)(zox zred) (5.8)

where e is the dielectric constant of the solvent. Although this
approach has rightly been criticized for failing to provide accurate
overall estimates of Asgc, the empirical analyses confirm the predicted
variations with 1/r and (22x - 22 ). The chief problem seems to lie

red
in the representation of the solvent.

295

 

 

 

 

 

(J l l I

0 ' 20 40 60
Solvent Acceptor Number

Figure 5.10. Reaction entrapy versus solvent acceptor number. Key to

reactants: (I) Cr<bpy)33+. (0) RU(en)33+. (o) Ru(NH3)63+. (e) c6(en)33+'

(A) Co(sepulchrate)3+. Solvents (and acceptor numbers): water (55),
formamide (40), N-methylformamide (31), nitromethane (20.5), acetoni-
trile (19), dimethylsulfoxide (l9), propylenecarbonate (18), dimethyl-

formamide (16), acetone (12.5).

296

 

 

 

 

 

i r I
200- .
O
O
A O
I. A o
E 100- 7
_ O
8’
-o
'5
,9
(D - ..
d
O J l l
0 20 4O 60

Solvent Acceptor Number

Figure 5.11. Reaction entropy versus solvent acceptor number. Key to

reactants: (C9 ferricinium. GD) Co(EFME-oxosar-H)2+, (A) Ru(NH3)5NCSZ+.

Solvents as in Figure 5.10, plus methanol (41).

297

 

 

 

Figure 5.12. Structure of Co(EFM£-oxosar-H)2+.

298

Evidently specific donor and acceptor interactions, rather than
nonspecific Bornian effects, are primarily responsible for the changes
in reaction entropy as the solvent is varied. We have previously
suggested that such interactions might involve solvent molecules as

291

acceptors and metal complexes as donors. Although plausible for

redox couples containing electron-rich ligands such as bipyridine or
cyclopentadiene, such behavior is hardly possible for couples such as
Ru(N83)g+/2+ which contain acidic ligands and act instead as electron
acceptors.*

The alternative to solvent-ligand interactions would be solvent-
solvent interactions, which would account for the insensitivity of the
reaction entropy-acceptor number correlation to the nature of the redox
couple. If strong donor-acceptor interactions induce significant
intermolecular solvent structuring, a straightforward explanation of
our empirical observations emerges. Highly structured solvents will
experience a loss of order. at substantial entropic expense, when
disrupted by short-range complex ion-solvent dipole (NOT donor-

276

acceptor) interactions. (Ion-dipole interactions are suggested in

order to account for the observed size and charge dependence of asgc).
The “entropy of disruption" will be greater with ions of higher charge,
yielding a negative contribution (for cationic couples) to A80 (.30

re red

- 32x)“ This contribution will be less significant in less structured

 

*The electron-accepting tendencies of ruthenium(III) ammine complexes
are demonstrated quite convincingly by Curtis, Sullivan and Meyer’s
solvatochromic experiments which show' that such complexes are
selectively solvated by strongly basic (i.e. electron-donating)
solvents when placed in mixed media (reference 305).

299
solvents. On the other hand, an increase in order and decrease in

entropy will result if a complex ion is able to orient disrupted
solvent molecules, again via charge-dipole interactions. Ions of
higher charge should be more effective and therefore, a positive

contribution to [18° (.3"

o . . .
to red 80‘) can be expected. This contribution

should be greatest in unstructured solvents where reorientation of
solvent molecules will be relatively unhindered. Note that together
the two effects predict that the largest positive Asic values will be
found with highly charged cationic redox couples in unstructured
solvents, while smaller or even negative values (e.g. ferrocene]
ferricinium in water)“ would be predicted in more structured solvents.
It is suggested that increases in solvent accepting capabilities are
parallelled closely by increases in solvent structuring and order,291
thereby accounting for the decrease of 43:6 with acceptor number. One
might expect that some combination of donor and acceptor properties
would lead to a better description of the degree of solvent ordering
and structure, but apparently this is not the case.

In light of these abervations and interpretations it is
useful to consider some specific reactions. Several authors have
commented on the very small reaction entropies for tris bipyridine
couples in water. Explanations are often sought in terms of
"hydrophobic" interactions between the aromatic ligands and water
molecules.306 However, the correlations in Figures 5.7 and 5.8
indicate that the small As:c values reflect simply the large size of
such couples in comparison to most others. One suspects that the

301,306

negative 433‘: values comonly observed for metalloproteins might

also represent size effects rather than hydrophobic interactions.

300

(This is not to deny the possible importance of hydrophobic
interactions to ionic salvation, but rather to point out that these

need not be invoked to account for entropy differences between oxidation

states).
On the other hand, the Ru(NH3)3+/2+ and Ru(H O)3+/2+ couples are
6 2 6
closely similar in size, yet exhibit very different reaction entro-

pies.55

From the empirical correlations of size with entropy change,
it is evident that it is the aqua couple which behaves anomalously. In
this case, as well as with other aqua couples, the idea of specific
ligand-solvent interactions (most likely, hydrogen-banding) probably
does represent the best explanation of the unusual behavior.

Various authors have noted that A8:c values for "mixed ligand"
complexes can be estimated approximately by linearly interpolating from
the values for complexes possessing homogeneous ligand. compo-
sitions.84’281'282 It has been suggested that each ligand provides an
additive contribution to A826. It now seems that the influences of
individual ligands can best be understood in terms of their effects on
the overall charge and size of a redox couple. Such effects may not be
strictly additive and the influence of a particular ligand should
depend on the size and charge of the complex to which it is being
added. A stringent test of these ideas would be to compare reaction
entropies for redox couples of diverse charge, size and structure. In

Figure 5.13 A3:c values for Fe(CN)63-/4P, Fe(CN)4bpy-/2-,

ferricinium/ferrocene, Ru(NH3)5NCSZ+/+, Ru(NH3)SClZ+/+, Ru(en)§+/2+.

3+/2+ 3+/2+

Ru(NH3)6 and Cr(bpy)3 , all in water. are plotted against (z:x -

Zied)/r. A.fairly reasonable linear correlation is observed, which

301

suggests that reaction entropies can be predicted with useful accuracy.
merely from the size and charge of a redox couple. Note that a nega-
tive. rather than a zero, entropy intercept is found, in disagreement
with the expectations from continuum models.

The llr dependence of Asgc appears also to account at least
partially for the approximate correlation between the reaction entropy

and self-exchange reactivity for several redox couples in water.288

13 the outer-shell reorganization

Thus, according to Narcus’. theory
energy for electron transfer also depends on l/r. An increase in r
will decrease the outer-shell barrier thereby increasing the exchange
rate and yielding an inverse correlation between the log of the
exchange rate constant and the reaction entropy, when other factors can
be neglected. With regard to electron transfer energetics the findings
concerning the solvent dependence of As:c and the shortcomings of the
Born model hint that a molecular approach is needed for understanding
not only the thermodynamics of solvent reorganization but also for the
static component, at least, of the nonequilibrium kinetic process.
Finally, it is worthwhile to consider the cobalt(III)/(II)
couples. Although the number and variety of such redox couples are
necessarily limited, the indications are that these also exhibit a
charge and size sensitivity. Thus, the A8:c values for Co(en)?”2+ are
)3+/2+ )3+/2+
3 3
/

greater than those for the larger Co(phen and Co(bpy

couples. The values for Co(EFME-oxosar-H)2+ + generally are smaller
than for the more highly charged parent sepulchrate complex. Reaction
entropies for these complexes are essentially always about 80 J deg-1

mole“1 greater than those for similar complexes containing different

302

 

 

 

 

1' i()()" 8 d
‘75 6 7 "
£5 5:; (e
To: 4
{‘3’ O " 3 . ‘
.1, 1.
Omb
<1 -100- 2 .,
1.
1
1.
-42CK)- ‘
-2 1—1 O i 2
2 2
(zox- Zred)/r

2 -22 )/radius. Key

Figure 5.13. Reaction entropy in water versus (Zox red

to reactants: (1) Fe(CN)63-; (2) Fe(CN)4bpy2-; (3) ferricinium;

(a) Cr(bpy)33+; (5) Ru(NH3)SC12+; (6) Ru(NH3)SNCSZ+; (7) Ru(en)33+;

3+
(8) Ru(NH3)6 .

303

metal centers. The distinguishing feature of the cobalt(III)/(II)
reactions is the change of spin multiplicity. One is almost forced to

1 l

conclude that the extra 80 J deg- mol- of reaction entropy must be

associated directly with the change of electronic spin state.294
Inter93t108171 the cobalt(II)/(I) bipyridine couple, which is low-spin

in both oxidation states, yields the same reaction entropy in

acetonitrile as the Cr(bby)§+/+ couple.

4. Conclusions

Empirical correlations of the reaction entropies for transition
metal couples are found with the inverse of the effective radius of the
redox couple and with the difference of the squares of the charges on
the two ions comprising the redox couples. The solvent dependence of
Asgc is described by Gutman’s acceptor number. The size and charge
correlations are expected on the basis of the Born model. However. the
solvent dependence requires a molecular rather than a continuum repre-
sentation of the solvent. Solvent disruption and reorientation effects
can adequately account for most of the entropy data. A number of
interesting observations can be rationalized. Little evidence is found
for specific donor-acceptor (hydrogen bonding) interactions between
redox couples and solvent molecules, except in the case of aqua couples

in water.

304

CHAPTER VI

APPLICATIONS OF THE RELATIVE ELECTRON-TRANSFER THEORY

Bennett has emphasized that there are actually two "electron-
transfer theories“: the absolute theory which enables rate constants
to be calculated from structural parameters and the relative theory

307 The

which emphasizes the predictive power of reactivity patterns.
latter approach exploits the conceptual similarities between related
homogeneous and electrochemical reactions and between homogeneous self-
exchange and cross reactions, and is embodied in the so-called Marcus
cross relations. The success of this approach relies on a type of
ideal behavior - sometimes termed the “weak overlap limit" - in which
isolated reactants retain their same degree of intrinsic reativity when
combined together in bimolecular reactions. The weak overlap limit
appears to provide a fairly reliable guide to outer-sphere reactivity
in various environments, although it sometimes breaks down for homo-
geneous inner-sphere reactions. on the assumption that outer-sphere
electrochemical reactions at mercury approximately achieve this ideal

behavior. the relative electron-transfer theory is applied to some

problems involving homogeneous redox reactions.

305

A. Electrochemical and Homogeneous Exchapge Kinetics for

Transition-Metal Aguo Couples: Anomalous Behavior
2; gexaaguo IrongIIIMSII)

[Originally published in Inorg. Chem., 2;, 2557 (1983)]

1. Introduction

Recent developments in the theory of outer-sphere electron
transfer have focused attention on the contribution of inner-shell
reorganization, A623, to the intrinsic free energy barrier for electron

14,2315691250’307'308 Transition-metal redox couples

exchange Mix.
containing only aqua ligands, of the type 11:; + (+11: :1 where N I Ru,
Fe, Ca, V, and Eu form an especially interesting series in which to
compare the theoretical predictions with experiment since they exhibit
large differences in redox reactivity which are likely to be due
primarily to the influences of the metal electronic structure upon
AOL. Some of these reactions exhibit remarkably small exchange
rates.309

Experimental estimates of AG:x can be obtained from the rates of
homogeneous self exchange or electrochemical exchange, or, less
directly, from the kinetics of suitable homogeneous cross reactions
with other redox couples having self-exchange kinetics that are known

13,180

or can be estimated. Relationships between the kinetics of these

reactions are given by the well-known equations derived from an

adiabatic electronrtransfer model:13’180

306

(k: ”/3 ) < (kgx/Ah)1/2 (4.15)
and

kgz- (k:1 ngxlzr)1/2 (6.1a)
where

log f - (108 K1,) 2/[4 108(k11k 221613)] (6.1b)

In Equation 4.15. kg: and kzx are the corresponding rate constants for
electrochemical exchange and homogeneous self exchange for a given
redox couple, and Ae and Ah are the electrochemical and homogeneous
frequency factors, respectively. In Equation 6.1, kIl and kgz are the
rate constants for the parent self-exchange reactions corresponding to
a homogeneous cross reaction having a rate constant kIl and equilibrium
constant K12. All these rate constants are presumed to be corrected
for work teams. A given rate constant for homogeneous self-exchange
13,23

*
k:: can be related to the corresponding intrinsic barrier AGex by

k Ahexp(-AG:x/RT) (6.2)

where Kis an electronic transmission coefficient. The rate constant

e
for electrochemical exchange kg: can also be related to AC x by

k3, ' A ,Gem- ,IRT) (6.3a)

307

e _ _ *
kex Aeexpl (Acex + c)/2a'r] (6.3b)
where AG* is the intrinsic electrochemical barrier. The factor 2

ex,e

arises in Equation 6.3b because only one reactant is required to be
activated in the electrochemical reaction, rather than a pair of
reactants as in the homogeneous case. The contributions to Aczx and
Aczx arising from inner-shell (i.e. metal-ligand) reorganization, AG:s
andAG:a e‘ respectively. are therefore related by AG:B I 2A6? . The

8 15.3

e *
relationship between the components of AG ex and AGex due to outer-
9

e *
shell (i.e. solvent) reorganization, AGO8 and AGO8 , is somewhat less
9

straightforward. According to a dielectric cantinuum.treatment these

quantities are given by13.180
AG* . ez(_1_ -l )(l_ l (6 4a)
0’ 4 a Rh Eop es
Ac* . 9_Z(1_.. l.)(1__ _1_) (6 4b)

where e is the electronic charge, a is the (average) reactant radius,
gap and 68 are the optical and static dielectric constants, Rh is the
close contact distance between the homogeneous redox centers, and Re is
twice the distance between the reactant and the electrode surface.
Since it is generally expected for outer-sphere reactions that Re >
Rh, therefore “6:. < 2Ast’e. The quantity C in Equation 6.3b accounts

for this inequality which is also the origin of the inequality sign in

Equation 4.15; from Equations 6.3 and 6.4

308

2
e l l 1 l
c - ( --— )(- - —) (6.5)
'4 'Eh Re Ebp E:s
Equation 4.15 can therefore be written in the more general form
e I h -
2 108 (hex/19) log (kex/Ah) c/2.303 RT (6.6)

For a series of reactants having a similar size and structure, as for
the present aqua couples, Rb and Re and hence C should remain
approximately constant.

It is desirable to obtain a self-consistent set of experimental
values of kg: or kg: for comparison with theoretical predictions
obtained from calculated values of Aczx using Equations 6.2 and 6.3.

This task is less straightforward than is commonly presumed for two

reasons. Firstly, the experimental values of kzx or kzx may not refer

3+/2+

) at least one of the
“Q

to outer-sphere pathways since (except for Ru
aqua reaction partners is substitutionally labile so that more facile
inner-sphere pathways may provide the dominant mechanism. Secondly,
values 0f 1&2: derived using Equation 6.1 from rate constants for
appropriate outer-sphere cross reactions rely not only on the
availability of values of k:x for the coreacting redox couple along
with values of K12, but depend also on the applicability of this

relation.3m‘311

The resulting estimates of kg: for different redox
couples are often difficult to compare since large systematic errors
can be introduced by the use of cross-reaction data involving

structurally different coreactants, inappropriate electrode potential

data. etc.

309

In spite of their direct relationship to the desired intrinsic
barriers, electrochemical exchange rate data have seldom been utilized
for this purpose. One reason is that these data have commonly been
gathered at ill-defined solid surfaces where the work terms arising
from double-layer effects are large and unknown, precluding quan-
titative intercomparison of the results. However, Weaver and coworkers
have determined accurate electrochemical rate data for Vi+l2+, Cr3+/2+

8Q
Buzz/2+. Ruiz/2+ and Feizn+ at the mercury-aqueous interface under

conditions where the work terms are small and can be estimated with

228’312‘313 The interactions between the reactant and the

confidence.
metal surface are likely to be weak and nonspecific, so that the
electrode can be viewed as an inert electron source or sink that does
not influence the electronrtransfer barrier. This allows information
on the electronrtransfer barriers to be gathered for individual redox
couples as a function of the thermodynamic driving force. Such
information is largely inaccessible from the kinetics of homogeneous
electron transfer.

In the present section, suitable rate data for electrochemical
and homogeneous reactions involving aqua redox couples are analyzed and
compared using Equations 6.1 and 6.6 in order to ascertain as unambig-

uously as possible how the kinetics of outer-sphere electron exchange

depend on the metal redox center.

2. Rap; Constants for Electrochemical Exchapge

Table 6.1 contains a summary of rate parameters for the

electrochemical exchange of Roy/2+, V3+l2+, Fey/2+ u3+l2+.
aq aq as a

, E and

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311

Orig/2+ at the mercury-aqueous interface, using potassium hexa-
fluorophosphate and lanthanum perchlorate supporting electrolytes.
These experimental conditions minimized the extent of the electrostatic
double-layer effect upon the apparent rate constants for electro-
chemical exchange k:x(app) (i.e. the "standard" rate constants measured
at the formal potential Ef for the redox couple concerned), enabling
values of the work-corrected rate constants kg: to be evaluated with

confidence using314

e F
ln kex ln ke‘(app) +RT(zr-acorr) 4) d (6.7)
where z r is the reactant charge number. a corr is the work-corrected

cathodic transfer coefficient, and ¢ (1 is the potential drop across the
diffuse layer. Details of this procedure are given in references 312
and 313. The KPF6 electrolyte provides an especially suitable medium
for this purpose. This is because (id is small over a wide potential
range positive of the potential of zero charge (-440 mV vs. -s.c.e.)
since the positive electronic charge density at the electrode is
matched approximately by the charge density due to specifically
adsorbed PF; anions.313’314 The rate data in Table 6.1 all refer to
acid-independent pathways. In contrast to homogeneous reactions
between aqua cations, the rates of most of these reactions are
independent of pH at values (<pH 2.5) below which the formation of
hydroxo complexes is unimportant. The single exception is Viz/2+,

which exhibits a significant inverse acid-dependent pathway at pH>l.315

312

3+/2+

The formal potential for Feaq

is too positive (495 mV vs.
s.c.e. in 0.4 NLKPFé) to allow rate measurements in the vicinity of Ef

to be made at mercury since anodic dissolution of the electrode occurs

3+
aq

beyond about 375 mV. However. the electroreduction of Fe was found

by Tyma313 to be sufficiently irreversible so that cathodic d.c. and

normal pulse polarograms were obtained over the potential range 300 to

4 l

0 mV, yielding values of kapp in 0.4 §,KPF to 4 x 10- cm s- at 300

l

6

mV, and 0.1 cm s- at 0 mV vs. s.c.e. Extrapolation of the cathodic

Tafel plots, (i.e. ln k: vs. E, where k: is the apparent cathodic

PP PP
rate constant), was therefore required in order to extract k:‘(apP).

However. this procedure can be applied with confidence: the work-

corrected cathodic transfer coefficients “cart for several other aqua

couples are close to 0.50 (10.02) over a wide range of cathodic over-

313 3+

potentials. aq

The observed transfer coefficientsapp (0.48) for Fe
reduction in 0.4,§._KPF6 indicates that the potential dependence of the
double-layer effects is likely to be small, as expected. Consequently.
the resulting value of k:‘(app), 2x10.5 cm s-l, is likely to be within
a factor of 2- to 5-fold of kg‘; we have therefore set an upper limit

-1

of lxlO-4 cm s for kg: in Table 6.1.

These values of k:x(app) and k:x are smaller than those commonly

i;l2+ at platinum.and gold electrodes.316a However.

reported for Fe
cathodic voltammograms that are highly irreversible (half-wave
potential E1/230 mV vs. s.c.e.), yielding similarly small values of
k:x(app) ("10"5 cm s-l), have recently been obtained at platinum and
gold in perchlorate media from which halide impurities had been

316b

rigorously excluded. The larger values of k:x(app) are therefore

313

due to the presence of halide-catalyzed, possibly inner-sphere
pathways. Rate measurements at dropping mercury electrodes are not
susceptible to such difficulties since the surface is continuously

renewed and adsorbs most anions much more weakly than do noble metals.

3+/2+ 3+/2+

The electrochemical reactivities of Feaq and Ruaq provide
an interesting comparison. At the formal potential for Ruiz/2+ in 0.4
.§,KPF6. -20 mN vs. s.c.e., the observed rate constant for electro-

3+

aq is 0.15 (10.05) cm s-l. This value is only mod-

3+
aq

reduction of Fe

erately (30-fold) larger than that for Ru

potential, 5(32) x 10-3 cm s-l, despite the enormous cathodic over-

reduction at the same

potential (515 mV, corresponding to an equilibrium constant of 5 x 108)
for the former reaction. Since these rate constants were obtained
under the same conditions and the reactants are of very similar

structure, the work terms should be essentially identical. Any reason-
able driving force correction for Fez; reduction therefore must yield a
value of kzx ca. 103-fold smaller than for Rug; reduction, irrespective

of the work term corrections upon the individual rate constants. From

h

Equation 6.7, this results in a corresponding estimate of kex

that is

ca. lOS-fold smaller for the former reaction (vide infra).

 

 

3. Rate Constants for Electron Exchange from Homogeneous

Cross-Reaction Kinetics

Table 6.2 summarizes pertinent rate and equilibrium data for the

3+/2+

acid-independent pathways for cross reactions involving Feaq ,

Ru3+l2+. V3+l2+, Eu3+I2+ and Cr3+l2+
aq aq a a

for these couples resulting from the application of Equation 6.2. The

, together with the values of k2:

314

 

 

 

 

 

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316

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317

rate constants for the cross reactions, k?2’ and for self exchange of

the various coreactants, 1:32, listed in Table II are taken from
literature data; they are corrected for Debye-Huckel work terms as
described in reference 311 (see the footnotes to Table 6.2 and
reference 311 for the data sources). The measured values of kEZ’
k?2(app), are also listed, with the ionic strength at which they were
determined in parentheses. The equilibrium constants K12 given in
Table 6.2 were obtained from measurements of formal potentials for the
appropriate redox couples at ionic strengths comparable (0.1 <p<1) to
those at which the kinetic data were gathered. (Most of these formal
potentials were obtained in our labotatory using cyclic voltammetry.
The uniform use of these data avoids the substantial errors that can

arise if literature electrode potential values gathered under disparate

conditions are employed to calculate K12, as is commonly done.)

Although values of kg: for some of the polypyridine redox couples

3+l2+ 3+/2+
3 3

Cr(bpy)§+/2+ (bpy - 2,2’bipyridine. phen - l, lO-phenanthroline)] have

utilized in Table 6.2 [Fe(bpy) . Pe(phen) , Os(bpy)§+/2+, and

not been determined, they are assumed to be comparable to that obtained
for Ru(bpy)g+/2+, ca. 1 x 109 !:1.310’317 The inner-shell reorgan-
ization energy for all these couples are likely to be close to zero
since they uniformly involve the transfer of a delocalized t

28
electron.310

Detailed inspection of Table 6.2 reveals that work-corrected
values of k2: are obtained for each aquo redox couple that are reason-
ably constant (mostly within ca. tenfold of each other), at least using

cross reactions having relatively small driving forces (say K12 (l x

318

106, 90.2).311

There are good reasons to prefer such weakly exoergic
cross reactions for extracting values of k2: using Equation 6.2. The
assumptions made in deriving Equation 6.2 that the reactions are adia-
batic ( K :1), the work terms are nonspecific and the reactant and
product free energy profiles are harmonic are questionable, especially
for reactions involving aquo cations. However, these factors have only
a minor influence upon its applicability for reactions having small
driving forces providing that the work terms and are comparable for

310,311

the corresponding self-exchange and cross reactions. Indeed,

progressively smaller estimates of k2: are generally determined using
Equation 6.1 from cross reactions having increasingly large driving
forces. These discrepancies have been attributed to the influence of

unfavorable specific work terms,311 249

310

anharmonicity. and

nonadiabaticity.

2+/2+, and

The extraction of relative values of kg: for Viz/2+. Eu
Griz/2+ is facilitated by the proximity of the formal potentials for
these couples (~472 my, -625 mV, and -655 mv vs. s.c.e.,u-‘O.5, Table

6.1). Thus especially reliable values of k2: can be determined from

cross reactions with similarly small driving forces involving common

318

coreactants having known self-exchange kinetics. Suitable oxidants

for this purpose are Co(en)3+, Ru(NE3)2+. Viz. Careful inspection of

Eu:+/2+ 3+/2+

Table 6.2 yields average values of kh for V3+l2+,
ex aq aq
of ca. 5(34) x 10-2, 2(31.5) x 10-4, and 2 x 10D6 11-1 s-1

, and Cr

respectively.

3+/2+

The estimation of kh for Ru
ex aq

self exchange in the same

manner is slightly less straightforward on account of the more positive

319

formal potential for this couple (-20 mV vs. s.c.e., u - 0.4, Table

6.1). values of k2: obtained from suitable cross reactions (f>0.2).

- . 2+ 2+ 3+ 3+ . 3+

1nvolv1ng Vaq, Ru(NE3)6 , Co(phen)3 Ru(NE3)5py , and Ru(Nfl3)sisn as

coreactants, vary by a factor of almost 200 (Table 6.2). However, the
l

relatively low value (0.4‘gf s-l) obtained using Co(phen);+ is also

characteristic of cross reactions involving this oxidant with Viz,

Eu2+. and Cr2+ (Table 6.2). Also, the estimate of kh obtained from
aq aq ex

1

the Ru3+ - Vi; reaction (z‘gf s-1) is likely to be too small since the

3Q

free energy barrier for Vi; oxidation appears to respond to changes in
driving force to a noticeably smaller extent than predicted from
Equation 6.2. The remaining cross reactions yield a reasonably consis-

tent estimate of kgx for Ruiz/2+ of ca. 50(120)§._-1 s-l. (A somewhat

larger estimate of kzx, ca. 200 gfl s-1

319

. was arrived at previously from

but involving extrapolation of values of k:

tained from cross reactions having highly varying driving forces.304)

related arguments, x ob-
The "observed" value of kzx for Viz/2+. 3 x 10-2 firl s-l, (i.e.
that obtained directly from the observed self-exchange kinetics) is
close to the estimates obtained from cross reactions with Co(en)§+ and
3+/2+
D

Ru(NE )3+. In contrast, the "observed“ value of kh for Fe
3 6 ex aq

. is considerably (104- to 105 fold) larger than those derived

15
”-1 s-1

from cross reactions having suitably small driving forces (f>0.2)

(Table 6.2). Equation 6.1 can be rewritten as

h
12

h

h
log k 0.5(log k11 + log kzz) + 0.5 log K

2
12 + (log K12) /

h h 2
4 log(k11k22/Ah) (6.8)

320

Figure 6.1. Plot of (2 log kh - log'khz ) vs. [log K + (log K12)2/

12
4 log (kh / )] for homogeneous cross reactions involving Fe3+l2+
11k 22 2h aq ’

12

calculated as Fe:q oxidations. k2x for low spin M(III)/(II) polypyridines

taken as l x 109 £71 sec.l (see text). kh refers to Fe:+/2+self exchange;

11
kgz to self exchange for coreactant couples. Zh assumed to equal 1 x 1012

gfl sec-l; value of kgl required for plot obtained by iteration: kh -

11
10'.3 M71 sec-1. Data sources given in Table 6.2 or reference 311 unless

otherwise stated. Closed points refer to cross reactions; Open point to

3+/2+ 3+ 3+,

"observed" Fea self exchange. Key to oxidants: l. Ruaq; 2. Euaq,

3. or3+; a. 3+82+; 5. v3+; 6. Ru (NH 3)2 ; 7. Ru(en)2+ ; 8. Ru(NH3)spy3+;

aq
3+

9. Ru(bpy)3+ 10. Ru(NH3)Snic3+, reference 320; ll. Ru(NH3éisn ;

3 ;
12. Ru(NH3)4bpy3+, reference 310; 13. Os(bpy)g+, reference 334;

14. Fe(bpy)§+, reference 335; 15. Fe(phen)2+, reference 336;

16. Co(phen)2+, reference 337; [17-19 from reference 330];

17. Ru(terpy)3+; 18. Ru(phen)§+; 19. Ru(bpy)2(py):+; [20-23 from refer-
ence 338] 20. Os[5,5'-(C33)2bpy]§+; 21. Os(phen)2+; 22. Os(5-Cl-phen)2+,
23. Ru[5,5'-(CHB)2bpy]§+; [24-31 from reference 339]; .

24. Ru[3, 4, 7 ,8-(CH3)4phen]2 ; 25. Ru[3,5,6,8-(CH3)4phen]2+,

26. Ru[4, 7-(CH 3)2phen]: +; 27. Ru[4,4'-(CH3)2bpy]§+

28. Ru[5,6-(cu3)2phen]2 ; 29. Ru(S-cu3phen)2+; 30. Ru(s-cénsphen)2+,

31. Ru(S-Cl-phen)3+ ; 32. Fe:+.

321

 

 

 

 

~30-
-20.
.c a
A:
O
s -10 -
gxg :2
5;, IS 20
- o - 22" ’2'
N I7,l§39gfi‘ :22 £32
9 3° 23
{a
IO+ l l I 1 I
IO 0 -lo -20 -3o
2 h h 2
log Klz-t-[Oog K'Z) /4log(k"k22/Zh)]

Figure 6.1

 

322

h
Therefore a plot of (2 108 klz 12
h h

log(k11k22[A§)] should yield a straight line of slope l.0 with an

intercept equal to log kgl’ Figure 6.1 shows such a plot for 32

i;/2+, formally expressed as Pei; oxidations.

h 2
- log kzz) vs. [log K + (log K12) [4

reactions involving Fe

(The data sources are summarized in the footnotes; Ah is assumed to

12 “fl -1

equal 1 x 10 s . The value of kgl in the last term in Equation

6.8 was obtained by iteration; for most reactions choosing any reason-

4 l -l

able value of Rh in the range ca. 10- to 10 gf s led to essen-

ll

tially identical results.) The straight line of slope 1.0 in Figure

6.1 provides the best fit to the solid points which refer to the cross

reactions. The intercept, which equals log kg: for Fe:;/2*

ponds to k2: - 7 x 10'4‘gfl s-l. The open point, which refers to the

, corres-

"observed" value of k2: (ls‘gfl s-l), is clearly at variance with the
other points; yielding a discrepancy of over 104 -fold in k:x° Figure
6.2 shows a similar plot for cross reactions involving Viz/2+.

Although the data points are less numerous, the open point for Viz/2+

1

self exchange (k:x - 3 x 10-2.§f s-l) is consistent with the remaining

entries. Admittedly, the slope (0.9) of the best fit line in Figure

6.2 differs somewhat from unity; possible causes are discussed

elsewhere.249

3+/2+

Such a behavioral difference between Vi+l2+ and Feaq

310

has been

noted previously.
3+/2+
Fea

The striking discrepancies with Equation 6.2 for
were ascribed in part to especially facile reaction pathways
for self exchange of the cross-reaction partners that contain pyridine-
type ligands arising from interpenetration of the pyridine rings.

Since such interactions will be absent for the FeiZIZ+ cross reactions,

323

 

h
k22

h
'z-log

2log k

 

 

 

no . 5 <3 -5 -IO
log Kl2+[(log Kl2)2/ «og(lfi'|k;2/z§ )]

 

++
Figure 6.2 As for Figure 6.1 but for cross reactions involving v2 /2 ,
expressed as Vi: oxidations; kql refers to Viz/2+ self exchange. Data

sources From Table 6.2 or reference 311 unless otherwise stated. Key to

. 4+. 3+. 3+.
oxidants. 1. an, 2. Rnaq’ Ru(Nl-l3)6 , 4. Ru(NH3)5py

6. Co(bpy)2+; 7. Ru(NH3)Sisn3+, reference 321; 8. Co(phen)2+; 9. V

3+; 5. Co(en)2+;

3+
aq'

324

the resulting estimates of *2: for Fei+l2+ (ca. 10"4 to 10-.3 §_s-1)
were considered to be falsely small, the observed self-exchange rate
310

constant being presumed to reflect a “normal" outer-sphere pathway.
A difficulty in comparing data for cross reactions involving

FeiZIZ+ with the other aquo couples is that the formal potential for

Fei+l2+ is substantially more positive of those for the remaining aquo
couples. Therefore cross reactions having suitably small driving
forces inevitably involve different coreactants with the likelihood

that systematic errors in the applicability of Ehuations 6.1 could

occur. These errors could vitiate its use for obtaining even relative

3+/2+

values of the self-exchange kinetics of Feaq

with respect to the
other couples. However. the comparison of the kinetics of the
Os(bpy)g+-Fei; and Viz-Cflbpy);+ cross reactions provides a way of

circumventing this problem. Both these reactions have suitably small

driving forces [equilibrium constants of 80 and 1.2, respectively

(Table 6.2)] and the coreacting redox couples, Os(bpy)2+/2+ and
Cr(bpy)§+/2+. have not only the same ligand composition but are also

likely to have similarly small barriers to electron exchange since they

both involve electron acceptance into a delocalized t28 orbital.310

3+/2+
3

self exchange, k22 and k44, are equal and noting that the

Assuming therefore that the rate constants for Os(bpy)
)3+/2+
3

and
Cr(bpy
f terms are essentially unity, the ratio of the rate constants for

3+/2+ h h
Fea ex,Fe/kex,V’ can be found from the

and Viz/2+ self exchange, k
kinetics and thermodynamics data for the corresponding cross reaction

using (cf. Equation 6.1]:

325

you uonu ou uooamou new: xmx mo moans“ mum numosumvas mononucouma a“ mosao> .moauosax mwcmnoxolmamm aoum
:zauoouwv: vocamuno mumxomun 3% N Wmom was wm> now mosaw> .Auxmu momv H.o cowumsvm wcams A~.o_manmav
sump coauomoulmmouo Bonn vocaawm 0v .owsmnwmm mama msomaowoao: you usmumaoo sump vouuouuoolxuoz mwmuo><

a

.H.o manna aoum moomwuousw msoosvml>usouoa um swamnoxo Hmoaaozoouuomao you unnumsoo mush cmuoouuoolxuo3o

 

 

an an an
new Aoauwm x we Am-mm x “MN Amuwm x we oH x N cmuo
s- a- o- o- +N\+m
com Amuowaxxmwav Ac-mm H We OH x m swam
«I «I ml +N\+m
Aquoa x m.mV finned x m-v Am ov .AmnonN-v cm
x a . x : a
owe as suoa ov Hmfl_ muoa a sled x av +N\+mmm
A soaxm.~v As oaxov.x uoaxav as
N N N x
mma Naca o --oa Me «nos n m-oa a +~\+m>
. adv Aav Aav
oma as x m.~ mm on oH x swam
m N- N +~\+m
HIaoa Hummv 5 Hum Hmm. Hum mm. Hum Mm. Hum Eu maasoo momma
on o so am . no no as
m owe ex a v as o A“ we as a as 8 we

 

.maoauowvoum Hmofiuouomca
sues comaumaaoo was .00mm on mwsmnuxm souuomHm now museumsoo oumm wo >umaasm .m.o magma

326

.mm.oomwuowwu scum nwxmu mosHm> .m.o: : nuwsouum oHsOH um wosHa

Iumuov .oHasoo xovmu now haouuao GOHuomomm . + am How many on uomammu nuH3 Aonov xmx mo moHumu mum

numoauovaa mommsumwuma aw moch> .Auxmu momv mmHMMmu moamou .mm.o was m.o m:0Huo=vm Bonn uoaHsuno mum mou<
«

can mwo< skews .m ou< + mHo< I xou< use .HI§ HI: NHOH x m. m I < .H u u umnu awsHasmmm N. o coHuwsvm Bonn

vmustono xmx moa mosHa>m .+~\+m:m now umsu ou uomamou SUHS Ac. 0 GOHumsamvx ms mo mOHumu mum snowshoes:
mmmwnuamuma sH mous> .Auxmu ommv H Hoam .Hmox o. m u M was HIoom so nOH x H n o<. Hloom HIz NHOH N m. m
I dd uosu wsHasmms e. o soHumswm msHmsx Mx mo mous> waHvsonmouuoo aouw xmx mo mmumEHummo .+~\+m:m

327

h h _ ,
kex’Fe/kex’vhku/k”) (klz)zqu34/<k34>’-unxu (6.9)

3 1 -l

a" s 3

Inserting the experimental values k12 . 3 x 10 , k34
H-1 s-1

._ , K12 - 80, K34 - 1.2 (Table 6.2), into Equation 6.9 yields

h h h l

- ’ -2 - -1 .
ex,Fe ex,V 0'1," Taking kex’v t° be 5 ‘ 10 E 3 (9-4—9 m}
h 3

. _ - -l -l 322 . h
yields the result kex,Fe 7 x 10 g s . This value of kex for
+/2+

q
self-exchange kinetics.

. l x 10

/k

Fe: is again over 103 -fold smaller than that obtained from the

4. Cogparisgn Between Electrochemical and Homogeneous Exchaggg

We
Strong evidence supporting the validity of such smaller estimates

0f kg‘ Fe is obtained from the rate constants for electrochemical
9

exchange, kgx. Table 6.3 contains a summary of the "best fit" values

of kg: and kg: for each redox couple, along with estimates of kh kh

ex’ ex
(Equation 6.6), obtained from the corresponding values of kg: using

Equation 6.6. The values given in parentheses are ratios of kg: and

k2x (Equation 6.6) with respect to those for Ruiz/2+. (kh lkh

ex ex,Ru

).
The frequency factors Ab and Ae required in Equation 6.6 were estimated

10,18,186,323

from an "encounter preequilibrium" model using the

. 2 3 186 .
expressions Ah - 41rth 6rh vn/lo and Ae - Grevn, where N 18
Avogadrofls number, rh is the average distance between the homogeneous
redox centers in the transition state, Grh is the approximate range of
encounter distances (“reaction zone thickness") within which electron

transfer occurs, are is the corresponding reaction zone thickness close

to the electrode surface, and “n is the effective nuclear activation

328

 

 

 

 

 

Figure 6.3. Comparison of rate constants for electrochemical exchange
at mercury-aqueous interface, kzx (cm sec-1), with corresponding rate

1sec-1), taken from

constants for homogeneous self exchange, RZx (yf
Table 6.3. Closed points refer to values of REX obtained from homo-
geneous cross reactions; open points to those obtained from measured

self-exchange kinetics. The straight line is the relationship between

log ke and log kh expected from Equation 6.6.
ex ex

329

frequency.23

18,10’186

Inserting the anticipated values rh I 78, 5th I 5ré

_ 1 x 1013 -1 10,23

vn s for the present aquo couples into

these expressions yields Ah I 3.5 x 1012,§f1 s"1 and Ae I l x 105 cm

8-1. The value of C in Equation 6.6 was estimated to be 3.0 kcal.
mol-1 by inserting the values Rh I 7 2, Re I 13 2324 into Equation 6.5.
(Note that although there is some uncertainty in the appropriate
absolute values of both Rh and Re. this partially cancels in Equation

6.5).

The absolute as well as relative values of k:x (Equation 6.6) are

seen to be uniformly in good agreement with those values of 1‘21:

(Equation 6.6) obtained from homogeneous cross reactions. This is also
illustrated in Figure 6.3 as a plot of log k:x against log kzx. The
straight line represents the correlation predicted from Equation 6.6.
The solid points refer to values of kzx obtained from cross reactions,
and the open circles represent the values of kg: for Viz/2+ and Fei+l2+
obtained from the self-exchange kinetics. Although all the other
entries are consistent with this correlation, that obtained from the
Fe3+l2+ self-exchange kinetics is again about 104 -fold larger than

expected.

5. Qorrelation‘g§_ln§rinsic Barriers gig; Regctant Structure

As noted above, it is instructive to compare the variations in the
experimental values of kg: and kg: with the structural properties of
the redox couples. To a certain extent, the observed reactivity
sequence Eu“,2+ > viz/2+ > Fe3+l2+ > Eu3+l2+ ) Cr3+’2+ is consistent

aq aq aq 8Q
with structural expectations; the three most reactive couples all

330

involve the acceptance of the transferring electron into a t28 orbital
for which the required distortions of the metal-ligand geometry, and
hence AGES. are anticipated to be relatively small.309

It is stressed that the calculations to follow are only
approximate. with more detailed calculations to appear in Chapter VII.
The primary purpose here is to discover the relative reactivity se-
quence that is predicted by electronrtransfer theory on the basis of
structural information. The absolute values of calculated rate con-
stants may well be in error by one to three orders of magnitude due to
uncertainties in the values of critical parameters (e.g. metal-ligand
'stretching force constants) as well as to simplifications introduced in
the calculations. Nevertheless, such calculational errors should be
closely similar for each redox couple, so that a useful prediction
concerning the sequence of reactivity can still be obtained.

The calculation 0f AG; and hence ktelx or ke from electron-

ex

transfer theory requires quantitative information on the changes in the
metal-ligand bond distances, Aa. accompanying electron transfer.13’23

Although sufficiently reliable determinations of As are sparse, recent

X-ray diffraction measurements have established values for Ru(0n2)2+/2+
3+IZ+ °f 0'09 211 and 0'14 2.325’326 respectively. An

2)6
effective value of As for Cr(OHz)g+/2+ equal to 0.202 has also been

23

and Fe(OH

determined from solution EXAFS measurements.10 The relation

as; - nf2f3 mnznuzws) (6.10)

331

where n is the nwmber of metal-ligand bonds involved (six) and f2 and
f3 are the metal-ligand force constants in the divalent and trivalent

oxidation states, assuming that the corresponding metal-oxygen stretch-

ing frequencies are 390 cIII-1 and 490 cmfl, respectively, yields values
of AG? of 14.6, 35 and 72 kJ mol"1 for Ru3+l2+ Fe3+l2+, and Cr3+/2+,
is aq aq aq

respectively. Errors in these values may arise both from anharmonicity
of the potential-energy surfaces as well as from uncertainties in the

. . *
appropriate force constants. The effective values Of‘AGia are slightly

smaller as a result of nuclear tunneling23; the nuclear tunneling fac-

3+/2+, Fe3+/2+, and Cr3+/2+
aq 3Q 3Q

yield effective values of A028 equal to 13.6, 33.5 and 67.4

tors, Tn, of 1.5. 2 and 6 for Ru
10,23

, respect-

ively,
-l . . . a *
kJ mol , respectively. The outer-shell contribution to AGex’ A603,
can be estimated from Equation 6.4a again using the values a I 3.5 X,
a -
Rh I 7 8 yields AGO8 I 27 kJ mol 1. Inserting the resulting estimates

of Aczx into Equation 6.2 along with the above estimate of Ab, 3.5 x

lOlngrl s-l, and assuming that K I 1 yields the calculated values of

kh

to that for Ru

k:x(calc), listed in Table 6.3. Ratios of k:x(ca1c) with respect

3*’2*.<kh lkh

aq ex ex,Ru) (calc)’ are also listed in parentheses

in Table 6.3 alongside the corresponding experimental rate ratios.

For all three couples it is seen that k2: < k:x(calc), the

calculated values being about 3-4 orders of magnitude larger than both

kg: and k2: (Equation 6.6). The "chewed" value of kh for Fe3.~/2+

ex aq
h

h .
self exchange, k e‘.1;e(calc), which

ex,Fe‘ is substantially closer to k
might be viewed as evidence that this value corresponds more closely to

3+/2+ 10

the “true” intrinsic barrier for Feaq . However, the overriding

evidence suggests otherwise. Thus the calculated rate ratio,

332

3+/2+ 3+IZ+

h / h ) versus Euaq couples (3.5 x

(kex,Fe for the Fe

ex,Ru calc’ aq

10-4) is roughly comparable to the corresponding experimental rate
ratio obtained from cross reactions ( 2 x 10-5) and electrochemical
reactivities ( 3 x 10-5), but substantially smaller than that obtained

using the “observed” value of kh (0.3) (Table 6.3).

ex,Fe
3+/2+

Recent calculations suggest that the Feaq

self-exchange

reaction is significantly nonadiabatic (Kz 0.01) at the normal ion-ion

X 14.15

"contact" distance of 6.9 although some overlap of the reactant

15.17

cospheres seems feasible. Since the values of K will depend upon

the extent of electronic coupling with the coreactants, and hence upon

the electronic and ligand structure. such nonadiabaticy (K<<l) may

h

account for some disparities in. kex

obtained using different
coreactants; for example, the low values obtained here using

Co(phen)g+/2+ (vide supra). Nevertheless, although nonadiabatic

effects could account in part for the discrepancies between k:

 

x and
k:x(calc), their inclusion. is unlikely to increase the 'ratio

(uh lkh )

ex,Fe ex,Ru since the 4d ruthenium orbitals are more likely to

calc

couple effectively than the more compact 3d iron orbitals, thereby

yielding larger values for the Ruiz/2+ reactions. Consequently, the
h h

rate ratio k /k (0.3) obtained from the Fe3+l2+

ex,Fe ex,Ru aq self-exchange

kinetics is not consistent with these theoretical expectations.

It might be argued that the discrepancies k2! (Equation 6.6) <
k:x(calc) arise at least in part from nonadiabaticity of the electro-
chemical reactions, especially since current physical models of the
double layer suggest that the reactant-electrode distance at the plane

of closest approach may be approximately the same as the distance

333

separating the reacting pair in homogeneous outer-sphere

178,221.312

reactions. Although such reactions may be marginally

nonadiabatic, it is unlikely that K (<1. Thus the frequency factors

determined from the temperature dependence of rate constants at mercury

18,249 3+/2+
3‘!

are not especially anomalous. The Cr electrochemical
exchange reaction at mercury has been ascertained to be only marginally
nonadiabatic, K: 0.2 -0.5. from a comparison with the rates of closely
related inner-sphere electrode reactions (Section IV. 0). Even if the
electrochemical as well as the homogeneous reactions are indeed
nonadiabatic, the K values would appear to be roughly (to within, say,
a factor of ten) independent of a reaction environment for each redox
couple in order to account for the consistently good agreement with the
predictions of Equations 6.1 and 6.6. Since r is expected to be
sensitive to the nature of the donor and acceptor orbitals in each

3

reactant pair. this implies than: is unlikely to lie below circa 10- .

especially in view of the variety of orbital symmetries (tzg, e f)

180

8,
involved in the present systems.

A major reason for the observed discrepancies between kg! and k2:

(calc) could arise from a deficiency in the theoretical models used to
calculate AG:x and/or the work terms. In addition, the outer-shell
barrier AG:a could be larger than calculated from Equation 6.4 in both
homogeneous and electrochemical environments as a result of alterations
in short-range solvent polarization that are known to accompany

55.222.230

electron transfer. (See also Sections V. A).

h

Finally. it is instructive to compare the relative values of kex

and kg: with the reaction entropies asgc for the redox couples (Table

334

6.3).288 The latter values provide a monitor of the changes in solvent
polarization that accompany formation of Hi; from Hi;. The magnitude
of Asgc might be expected to parallel AG:8 and hence Aczx; larger

values of a should yield greater alterations in the electron density

327

of the aquo hydrogens and hence more extensive changes in the extent

of ligand-solvent hydrogen bonding induced by electron transfer.

Indeed, the observed sequence of AS0 values Buy/2+ < v3+l2+ < 393+Iz+
< Buzz/2+ < Cri+l2+ uniformly parallels the observed reactivity

3+IZ+

sequence. (Table 6.3), although yet again the value of kg: for Feaq

obtained from self-exchange data is not consistent with this trend.

6. Conclusions and Mechanistic Implications

Taken together. the above results provide strong support to the

328,329

suspicions noted previously that the intrinsic barrier to outer-

3+/2+
aq
anticipated from the measured self-exchange kinetics. Persuasive

sphere electron exchange for Fe is significantly greater than

evidence is provided by the observation that the electrochemical

3+/2+ is noticeably smaller than that of 8113+”+ a

reactivity of Fe
aq aq

nd

even Viz/2+ under the same or comparabha conditions at the mercury-

h

aqueous interface. and by the uniformly good agreement between kex

(Equation 6.6) and the values of k2: extracted from homogeneous cross
reactions having suitably small driving forces and structurally similar

coreactants. On this basis, the reported self-exchange rate constant

3+/2+
8q

for Fe may well be ca. 104 -fold larger than that corresponding to

the ”normal" outer-sphere intrinsic barrier for this couple in other

homogeneous and also electrochemical environments.

335

It remains to consider physical reasons for the enhanced

3+/2+

reactivity of the Fe
3Q

couple when undergoing self exchange. One
possibility is that the "observed“ value of k2! refers to a "normal"
outer-sphere pathway. the electrochemical and homogeneous cross-
reaction data corresponding to "abnormal", for example strongly
nonadiabatic, pathways that are all less favorable by about 103 to 105
-fold in k2x° While not beyond the bound of possibility, the weight of
evidence presented above would seem to disfavor strongly this explana-
tion. A more likely possibility is that the self-exchange reaction

proceeds via a more facile inner-sphere pathway. Although the presence

3+/2+

of an acid-independent pathway for Feaq

self exchange has recently
been confirmed by using mixed LiCloafEClOA electrolytes,10 this result
by no means eliminates the possibility that the observed pathway

involves a water-bridged inner-sphere transition state. The observed

acid-independent rate constant for 0033/2+

been demonstrated to be ca. 1012 -fold larger than that obtained from

self exchange has recently

cross-reaction data. and attributed to the presence of a water-bridged

pathway. It was suggested that the dominant presence of such pathways

3+/2+

may be confined to redox couples such as Co with extremely

positive standard potentials on the basis of a model where the inner-

sphere reaction coordinate involves metal-bridging ligand

homolysis.320‘330 Such a pathway would, all other factors being equal,

3+/2+
3Q

be less favorable for Fe self exchange since Fea:+ is considerably

less strong an oxidant than C031".320 However. this could easily be

offset by the manifold other factors that control the relative rates of

competing inner- and outer-sphere pathways.331 In any case, on this

336

basis water bridging is more likely for Fe:;/2+

332

self exchange than for

3+/2+
3q

the other aquo complexes considered here; aside from Ru which

is constrained to follow outer-sphere pathways all the other couples
considered here are ca. 1 V less strongly oxidizing than Fei+l2+.
Such pathways are clearly unavailable for cross reactions involving
substitutionally inert coreactants such as those in Table 6.2, so that
the Fei§l2+ reactivity within these environments should reflect that
for a "normal" outer-sphere pathway. Water- or hydroxo-bridged
pathways are also unlikely within electrochemical redox environments,
especially at mercury electrodes in view of the weak interaction

between water molecules and this surface.333

Regardless of the detailed reasons for the anomalous behavior of
Fei+l2+ self exchange it can be concluded that this couple is in some
respects a nonideal choice for the detailed comparisons between experi-
mental rate parameters and the predictions of contemporary theory.23
Nevertheless, the required structural information is becoming available
for a number of other redox couples,10 enabling such comparisons to be

made not only for self-exchangereactions,10 but also for a variety of

cross reactions and electrochemical processes.

337

B. Some Comparisons between the Energetics 2f Electrochemical and

Homogeneous Electron-Transfer Reactions
[Excerpted from ACS Symposium Series, 128, 181 (1982)]

1. .l2££2§222222
The kinetics of inorganic electrode reactions have long been the

subject of experimental study. The advances in methodology, both in

the precise treatment of mass transfer effects and the evolution of
electrochemical relaxation techniques, have allowed the kinetics of a
wide variety of electrode reactions to be studied. In addition,
double-layer structural data are becoming available for a wide range of
metal-electrolyte interfaces, which is enabling the kinetics of elec-
trode reactions to be explored quantitatively in a variety of inter-
facial environments. However. the electrode kinetics area is notice-
ably underdeveloped in comparison with homogeneous redox kinetics, not
only in terms of the availability of accurate kinetics data, but also
in the degree of molecular interpretation.

Nevertheless, simple electrochemical processes of the type
0x + e- (electrode,¢m)*'Red (1.12)

where both 0x and Red are solution species, form a valuable class of
reactions with which to study some fundamental features of electron
transfer in condensed media. Thus such processes involve the activa-

tion of only a single redox center, and the free energy driving force

338

can be continuously varied at a given temperature simply by altering
the metal-solution potential difference ¢In by means of an external
potential source. In addition, electrode surfaces may exert only a
weak electrostatic influence upon the energy state of the reacting
species, so that in some cases they could provide a good approximation
to the "outer-sphere, weak overlap“ limit described by conventional
electron-transfer theory. Electrochemical kinetics measurements
therefore provide a unique Opportunity to examine separately the
reaction energetics of individual redox couples ("half-reactions"),
which can only be studied in tandem in homogeneous solution. In this
section, some relationships between the kinetics of heterogeneous and
homogeneous redox processes are explored in order to illustrate the
utility of electrochemical kinetics and thermodynamics for gaining
fundamental insights into the energetics of outer-sphere electron

transfer.

2. Elgcprochemical Rate gormulppions

Similarly to homogeneous electron-transfer processes, one can
consider the observed electrochemical rate constant kob to be related
to the electrochemical free energy of reorganization for the elgppntary
electronrtransfer step AG* by

kob I A exp(-Hp)exp(-AG*/RT) (6.11)

where A is a frequency factor. and up is the work required to transport

the reactant from the bulk solution to a site sufficiently close to the

339

electrode surface ("precursor" or "pre-electrode" state) so that
thermal reorganization of the appropriate nuclear coordinates can
result in electron transfer. Also, for one-electron electroreduction
reactions (Equations 1.12) 136* can usefully be separated into

"intrinsic" and “thermodynamic" contributions according “30,245,340

AG* Acint + a[F(E E ) + We II ] (6.12)

se P

where E is the electrode potential at which k0b is measured, Ef is the
formal potential for the redox couple concerned, we is the work
required to transport the product from the bulk solution to the
"successor" state which is formed immediately following electron
transfer, a, is the (work-corrected) electrochemical transfer
coefficient, and AG? is the "intrinsic" free energy of activation

int,e
30

for electrochemical exchange. This last term equals AG* for the
particular case when the precursor and successor states have equal
energies, i.e. when the free energy driving force for the elementary
reaction [F(E - E) + we - Hp] equals zero. The electrochemical
transfer coefficient 0: reflects the extent to which AG* is altered when
this driving force is nonzero; a therefore provides valuable infor-
mation on the syImsetry properties of the elementary electron-transfer

barrier.341

It is conventional (and useful) to define a "work-corrected“ rate

constant kco that is related to kob at a given electrode potential by

rr

kcorr - kob e‘Pflwp "’ (ws " “pH/RT} (6.13)

340

This represents the value of kob that (hypothetically) would be

obtained at the same electrode potential if the work terms Np and We
both equalled zero. For outer-sphere reactions, the work terms can be
calculated approximately from.a knowledge of the average potential on
the reaction plane ¢rp’ since Np I ZF¢rp and Ha I (z - l)F¢rp, where z

is the reactantfs charge number. Equation 6.13 can then be written as
kcorr I k0b exp([(Z <1)F¢rp]/RT} (6.14)

Usually ¢rp is identified with the average potential across the diffuse
layer ¢2 as calculated from Gouy-Chapman theory using the diffuse-layer
charge density obtained from thermodynamic data. In view of the use-

fulness of kc it is also convenient to define a “work-corrected“

orr’

. . * . . . .
free energy of activation Ace at a given electrode potential, which is

related to kcorr by [cf. Equation 6.13]:

*
icon, - A exp(-Mela!) (6.15)

so that Equation 6.13 can be written simply as

t e f
A U A . - 0
Ge out”: + am E ) (6 16)
Therefore the value of k measured at Ef. i.e. the "standard" rate

corr

s . . . . . . *

c us an k . . .

o t t corr is directly related to the intrinsic barrier AGint,e
Consequently. information on the energetics of electron transfer

for individual redox couples ("half-reactions") can be extracted from

341

measurements of electrochemical rate constant-overpotential relation-
ships. (Similarly. thermodynamics measurements for half-reactions,
e.g. reaction entropies, can also yield information on electron-
transfer energetics for individual redox couples as shown in Chapter
V). Such electrochemical measurements can therefore provide informa-
tion on the contributions of each redox couple to the energetics of the
bimolecular homogeneous reactions which is unobtainable from ordinary

chemical thermodynamics and kinetics measurements.

3. Relation between Electrochpmi cal 5951 Hoggengus Reaction
W

Consider the following pair of electrochemical reduction and

oxidation reactions
0x1 + e" (electrode,¢m)+Redl (6.17s)

Red +0x + e- (electrode,¢m) (6.17b)

2 2

and the corresponding homogeneous cross reaction

0xl + Red2+Red1 + 0x2 (6.18)
Providing that the interactions between the reactant and the
electrode in the electrochemical transition state, and between the two

reactants in the homogeneous transition state, are negligible (“weak

342

overlap" limit), the activation barriers for reactions (6.17) and
(6.18) will be closely related.

At a given value of 4,111 (and hence electrode potential E), the
thermodynamics of reactions (6.17) and (6.18) are identical since the
energy required to transport the electron across the metal-solution
interface in the half reactions (6.17s) and (6.17b) will then cancel.
The overall activation free energy ACLU for reaction (6.18) can be
considered to consist of separate contributions, AG;’1 and 662,2.
arising from the activation of 0x, and Red, respectively. Although a
multitude of different transition-state structures may be formed,

at

at
corresponding to different individual values of AG ,1 and ACh 2, the
0

predominant reaction channel will be that corresponding to a minimum in

) 343

e *
O O .
the activation free energy ( [161,1 + AG i . In the 'weak

h,2
overlap" limit, each pair of values of AGE,1 and AG;,2 satisfying the
thermodynamic constraints of reaction (6.18) will be identical to the
corresponding pair of electrochemical free energies of activation,
AG:.1 and AG:,2, for reactions (6.17s) and (6.17b), respectively,
having the same transition-state structures. Therefore the energetics

of reactions (6.17) and (6.18) are related in the "weak overlap" limit

by

E

a *
h (66* + A * )E f h ' 1 1 d ° 1
w ere e,l Ge,2 min re ers to t e particu ar e ectro e potentia
I: 'k
where the sum of ace 1 and Ace 2 is a minimum. Although only the sum
9 0

* * . . .
(Ach’1 + Ach,2)min can be determined experimentally for a given homo

343

 

AG;',(OII,4' 0‘ .. Rem)
AG‘Ade, - r -' 0:2)

3 2'
(66:, moms 3.

 
  

 

 

 

 

 

AGE“!
65.3, act...
2 .
I
45.- -FE—I -FEa°

Figure 6.4. Schematic illustraion of general relationship between
electrochemical and homogeneous reaction energetics. Curves 11' and 22'
are plots of activation free energy AG: versus thermodynamic driving

force -FE for an electroreduction and electrooxidation reaction (reactions
6.17s and 6.17b), respectively. E: and E: are the standard electrode

potentials for these two redox couples. Curve 33' is formed by the sum

(AG: 1 + AG: The corresponding homogeneous activation barrier
S 9

E
2)-

AGfi 12 is, in the "weak overlap" limit, given by the minimum in this curve.
2

344

dually as a function of the free energy driving forces AG: and AG; for
these two half reactions (6.17s) and (6.17b), which equal F(E -E{) and
F(E - Eg), respectively. where E: and E; are the corresponding standard
electrode potentials.

This relationship is illustrated schematically in Figure 6.4.
Curves 111 and 222.represent plots of 66: against the reaction free
energy F(E -Ef) for a pair of cathodic and anodic half reactions on a

coupon scale of electrode potential FE; such curves are generally

expected to be curved in the manner shown (vide infra) so that a shal-

 

 

. . . * * . .
low minimum.in the plot of (AGe 1 + Ace 2) versus FE will be obtained.
0 D
In practice, unless AG: is small (<3-4 kcal mol-1) the slopes of these
plots, i.e., the cathodic and anodic transfer coefficients, are often

found to be equal and close to 0.5 so that to a good approxima-

tion182.344.348

* *
2AGe’12 . AG (6.20)

11.12

a a *
where AG is the value of AC at the intersection of theISG - E
9,12 e e,l

*
and Ace - E plots.

2
0
For the special case where the cathodic and anodic half-reactions
*
are identical. since the two AGe - E plots must intersect at Ef for the

redox couple, then Equation 6.20 can be written in terms of the elec-

trochemical and homogeneous intrinsic barriers:

*
2AG

int,e I (6.21)

int,h

345

Provided that the reactions are adiabatic and the conventional
collision model applies, Equation 6.21 can be written in the familiar
form relating the rate constants of electrochemical exchange and

homogeneous self-exchange reactions:180

s 2 h,ex
(kcorr/Ae) - (keen/Ah) (6.22)

where k:;:: is the (work-corrected) rate constant for homogeneous self
exchange, and A e and Ah are the electrochemical and homogeneous
frequency factors, respectively. (See Section IV. A).

In the following subsections, we shall explore the applicability
of such relationships to experimental data for some simple outer-sphere
rections involving transition-metal aquo complexes. In keeping with
the distinction between intrinsic and thermodynamic barriers (Equation
6.6), exchange reactions will be reviewed first, followed by a com-

parison of driving force effects for related electrochemical and

homogeneous reactions.

4. Elecpron Exchapge

In the previous section (VI. A) a good correlation was found
between homogeneous and electrochemical electron exchange kinetics for
reactions of aquo redox couples (e.g. see Figure 6.3). However, the
correlation does not strictly follow Equations 6.21 and 6.22. Instead,

an equation of the form:

s 2 _ h,ex
(kcorr/Ae) (kcorr/Ah) + C (6'23)

346

is followed where C is a constant. This constant evidently takes
account of the differing imaging conditions in homogeneous versus
electrochemical environments (see Sections IV. A and V. A for further

discussion). There are few additional reactions which might be used to

test Equations 6.21 and 6.22.345’346 However. we have recently found
that the exchange reactions of Ru(NH3)2+/2+ also obey Equation 6.23,

provided that a relatively large reaction zone thickness (6re I 2-5 X)
is used in estimating Ae. (In Section IV. C it is shown that are

values of this magnitude are appropriate for ammine couples).

5. Influence pf Thermodxpamic Driving Force

 

Given that the reorganization parameters for electrochemical
exchange of various aquo redox couples are in acceptable agreement with
the corresponding homogeneous rate parameters on the basis of the weak
overlap model, it is of interest to compare the manner in which the
energetics of these two types of redox processes respond to the appli-
cation of a net thermodynamic driving force.

For one-electron electrochemical reactions, the harmonic
oscillator (“Marcus") model13 yields the following predicted dependence

*
of Ace upon the electrode potential:

* * + f f 2 *
- A . - s - - . s
AI;e s1" 0 5 F(E E ) + F(E E ) I16AGmt’e (6 24)
where the plus/minus sign refers to reduction and oxidation reactions,
respectively. The transfer coefficient (Equation 6.16) is therefore

predicted to decrease linearly from 0.5 with increasing electrochemical

347

driving force 1 F(E - Ef). The derivation of Equation 6.24 involves
the assumption that the reactant and product free energy barriers are
parabolic and have identical shapes, and that the reactions are
adiabatic yet involve only a small "resonance splitting“ of the free
energy curves in the intersection region.13
A number of experimental tests of Equation 6.24 have been

made.313’347

Generally speaking, it has been found that a=0.5 at small
to moderate overpotentials, in agreement with Equation 6.24. Tests of
this relation over sufficiently large ranges of overpotential where the
quadratic term becomes significant are not numerous. A practical
difficulty with multicharged redox couples is that the extent of the
work term corrections is frequently sufficiently large to make the
extraction of kcorr‘ and hence AG: and a, from the observed rate-
potential behavior fraught with uncertainty. However, weaver and Tyma
have recently obtained kinetic data for Griz, Euiz and Vi; electro-
oxidation over wide ranges of anodic overpotential (up to 900 mv) under

conditions where the electrostatic work terms are small.313

The anodic
transfer coefficients aa for all those reactions were found to decrease
with increasing anodic overpotential, but to a greater extent than pre-

dicted by Equation 6.24.313’347

This behavior contrasts that found for
cathodic overpotentials, where the cathodic transfer coefficients ac

remain essentially constant at 0.5, even over regions of overpotential
where detectable dcreases in ac are predicted by Equation 6.24. These
aquo redox couples therefore exhibit a markedly different overpotential

dependence of the anodic and cathodic rate constants; this contrasts

with the symmetrical dependence predicted by Equation 6.24. An example

348

 

B
I

      

I

 

 

l3 1 /

o -260 «160 £60 860
E/mv vs. ace

 

*
Figure 6.5. The electrochemical free energy of activation,IAGe,
9
3+/2+
for Cr(H20)6 at the mercury-aqueous interface, plotted against
the electrode potential for both anodic and cathodic overpotentials.
Solid lines are obtained from the experimental rate constant-overpoten-

tial plot in reference 313, by using Equation 6.15. Dashed lines are

the predictions from Equation 6.24.

349

of this behavior is shown in Figure 6.24 which is a plot of AG: versus

3+/2+

f
(E E ) for Craq

at the mercury-aqueous interface at both anodic
and cathodic overpotentials. The solid curves are obtained from the
experimental data, and the dashed lines show the overpotential depen-
dence of AG: predicted from Equation 6.24.

The prediction corresponding to Equation 6.24 for driving force

effects upon homogenous kinetics is13

* *

*
. A . o o 2

e
where AG. is the mean of the intrinsic barriers for the parent
int,h.12

* 66* )1 666° ' h
int,h,l+ int,h,2 “ 12 ““3

free energy driving force for the cross reaction. Equation (6.25) has

self-exchange reactions [0.5(AG

been found to be in satisfactory agreement with experimental data for a
number of outer-sphere cross reactions having small or moderate driving
forces. However there appear to be significant discrepancies for some
reactions having large driving forces (where the last term in Equation
6.25 becomes important) in that the rate constants do not increase with
increasing driving force to the extent predicted by Equation 6.25; i.e.
the values of AG;,12 are larger than those calculated from the corres-
ponding values of AG*

int h 12 and. Acgz using Equation
’ s

It has been suggested that these apparent discrepancies could be

e e .
due to the values of AGh,12 and‘AGint,h,12 that are obtained from the
experimental work-corrected rate constants being incorrectly large due

to nonadiabatic pathways, or to the presence of additional unfavorable

350

work terms arising from solvent orientation required to form the highly

charged precursor complex.207

An. alternative, or additional.
explanation is that the free energy barriers are anharmonic so that the
quadratic driving force dependence of Equation 6.25 is inappropriate.
It is interesting to note that the form of the discrepancies between
the kinetics data for the electrooxidation of aquo cations and Equation
6.24 is at least qualitatively similar in that both involve unexpec-
tedly small dependencies of the rate constants upon the thermodynamic
driving force. Moreover. the large majority of homogeneous reactions
for which such discrepancies have been observed involve the oxidation

of aquo cations.207’310

However. nonadiabaticity effects cannot
explain the asymmetry between the AG: - E plots at anodic and cathodic
overpotentials (Figure 6.5). Also, any specific work term effects
should be different (and probably smaller) at the mercury-aqueous
interface compared with homogeneous reactions between multicharge
cations, yet any anharmonicity of the free energy barriers should be
similar, at least on the basis of the weak overlap model. A
quantitative comparison of the driving force dependence of the kinetics
of related electrochemical and homogeneous reactions should therefore
shed light on the causes of the observed discrepancies for the latter,
more complicated processes.

One can generally express the free energy barrierslIC: for the

pair of cathodic and anodic electrochemical reactions 6.17s and 6.17b

as (cf. Equations 6.16 and 6.24):

(6.26s)

351

and

- 66* +6 Ac; (6.26b)

*
8,2 int,e,2 2

wheres1 and o2 are the transfer coefficients for these two reactions
at a given electrode potential. A similar relationship may be written

. * .
for the free energy barrier ash 12 of the corresponding homogeneous
3

cross reaction 6.18 (cf. Equation 6.25):

where 012 is a “chemical“ transfer coefficient. Although 01 and 02 are
determined only by the shapes of the free energy barriers for the indi-
vidual redox couples at a given driving force. “12 is a composite
quantity which is determined not only by a1 andez but also by the
* d *
t,h,l’ AGint,h,2 ‘“ AGb.12'
*

Nevertheless, comparisons of values of 46h 12 for a series of
9

*
relative magnitudes of Agin

related cross reactions having systematically varying driving forces

a
can yield useful information. Figure 6.6 is a plot ofAGh 12/
’

Ac" c° I 6* £ ' f ’
int,h.12 Vera“ A 12 A int,h.12 or a aerles 0 CtOIS reaCtlon‘

*
12

e
and AC. were obtained from the measured homogeneous rate con-
int,h.12

involving the oxidation of various aquo complexes. (The values of AG

stants by using equations exactly analogous to 6.11. Details are given

in reference 311). The graphical presentation in Figure 6.6 has the

*
virtue that the values ofAGh 12 for different cross reactions are
0
*
normalized for variations in the intrinsic barriers AG. ; the
int,h,12

352

driving force dependence predicted by the Marcus model is such that all

*
AGh 12 values fall on a common curve (shown as a solid line in Figure
8

351

6.6) when presented in this manner. (Omitted from Figure 6.6 are

3+/2+

aq since there is evidence that the measured

352

reactions involving Co
self-exchange rate does not correspond to an outer-sphere pathway).
It is seen that the experimental points deviate systematically from the
Marcus predictions in that the apparent values of 012 (Equation 6.27)
are significantly smaller than predicted from Equation 6.25 at moderate
to high driving forces. Figure 6.7 consists of the same plot as Figure
6.6 but for a number of outer-sphere cross rections involving reduc-
tants other than aquo complexes.351 In contrast to Figure 6.6 reason-
able agreement with the Marcus prediction is obtained (cf. reference
351). The data in Figure 6.6 are also shown in Figure 6.8 as a plot of
*

*
[A612 - AG

*
int,12 '

int,12
plot is an expression of Equation 6.25, the Marcus model predicts a

] versus -[0.54622 + (46:2)2/1646 ]. Since this
slope of unity (the solid line in Figure 6.8). However, the points are
almost uniformly clustered beneath this predicted line, and increas-
ingly so as -AG:2 increases, again indicating that “12 tends to be
smaller than predicted.

It therefore seems feasible that these anomalously small values
of “12 noted from Figures 6.6 and 6.8 have their primary origin in
the oxidation half-reactions which uniformly involve aquo complexes.
This possibility was explored by converting the electrooxidation data
into a form suitable for direct comparison with the homogeneous data in
Figure 6.8 in the following manner. As noted above, the free energy

a
barrier Ash 12 for each outer-sphere cross section will consist of
3

1.0

A619/136“-

353

 

 

 

I I I I I I
M
r m
25
s
as
a
Is 9
s
3'
h s 2
‘l ' .
o
I: .
6
l2. '9
__ Ie 7. 7
o I V
:1 ..
m
:93? s ' V
20
.s
_ Is
24 ..
s Marcus prchcuon
n
s
)—
l I 1 l 1 l
15 BO 25 20 15 IO 05
" Barf/40¢. 12'

Figure 6.6

 

354

* * . o a
Figure 6.6. Plot of AG12/A61,12 against -A612/AG1,12 for
homogeneous cross reactions involving oxidation of aquo
2+ 2+ 2+
cations. Reductants. (s) Euaq, (A) Craq, (v) vaq’ and
+
(I) Ruiz. Key to oxidants and data sources: 1, Feiq; 2,
3+_ 4+. 3+, 3+. 3+. 3+,
Ruaq’ 3’ Npaq, a, vaq’ 5’ Euaq’ 6’ Ru(NH3)6 9 7’ Ru(NHB)5py 9

8, Co(en)§+; 9, Co(phen)§+; lO, Co(bpy)§+(data sources for
1-10 listed in reference 311); ll, Ru(NHB)Sisn3+ (reference
321); 12, Co(phen):+ (reference 310); 13, Co(phen);+
(reference 337); 14 to 17 and 25 are from reference 388;

3+

14, Co(phen)§+; 15, Ru(NH isn ; 16, Os(bpy)2+; 17,

3)5
Ru(bpy)2+; 18 to 22 are from reference 304; 18, *Rul4,4'
(CH3)2bpy]§+; l9, *Ru(phen)§+; 20, *Ru(bpy)§+; 21, *Ru(5-
c1 phen)§+; 22, *Ru[4,7-(CH3)2Phen]§+: 23, *Os(5-Cl phen);+
(reference 386); 24, Ru[4,7-(CH3)2 phen]:+ (reference 387);
25, Ru(NH3)5py3+. An asterisk(*) indicates the oxidant is

a photoexcited state reactant.

355

 

I I I r I I I
5
O
1.0»-
r
C 2
O
9— kg . J
a
O C
. O
Q
.: (hr-l . q
c.
4
3 ° 0
L:
4

\ Marcus prchcuon

is

 

 

 

3.5 3.0 2.5 2.0 LS I 0 0.5 0
-aG.;°/AG., .3'

Figure 6.7. Plot as for Figure 6.6, but involving reactants
other than aquo complexes. Key to reactants and data sour-
ces: (O) Co(III)/(II) macrocycle oxidants (data are given
in Figures 2. S and 6 of reference 351); (0) other nonaquo
oxidants; 1, Ru(NH3)Spy3+ + Ru(NH3)2+; 2, Rug: + Ru(NH3)62+;
3, Co(phen)2+ + Ru(NH3):+; 4, Co(bpy);+ + Ru(NH3)2+; 5.

2+ + Ru(NH3)5py2+ (data sources for 1-5 are listed

in reference 311); 6, horse heart ferricytochrome c +

Co(phen)

Ru(NH3)2+ (reference 389); 7,Co(phen)2+ + horse heart ferro-

cytochrome c (reference 390); 8. Ru(NH3)hbpy3+ + Ru(NH3)5‘

pyz+ (reference 310).

356

 

I I I I I I I

RedmzmmsI O 0 Eu?“
‘9
A A Cr,"

vv W’"

to
IRu“

 

l J l I l I

I
I2! £3 41 C)
-[ I/2AG°,, + (A61, )1/ I6AG;,,]. kcol-mol"

 

 

 

Figure 6.8. Plot of (A622 - AGg’lz) for homogeneous cross reactions
involving oxidation of aquo complexes given in Figure 6.5, against the
thermodynamic driving force function {0.56622 + (AGiz)z/16AGL12].
Closed symbols are obtained from homogeneous data; key to points as in
Figure 6.5. Open symbols are corresponding points obtained from elec-

trochemical kinetic data for oxidation of aquo cations (see text for

details).

357

'A' I:
contributions AGh 1 and 46h 2 from the oxidant and reductant, respect-
9 9

I: I: ,
h,1 and AGh,2 will equal the free

I: I:
energy barriers Ase 1 and we 2 for the corresponding electrochemical
9 9

ively. In the "weak overlap" limit AC

I: I:
reactions at an electrode potential where the sum (A69 1 + AGe 2) is a

9 9
I:
minimum (Equation 6.19 and Figure 6.4). Estimates of ACh 2 for Eu:;,
9
2+

Craq, and Vi; oxidation as a function of the half-reaction driving

*
force G;[ I -F(E -E§)] were obtained from the corresponding AGe - E
plots (see Figure 6.5 and reference 313) by assuming that they have the

I:
O I 0 (i.e. AG. ) by

. I:
same shape but replacing the value of we at A62 int,e

I:
0.5 Acint h (this procedure corrects for the differences between
9

I: I: . . . .
AGint,e and 0.5 AGint,h resulting from the limitations of the weak

' I:
overlap model (Equation 6.23). The accompanying plots of AGh 1 versus
9

A6? for the reduction half reactions involved in Figure 6.8 were
I:

constructed using the experimental value of AG. by assuming that
int,h,12

the harmonic oscillator model applies, i.e. by utilizing Equation 6.24

written for homogeneous half reactions:

*

* * o o 2
AG 0.5AG. + 0.51.\.G1 + (A61) l8AGmt,h,12

hsl int,h,12 (6.28)

. * o * ,.0
These pairs of AGh,l A61 and M511,2 Aoz curves were plotted on a

o-
l

for each cross

common driving force (i.e. electrode potential) axis such that AG
0 o . . *
AG2 A612, and the required estimates of AGh,12
reaction were then obtained from the sum (AG; 1 + AG; 2) at the value
9 9
of AG0 where the quantity has a minimum'value (Equation 6.19). These
I:
estimates of AGh 12 are plotted as open symbols in Figure 6.8 for the
9

reactions having moderate to large driving forces (-AC:2>8 kcal mol-1),

358

*
alongside the corresponding experimental values of ACh 12. It is seen
9
I
that the estimated values of AG diverge from the straight line

h,12

predicted from the harmonic oscillator model to a similar. albeit
slightly smaller. extent than the experimental values. Admittedly.
there is no particular justification for assuming that the reduction
half reactions obey the harmonic oscillator model. However, it turns
out that the estimates of AG;,12 are relatively insensitive to altera-
tions in the shapes of the mil - AG: plots. It therefore seems
reasonable that the deviations of the activation free energies for
highly exoergic electrochemical and homogeneous reactions illustrated
in Figures 6.5 and 6.8 may arise partly from the same source, i.e. from
values of oz for the oxidation half reactions that are unexpectedly
small. That is not to say that other factors are not responsible, at
least in part, for these discrepancies. Nonadiabaticity, work terms,

specific salvation, and other environmental effects may all play impor-

tant roles depending on the reactants. For example, there is evidence

3+/2+

to suggest that the true rate constant for outer-sphere Feaq

self-
exchange is significantly smaller than the directly measured value (see
Section VI. A). This can account for a good part of the unexpectedly
slow rates of cross reactions involving this couple.

It is remarked here that in the original publication we were
unable to uncover the reasons for the peculiar driving-force dependence
of the aquo oxidation kinetics, although a number of ideas were
advanced. However. absolute calculations reported in Chapter VII

indicate that the driving-force anomalies on both the anodic and

cathodic sides for electrochemical reactions of Fig/2+ and Griz/2+ can

359

be attributed in large measure to differences in inner-shell force
constants for the M(III)-L versus the M(II)—L state. Such differences
also should account for the homogeneous reactivity patterns. Rather
than indicating a failure of electronrtransfer theory, these findings
point to the unsuitability of the "equal force constant" approximation
which is employed in deriving Equation 6.24, as well as 6.25 and 6.28.
This discussion therefore illustrates the inadequacy of the relative
theory in comparison to the absolute theory for solving one aspect of
the problem. However, in another regard the relative theory is the

superior approach in that the reaction energetics of redox couples such

as Eu3+/2+
sq

can still be examined even though structural data (required
in the absolute approach) are lacking. Furthermore the relative
electronrtransfer theory appears to represent the only method of inter-
preting kinetics data for chemically irreversible reactions for which
thermodynamic data are lacking. Thus, the two approaches are comple-

mentary in generating insights and understanding concerning the details

of electron-transfer kinetics.

6- mm

It seems clear that kinetics as well as thermodynamics data
gathered for simple electrode reactions can contribute significantly
towards the development of our fundamental understanding of electron
transfer in condensed media. In particular, detailed studies of
electrochemical kinetics with due regard for work term corrections can
yield information on the shapes of free energy barriers and the

isolated “weak overlap" reactivity of metal complexes.

360

C. Entropic Drivipg Fprcg Effects Upon Preegppnential chtors for
Intramolecular Electron Tppnsfer: Ipplications for the
Assessment,p§,§pnadiabaticity

(Accepted for publication in Inorg. Chem.)

Considerable effort has been directed recently towards measuring
rate parameters for intramolecular redox reactions, typically involving

electron transfer between a pair of transition metals linked by an

353-355

organic bridge. Such reactions offer important advantages over

bimolecular processes for investigating detailed aspects of electron
transfer. since the uncertainties concerning the energetics of forming

the precursor complex are absent and the geometry of the transition

state is relatively well defined.355

The study of intramolecular electron transfer should be parti-
cularly useful for examining electronic coupling effects; i.e. the

factors influencing the occurrence and. degree of

353b-d,355b

nonadiabaticity. Since the rate constant, k for such

et’
processes should refer to an elementary electronrtransfer step, it can

be expressed as342

It“ wn Kelexp(As;C/R)exp(-AH;CIRT) (6.29s)

ket I\hexp(A8:/R)exp(-AH;c/RT) (6.29b)

where vn is the effective nuclear frequency factor, K e1 is the

*

electronic transmission coefficient, and ASFC

*
and AflFC are the entropic

361

and enthalpic components of the Franck-Condon barrier. Values of Kel

can therefore be extracted from rate measurements as a function of
I:
temperature provided that vn and A8 are known. For many reactionsv 11

PC
is numerically very similar to the conventional factor kT/h.9’23

Since
work terms are absent for intramolecular reactions, the observation of
preexponential factors that are substantially less than vn, or equi-

I:
valently, of negative apparent entropies of activation ASa is often

taken as prima facig evidence of nonadiabaticity (i.e. meld).
I:
Implicit in this interpretation is the assumption that ASFC is essen-

tially zero. However, contrary to common belief this is probably
incorrect for most nonsymmetrical intramolecular reactions, including
those featuring charge symetry. The purpose of this section is to
explore the likely magnitudes of AS* for some representative intra-
molecular reactions and to examine the consequences for the interpre-
tation of unimolecular preexponential factors in terms of nonadia-
baticy.

The origin of the Franck-Condon activation entropy can be clari-

13 for the activation free

356

fied by considering Marcus’ expression
I:
energy, AGFC’ and its temperature derivative. Thus,

* *
AG I AG. t + 0.5 (1 +OL)AGO (6.30)

FC in
I:
where Mint is the intrinsic barrier, AG0 is the thermodynamic free
I:
energy change, and o I AGO/82361“. The temperature derivative of

Equation 6.30 is

362

* * 2 O
Ach I AShun-4o ) + 0.5 (1 + 2a)AS (6.31)

* . .
where ASint is the intrinsic activation entropy and AS0 is the entropic
driving force. For reactions having small or moderate values of Co,

I 0 so that Equation 6.31 simplifies to

* * o
ASFC Asiat + 0.5AS (6.32)

*
For reactions in aqueous media, Asin
1

t is usually assumed to be

close to zero (within ca. 2 J deg- mol-1) as predicted by the

conventional dielectric continuum expression,203 although as shown in

Section V. A there is good reason to suspect that small positive values

1

I _ -
of ASint (ca. 4-12 J deg mole 1) are common.357 However, A80 can

often be sufficiently large to yield a substantial contribution to
*
Ach' even for reactions leading to no net change in ionic charge. For

example, the reduction of Fe(phen)2+ (phen I phenanthroline) by

Fe(OH2):+ yields a net entropy change of -l67 J deg”1 mole-1,55

l

* - -
corresponding to a value of SPC of ca -67 J deg mole 1 from Equation

6.31.
The values of AS0 can be related to the component entropic chan-
ges (the so-called "reaction entropies" As:c)55 at the redox centers

undergoing reduction and oxidation, As: and As:c ox’ respectively,
9

c,red

by

O O 0

AS - Asrc’red -ASrc ox (6.33)

363

Such large entropic driving forces arise because the component reaction
entrapies are often large and extremely sensitive not only to the
charge type of the redox couple, but also the chemical nature of the

coordinated ligands, the metal redox center, and the surrounding sol-

52.82.84

vent. This is due to the large changes in the degree of

specific ligand-solvent interactions, and hence the extent of local

solvent polarization, brought about by electron transfer.55’82’222

I
PC

were calculated for thirty outer-sphere bimolecular reactions and com-

In order to check the validity of Equation 6.31, values of AS

pared with the experimental activation entropies, Asz‘p. The choice of

reactions was limited to those for which values of 48° and AG0 are

available and sufficient structural information exists to permit the
calculation of AGE“. Details will be presented in chapter VII. All
these reactions involve reactant pairs having charges of +3 and +2.

* * 358

against AS . This plot

Figure 6.9 is the resulting plot of ASm exp

demonstrates that the experimental activation entropies do indeed

respond to the entropic driving force roughly in the manner predicted
I

by Equation 6.31. although the values of Asexp are about 60 to 100 J

-1 1

- I
deg mole smaller than AS . (Also note that reactions involving

PC

high-spin Co(III)/(II) couples do not display noticeably different
behavior in spite of the spin state change that occurs for these
reactions.) This discrepancy could be taken as an indication of
nonadiabaticity (Kel (<1), but more likely it arises chiefly from an
additional component of [58sz associated with the unfavorable entropic

work of forming the highly charged precursor complex from the separated

cationic reactants.

364

Figure 6.9. Plot of the experimental activation entropy, ASpr’ against

the activation entropy, AS§C, calculated from Equation 6.32 for bimolec-

ular reactions involving 3+/2+ redox couples. Data sources quoted ref-
erence 311 unless otherwise noted (aq I 032, an I ethylenediamine, bpy -

2,2'-bipyridine, phen I 1,10'Iphenanthroline, terpy I 2,2',2"-terpyridine

py I pyridine). Reactions: 1. Co3+ + Fe2+; 2. Co3+ + Cr2+;
aq aq aq aq
3. Fe3+ + 1162"; a; 5.3+ + v“; 5. $433+ + Fe2+; 6. v3+ + v“;
a aq aq aq’ aq 8Q aq 3Q
3““ 2+. 3+ 2+. 3+ 2+.
7. Feaq + Ru(NH3)6 , 8. Fe q + Ru(NH3)5py , 9. Feaq + Ru(en)3 .

10. Ru(bpy)3+ + Fe2:§; 11. Ru(phen)3 +Ru 2:; 329 12. Fe(bpy)3+ +

Fe 2+ 359 13. Fe(phen)g+ + Fe::;335 14. Ru(terpy)2 + Fe 2:; 329

W
3+ 2+.329 2+. 3+
15. Ru(bpy)2(py)2 +Feaq, 16. Ru(NH3)6 +Vaq.17. Ru(NH3)5py *-

2+ 3+ 2+. 3+ 2+.31o 3+
Vaq’ 18. Co(en)3 + Vaq, 19. Co(bpy)3 + vaq’ 20. Co(phen)3 +

2+.310 3+ 2+.360 3+ 2+.28
vaq’ . 21. Ru(NHa)5py + Crag, 22. Ru(NH3)4bpy + Ru(NH3)4phen ,

3+ 2+. 3+ 2+. 3+
23. Ru(NH3)6 + Ru(NH3)6 , 24. Ru(en)3 + Ru(NH3)6 . 25. Co(en)3

Co(en)§+; 26. Co(bpy)3+ + Co(bpy)§+; 27. Co(phen)§+b+ Co(phen)§+,

28. Co(en)3 + Cr(b py)§+;361 29. Co(phen);+ + Ru(NH3)§+; 30. Co(phen)§+
+ Ru(NH3)5py2+.

deg" mole"

asap, J

365

 

 

4.0%

 

 

(18%;, J deg"mole"

Figure 6.9

€53
s
5 O
6
20
O in
'5 02: ITO
0 019
Ono 27
n O
' '3()0 l6
0
:2
o 0'4
L 1 1
-25 O 25

 

366

Most nonsymmetrical intramolecular reactions will also have
nonzero entropic driving forces, thereby yielding nonzero values of
38:0 on the basis of Equation 6.31. This is most obviously the case
for systems where electron transfer leads to a net charge decrease at
the redox centers, such as the pentaamminecoba1t(III) -
pentacyanoiron(II) reactions examined by Haim and coworkers.354 Here
the values of Ass-c,red and Asicmx will be positive and negative,
respectively (Table 6.4), yielding large positive values of A80
(Equation 6.32) and hence A8;c (Equation 6.31), due to the decrease in

solvent polarization attending such charge neutralization.354C

Quanti-
tative values of AS0 for these systems are difficult to estimate from
values of As:c for the isolated redox centers. This is because As:c
for cyano Fe(III)/(II) couples are known to be extremely sensitive to
the cationic environment (Table 6.4)286 so that the proximity of the
cationic Co(III)/(II) redox center is expected to influence the solva-
tion around Fe(III)/(II). Nevertheless, from the As:c values in Table
6.4, we estimate that As:c equals roughly 125 J deg.1 mol.1 and I45 J
-1 mol.1 for the Co(NH3)5L-3+/2+ and ILFe(CN)§-/3- fragments, where

L is a pyridine-type ligand, yielding A8°==170 J deg.-1
1

deg

mol.1 (Equation

mol.1 (Equation 6.31s). Indeed values of

I .
Asa of this order are obtained for (CN)5FeII-L- CoIII(NHB); reac-

354

6.32) and AS;C :80 J deg-
tions. Similarly, large values of A30 are expected for the intra-
molecular reduction of pentaamminecobalt(III) by the p-nitrobenzoate
radical anion since both redox centers will again contribute positive
components to A80, thereby explaining the large preexponential factors

I
(i.e. positive values of A88) observed for these reactions.362

Reaction Entropies, A30 (J deg-

367

Table 6.4
l

mol-1) for Various

Redox Couplesrin Aqueous Solution.8

Redox Coupleb

M(Z-bpy)g+/2+

u(puen)g*’2*
)3+/2+
2 6
)3+/2+
3

u(os
M(en
3+/2+
u(nn3)6
M(NE3)5(bpy)3+/2+
M(Nfl3)5Py3+/2+
n(nu3)50s2
M(Nfl3)5NCS
2+/+
M(Nfi3)561
2+l+

c-H(Nfl3)4Cl2
u(cu):'/4'

3+/2+
2+l+

H(¢.‘.N)4bpy-/-2

Co
92
92

250°

155

190

Metal Center

Ru Fe

4 8

150 180
54
75

71
105
63
42
42
-l76f, - 96"8

I120

368

Notes to Table 6.4

a Ionic strength uI0.1; data taken from references 55 and 84, except

1 1

where noted. Uncertainties estimated to equal 2-6 J deg- mol- for

directly measured values. bl! denotes metal redox centers. Ligand
abbreviations: 2-bpy I 2.239bipyridine; bpy I 4,4’Ibipyridine; phen I
l. lO-phenanthroline; en I ethylenediamine; py I pyridine. cEstimated
from thermodynamic data given in reference 398by correcting for tem-

perature dependence of reference electrode and likely ionic strength

1

effects. A similar value (ca. 1 20 J deg- mol-1) is obtained from a

correlation of 83c with the self-exchange rate constant kex’ assuming

that her 10-10 5.1 sec-1. dEstimated by comparison with corresponding

3+/2+
)3

value for Co(en . eJ. T. Eupp, unpublished experiments. ronic

strength uIO. 8Determined in 0.2 g La(CLO4)3.

369

Substantial values of A80 are expected also for intramolecular
reactions between pairs of metal redox centers having the same charge
type and similar ligand environments but with different electronic
conf igurations. Important examples of this type are the pentaammine-
cobalt(III) - aquotetraamine- ruthenium(II) reactions bridged by

nitrogen heterocyclic ligands that have been studied by Taube and

353

coworkers. Since these reactions are charge symmetric, it is

usually tacitly assumed that A80 I0 and hence As;c . 0.353b.355b

However. since reaction entropies for high-spin Co(III)/(II) couples

are generally found to be markedly larger than for Ru(III)/(II) and

55.84

other low-spin couples (Table 6.4), large positive values of AS0

and hence 138:6 are therefore anticipated for these reactions.

As an illustrative example, we consider the intramolecular
reduction of Co(III) by Ru(II) in (£120) Ru(m3)4(bpy)Co(NH3)g+ (where
bpy I 4,4’Ibipyridine). The reaction for the corresponding dipenta-

ammine complex Ru(m3)5(bpy)0o(un3)g+ is liable to involve a net

1 mil-1

entropy change of ca 85-105 J deg- since the reaction entropies
of Co(III)/(II) and Ru(III)/(II) couples having similar ligands uni-
formly differ by this amount (Table 6.4), (references 55 and 82). The

effect of replacing one ammonia on ruthenium by an aquo ligand can be

gauged from the 30 J deg.1 mol.1 larger reaction entropy for
Ru(lfil3)50n§+/2+ than for Ru(NH3)2+/2+ (Table 6.4). (To a first

approximation, the reaction entropies for mixed-ligand complexes appear
to arise from linear additive contributions from each ligand).84 Thus
the entropic driving force for electron tranfer in

II III
EZORu (N33 ) 4( bpy)-Co 1

(m3)? is estimated to be ca. +65 J deg-

370

mol-1. Bearing in mind the likely value of A821"; (3-8 J deg-1
mol-1) ,357 a value of 188:0 of ca. 35 J deg.1 mol.1 is obtained from

I
Equation 6.31. Therefore the "measured" activation entropy, Asa’ of 10
J deg”1 mol-1 for this system is suggestive of a significantly
nonadiabatic pathway (Ke1 3 0.1; Equation 6.29) rather than the

adiabatic pathway that has been inferred without consideration of the

35“": Since similar values of 133* should 8180

PC
III ( m3 ) 2+ react ions with

entropic driving force.
. II
be appropriate for other nzoau (1013) 4LCo

related bridging ligands L, the smaller or negative values of As: seen,

1

for instance, with L I l,2-bis(4-pyridyl)ethylene (5 J deg- mol-1) and

1

l.2-bis(4-pyridyl)ethane(-40 J deg- mol-1) infer the presence of

I
decidely nonadiabatic pathways. Thus from Equation 6.4. if ASFC I 35 J

-1 2 4

~3 x 10- and l x 10- for these two reactions,

mol"1 then K ..
e1

deg
respectively.

Naturally. these resulting values of Kel should be regarded as
being only approximate. Since the values of A8:c for the
intramolecular binuclear reactions are inferred from data for
structurally similar mononuclear couples, the reliability of the
resulting estimates of AS0 and hence 138:0 may be called into question.
Unfortunately. values of AS:0 cannot be measured directly for these and
other thermal intramolecular reactions on account of the rapid equation
that follows the formation of Co(II). However, Steggarda et al. have
shown that the values of Asgc for mononuclear Ru(III)/(II) couples
containing pyrazine ligands are essentially the same as in binuclear
complexes where the pyrazine ligand is also bound to another ruthenium

363

redox center. This result therefore provides strong support to the

371

present method for estimating ASo values for binuclear complexes.

Providing that the present estimates of A80 are accurate to within ca.

1 I1

20 J deg- mol . which seem reasonable, then the corresponding

estimates of Kel are reliable to within ca. 4 fold.

Such entropic driving force effects also provide a rationalization

1 mol"1 for

3+ 353d

of the substantially more positive value of As; 44 J deg-

electron transfer in (SO3)Run(NH3)4(pyrazine) CoIn(HH3)

S 9
. . . . II III 5+
especially in comparison with that for HZORu (8H3) 4(bpy)Co (m3) ,
11 J deg.1 mol-1. This increase is difficult to explain on the basis

of electronic coupling effect3,353d

but can easily be understood in
terms of the influence of nonbridging ligand composition on A80 and

hence AS;C.55'84 The presence of anionic ligands generally yields
substantially smaller values of Asgc. The influence of substituting an

aquo by a sulfite ligand on the reaction entropy of Hu(III)/(II) can be

gauged roughly from the decrease in Asgc, 65 J deg.1 mol.1 between
Ru(NH3)SOHg+/2+ and c-Hu(NH3)4Clz+/°, (Table 6.1).55 corresponding to

1

an increase in AS“r of ca. 35 J deg- mol.-1 (Equation 6.3l) since the

PC
ruthenium undergoes oxidation.

Similar considerations can also be applied to a number of other
intramolecular reactions, such as those involving nonsymetrical

bridging ligands.353’365

Weaver and co-workers have also recently
analyzed activation parameter data for a number of intramolecular redox
reactions at metal surfaces (i.e. electrochemical processes involving
adsorbed reactants) in a similar manner. Entropic driving force

effects are generally important for these processes since only one

redox center is involved, so that ASo equals asgc. Although the

372

estimation of As;c for intramolecular systems should be approached with
caution since the factors influencing the thermodynamic entropy changes
are incompletely understood, it is clear that careful consideration of
the ligand composition and the chemical as well as electrostatic nature

of the redox centers is required in order to evaluate the contribution

of the Franck-Condom entropy to be measured preexponential factors.

373

CHAPTER VII
COHEARISONS BETHEEB EXPERIMENTAL KINETICS BARAMETERS AND THE

ABSOLUTE PEEDICTIONS OF ELECTRON-TRANSFER THEORY

A. Introduction

The kinetics of inorganic electron transfer reactions have been

366

widely investigated over the past thirty years. Experimental work in

this area has been spurred by the ongoing development of detailed and
sophisticated theoretical treatments of these reactions. The theore-
tical descriptions of homogeneous outer-sphere reactions of transition-
metal complexes in particular have reached a high level of refine-

.56 .57

ment It is probably fair to say that theory has remained one step

ahead of experiment since the original work of Marcus.367

Thus, in the
absence of crucial structural and spectroscopic information the predic-
tions of electron transfer theories in an absolute sense have in large
measure remained untested. Researchers have had to be content with
relative evaluations of theory. such as tests of the so-called cross
relations, relative rate comparisons, etc.

Very recently this state of affairs has changed. Sutin and
coworkers have determined from EXAFS measurements the magnitude of
metal-ligand bond distance alterations accompanying changes of oxidation
state for several transition metal complexes.10 These data together

11 . 12 .325 .326 ,368-371

with previous structural measurements have been

374

used by Sutin to estimate inner-shell Franck-Condon barriers and abso-
lute electronItransfer rate constants for a number of homogeneous self-
exchange reactions.10 Broad agreement is found between theory and
experiment.

It would be interesting and useful to extend this study to
electrochemical reactions. Indeed this is possible by using the newly
published EXAFS data. Also it may be enlightening to examine homo-
geneous crosa reactions. In comparison to directly measured self-
exchange rate constants, the number of cross-reaction rate data avail-
able for comparison with theoretical predictions is quite large. Thus a
comprehensive evaluation of contemporary theories should be possible.
Besides enabling a considerable expansion of the data set, a virtue of
cross reactions is that they enable important supplemental features of
the theoretical treatments to be tested, most notably the predictions
concerning the response of rate constants to changes in thermodynamic
driving force. Comparisons between experimental and theoretical
activation parameters provide further tests of theories.

In this chapter, kinetics parameters for approximately ten
electrochemical and fifty homogeneous electron-transfer reactions are
calculated using current theories and are compared with the
corresponding experimental parameters. It is hoped that such
comparisons, by uncovering the areas of agreement as well as
disagreement, will indicate which aspects of electronItransfer processes
are well understood while identifying problems which merit further

investigation.

375

3. Calculation 9; Kinetics Pargmeters

In so far as this work overlaps with that of Brunschwig and

10

Sutin, their semi-classical approach is largely retained. (Neverthe-

less, in instances where commonly employed approximations produce signi-
ficant errors, more rigorous calculations are made). More sophisticated
(and difficult) quantum mechanical analyses are certainly possible, as

63

shown for example by Marcus and Siders. However, it has been demon-

strated convincingly that the semi-classical and quantum approaches
yield virtually identical results when the same assumptions are used in
each calculation.23
The basic elements of electron-transfer theory have been outlined
in Chapter II. According to the semi-classical treatment the overall

rate constant can be written «:23

o,
k - KAvnrnKelexp(-AG*/RT) (1.24)
or
o *
k I KAvnK e1exp[--AG(T) IRT] (7.1)

where K: is the precursor formation constant, Vn is the nuclear
activation frequency Kel is the transmission coefficient for electron
tunneling, n is the nuclear tunneling correction, 136* is the activation
free energy representing the height of the classical Franck-Condom
barrier. and AG*(T) is the quantum mechanical equivalent of this

barrier. An additional factor which should appear in the rate formula-

376

tions is the electrostatic work of assembling charged reactants.
However, work terms in both the electrochemical and homogeneous reac-
tions are taken into account instead by correcting the experimental rate
parameters for such work. This enables experimental results obtained
under dissimilar conditions to be intercompared in a straightforward

manner 0

l. Pre-expgnential Terms
The precursor formation constant for electrochemical reactions is

equivalent to the effective thickness of a heterogeneous reaction zone

9

(Section IV. A). A value of 6 x 10’ cm is deduced from considerations

relating to the probable distance dependence of the tunneling coeffi-

15

cient Kel’ (This is discussed further in Sections IV A and C).

Vhlues of I: for homogeneous reactions were estimated from:9

o 2
EA 41rN(a1 + a2) 6r (7.2)

where N is Avogadrofis number. a1 and a2 are the radii of the reactants

(Table 7.1), and 6r is the homogeneous reaction zone thickness, again

taken as 6 x 10.9cm.372

The activation frequency vn was calculated from Equation 7.3:9

2 I I

I I
O I A .
AGin,l+ in,2 Gin,2

A
in,1+ G

vn (v AG in,2

I
A
out out+vin,l )/(AGout+ G

) (7.3)

I . .
where v out and Acout are the frequency and activation free energy

associated with solvent (outer—shell) repolarization, v. and ‘V.
in,l in,2

377

are the inner-shell bond vibration frequencies for each of two

I I . .
reactants, and AG- and AG. are the corresponding contributions to
1n,l in,2

the overall activation free energy. (Only one inner-shell term appears

in the calculation of v n for electrochemical reactions). A value of 9 x

was used for v .9 The values of vin for the most part are

out

not known with certainty. since the necessary spectroscopic data
concerning symetrical metal-ligand bond vibrations are lacking. The
estimates listed in Table 7.1 were obtained by extrapolation from
measurements on closely related systems as summarized in the footnotes.

The uncertainties in v in values lead directly to only minor

uncertainties in calculated rate constants. For almost all of the

reactions vn values in the range from 9 x 1012 s.1 to 1.3 x 1013 a”1

were calculated. (See Table 7.2). Although some controversy exists

concerning the correctness of Equation 7.3, alternative formulations17

were found to yield similar values of the frequency factor.

Since values of the transmission coefficient can be calculated

14,15

satisfactorily only by means of 31; initio techniques, this

 

parameter was simply assumed to be unity at the closest separation

distance of the reactants. While this assumption represents a possible
source of significant error. some justification is provided by Newton’s
“"15 2+,” and Ru(NH3)g+/2+ self-exchanges

as well as our own experimental work (Sections IV. C and V. C, and

calculations for the Fe(H20)
reference 373). These studies indicate that Kel values for homogeneous

as well as electrochemical electron-transfer reactions probably lie

between 0.2 and l.

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A:ammuv:m mom

fiHHvz mu ummumcou momom magnoumuum ommwfianamuma Hmofiuumaamm mo mmam> mommaaumw .n .mouoc mm ummuxm .MH.N
mowummvm mean: he omumasoamu .oumum AHHHVZ ma ummummoo mouow wmfinoummum mammNHINMuma Hmowuumaahm mo o=Hm>
mmumafiuwm .w .mumum AmHvz ma zommsumum wowsououum vmmeHIHMuoa Homauuoaamm mo o=Hm> mommafiumm .m .mumum
aHHHvz aw hoammvmuu wagsuummum vmmwfialamuma Hmuuuumaazm mo unam> mommawumm .m .mumum moaummfixo mo owmmnu
wmfixmmmaooum summed once mmmmNHIHmuma ma ommmmu .m .ON menopause Eoum mum mommaflumm .msfimmu ummuommm .u
. Hoe

HI HI
Ion omomuumam on umm>aoo ou >8 mQN mvmvmmomuooam Hmaoamo menopaumm .m> mammoo women no Hmfiummuom Nmauom .m

mom h «IN mum moaucamuumomn .mmuoa mafiamonuo woman: mm oommuowom aoum mum mama .Ammz .m> mamfiucmu

380

2. FranckICondon Barriers

Three factors contribute to the classical Franck-Condon barrier,
namely, the solvent reorganization energy vout’ the metal-ligand bond
(inner-shell) reorganization energy vin’ and the free energy driving
force AG°.13 The reorganization energies correspond to the energy
required to displace the nuclear (solvent and bond) coordinates of the

reactants so that they match those of the products (Figure 7.1) under

conditions where AGo I 0. For a cross reaction such as

A + B + A + B (7.4)

the total intrinsic reorganization energies f and r for forward and

reverse reactions are given by Harcusfl additivity rules as13

A 3+ B 2+
1 :- 1 ’ A ! A
f ( in + in + out) (7'5)
and
A2+ 33+
)\ II A.’ A,’ 1
r ( in + in + out) (7'6)

where the superscripts on kin designate the particular reactant being
reorganized. Equations 7.5 and 7.6 are sometimes reformulated using a
pair of kin values derived from reduced force constants for metal-ligand
vibrations rather than the four kin values obtained using the individual
force constants for inner-shell modes. This approximation was avoided

since it yields substantial errors in AG* for certain reactions, e.g.

10
11
12
13
14
15
16
17
18
19
20
21

22

Table 7.2.

381

Experimental and Theoretical Rate

Constants (M-ls-l) for Homogeneous ElectronITransfer Reactions.

oxidant

3+
V0120)6

3+
Fe(H20)6

¥

Co(HZO)6
Co(HZO)
Co(HZO)
Co(HZO)

Fe(H 0)

2
Fe(H20)
Fe(H20)
Ru(HZO)
Fe(H20)

Fe(H20)

¥°$°$°¥°¥°¥°¥°¥°$°$

Ru(HZO)6

2;:

Ru(NH3)6

Co(en)?+
3+
Ru(NH3)6
Co(en)?
3+
Co(HZO)6

Ru(bpy) 3"

Ru(phen):+
Ru(terpy)§+

Fe(phen)§+

reductant

«1120) 2"”
Fe(H20) (2:
Co (1120) 2"
@0120):+
@0120):+
V(H20)§+
Ru(HZO) :1"
Cr(HZO)§+
«1120)?
«1120)?
Ru(NH3) 3"
Ru(en) :2:

2+
Ru(NH3)6

2+
V(H20)6

Fe(phen)§+

2+
Fe(HzO)6
2+
Fe(H20)6

2+
Fe(H20)6

2+
Fe(H20)6

1.5x10'
4
8

50

1.3x10“

9x10S

2.3x103

2.32.103

1 . 23x104

2.81.102

3.51.105

1.4X105

1.46.10“

2x102

3x10"4

1.52.103

4.6x10’

1.4x104

6.4x105

8x105

7 .2x105

3.7x104

(2)
(0.55)
(3)
(l)
(3)
(3)
(1)
(1)
(1)
(1)
(0.1)
(0.1)
(1)
(0.2)
(1.0)
(0.1)
(0.1)
(3)
(l)
(l)
(1)

(0.5)

corr

8x10-
52

33.

4.22.102

5.3x104

3. 7x106

1.9x104

1.91.104

1.5x105

2.3x103

1.5x107

3.6..106

1.11.105

5.61.103

1.6x1o'

6.5x104

1.2x10’

3.8x104

1. 7x106

2.2x106

2x106

7.4x104

k (T) Reference

 

1.7
9.7

2 . 2x10"

21.105

5.2x107

3.3x109

1.1x106

4.4.405

5.61.107

3.2x10S

8.2x108

1 . 1x109

5.5x106

2.11.104

1 .4x10’

4.8x105

5 . 2x10-

3.3x109

10

1.3x10

10

1.3x10

10

1.3x10

4x108

267
266
399
400
401
401
304
402
379
304
268
268
304
403,404
337
405
337
406
329
329
329

335

382

Table 7.2 (continued)

 

2§$g§2£_ .EEEEEEEEE. .3L. .JL. .EEEEE k (T) Reference
Fe(bpy)§+ Fe(H20)§+ 2.7x104 (0.5) 9.6x104 1.3x109 335
Os(bpy)§+ Fe(H20):+ 1.4x103 (0.5) 5x103 3.22.107 334
Fe(820)2+ Co(phen)§+ 5.3x103 (1) 1.4x104 1.9x104 337
Co(phen);+ @0120):+ 3O (1) 81 4x103 337
Ru(bpy)§+ Ru(HZO):+ 1.9x109 (1) 5.1x109 3x1012 317
Os(bpy)§+ Ru(HZO)§+ 2.9x108 (1) 7.8x108 9.8x1011 304
Co(phen):+ Ru(HZO)§+ 53 (l) 1.4x102 1.3x105 304
Cr(bpy)§+ v(n20)§+ 4.2x102 (1) 1.1x103 2.6x106 310
Co(phen)§+ «320):+ 4x103 (1) 1.1x104 7.3x105 31o
Co(bpy)§+ «1120):+ 1.1x103 ,(2) 2.3x103 2.3x105 407
Ru(NH3)g+ Ru(NH3)§+ 4.3x103 (0.1) 1.9x105 2.2x106 268
Ru(en)§+ Ru(NH3):+ 2.7x104 (0.1) 7.1x1o5 1.6x108 28
Egan):+ Ru(en)§+ 2.8x104 (0.75) 1.3x105 2.5x107 12
Co(en)§+ Co(en)§+ 8x10"5 (1) 3x10-4 1.1x10’4 269
Ru(bpy)§+ Ru(phen)§+ 4.2x108 (0.1) 1.6x109 7.2x108 317
Ru(bpy)§+ Fe(phen)§‘+ 1.8x109 (1) 3.2x109 3.4x1010 317
Ru(bpy)§+ Co(bpy)§+ 2.4x108 (1) 4.3x108 2.1x10lo 408
Ru(bpy)§+ Co(phen):+ 1.4x108 (1) 2.5x108 1.1xlOlo 408
Co(phen):+ Co(phen)§+ 40 (0.1) 1.5x102 5x10”1 409
Co(bpy)§+ Co(bpy)§+ 20 (0.1) 74 5x10’1 409
Co(bpy)§+ Ru(NH3):+ 1.1x104 (0.1) 9.2x104 4.62.106 310
Co(phen);+ Ru(NH3)§+ 1.5x104 (0.1) 1.2x105 1.7x107 310

00(31):;+ Cr(bpy)§+ 35 (0.1) 2.2x102 4.9x103 361

383

those involving considerable inner-shell reorganization around one metal
center and none around the other.

The free energy driving force can be represented as a displacement
of the wells of the free energy curves which are generated in calcula-
ting Af and Ar' If both outer- and inner-shell reorganization energies
are taken to be quadratic functions of the nuclear coordinates, the
energy at the intersection point of the free energy curves, i.e. AG*,
can be calculated from

AG IX Af (7.7)
where the value of the nuclear coordinate X is found by solving Equation
7.8:

2

A X

_ _ 2 o
f Ar (1 x) + A6 (7.8)

Values of AG0 were determined from the formal potentials listed in Table
7.1. Activation free energies for electrochemical reactions were calcu-
lated in a similar manner.

Solvent reorganization energies for homogeneous reactions were
calculated. by ‘using the ellipsoidal cavity' model described. by

374,375

Cannon. The ellipsoidal cavity model allows for interpenetration

of reactant coordination spheres, whereas as the conventional two-sphere
model of Marcus in principle does not. In light of recent theoretical

14 15

work ’ suggesting that reactant interpenetration is generally re-

quired in order to achieve significant coupling of donor and acceptor

384

 

ENERGY

 

 

NUCLEAR REACTION COORDINATE

Figure 7.1 Schematic representation of the relationships between the
nuclear reaction coordinate, forward and reverse reorganization energies,

and the classical free energy barrier to electron transfer.

385

electronic orbitals, the Cannon model was considered to be more appro-
priate than the two-sphere model. Reorganization energies were calcu-
lated according to Equation 7.9:

2 2 2 2
Aout I (Ne r [2131b)(1/sop - 1/E8) S ( )5) (7.9)

where e is the electronic charge, r is the distance between the redox

centers, 1a and 1b are the semi-major and semi-minor axes of an
ellipsoid encompassing the reactants, Sop and 68 are the optical and
static dielectric constant of the solvent and S( :3) is the "shape

factor.“ This last term depends in a complicated manner on the

eccentricity of the ellipsoid, but can be reasonably approximated as

3 (A0) 2‘. 1.19 - 0054 e (7.10)

for 0.8§_e:1.

The major semiaxis la equals (a1 + a2 + r)/2 with radii a1 and a2
listed in Table 7.1. In the absence of additional information, coor-
dination spheres were assumed to interpenetrate such that redox centers
were 1.25 X closer than expected from close contact of reactants. Such
an estimate is appropriate at least for reactions between aquo

14.15.17

complexes. The minor semiaxis 1b were calculated by taking the

volume of the ellipsoid to be equal to that of the two isolated reactant

spheres. The values of cop and 68 respectively, at 25°C are 1.78 and

78.3.

386

Solvent reorganization energies for electrode reactions were

calculated from13

2
l a — e - e
out (Ne Ir)(1/a IIRQ)(1/ op 1/ 8) (7.11)
where Re is twice the distance from the redox center to the electrode
(i.e. the distance from the charged reactant to its image in the
electrode). A value of Re I 78 is obtained if the reaction site is
assumed to correspond to the outer Helmholtz plane.

Inner-shell reorganization energies were calculated from

A $3" - 3fu’m'Ad (7.12)
where fM,n+ is the force constant for the symmetrical breathing motions
of six identical metal-ligand bonds and Ad is the difference between
the equilibrium bond distances for the two oxidation states. Intra-
ligand bond vibrations and contortions were ignored since these are
anticipated to provide only minor contributions to Ain' values of Ad
are listed in Table 7.1. Since structural data are lacking for hexaaquo
vanadium(II), the change in the metal-oxygen bond length for the
V(H20)2+/2+ couple was estimated from the vanadium-oxygen bond lengthi
in related oxide compounds. Beattie, et al. have shown that there is a
close correspondence (within circa 0.02 3) between the metal-oxygenIAd
values found in oxides and those found in hexa-aquo metal complexes.326

Given the structural similarities, the low spin Fe(bpy

g+l2+ g+l2+ and Ru(terpy)§+/Z+ couples were each assumed

3+/2+
)3 .

Ru(bpy) , Ru(phen)

387

to require negligible inner-shell reorganization, as for the low spin
Fe(phen)g+/2+ couple (Ad I 0.08).376
Force constants were calculated from

. 2 2
f AflVinc u (7.13)

where c is the velocity of light and u is the reduced mass of the li—
gand. For cobalt complexes containing large bidentate ligands, force
constants for the cobalt-nitrogen stretch were taken to equal those for
hexaammine cobalt (III) and (II). Although values of vin are available
for the bidentate ligand complexes suggesting that Equation 7.13 could
be employed to calculate force constants, it is not clear what values of
the reduced mass should be used since the metal-nitrogen vibration may
be partially decoupled from the motion of the ligand as a whole.
Nuclear tunneling factors (Tn) can be expressed as AG*(Q) values , thus
emphasizing that these represent the difference between the average
quantum barrier AG*(T) and the classical Franck-Condon barrier AG*.

19,20,377,378

From Holsteinfis work on small polaron motion, , Sutin, et

al. have derived expressions for AG*(T) for reactions without (Equation
9,23

7.14) and with (Equation 7.15) a free energy driving force. The
expressions for AG*(T) are:
AG*(T) - 1(kT/hv) tanh(hv/AkT) (7.14)

and

388

AG*(T) I 0.5AGO’ + (kT/hv){coth(hv/2kT)-
[(AG°’)2/Az + csch2(hv/2kT)]1/2 +
( c°’/1)ainh'1[(Ac°'/A)sinh(h»/2kr)]} (7.15)

Equations 7.14 and 7.15 were derived for tunneling processes involving a
single vibrational mode characterizd by equal force constants and
vibrational frequencies in the two oxidation states. Accordingly AG*(T)
and refer to only one mode in each calculation. Although AG*(T)
values which are calculated assuming equal force constants will
certainly differ from those obtained using unequal force constants and
frequencies, the difference, AG*(Q), between quantum and classical
barriers under simplified conditions should be similar to that obtained
when quantum and classical barriers are both calculated more rigorously.
Thus, simple single-frequency modes were assumed in calculating AG*(T)
and AG*(Q), with the tunneling corrections then being applied to AG*
values which had been calculated in a more complicated manner (m
m). Tunneling corrections for solvent repolarization were

considered to be negligible since v is probably significantly less

out
than kT/h. Tunneling corrections for each inner-shell mode were
calculated separately. Thus the driving force AGO’ appearing in
Equation 7.15 was taken in each calculation to be an appropriate
fraction “Mm-PAR of the total driving force AGO. This insures that

AG*(Q) reaches zero at the same AGO value at which AG* is zero. A point

of confusion concerns the use of 1360’ given that Equation 7.15 is

389

'derived using a temperature independent driving force AE°’,. Although
the substantial ASo values found for many reactions reflect chiefly the
solvation changes accompanying the overall electron transfer process,
this entropy change nonetheless would seem to provide a driving force
for reorganization of both inner- and outer-shell modes. At least for
the present then, A602 is used in place of AEOT.

The methods outlined thus far enable rate constants to be
calculated using Equation 7.2. However. it is also of interest to com-
pare the theoretical predictions concerning activation enthalpies and
entropies with the corresponding experimental quantities.

Activation entropies (classical limit) were calculated for

homogeneous reactions according to:

I
AS

0 o o
calc 0.5AS '0' AG AS [If (7.16)

Contributions to AS* arising from the temperature dependence of 1 out
were neglected since these amount to 2 J deg-1 11101”1 at most when cal-
culated by Marcus’, theory.13 (However, see Section VI. A). Equation
7.16 was derived by assuming equal force constants for inner-shell

reorganization in different oxidation states. Nevertheless, this

approximation yields little error since the purely thermodynamic term

*

(0.5 43°) usually provides the dominant contribution to Ascalc'

Also,
the use of If rather than a "reduced“ reorganization energy nullifies to
some extent the errors resulting from the force constant approximation.

The required values of A30 were obtained from the thermodynamic reaction

390

entropies 433‘: for the individual half-reactions (Table 7.1). For

. . * . .
electrochemical reactions ASCalc is given by:

* o
Ascalc oAsrc (7.17)
where a is the electrochemical transfer coefficient and the contribution
from Idlf/dT again has been ignored.
Classical activation enthalpies were determined from the relation:

* t

*
AHCalc - 11ccalc + 'rAscalc (7.18)

where AH* is defined as -R(d1nk/d(l/T)).

Nuclear tunneling corrections, An*(Q) and AS*(Q), to activation
parameters were obtained by calculating AG*(Q) at various temperatures
(Equations 7.14 and 7.15) and noting that - AG*(Q)/ r - AS*(Q), while
AG*(Q) + TAS*(Q) I AH*(Q). The resulting tunneling-corrected parameters

are designated as AS*(T) and An*(r).

3. Kinetics Formulations and Work Corrections for Experimental

W
Activation parameters are usually extracted from experimental rate
data by assuming a pre-exponential factor of kT/h rather than szn.
These "Eyring" (kT/h) activation parameters (designated using

superscript daggers) can be converted to "pre-equilibrium” (szn

prefactor) activation parameters using:

391

13* - AS #-8 + a 1n(xzvnh/kr) (7.19)

and
111* .. AH#+ RT (7.20)

thereby enabling straightforward comparisons to be made between theory
and experiment.

Experimental rate constants for homogeneous electron transfer were
corrected for the electrostatic work of precursor formation by way of
the Debye-Huckel treatment. This approach is embodied in Equation 7.21:

log k

corr I logk + zAzBezN/2.303RT ssr(l+8r111/2) (7.21)

where 2A and 2B are the reactant charge numbers,8»is the Debye-Huckel
parameter,11is the ionic strength and Rear: is the rate constant which
is expected in the absence of electrostatic interactions. Since several
kinetics studies have indicated that activation enthalpies decrease
systematically with increasing ionic strength while activation entropies
remain essentially unchanged,28’379-381 it was deemed appropriate to
incorporate work corrections wholly in AH*.

Electrochemical rates were corrected for electrostatic diffuse

double-layer effects by using the Gouy-ChapmanISternIFrumkin theory as

described in Sections 1.0 and 4.0.

392

C. Results

Calculated rate constants are compared in Figures 7.2 and 7.3 with
work-corrected experimental rate constants. The reactions are separated
into two groups: those involving a pair of reactants possessing iden-
tical ligand compositions (Figure 7.2) and those involving reactants
possessing dissimilar ligand compositions (Figure 7.3). In order to
compare electrochemical and homogeneous reactions on a common basis,
rate constants for the former (kE ) were converted to second-order

calc
E

rate constants by multiplying k values by the ratio of statistical

calc
factors for the two types of reactions, i.e. Rz/Gr. values of kcalc and
kcor r are listed in Tables 7.2 and 7.3 together with observed
experimental rate constants. values of AC*(T), AG*(Q), AC°,Af, E2 and

Vn are given in Table 7.4. The calculated rate constants plotted in
Figures 7.2 and 7.3 were obtained using individual force constants and
tunneling corrections for the inner-shell, with the Cannon model being
used to calculate the outer-shell barrier for the homogeneous reactions.
The use of individual force constants yielded rates that were typically
three-fold, but in a few cases as much as twenty-fold, smaller than
calculated using reduced force constants. Allowing for reactant
interpenetration usually decreased. .AG*(T). Nuclear tunneling
corrections increased the calculated rate constants. Values of In range
from 1 to 17, the largest values being found for low driving force
reactions requiring substantial inner-shell reorganization. values of
l 1

K0 range from 0.13 M-

A for reactions between aquo complexes to 0.9 M-

for reactions between polypyridine complexes.

393

 

 

 

 

l I T I
//
'0’ /18 w
3’. / 6
)9
/ 405°
/
/
/ g
)1
A: 7
5- // . 359:9 _.
t / I'oo 5
x5 /
/ I0
4| / O
.5 3 3 g / ‘
o I? /
//
0e / _
/ I
/ o
/
/
36 /
5 L l l l
-5 0 5 IO
RMIMT)

Figure 7.2 The log of the work-corrected experimental rate parameter
kcorr plotted against the log of the corresponding theoretical parameter
k(T) for electron transfer between reactants possessing identical ligand

compositions. Key: (C» aquo complexes: (A) polvpvridine complexes:

(O) ammine complexes. Reactions and rate constants listed in Table 7.2.

394

 

 

 

 

I
no» ‘
35
S _
:5 5"
!.
3
5 _
.1 0’
3
-5 1 l l

 

'5 0 5 l0
log k(T) and Iog[K;km/sr]

Figure 7.3 The log of the work-corrected experimental rate parameters

0 .

kcorr and KAkcorr/ér plotted against the log of the corresponding theo
retical parameters k(T) and K:k(T)/6r for electron—transfer between re-
actants possessing dissimilar ligand compositions. Key to cross-reactions:

GB) aquo-ammine; (A) aquo-polypyridine: (A) ammine-polypyridine; (-—9

electrochemical. Reactions and rate constants are listed in Tables 7.2

and 7.3.

46

47

48

49

50

51

Table 7.3.

395

Experimental and Theoretical Rate

Constants for Electrochemical Electron-Transfer Reactions

k

(kJ-mol-l) (cm 5.1)

 

AG°
Reactant
Ru(NHB) 2" o
410
at mercury
Ru(H20)2+ o
313
at mercury
Fe(320)2+ -20
313
at mercury _30
-40
—so
«1120)? +30
313
at mercury +20
+10
0
-1o
v<n 0)3+ -2o
2 6
136
at lead _30
-40
3+
Cr(HZO)6 80
313
at mercury 70
60
so
40

0.3

52:10‘3

3 . 0x10’
2 . 3x10“
1 . 7x10“
1.3x10"
2.1x10-
4 . 7x10-
1x10'4

Léflfl-
L3mo‘
LSflD-
4.7x1o'

2.8x10’

4

3

2

l

7

6

3

2

4

3

2

 

(KP/5r)x (KP/5r)x
corr corr k (T)
(cm 8.1) (Mile-l) I (Mrls-l)
1.5 3.4x107 5.8x107
2x10"2 4x105 7.2x106
~1.2x10'3 2.5x104 1.3xlo7
~9x10‘3 2x105 8x107
~7x10"2 1.5x106 4x108
"52:10"1 1x107 1. 7x109
2.6x10’7 5.4 4.9x102
6x10’6 1.3x102 2.6x103
8.7x10‘5 1.8x103 1.4x104
5.7x10‘4 1.2x104 8.3x104
4.7x10‘3 1x105 6.8x105
~1.8x10‘4 ~3.8x103 4.9x106
~1.3x10"3 ~2.7x104 3.1x107
~9x10’3 ~2x10S 1.7x108
1.6x10'16 3.3::10‘9 2.4x10'
4.6x10'15 9.5x10'8 7.4x10’
1.3::10‘13 2.7m)"6 2.6x10‘
3.23.10“12 6.6x10"S 9.5x10'2
8.1x10"ll 1.7x10’3 3.8x10'

52

53

54

55

396

 

Table 7.3.
AG° k
-1 --1
Reactant (kJomol ) (cm s )
30
20
1o
0 1x10"5
-10 1.11110"4
-20 1.21.10"3
-30 1.2x10'2
Cr(H o)3+ -40 1.8x10'4
at liad6136 -3
-50 2x10
-60 2.62.10“2
Cr(H 0)3+ -6o 1.51:10"3
2 6 136
at gallium
Cr(H20)2+ -20 1.7x10'4
at UPD lead/ 30 1.6x10-3
silver
-40 1.5x10"2
«(1120):+ -30 1.7x10'4
at UPD -3
thallium/silver -40 1'3Xl0
-50 1x10"2

(continued)

COI‘I‘

1.9x1o'
$Sflfl-
3.7x10'
LGo'
L7mo‘
LOflD-
1.6x1o'
1.9x10-
1.7x10'
1.6x10‘

3xlO-S

5x10‘5

51:10‘4

4.5x1o'
1.7x1o‘
1.3x1o'

1210'2

1)

(cm s-

9

8

7

6

5

4

3

5

4

3

3
4

3

 

(KP/6r)x (KP/5r)x
kcorr k (T)
(M71S-1) (M518-1)
4.1x10'2 1.7
7.2x10"l 8.0
7. 7 4.22.101
6.6x1ol 2.4x102
5.611102 2.22.103
4.2x103 1.8x1o4
3.3x1o4 1.3x1o5
4xk92 8. 51:105
3.6x1o3 5.1x106
3.3x1o4 2.7x1o7
6x102 2.7x1o7
1.1x103 1.8x104
11:104 1. 3x105
9x104 8.5x1o5
3.5::103 1. 3x105
2.7x1o4 8.5x1o5
2x105 5.1x1o6

397

 

Naoaxo.o moa.o on m.o sea mod m.mH ma
NHonm o~.o m.mH q.H and com o.qm NH
Naoaxa mea.o «.ma m.H “NH mHN o.mo as
Naoaxm «NH.o n.5m o.~ Nam wHN e.ss oH
«HonH mNH.o ms s.~ cow mnN H.sm a

maoaxa.a m~H.o mm m.¢ hem mam m.HHH m

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a; we Aav«u< ongoqn pa we ooqn coauummm

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.s.e magma

398

NHonm.m o¢.o Hm o.q mmm mmm o.wo om

 

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399

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403

The use of individual rather than reduced force constants for
electrochemical rate calculations lead to a significant asymetry
between the calculated oxidation and reduction rate-potential plots.

ThUB Tafel 910‘s 0f 108 kcalc versus E exhibit a marked curvature on the

anodic side while cathodic plots are nearly linear over several hundred

3+/2+

millivolts. For the Cr(H2 0)6 couple for example, the inner-shell

asymmetry yields a calculated reduction transfer coefficient of 0.48 at
an overpotential of -300 my, while the oxidation transfer coefficient
equals 0.38 when E-Ef - +300 mV. Despite assertions to the
contrary,382’383 neglect of solvent contributions to the reorganization
energy results in rather poor agreeement between experiment and theory.

work-corrected experimental activation entropies are compared with
calculated values in Figure 7.4 for the available electrochemical and
homogeneous reactions. A similar comparison of activation enthalpies is
made in Figure 7.5. *Listings of AH*(T), AH*(Q), AB* AH, a£1311"

corr’

AS*(T), AS*(Q), AS and AS0 are assembled in Tables 7.5 through

scorr’
7.8. Nuclear tunneling contributions to AH*(T) and As*(T) were found to
be larger than the contributions to AG*(T). In each case such contri-

butions yielded smaller activation enthalpies and larger (less negative)

. O O I I I *
activation entropies than obtained Via the classical model.

 

*It is stated in reference 17, that AH*(Q) represents the amount by
which the classical Franck-Condon barrier is lowered by nuclear
tunneling, while As*(Q) represents a dynamical correction to the rate
process. In fact both AH*(Q) and AS*(Q) describe the direct influence
of tunneling on the Franck-ffpdon barrier. A "dynamical correction"

{ZkT sinh (h\;. l2kT)/h\;i,}n appears as a pre-exponential factor in
some formulations of \) (reference 377), but is essentially negligible,
varying between 1 in the low temperature limit and 1.05 at 298K.

404

 

 

 

 

Ti l I
/ *A'
/ 51
/
37 / /
49
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/
/
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‘ ‘01
12 32
A -848 :'
20
23. ‘
-150 ’- 16 —1
O
23 111
1 L 1
-50 0 50 100
MW)

Figure 7.4. Experimental activation entropies versus theoretical acti-
vation entropies. Key to reaction tvpe: (ikfl eleccrochemical; others as

in Figures 7.2 and 7.3. Results listed in Table 7.4.

405

 

 

 

 

 

3 ..
O
‘5
E
a
x
E
O
c U
:1:
<1
I l l l l
0 so 100

AH‘ (T). kJ mor'

Figure 7.5. Work-corrected experimental activation enthalpies versus
theoretical activation enthalpies. Key to reaction type as in Figures

7.2 and 7.3. Results listed in Table 7.5.

406'

Table 7.5. Experimental and Theoretical Activation Entrapies and

Entropy Driving Forces (J deg"1 mol-1) for Homogeneous Electron Transfer.

 

 

Reaction No. AS° AS AS* AS*(Q) AS*(TL
l 0 -105 -83 -l4 -l4
2 0 -105 -83 -20.5 -20.5
3 0 -88 -67 -42.5 -42.5
4 71 -96 -74 -25 -2.5
5 46 -33 -12 -14 -5
7 29 -84 -62 -14 —3
9 25 -109 —87 —6.5 2
11 105 -92 -71 -9.5 17
12 126 -92 -73 -9.5 37
16 -79 -l76 -155 -8.5 -40
17 0 -109 -91 -22 ~22
19 —l76 -138 -125 -8.5 -60
20 -l76 ~155 -l42 -8.5 -60
21 -l76 -176 -163 -8.5 -60
22 -l67 -l76 -l63 -6.5 -69
23 -l72 -159 -l46 -6.5 —66
28 -l46 -92 —85 0 -30
31 -63 -121 -108 -15 -40
32 -63 -l38 -125 -15.5 -41
33 0 -54 -25 -l.5 -2
34 -21 -113 -87 -l -10

36 0 -92 -75 -30 -30

407

Table 7.5 (continued)

 

Reaction No._ AS° AS+ AS* AS*(Q) 43*(T)
37 0 29 53 0 0
41 o -142 —134 -15 -15
42 o -113 -105 -15 -15
43 17 -105 -90 -8 -1
45 138 ‘ ~63 -52 -15 61
44 17 -105 -90 -8 0
39 -88 -124 -l3l -4 -24

40 -88 -104 -111 -4 ~25

408

0mmH am: um

 

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.0.m oHan

409

 

m.m1 m.m m.H1 n.H1 «oH1 +MAoN=van +MAsanvam aH
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.50 mHan.

410

 

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«m «H «.mm n.0m o +mAaovou +MAamvoo on

NH «.o m.« OH mH1 +wAmmzv=m +MAamV=m «m

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+N is

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411

 

N N.H «61 m.«1 81 +MA5e38 M2303 o«
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412

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413

 

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414

D. Discussion

The overall impression that is conveyed by the rate comparisons in
Figures 7.2 and 7.3 is that the theory of weak-overlap electron transfer
is remarkably successful in describing the energetics of such reactions.
The experimental rate constants do tend to be smaller than the calcu-
lated values. This might be taken as an indication that nonadiabaticity
prevails in most reactions even at the closest approach distance.
Nevertheless, it is worthwhile to search further for explanations of
rate disparities.

A useful approach is to examine the reactivity of metal complexes
as a function of ligand and electronic structure. Reactions of aquo,
ammine and electrode co-reactants are examined in Figure 7.6 by plotting

the difference between log k and log k(T) against a normalized free

corr
energy driving force coordinate. Although the data for the homogeneous
reactions are scattered, the overall indication is that the reaction
rates increasingly deviate from theory as the driving force increases.
This trend is reminiscent of earlier observations based on relative rate

comparisons via the Marcus cross relation.3m’311

(See Section VI. B).
However. explanations in terms of the shortcomings of the cross relation
are inapplicable here, since the various approximations inherent in that
approach (cancellation of electrostatic work terms, additivity of
outer-shell reorganization energies, etc.) are not made in the present
analysis.

The driving force dependence of the breakdown of theory suggests

that the explanation lies in the calculation of Franck-Condon barriers,

rather than in neglect of work terms, nonadiabaticity or other pre-

415

 

 

 

 

 

l I l
5
30 H
S%
4
o 9
12 o
O
‘8
2.. .1
.5.
x
T;
v
5
8 I’- —
Or- 15 H
2
o
111 l L
O I 2 3

- 8m0/(Af‘b'lr)

Figure 7.6. Log [k(T)/kcorr] for reactions involving aquo complexes

0 . .
plotted against the reduced driving force AC /(lf+ir). Key to reaction

type as in Figures 7.3 and 7.4.

416

exponential factors. Interestingly. the apparent breakdowns for homo-
geneous reactions are paralleled in the electrochemical oxidations of
“320):+ and Cr(H20)§+. but not in the corresponding electrochemical
reductions. Nearly all of the homogeneous reactions included in Figure
7.6 also involve aquo oxidations. A reasonable hypothesis is that the
driving force dependent rate discrepancies for both the homogeneous and
electrochemical reactions arise because the rate calculations over-
estimate the extent to which the activation free energy for the aquo
complex is diminished by increasing the driving force. A speculative
explanation for this centers on hydrogen bonding between aquo ligands
and the surrounding solvent. It is usually assumed that Aout possesses
the same value for forward and reverse reactions. Nevertheless there is
ample evidence of enhanced hydrogen bonding in the M(EZO):+ state com-
55,222 3+

This suggests that A could

. 2+
pared with the M(HZO)6 state. out

exceed Aoutzfi which in turn would yield an even greater asymetry
between aquo oxidation and reduction reactions than is expected from
inner-shell considerations alone.

A possibility which cannot be dismissed is that the driving force
dependence of log [(kcorr/HTH represents merely a coincidental
ordering of experimental and calculational errors, at least for the
homogeneous rate data. Indeed, the calculated rate constants could be
in error by as much as a factor of twenty in some cases due to errors in
estimating An and therefore xin' Such uncertainties emphasize the need

for accurate measures of symmetrical metal-ligand stretching frequencies

in addition to data concerning metal-ligand bond lengths.

417

3+/2+
6

since it has often been suggested that these are somehow anomalous. The

The cross reactions involving Co(HZO) are of special interest
rate comparisons summarized in Figures 7.2, 7.3 and 7.6 indicate that
while such cross-reaction rates do indeed deviate frmm the predictions
of theory, these deviations are comparable to those found for other
reactions. The Co(HZO)2+’2+ self-exchange rate however, exhibits
enormous deviations from theory. Endicott and coworkers have suggested
that the hexaaquo cobalt (III)I(II) self-exchange follows a catalytic
inner-sphere pathway, a conclusion which is strongly supported by the

present calculations of outer-sphere rates.384

Curiously, the activa-
tion entropy for this reaction is typical of that found for genuine
outer-sphere reactions (Figure 7.4), suggesting that the magnitude of
AS* is not particularly diagnostic of the reaction mechanism.

Reactions between aquo and polypyridine complexes form an
interesting group. On average the cross reactions involving low-spin
polypyridine complexes with aquos are nearly four orders of magnitude
. slower than predicted (Figure 7.7) while the rates of corresponding
reactions of pairs of aquo complexes and pairs of polypyridine complexes
evidently agree more closely with theory. The discrepancies between
kcorr and k(T) appear to be independent (or nearly so) of the reduced
driving force for the reaction. These observations suggest that some
type of barrier to precursor formation, perhaps arising from an
energetically unfavorable interaction between aquo and polypyridine
ligands, might be the source of the rate discrepancies.

Alternative explanations of rate disparities in terms of

miscalculation or neglect of factors contributing to nuclear reorgani-

418

 

23 13
_ 19. 20.21 _
a a
32
29 22
a

log 111(T)/11.°,,1

 

 

1 1 1 1 1
O I 2 3

-8 AGOI (Ag Ar)

 

Figure 7.7. Log [k(T)/Reorr] for reactions between polypyridine and aquo
complexes plotted against the reduced driving force AGO/(Af+lr). Key:
(A) low-spin polypyridines with aquos; CV) high-spin polvpyridines with

aquos.

419

zation energies appear to be untenable. The effects of such errors
should be attenuated with increasing exoergonicity, yet rate discrep-
ancies persist even in the 8110120):+ - Ru(bpy);+ cross reaction for
which AGO approaches if.

The behavior of aquo complexes with polypyridine reactants con-
trasts with the behavior of aquo complexes with other coreactants. One

might speculate that some of the pecularities of 110120)?”2+

reactivity,
which are tentatively attributed here to hydrogen bonding interactions
with the solvent, are somehow circumwented when the aquo reactant is

placed in close proximity to a polypyridine coreactant. Interestingly
solvent "structure breaking“ capabilities have often been attributed to

the latter.306

The activation parameters for the aqua-polypyridine reactions are

characterized by AS* values which differ from theory by some 85 J deg-1

mol-l, a discrepancy similar to that found for a variety of other cross

*
reactions (Figure 7.4). In contrast, AH

values are quite similar to
corr

the predicted values. Such enthalpic behavior.distinguishes these cross

values that are

*
01':

reactions from most others, which instead exhibit AH.c
smaller than expected (Figure 7.5). Evidently, AH* rather than AS* is
the parameter signaling a typical rate behavior. It is perhaps not use-

ful to speculate concerning the details of the postulated work term for

aqua-polypyridine reactions, beyond suggesting that it is partly

enthalpic.
. . . 3+/2+
Since cross reactions between aquo couples and either Co(bpy)3
or Co(phen)§+/2+ agree more closely with theory than do the corre-

sponding reactions involving low-spin polypyridine couples, it is worth-

420

while to ask whether these differences are related to the differences in
electronic structure between the cobalt couples and the others. Although
this is an intriguing possibility, the most likely explanation is that
the reactivity differences are due to miscalculations of metal-ligand

bond reorganization energies for the Co(phen)§+/2+ and Co(bpy)3+/2+

3
couples. Substantial calculational errors are certainly possible for
these couples given the tentative nature of the force constant estimates
and the 18r8e uncertainties (10.027 8) in the EXAFS estimate of Ad. For
reactions requiring very large adjustments of bond lengths even small
uncertainties in Ad become significant in the rate calculations. This
suggestion also accounts for the otherwise puzzling result that kcorr
exceeds k(T) for the Co(bpy)g+/2+ and Co(phen)g+/2+ self-exchanges.
Note that catalytic inner-sphere pathways which might cause kcorr to
exceed the calculated outer-sphere rate constant are not available since
both the Co(III) and Co(II) centers are substitutionally inert.

“Cross reactions“ between metal complexes and electrode surfaces
form a distinctive set of electronrtransfer reactions, one in which the
thermodynamic driving force can be varied without altering the chemical
identity of the reactants. Complexes exhibit different degrees of
outer-sphere reactivity with different electrode co-reactants, even
after correcting rate constants for electrostatic double-layer effects.
Such differences are intriguing since they are unexpected from electron

transfer theory. Despite the differences in kco values, the varia-

rr
tions of rate constants with driving force are essentially the same for

the reduction of Cr(HZO):+ (and other reactants) at different surfaces.

Liu has suggested that differences in kcorr can be understood in terms

421

of the degree of hydrophilicity and corresponding interfacial solvent

ordering that is exhibited by each metal surface.136

The least hydro-
philic surface, mercury. appears to correspond most closely to an ideal
“weakly interacting” coreactant, while the most hydrophilic surface,
gallium. strays furthest from ideal behavior. Activation parameters are
markedly more sensitive to electrode composition that are the kcorr
values. The electrostatic effects of double-layer structure an
electrochemical kinetics parameters are better understood than the
corresponding salt effects on homogeneous kinetics. Thus it can be
claimed fairly confidently that the variations in activation parameters,
as well as rates, represent something other than uncertainties in
electrostatic work terms. The apparent importance of electrode-solvent
interactions suggests that specific solvent interactions with homo-
geneous co-resctants, most notably polypyridine complexes, may also bear
some relation to electronrtransfer reactivity.

There are too few examples of each of the other groups of cross
reactions to be able to isolate ligand-specific reactivity patterns. A
general observation however is that a rather less optimistic view of the
success of electronetransfer theory emerges when cross reactions are
included in rate comparisons.

Comparisons between calculated and experimental activation
parameters, which have been alluded to in the foregoing discussion, form
a more demanding test of theory than do the rate comparisons. Overall

the results summarized in Figures 7.4 and 7.5 are nothing short of re-

markable. Clearly. it is possible to account quite adequately for

422

variations in AB? and especially, AS* as the thermodynamic and struc-
tural properties of the reactant are varied.

On an absolute level however. the agreement between experiment and
theory is less satisfactory. we noted above that a discrepancy of 85 1

1

25 J‘deg- mol-1 is observed between A3* and As:alc for nearly all

corr

the homogeneous reactions. One explanation is that outer-sphere elec-
tron transfer reactions are in most cases strongly nonadiabatic, such
that the value of :31 is about 10‘s. Besides being inconsistent with
the few 52 inigig studies of electronic coupling between outer-sphere

14,15

reactants, such a value is simply too small to account for the

observed rate constants.

Various other possibilities were considered in a recent examina-
tion of the effects of reactant-solvent hydrogen bonding on activation
entropies (Section V. A). There it was concluded that the differences
between As:orr and As:alc almost certainly cannot arise from the tem-
perature dependence of the Franck-Condom barrier, i.e. AG*(T), but
instead must represent a work term associated with precursor formation.
Friedman has arrived at a similar conclusion on the basis of his studies
of ion-pair correlation functions for aquo complexes.17 Specifically,
the negative activation entropy for the Fe(H20)2+/2+ self-exchange is
assigned in his work to coulombic repulsion between the charged reac-
tants. A significant finding is that the entropic component of such
electrostatic interactions is expected to be nearly independent of ionic
strength. The common feature of the set of homogeneous reactions exam-

ined here is that each involves a reactant of charge 2+ and another of

charge 3+. Provided that electrostatic interactions are controlled

423

essentially entirely by the charges of the reactants, with the reactant
size and ligand composition being of little importance, such inter-
actions provide a very plausible explanation. various other factors
such as steric requirements, marginal nonadiabiticity, variations of Kel
with temperature. etc. likely contribute, at least to a minor extent, to
the observed discrepancies between experiment and theory.

Returning to the adiabaticity question, an unexpected finding is
that the 88* values for reactions involving cobalt centers are no more
discrepant than for other reactions. For the cobalt couples electron
transfer is a formally spin-forbidden process. Jortner and co-

385

workers have noted that the spin restriction an be partially overcome

through spin-orbit coupling, leading to an electron transfer probability
in the case of Co(NH3)2+’2+ self-exchange that is 10-4 smaller than
expected in the absence of spin restrictions. Since evidence for such
spin-related nonadiabaticity is wholly lacking in the rate and activa-
tion entropy comparisons (note also the conclusions for Section VI. C),
one suspects either that spin-orbit coupling is a more successful reac-
tion scheme than previously believed or that another mechanism is
available to circumvent the spin restriction.

Offsetting the unfavorable experimental activation entropies are
activation enthalpies which tend to be more favorable than predicted.
Overall, the differences between1Aflzorr and1AR*(T) may possibly have the
same origin as the AS*(T) - As:orr differences, namely. electrostatic
interactions that differ from those assumed in making work term correc-

tions. The larger scatter in AH* values compared with AS* values is not

unexpected, since calculations of activation enthalpies are more sensi-

424

tive to errors in estimates of reactant structural parameters. However,
scatter in the (18* data could additionally result if breakdowns in
theory arising from inter-reactant and reactant-solvent interactions, as
well as departures from purely outer-sphere reaction pathways, are

manifested chiefly as enthalpic effects, as has been suggested above.

E. Conclusions

In a global sense electron-transfer theory is reasonably
successful in predicting the rates of self-exchange and cross reactions
as well as those for electrochemical reactions. The lingering discrep-
ancies of 10 to 104 -fold seem to arise chiefly from specific, non-
electrostatic reactant-reactant and reactant-solvent interactions.
Cross reactions between aquo and polypyridine couples are the best
example of the former, while electrochemical reactions are the strongest
example of the latter. Nonadiabaticity evidently is not exceptionally
important in reactions requiring formally spin-forbidden electronic
transitions, although in general nonadiabaticity certainly could be
significant. The variations of activation parameters with thermodynamic
and structural factors generally are well described by theory. However,
the theoretical estimates of AH" and 118* are substantially different to
the experimental values. Such errors, at least in homogeneous
reactions, probably represent mainly miscalculations of electrostatic

work terms rather than failures of electron-transfer theory itself.

425

CHAPTER VIII

CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK

A. Conclusions

Differential capacitance experiments indicate that silver is
capable of adsorbing several simple inorganic anions to the extent of a
monolayer at the most positive accessible potentials. Electrochemical
toughening which is required in order to observe SERS signals from
adsorbed ions was found to produce only minor changes in the average
surface concentrations of such anions at silver.

Underpotential deposition of a single atomic layer of lead on to a
silver electrode was sufficient to transform the double-layer structure
and adsorption properties of the surface essentially to those of a bulk
lead electrode. Thus, adsorption is greatly diminished at the UPD
lead/silver. as well as UPD thallium/silver surfaces in comparison to
unmodified silver. The differences in anionic adsorption properties
between these three surfaces as well as mercury appear to be related to
differences in the degree hydrophilicity of the metal electrodes.

In order to evaluate electrochemical kinetics at a molecular level
and to correlate electron-transfer reactivity in different environments
an "encounter preequilibrium" treatment was refined and further devel-
oped. This treatment provides a more satisfactory theoretical descrip-

tion of redox processes than is obtained from the conventional colli-

426

sional model. In particular, a clearer picture of the physical
signifiance of the frequency factor emerges from the former.

The preequilibrium treatment was applied to the problem of the
influence of the metal surface composition on redox reactivity. Rates
of inner-sphere electron transfer reactions at silver are diminished by
three to five orders of magnitude when the surface is modified by
underpotentially depositing a monolayer of lead or thallium on silver.
An analysis on the basis of the encounter pre-equilibrium model
indicates that the reactivity differences are associated with a change
in reaction mechanism from inner- to outer-sphere and an accompanying
decrease in the precursor stability constant. Surprisingly, rate
constants for the elementary electron-transfer step for several
chromium(III) reductions are closely similar at four different electrode
surfaces.

A careful consideration of entropic driving forces for intra-
molecular electron-transfer indicates that, contrary to common belief,
such reactions typically involve a significant Franck-Condon activation
entropy. This complicates the analysis of frequency factors from the
point of view of monitoring nonadiabaticy. Nevertheless when carried
through, such an analysis can account for peculiarities such as positive
activation entropies and the marked sensitivity of apparent frequency
factors to nonrbridging ligands.

From an evaluation of rate responses to systematic alterations of
double-layer structure together with rate comparisons for parallel
inner- and outer-sphere reaction pathways it is concluded that

chromium(III) aquo reductions are mildly nonadiabatic at the mercury-

427

aqueous interface. The corresponding ammine reductions are decidely
adiabatic.

A generalized comparison of electrochemical and homogeneous redox
reactions of aquo complexes provides good evidence that driving force
dependent breakdowns of the relative electronetransfer theory for the
two data sets have a common origin. Subsequent absolute calculations
indicated that such breakdowns result in part from differences in force
constants for inner-shell reorganization between different oxidation
states.

Comparisons between self-exchange and cross reactions as well as
between homogeneous and electrochemical exchange reactions provide
substantial evidence that the homogeneous self-exchange reaction of the
hexaaquo iron(III)/(II) couple probably follows an extraordinary
reaction pathway in comparison to other reactions involving aquo com-
plexes. It was speculated that this pathway might involve water as a
bridging ligand. Since the Fe(H20)2+l2+ self exchange has served as the
prototype reaction for evaluating outer-sphere electronetransfer
theories, information concerning the reaction mechanism should be of
considerable interest.

Absolute theoretical estimates of rate constants for nearly sixty
outer-sphere homogeneous and electrochemical reactions exhibit tolerably
good agreement with the experimental rate constants. However. the
agreement between theoretical and experimental activation enthalpies and
entropies is considerably worse. Nevertheless, this poor agreement
appears to be due to the inadequecies of the treatment of elecrostatic

work terms rather than the electron-transfer theory itself.

428

1!... W m 2.113322; M

There are a number of experiments involving ionic adsorption which
might be expected to yield interesting surface chemistry. Because of
the enormous range of positive electrode charge available at silver in
adsorbing electrolytes diffuse-layer potentials should be an important
factor in anionic specific adsorption. The influence of the diffuse
layer could be investigated by examining specific adsorption over a
range of ionic strengths. One suspects that diffuse-layer effects at
very large electrode charges might be coupled to electrosorption
valencies, Frumkin g parameters and other isotherm parameters in
peculiar and interesting ways.

The simultaneous ionic adsorption analysis scheme outlined in
Section III. C could be tested by monitoring specific adsorption from
chloride-bromide-perchlorate or chloride-aside-perchlorate mixed elec-
trolytes at silver. If successful, such experiments would be useful for
establishing more satisfactorily the connections between SERS intensi-
ties and average surface concentrations of Raman scatterers. A fault of
the experiments described in Section III. A is that the electrode was
roughened in one electrolyte, soaked in a second solution and subjected
to capacitance measurements in a third electrolyte. All of these steps
could be performed with a single electrolyte solution, thereby parallel-
ing more closely the experimental protocol employed in SERS experiments,
if the proposed analysis scheme were used in place of the conventional
Hurwitz-Parsons‘method.

Another sequence of experiments might focus on the thermodynamic

properties of metal electrodes which are covered only partially by an

429

underpotentially deposited metal. A nonlinear relationship was found
between the degree of surface coverage of silver by UPD lead, and the
magnitude of the differential double-layer capacitance. Such nonlinear
relationships might be observed with regard to the ionic adsorption pro-
perties of such surfaces as well and would have interesting consequences
in connection with theories of adsorption.

Additional insights concerning electron-transfer energetics at
underpotentially deposited lead and thallium electrode surfaces might be
gained if suitable inner-sphere reactions could be found. Chromium(III)
thiophene reductions represent one possibility. An investigation of
solvent isotope effects on outer-sphere reductions at UPD lead/silver
and UPD thallium/silver might provide clues as to the importance of
specific interactions between reactants and the solvent inner-layer
since hydrogen-bonding interactions generally are stronger in D20 than
H20.

The relative electron-transfer theory could be used to elucidate
various features of the electrochemical reduction kinetics of
chromium(III) complexes if corresponding data could be gathered for
homogeneous outer-sphere reactions of such complexes with a common
reductant such as Os(NH3):+ or Co(bpy)3+.

An improtant conclusion from the study 'of absolute electron-
transfer reactivity (Chapter VII) is that the usual treatment of elec-
trostatic work terms is inadequate for homogeneous redox reactions. In
order to develop a more satisfactory treatment it would be useful to

measure activation parameters over a range of ionic strengths for a set

430

of reactions involving a number of different reactant charge combina-
tions. .An improved description of electrostatic effects on kinetics
parameters would be extremely valuable if it enabled such effects to be
separated from additional subtle factors (e.g. nonadiabaticity) which
may well be masked in many circumstances by the current (inadequate)
treatment.

Finally, it was noted that the absolute theory of electron
transfer predicts that negative activation enthalpies will be observed
under certain conditions. Indeed negative values have been found for
the Fe(H20)§+-Fe(bpy)g+ and other cross reactions. It would be worth-
while to search out electrochemical rections which might exhibit such
behavior. The chief requirement according to theory is that the entropy
driving force (-TAS°) for electron transfer must exceed in absolute
value the intrinsic reorganization energy (If). Oxidations of ammine
complexes in nonaqueous solvents such as acetonitrile involve enormous

entropy changes (circa 200 J deg-1

mol-l) or more; (see Section V. C).
The entropy driving force effects might be sufficiently large to reveal
not only negative ideal activation enthalpies, but also an “inverted“
enthalpy region where AH* increases with increasing exothermicity. An
enthalpic inverted region is expected under extreme circumstances from
theoretical considerations. Rate behavior in this region might be in-
fluenced by anomalous nuclear tunneling effects or other unusual fac-

tors. In any case this problem appears to be worthy of additional

cogitation, as well as experimental investigation.

431

APPENDIX I

Reaction Entropies for Transition-Metal

Redox Couples in Various Solvents

This appendix consists of a listing of previously unreported As:c
data which were obtained in connection with the work described in

Section V. C.

Table A.1. Reaction Entropies (J deg-

432

1

Couples in Various Solvents.

Redox Couple

)3+/2+
3 6
3+/2+
Ru(NH3)6
3+/2+
3

Ru(N83)5pz

Ru(NH

Ru(en)
3+/2+

2+
Ru(NH3)5pz3+/

3+/2+
Ru(NH3)4bpy

3+/2+
Ru(NH3)4phen

3+/2+
Ru(NH3)4phen

Ru(NH3)4phen3+/2+

3+/2+
Ru(NHS)2(bpy)2

3+/2+
2

3+/2+
Ru(NH3)2(bPY)2

3+/2+
Ru(NH3)2(bpy)2

Ru(bpy)§+/2+

Ru(bpy)§+l+

Ru(NH3)2(bPY)

Ru(bpy)}:l0

Cr(bpy)g+/2+

Cr(bpy)§+l+
Cr(bpy);/o

Co(bpy)§+/2+

Solventa

acetonitrile
acetone

formamide
nitromethane
propylene carbonate
acetonitrile
nitromethane
dimethylsulfoxide
propylene carbonate
nitromethane

acetonitrile

dimethylsulfoxide
propylene carbonate
acetonitrile
acetonitrile
acetonitrile
acetonitrile
acetonitrile
acetonitrile

acetonitrile

mol-1)for Transition-Metal Redox

Asrc
185
200

90
165
155
155
120
125
150
115

130

110
135
115
70
25
105
65
20

190

433

Table A.l (continued)

I'C

Redox couple Solvent ASO
Co(bpy) :14" acetonitrile 60
Co(sep )3” 2+ acetonitrile 210
Co (EFME—Oxo sar-H) 2+“ water 30
Co(EFME-Oxosar-H) 2+, + N-methylformamide 9O
" acetonitrile 165
" formamide 75
" ' dimethylformamide 175
" dime thylsulfoxide 160b
" propylene carbonate 150b
" methano l 16 5b
" ni trome thane l40b
1' ethano 1 1051’

a. 0.1 .11 RPF6 supporting electrolyte

b. measured by Dr. Peter Lay

434

APPENDIX II

Negative Activation Enthalpies

In comparing experimental and theoretically calculated activation
parameters (Chapter VII) it was noticed that in a few instances negative
activation enthalpies have been observed for homogeneous outer-sphere

329,359

reactions. Marcus and Sutin have pointed out that this peculiar

result is actually expected under certain conditions from.the relative

246 In connection with the work described in

electronrtransfer theory.
Chapter VII it was of interest to ascertain under what conditions the
absolute electronrtransfer theory would predict negative values of AH*.

The problem is somewhat simplified by assuming equal inner-shell
force constants for the oxidized and reduced states of each reactant,
and a value of the intrinsic activation entropy equal to zero. It is
found that negative 1111*~ values can be expected within the so-called
"normal" free energy region (AG°<A), provided that the value of the
thermodynamic factor TASo exceeds the intrinsic free energy barrier
Acint ( ' X/4). This requirement is fulfilled, for example in the cross
reaction of Fe(H20)§+ with Ru(bpy);+. Furthermore it is found that the
range of free energy driving force over which AH* should be negative
extends from AGO - -l- ZTASo to 86° -—1. Under all circumstances AH* as
well as AC* and 83* are expected to equal zero at AGo -().

Figure A.l illustrates a sample calculation for a hypothetical

reaction involving an intrinsic free energy barrier of 44 kJ mol.1 and a

435

1 mol-1. The most interesting

thermodynamic entropy change of 180 J deg-
result of the calculation is the appearance within the "normal" free
energy region of an enthalpic "inverted" region where AH* increases with
increasing free energy or enthalpy driving force. The threshhold of the
inverted enthalpy region is at AGo - -A.- TASO.

The $323 gggggy inverted region has been the subject of much

391-393

experimentation and speculation. Part of the interest stems from

the possibility that nuclear tunneling phenomena may be of greater

393 One difficulty in probing reacti-

importance in inverted reactions.
vity in this region is that reactions are usually extremely fast since

AG* is small in the vicinity involving sufficiently large thermodynamic
entropy changes it appears that the analogous enthalpic inverted region
may be more easily studied, since AG* will be significantly greater than
zero through most of the region. we note also that the free energy in-
verted region is expected to be inaccessible in electrochemical reac-

tions at ‘metal electrodes for reasons outlined. by' Marcus.394

Nevertheless the enthalpic inverted region should be observable.

Currently experimental results along these lines for the electrochemical

o l

. . 2+
oxidations of Cr(HZO)6 (A81.
1

_ - -1 2+ 0 _
c 205 J deg mol ) and V(H20)6 (ASrc

155 J deg- mol-1) are being compared with theoretical predictions.

436

 

 

 

 

I I l l l l T
40-— -
20 — —
T.
O
E r '06“ I
,
.w
o - ___________________________ .1
\ *
_ AH a
K-TAS'
-201. H
1 l g l L L 1_
-240 -160 -80 0

250°. kJ mol"

Figure A.1. Variation of calculated activation parameters with free

energy driving force, A00, for a reaction for which AG
-1

* —1
- 44 kJ mol ,
int

* -l -l o -1
Asint. 0 J deg mol and AS - -180 J deg mol

1.

9.

10.

11.

12.
13.
14.
15.

437

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D. Larkin, K. L. Guyer, J. T. Hupp and M. J. Weaver, J.
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M. J. Weaver, unpublished observations.

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96.

97.

98.

99.
100.

101.

102.

103.

104.

105.

106.

107.

108.

109.

110.

111.
112.

113.

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442

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M. J. Weaver, F. Barz, J. G. Gordon II, M. R. Philpott, Surface
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G. Valette, J. Electroanal. Chem., 122 (1981).
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J. T. Hupp, D. Larkin, H. Y. Liu and M. J. Weaver, J.
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M. J. Weaver, F. Bars, J. G. Gordon II, J. T. Hupp and M. R.
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M. J. Weaver and F. C. Anson, J. Electroanal. Chem., 82, 737
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Analogous methgds to I and II, but involving analyses carried out
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Reference 45, p. 103.

J. W. Schultze and K. J. Vetter, J. Electroanal. Chem., 44, 63
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M. J. Weaver and F. C. Anson, J. Electroanal. Chem., 58, 95
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B. B. Damaskin, Sov. Electrochem. Z, 776 (1971).

R. L. Guyer, S. W. Barr and M. J. Weaver, in "Proc. 3rd Symp. on
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S. W. Barr, R. L. Guyer and M. J. Weaver, J. Electroanal. Chem.
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G. valette, J. Electroanal. Chem., A48, 439 (1983).
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J. F. Evans, M. G. Albrecht, D. M. Ullevig and R. M. Hexter, J.
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115.

116.

117.

118.

119.

120.

121.

122.

123.

124.

125.

126.

127.

-128.

129.

130.

131.

443

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W. J. Plieth, J. Phys. Chem. 88, 3166 (1982).

J. W. Owen, T. T. Chen, R. K. Chang and B. L. Laube, paper
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Note that the electrode potentials in reference 97 are quoted
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mV on the s.c.e. scale employed here.

C. S. Allen and R. P. Van Duyne, Chem. Phys. Lett. 88, 455
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F. Barz, J. G. Gordon II, M. R. Philpott and M. J. Weaver, Chem.
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S. H. Macomber, T. E. Furtak and T. M. Devine, Chem. Phys. Lett.,
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T. Watanabe, N. Yanagihara, R. Honda, B. Pettinger and L. Moerl,
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J. J. Rester, J. Chem. Phys., 18, 7466 (1983).

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B. Pettinger, M. R. Philpott and J. G. Gordon III, J. Phys.
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R. L. Guyer, S. W. Barr and M. J. Weaver in "Proc. Symp. on
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8. Farquharson, K. L. Guyer, P. A. "Bruce" Lay, R. H. Magnuson
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132.

133.

134.

135.

136.
137.

138.

139.

140.

141.
142.

143.

144.

145.
146.
147.
148.

149.

444

S. Farquharson, M. J. Weaver, P. A. Lay, R. H. Magnuson and H.
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For reviews, see: (a) A. N. Frumkin, O. A. Petrii and B. B.
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D. M. Rolb, M. Przasnyski and H. Gerischer, J. Electroanal. Chem.
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M. Bratoeva, Sov. Electrochem., 28, 19 (1980).

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L. Pauling, "The Nature of the Chemical Bond", 3rd edition,
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J. E. Huheey, "Inorganic Chemistry", 2nd edition, Harper and Row,
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N. B. Grigorev, V. A. Bulavka and Y. M. Loshkarev, Sov.
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150.

151.

152.

153.
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155.
156.

157.

158.

159.

160.

161.

162.

163.

164.

165.

166.

167.

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V. A. Bulavka, N. B. Grigorev and Y. M. Loshkarev, Sov.
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J. W. Rauffman, R. H. Hauge and J. L. Margrave, ACS Symp. Ser.,
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"CRC Handbook of Chemistry and Physics", 56th edition, R. C.
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L. A. Avaca, E. A. Gonzales and R. C. Rocha Filho, J.
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170.
171.

172.
173.

174.

175.

176.

177.

178.

179.
180.
181.
182.
183.

184.

185.

186.

187.
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189.

446

B. B. Damaskin, Sov. Electrochem., 1, 776 (1971).

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J. Reiss, J. Chem. Phys. 28, 996 (1950).

Reference 29, pp. 37-9.

T. T-T Li, M. J. Weaver and C. H. Brubaker, J. Am. Chem. Soc.
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N. B. Slater, "Theory of Unimolecular Reactions", Cornell
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E. Buhks, M. Bixon and J. Jortner, J. Phys. Chem. 82, 3763
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J. M. Hale, J. Electroanal. Chem. 22, 315 (1968).

M. J. Weaver and T. L. Satterberg, J. Phys. Chem. 81, 1772
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L. I. Rrishtalik, Sov. Electrochem.,Ll, 174 (1975).

R. A. Marcus, J. Phys. Chem. 81, 853 (1963).

M. J. Weaver, J. Phys. Chem. 84, 568 (1980).

J. T. Hupp and M. J. Weaver, Inorg. Chem., 22, 2557 (1983).
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V. Srinivasan, S. W. Barr and M. J. Weaver, Inorg. Chem. 2;, 3154
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J. E. B. Randles and R. W. Somerton, Trans. Far. Soc. 48, 937
(1952).

J. T. Hupp and M. J. Weaver, J. Electroanal. Chem., 142, 43
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H. Y. Liu, J. T. Hupp and M. J. Weaver, to be published.

J. R. Farmer and M. J. Weaver, to be published.

See for example, C. M. Torrie and J. P. Velleau, J. Phys. Chem.
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Chem. Phys. 18, 4615 (1982); D. Henderson and L. Blum, Can. J.
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190 .
19 1 .
192 .
193 .
194 .

195 .

196 .

197 .

198.

199 .
200 .
201.
202.

203 .

204.

205 .
206 .
207 .

208 .

209.

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447
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R. F. Lane and A. T. Hubbard, J. Phys. Chem., 11, 1401, 1411
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N. Oyama, T. Ohsaka, M. Raneko, K. Sato and H. Matsuda, J. Am.
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A. P. Brown and F. C. Anson, J. Electroanal. Chem., 22, 133
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M. Sharp, M. Petersson and R. Edstrom, J. Electroanal. Chem.,
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M. Sharp and M. Petersson , J. Electroanal. Chem., _1_22, 409
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R. L. Guyer and M. J. Weaver,J. Phys. Chem., in press.
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M. J. Weaver, J. Israel Chem., _1_8_, 35 (1979).

R. A. Marcus, Electrochim. Acts, 41, 995 (1968).

W. L. Reynolds and R. W. Lumry, "Mechanisms of Electron
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S. W. Barr, R. L. Guyer and M. J. Weaver, J. Phys. Chem.,
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W. H. Reinmuth, J. Electroanal. Chem., A, 425 (1969).
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J. A. Riddick and W. B. Bunger, "Organic Solvents", Wiley, New
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G. J. Janz and R. P. T. Tomkins, "Nonaqueous Electrolyte
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211.

212.
213.

214.

215.

216.

217.

218.

219.

220.

221.

222.

223.

224.

448

J. T. Hupp and M. J. Weaver, J. Electroanal. Chem., 122, 1
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T. T-T. Li and M. J. Weaver, manuscript in preparation.
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T. T-T. Li, M. J. Weaver and C. H. Brubaker, Jr., J. Am. Chem.
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J. Logan and M. D. Newton, J. Chem. Phys., 18 (Part II), 4086
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T. L. Satterberg and M. J. Weaver, J. Phys. Chem., fl, 1772
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R. L. Guyer, T. L. Satterberg and M. J. Weaver, unpublished
experimental results.

8. W. Barr, R. L. Guyer, T. T-T. Li, H. Y. Liu and M. J. Weaver
in ”The Chemistry and Physics of Electrocatalysis", J. D. E.
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Society, Pennington, N.J., in press.

On the basis of crystallographic data for bridging azide and
thiocyanate (Wells, A. F. "Structural Inorganic Chemistry",
Clarendon Press, Oxford, 4th ed., 1975, pp. 746, 648)
Cr([ID;-_surface bond_ distances of 6.5 - 7 X are obtained. However,
the N -Hg and NCS -Hg bond angles are likely to be less than

180°,3thereby decreasing somewhat the Cr([ID- surface separation.

M. J. Weaver, J. Phys. Chem., &, 568 (1980).

M. J. Weaver, H. Y. Liu and Y. Rim, Can. J. Chem., 22, 1944
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M. J. Weaver and S. M. Nettles, Inorg. Chem., _1_2_, 1641 (1980).

rn was fpfiplated from the nuclear tunneling factor, I’ h, for the
Cr(OH2)6 selfl-zexchange reaction (6.0) given in refnerence 10
using .. (1‘ ) (reference 211); n equals the corrresponding

value for the nself--exchange reaction.

Equivalently, A can be determined by evaluating the so-called

"rsal" activation enthalpy from the temperature dependence of

k0 measured at a fixed overpotential, and assuming that the

"€351" activation entropy (i.e. the activation, entropy corrected
for the entropic driving force ASZc) equals Asint'

225.

226.

227.

228.

229.

230.

231.

232.
233.

234.
235.

236.

237.

238.

239.

449

The major component of as? is associated with she temperature
dependence of n3 a value of ca. -15 J deg. mol is obtained
from the corrfsponding quantity determined for self-exchange
reactionp, AS. t ,*using the relations in reference 23, given
that AS at '1‘0.§ sin h' (The factor 0.5 arises because only
one redox center has t1:!) be activated for the electrochemical
reaction, rather than two as in the homogeneous case.)

The effective value of AB: is slightly (0.3 kcal. mol-1) larger
than the values given in reference 54 since these were determined
by assuming that the frequency factor is slightly temperature
dependent as expected from the collision model utilized in
reference 54.

Evidence supporting such differences in the reactant solvation at
mercury electrodes and in homogeneous solution is provided by the
markedly different solvent deuterium isotope effects observed for
aquo complexes in these reaction environments (references 228 and
229).

M. J. Weaver, P. D. Tyma and S. M. Nettles, J. Electroanal.
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11. J. Weaver and 1'. 'r-r. Li, J. Phys. Chem., .81. 1153 (1983).

J. F. Endicott and J. Ramasami, J. Am. Chem. Soc., _1_84, 5252
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J. M. Hale in "Reactions of Molecules at Electrodes”, N. 3. Bush
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J. M. Hale, J. Electroanal. Chem., 42, 315 (1968).

T. T—T. Li, H. Y. Liu and M. J. Weaver, manuscript submitted for
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A. Capon and R. Parsons, J. Electroanal. Chem., 48 215 (1973).

A. N. Frumkin, N. V. Fedorovich, N. P. Berezina and R. E. Ries,
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N. V. Fedorovich, A. N. Frumkin and H. E. Case, Collect. Czech.
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A. N. Frumkin, N. V. Fedorovich and S. I. Rulakovskaya, Sov.
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240.

241.

242.

243 .
244.
245 .
246 .
247 .

248 .
249 .
250 .

251.

252 .

253.

254.

255.
256.

257.

258.

450

M. J. Weaver and F. C. Anson, J. Electroanal. Chem., §_8_, 81
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M. J. Weaver and P. C. Anson, J. Electroanal. Chem., .91, 4403
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M. J. Weaver and T. L. Satterberg, J. Phys. Chem., 8;, 1784
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R. A. Marcus, J. Phys. Chem., 1;, 891 (1968).

R. A. Marcus and N. Sutin, Inorg. Chem., 14, 213 (1975).

The value of 3 is related to both the magnitude of the entropic
and free energy driving forces (see reference 2 ). However, for
reactions having small or moderate values of AG , B=0.5 - 0.05.
M. J. Weaver, Israel J. Chem., g, 35 (1979).

J. T. Hupp and M. J. Weaver, ACS Symp. Ser., 128, 181 (1982).

P. P. Schmidt in "Electrochemistry - A Specialist Periodical
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R. A. Marcus in "Special Topics in Electrochemistry", P. A. Rock
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See, for example: R. M. Noyes, J. Am. Chem. Soc., .84, 513
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S. Sahami and M. J. Weaver, J. Electroanal. Chem., 122, 155
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S. Sahami and M. J. Weaver, J. Electroanal. Chem., 11;, 171
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S. Sahami and M. J. Weaver, J. Sol. Chem., _1_9_, 199 (1981).
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N. Sailsuta, F. C. Anson and E. B. Gray, J. Am. Chem. Soc., m,
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In reference 23 the temperature depfndence of I‘ n is formally
considered to be a constituent of AS. t’ Since decreases with
temperature, this yields a significane negative contribution to
A8. t’ For the present purposes, we prefer to correct for the
nué‘Pear tunneling factor separately since this forms par; of the
inner-shell, rather than the solvent, contribution to ASint

259.
260.
261.
262.
263.

264.

265.
266.

267.

268.
269.

270.

271.

272.

273.

274.

275.

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Reference 45, Chapters 7 and 9.

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282.

283.

284.

285.

286.

287.

288.

289.

290.
291.
292.

293.

294.

295.
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P. George, G. I. H. Hanania and D. H. Irvine, Recl. Trav. Chim.
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P. George, G. I. H. Hanania and D. H. Irvine, J. Chem. Soc., 2448
(1959).

P. George, W. A. Eaton and M. Trachman, Federation Proc., 21, 526
(1968) .

P. George., G. I. H. Hanania and W. A. Eaton in "Hemes and
Hemoproteins", B. Chance, R. W. Estabrook and T. Yonetani, Eds.,
Academic Press, New York, 1966, p. 267.

G. I. H. Hanania and D. H. Irvine, W. A. Eaton and P. George, J.
Phys. Chem., 1;, 2022 (1967).

J. Lin and W. G. Breck, Can. J. Chem., 4;, 766 (1965).

N. Sutin, M. J. Weaver and E. L. Yee, Inorg. Chem., 12, 1096
(1980).

E. L. Yee, O. A. Gansow and M. J. Weaver, J. Am. Chem. Soc., 102,
2278 (1980).

J. T. Hupp and M. J. Weaver, Inorg. Chem., in press.
J. T. Hupp and M. J. Weaver, J. Electrochem. Soc., in press.
Y-M. Tsou and F. C. Anson, J. Electrochem. Soc., in press.

J. E. J. Schmitz and J. J. Steggerda, manuscripts submitted for
publication.

K. M. Radish, K. Das, D. Schaeper, C. L. Merill and B. R. Welch,
Inorg. Chem., 12, 2816 (1980).

K. M. Kadish and D. Schaeper, JCS Chem. Com., 1273 (1980).
H. Ogino and K. Ogino, Inorg. Chem., 22, 2208 (1983).
J. T. Hupp and M. J. Weaver, J. Phys. Chem., in press.

M. P. Youngblood and D. W. Margerum, Inorg. Chem. , )1, 2816
(1983).

J. M. Anast, A. W. Hamburg and D. W. Margerum, Inorg. Chem., 22,
2139 (1983).

300.

301.

302.

303 .

304.

305.

306.

307 .
308.

309.

310.

311.
312.

313.

314.

315.

316.

317.

453

V. T. Taniguchi, 8. G. Malmstrom, F. C. Anson and H. B. Gray.,
Proc. Natl. Acad. Sci., usa, 19, 3387 (1982).

V. T. Taniguchi, N. Salisuta-Scott, F. C. Anson and H. B. Gray,
Pure and Appl. Chem., 5;, 2275 (1980).

V. T. Taniguchi, W. R. Ellis, Jr., V. Cammarata., J. Webb, F. C.
Anson and H. B. Gray, Adv. Chan. Ser.

E. F. Wawrousek and J. V. Mcardle, J. Inorg. Biochem., g, 169
(1982).

W. Eottcher, G. M. Brown and N. Sutin, Inorg. Chem., 1.8, 1447
(1979).

T. Curtis, E. P. Sullivan and T. J. Meyer, Inorg. Chem. ,2}, 224 (1983).

N. Sailasuta, F. C. Anson and H. B. Gray, J. Am. Chem. Soc., 22,
2786 (1977).

L. E. Bennett, Prog. Inorg. Chem., 18, l (1973).
P. Siders and R. A. Marcus, J. Am. Chem. Soc. _1_Q§_, 741 (1981).

See for example, R. G. Linck, in "Homogeneous Catalysis", G. N.
Schrauser, ed., Marcel Dekker, New York, 1971, Chapter 7.

M. Chou, C. Creutz and N. Sutin, J. Am. Chan. Soc. 22, 5615
(1977).

M. J. Weaver and E. L. Yee, Inorg. Chem. 12, 1936 (1980).
M. J. Weaver, J. Phys. Chem. 84, 568 (1980).

P. D. Tyma and M. J. Weaver, J. Electroanal. Chem. ALL, 195
(1980).

M. J. Weaver, J. Electroanal. Chem. 9_3_, 231 (1978).

J. Lipkowski, A. Czerwinski, E. Cieszynska, z. Galus and J.
Sobkowski, J. Electroanal. Chem. 112, 261 (1981).

(a) For example, see D. H. Angell and T. Dickinson, J.
Electroanal. Chem. 12, 55 (1972). (b) D. C. Johnson and E. W.
Resnick, Anal. Chem. 42, 1918; J. Weber, 2. Samec, V. Marecek, J.
Electroanal. Chem. 81, 271 (1978).

R. C. Young, F. R. Keene and T. J. Meyer, J. Am. Chem. Soc. 22,
2468 (1977).

318.

319.

320.

321.

322.

323.

324.

325.

326.

327.

454

Under these conditions it if likely that any residual work terms
will affect each value of k to an essentially equal extent. In
addition, the predominant fghction channel for cross reactions
having similarly small driving forces will involve transition-
state structures that closely resemble those for the parent self-
exchange reactions (reference 249).
The estimate of kh , 60 571 sec-1, for Ru3+l2+ given in reference
304 refers to ane ionic strength of l M: and is therefore
uncorrecfied for ‘1qu terms. Application of this correction
yields kex 200 M_ sec . (reference 311)

J. F. Endicott, B. Durham and K. Kumar, Inorg. Chem. 21, 2437
(1982).

G. M. Brown, H. J. Erentzien, M. Abe and H. Taube, Inorg. Chem.
1;, 3374 (1979).

Admittedly2+ the self-exchange rates for Os(bpy)3+/2+’ and
Cr(bpy) may differ somewhat. However, if agything the
former ieaction should have faster self-exchange kinetics as a
result of greater delocalization of the tragsferred electron,

which would yield an even smaller value of k from Equation
6.6. ex,Fe

Although a frequency factor formulation based on the simple gas-
phase collision model has conventionally been employed to
estimate and A (reference 13), the use of an "encounter
preequilibrium" 665.1 where activation is considered to occur
chiefly via unimolecular activation within a statistical dis-
tribution of encounter complexes appears to be physically more
appropriate for both homogeneous (references 10,18,28) and
electrochemical (reference 186) electronrtransfer processes.

Since it is likely that the outer-sphere transition state for

electrochemical reactions is separated from the electrode by a
layer of water molecules (reference 312), Re is estimated to be
twice the sum of the reactant radius ( 3.5 fiend the effective
diameter of a water molecule (3 ).

J. Strouse, S. W. Layten and C. E. Strouse, J. Am. Chem. Soc.,
22, 562 (1978).

N. J. Hair and J. K. Beattie, Inorg. Chem. 19, 245 (1977); J. K.
Beattie, S. P. Best, B. W. Skelton and A. H. White, J. Chem. Soc.
Dalton, 2105 (1981).

J. A. Jafri, J. Logan and M. D. Newton, Israel J. Chem. 12, 340
(1980).

328.

329.

330.

331.

332.

333 .

334.

335.

336.
337 .
338.

339.

340.
341.
342.
343 .

455

T. D. Hand, M. R. Hyde and A. G. Sykes, Inorg. Chem. 13, 1720
(1975).

J. N. Braddock and T. J. Meyer, J. Am. Chem. Soc. 25, 3158
(1973).

J. F. Endicott, C.-L. Wong, J. M. Ciskowski and K. P.
Balakrishman, J. Am. Chem. Soc. 192, 2100 (1980).

Footnote 54, reference 320.

It was asserted (reference 330) that the Co3+ - Fe2+ reaction
follows an outer-sphere pathway on the basis 863E t5; ggeanent with
the rates of other cross reactions involving Coa and outer-
sphere redactants using Equation 6.1 and utilizing the me sured
value of k x . However, using the present estimate33f k x210:
outer-sphefe’gs self exchange instead yields a Co - Fe
reaction rate c3? 10 -fold larger than predicted 6:83 Equagiqon
6.1, indicative of a water-bridged pathway for this reaction as
W911.

For example, see 3. Trassati, J. Electroanal. Chan. 222, 121
(1981).

B. M. Gordon, L. L. Williams and N. Sutin, J. Am. Chem. Soc., _8_2,
2061 (1961).

M. H. Ford-Smith and N. Sutin, J. Am. Chem. Soc., _8§_, 1830
(1961).

N. Sutin and B. M. Gordon , J. Am. Chem. Soc., s3, 70 (1961).
T. J. Przystas and N. Sutin, J. Am. Chem. Soc., 22, 5545 (1973).

Y. Ohsawa, T. Sajiand and S. Aoyagi, J. Electroanal. Chem., 106,
327 (1980).

C. T. Lin, W. Bottcher, M. Chou., C. Creutz and N. Sutin, J. Am.
Chem. Soc., 28, 6536 (1976).

N. Sutin, Acc. Chem. Res. _1_, 225 (1968).
E. Parsons, Croat, Chim. Acta 42, 281 (1970).
N. Sutin and B. Brunschwig, ACS Symp. Ser., 12g, 106 (1982).

T. W. Newton, J. Chem. Educ. 22, 571 (1968).

344.

345.

346.

347.

348.

349.
350.

351.

352.
353.

354.

355.

356.

456

In the Appendix to reference Equation 6.20 was claimed to be
generally applicable in the “weak overlap" limit when AG and
AG exhibit a quadratic dependence upon (E-E ) as pred161ed by
thg’flarmonic oscillator model. In fact, the proof given is not
entirely correct inasmuch as the energy minimization condition I
[(AG 1 + AG )Iae ' 0] employed in reference 182 leads to
Equaffon 6.21e,’2Equation 6.22 only being obtained as a special
case when the slopes of the AG 1 - E and AG - 1!: plots are
equal and of opposite sign at tfie intersection int.

J. F. Endicott, E. E. Schroeder, D. H. Chidester and D. R.
Perrier, J. Phys. Chem. 11, 2579 (1973).

T. Saji, T. Yamada and S. Aoyagui, J. Electroanal. Chem. 2;, 147
(1975); T. Saji and Y. Mariyama, S. Aoyagui, ibid, 8_6, 219
(1978).

M. J. Weaver and F. C. Anson, J. Phys. Chem. g9, 1861 (1976).
M. J. Weaver, Inorg. Chem., 22, 1733 (1976).
See reference 313 for references to earlier work.

V. Balzani, F. Scandola, G. Orlandi, N. Sabbatini and M. T.
Indelli, J. Am. Chem. Soc. 193, 3370 (1981).

J. F. Endicott, B. Durham, M. D. Glick, T. J. Anderson, J. M.
Kuszaj, W. G. Schmonsees and E. P. Balakishnan, J. Am. Chem. Soc.
292, 1431 (1981).

J. F. Endicott, B. Durham and K. Kumar, Inorg. Chem. in press.

(a) S. S. Isied and H. Taube, J. Am. Chem. Soc., 22, 8198
(1973); (b) H. Fischer, G. M. Tom and H. Taube, ibid,, 22, 5512
(1976); (c) K. Eieder and H. Taube, ibid,, 22, 789 (1977). (d)
S. K. S. Zawacky and H. Taube, ibid,, 292, 3379 (1981).

(a) J.-J. Juro, P. L. Gaus and A. Haim, J. Am. Chem. Soc., 19;,
6189 (1979); (b) A. P. Szecsy, and A. Haim, ibid,, 103, 1679
(1981); (c) A. P. Szecsy and A. Haim, 194, 3063 (1982).

For reviews, see: (a) A. Haim, Acc. Chem. Res., g, 264 (1975);
(b) H. Taube, Adv. Chem. Ser., 162, 217 (1977); (C) T. J.
Meyer, ACS Symp. Ser., 12§, 137 (1982).

E. A. Marcus and N. Sutin, Inorg. Chem., 14, 213 (1975).

357.

358.

359.
360.

361.

362.

363 .

364.

365.

366.

367 .
368.

457

The nonzero, albeit small, values of S": that are obtained
using the dielectric continuum model arise from the prediction
that the polarization entrqpy will depend on the square of the
ionic charge 2. Since 8. - 0 when the transition-state
entropy is exactly midway Eiween that for the reactants and
products,* any such nonlinearity between the entropy and 2 will
yield 3. - 0. However, the entropy difference between
reactants and products is commonly found to be substantially
larger than predicted from the Born model, yielding more positive
values of S. t.' For the systems cogiidergti here,*we estimate
that this conlf‘ributes ca. 12-16 J deg mol to sin , although
partly compensated by the. influence of nuclear tunaeling

(reference 5), yielding A8111t = 4-9 J deg-1 mol'l.

The values of 118* were extracted from literature experimental
data using the profer bimolecular frequency factor' K v , where R
is the precursor equilibrium constant, rather thanP e uni-
molecular frequency factor kT/h. (Note that the compilation
given in reference 311 contains values of AS* that are ca. 10
e.u. less negative resulting from the use oefxpa collisional,
rather than a preequilibrium, model for the frequency factor).

J. L. Cramer and T. J. Meyer, Inorg. Chem., 2}, 1250 (1974).
R. G. Gaunder and H. Taube, Inorg. Chem., 2, 2627 (1970).

J. E. Beattie, E. A. Binstesd, and M. Broccardo, Inorg. Chem. 21,
1822 (1978).

J. V. Beitz, J. R. Miller, H. Cohen, R. Wieghardt and D.
Meyerstein, Inorg. Chem., 12, 966 (1980).

J. E. Schmitz, P. J. Eoonen, J. G. M. van der Linden, and J. J.
Steggarda, Extended Abstracts, Electrochemical Society meeting,
Washington, D. 0., October 1983; J. J. Steggarda, personal
communication.

An attempt to evaluate As:c for RtsgIII)/(II) in the binuclear
complex H 0 Ru( )4(bpy) Co(NH) was thwarted by minor
distortio in the cyclic voltamgozrams, most likely caused by
small amounts of the mononuclear ruthenium complex as an
impurity.

S. S. Isied, ACS Symp. Ser., 125, 221 (1982).

For reviews of experimental work see: R. D. Cannon in "Inorganic
Reaction Mechanisms-Specialist Periodical Reports", vol. 7, Royal
Society of Chemistry, 1981, Chapter 1 and chapters in previous
volumes.

R. A. Marcus, J. Chem. Phys., 22, 966 (1956).

H. C. Stynes and J. A. Ibers, Inorg. Chem., _l_0_, 2304 (1971).

369.
370.

371.

372.

373.
374.

375.
376.
377.

378.
379.

380.

381.

382.
383.
384.
385.
386.

387.

458
J. K. Beattie and C. J. Moore, Inorg. Chem., 22, 1292 (1982).

A. Wheeler, C. Brouty, P. Spinat and P. Herpin, Acta Crystallogr.
Sect. 3., L1. 2069 (1975).

D. J. Szalda, C. Creutz, D. Mahajan and N. Sutin, Inorg. Chem., 22,
2372 (1983).

In a number of recent publications (e.g. reference 9) It0
underestimated by a factor of two, evidently because eh
coefficient describing the distance dependence of the electronic-
coupling matrix element H has been mistakenly identified with a
related factor describingAEhe variation of Kel with distance.

is
e

E. L. Yee, J. T. Hupp, and M. J. Weaver, Inorg. Chem., in press.
R. D. Cannon, Chem. Phys. Lett., 42, 299 (1977).

Solvent reorganization models are reviewed in: E. D. German and
A. M. Kuznetsov, Electrochim. Acts, 24, 1595 (1981).

(a) A. Zalkin, D. H. Templeton and T. Ukei, Inorg. Chem., 42,
1641 (1973); (b) J. Baker, L. M. Englehardt, B. N. Figgis and
A. H. White, J. Chem. Soc., Dalton Trans., 2105 (1981).

T. Holstein in reference 59, p. 129.

T. Holstein and H. Scher, Philos. Mag. 8, 44, 343 (1981).

A. Ekstrom, A. B. McLaren and L. E. Smythe, Inorg. Chem., 22,
2853 (1976).

A. Ekstrom, A. B. McLaren and L. E. Smythe, Inorg. Chem., )4,
2899 (1975).

M. J. Burkhart and T. W. Newton, J. Phys. Chem., 12, 1741 (1969).

Bruce Bockris and S. U. M. Khan, "Quanttnn Electrochemistry",
Plenum Press, New York, 1979.

S. U. M. Khan and J. O’M. Bockris, J. Phys. Chem., _8_7_, 2599
(1983). '

J. F. Endicott, B. Durham and K. Kumar, Inorg. Chem., 22, 2437
(1982).

E. Buhks, M. Bixon, J. Jortner and G. Navon, Inorg. Chem., 22,
2014 (1979).

C. Creutz, M. Chou, T. L. Netzel, M. Okumura and N. Sutin, J. Am.
Chem. Soc., 102, 1309 (1980).

C.-T. Lin, W. Bottcher, G. M. Brown, C. Creutz and N. Sutin, J.
Am. Chem. Soc., 22, 6536 (1976).

459
388. C. Creutz, Inorg. Chem., 21, 1056 (1978).
389. R. X. Ewell and L. E. Bennett, J. Am. Chem. Soc., 24, 940 (1974).

390. J. V. MeArdle, H. B. Gray, C. Creutz and N. Sutin, J. Am. Chem.
Soc., 24, 5737 (1974).

391. F. P. Dwyer, N. A. Gibson and E. C. Gyarfas, J. Proc. R. Soc., N.
s. W., 24, 80 (1950).

392. D. N. Huchital, N. Sutin and B. Warnquist, Inorg. Chem., 6, 838
(1967).

393. A. A. Schilt, "Analytical Applications of 1,10—Phenanthroline and
Related Compounds," Pergamon Press, New York, 1969, p. 120.

394. F. P. Dwyer and E. C. Gyarfas, J. Am. Chem. Soc., 16, 6320 (1954).

395. P. J. Smoelnars, J. K. Beattie and N. D. Hutchinson, Inorg. Chem.,
29, 2202 (1981).

396. K. Nakamoto, "Infrared and Raman Spectra of Inorganic and Coordin-
ation Compounds," 3rd ed., Wiley-Interscience, New York, 1978.

397. K. H. Schmidt and A. Muller, Inorg. Chem., 14, 2183 (1975).

398. A. A. Noyes and T. J. Deahl, J. Am. Chem. Soc., 52, 1337 (1937).
399. H. S. Habib and J. P. Hunt, J. Am. Chem. Soc., 88, 1668 (1966).
400. L. E. Bennett and J. L. Shepherd, J. Phys. Chem., 66, 1275 (1962).

401. M. R. Hyde, R. Davies and A. G. Sykes, J. Chem. Soc., Dalton Trans.,
1838 (1972).

402. G. Dulz and N. Sutin, J. Am. Chem. Soc., 86, 829 (1964).

403. J. F. Endicott and H. Taube, J. Am. Chem. Soc., 86, 1686 (1964).
404. K. Ohashi, T. Amano and K. Yamamoto, Inorg. Chem., 16, 3364 (1977).
405. C. A. Jacks and L. E. Bennett, Inorg. Chem., 1;, 2035 (1974).

406. R. J. Campion, N. Purdie and N. Sutin, Inorg. Chem., 2, 1091 (1964).

407. R. Davies, M. Green and A. G. Sykes, J. Chem.Soc., Dalton Trans.,
1171 (1972).

408. R. Berkoff, K. Krist and H. D. Gafney, Inorg. Chem., 12, 1 (1980).

409. H. M. Neumann, quoted in R. Farina and R. G. Wilkens, Inorg Chem.,
_1, 514 (1968).

460

410. T. Gennett and M. J. Weaver, Anal. Chem., in press.

3.3A

3.38

3.4

3.5A

3.58

3.50

Differential capacitance versus electrode potential for
electropolished polycrystalline silver (RF 1.2) in NaCth-
Na61 ‘mixtures at ionic strength 0.5. Keys to chloride
concentrations: 1, 0 mg; 2, 1 mg; 3, 2.5 mg; 4, 6 mg;

5’ 15 mg; 7, 100 mg; 8’ 200 “E’OOOOOOOOOO0.00.00.00.00... 58

As in Figure 3.3A, but for electrochemically roughened

silver (RF 1.9)....00000000000.0.0.000...OOOOOOOOOOOOOOO. 59

Surface concentration of specifically adsorbed chloride
(per on2 real area) at electropolished (solid curves) and
roughened silver (dashed curves) versus electrode potential
for bulk chloride concentrations of 0.015 M, and 0.10 g,

analyzed from Figures 3A and B as outlined in text ........ 60

Differential capacitance vesus electrode potential for
electropolished polycrystalline silver in NaClOa-NaBr mix-
tures of ionic strength 0.5. Key to bromide concentration
(solid curves): 1, 0 mg; 2, 0.6 mg; 3, 2 mg; 4, 5 mg;

5, 15 mg; 6, 40 mg; 7, 100 mg. The dashed curve is the
cell resistance for 100 mg bromide plotted on the same

potential scale.00......OCOCOOOOOOOOOOOOOOOOC000......O... 61

As in Figure 3.5A, but for roughened silver (RF-1.9)...... 62

As in Figure 3.5A, but for roughened silver (RF-4.2)...... 63

xiv

3.6A

3.68

3.7A

3.78

3.8A

3.88

3.9

Surface concentration of specifically adsorbed bromide (per
cm2 real area) at electropolished (solid curve) and rough-
ened silver (dashed, dotted-dashed curves) for bulk concen-

tration x at 15 m!_obtained from Figures 3.5A and B........ 64
A8 in Figure 306A but for :-100 @000000000OOOOOOOOOOOOOOO. 65

Differential capacitance versus electrode potential for
electropolished silver in Na0104-NaI mixtures of ionic
strength 0.5. Key to iodide concentrations: 1,0 m!; 2,

0.2 mg; 3, 0.5 mg; 4, 1.2 mg; 5, 2.2 mg; 6, 5 mg; 7, 10 mg. 70
As in Figure 3.7A but for roughened silver (RF 4.6)........ 71

Differential capacitance versus electrode potential for
electropolished silver in NaClOa-NaNCS mixtures of ionic
strength 0.5. Key to thiocyanate concentrations: 1, 0 mg;

2, 0.3 mg; 3, 1 mg; 4, 3 mg; 5, 10 mg; 6, 30 mg; 7, 100 my, 72
As in Figure 3.8A but for roughened silver (RF-2.1)........ 73
Differential capacitance versus electrode potential for
roughened silver in KCl-KSCN mixtures of ionic strength

0.1. Key to thiocyanate concentrations: 1, 0 mg; 2, 5 mg;

3’ 10 m; 4’ 23 m; 5’ 50 MOOOOCOCOOOOOOOOOOOOOOOOO0.00... 74

XV

3.10A

3.108

3.11

3.12

3.13

3.14

Differential capacitance versus electrode potential for
electropolished silver in NaCloé-NaN3 mixtures of ionic
strength 0.5. Key to azide concentrations: 1, 0 mg; 2,

1 mg; 3, 3 mg; 4, 10 mg; 5, 30 11121; 6, 100 mg.............. 77

As in Figure 3.10A, but for roughened silver (RF-2.3)..... 78

Surface concentration of specifically adsorbed azide

(per cm2 real area) at electropolished (solid curves) and
roughened silver (dashed curves) versus electrode potential
for bulk azide concentrations of 0.01 g and 0.1 g analyzed

from Figures 3.10A, 3.10B as outlined in the text.......... 79

Atomic positions of the (100). (110), and (111) faces in

the fcc structure (after Hamelin, et a1., reference 1)..... 85

Fractional coverage, a, of chloride anions (solid curve)
and normalized Raman peak intensity (dashed curve) for
Ag-Cl- stretching mode, both plotted against electrode
potential. Electrolyte is 0.4 g NaClO4 + 0.1 2 NaCl. Raman

data from reference 102.00.000.00000000000.00.0.0...0...... 89

Fractional coverage, 6, of bromide anions (solid curve) and

normalized Raman peak intensity (dashed curve) for Ag-Br-

3.15

3.16

3.17

stretching mode, both plotted against electrode potential.
Electrolyte is 0.45 g Na0104 + 0-05 g NaBr. Raman data from

reference 102.00.00.00000000000000000000000000000COOOOOOOOO 90

Cathodic-anodic cyclic voltammogram for adsorbed

III/II redox couple at roughened silver-aqueous

08(NH3)5py
interface in 0.1 21 NaBr + 0.08 1! NaCl + 0.02 _M HCl.
Electrode roughened by means of an oxidation-reduction cycle
in this supporting electrolyte; (roughness factor ca. 1.8 on
III

(N83)5py

concentration - 50 uM; sweep rate . 100 V sec-1............ 94

basis of capacitance measurements). Bulk 0s

Cyclic voltammetry of surface bound Os(NH3)5pyIn/II in

80 m§_NaCl + 20 m! HCl. Silver electrode was roughened by
means of an oxidationrreduction cycle. Initial potentials

for each sweep are indicated. Sweep rate - 50 V s-l. Bulk

OBIII (m3)5py concentration . so “£00....OOOOCOOOOOOOOOOOO 96

Linear sweep current-potential curves for anodic removal of
lead layer deposited on silver. Sweep rate was 5 mV sec-1,
electrode rotated at 600 r.p.m. Electrolyte was 0.5,!
NaClOa adjusted to pH 3.5 with H0104, containing 0.6 p!
Pb2+. Lead deposits corresponding to anodic stripping

curves (s)-(c) formed as follows: (a) Electrode rotated at

xvii

3.18

3.19

3 .20

600 r.p.m., held at -0.585 V for 10 mins. (b) As (a), but
held at -l.20 V: for additional 20 mins. with no rotation. (c)
Electrode rotated at 600 r.p.m., held at -0.65 v for 10

Minute. 101.000.000.000...0......OOOOOOOOOOOOOOOOOOOOOOOOO 101

Differential electrode capacitance c versus electrode
potential E for: (a) polycrystalline silver; (b) silver
containing a monolayer of UPD lead; (c) as in (b) but with
additional ca. 2 monolayers of bulk lead deposit; (d)
polycrystalline lead. Lead layer prepared as indicated in
caption to Figure 3.17 and the text. Electrolyte was 0.5 !
Ra0104 pE - 3.5; for (b) and (c) additionally contained

0.6 u! Pb“

............................................... 104
Differential electrode capacitance 0 versus electrode
potential E for a mono layer of UPD lead on polycrystalline
silver (curves a-d) and for polycrystalline bulk lead

(curves a’-d’) as a function of ionic strength of sodium
perchlorate, adjusted to pH 3 .5 . Concentrations of NaClOa:

(a,a’) 0.5 1!; (b) 0.1 g; (c) 0.05 11; (d,d’) 0.01 g.........105
Differential electrode capacitance 0 versus electrode

potential E for UPD lead layer on polycrystalline silver

having various coverages of lead. Electrolyte was 0.5 g

xviii

3.21

3.22

3.23

3.24

3.25

NaClOa, pH 3.5, with 0.6,.5 Pb2+. Lead coverages

(percentage of monolayer): (a) 02; (b) 222; (c) 352;

(d)611; (e)93:; (f)1001.00.00.000000000000.0.00000000000107

Differential capacitance vs. electrode potential for UPD
lead/silver in NaCl + NaF mixed electrolytes at an ionic
strength of 0.5 :3. Key to chloride concentrations: (0)

0 mg; (0) 8 mg; (v) 20 mg; (A) 60 mg; (I) 200 mg..........113

Differential capacitance vs. electrode potential for UPD
lead/silver in NaBr + Na? mixed electrolytes at an ionic
strength of 0.5 5. Key to bromide concentrations: (0)

01-5: (O) 3 mg: (A)10 m5; (v) 301-£114

Differential capacitance vs. electrode potential for UPD
lead/silver in NaI + Ear mixed electrolytes at an ionic
strength of 0.5 5. Key to iodide concentrations: (0) 0 mg;

(0) 0.3 mg; (v) 1 mg; (A) 3 mg; (A) 10 mg; (D) 30 all:

(.)100 $.00...OOOOOOOOOOOOOOOOOOOOOOOOOOCOOOO.0.00.00.00.0115

Surface concentration of chloride vs. electrode potential

for UPD lead/silver. Conditions as in Figure 3.21.........ll6

Surface concentration of bromide vs. electrode potential

for UPD lead/silver. Conditions as in Figure 3.22.........117

xix

3.26

3.27

3.28

3.29

3.30

3.31

Surface concentration of iodide vs. electrode potential for
UPD lead/silver. Solid line: NaI + NaF. Dashed line:

Hal + NaClOA. Concentrations as in Figure 3.23............118

Surface concentration of chloride vs. electrode charge

density for UPD lead/silver. Conditions as in Figure

3.21.0000000000000000000COOOOOOOOOOOOOOOOOOOO0.0.0.00000000119

Surface concentration of bromide vs. electrode charge

density for UPD lead/silver. Conditions as in Figure

3.22.00.00.0000000000000000000000000000000000000000000000.0122

Surface concentration of iodide vs. electrode charge
density for UPD lead/silver. Conditions as in Figure

3026......0.00.0.0...0.00.0.0...0.0COOOOOOOOOOOOOOOOO0.0.0.0121

Differential capacitance vs. electrode potential for UPD
lead/silver in NaBr + NaC104 mixed electrolytes at an
ionic strength of 0.5 5. Key to bromide concentrations:

(0) 0 mg, (0) 1 mg, (0) 3 mg, (A) 10 mg, (v) 30 mg, (n) 100

QE’OOOOOOOO00......OOOCCOCOOOOOOOOOOOOOOO0.00.00.00.00000000120

Surface concentration of thiocyanate vs. electrode potential
for UPD lead/silver in NaNCS + NaF mixed electrolytes at an
ionic strength of 0.5 g. Key to thiocyanate concentrations:

(A) 1 mg, (0) 3 mg, (0)10 mg, (I) 30 1115,03) 100 mg........124

3.32

3.33

3.34

3.35

3.36

3.37

Surface concentration of aside vs. electrode potential for
UPD lead/silver in NaN3 + NaF mixed electrolytes at an ionic
strength of 0.5 5. Key to azide concentrations: (D) 3 m5,

(.)10 “fl. (A) 30 13!, (‘) 121 flooocco......................125

Surface concentration of perchlorate vs. electrode potential
for UPD lead/silver in NaClO4 + NaF at an ionic strength of
0.5 11. Key to perchlorate concentrations: (I) 10 mg, (0)
30 mg, (C) 80 mg, (4) 200 mg, (l) 500 all; (data at 500 mg

were extr‘WIated)OOOO0.0.0.0....00.000.00.000...00......00.126

Surface concentration of thiocyanate vs. electrode charge

density for UPD lead/silver. Conditions as in Figure 3.31..127

Surface concentration of azide vs. electrode charge density

for UPD lead/silver. Conditions as in Figure 3.32..........128

Surface concentration of perchlorate vs. electrode charge

density for UPD lead/silver. Conditions as in Figure 3.33..129

Linear sweep current-potential curves for anodic of thallium
layer deposited on silver. Sweep rate was 20 mV/s. Electro-

lyte was 0.5 g NaF, containing 0.8 1.131 1.10104. Curve (a):

3.38

3.39

3.40

3.41

electrode held at -960 mV for 12 minutes while rotating at
600 RPM; (b) electrode held at -960 mV for 12 minutes at

600 RPM and then held for 15 minutes at -l300 mV without

rot‘tionOOOOOCOOOCCOO00......OOOOOOOOOOOOCOOO...0.00.00.00.0137

Differential capacitance versus electrode potential for: (a)
a monolayer of underpotentially deposited thallium on poly-
crystalline silver in 0.05 g NaClOa; (b) the equivalent of
circa. 3 monolayers of thallium on polycrystalline silver in

0.05 §_Na0104; (c) as in (a), except in 0.2‘§,Ra0104........139

Differential capacitance vs. electrode potential for UPD
thallium/silver in NaCl+NaF mixed electrolytes at an ionic
strength of 0.5 3;. Key to chloride concentrations:

(0) 0 mg; (0) 1 mg; (A) 10 mg; (V) 50 In! (I) 100 mg; (0) 200

“.00...OI...0..OOOOOOCOOOIOCOOOOOOOIO0.0.0.0...0.00.00.00.00141

Differential capacitance vs. electrode potential for UPD
thallium/silver in NaBr + NaF mixed electrolytes at an ionic
strength of 0.5 5. Key to bromide concentrations: (0) 0 mg;

(I) 20 mg: (A) 50 mg; (v) 100 mg; (O) 200 @1142

Differential capacitance vs. electrode potential for UPD
thallium/silver in NaI + NaF mixed electrolytes at an ionic
strength of 0.5 5. Key to iodide concentrations: (A) 0 mg;

(.)1 “213(075 m; (.)10 “1!; (a) 20 In! (.) 50 “flossosssooss143

3.42

3.43

3.44

3.45

3.46

3.47

3.48

3.49

Surface

concentration of chloride vs. electrode potential for

UPD thallium/silver. Conditions as in Figure 3.38...........144

Surface

concentration of bromide vs. electrode potential for

UPD thallium/silver. Conditions as in Figure 3.39...........145

Surface

concentration of iodide vs. electrode potential for

UPD thallium/silver. Conditions as in Figure 3.40...........l46

Surface

for UPD

Surface

for UPD

Surface

for UPD

Surface

for UPD

concentration of chloride vs. electrode charge density

thallium/silver. Conditions as in Figure 3.38.......147

concentration of bromide vs. electrode charge density

thallium/silver. Conditions as in Figure 3.39.......148

concentration of iodide vs. electrode charge density

thallium/silver. Conditions as in Figure 3.40.......149

concentration of perchlorate vs. electrode potential

thallium/silver in NaClO4 + NaF mixed electrolytes at

an ionic strength of 0.5 g, Key to perchlorate concentra-

tionfl: (0)10 m; (A) 60 mg; (a) 200 mssosososssssossssssealso

Surface

concentration of thiocyanate vs. electrode potential

for UPD thallium/silver in NaNCS + NaF mixed electrolytes at

an ionic strength of 0.5 .11. Key to thiocyanate concentra-

tions: (0) 1 mg, (I) 5 mg, (A) 20 mg, (I) 50 mg, (D) 100 InflulSl