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ELECTROCHEMICAL STUDIES OF SOME TRANSITION METAL CRYPTATES
AND AN INVESTIGATION OF THE REACTION ENTROPIES

OF TRANSITION METAL REDOX COUPLES

By

Edmund L. Yes

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1980

ABSTRACT
ELECTROCHEMICAL STUDIES OF SOME TRANSITION METAL CRYPTATES
AND AN INVESTIGATION OF THE REACTION ENTROPIES
OF TRANSITION METAL REDOX COUPLES
By

Edmund L. Yee

Electrochemical methods were used to monitor the aqueous
complexation and redox chemistry of the trivalent and divalent
europium and ytterbium (2.2.1) and (2.2.2) cryptates. All of these
complexes were electrochemically reversible and substitutionally
inert on the cyclic-voltammetric time scale. The substantially
lower thermodynamic stabilities of the trivalent cryptates (com—
pared to the corresponding divalent complexes) arise from large
enthalpic destabilizations which outweigh smaller entropic stabili-
zations. These differences can be explained in terms of the relative
hydration of the uncomplexed trivalent and divalent ions and comply
with the trends observed for alkali metal and alkaline earth cryp-
tates having comparable ionic radii. The strong associations of the
trivalent Eu and Yb cryptates with hydroxide and fluoride were found
to be comparable to those reported for the aquated cations and these
anions.

For cations of approximately constant size, striking decreases
in the cryptate formation and dissociation rates are observed as
the cation charge increases from one to three. These rate varia-

tions and their enthalpic and entropic components reflect the

Edmund L. Yee

increased changes in cation hydration which accompany cryptate
formation when the cation charge increases. The dissociation of
the lanthanide cryptates is substantially catalyzed by acid, and
this can be attributed to the need to alter the ligand conformation
before or during the removal of the cation from the cryptate cavity.

The homogeneous self-exchange rate constants of Eu(2.2.l)3+/2+

and Eu(2.2.2)3+/2+

were calculated by using the Marcus cross-relation
and rate data from cross reactions involving these cryptates. The
greatly enhanced self-exchange rates of the cryptates relative to

that for Hui.”2+

were ascribed to smaller solvent- and ligand-
reorganization components in the activation barriers of the cryptate
systems.

The half-cell reaction entropies As;c, for some M(III)/(II)
redox couples in aqueous solution were determined with a non-
isothermal cell arrangement. Reaction entropies in the 36-49 e.u.
range were found for aquo couples. Several substitutionally inert
ruthenium (III)/(II) complexes were employed to scrutinize ligand
effects on A8;c. The sizable decreases in A8;c which accompany the
replacement of aquo ligands by ammines were attributed to less ex-
tensive solvent-ligand hydrogen bonding by the latter ligands. Large
reductions in As;c resulted when aquo and ammine ligands were
replaced by anionic ligands. Examinations of couples containing
chelating ligands disclosed substantial differences in Aséc between
Co(III)/(II) couples and the corresponding Ru(III)/(II) and

Fe(III)/(II) systems. These could be attributed to ligand-

conformation and electron-delocalization factors.

Edmund L. Yee

A correlation between the self-exchange rate of a redox couple
and its reaction entropy was found. This relation arises since
both A8;c and outer- and inner-shell reorganization energies are
related to solvent-structuring effects.

Reaction entrOpies and formal potentials were employed to
calculate the free-energy, enthalpic, and entrOpic driving forces
necessary for a thorough comparison of the predictions of the con-
ventional Marcus model of outer-sphere electron transfer with the
experimental rate constants and activation parameters for cross
reactions involving cationic transition metal complexes. The con-
siderable disagreements between the experhmental and predicted
values found at very large driving forces were compatible with the
presence of sizable, unfavorable work terms in the formation of the
binuclear collision complex. An analysis of the activation parame-
ters indicated that these work terms may be due to the reorganiza-
tion of solvating water molecules that is needed in order to form

the highly charged collision complex.

ACKNOWLEDGMENTS

The author would like to express his appreciation to all the
peOple, big and small, who helped him travel the long and winding
road to the doctoral degree. The contributions of the following
individuals deserve special mention:

Dr. Michael Weaver has provided invaluable advice, guidance,
and support. His insights have been shafts of gold amid the
Stygian darkness of scientific uncertainty.

My colleagues in the Weaver group have come through like pros
in giving me some much-needed assistance in the laboratory. MOre
importantly, they have been good and true friends.

Special thanks go to Rose Manlove for transforming a scrawled,
unreadable mass of dubious prose into a well—typed, legible mass of
dubious prose.

The staff of the Department of Chemistry has been an important
factor in much of my progress as a graduate student.

The Air Force and Navy have supported the author through a good
deal of his graduate career. One can only marvel at the manner in
which money is expended in the never-ending struggle against the
spread of communism.

Financial support by the Department of Chemistry and the General

Electric Foundation is also gratefully acknowledged.

ii

iii

The encouragement given to me by my brothers and sisters is
profoundly appreciated. MOst importantly, I would like to thank

my parents, without whom this all would have been impossible.

TABLE OF CONTENTS

LI S T OF T“ LES O O O O O O O O O O O O C O O O C O O O 0

LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . .

INTRODUCTION .

CHAPTER I

CHAPTER II

HISTORICAL BACKGROUND. . . . . . . . . . .
A. Historical Background - Cryptates.
B. Reaction Entropies . . . . . . . . . .
C. Homogeneous Electron-Transfer Kinetics
of Transition Metal Complexes . . . .
EXPERIMENTAL ASPECTS . . . . . . . . . . .
A. Apparatus. . . . . . . . . . . . . . .
B. Electrochemical Techniques . . . . . .
1. Common Features. . . . . . . . . .
2. Cyclic Voltammetry . . . . . . . .
3. Polarography . . . . . . . . . . .
4. Preparative Electrolyses . . .

S. Potentiometric Methods . . . . . .

iv

Page

viii

xii

16

20

33
34
34
34
35
36
36

37

CHAPTER III

C.

D.

Materials . . . . . . . . . . . . . . .

1.

2.

3.

Cryptates . . . . . . . . .
Reagents and Solvents . . . .

Transition Metal Complexes
and Compounds . . . . . . . . . . .

Kinetic Methods . . . . . . . . . . . .

1.

2.

3.

Cryptate Aquation Studies .

Homogeneous Redox Reactions
Princ iples O O O O O O O O O O O O O

Homogeneous Redox Reactions
Experimental Specifics. . . . . .

ELECTROCHEMICAL STUDIES OF EUROPIUM AND
YTTERBIUM CRYPTATE FORMATION IN AQUEOUS
SOLUTION. O O O O O I O O O O O O O O O O O

A.

B.

IntrOduc t ion 0 O O O O O O O O O O O O O

Res‘llt s O O O O O O O O O O O O C O O O

1.

2.

Complexation Thermodynamics . . . .

:Conplexation and Dissociation

Kinet ics O O O O O O O O O O O O O O

DisCUSSiono O I O O O O O O O O O O O O

1.

2.

The Effects of varying the Cation

Charge upon Cryptate Thermodynamics .

Association of Lanthanide Cryptates
Wit h smal 1 An ion 8 O O O O O O O O O

The Effects of Varying the Cation
Charge upon Cryptate Substitution
Kinetics O O O O O O O O I O O O O O

37

. 37

38

38
42

42

43

45

48
49
49

49

63

67

67

74

. 7S

CHAPTER IV

CHAPTER V

vi
REACTION ENTROPIES OF TRANSITION METAL
Rwa COUPLES O O O O O O O O O O O I O 0
A. Introduction. . . . . . . . . . . .
B. Determination of Reaction Entropies .

C. Practical Aspects of Nonisothermal
cells C O I C O O O O O O O O O O O O

D. Results . . . . . . . . . . . . . . .
1. Aqua Couples. . . . . . . . .

2 . Mthenium(III)/ (II) and

Osmium(III)/(II) Couples Containing
Mbnodentate Ligands . . . . . . . . .

3. Redox Couples Containing
Chelating Ligands . . . . . . . . . .

Discussion. . . . . . . . . . . . . . . .

1. Ligand Effects on Reaction Entropies.

2. Correlations between Outer-Sphere
Self-Exchange Rates and Reaction
Entropies . . . . . . . . . . . .

ACTIVATION PARAMETERS FOR HOMOGENEOUS

OUTERPSPHERE ELECTRON-TRANSFER REACTIONS:

COMPARISONS BETWEEN SELF-EXCHANGE AND

CROSS REACTIONS USING THE MARCUS THEORY .

A. Introduction. . . . . . . . . . .

B. Kinetic Formulations and Results. . .
1. Free Energies of Activation . .

2. EntrOpies and Enthalpies of
Activation. . . . . . . . . . .

C. Discussion. . . . . . . . . . . . . .

.102
.106

.106

.119

.133
.134
.136

.136

.156

.173

vii

CHAPTER VI HOMOGENEOUS REDOX KINETICS OF THE
EUROP I W cm TATES O I O O O I O O O O O O O O

A.

B.

C.

Introduction. . . . . . . . . . . . . . .
Resu1ts O O O O O O O O O O O O O O O O 0
Discussion. 0 O O O O O O O O O O O O O O

1. Reliability of the Self-Exchange
Parameters of the Eu Cryptates. . .

2. Mechanistic Implications of the
Inverse-Acid Dependence of k12' . .

3. The Effect of Cryptate Formation
on the Intrinsic Redox Reactivity
of the Eu(III)/(II) Couple. . . . . .

CHAPTER VII MISCELLANEOUS EXPERIMENTS . . . . . . . . . .

A.

Electrochemistry of Other Transition
Metal Cryptates . . . . ; . . . . . . .

The T1(2.2.2)+/T1(Hg) Couple as a Probe
of Cryptate Thermodynamics in Nonaqueous
Solvents. . . . . . . . . . . . . . . . .

CHAPTER VIII CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK.

A.

B.

LIST OF REFERENCES .

COnclusions . . . . . . . . . . . . . . .
1. Transition Metal Cryptates. . . . . .
2. Reaction Entropies. . . . . . . . . .

Suggestions for Further WOrk. . . . . . .

O O O O O O I O O O O O O O O O O O O O

180

. 181

182

190

190

191

197

202

203

207

214
215

215

. 217

220

224

LIST OF TABLES

Table Page

1 Thermodynamics of some Europium and
Ytterbium.Cryptate Redox Couples in
NOncomplexing Media at 25°C. . . . . . . . . . . . . . . .54

2 Chmulative Formation Constants for
Association between Europium and
Ytterbium Cryptates and Fluoride and

Hydroxide Ions . . . . . . . . . . . . . . . . . . . . . .62

3 Rate Constants and Activation
Parameters for Dissociation of Europium

and Ytterbium.Cryptates at 25°C. . . . . . . . . . . . . .64

4 Acid Acceleration of Europium Cryptate
Dissociation at 25°C . . . . . . . . . . . . . . . . . . .66

5 Anion Effects on Eu(2.2.l)3+ Dissociation. . . . . . . . .67
6 Free Energies, Enthalpies, and EntrOpies

of Complexation of various Metal Cations

with (2.2.1) and (2.2.2) Cryptands at 25°C . . . . . . . .69
7 Rate Constants and Activation Parameters

for Formation and Dissociation of some
(2.2.1) Cryptates at 25°C. . . . . . . . . . . . . . . . .77

viii

Rate Constants and Activation Parameters

for Formation and Dissociation of some

(2.2.2) Cryptates at 25°C. . . . . . . . . . . . . . . . . .79
Reaction Entropies for various M(OH2):+/2+

Redox Couples. . . . . . . . . . . . . . . . . . . . . . . .96

Reaction EntrOpies for Various Ru(III)/(II)
and Os(III)/(II) Redox Couples Containing
Unindentate Ligands. . . . . . . . . . . . . . . . . . . . .98

Reaction Entropies for various M(III)/(II)
Couples Containing Chelating Ligands . . . . . . . . . . . 103

Reaction Entropies for various Redox
Cbuples of the Type MLéLKCIII)/(II). . . . . . . . . . . . 113

Comparison between Activation Free Energies

for Some Ruthenium(III)/(II) Self—Exchange .

Reactions and the Reaction Entropies for the

Corresponding Redox Couples at 25°C. . . . . . . . . . . . 122

Comparison between Kinetic Parameters for

Selected Outer-Sphere Self-Exchange Reactions

and Reaction Entropies for Corresponding

Redox Couples at 25°C. . . . . . . . . . . . . . . . . . . 125

Kinetic and Thermodynamic Parameters of
Some Self-Exchange Reactions at 25°C . . . . . . . . . . . 140

Kinetic Parameters of Some Self-Exchange
Reactions at 25°C. . . . . . . . . . . . . . . . . . . . . 142

Driving Forces and Rate Constants for

Cross Reactions Between Aquo Cations
at 25°C. 0 O O O I O I O O C O O O O O O O O O O O O O O O 145

Free Energies of Activation and Driving
Forces for Cross Reactions Between Aquo
“tions at 25°C. 0 O O O O O O O O I O O O O O O O O O O O 147

Driving Forces and Rate Constants for

Cross Reactions Involving NOn-Aquo Cations
at 25°C. 0 C O O O O O O O O O O O O O C O O O O O O O O O 150

Free Energies of Activation and Driving
FOrces for Cross Reactions Involving
Non-Aqua cations at 25°C 0 o o o e o o o o e e o o o e o o 152

Thermodynamic Parameters for Some
Cross Reactions at 25°C. . . . . . . . . . . . . . . . . . 159

Comparison of Activation Parameters for
Cross Reactions Determined at 25°C with
Those Calculated from Marcus Theory. . . . . . . . . . . . 161

Activation Parameters for some Cross
Reactions Having Large Driving Forces.
Effect of Including Entropic and Enthalpic

Werk Terms on Marcus Predictions . . . . . . . . . . . . . 166

Rate Constants and Equilibrium Constants
for Cross Reactions Involving Europium
cryptates O O O O O O O O O O I O O O O O O O O O O O O I O 187

Self-Exchange Rate Constants for Selected
NOncryptate Couples. . . . . . . . . . . . . . . . . . . . 188

26

27

.xi

Self-Exchange Rate Constants for

Eu(2.2.1)3+/2+- and Eu(2.2.2)3+/2+.

. . . . . . . . . . . . 190
Formal Stability Constants (log Ks)
for Univalent (2.2.2) Cryptates in
Dimethylsulfoxide and Acetonitrile
at 25°C. 0 o ‘0 o o o o e o e o o e o o o o o o o e o e e o 213

Figure

LIST OF FIGURES

Structures of Some Common Cryptands . . . . . . . . . . . 6

Cathodic-anodic cyclic voltammograms of

BuzzandEu(2.2.l)3+...................52

Plot of Bi, the formal potential for the
Eu(2.2.l)3+/2+ couple in the presence of

fluoride anions, vs. the logarithm of the
fluoride concentration. Ionic strength

held constant at 0.5 with sodium perchlorate. . . . . . . 60

Plot of the activation energy for some outer-
sphere self-exchange reactions vs. the reaction

entropy for the redox couples . . . . . . . . . . . . . . 127

Experimental (AH: for cross reactions

2)corr
corrected for Debye—Hfickel work terms

plotted against (AHf2)calc. . . . . .l. . . . . . . . . . 170

Experimental (ASE for cross reactions

2)corr
corrected for Debye-Hflckel work terms plotted

against (A8i2)calc. . . . . . . . . . . . . . . . . . . . 172

xii

INTRODUCTION

Electron-transfer reactions that involve transition metal ions
form the basis of many scientifically and technologically important
processes, so it is crucial that attempts be made to reach a better
understanding of the factors which govern the redox thermodynamics
and kinetics of these cations. One effective method to scrutinize
the behavior of a metal ion redox couple is to vary its ligand
environment in some appropriate fashion. This variation is often
accompanied by changes in the standard potential of the couple and in
the lability of the respective oxidation states, by modifications in
the degree and type of solvent structure around the metal complex, and
by alterations in the mechanism by which electron transfer occurs.

Among the numerous ligands and complexes which have been or could
be employed for such studies are cryptands, a class of polyoxadiaza-
macrobicycles, and cryptates, the complexes which these cryptands form
with metal ions. In these complexes, the cation is encapsulated
‘Within a molecular cavity defined by the ligand, and this uncommon
structure is expected to modify the redox prOperties of the enclosed
Inetal ion couple in an interesting and unusual fashion. The elucida-
tion of these ligand effects will require the investigations of the
cryptate ligation processes and redox thermodynamics that are re-

counted in Chapter III along with the studies of the redox kinetics

of these complexes which are described in Chapter VI. While little
work has been done on electroactive cryptates, some of the previous
findings regarding cryptate complexation thermodynamics and kinetics
do have a bearing on the results detailed in Chapter III. Therefore,
this earlier material is summarized in Section A of Chapter I.

‘While some useful trends concerning ligand effects may be
gleaned from the transition metal cryptates, their unusual structure
may restrict the conclusions of this study to a number of specific
points. One way to achieve a broader view of ligand influences is
to determine the ionic entrOpies of transition metal ions and the
factors which affect these quantities. Such measurements can be
utilized to obtain information on the degree of hydration around an
ion and on the extent and type of interactions between the ligands
and the solvent. In Chapter IV, entrOpy data for a series of related
transition metal couples are used to ascertain the ligand effects on
the redox thermodynamics of these complexes. 1

Since solvent reorganization is a major component of the acti-
vation barrier in electron transfer reactions, one would expect that
the solvent structure around an ionic reactant would have a strong
influence on the redox kinetics of this species. Ionic entrOpies
should provide insights into this structure and thus will be helpful
in explaining trends in redox reactivity. In fact, a correlation
between entrOpies and redox rate parameters is found and discussed
in Chapter IV.

The entrOpies of some transition metal ions have been determined,

but very little has been done in a systematic manner, especially with
respect to the ligand dependences of these entropies. Section B of
Chapter I attempts to introduce some unity to the field of ionic
entropies. It is especially important to make some sense of the
various scales of entropy that are employed since no standard con-
vention has been agreed upon.

The combination of the appropriate ionic entrOpies will yield
the entrOpy of reaction for complexation and redox processes. The
former application will prove useful in Chapter III when cryptate
stabilities are discussed in entropic and enthalpic terms. The
latter employment of entropies will be essential in Chapter V when
the limitations of the contemporary theories of homogeneous electron-
transfer kinetics are examined. As a preparation for this analysis,
Section C of Chapter I briefly summarizes these theories. An over-
view of some of the principal electron-transfer mechanisms is also
given since these concepts and the theories mentioned above will be
needed to analyze and to understand the kinetic data for cryptate

redox reactions presented in Chapter VI.

CHAPTER. I

HISTORICAL BACKGROUND

A. Historical Background - Cryptates

Cryptands were first synthesized by Lehn and co-workers in 1969
(1,2). (The macrobicyclic ligands used in this study are more pre-
cisely designated [2]-cryptands to distinguish them from the tri—
cyclic [3]-cryptands and the tetracyclic [4]-cryptands.) All of these
cryptands contain three polyether strands attached to two bridgehead
nitrogens. The common names of the cryptands consist simply of the
number of ether oxygens in each of these strands. Figure 1 gives the
structure and the common names of the cryptands used in this work.

The polyether strands of a cryptand define a three-dimensional
molecular cavity. Thus, the cryptands (2.2.2), (2.2.1), and (2.1.1)
have a cavity radii of 1.4, 1.1, and 0.8 A, respectively (3). A metal
ion of the proper size can be encapsulated within this cavity to yield
an inclusion complex (3). In actual practice, these cryptate complexes
are frequently made simply by mixing the ligand and a suitable salt of
the metal ion in an appropriate solvent.

The crystal structures of a number of these cryptates have been
determined, principally by Weiss and co-workers (4,5). The inclusion
complexes have several structural features in common. The metal ion
lies within the molecular cavity of the ligand. The cryptand has an
endb-endb conformation; i.e., the lone pairs on the nitrogen atoms
point into the cavity. It is also possible for the ligand to adopt
configurations where both electron pairs are directed away from the

cavity (exo-exo} or where one pair points into the cavity while the

= l, n = 1 (2.2.2)
m = l, n = 0 (2.2.1)
= 0, n = 1 (2.1.1)

Figure 1. Structures of some common cryptands.

other points away (endb-exo) (3). When the size of the metal ion
closely matches that of the cryptand cavity, the heteroatoms of the
ligand tend to be symmetrically distributed around the ion. On the
other hand, some distortion of the ligand occurs when the metal ion is
too large or too small for the cryptand cavity (5).

A good deal of the interest in and the work on cryptands has
stemmed from the discovery that these ligands are capable of forming
stable complexes with alkali metal cations in aqueous and nonaqueous
solutions (3,6). Along with the crown cyclic polyethers first synthe-
sized by Pederson (7-10), the cryptands gave rise to the new field
of the study of alkali metal complexation. Investigations of the in-
fluences on the stability of the alkali metal cryptates have found that
each cryptand has its own size selectivity. Simply stated, for a
given cryptand, the most stable alkali metal cryptate has the closest
match between the radius of the cation and that of the ligand's
molecular cavity. The other alkali metal complexes of this particular
cryptand would be less stable and would contain a cation which was
unsuitably large or small with respect to the ligand cavity. Thus Na+
with a radius of 1.02 A.(1l) forms the most stable alkali metal complex
with the (2.2.1) cryptand (cavity radius - 1.1 A.(3)), while K+
(radius - 1.38.A) plays the corresponding role for (2.2.2), which has
a cavity radius of 1.4 A.(3). However, it has been noted (12) that one
cannot think of this selectivity as a simple consequence of one cation
having a better steric fit into the ligand cavity. The observed selec-
tivity is the product of a combination of several factors, including
the change in enthalpy which results when the solvent molecules in the

metal ion's first coordination sphere are replaced by the ligand and

the entropic effects arising from the desolvation of the cation and
cryptand which accompanies the formation of the complex (12) .

Alkaline earth cations also form stable cryptates and several
studies of the formation thermodynamics of these complexes have been
published (3,6,12). For alkali metal and alkaline earth ions of
comparable radius, the divalent ion has generally been found to form
more stable cryptates (6,13). The greater stability of the divalent
cryptates is not merely due to greater ion—dipole interactions between
the cation and the bonding sites of the ligand. An examination of the
complexation thermodynamics for alkali metal and alkaline earth
cryptates (12) revealed that the enthalpies of complexation for the
divalent ions were comparable to or less favorable than those for
monovalent ions of similar size. Thus, the increased stabilities of
the alkaline earth complexes arise from entropic factors. The hydra-
tion around one of these divalent ions is more extensive than that for
the corresponding monovalent ion, so one expects a more favorable
entropy of complexation when the divalent hydration shell is renoved
upon the encapsulation of the metal ion within the cavity of the
cryptand.

Studies of cryptate thermodynamics have not been limited to alkali
Inetal and alkaline earth complexes. The formation constants for a
n‘J-‘lnber of transition metal cryptates have been determined (14-17) with
the same potentiometric methods which were used for most of the alkali

2+, P 2+

metal and alkaline earth systems. While Cd 13 a 382+: Ag+. and

+
T1 form a number of stable cryptates, the d-block transition metal
10118, Niz+, an+, and Coz+, do not show much tendency to follow suit.

of the d-block ions, only Cu2+ forms cryptates with appreciable

Q“

IAI

DU.

 

stability constants. The relative instability of the cryptates of the
d—block ions has been explained as a combination of size incompati-
bility and hard-soft acid-base considerations (15) . The d-block
cations studied all have radii of about 0.70 to 0.75 A (18) which
would be too small for the (2.2.1) and (2.2.2) cavities. The (2.2.1)
cryptand has a cavity radius of 0.8 Awhich would be a good match for
the cation radii given above. However, the (2.2.1) complexes are still
not very stable. The majority of the donor groups of this ligand are
OXygen atoms, but the d-block ions have a greater affinity for nitrogen
donors. Therefore, the instability of these (2.1.1) complexes is at
least partly the result of relatively unfavorable interactions between
the cations and the donor atoms of the ligand. To verify this, Lehn
and Montavon synthesized a ligand which was analogous to (2.1.1) except
that the two oxygen atoms in the diether strand were replaced by two
N—CH3 groups. This tetraaza ligand formed complexes with d-block ions
which were at least five orders of magnitude more stable than the cor-
responding (2.l.l) cryptates (15).

While cryptate thermodynamics has received a good deal of attention,
the kinetics of cryptate formation and dissociation has not been
neglected either. Studies have been performed in aqueous and non-
a><1ueous solution, and several kinetic techniques have been employed,
ineluding proton and alkali metal NMR (nuclear magnetic resonance)
(19-23), temperature jump (24), and stopped-flow (25-28). Investiga-
t343113 of cryptate ligation kinetics have concentrated almost exclu-

31\iely on the alkali metal and alkaline earth complexes. (The only
Q‘ertions to this pattern are T1(2.2.2)+ (19) and Ag(2.2.l)+ (27).)

Ho"Never, much useful information may be gleaned from these studies.

10

What follows will be concerned with aqueous solutions since these are
the media employed in the kinetic studies of the present work.

Within a given charge and ligand type, cryptate stability is more
clearly reflected in the dissociation rate constant kd rather than the
formation rate constant kf. The most stable cryptate will generally

have the smallest kd (26) although a number of anomalies

(e.g. Sr(2.2.l) 24-) are present. A comparison between the k and k

f d
values of alkali metal and alkaline earth cryptates reveals that greater

differences tend to be found between charge types than within a given
charge type. The kf and kd for alkaline earth cryptates are consis-
tently much smaller than those for the alkali metal complexes.

Some insight into the factors which determine these relative values
has been gained by examining the activation entropies of the formation
and dissociation reactions. In particular, the activation entropies
for the dissociation process usually are negative quantities having
appreciable magnitudes. This has led a number of workers (13,26) to
conclude that the activated complex for the formation and dissociation
reactions bears a greater resemblance to the separated reactants
(i.e., metal ion and ligand) than it does to the cryptate product.
One can think of the dissociation reaction as proceeding via a transi-
tion state in which some water molecules coordinate to the metal ion.

while the metal ion is still bound to the cryptand, there is a good
Possibility that the cation is no longer inside the molecular cavity
(25). The alkaline earth ions, as a consequence of their greater charge,
will tend to structure the solvent to a greater extent in this transi-
tion state than alkali metal ions would. This is supported by a study

Of the activation entropies for the dissociation of the (2.2.2)

11

cryptates. The Na+ and K+ complexes have positive activation entropies

2+ 2+ 2+
9

while the corresponding quantities for the Ca Sr , and Ba

cryptates are considerably more negative. Similar trends may be seen
for the (2.2.1) complexes, although the data for the K+ complex are

not available.

It has also been noted (13,26) that the kf values for the alkali
metal and alkaline earth cryptates follow the same pattern as the rates
of water exchange in the hydration shells of the cations although the
actual formation rates are several orders of magnitude slower than
those for the water exchange. The correspondence between kf and the
VPater exchange rates has led to the conclusion that partial or complete
desolvation of the cation occurs prior to or during the rate-determining
step of cryptate formation (26). Of course, the extent of desolvation
which is thought to occur here must be in harmony with the general
description of the activated complex given in the preceding paragraph.
The much smaller kf values relative to the water exchange rates have
been explained in terms of two alternative or possibly coexistent
effects (26). To begin with, the concentration of the reactive con-
formation of the cryptand may be quite low. This would imply an

.unfavorable pre-equilibrium before the rate-determining step. In
a~<1vziition, a large steric barrier to cryptate formation may exist as
a. consequence of the possibility that first bond formation may be
difficult with the relatively rigid cryptands.

Under a given set of conditions, cryptate formation and dissocia-
t10:)n follow simple second-order and f irst-order rate laws, respectively

(26). However, kinetic studies of these processes are not completely

free from complications. Cox and Schneider have found that the

12

dissociation rates of many cryptates are acid catalyzed (27). The

experimental dissociation rate constants Rd had acid dependencies of

the form

k3 - kd + ka [3*] (1)

The presence of the acid catalyses furnishes one with information on
the mechanism of the dissociation reaction. Cryptates can be con-
sidered to exist in solution as an equilibrium mixture of three
conformations: the endo-endo, exo-endo, and exo-exo. The endo-endo
is the most stable thermodynamic form and is the conformation found
in the solid state, as mentioned previously. However, dissociation
can occur when the cryptate is any of these three forms (19). Because
at least one nitrogen in the exo-endo and exo—exo forms is directed
away from the interior of the cavity, one can conceive of a mechanism
in which protonation at this site occurs prior to the rate-determining.
removal of the metal ion from the cryptate cavity (27) . Such a pre-
ceding step could give rise to the 1:3 [H+] term in eqn (1). The kcl
term can then be seen to stem from a mechanistic pathway in which the
metal ion exits the cryptate cavity while the ligand remains in an
endo-endo conformation. Because the lone pairs on the nitrogens
renain directed into the cryptate cavity, there is little chance for
Protonation to occur prior to the rate-determining removal of the
Qation. The combination of the two pathways described here results
in the two terms of eqn (1). The observed acid catalyses originate
in the stabilization of an intermediate (the exo-endo or era-3x0
c(Intiformation) via the protonation of one of the nitrogens on the ligand.

It is also enlightening to note that the dissociation rates of several

13

cryptates (e.g. Sr(2.2.1)2+) show little, if any, acid dependence.
This has been interpreted as an indication that the conversion of the

endo—endo to exo-endo and era-9x0 forms is the rate-determining step

in the equation of these cryptates (27). Thus no protonation preceding

the rate-controlling step is found for either the endo-endo or the

exo-endo/exo—exo dissociation pathways.
Several electrochemical studies of cryptates have been performed.

The great majority of these have been carried out by Peter and Gross

neith propylene carbonate as the solvent (29-36). Britz and Rnittel

have produced the only electrochemical study conducted in aqueous

solution (37). This work dealt with an investigation of the adsorption

of K(2.2.2)+ at a dropping mercury electrode. A combination of drop-
time measurements and a.c. polarography was employed to determine the

maximum surface coverage and the diffusion coefficient of the K(2.2.2)+

ion.
Most of the work of Peter and Gross has been devoted to the electro-

chemistry of alkaline metal and alkaline earth cryptates in propylene

carbonate (29-33,35). Several points have emerged. The electroreduc-

tions of the cryptates occur at considerably more negative potentials

than those for the corresponding solvated metal ions. This result is

not unexpected. The oxidized form of the alkali metal or alkaline

earth is stabilized by complexation with the cryptand. An electrode

of greater reducing power (i.e., more negative potential) is required

to reduce such a species to the metallic state. Prolonged electrolysis

of solutions of K(2.2.2)+ has shown that the products of the electro-
reduction of this complex are the amalgam K°(Hg) and the free ligand (35) .

Gisselbrecht, Peter, and Gross have also studied the electrochemistry

14

of other cryptates. In cyclic voltammograms of the Tl(2.2.2)+ complex,
one could observe peaks corresponding to the electroreduction and
electrooxidation of the cryptate (34). In contrast, the voltammograms
of the alkali metal and alkaline earth cryptates have no anodic peaks
which can be attributed to an oxidation which involves the complex
(32,35).

The complexes of (2.2.1) and (2.2.2) with Fe2+, Coz+, and Ni2+ have
been investigated as well (36). The electrochemical reductions of these
(:ryptates in prOpylene carbonate were found to be irreversible deposi-
tion reactions and to occur at'more negative potentials than the
Ireductions of the corresponding solvated metal ions. (The Ni(2.2.l)2+
complex is the sole exception to this second trend.) While these
alkali metal, alkaline earth, and transition metal cryptates which
tindergo deposition reactions are interesting electrochemical systems
for several reasons, they are far from ideal for studies of the funda-
muental parameters which affect redox reactivity. For these complexes,
the analysis of electron-transfer kinetics is complicated by the
ligation reactions which are involved in the deposition process.

Given the above considerations, one comes to the conclusion that
a more suitable cryptatewould be one in which both halves of a metal
ion redox couple would remain in complexed form and in solution. These
Properties would yield mechanistically simpler systems which are free
from the preceding or following chemical steps that accompany a

deposition. These simple, straightforward systems would be a crucial
c<>tnponent in a fundamental and unambiguous investigation of the factors
‘thich control redox thermodynamics and kinetics.

One may examine the preceding information and trends and speculate

15

that a europium or ytterbium cryptate couple might have the desirable
features which were described above. The radii of Euz+ (1.17 A (11))
and Yb2+ (0.93 A (18)) are such that one would expect these ions to
have stable (2.2.1) and (2.2.2) complexes, if the size criteria for-
mulated from the alkali metal and alkaline earth systems are valid for
these lanthanide ions. It has been stated the chemical properties of
divalent lanthanide ions are similar to those of the alkaline earth
ions having comparable radii (38) . Thus, there is no reason to suspect
that the factors which govern the thermodynamic and kinetic stabilities
of the alkaline earth cryptates will be qualitatively different to
those for the Eu2+ and Yb2+ cryptates.

In aqueous solution, both of these divalent lanthanide ions can be
converted to the trivalent ion in a simple one-electron oxidation.
Whether a similar one-electron interconversion in aqueous solution can
be found for the cryptates of Eu and Yb is one of the objectives of
this investigation. In addition, the kinetic stabilities of the tri-
valent and divalent forms of these complexes will be determined. This
information would be useful in determining the suitability of the
lanthanide systems for redox studies since complexes with slow dissoci-
ation kinetics are required in order to minimize the mechanistic com-
Plications arising from ligation reactions which coincide with the
electron-transf er proc es s .

Furthermore, a study of the thermodynamic stabilities of the Eu
and Yb cryptates will be necessary. This will be needed for determin—
ing the thermodynamic contributions to the redox reactivity of the Eu
and Yb complexes. As usual, an analysis in terms of enthalpies and

er1t:ropies will provide more detailed information than could be acquired

16

solely from free energies. One effective method for obtaining such

entropic and enthalpic data is through the employment of ionic

entrOpies and particularly reaction entropies as is now discussed.

B. Reaction EntrOpies

Partial molal entropies can be exceedingly useful when one investi-
gates the thermodynamic properties of a species in solution. The
partial molal entropy of the i th component of a system §i can be
defined as the temperature dependence (under constant solution com-

position and pressure) of Hi, the chemical potential of component 1
(3111/31)? - S1 (2)

In solution studies one frequently focuses attention on. one ion. Under
these circumstances, it would seem appropriate to think in terms of
partial molal entrOpies for single ions. Strictly speaking, only the
partial molal entropy of a neutral salt (i.e., the sum of the ionic
entrOpies for the cation and anion) is experimentally accessible (39).
However, methods are available for estimating single ion entropies.
One may arbitrarily assign an entrOpy value to some ion and thus
Obtain a relative scale of single ion partial molal entropies. It is
conventional to use the scale based on the assumption that SE... - O (39).
An alternative approach is to estimate the absolute entropy of an ion
in solution by means of an extrathermodynamic assumption. Several

different assumptions have yielded similar values of S12... (40). An
cal deg-1 mol-l) was obtained

approximate value of SE... - -5 e.u. (e.u.

by each of these methods (40). Once one absolute ionic entropy is set,
the relative scale of entropies mentioned above can be easily converted

into an absolute scale. While the scope for interpretation is somewhat

17

smaller with the relative scale, an ionic entropy from either scale
contains valuable information on the interactions between an ion and
the solvent molecules surrounding it.

Because transition metal ions are used extensively in studies of
redox thermodynamics and kinetics, the entropies of such ions are of
especial interest to the present work. Some investigations of these
systems have already been conducted. The entropies of these ions have
been found to depend markedly on the ionic charge and on the types of
ligands coordinated to the metal ion (41-44). In examining the various
possible reasons for this wide variation in entropies, one can see how
electrostatic effects (45,46) contribute to the first dependence. The
"structure-making" and "structure-breaking" ability of the ion in
solution will also have a decided influence on the ionic entropy (47-49) .

While individual ionic entropies have a great deal of utility, a
variation of this concept is particularly appropriate when dealing with
redox systems. One can consider 115:1, the difference in partial molal
entropies of the reduced and oxidized forms of a redox couple. Under
this definition AS]; is the reaction entropy for a half-cell reaction
and can be regarded as the entrOpic analog to the half-cell electrode
potential. One can employ 138;: in a variety of circumstances for a
substantial number of purposes. A relative measure of the degree of
Solvent ordering around the oxidized and reduced forms of a metal ion
complex is available through the use of As;c. The entropy of reaction
for a homogeneous redox reaction can be calculated by taking the dif-
ference between the Asgc values of the two redox couples involved.

I—-arge AS:c values are frequently found, so that this term can be the

 

18

predominant constituent of the free-energy driving force for an
electrode reaction (50). By combining the appropriate values of
As;c one can obtain the difference in the entrOpies of formation
between the oxidized and reduced forms of a metal ion complex. This
will definitely be applicable when comparing the complexation thermo-
dynamics of the trivalent and divalent lanthanide cryptates.

As evidenced by the above discussion, a knowledge of As;c values
can be very useful. Unfortunately, a comparative shortage of data
wists in the literature. Moreover, some of these data are contradic-
t°tY . possibly as a consequence of the different entrOpy scales which

are in use. Nonetheless, it has been determined that the A31; values
°f transition metal couples are considerably influenced by the ligands
which are coordinated to the metal ion. For one-electron aquo cation
redox couples, Asa: values in the 40 to 50 e.u. range are found, pro-
vIlded that one enploys the normal convention and writes the half-cell
reaction as a reduction (41,45,49). In contrast, metal ion complexes
cuntaining large organic ligands have been found to have much smaller
481°C values (51). This effect has been ascribed to the efficacy of the
Organic ligands in shielding the cation from the surrounding water
tllolecules (51).

While some helpful points have emerged from the results described
above, it is clear that a study of wider scape and greater depth is
1leaded. In particular, Asgc values are required for transition metal
QDuples containing other ligands besides aquo or large organics. This
QOuld enable ligand influences on A911: to be discerned. The dearth of
data on other ligand systems can be traced to a lack of suitable redox

couples. In aqueous solution, most transition metal ion complexes

.19

which contain common unidentate ligands are not stable in both oxida-

tion states. In at least one of these states, the complex will be

unstable with regard to dissociation, chemical oxidation, or other
reactions. These circumstances preclude the measurement of the equili-
briuxn cell potentials of these couples. Furthermore, the determination

of ionic entropies is prevented since these quantities are frequently
obtained from the temperature dependences of cell potentials (40).
This situation has recently been significantly improved when it
was determined that several ruthenium(III)/ (II) complexes were substi-
tutionally inert in both oxidation states, at least on the time scale
of cyclic voltammetry and other electrochenical perturbation techniques

(52-56). Some osmium(III)/(II) amine couples have been found to have

8ll-Inilar properties (57,58). The Ru and the Os couples are electro-

c1lenically reversible (i.e., have fast heterogeneous charge-transfer
k1|.netics), and as a consequence, cyclic voltammetry can be used to

0'btain accurate values of the equilibrium cell potentials.

Thus, the As;c values for these Ru(III)/(II) and Os(III)/(II)

couples should be experimentally accessible. Aside from their intrinsic

Value these data could also find application as a means of estimating

AS°C values for other systems. The reaction entropies for Cr(III)/(II)

end Co(III)/ (II) complexes (which cannot be determined directly) might

be roughly approximated by the As:c for the analogous Ru or Os system.

This procedure is not without some justification. Empirical correla-

tions (41-44) suggest that As:c values for simple transition metal
c=<>uples are chiefly functions of the type of ligand involved and of
the charge on the cation. The actual identity of the central metal.

ion appears to be of less significance. Given the paucity of data at

20
present, it is obvious that a generalization such as this cannot be

either verified or refuted until further investigations are carried out.
C.

Homogeneous Electron-Transfer Kinetics of Transition Metal Complexes
The great practical and theoretical importance of homogeneous redox

reactions has encouraged extensive research in this area.
lat ,

In particu-
much attention has been devoted to the kinetics and mechanisms of
these reactions.

Significant progress has been made for those reac-

tions where oxidation-reduction is actually brought about by electron

transfer (as opposed to reactions which occur via atom transfer or

some, other complex mechanism).

The mechanistic simplicity of such
e‘-1-ec:tron-transfer reactions has been one of the primary reasons behind
the progress in this field (59,60).

Probably the most significant and useful advancement in the area of

electron-transfer mechanisms is the concept of outer-Sphere and inner-
Sphere processes (59).

The basic notion here is that an electron-

transfer reaction can be placed in one of two broad classifications on
the basis of the geometry of the transition state. In an outer-sphere
Inechanism, the coordination shells of the oxidant and reductant remain

intact while electron transfer occurs.

Although some perturbation of
the coordination shell is allowed (and usually necessary as will be

Seen later), no bond rupture takes place in an outer-sphere process.

In contrast, an inner-sphere mechanism involves definite changes in
the coordination shells of the reactants.

For transition metal com-
Plexes, these changes often yield a species in which one ligand is

eoznmon to the coordination spheres of both reactants. In this arrange-
tllent, the ligand serves to connect the two metal centers.

Consequently,

21

the mechanisms of reactions which proceed via such transition states
or intermediates are frequently termed "ligand-bridged mechanisms",
and one commonly finds that "inner-sphere" and "ligand-bridged" are
used synonymou sly.

Because the understanding of a redox reaction can be greatly
improved by a knowledge of its mechanism, one finds that the charac-
terization of a reaction as inner sphere or outer sphere is one of the
main concerns of the redox kineticist. The assignment of a mechanism
can be straightforward, difficult, or impossible, depending on the
ehethical properties of the reactants (59) . When both reactants are

a\lbetitutionally. inert on the timescale of the redox reaction, an
ontier-sphere mechanism is the only one possible (60). The same res-
tit‘iction applies to a reaction containing one inert and one labile
I'-’eactant as long as the inert species has no unoccupied binding sites
which could coordinate with its labile reaction partner (60) . Thus a
cognizance of the ligation kinetics of a transition metal complex is
a crucial requirement for an investigation of its electron-transfer
I‘eactions. In the case of the present study of the redox kinetics of
the Eu cryptates, one sees that a knowledge of the dissociation kinet-
ics of these complexes will have both intrinsic worth and a direct
bearing on the characterization of the electron-transfer mechanism.

A consideration of the substitutional inertness of the reactants
is not the only means by which one may distinguish outer-sphere and
inner-sphere reactions. One of the first methods employed was the
meanination of the products of the redox reaction. Taube's pioneering
efforts in this field (61-63) and also in the general area of redox

1(Zinetics and reaction mechanisms must be acknowledged at this point.

22

The product analysis method takes advantage of the substitutional
inertness or lability of the reactants and products of certain redox

reactions. This can be best illustrated by examining one of these
2+ with Cr2+:

reactions; e.g. the reduction of Co(NH3)SCl

+
Ce(x~ui3)5c12+ + Cr2+ EL Co“ + 01:012+ + 5NH4+ (3)

This is a rapid reaction. The Co(NH3)SClz+ and CrC12+ complues are

substitutionally inert and only aquate slowly. On the other hand,

2+
Co and Cr2+ are labile species. The preceding points lead one to

conclude that this is an inner-sphere reaction which involves the for-
mation of a chloride-bridged intermediate or transition state 13:

2+ 2+ ' 4+
(NH3)SCoCl + Cr(OHz)6 -> (NH3)SCo§lCr(OHZ)5 +1120 (4)

The lability of the or2+ ion allows the bridged species B to be pro-

duced. An intramolecular electron transfer occurs within B to yield

Co(II) and Cr(III). This followed by the cleavage of B to yield the

reaction products:
(NH ) CoClCr(OH )4+ 3+ c 2+ + Cr(OH ) c12+ + 51m+ (5)
3 5 2 5 * ° ' 2 5 4

The lability of the Co(II) ion combined with the substitutional inert-

tless of Cr(III) leads- to the rupture of the Co-Cl bond and to the

chloride being frozen into the coordination sphere of the Cr(III) ion.

F'or the sake of argument, one can imagine that reaction (3) could pro-

2+, Cr3+ and free

Qeed via an outer-sphere electron transfer to yield Co
2+

(31-. A subsequent ligand-substitution reaction yields the CrCl
QOrnplex which is found in the product analysis:

Cr3+ + c1” + c::c12+ (6)

 

23

Such an outer-sphere route can be ruled out because of the substitu-
tional inertness of Cr3+ and rapidity of the redox reaction. The
elec tron-transfer reaction would occur and the analysis of the products
would be carried out well before reaction (6) could occur to any sig-
nif 1cant extent. Thus an inner-sphere mechanism is the only one which
is consistent with the properties of the reactants and products.
Add itional confirmation was obtained when the reaction was carried out
in the presence of 3601-. None of this isotope was found coordinated
t0 the Cr3+ ion (59,61-63) which reaffirms the assertion that equation
(5) and hence the outer-sphere pathway do not contribute significantly

to the production of CrClz+.

Because many redox reactions involving Cr2+ are rapid and since
Cr3+ is substitutionally labile, several other mechanistic studies have
elllployed Cr2+ as a reducing agent for Co(III), Cr(III), and Fe(III)

c(unplexes (59,64) . These reactions can be written as

Cr2+ + MIII xn+ + Crux Xn+ + M2+ (7)

‘ihere X is the potential bridging ligand. In each case the detection
<>f (:an+ has served to verify the presence of an inner-sphere pathway.

Reducing agents other than Cr2+ have also been used. Co(CN)§- and
5hu(N83)5 OH:+ are similar to Cr2+ in that they aquate very slowly after
<>lxidation. Thus the technique of separation by ion exchange followed
13}? spectral identification. which was used on the products of the Cr2+
ITeactions, can be applied equally well to the products of the Co(CN)§-
Eltld Ru(NH3)SOH§+ reactions (59,65,66).

Other mechanistic studies have employed Fe2+ or V2+ as reductants.

while the product species (Fe3+ or V3+) hydrolyze rapidly, the

24

appropriate remnants of the ligand-bridged intermediates can be

observed using flow techniques (59).
Another method for identifying an inner-sphere mechanism is to

detect the bridged species itself. This is unambiguous evidence only

when both of the products are substitutionally inert (59). Thus the

reactions of Cr2+ with Ru(III) chloride complexes have yielded

I
Cr II - Cl - Run species (67,68). Other reactions have been charac—

ter :Lzed as inner-sphere by this method (59).
One may sometimes diagnose reaction mechanisms from a careful

inspection of rate constants or activation parameters. For example,

the activation parameters for ligand substitution on the V2+ ion are
eJamparable to those found for the reactions between V2+ and certain

OXidizing agents. It is strongly suspected that an inner-sphere

electron—l:ransfer mechanism is Operative and that the rate-determining

Step is the displacement of an aquo ligand on V2+ by the bridging

ligand (69,70).

Second-order rate laws of the form:
Rate - k [reductant] [oxidant] (8)
are usually found for both inner-sphere and outer-sphere reactions.

Thus no mechanistic assignments can be made when such experimental

behavior is observed. However, when dealing with reactions where at

least one of the reactants has an aquo ligand, one frequently finds

that the rate constant k shows an inverse dependence on [11+]. Often

One

the rate constant is found to be the sum of two contributions.
-1

Of these is an acid-independent term while the other is the [11+]
term. This inverse term can be a very significant fraction of the

mcperimentally observed rate constant, and in some reactions, the

25

contribution of the [H+] term is so large that the [fin-independent
term cannot be reliably determined. The inverse-acid term is generally
attributed to the presence of an inner-sphere mechanism which proceeds
via a hydroxide bridge (71-73). In more explicit terms, the reaction

involves the deprotonation of one of the aquo ligands (this pre-

1

equilibrium step is the source of the [11+] term), followed by the

formation of the hydroxide-bridged intermediate or transition state,
and finally concludes with the electron-transfer step and any other
following chemical process. Therefore, the [Kn-dependence of the
rate constant may provide a clue to the mechanism as long as an aquo
ligand is present on one of the reactants.
While determinations, of mechanism are very important to the study
of redox reactions, one is still left with the problem of formulating
quantitative explanations for the experimentally observed rate constants.
Considerable theoretical work has addressed this question and signifi-
cant progress has been made, especially in the case of outer-sphere
reactions (59,60,74). The calculation of the energy barrier to electron
transfer is simplified for outer-sphere processes because of the absence
of bond formation and breakage (59,60). Several theories dealing with
this area have emerged (74) including those of Hush (75) and Levich
and Dogenadze (76), but the treatment of Marcus has been the most suc-
cessful and widely used (77-79). The Marcus theory may be approached
on two levels. It can be used in an absolute sense to calculate outer-
8phere electron-transfer rate constants without the use of other rate
data, or it can serve as a relative theory which begins with some

eatisting rate parameters and employs these to calculate the corres-

ponding values for other reactions (60).

26

A summary of the Marcus theory can be started by considering a
self-exchange reaction, i.e., one in which the oxidant and reductant

are from a single redox couple.

kll

mums):+ +'m(m3)§+ + Ru(NH3)§+ + Ru(NH3):+ (9)
is an example of such a reaction. Obviously no net chemical reaction
occurs, and consequently AG° - O for self-exchange reactions. The
absolute Marcus theory attempts to obtain the self-exchange rate
constant k11 by calculating the associated free energy of activation.
In order to do this, the reaction is assumed to be weakly adiabatic.
This implies two conditions. First, the interaction between the two
reactants is sufficiently small so that neither is directly perturbed
by the presence of the other. Thus the activation of the two reaction
centers occurs independently. The second condition requires that the
interaction between the reactants is strong enough to allow K, the
probability for electron transfer in the activated complex, to approach
unity. The requirement of weak adiabaticity may seem overly restric-
tive, but fortunately most redox reactions which have been studied do
indeed satisfy this condition (80).

Before continuing on, it should be pointed out that this section
is only considering thermal electron-transfer reactions, i.e., those
which occur without the absorption of light. Only thermal energy is
used in surmounting the activation barrier. Furthermore, the Franck-
Condon principle applies in these systems. In more specific terms,
because electronic motion occurs much more rapidly than nuclear vibra—

ttion, the reactants have no time to change their nuclear geometry while

27

the electron-transfer act takes place. Consequently, there are actually
two activated complexes, one for the reactants and the other for the
products. These are isoenergetic states which differ only in the loca-
tion of the transferred electron. In the reactants' activated complex,
this electron is located on the reducing agent, while in the products'
activated complex, the electron is found on the reduction product.

The laws of thermodynamics require electron transfer between iso-
energetic states. In principle, a nonisoenergetic transfer could be
employed to circumvent the First Law (74).

Having dealt with these preliminaries, one can now consider the
free energy of activation. It is helpful to think of this quantity as
a sum of contributions which stem from the individual events which
make up the activation process. The sequence begins with the diffusion
of the reactants toward each other to form the so-called precursor
complex. The energy involved here is the electrostatic work (and
other forms of work) which must be done when bringing the reactants
together from an infinite separation to one which corresponds to the
internuclear distance in the activated complex. This term can be quite
appreciable, particularly in those cases when like-charged species are
being brought together.

Once the reactants are in the precursor state, the remainder of
the activation process occurs. Changes in the nuclear coordinates must
occur in order to reach the activated complex. The isoenergetic con-
straints which were described above dictate the nature and the degree
of these changes. Both the inner coordination sphere of the reactants
and the surrounding solvent must be modified appropriately. The

ennergies associated with these processes are respectively designated

28

the inner-shell and the outer-shell terms. The inner-shell term has
its origins in the distortions in the metal-ligand bond distances and
the changes in ligand conformation which are necessary to attain the
activated complex. For example, in the reduction of a trivalent ion
to its divalent form, the M(III)—L bond would be required to stretch
to a length which was intermediate to those for M(III)-L and M(II)-L.
The same considerations apply to the other reactant. The outer-shell
term may be regarded along similar lines. The solvent orientation in
the activated complex will be intermediate to those surrounding the
reactants in the precursor complex and the products in the analogous
successor state.

One can summarize the preceding discussion by beginning with the
relationship between the self-exchange rate constant k11 and the free

*
energy of activation AG :

kll I Z exP(-AG*/RT) (10)

where Z is the bimolecular collision frequency, R is the gas constant,
and T is the absolute temperature. As discussed previously, the free
energy of activation can be expressed as the sum of contributions from

each of the steps in the activation process

* *
in + AGO“It (11)

*
AG 8 w ‘+ AG
r
where wk is the work required to form the precursor state from the
a *
separated reactants, and AGin and AGout are the inner-shell and outer-
shell terms described above.

It is important to note that the Marcus formulations are associated

vwdth a kinetic model in which each collision between reactants of the

29

proper energy and orientation leads to reaction. This differs from
the transition-state or ion-pair pre-equilibrium model which is widely
applied to solution reactions. The Marcus model is concerned with the
non-equilibrium.energy fluctuations of the system rather than a formal
equilibrium.between the transition state and the reactants. A recent
paper by Brown and Sutin (81) has addressed these two models with
regard to the self-exchange rate parameters for some Ru(III)/(II)
complexes. The Marcus effective-collision model was found to agree
more closely with the experimental results.

Analytical expressions for each of the terms in equation (11) have
been derived. If the work in forming the precursor state is considered
to be purely electrostatic in nature, the appropriate expression for wf
can be obtained from Debye-Hfickel theory (81):

z z ezN

w - ‘ 1 2 1 (12)
583(1+B§u l2)

 

where 21 and 22 are the charges on the reactants, e is the electronic
charge, N is Avogadro's number, 88 is the (static) dielectric constant
of the solvent, 3 is the distance between the centers of the reactants
in the precursor state, B is the Debye-Hfickel parameter (82) and u is

ionic strength. The inner-shell term is given by (74,79,81):

* 6 f1f2(Aa)2

Acin - 2(f1+f2)

 

(13)

where f1 and f2 are the force constants for the metal-ligand bonds
in the reactants and As is the difference between the equilibrium M-L
bond distances in these two Species. The metal-ligand bonds are

assumed to behave harmonically. The expression for the outer-shell

3O
term is (74,79,81):

* e2 1 1 1 1 1
Acout T (2a; + 232 " 3') (cap ’ F8) (14)

where a1 and a2 are the radii of the reactants, sop is the optical
dielectric constant (equal to the square of the refractive index of
the solvent) and the other terms have already been defined above.
In this case, the solvent is assumed to be a dielectric continuum.

As formulated here, the absolute Marcus theory is a remarkable
achievement since it allows the calculation of a rate constant from
non-kinetic quantities. The theory has performed well in qualitative
comparisons between related complexes and has proved useful as a con-
ceptual structure within which activation barriers may be treated in
terms of separate components (60). As a tool for making quantitative
predictions, the Marcus theory has met with partial success (60).
Reasonable agreements between calculated and experimental self-exchange
rate constants have been observed for some systems (81) but this is
not always the case (60). One point which is of particular interest
to the present work involves the entropy of activation. Many outer-
sphere self-exchange reactions have been found to have considerable
entropies of activation (AS*‘¥ -30 e.u.) (83). A AS$ value of about
-20 e.u. is expected at zero ionic strength but theory predicts an
essentially negligible value at the ionic strengths employed for most
experimental studies (60). This question will be discussed in Chapter V
and a possible explanation will be advanced.

While the Marcus theory has encountered problems when used in

absolute terms, it has been quite successful when employed in a relative

 

J 31

sense. For the purpose of illustrating the relative theory the

following reactions will be considered.

k11
3+ 2+ 2+ 3+
Ru(NH3) 6 + Ru(N'H3) 6 + Ru(NHB) 6 + R11(NH3) 6 (15)
' 3+ 2+ 1‘22 2+ 3+
Fe + Fe + Fe + Fe (16)
3+ 2+ 1‘12 2+ 3+
Fe + Ru(NH3)6 +’ Fe + Ru(NHB)6 (17)

Reactions (15) and (16) are examples of the previously described self-
exchange reaction. Reaction (l7) differs from the other two in that
its reactants are from two different redox couples. Such reactions
are designated "cross reactions" (59). The self-exchange rate
constants can be considered to reflect the intrinsic reactivity of

the single couple which is involved. Rate constants for cross reactions
are a less clear-cut measure of reactivity since they are composites
of the intrinsic reactivities of the oxidant and reductant combined
with the effect of the non—zero AG° on the activation barrier. The
achievement of the relative Marcus theory has been its ability to
relate the cross-reaction rate constants to those for the constituent
self-exchange reactions. The activation free energy of the cross
reaction, AGtz, can be considered to consist of contributions from the
intrinsic (self-exchange) barriers, 0.5Ath and 0.5AG22, combined with
0.5AG:2, the lowering of the activation barrier due to the favorable
free energy of reaction, AGiz. The assumption of weak adiabaticity
removes the need for any coupled activation of the redox centers.
Thus, one finds independent additive contributions from each redox

couple involved. In summary, the activation barrier of a cross

32

reaction can be given by (60):
* * * o
A612 I 0.5(AG11 + AGzz + AG12) (18)

This equation can also be expressed in terms of rate constants

(59,60,74,79), which yields the Marcus cross-relation:

1.
a 2
R12 (kll k22 K12 f), (19)

where K12 is the equilibrium constant for the cross reaction and f is

given by
log f = (103 K12)2 4/ 4 108 (k11k22/22) (20)

The f term is a correction factor which accounts for the parabolic
shape of the activation barrier. (Equation (18) is a simplified form
which does not include this term.)

The cross-reaction rate constants calculated using equation (19)
have generally agreed quite well with experimental results (59,60).
While this is a substantial accomplishment, some difficulties remain.
The cross-relation does not work well for reactions involving
Co(OHz):+ (60). Detailed studies have isolated the thermodynamic
(A632) and intrinsic (Ath and A622) factors which.make up Actz.
While the thermodynamic term tends to comply with equation (18), some
exceptions have been found for the intrinsic contributions (60).
A possible explanation is that nonelectrostatic work terms resulting
from the formation of precursor state may not always cancel each other
completely (77). The limitations of the relative Marcus theory will
be examined in greater detail in Chapter V and some possible sources

of these problens will be discussed.

CHAPTER II

EXPERIMENTAL ASPECTS

33

A. Apparatus

Electrochemical measurements were carried out in conventional glass
cells having working and reference compartments separated by a "very
fine" or "ultrafine" grade frit obtained from Corning Inc. These frits
had an average porosity of 1-3 um and prevented significant mixing of
the solutions in the two compartments in the 2-3 hours required for
most experiments. The capacity of the working compartment was in the
5-10 ml range. The working compartment, the liquid junction formed in
the frit, and part of the salt bridge between the working compartment
and the reference electrode were surrounded by a jacket through which
water from a Braun Melsungen circulating thermostat could be circulated.
This enabled the temperature within this region to be controlled to
within i0.0S°C. A separate jacket around the reference compartment
enabled the temperature of this part of the cell to be controlled inde-
pendently when necessary.

Bulk electrolyses were performed in cells in which the counter
electrode was given its own compartment adjacent to the working compart-
ment but separated from the latter by means of a frit. For applica-
tions which required substantial current densities, a similar cell with

more porous frits was employed to minimize resistance problems.

B. Electrochemical Techniques

 

1. Common Features

 

All solutions were deoxygenated by bubbling with prepurified
nitrogen or argon which had been passed through a series of wash
bottles containing V(II) solution. The final bottle in the series

34

35

contained water and served to humidify the gas so that prolonged
bubbling would not result in extensive evaporation of the solution in
the working compartment.

Saturated aqueous calomel reference electrodes (SCE) were used
throughout this work, although the choice of the fill solution was dic-
tated by the experimental conditions. In.most cases, a SCE filled with
saturated KCl (KSCE) was used. In concentrated perchlorate media, a
KSCE suffers from problems arising from the limited solubility of
KClOA. Precipitation of this salt in the liquid junction of the
electrode can lead to spurious potentials. Therefore a SCE filled
with saturated NaCl (NaSCE) was employed when working with perchlorate
solutions. The potentials in this work are quoted versus the KSCE
although in a good many cases they were actually measured versus a
NaSCE.

A platinum wire served as the counter electrode for most experi-

ments. When larger currents were passed, a platinum gauze was used.

2. Cyclic VOltammetry

 

Cyclic voltammograms were obtained by means of a PAR l74A
polarographic analyzer (Princeton Applied Research Corp.) coupled with
a Hewlett-Packard HP 7045A X-Y recorder. Sweep rates in the 50-1000
mV/sec range were used in this work. Under these conditions and with
the above instrumentation, peak potentials could be measured with a
precision of il-Z mV.

Several working electrodes were used in the cyclic voltammetric
studies. For redox couples with relatively negative standard poten-

tials, a commercial (Metrohm Model E410, Brinkmann Instruments)

36

hanging mercury drop electrode (HMDE) was used. An electrode area of
about 0.03 cm2 was employed for most studies. Mercury has a limited
anodic range, so for work at positive potentials (E > +200 to 300 mV
vs. KSCE), a glassy carbon or platinum "flag" electrode was employed.
The latter consisted of a 2 mm2 sheet of platinum foil spot-welded to
a fine platinum wire. The Pt flag electrode was pretreated by immer-
sion in warm l:l BN03 followed by activation by passage over a Bunsen
burner flame. The literature procedure for the pretreatment of glassy
carbon electrodes (84) was found to do very little to improve their
performance as measured by the residual current and the anodic limit.

Therefore, these electrodes were not chemically pretreated before use.

3. Polarggraphy

 

The instrumentation used for cyclic voltammetry was also
employed to obtain dc polarograms. Sweep rates between 0.5 and 5 mV/s
were utilized. The working electrode was a DME (dropping mercury
electrode). Mercury flow rates of l-2 mg/s and column heights of
v~50 cm were used. The drop time could be mechanically controlled to

be 1, 2, or 5 sec. by means of a PAR 174/70 drOp timer.

4. Preparative Electrolyses
Constant-potential electrolyses were performed with a PAR 174A
or a PAR 173 potentiostat. The latter was used when currents greater
than 10 mA.were required. A stirred Hg pool served as the working
electrode while a Pt gauze electrode was used as the counter electrode.
The electrolyzed solution was kept under a blanket of deoxygenated

nitrogen or argon to prevent any reactions with atmospheric 02.

37

5. Potentiometric Methods

 

A Corning Digital 109 pH meter combined with a regular glass
electrode or a miniature combination pH electrode was used for pH
measurements and potentiometric titrations. The study of fluoride ion
association employed an Orion Model 94-09A fluoride ion-selective

electrode.

C. Materials

1. Cryptates

Solid cryptate samples were obtained from Dr. 0. A. Gansow‘s

group. The syntheses of Eu(2.2.l)Cl Eu(2.2.2)C13, Eu(2.2.l)(NO

3' 3’3'
Eu(2.2.2)(N03)3, Yb(2.2.l)C13, and Eu(23.2.l)Cl3 involved mixing the
appropriate Eu or Yb salt with the cryptand in a nonaqueous solvent
such as acetonitrile. After this mixture is refluxed for a few hours,
the complex is precipitated from solution by the addition of ethyl
ether (85).

The Eu(2.2.l), Eu(2.2.2), and Yb(2.2.l) complexes can also be syn-

thesized in aqueous solution. One can prepare aqueous Eu2+ and Yb2+

3+ 3+

by electrolyzing solutions of Eu and Yb in 0-1.§.Et NClO4 (tetra—

4
ethylammonium perchlorate) at -l.l and -l.4 V, respectively. A slight
excess of the appropriate cryptand is then added to yield the cryptate
complex. This divalent form of the cryptate can then be oxidized to
the trivalent state by an electrolysis at the appropriate potential
(about 100-200 mV anodic of the formal potential of the cryptate given
in Chapter III). The complexes made in this fashion display the same

electrochemical behavior as that found for the solid cryptate samples.

The yield of Yb(2.2.l)2+ was somewhat low due to the hydrolysis of Yb2+

38

which occurs in the basic solution produced by the addition of the
cryptand. The (2.2.2), (2.2.1), and (2.1.1) cryptands were obtained
from PCR, Inc.

2. Reagents and Solvents

 

Except where otherwise noted, the reagents employed in this
work were analytical grade and used without further purification.
Samples of Et4N0104 (from Eastman or G. F. Smith) used in making sup-
porting electrolytes were recrystallized from water. This resulted
in substantial reductions in the background polarographic current.
water was purified by the use of a Milli-Q purification system
(Millipore Corp.). Spectral-grade nonaqueous solvents dried over
molecular sieves were used in the nonaqueous studies. Mercury which
had been triply distilled under vacuum (Bethlehem Apparatus Co.) was
employed in the HMDE and DME and as the pool electrode in electrolyses.

3. Transition Metal Complexes and Compounds

 

Solutions of V2: were synthesized as follows: a solution of

V0; is made by dissolving V20s in an appropriate acid (usually HC104).

At least a five-fold excess of acid must be used if the dissolution
of the V205 is to occur in a reasonable length of time (i.e., less
than l-2 days). Electrolysis of the V0:

atmosphere at a mercury pool yields Viz. The reduction consumes H+,

at -1100 mV under a nitrogen

and this must be remembered when a knowledge of the H+ concentration
is necessary. The V2: solution produced using this method can be
converted to V3: by electrolyzing at -300 mV.

The preparation of solutions of U2: involved electrolysis of
uranyl perchlorate (obtained from G. F. Smith Co.) in 0.5 M HClO

4
at a stirred mercury pool at ~1000 mV. A subsequent electrolysis

39

at -650 mV yielded ng. A.nitrogen atmOSphere was maintained over

the solutions throughout the electrolyses.

Solutions of Eng: were obtained by dissolving Eu203 in an
appropriate acid. In order to dissolve the oxide within a reasonably
short period of time, a stoichiometric excess of acid must be used.

The Fe(bpy) §+, Fe(phen) §+, Co(bpy) 3+, and Co(phen) §+ complexes
(where bpy - 2,2'-bipyridine and phen - l,lO-phenanthroline) were pre-
pared by mixing the metal ion with at least a five-fold excess of the
ligand. These complexes were generally studied in a KCl supporting
electrolyte since they were only slightly soluble in perchlorate media.

Ru(NH3)6-Cl3 (obtained from Matthey BishOp) served as the starting
material for the syntheses of several other ruthenium ammine complexes.
3)6-Cl3 in 1:1 HCl for 4-5
ncs-(c104)2 was synthesized using the procedure of

Ru(NHB)5Cl°Cl2 was made by refluxing Ru(NH

hours. Ru(NH3)5

Lim et a1. (52). A 10-15 m_l_4_ solution of Ru(NH Cl-Cl in

3’5 2
300011 was made. An electrolysis at -700‘mV under

2+
2

C1. by H20 occurs at an appreciable rate with divalent Ru. The ten—

o.1 g NaCF coo, 1m._M_ CF

3

an argon atmosphere yielded the Ru(NH3)50H complex. (Substitution of

dency of Ru(II) to form dinitrogen complexes necessitates the use of
the argon atmosphere.) A ten-fold excess of NaSCN was added, and the
color of the solution changed from faint yellow to red~orange, indicat-
ing the formation of Ru(NH3)5NCS+. An electrolysis at -80 mV converted

2+

this divalent complex to Ru(NH NCS The solution was then removed

3’5
from the electrolysis cell and Ru(NH3)5NC8-(C104)2 was precipitated
out by the dropwise addition of a slight excess of 5 M NaClO4.

The Ru(NH3)SOH3+ complex was prepared by two methods. The procedure

40

of Lim.et al. (52) begins by electrolyzing a solution of 2-3 WE.

Ru(NHa)5Cl-Cl in 0.2 _M_ NaCF

2 3
to yield Ru(NH3)SOH§+. This solution was removed from the electrolysis

C00, 1 my CFBCOOH under argon at -700 mV

cell and added to a solution of AgCFBCOO which had been made by dis—

solving Ag20 in CF COOH. The silver solution had sufficient Ag+ to

3
precipitate out all Cl- and to oxidize the divalent Ru complex to

Ru(NEB)SOH§+. The resulting solution was filtered to remove AgCl and

Ag. The electrolysis cell was cleaned in order to remove any residual
01-, and the filtrate was returned to the cell. An electrolysis at
0.0 V reduced any excess Ag+ in solution. Thus, one should obtain a

solution of Ru(NH3)SOH§+ in NaCF COO at the end of this procedure.

3
A.nodification of this method served as the second means for synthe-

3+
2

the solution was kept in the cell. A second electrol—

ysis was then carried out at -50 mV. This oxidized the Ru(NH3)50H§+
produced in the first electrolysis to Ru(NH3)SOH§+, Neither method was
2+

sizing the Ru(N33)SOH complex. After the initial electrolysis of the

Ru(NH3)5Cl-C12,

ideal since both yielded solutions containing a slight Ru(N33)5C1

contamination. Fortunately, the standard potentials of the

2+/+ 3+/2+

Ru(N83)5Cl and Ru(NHB)SOH2 couples are sufficiently well

separated so that the latter complex can be studied without significant

2+

interference from the former. Solutions of the Ru(NH3)SOH complex

3+
2 O
The Ru(NHa)5 py2+ complex (where py - pyridine) was synthesized as

were made by adding base (NaOH) to solutions of Ru(NH3)50H

follows: a 1 mg solution of Ru(NH3)5Cl-Cl in 0.1 M NaCF C00,

2 3
l mM_CF3COOH was electrolyzed at -700 mV under an argon atmosphere

for about 3% hours. The resulting light yellow solution of

Ru(NH3)SOH§+ turned a bright yellowbgreen upon the addition of

41

2.5 mMprridine. The color change indicated the formation of

2+

The cis- and trans-Ru(OH2)2(bpy)§+ complexes were made by starting
with Ru(bpy)2003-ZHZO (obtained from Dr. G. M. Brown). The carbonate
complex was dissolved in l M HpTS (pTS- - p—toulenesulfonate) to yield
cis-Ru(032)2(bpy)§+. The high acid concentration is necessitated by
the large pKa value of the trivalent form of the cis complex [pKa «a1
based on pKa - 0.85 for the closely similar Ru(bpy)2(py)OH§+ complex
(86,87)]. Because it is photosensitive, the cis-Ru(OHZ)2(bpy)§+
complex must be shielded from light. The trans isomer was obtained by
exposing the cis form to sunlight.

Solutions of the Ru(NHB)4bpy2+ complex were made as follows:

a 1.5 my solution of sis-Ru(NH °Cl (obtained from Dr. F. B. Anson)

3)AC12

in 0.1 M NaCF C00, 5 mg! CF3COOH was electrolyzed under argon at ~550 mV

3
for about two hours to yield Ru(NH3)4(OH2)§+. The solution was then

made 14 mg in bpy. About 30 minutes are required for the Ru(NH3)4bpy2+

complex to form quantitatively.

Several complexes were obtained from outside sources.
Ru(NH3)2(bpy)2'(0104)2, dis-Ru(NH3)4(H20)2°(CF3SO3)2, Ru(en)3-Br3, and
Ru(bpy)2C03-ZHZO were kindly provided by Dr. Gilbert Brown. Professor
Larry Bennett's group supplied a sample of Ru(NH3)4phen-(CFBSO3)2°3H20.
Co(sep)Cl3 (where sep - sepulchrate, see reference 88 for further
details) was obtained from Professor John Endicott. Ru(bpy)3°C12°6H20
was purchased from.G. F. Smith.

The procedure of Bottomley and Tong (89) was used to synthesize

Os(NH3)6:I3. A solution of 0.5 g of Na 08016 (from Matthey BishOp Inc.)

2

in 15 ml of concentrated ammonia was made and 0.12 g of NHACl was added

42

to duplicate the solution conditions of the Bottomley preparation.
After the addition of 0.5 g of Zn dust, the solution was refluxed for

five hours under an argon atmOSphere. (The reaction is

2- 2

208016 + 12NH + Zn +20301113):+ + Zn +I+ 12Cl-). The solution was

3
allowed to cool to room temperature and 3 ml of ammonia were then added.
The solid Zn and Os which had been produced were filtered off. The
filtrate was cooled in an ice bath, and 2 g of KI were added. The
yellow precipitate of Os(NHB)6°I3 was isolated by filtration. The
solution cannot be cooled excessively since KI will precipitate along
with the Os complex.

The synthesis of T10104 involved combining stoichiometric quantities

of T12C03 (Alfa Ventron) and HClOa. A solid brown impurity remained

after the T12C03

and the filtrate was concentrated to about a quarter of its original

had dissolved. This solid was removed by filtration,

volume. The solution was cooled and TlC104 was allowed to crystallize

from solution. The yield is about 80% of the theoretical value.

D. Kinetic Methods

 

l. Cryptate Aquation Studies

Aquation rate constants for the Eu and Yb cryptates were
obtained by recording polarographic limiting currents or cyclic
voltammetric peak currents as a function of time. This yields concen-
tration vs. time.data since these two currents (after correction for
background) are proportional to cryptate concentration. The timescale
of these electrochemical techniques is sufficiently short relative to
those of the aquation reactions so that each recorded current is

essentially a measure of the instantaneous concentration. The

43

polarographic method has slightly better precision since reproducible

electrode areas are more easily attained with a DME than with a HDME.

2. Homogeneous Redox Reactions - Principles

 

Polarography was used in a similar fashion to follow the
homogeneous redox reactions of the cryptates. In this case, the
measurement is complicated by the presence of the coreactant, which
is also an electroactive species. In fact the limiting current cannot
serve as a practical measure of the cryptate concentration when both
the cryptate and its coreactant are electrochemically reversible.
Uhder these conditions, the current recorded at a given potential
would contain contributions from both of the redox couples present.
For example, difficulties would be encountered if one wanted to
follow the progress of a reaction between Ru(NH3)g+ and Eu(2.2.1)2+ by
monitoring the polarographic oxidation current of Eu(2.2.1)2+.
Depending on the potential region employed for the measurement, the
observed current will be either the difference between the Ru(NH3)2+
reduction current and the Eu(2.2.l)2+ oxidation current or the sum.of
the oxidative currents of Eu(2.2.l)2+ and the reaction product,
Ru(NH3)§+. Similar considerations would apply if one attempted to
watch the formation of the Eu(2.2.l)3+ reaction product. In any case,
one cannot obtain unambiguous determinations of the concentrations of
the reactants or products.

‘ Fortunately, a number of well-characterized inorganic redox

couples are electrochemically irreversible since they have, or can be

made to have, slow heterogeneous charge-transfer kinetics. If the

44

degree of irreversibility is sufficiently large, one will find a
potential region within which neither the oxidized form of the couple
is reduced nor the reduced form is oxidized. The appropriate form
of this couple can then serve as a reductant or oxidant for a cryptate
since the concentration of this last species can be monitored in the
potential region described above without any interference from the
other couple.

Polarography appears to be the best method available for the study
of the homogeneous redox reactions in this work. The cryptates and
a number of the other coreactants do not have any absorption bands in
the visible or near ultraviolet regions of the spectrum which would
be suitable for spectrophotometric use. The nature of cyclic volt-
ammetry is incompatible to the present situation where both forms of
a redox couple are found in solution during the course of the redox
reaction. In cyclic voltammetry, one can calculate a concentration
from the appropriate peak current; i.e., a potential sweep is required
to obtain the desired information. Obviously, a potential scan for
this purpose will require the selection of an initial potential which
is well anodic or cathodic of the standard potential of the redox
couple under study. If, for example, an anodic initial potential is
imposed on the system, all of the reduced form of the couple in the
vicinity of the working electrode will be converted to the oxidized
form. The ensuing sweep will reflect this concentration distribution
and not the one existing in bulk solution. Analogous considerations
hold if a cathodic initial potential is chosen. The superiority of

Polarography stems from the fact that each current measurement is made

45

under essentially constant-potential conditions. The perturbation
applied to the system is the growth of the drOp on the DME. In con—
trast, cyclic voltammetry perturbs a system by applying a linear ramp

of potential.

3. Homogeneous Redox Reactions - Experimental Specifics

 

Five homogeneous electron-transfer reactiens involving Eu

+
cryptates were studied: Eu(2.2.l)3+ + Euiq, Eu(2.2.l)3+ + Viz,

Eu(2.2.l)2+ + Co(NH3):+, Eu(2.2.2)3+ + Ruiz, and Eu(2.2.2)3+ + viz.

All kinetic runs were made at 25.0 i 0.1°C. Efforts were made to
minimize the amount of cryptate sample used in each run in order to
conserve the rather limited supply. In order to facilitate data
analysis, pseudo first-order conditions were employed with the non-
cryptate reactant present in at least a ten-fold excess.

Experiments involving Euiz began with an electrolysis of

1-5 mg Eng: in 20 mg EtéNClOA (u - .l or .2) at -l.2 v to yield a

solution of the divalent cation. The adsorption of the Et N+ cation

4
gives rise to kinetic double-layer effects which slow the charge-

transfer rates of Eng: and Eu2+. Polarograms of the Eu3+ and Eu2+

aq 89 39
solutions show no faradaic current between -200 and -800 mV. The
concentration of the cryptates can be monitored by recording the

polarographic current at some potential in this region. After the

electrolysis, the homogeneous reaction was initiated by adding a known

quantity of solid Eu(2.2.l)(NO3)3 or Eu(2.2.2)Cl3 to the Bug: solution.
The latter was stirred during this process and kept under a N2 or Ar

blanket. The solid cryptate dissolved in at most 1-2 seconds. This

mixing time is not significant relative to the half lives of the

46

redox reactions. The reduction of Eu(2.2.l)3+ and Eu(2.2.2)3+ was
followed by recording the polarographic current at -700 mV as a
function of time. A correction for the background (charging) current
was applied to the recorded data. The potentials used for these
measurements were in the diffusion-limited regions of the polaro-
graphic waves of Eu(2.2.l)3+ and Eu(2.2.2)3+. From the Ilkovic
equation (see Chapter III), the corrected current is directly pro-
portional to the cryptate concentration. The acid dependence of the

3+/ 2+

rate constant was studied for both the Eu(2.2.l) and the

3+/ 2+

Eu(2.2.2) reactions.

A.similar procedure was employed in the Vi: reactions. A solution
of 2-5‘mMLVO; was electrolyzed at -l.2 V to yield Viz. The solution
was then made 20 mM in EtaNC104. The Et4N+ ion could not be present
initially because the resultant double-layer effects would slow the
electroreduction rates of V0; excessively. Ionic strengths of 0.1
were used in the. Eu(2.2.l)3+ - v: and Fu(2.2.2)3"' - vi: reactions,
although the former was carried out in a KPF6/HCl/Et4NClO4 electrolyte
and the latter reaction used a LiClOl‘lHClOAIEtANClO4 electrolyte. It
was not possible to obtain reliable kinetic data for the
Eu(2.2.l)3+ - Vi: reaction in concentrated perchlorate media. The
homogeneous redox reaction was initiated by adding a known quantity
of Eu(2.2.l)(N03)3
V::. The loss of the trivalent cryptate was monitored by recording

or Eu(2.2.2)Cl3 (with stirring) to the solution of

the polarographic limiting current at about -6OO mV. Significantly
more cathodic potentials could not be used since appreciable currents

due to V3: reduction were observed in this region.

 

47

The preparation for the Co(NH3)2+ - Eu(2.2.l)2+ reaction began
with the electrolysis of a ~l my solution of Eu(2.2.l)(NO3)3 at
-900 mV to obtain Eu(2.2.1)2+. A (C3H7)4NC104/LiC104/HC104 electro-
lyte (u - 0.05) was used because the double-layer effects arising
from the strong adsorption of the (03H7)4N+ ion slowed the electro—
reduction of Co(NH3)z+ to a point where it would not interfere with
the observations of Eu(2.2.l)2+. The double-layer effects resulting
from Et4N+ adsorption were not sufficient for this purpose. The
electrolysis of the cryptate had to be carried out quickly in order
to minimize the loss of Eu(2.2.1)2+ which occurs through the dis-
sociation reaction: Eu(2.2.1)2+'+ Bug: + (2.2.1). Fortunately, the

reduction could be completed in 20-30 minutes. While the electrolysis

was being carried out, a 20 my solution of Co(NH °(C104)3 was

3)6
deoxygenated by bubbling with nitrogen. A known volume of this
solution was transferred to the Eu(2.2.l)2+ solution by means of a
gas-tight syringe. The ensuing reaction was followed by recording
the Eu(2.2.l)2+ polarographic limiting current at -300 mV as a
function of time. More anodic potentials could not be used because
the Eu:: which was produced from the Eu(2.2.l)2+ dissociation would
also yield an oxidation current in this region. The low solubility
of Co(NH3):+ in perchlorate media coupled with the need to maintain
pseudo first-order conditions limited the Co(NH3):+ concentration to
the 5-10 m_M_ range.

The cryptate redox rate constants and those for cryptate aquation

were generally found to have relative precisions of 5-102 as deter-

uuned from the average deviation of 3-10 kinetic runs.

CHAPTER III
ELECTROCHEMICAL STUDIES OF EUROPIUM AND YTTERBIUM

CRXPTATE FORMATION IN AQUEOUS SOLUTION

48

A. Introduction

 

The study of the eurOpium and ytterbium cryptates has several
interesting features. The trivalent Eu and Yb complexes are the first
trivalent cryptates to be scrutinized in aqueous solution (with the

3+’- macrotricyclic cryptate (90)). The comparison

exception of a La
of the complexation thermodynamics and kinetics of these trivalent
cryptates with the corresponding quantities for monovalent and
divalent cryptates should provide useful insights into the factors
(such as ionic charge and size) which influence cryptate formation.

If the electrochemistry of the Eu and Yb cryptates is anything like
that for the aquo ions, these complexes will be the first electro-
active cryptates to be studied in aqueous solution, and electro-
chemical techniques should prove to be quite effective for the
investigation of their properties. These methods can be employed to
determine the relative stabilities of the trivalent and divalent forms
of the Eu and Yb cryptates and to analyze these stabilities in terms
of enthalpic and entropic components. In addition, the electrochemi-
cal behavior of these complexes will yield some general information
on the effect of cryptate formation on the electrochemical reactivity

of europium and ytterbium,

B. Results
1. Complexation Thermodynamics
Cathodic-anodic cyclic voltammograms were obtained for each
of the trivalent cryptates. Substantial changes in the electrochemi-

3+/ 2+ 3+/ 2+

cal behavior of the Eu and Yb couples are found when these

49

50

ions are encapsulated within cryptate cavities. The cyclic voltam-
mograms shown in Figure 2 are typical. Figure 2a contains the
voltammetric waves observed for the reduction and subsequent oxi-
dative regeneration of Euzz in acidified 0.5 M NaClO . The results
are independent of acid concentration for pH 1-4. The separation
between the cathodic and anodic peak potentials is large and sweep-
rate dependent, reflecting the electrochemical irreversibility

(i.e., the slow heterogeneous charge-transfer kinetics) of the
3+] 2+
aq

mograms for Eu(2.2.1)3+ in 0.5‘1\_‘I.NaC104 shown in Figure 2b have a much

Eu couple (91). On the other hand, the corresponding voltamr
smaller, sweep-rate independent peak separation of 65 mV, which is
quite close to the 57~mV value expected for electrochemically revers-
ible, one-electron couples at room temperature (92). Similar peak
separations are found for the other cryptate couples studied.

The observed peak separations demonstrate that the heterogeneous
electron-transfer rate constants of the Eu cryptates are at least
two orders of magnitude larger than those for Buzz/2+. In order to
yield reversible cyclic voltammograms with the sweep rates used in
these studies, the Eu cryptates must have kS values of at least

»~.01 cm.s-1, where k8 is the electrochemical charge-transfer rate

constant at the standard potential of the couple of interest (92,93).

3+/ 2+

In contrast, Eu
aq

5

has been found to have a ks value of

~8 X 10- cms-1 (94). Thus a considerable alteration in the

electrochemical reactivity of the europium couple occurs when it is

encapsulated within a cryptate cavity.

3+/ 2+ 3+/ 2+

The formal potentials E for Eu(2.2.1) and Eu(2.2.2)

f

51

Figure 2. Cathodic-anodic cyclic voltammograms
of so: and Eu(2.2.l)3+.

Key: (a) 0.4 mM Eu3+ in 0.5 M NaClO4 at pH 2.
(b) 0.35 m! Eu(2.2.1)3+ in 0.5 M NaClOa at pH 7.

Electrode area - 0.032 cm3.
Sweep rates: (1) 200 mV 5-1; (2) 50 mV s-l.

52

 

 

 

 

 

 

 

 

 

I . T r l l T I l
o) Eu3+(oq)/Eu2+(oq) (I)
2*- ..
(2)
| _ —
O- _
-l ..... _.
i(/.LA)
2 b) Eu(2.2. I)3+/Eu(2.2.|)2+ _‘
(I)

'” (2) -

I
0 " '1
-l >— _
l I 1 I L l l L
200 400 soo 800 900

-E (mv. vs. S.C.E.)

Figure 2.

53

obtained from the mean of the cathodic and anodic peak potentials (95)

were found to be markedly more positive than for Fuzz/2+. Analogous

3+/2+ (96) and Yb(2.2.l)3+/2+

behavior was also observed for Eu(23.2.l)
redox couples. These results are summarized in Table 1, along with
data for the Eu and Yb aquo couples. Identical cyclic voltammograms
with peak separations of about 60 mV were obtained for the cryptate
couples in the absence and presence of added cryptand over the range
of sweep rates 50-500 mV 8-1, for solutions in the pH 1-9 range, and
for supporting electrolytes containing sodium, barium, or tetra-
ethylammonium cations which have widely differing abilities to compete
for the cryptand (6). These results indicate that both the trivalent
and divalent cryptates are substitutionally inert on the cyclic
voltammetric timescale. Since the difference in formal potentials E

f

between the complexed and uncomplexed one-electron redox couples can
be related directly to the difference in the free energy driving

force for these processes, we can write

AE - (2.303 RT/F) log(KII/KIII) - (AG

f AGEI)/F (1)

o -
III
where AG° and AG° are the free energies of complexation for the

III II
trivalent and divalent species, respectively, and KIII and KII are

the corresponding stability constants. The resulting values of

(AG are listed in Table l.

O 0
III ' AGII)
It is seen that the stabilities of the trivalent cryptates are

consistently and substantially smaller than those of the corres-

ponding divalent cryptates; i.e., the values °f (A6111

0
AGII) are

54

 

c

 

mm case oeooz.mw.o
e e ..
O O u 0 O O
me o as e a nose oeoz m ea c +N\+mae N Moo»
0 u. m
n— o
as muse in m V was 2H o +~\+mow
s .. m
0 m 0 O O D
o m ohm sec 2 2m o +~\+mAH N No m
Hm mHN ooaooz.mn.o
o a *V q ll.
u 0 O O :
e~ a he a s new oeoz m ea c +N\+mA~ N No a
m.e~ one is-“ mevooeooz.mw.o
mNo ooeooz m:.o
. . e o I.
. u . . . a
m cm a co o s «we oeoz m as o +N\+mAH N No m
I. oo
we ems Am mesmeeoz ze.o +~\+m=m
.:.o Hhaoa.Hoox HhHoE.Hoox .=.o >6 ouhaouuooam oHoaoo Nooom
HHH HH HHH HH HHH on
Q Om<l v Gad!- OmQ U OUQI OQQ n °m< “Hm-I

 

.oomN as warm:

mcaxoaoaoocoz aw moHosoo Nooox oumuohuo Emanuouuw one asueouam mace mo mouamczooahore 2H manna

55
Notes to Table 1.

8Formal potential of redox couple vs. KSCE, obtained from average

of cathodic and anodic peak potentials of reversible cyclic

voltammogram.

bReaction entropy of redox couple; determined using nonisothermal

cell arrangement. (See Chapter IV.)

cDifference between free energies of complexation of corresponding

trivalent and divalent cryptates, obtained from E values using

f
eqn(l).

d o o a O _ 0 d - °
Obtained from (AHIII AH II) (AGIII AGII) + T(ASIII ASII)°

eObtained from As;c using eqn(2).

56

large and positive. In order to ascertain the factors responsible
for this behavior, it is desirable to evaluate the enthalpic and
entropic contributions to (AGIII - AGII)° This can be achieved by
determining the reaction entropies As;c for the various redox
couples. This quantity is discussed in great detail in Chapter IV.
For the present, it is only necessary to note that As;c is essen-
tially equal to the difference (gged - ng) between the partial molal
entropies of the reduced and oxidized halves of the redox couple.

The values of As;c are also given in Table 1. These values can be
used to determine the difference in the entropies of complexation

between the trivalent and divalent species (ASIII - ASII) from

O O - O _ 0
III - ASII (Asrc)aq (Asrc)crypt (2)
O 0

where (Asrc)aq and (Asrc)crypt are the reaction entropies for the

corresponding aquo and cryptate redox couples, respectively (97).

III

also positive, so that the large positive values of (AGIII - AGII)

are associated with still larger values of the corresponding enthalpic

“m9 (min ' AHII)'

It is clearly desirable to obtain absolute values of these thermo-

Inspection of Table 1 reveals that the values of (AS - A821) are

dynamic complexation parameters for either the trivalent or divalent
cryptates. Since both the nitrogens on the (2.2.1) and (2.2.2)
cryptands have a strong tendency to be protonated, the required
stability constants could in principle be determined by the
pH-titration method used by Lehn and Sauvage (6). For the present

systems, a slightly more direct method can be employed since the

57

concentration ratios of lanthanide cryptate to free lanthanide
[LnC]/[Ln] in equilibrated solutions containing a stoichiometric
excess of the cryptand can be directly determined from the magnitude

3+l2+ couple combined

of the reversible voltammetric peaks for the LnC
with the known total lanthanide concentration. These ratios [LnC]/[Ln]
determined as a function of pH yield the required cryptate stability
constants when coupled with the known pK.a values for the free
cryptands (6).

Attempts to determine KIII for either Eu(2.2.l)3+ or Eu(2.2.2)3+
in this manner failed for two reasons. First, the rate constant for
ligand exchange with Eu(2.2.l)3+ is extremely small (vids infra) so
that times of the order of a few months would be required to attain
equilibrium (98). Second, neither trivalent complex is sufficiently
stable to form significantly at pH values below those where precipi-
tation of Eu(III) as Eu(OH)3 occurs (pH-7). NOnetheless, approximate
values of KII for the divalent cryptates Eu(2.2.l)2+ and Eu(2.2.2)2+
could be obtained using the above procedure since Eu2+ is stable over
the pH range (ca. 5-9) required for the pH titration. The ligand
exchange rate for Eu(2.2.2)2+ is rather small (half-life for aquation
1'15 ~6- hours at 25°C) which necessitated long equilibration times
(Klday). Coupled with the sensitivity of Eu2+ to air oxidation,
this difficulty limited somewhat the accuracy of the resulting stabil-
ity constants and precluded determination of KII for Yb(2.2.1)2+.
at 25°C are

4
for Eu(2.2.l)2+ and 3 r 2 x lo10 [1

The values of KII obtained in 0.1 M_Et4NC10

1

2 i l x 109 g‘ for Eu(2.2.2)2+.

Unfortunately, the reproducibility of these values was not sufficient

58

to allow useful estimates of AHII and As;I to be obtained from the
temperature dependence of XII“

The trivalent cryptates exhibit a surprisingly strong tendency
to associate with fluoride and hydroxide ions. The addition of F- or
0H- ions to NaClOA at a constant ionic strength results in shifts in
the formal potentials of the lanthanide cryptate couples to markedly
more negative potentials. Significant potential shifts to more
negative values were also observed upon the addition of oxalate,
phthalate, and phosphate anions, although small or negligible effects
were observed with sulfate, chloride, azide, and nitrate anions up to
concentrations around 0.1 M. The magnitude of these potential shifts
AH: were found to be linearly dependent upon log[X], where [X] is the
concentration of fluoride or hydroxide anions. Some typical results
are shown in Figure 3, which is a plot of -E§ versus 1°310[F-] at

u - 0.5 for the Eu(2.2.l)3+/2+

couple. Two linear regions are
obtained, with slopes of 57 mV and 116 mV. (Virtually identical slopes
were observed if the logarithm of the fluoride activity, obtained

using a fluoride ion-selective electrode, was plotted instead of
log10[F-].) This behavior is expected if Eu(2.2.l)3+ and possibly

also Eu(2.2.l)2+ associate with fluoride anions to form substitution-

ally labile complexes. Then we can write (99)
-AE: - (2.303RT/F) {logBMx - logBMx + (p-q)1og[X]} (3)
P q
or

-AE: - (2.303RT/F) {logsMxp + p log[X]} (4)

59

Figure 3. Plot of Bi,
l':'.u(2.2.l)3+/2+

couple in the presence of fluoride
anions, vs. the logarithm of the fluoride concentration.

the formal potential for the

Ionic strength held constant at 0.5 with sodium perchlorate.

60

 

 

 

700-
r.
) -
E. .-
u... '
“3 600—
: o
_ O
500 —
I l ' I
-3.0 -2.0 - LO
loglo [F ']

 

 

61

where gMX and aMX are the (cumulative) stability constants of the
oxidized End reducgd complexes having the stoichiometry p and q, res-
pectively. Equation (3) will apply when both the oxidized and reduced
species form stable complexes with X, and equation (4) when only the
oxidized species undergoes complexation under the conditions of the
experiment (100). To decide between these two possibilities for the
present systems, absolute values of BMX were determined for the
association of trivalent cryptates with?F- and OH- by means of
potentiometric titrations using a fluoride ion-selective electrode
and a pH electrode, respectively. The values of BMX were found to
agree consistently with the values determined using Equation (4). It
was therefore deduced that only the trivalent cryptates significantly
associate with F- and OH- under the experimental conditions employed

3

([F-],[0H-] ~'10- to 0.5 M). Consequently, from equation (4) the

two linear regions in Figure 3 with slopes of 57 and 116 mV correspond

3 -
to Eu(2.2.l) +FF stoichiometries of 1:1 and 1:2, respectively, with

cumulative stability constants 81 and 82 equal to 2 x 104.M71

3 x lOé'MfZ. Large values of 81 and 82 were also deduced for the

and

association between Eu(2.2.l)3+ and on". Although Eu(2.2.2)3+ was
also found to associate strongly with OH-, quantitative studies were
precluded by a very rapid base-catalyzed cryptate dissociation
observed for this system. A summary of 81 and 82 values determined
for the trivalent cryptates with F- and OH- is given in Table 2,
along with the corresponding data for Eng: and YbZ: taken from the

literature.

62

Table 2. Cumulative Formation Constants for Association between

Europium and Ytterbium Cryptates and Fluoride and Hydroxide Ions.

 

 

~18 -2b
Cation-Anion pair Medium 81,11 82,!
C C
Eu(2.2.1)3+-F" 0.5}; NaC104-NaF 2x104 3x106
3+ - 5° 6°
Eu(2.2.1) -OH 0.5g NaClOa-NaOH 3x10 7x10
3+ — 4° 6c
Eu(2.2.2) -F 0.5M'NaC104-NaF 3x10 7X10
3+ - 6°
3+ - 3d 6d
Euaq-F 0.5! NaC104-NaF 2.5xlo 3x10
e
Buzz-OH- 0.3M_NaClO4-Na0H ~5X105 -
f
Yb3+-F" 0.5M NaClO -NaF 4x103 -
aq - 4

 

aFormation constant for 1:1 complex.

bFormation constant for 1:2 complex.

x
f

complexing anion X using eqn(4), and by potentiometric titration

cDetermined from shift in formal potential AE upon addition of

(see text).
dFrom A. Aziz, S. J. Lyle, Anal. Chim. Acta., 41, 49 (1969).
eCalculated from pKa - 8.3, given in U.K. Frolova, V.N. Kumok,

V. V. Serebrennikov, Izvest. V.U.Z. Khim., 9, 176 (1966).

fFrom B.N. Ivanov-Emin, V.A. Zaitseva, A.M. Egorov, Russ. J.

Inorg. Chem., 13, 1368 (1968).

The 81 and 82 values have relative precisions of 5% as obtained from

the average deviation of five determinations.

63

2. Complexation and Dissociation Kinetics

Since both the trivalent and divalent lanthanide cryptates
are substitutionally inert on the timescale of cyclic voltammetry,
this technique can be used to monitor the kinetics of ligand exchange.
In principle the kinetics of either complexation or dissociation could
be monitored. However, in practice it is only feasible to monitor
the latter processes in this manner because the rates of complexation
are typically rapid, and in any case the trivalent cryptates cannot
be produced in aqueous media from the cryptand and the metal cation
due to hydrolysis of the latter Species at the pH values (>8) where
the cryptand is significantly deprotonated.

Table 3 summarizes rate constants and activation parameters for
the dissociation of five europium and ytterbium cryptates. The
progress of the dissociation reactions was followed by periodically
monitoring the cathodic or anodic voltammetric peak heights for the
trivalent and divalent cryptates, respectively. Concentration-time
data for the dissociation reactions were obtained from these cyclic

voltammetric peak currents by using (92b):
ip - .4463 (nF)3/2 A (Dv/RT)J§ c (5)

where i.p is the peak current (corrected for background) in uA, A is
the electrode area in cm2, D is the diffusion coefficient of the
reacting species in cmzs-l, c is the bulk concentration in mg,

v is the scan rate in V-s-1 and the other terms have their usual

meaning. Concentrations were also calculated from polarographic

64

Table 3. Rate Constants and Activation Parameters for Dissociation

of Europium and Ytterbium Cryptates at 25°C.

 

 

kda AHIb AS#C

-1 d -1 d

Cryptate Medium sec. kca1.mol. e.u.

3+ -7

Eu(2.2.1) 0.5g Na0104(pH ~7) 4.1X10 - -
0.5g NaClO4(pH 2.5) 3.0xlo'7 18.9d -25.5d
0.1g Et4NC104(pH 2.5) 4.0xlo'7 19.2d -23.5d

Eu(2.2.1)2+ 0.1g Et4Nc1o4(pH 2.5) 2.0xlo’4 15.1e -25.oe

Eu(2.2.2)3"' 0.1g Et4NC104(pH ~7) 1.1x1o’3 13.8e -26.oe

Eu(2.2.2)2+ 0.055 Et4NC104 3.0xlo’5 18.3‘1 -16.0(1

+ 0.033_M_Ba(NO3)2
Yb(2.2.l)3+ 0.5g NaClO 1.3xlo'6 22.3d -1o.5d

4

aFirst-order rate constant for cryptate dissociation at 25°C, obtained
at pH values where the acid-independent pathway dominates, i.e., where

k& = kd (eqn(9)).

bEnthalpy of activation for cryptate dissociation, determined from
eqn(7).

cEntropy of activation for cryptate dissociation, determined from eqn(8).

dDetermined over the temperature range 25-60°C.

eDetermined over the temperature range 5-35°C.

65

limiting currents by using the Ilkovic equation (92b):

1 2 1
id=706nDl2 m/3 td/6 c (6)

where id is the (baseline-corrected) limiting current in uA, m is the

mercury flow rate through the DME in mg 3.1, t is the drop time in

d
seconds, and the other terms have the same meaning and units as in
equation (5).

The dissociation reactions of the trivalent and divalent cryptates
adhere to first-order kinetics. Plots of In c (or ln ip or 1n id)
vs. time are linear over several half lives. The enthalpy and entrOpy

i i

of activation (AH and AS respectively) were calculated from the

temperature dependence of the dissociation rate constant k.cl (101):

Mi .. -Rii‘fl‘i - n m
d(1/T)
AS? - AH*/T + R ln(kBT/h) - R lnflkd (8)

where kB is the Boltzmann constant, and h is Planck's constant.
(Strictly speaking, k.d is the acid-independent rate constant defined
in equation ( 9)) .

Experimental conditions were arranged so that the dissociation
reactions went essentially to completion. Initially, strongly acidic
solutions were employed to achieve this end. However, for some
reactions it was found that the apparent first-order rate constants
for dissociation ké increased significantly as the pH was lowered.
This pH dependence was determined to be in accordance with the rate

law:

k5 = Rd + half] (9)

66

Similar behavior has been noted previously for the dissociation kinet-
ics of univalent and divalent cryptates in aqueous media (27). The
derived values of k.d and kH for the dissociation kinetics of four

eurOpium cryptates are given in Table 4. The values of kd
given in Table 3 were obtained under conditions where the acid-
dependent term in equation (9) is unimportant, i.e., where k.d “’ké.
For some reactions, this condition could only be achieved by working
in neutral media where the cryptand released by the dissociation
reaction could prevent aquation of the remaining cryptate. In such
cases the occurrence of such back-(complexation) reactions was
prevented by adding a small stoichiometric excess of a metal cation
such as Ba2+ that will strongly complex the released cryptand (6).

Unlike protons, these complexing agents were not found to have any

significant effect upon ké.

Table 4. Acid Acceleration of EurOpium Cryptate Dissociation at 25°C.

 

 

a b

kd 1% kmfl‘d

Cryptate Medium sec-1 M-l sec.1 5.1
3+ £7 . -6

Eu(2.2.l) 1g LiClO4/HC104 3 x 10 ~1.o x 10 ~3
Eu(2.2.1)2+ 1g LiC104/HC104 1 x 10‘4 2.5 x 10"3 40
Eu(2.2.2)3"' 0.1g LiC104/HC104 l x 10"3 0.2 200
Eu(2.2.2)2+ 0.1gLic104/Hc104 -5 x 10'5 7.5 x lo‘3 ..150

 

aFirst-order rate constant for acid-independent aquation pathway,

obtained from intercept of plot of k5 versus [H+] (eqn (9)).

b
Second-order rate constant, determined from slope of plot of ké
versus [n+1 (eqn (9)).

67

The presence of significant concentrations of OH- and F- were
also found to have a marked accelerating effect upon the dissociation
rates of the trivalent lanthanide cryptates. For example, the
addition of 50 mg OH- or F- to 0.5-MNaC104 reduced T55 for the
aquation of Eu(2.2.l)3+ at 25°C from 27 days to 22 minutes and 2.8

days, respectively. Table 5 summarizes these effects.

Table 5. Anion Effects on Eu(2.2.1)3+

 

 

 

Dissociation.
Anion Anion Concentration kd a
09 (sail)
on‘ .02 1.6 x 10"
.046 5.2 x 10-4
.10 1.3 x 10"3
F‘ .01 1.0 x 10‘6
.05 2.9 x 10‘6
.50 1.4 x 10‘5

 

aDissociation rate constant for Eu(2.2.1)3fi
determined at 25 C. Ionic strength - 0.5.

C. Discussion

1. The Effects of Varying the Cation Charge upon Cryptate
Thermodynamics

 

In view of the substantial differences in the complexation
thermodynamics between corresponding divalent and trivalent
lanthanide cryptates (Table 2), it is of interest to compare these
results with the thermodynamic parameters for other divalent cations

of comparable size, as well as for univalent cations.

68

Table 6 summarizes values of the free energies (A63), enthalpies
(AHz), and entropies (A52) of complexation for alkali and alkaline
earth cations with (2.2.1) and (2.2.2) cryptands as well as for the

lanthanides considered here.and the post-transition metal cation Pb2+.

3+ cryptates were obtained from

The values AG; given for Eu2+ and Eu
the measured values of KII for the divalent complexes together with
the corresponding values of (AGIII - AGII) given in Table 1. Also
listed in Table 6 are estimates of the ionic radius of each cation.
These values were obtained using the method of Goldschmidt but are
from the more recent compilation of Shannon and Prewitt (11) and are
estimated for a coordination number of six. Although there are a
number of scales of ionic radii which differ to a greater or lesser
degree (102a), there is reason to believe that the scale used here
provides a reasonable approximation (i0.lA) to at least the relative
radii of the bare ions in aqueous solution (102b) as well as the
effective cation radii within the cryptate cavities. The use of
larger coordination numbers such as eight, which may be appropriate
f0r the larger cations considered here, increases the absolute values
of the ionic radii by ca. 0.1-0.15 A, but the differences between the
values remain relatively unchanged provided that the coordination
numbers do not greatly depend upon the cation size (11).

The large variations in AG; and particularly in the enthalpic
and entrapic components AH; and As; seen between different cations of
Group I and II have been discussed by Lehn and co-workers (6,12).

The markedly increasing values of -AGZ seen as the ionic radii of the

69

 

 

waomv sao.suv oo.s mam.mmv cao.e-v oH.m nn.o +msm
mime she.meo om.sa mimic cam.rv os.ma NH.H +Nsm
we.oe am.ma he.oa wH.H +~oe
no.3- oa.ea o.eo.NH oe.e om.o o.oo.w om.H +~on
oN om.oH o.co.oH oe.me oH.o o.oo.oH oH.H +~nm
om.oe o~.o o.oo.o oNN oo.~ o.om.s oo.H +~oo
was- w~.m so.~ o~.e +wu
om.seu om.HH o.oo.n om.o- os.m o.om.m no.H +ce
use- oo.HH o~.e oe.s- ow.o o.os.m mm.a +e
ox- os.a o.om.m o~.o os.n om.“ No.H +oz
we ss.H w~.e os.HH oo.o o.oo.m om.o +en
.=.o HhHoE.Hmox deoa.amox .=.o “Hoa.Hcox hHoE.Hmox M cowumu
wmc was- was: wmo was- was- on

osoocsso A~.~.NV

ocmuozuo Aa.~.~v
.oomu an consensus Au.~.~v and Aa.~.~o

mafia maoauoo Hmuoz maowum> we coauoxwfioaou mo moaoouucm one .moaoncucm .moamuocm monk .o marsh

70

Notes to Table 6.

annic radius for metal cation, taken from the compilation in ref. 11

for coordination number six.
bFrom ref. 6; determined at ionic strength u - 0.5.
cFromref. 12; u ~.0.05.

dDetermined from experimental value of KII (see text).

eCalculated from A62 for the corresponding Eu(II) cryptate combined

with the value of (1101"II - Ash) given in Table 1; u - 0.1.

fEstimated value; obtained by assuming A8; to be equal to As; for

the corresponding Sr cryptate (see text).

8Calculated by combining As; for the:corresponding Eu(II) cryptate

‘with the value of (ASIII - ASII) given in Table 1; mar 0.5.

hCalculated from the corresponding AG; and As; values.

iFrom M. H. Abraham, A. F. Danil de Namor, and W. H. Lee, extended
abstracts, 29th Meeting, I.S.E., Budapest, Hungary, 1978.

3From ref. 17.

71

cations approach the radius of the cryptate cavity (ca. 1.1 and 1.4 A
for (2.2.1) and (2.2.2) cryptates, respectively) could be due in part
to steric factors associated with the fit of the cation into the
relatively inflexible cryptate cavity (6). However, these variations
in AG; consist of marked alterations in As; as well as ABE. This
indicates that an important factor determining the sensitivity of
cryptate stability to cation size is the partial desolvation of the
cation that is required in order for cryptate formation to be accom-
plished (12). Thus the large increases in -AH; and -As; that generally
occur as the cation radius increases for a given cationic charge
(Table 6) undoubtedly stem in part from the decreasingly negative
hydration enthalpy and the decreasingly negative hydration entropy

of the aquated cations under these circumstances (12). Such influ-
ences also appear to be paramount in determining the dependence of
cryptate thermodynamics upon the cation charge. Thus cations such.as

2+, or K+ and Ba2

Na+ and Ca + have similar ionic radii, and yet the
divalent cations exhibit markedly less favorable values of -AH; for
formation of either (2.2.1) or (2.2.2) cryptates. These differences
are nevertheless counteracted by markedly more favorable values of
As; so that -AG; is typically larger for the divalent compared with
the univalent cations. The variations in AH; are in harmony with the
markedly more negative hydration enthalpies for divalent cations
compared with those for univalent cations of similar size (12,103).
Although increasing the cationic charge will result in increasingly

favorable ion-dipole interactions with the cryptate ether moieties,

72

these interactions are expected to be relatively less favorable
compared with those involving the solvent molecules on account of

the smaller dipole moment of dimethyl ether versus water (12). These
factors therefore result in a net increase in AH; as the cationic
charge increases. The more positive values of As; for the divalent
cryptates undoubtedly result from the more negative hydration
entrOpies of the divalent versus univalent cations. These stem from
a greater degree of "solvent ordering" (or less "solvent disordering")
in the vicinity of the former cations (103) which is markedly dimin-
ished or removed upon cryptate formation (12).

3+

The differences in both cation size and charge between Eu and

Eu2+ are therefore both expected to play important roles in determin-

ing the relative stabilities of their (2.2.1) and (2.2.2) cryptates.

Thus Eu3+ has a noticeably smaller ionic radius (0.95 A) than

Eu2+ (1.17 A) which, along with the increase in the cationic charge,
contributes towards a markedly more negative hydration enthalpy for
the former cation (103). These two factors taken together appear to
account nicely for the relatively more positive values of AH; and

As; seen for Eu(III) versus Eu(II) cryptate formation (Table 1).

The larger value of (AHIII - AHII) seen for Eu(2.2.2)3+/2+

3+/ 2+

(17.7 kcal. nol’l) compared with Eu(2.2.1) (10.7 kcal. nol'l)

could be anticipated since the AH; values of (2.2.2) cryptates appear

to be more sensitive to cation size than those of (2.2.1) cryptates,

3+

at least for cations of comparable size to Eu and Eu2+ (Table 6).

For example, AH; for Ca(2.2.2)2+ is 10.2 kcal. mol"1 less negative

73

than for Sr(2.2.2)2+, whereas AH; for Ca(2.2.1)2+ is ca. 4 kcal mol-1
less negative than for Sr(2.2.1)2+. One reason for the behavioral
difference between (2.2.2) and (2.2.1) cryptates may be the greater
size mismatch between these cations and the cavity of the former
ligand.

The similar effects seen for the complexation thermodynamics of

3+/ 2+ 3+/ 2+

the other lanthanide couples Eu(23.2.l) and Yb(2.2.l)

(Table 1) also support these arguments. The relative instability of
Yb(2.2.1)3+ versus Yb(2.2.1)2+, arising again.from a marked enthalpic
destabilization outweighing a smaller entrapic stabilization, is
expected in view of the smaller ionic radius of Yb3+ (0.86 A (11,18))
compared to r152“ (0.93 A(lS)).

Table 6 also lists the absolute values of AG; for Eu(2.2.l)2+ and
Eu(2.2.2)2+ obtained from the experimental values of KII’ along with
AG; for the corresponding Eu(III) cryptates calculated by combining
the divalent results with the data in Table 1. Although the ionic

2+

radii of Sr and Eu2+ are very similar, it is seen that the values

of -AG; for the latter cation are significantly larger with both
(2.2.1) and (2.2.2) cryptands (by 2.7 and 3.4 kcal mol-l, respectively).
These differences may arise from weak covalent bonding between the

oxygen and nitrogen atoms of the cryptand with the unfilled 4f-orbitals

2+

on Eu Although the chemistry of divalent lanthanide ions closely

resembles that for the corresponding alkaline earth cations having

2+

similar ionic radii (38), the stability constant of the Eu -EDTA

2

complex is also somewhat larger than for the Sr +-EDTA.complex (104).

74

It is interesting to note that Pb2+, which has an ionic radius close

to those for Sr2+ and Eu2+ (Table 6), exhibits a value of -A62 for

(2.2.2) cryptate formation which is markedly larger than for Sr2+
(Table 6). This difference can also be attributed to the influence
of covalent bonding.

Although absolute values of AH; and A8; were not determined for
the lanthanide cryptates, rough estimates are included in Table 6
for comparison purposes. The entropy values were obtained by assuming
that As; for Eu(2.2.l)2+ and Eu(2.2.2)2+ are approximately equal to
those for Sr(2.2.1)2+ and Sr(2.2.2)2+. values of AH; were then com-
puted by combining these estimates of As; with the experimental values
of A02. Although this procedure may be questionable, it is supported
by the similar activation entropies observed for the dissociation of

2+ 2+

corresponding Sr and Eu cryptates (vids infra).

2. Association of Lanthanide Cryptates with Small Anions

 

It has been asserted that cryptates provide an unusually
effective way of shielding the encapsulated ions from their environ-
ment. However, the strong ion association found between the tri-
positive cryptates and F- and OH' is a striking example of the
limitations of this argument. Particularly unexpected is the finding
that both the first and second cumulative association constants 81
and 82 are comparable to or even larger than the values for the
corresponding aquo lanthanide cations (Table 2). Therefore it is
possible that one or two anions are able to approach closely or even

contact Eu3+ or Yb3+ despite the encapsulation of the cations within

75

the cryptate cavity. Inspection of molecular models reveals that such
small anions could indeed fit between the polyether strands of the
cryptate, although a significant distortion of the symmetrical
endb-endb form is required in order to bring two anions into contact
with the encapsulated cation. Bearing in mind that the measured
association constants reflect the ability of the anion to compete with
water molecules for sites adjacent to the metal cation, it is plausi-
ble that the surprising stability of the cryptate-anion complexes
arises simply from the markedly weaker salvation of the cryptate
cation compared with the aquated metal cation. An alternative way of
rationalizing the experimental results is to regard the cryptand which
surrounds the cation as a region of saturated dielectric and therefore
of low dielectric constant which will enhance the coulombic attraction
between the encapsulated cation and the incoming anions. It is not
clear if the cation-anion associates involve direct cation-anion
contact or solvent-separated ion pairs. It therefore may not be
necessary for the complexing anions to be buried between the cryptate
polyether strands in order to achieve strongly favorable interactions

with the metal ion.

3. The Effects of Varyinggthe Cation Charge upon Cryptate

 

Substitution Kinetics

 

The kinetic parameters for the dissociation of lanthanide
cryptates summarized in Table 3 exhibit some surprising features.
Most prominently, the rate constant k.d for dissociation of Eu(2.2.l)3+

is almost three orders of magnitude smaller than for the corresponding

76

divalent cryptate Eu(2.2.l)2+ despite the substantially lower thermo-
dynamic stability of the former complex (Table 1). Even the enormously
larger driving force for the dissociation of Eu(2.2.2)3+ compared with

Eu(2.2.l)2+[(AG£II - AGII) - 9.7 kcal. mol.-1, Table 1] results in

values of ka that are only ca. 30-fold larger for the former cryptate.
As in the case of the corresponding thermodynamic parameters, some
insight into the factors responsible for this behavior can be obtained
by comparing the rate constants and activation parameters for Eu(III)
and Eu(II) cryptate dissociation and formation with corresponding data
for cryptates of alkali and alkaline earth cations having similar
ionic radii. Tables 7 and 8 contain such data for (2.2.1) and (2.2.2)
cryptates, respectively. The values of k.c for the europium cryptates
were obtained by combining the values of k

d
values of AG; listed in Table 1. Approximate estimates of the corres-

given in Table 3 with the

ponding activation parameters AH: and A8: are listed in Tables7'and 8;

these were similarly obtained using the estimates of AH; and As;

given in Table 6.
Inspection of Tables 7 and 8 reveals the importance of the charge

as well as the size of the cation in determining the kinetics of

cryptate substitution. Thus for the series of cations Na+, Ca2+,

Eu3+ which have comparable radii, both k.d and kt fall dramatically as

the cation charge increases. For example, Eu(2.2.l)3+ exhibits values

of kc and ka that are ca. 107-fold smaller than for Na(2.2.l)+ and

ca. 104-fold smaller than for Ca(2.2.1)2+ (Table 7). Since Eu
Srz+

2+ and

have very similar ionic radii and also exhibit comparable kinetic

parameters with both (2.2.1) and (2.2.2) cryptands (Tables 7 and 8),it

77

 

 

s OH- m «N so-onm H mm o +mo»
HoH V HH.H~V amm.o m.m- ~.oH sh-onH.s mo.o +msm
cl 0 II o o o D
HNH V Hm s V amons mm H mH ss-ono N NH H +N m
H.o- e.m onm.H m.mH- o.mH ons.o om.H om
H.cm H.e~ +~
e.o- m.m OHx~.m m.-- o.eH onmo.H oH.H um
H.es H.em +~
a m.mH con.H ow- H.mH OHx~.~
so am-
~.o- m.oH Cme.m ~.m- e.MH onm.o oo.H no
H.sm H.eo +~
fieonm fl$me mm H +s
o I. o o I. o o c m
m m o o . oons a HH m NH H.em nH No H + 2
H a
5.0 . Hoe. .233 . com fl 5.0 :33. Hoos— . oom < sown—mo
o H- o H- o H- o H- o H-n m
+m< +m< u— +m< +=< o— H
w m o o o o o

 

Aa.N.NV oaom mo

coaumaoommaa ocm coauoeuom

How muouosmumm coaum>auo< ocm mucmumcoo oumm

.Uomm um cosmonauu
.m oHome

78

Notes to Table 7.

 

aCrystallographic ionic radius, from Table 6.

bRate constant for cryptate dissociation, acid-independent pathway.

cActivation enthalpy for dissociation.
dActivation entropy for dissociation.
eRate constant for cryptate formation.
f

Activation enthalpy for cryptate formation.

gActivation entropy for cryptate formation.

Data Sources:

 

hB. G. Cox and H. Schneider, to be published; determined at
u~10"3 - 10‘2.
iRef. 27.
jRef. 24.

kRef. 26.

1From Table 3.

mCalculated from the kinetic parameters for cryptate dissociation
combined with the thermodynamic data in Table 6.

 

79

 

AmV Am.NHV Em.~ cm- w.mH HMIOHxH.H mm.o +mnm
' o o In 0 o o D
AoH V aH mV emOme e 0H m mH am-ono m NH H +N m
mHu m.o OHxn ml m.o~ IOHx~.~
so an
m.mHI ~.o Ome m.m| o.HN Owa.H om.H mm
H.es H.cm +~
Con OHxH
on oel
mHI «.5 Oon.H NHI n.hH Oon.H
as se-
a HHI m m n<Ome H H «HI m mH cm-OHxH m oH H +~nm
«HI m.n Ome.n mm- <.m om.o
Hm x
0 II 0 I I II I O O m
o m m 0H anon o w mm m 0H cmmH o oo H +N 0
fleOHxN am
a N m m H.noono m m mH m cm H.rm m an H +M
m.on 5.x nconm.H H.oH m.eH nan No.H +mz
.=.o .HoE.Hmox H.ummH.mw .=.o H.Hoa.Hnox Hwomm M nOHumo
o o o o
+m< +m< a was Men as u
w u o o o a m

 

Au.N.NV oaom mo

nOHumHoommHn onm nOHumeuom now

muouoamumm nOHum>Huo< one

.osmw on successes

munmumnoo comm

.m «Home

80

Netes to Table 8.

aCrystallographic ionic radius, from Table 6.
bRate constant for cryptate dissociation, acid-independent pathway.
cActivation enthalpy for dissociation.

dActivation entrOpy for dissociation.
eRate constant for cryptate formation.
f

Activation enthalpy for cryptate formation.

8Activation entropy for cryptate formation.

Date Sources

hB. c. Cox and H. Schneider, to be published; determined at
u ~10"3 - 10'”2 .

1Ref. 27.
jRef. 24.
"Ref. 26.
1

From Table 3.

mCalculated by combining the kinetic parameters for cryptate
dissociation with the thermodynamic data in Table 6.

“Ref. 22.

oRef. 3.

81

is likely that these behavioral differences between Eu3+ and the

lower charged cations result chiefly from the influences of ionic
rather than covalent bonding.

The first possibility to be considered in interpreting such
variations in the complexation kinetics is that they may be connected
with the rates of substitution of water molecules in the coordina-
tion sphere of the cations. However, the "characteristic rates" of
water substitution (105) for tripositive lanthanides, heavier alkaline

earth, and alkali metal cations are all extremely fast (107-108,

9, and ca. 109 sec-1, respectively (105). It has been sug-

108-10
gested (25) that the rate contents k.c for complexation of cryptands
with alkali metal and alkaline earth cations that are markedly below
these water substitution rates may be due partly to the need for the
complexing cation to bind with the endb-endb form of the cryptand,
whereas the exo-exo conformation could be the predominant form in

solution. However, since the 106- to 107-fold difference in kc

between Na+ and Eu3+

, for example, is much greater than the corres-
ponding variation in water substitution rates, it is likely that an
additional factor is chiefly responsible at least for the large energy
barriers observed in the complexation of multicharge cations. It is
plausible that an important component of these energy barriers arises
from the inability of the relatively rigid ligand to achieve a smooth
stepwise replacement of water molecules in the metal coordination
sphere by the cryptand coordinating groups, so that the loss of sol-

vating water molecules may not be immediately compensated by the

formation of cation-cryptand bonds (3,26).

82

As noted previously (3,25,26), the large changes in cryptate
thermodynamic stability induced by variations in the radii of uni-
valent or divalent cations (Table 6) are chiefly reflected in k.d
rather than kc (Tables 7 and 8), suggesting that the transition
states more closely resemble the separated cation and cryptand
rather than the cryptate (3,25,26,28). This latter result can be
reconciled with the larger sensitivity of kc to cation charge by
noting that the hydration energies of simple cations are much more
sensitive to ionic charge than to size (103). Therefore any partial
desolvation of the cation required to reach the transition state from
the separated reactants is expected to result in a greater dependence
of the activation barrier on the charge rather than the size of the
cation.

Further insight into the factors influencing these reactivity
patterns can be obtained by inspecting the corresponding activation
parameters (Tables 7 and 8). By and large, the marked variations in

As; seen for the univalent and divalent cryptates as the cation size

and charge varies (Table 6) are primarily reflected in the activation

1?
d.

activation entropies for complexation As: are.small and negative

entrOpies for the dissociation reactions AS The majority of the
(ca. -5 to -10 e.u.). Since an activation entropy of ca. -10 to

-15 e.u. is predicted on statistical grounds from the loss of trans-
lational entropy ASt attending such association reactions (106), the
values of ASc support the above contention that no drastic changes in
the solvent structure surrounding the cation are required in order to

form the transition state from the separated reactants.

83

In contrast, however, the large differences in As; between
Eu(2.2.1)3+ and Eu(2.2.l)2+, and between Eu(2.2.2)3+ and Eu(2.2.2)2+
are chiefly reflected in As: rather than as: (Tables 7 and 8).
Consequently, the markedly smaller values of kc for the Eu(III)
cryptates compared to Eu(II) and other divalent cryptates arise from
substantially larger enthalpic barriers AH: (Tables 7 and 8). The
differences in activation entropy suggest that the solvent surrounding
the Eu3+ cation in the transition state is substantially less
"ordered" than around the uncomplexed cation. Indeed, the ionic
entropy S9 of E113+ is large and negative (107), reflecting its large
solvent "structure-making" ability (108). A good deal of this exten-
sive hydration shell may be required to be removed in order to form
the transition state, resulting in the observed large positive values
of AH: as well as ASI. c

It has been pointed out that the significant acid catalyses
observed for the dissociation of a number of metal cryptates suggest
that it is often necessary to alter the cryptate conformation from
the stable endb-endb form to one where at least one of the two
bridgehead nitrogens has its lone pair pointing outwards (exo-endb
or exo-exo conformation) prior to or within the transition state (27).
The significant acid catalyses of the dissociation of the Eu(III)
and Eu(II) cryptates (Table 4) indicate that such a mechanism is
also applicable to these systems. Inasmuch as any clear trends can
be discerned in the ratios kH/kd of the rate constants for the acid-
catalyzed and uncatalyzed pathways, it appears from the data in

Table 4 and reference 27 that kH/kd increases as the charge/radius

84

ratio of the cation increases and as the thermodynamic (or kinetic)
stability of the cryptate decreases. The ratio kH/kd also is gener-
ally larger for (2.2.2) compared with (2.2.1) cryptates. The role
of the proton in accelerating the dissociation reaction may be to
aid the endb-endb to exo-endb conformational change by forming an
incipient nitrogen-proton bond as the nitrogen-metal bond is broken,
or it may simply provide a more favorable dissociation pathway once
this conformational change has occurred (27). The smaller or
undetectable acid catalysis in the more stable cryptates may arise
from the greater difficulty of accomplishing this conformation change
due to their tendency to be less flexible than cryptates having
cavities that are markedly larger than the encapsulated cations.

The very marked acceleration of the dissociation rates of tri-
valent lanthanide cryptates by OH- or F- is not surprising in view of
the strong binding of these anions to the cryptates. These anions
could lower the energy barrier to dissociation by distorting the
cryptate conformation and enlarging the space between the polyether
strands so that the encapsulated cation can exit the cavity more
easily. Additionally, the bound anions will act to lower the effec-
tive charge of the assembly, thus reducing the extent of the solva-
tion changes which must occur in order for the cation to be removed
from the cryptate cavity. It is interesting to note that these sub-
stantial accelerations of the cryptate dissociation rates occur in
spite of the considerable thermodynamic stabilization of the cryptates

by both fluoride and hydroxide (Table 2).

CHAPTER IV

REACTION ENTROPIES 0F TRANSITION METAL REDOX COUPLES

8S

A. Introduction

 

While the ionic entropies of a number of transition metal com-
plexes have been determined (see Historical Background), no systematic
study has been conducted on the dependence of the entrOpies on the
nature of the metal ion or on the ligands coordinated to the cation.
Ru(III)/(II) couples can be used to explore the ligand effects since
Ru(III) and Ru(II) complexes containing a variety of ligands are
known. As stated previously, the substitutional inertness of Ru(III)
and Ru(II) allows electrochemical determinations of the reaction
entropies of these systems. The influence of the central metal ion
on the ionic entropy of its complexes can be investigated by obtain-
ing the reaction entrOpies for a series of redox couples having the
same ligand structure and charge type. A.large number of
M(III)/M(II) couples is available so that complexes of this charge
type will be used to investigate the metal dependence. A wide
variation in metal ions is not possible with.most unidentate ligands
due to the instability of many of these complexes with regard to
aquation. Fortunately, many complexes containing chelating ligands
should be sufficiently stable to maintain their structural integrity
in both the oxidized and the reduced forms. Therefore the employment
of chelating systems will allow one to achieve the desired diversity

in the nature of the complexed metal ion.

86

87

B. Determination of Reaction Entropies

The reaction entropy has already been defined in Chapter I but
it is useful to present this definition more formally. One can con-

sider the reaction

M¢IIL'L" + e-(metal electrode) ‘+ MIIL'L" (1)
m'n m n

in which.M is a metal which has trivalent and divalent oxidation
states, L' and L" are neutral or anionic ligands, and the number of
L' and L" coordinated to M is given by m and n, respectively. The

reaction entropy A8;c of the MIII

L'L"/MIIL'L" redox couple can then
m n m n
be defined as the standard entrOpy change for reaction (1). This

definition can also be stated as

O . -0 - -0
ASrc 311 S111 (2)
where §° and §° are the absolute ionic entropies of MIIL'L" and
II III m n

MlllhéLg, respectively. The As;c definition can be expanded to
include other classes of redox couples but this work will only deal
with those of the M(III)/M(II) type.

Reaction (1) is only a half of a complete electrochemical cell
reaction. Consequently, any determination of the AS°for this reac-
tion must involve some sort of extrathermodynamic assumption.
Several of these have been proposed, and some reliable methods for
the quantitative estimation of ionic entropies and As;c values have
been developed from these assumptions. Criss and Soloman have

assembled a helpful summary of the various procedures (40).

88

Of the several methods-of determining ASrc which are available,
the most convenient from the viewpoint of the present study is the
one which utilizes a nonisothermal cell (109,110). This is
basically an electrochemical cell in which the reference and work-
ing compartments are.maintained at different temperatures. One
nonisothermal arrangement which was frequently employed in the
present work can be written as: '

CuIHgIHgZClz,
A

KCl(sat) I I KC1(3.5M) IKC1(3.5M_) I luau) ,M(II) IHg IHgI Cu
B c n

The temperature in regions AB and CD was maintained at the ambient

value, while that of region BC (i.e., the working compartment) was

ni

 

varied, usually between 3° and 60°C. The formal potential Ef across
this nonisothermal cell was determined as a fuzcgion of temperature.
The temperature coefficient of this potential :T can be divided into
three components

cuff11 Pill d¢tc do:l

71'— ' dT + 71'1— + “d? (3)

where ¢tlj is the Galvani potential difference across the thermal
liquid junction within the KCl salt bridge (point B in the cell
diagram), ¢tc is the "thermocouple" potential difference between the
hot and cold sections of the mercury working electrode, and 0: is
the Galvani metal-solution potential at the working electrode. The

important quantity in the present content is do:/dT since

0 m
ASrc a F(d¢f/dT) (4)

89

Therefore one can obtain A8;c values from.measurements of the formal

potential if the temperature coefficients of 0

tlc and ¢tc are known

or can be estimated.

Absolute values of the Thomsen coefficient, dotc/dT, are known
for several metals and are usually found to be on the order of a few
microvolts per degree. Mercury and platinum, which served as the
working electrode materials in this study, have Thomsen coefficients
of about 14 and 6 uV/deg, respectively, over the 0°-100°C tenperature
range (111). The experimental dEgildT values are usually 1-2 mV/deg
so it is clear that the contribution of the Thomsen coefficient to
the temperature coefficient of En1 is essentially negligible.

f

Relative rather than absolute values of d0 ldT are thermo-

tlj
dynamically accessible but there is good evidence to indicate that
this quantity is not greater than 50 uV/deg for most aqueous electro-
lytes (109,110). Much larger values are found with strongly acidic
or basic media (llOb), in analogy to the considerable isothermal
liquid junction potentials which can arise in such media. The use of
concentrated aqueous KCl in the region of the thermal gradient (in
effect, the employment of a "nonisothermal salt’bridge") has been

suggested by de Bethune et a1. (llOb) as a way to minimize d0 IdT.

tlc
Although reservations have been expressed concerning the exact sound-
ness of this assumption (llOc), there is little question that

dotlj/dT is at most 20 uV/deg and is quite likely to be considerably

smaller. htremely accurate measurenents of ASI‘:c will definitely be

hampered by the uncertainty in do /dT, but this uncertainty will be

tlj
essentially negligible within the context of the present study, in

90

which the precision of the dErflildT determinations was only :50 uV/deg.
An uncertainty in As;c of i0.5-l e.u. arises from this thermal liquid
junction problem but this is quite small compared to the measured
Asgc values, which can range up to 50 e.u. Furthermore, comparisons
of the As;c values of various systems will be unaffected by the
dotlj/dT effect since the value of this last quantity will depend on
the composition of the salt bridge and not on the individual redox
couple under study. The AS° values for homogeneous redox reactions,
which can be obtained from the difference in As;c values of the redox
couples involved, will be free of the above effects since the uncer-
tainties in As;c will cancel when this difference is taken. On the
basis of the preceding discussion it can be concluded that, for the

purposes of the present study:

0 ni
Asrc - F(d1:f MT) (5)

The formal potentials which are needed for determinations of As;c
can be conveniently obtained from measurements of Elh’ the (polaro-
graphic) half-wave potential. For a reversible cyclic voltammogram,
35513 the average of the cathodic and anodic peak potentials (92b).

A simple relationship exists between E and E1/2 (93,112):

f

1
21,2 - s + (RT/nF) 1n (DH/D ’2 (6)

f III)

where DH and DIII are the diffusion coefficients of the reduced and

oxidized species, respectively. The DII/D ratio is usually close

III

to unity so that E55 differs from E by only 2-3 mV. In the context

f

of the present work, the temperature coefficient of the

91

(RT/nF)ln(D >72 term must be negligible so that dEfitldT can be

II/DIII /
equated to dEgildT. The validity of this assumption was confirmed

by using the limiting polarographic and cyclic voltammetric peak

currents of the oxidized and reduced forms of the representative

- 3+/ 2+ 3+/ 2+
E“ 3)6~

as a function of temperature (113). These determinations allow one

and Ru(NH , to obtain values of D and D

couples, II III

to estimate that an error of less than 0.5-1 e.u. in As;c stems from

the equation of the temperature dependences of 3:1 and Efii. Therefore,

one is justified in writing
0 ni
Asrc F(dE5&/dT) (7)

The procedure described here yields As;c values which are absolute
entropy differences for redox couples. Other reaction entropy measure-
measurements and scales are found in the literature. Unfortunately,
little has been done to reconcile the conflicting conventions which
exist. One pOpular scale stems from the determination of reaction
entropies from the temperature dependence of the cell potential of an
isothermal cell (114); i.e., one in which no temperature gradient
exists between the working and reference electrode and in which the
temperature of the entire cell is varied. Reaction entropies for
single redox couples are then obtained by assuming As;c - 0 for the

reaction H+‘+ e ’5 )5 H In effect, a relative scale is established

2.
for As;c. Reaction entropies on the absolute scale can be obtained
by adding 21 e.u. to the relative values (40,114). (This conversion
is derived by recalling the estimated entropy of 5 e.u. for H+ in

aqueous solution (40).and by combining this with an entropy value of

92

31.2 e.u. for hydrogen gas at 25°C (115) to compute a value of
As;c - 21 e.u. (on the absolute scale) for the H+IH2 couple.)

As noted in Chapter I, a relative scale of ionic entrOpies can
be set up by arbitrarily assigning a value of zero to the ionic
entrOpy of hydrogen ion. Such assumptions will not affect the value
of the reaction entrOpy since they will cancel when the difference
between ionic entropies is calculated. However, care must be taken
to avoid mixing values from the relative and absolute scales when
taking such a difference. Both scales are used in the literature,
and in some papers it is not obvious which scale is being employed.
Furthermore, As;c values from.older papers often refer to oxidation
processes. Such values will have the apposite sign to those in this

work where the reaction entropy is defined in equations (1) and (2)

in terms of reduction reactions.

C. Practical Aspects of Nonisothermal Cells

 

As shown in the preceding section, the nonisothermal salt bridge
is a crucial component of the nonisothermal cell. In the present
experiments, 3.5 M_KC1 was employed in the salt bridge for most
experiments. The solution in the bridge must contain cations and
anions of similar, if not identical, mobilities (116). This require-
ment is met by K+ and Cl_ (116,117). The solubility of KCl is
slightly greater than 3.5 M even at 2°C, the lowest temperature
used in this study, so no problems with precipitation of KCl are
encountered.

The composition of the salt bridge can be important under some

93

conditions. With experiments performed in neutral supporting electro-
lytes, a negligible change in dEgi/dT was found when the supporting

electrolyte was used in the salt bridge in place of the 3.5 M KCl.

An appreciable alteration in the value of dB:

the analogous comparison was made with acidic supporting electrolytes

i/dT was observed when

(i.e. where [H+] >‘52 of the total ionic strength). In these par-
ticular cases, 3.5 M KCl was enployed to minimize the isothermal
junction potential between the salt bridge and the acidic supporting
electrolyte.

For experiments conducted in concentrated perchlorate electroq.
lytes, one cannot use 3.5 M KCl in the salt bridge since the precipi-

tation of the sparingly soluble salt, KClO in the liquid junction

49
will give rise to spurious potentials. This problem can be avoided
by employing 3 M NH4C1 in the salt bridge. The mobilities of NH: and

C1. are quite similar (117), so NH Cl is a good replacement for KCl.

4
Some attention.must also be devoted to the actual physical dimen-
sions of the salt bridge. The electrochemical cell was constructed
so that the temperature drop within the nonisothermal salt bridge
was confined to a short distance (<1 cm) within a glass tube having
an internal diameter of about 0.8 cm. These features prevented the
development of any significant concentration gradients due to thermal
diffusion (the Soret effect). Considerably larger values of dotlj/dT
would result from the presence of a Soret effect (109,110b) but the
lack of any significant thermal-diffusion effect was demonstrated

by the fact that cell potentials were found to be stable to within

1 mV'even after several hours under nonisothermal conditions.

D. Results

An orderly presentation of the results is facilitated by dividing
them into three categories on the basis of the ligand structure and
type of central metal ion. This arrangement will also allow one to
make comparisons within and between classifications to determine the
factors which influence the reaction entrOpy.

1 . Acme Couples

 

The reaction entropy As;c of six aquo couples of the type

3+/ 2+ , 3+/2+ 3+] 2+ 3+/ 2+
M(OH2)n were studied. Cr(OH2)6 , V(OH2)6 , Fe(OH2)6 ,
Ru(OH2)2+/2+, Eu(OHz):+/2+, and Yb(OH2):+/2+. These systems were

chosen because their formal potentials allow them to be conveniently
studied at mercury or platinum electrodes, and they exhibit sub-

stantial variations in the electronic structure of the central metal
ions. Previous determinations of As;c for these couples are sparse.

The absolute ionic entropies of Fe3+ and Fe2+ have been determined

(41), and an estimate of Asgc for Ril(032)g+/2+ has been reported from

electrochemical measurements (118).

3+/ 2+ 3+/ 2+

Most data for the Cr and Eu couples in perchlorate media
were obtained using potentiometry because the small heterogeneous
electron-transfer rates for these systems resulted in distinctly
irreversible cyclic voltammograms. However, in sodium p-toluene-
sulfonate (NapTS) media, the strong specific adsorption of
p-toluenesulfonate anions resulted in almost reversible cyclic

voltammograms for the Eu3+/2+

couple. Accurate values of E1, could
2
still be obtained from such "quasi-reversible" voltammograms in the

usual way provided that the cathodic-anodic peak separation lies in

94

95

the range 57 to ca. 90 mV (93). For the remaining aquo couples,
essentially reversible or quasi-reversible cyclic voltammograms were
obtained, at least after the addition of small quantities of NapTS.
The resulting values of As;c are listed in Table 9, together with
other pertinent information. It is seen that some limited dependence
of As;c upon the nature of the metal ion is observed. The dependence

3+/ 2+

of As;c upon ionic strength was investigated for Eu and was found

to be small (Table 9). Good agreement between the earlier and present

determinations are found for Fe(OHz):é+/2+ (Table 9), but a large

qualitative discrepancy is seen for Ru(OH2)g+/2+. A possible reason
for this disparity is that the isothermal cell measurements of ref.118.
were complicated by an unknown temperature dependence of the electrode
potential of the glass reference electrode used in that study.

2. Rutheniun(111)/(II) and Osmium(III)/ (II) Couples Containing

Monodentate Ligands

 

The reaction entropies of twelve Ru(III)/(II) couples containing
ammine, aquo, and simple anionic ligands were evaluated. These
systems were selected in order to scrutinize the effects of replac-
ing ammine by aquo ligands and of changing the charge type of the
couple resulting from substitution of the ammine and aquo ligands by
simple anions. The results are summarized in Table 10. For all.of
these systems, the heterogeneous electron-transfer rates were suf-
ficiently rapid so that the cyclic voltammograms were essentially
reversible even at the highest sweep rates (100 V s-l). However,

11 2+/+

the relative lability of the Ru state for the Ru(NH3)5Cl and

96

 

 

I- I. I a II o o N

HHmm V bosom 8 m 3 H2. m an o +~ \ +1 :85
a
o .- c a.
II II \I\ Q 0
Home 8 m 2.: us a V use an o a +N\+nH .18.;
Homes oo-m oao- o3 mnVaeHusz mH
one oo-m oao- u? rnVHmaeoz mic

. I I n on .I . n N
on as 8 m was on” m V H8. 2 :8 o H +~H+mH more

N” cl II ..o I. Q q II o o N
o 8 m as 8 H H e V 303 an o H +~H+nH no;
A ”Q m I” o I. Q Q I o 0 N 0
r V no 8 8... SHHHHVoHoHH one n+2+mH =8 e

o I o N
I I n o u
use 8 n one 8 m V on. z E H +~H+mH :8 o

and.” .00 >6
a cums omens case 3.383on oHesoo
.oamH

 

.moHenoo wooom

Ha
+N\+m

Ammovz mnOHum> pom moHnouunm noHuooom .m oHoma

97

Notes to Table 9.

 

aReversible half-wave potential in mV, determined at 25°C against a

KSCE held at ambient temperature (23 i 0.3°C); related to E by

f
equation (7).

bReaction entropy of redox couple in e.u. (cal K.1 mole-1) at 25°C;
determined by using equation (6). Experimental precision estimated
to be 11 e.u., accuracy within 1-2 e.u. (see text). values in

parentheses are from literature sources.

cDetermined using a combination of potentiometry and dc polarography.

l

dDetermined using cyclic voltammetry (sweep rates: 50 - 500 mV s- ).

e355 found to be unaffected by pH variations of at least one unit

around the pH value given.

fDetermined using HMDE.

gDetermined using Pt electrode.

hCalculated by taking the difference in the relative ionic entrOpies

given in ref. 41.

1Calculated from isothermal cell data given in ref. 118 by noting that

Asgc .- 21 e.u. for the reaction H+ + e- e 1le2 (40,114).

i

pTS - p-toulene sulfonate.

kDetermined by P. D. Tyma.

 

98

 

 

s.os H oH ra-H Hanan.ussmVonm soHuoz.mH oH+HHoon=zVse-ouo

H.n~ H oH ms-H AnorHVmoH HOHonz_mw.o +H+HHQHHH=zVsm
on.o H mH oo-m HmHHHVoaH cans ms.o

om.o H «H oo-m omH oaas_muo.o +x+~wozmHH=zVse

uH H NH oo-m me- ozooommo mH.o H +H\+msanamszsm

on H o om-m Hmooonmo mooz_mw.o +H+Hmomam=anm

oH H om oo-m HeooH.mosHmeH mene_mH.o +~H+Ma~moVsAmsz:m-uoo

HHm.aHVoH H mm oo-m HessH.mmaHV~oH mooomm0.m~.o +HH+MeomHmszsm

om.o H mH mo-m HHoHoHVoss ozooomno.mwo.o +~H+mhmszno
on.o H sH oo-H er ozooomao_mn.o
om.o H n.oH oo-m onH ozooomno_mu.o
HHaVo.om.o H oH oo-mH naH soHusz.mH.o
on.o H HH oo-HH HeerH.nmaHerH some me.o

om.o H sH oo-m HHH seas moo.o +~H+MHHansm

A. 9.3 omm< Us . owned 95 “Hal 3.30.50on oHnnoo
A . neon. n

 

mcHdHousou ooHnsoo noose HHHHHHHVoo one HHHHHHHVne

.oncanH ououcoana

mnOHuo> you moHoouunm noHuonmm .CH 038.

99

 

 

- I . - . N a N I
area + 2 mm s 8H mean an e or. Ho A more one
- n I m N
on H mm 3 m 3H 2. m 2H +\+~Hu A more
ram-Vatom H on om-m romVoH menu mic AEMHHHSVSH
on
a.c.mv omo 0o .ownwm 35 WT oumHowuumHu oHonoo
A .93“. n

 

A3363 .3 Home

100

Notes to Table 10.

 

aReversible half-wave potential in mv vs. ambient KSCE; determined

at 25°C by cyclic voltammetry at a HMDE (except for Ru(NH3)5py3+/2+

in which a Pt electrode was used). See notes to Table 9 for further
details. values in parentheses are from the indicated literature

sources and correspond to comparable experimental conditions.

bReaction entrOpy of redox couple in e.u. at 25°C (see notes to
Table 9). Stated precision was estimated from the scatter of
experimental points in the vicinity of 25°C. Values in parentheses

are from the indicated literature sources.

cDetermined using cyclic voltammetry sweep rates in the range
50-500 mV s-l.

dDetermined using sweep rates in the range 1-100 V 3‘1.
epTS - p-toluenesulfonate.

fpy - pyridine.

8Reference 52.

hReference 54.

1Reference 56.
1Reference 119.
kReference 118.

lDetermined by K. L. Guyer.

mDetermined by P. D. Tyma.

101

Ru(NH3)l‘Cl-;/o couples necessitated the use of large sweep rates

II

(1-100 V sec-1) in order to avoid significant aquation of Ru during

the potential scan. Estimates of As;c have previously been obtained
i+l2+ §+IZ+ and Ru(en)§+/2+ from the tempera-

ture dependence of the equilibrium constants for reduction of the

for Ru(NH3) , Ru(NH3)50H

Ru(III) complexes by Npiz coupled with an estimate of As;c for the

4+/ 3+

Npaq

couple (119). While reasonable agreement between the present
and earlier determinations is found for Ru(en)§+/2+, substantial dif-
ferences are seen for the other two systems (Table 10). These dis-
crepancies may arise from systematic errors in the kinetic analysis
employed to determine the equilibrium constants in ref. 119 when
these quantities are much larger than unity. Significant differences

are also seen between present and earlier determinations of E for

72
some Eu(III)/(II) couples (Table 10). In view of the care taken to
minimize liquid junction potentials in the present work, these dif-
ferences probably arise chiefly from the presence of such potentials
in the earlier work combined with the slightly different thermal con-
ditions employed here.

Two main trends are seen upon inspecting the data given in
Table 10. Firstly, the stepwise replacement of ammonia by aquo
ligands results in large increases in As;c. Secondly, the substi-
tution by anionic ligands results in significant decreases in us;c.

A value of A8;c for Os(NH3)2+/2+ is also given in Table 10 for com-

parison with Ru(NH3)g+/2+. The reaction entropies for these two

systems containing the same ligands are in close agreement.

102

3. Redox Couples Containing Chelating Ligands

In contrast to redox couples containing only simple unidentate
ligands as considered above, a significant quantity of information
has been gathered previously on the reaction entropies of couples
containing chelating ligands. One reason is that such complexes are
often sufficiently stable so that even labile oxidation states can
remain in the complexed form in the presence of small stoichiometric
excesses of the chelating ligands. Consequently it becomes feasible
to determine values of As;c for redox pairs such as Co(III)/(II) when
bound to some chelating ligands. The lability coupled with the weak
complexing ability of the Co(II) state precludes such studies of
Co(III)/(II) couples containing only unidentate ligands. The simplest
example is Co(en)3+/2+

3
3+/ 2+
with Ru(en)3 . The relevant data for these couples are given in

, which provides an interesting comparison

Table 11. It is seen that the values of A8;c for these two redox

couples are strikingly different. The large value of As;c for

3+/ 2+
3

smaller values seen for other amine complexes (Table 11), but is

Co(en) (37 e.u.) is surprising in view of the markedly

in accord with an earlier determination (120).

Consequently, we decided to evaluate As;c for some other related
Co(III)/(II) couples. The tris-l,lO-phenanthroline and
tris-2,2'-pipyridine Fe(III)/(II) and Ru(III)/(II) couples have

previously been found to exhibit values of As;c that are close to

zero (51,121). The present determinations of As;c for Fe(phen)§+/2+

3+/ 2+

and Fe(bpy)3

are in close agreement with these earlier values.

103

 

m

 

 

88 no man +
H H m.¢H oolm mwu mzooommo meo.o +N\+mnoroonmsznm
w.o
H H m.~H oolm «mm ozooommo.mH.o +~\+mhooeAmsznm
wao
H H o oo-m Hos once HHH +HH+MHxnoVHHm=zVse-nso
w.o
rooomeo maoH
H H no oo-m «so .ozooomeo mic HHxaoVHHmszsH-uso
To +~\+m
- m I. m
an. I "1. . sham HH
fo 8 H + H on m fa 2:333 58 no 5 c we +HH+HH 5 a
sec mama +
- .I m
I:- I o ".1. o sham
HHJHo V H + N 3 .H f8 o 389% Hon :3 o in +Q+HH oVue
core mafl +
I. m
"1. I I o "1. o a m
HHJHo 8 N + n no .H Heme o moonS Us :3 o no +N\+m?uc V n
so maom +
I a .V I. m
ens. HV H + R oo-m H as nos-V8.7 332 E 338
ho ho 0.0 +~\+m
. - I - o l . n no
HHHHV m o + H 8 m 8 no. 2H o oJ +H\+HH Van
A.n.oV ouom< .oo .ownnx A>nV mam ouhHoHuooHu oHonoo
.monmeH wnHumHonu wanHmunoo mmHonoo aHHv\AHHHVz mnoHuo> How moHoouunm noHuomom .HH oHomh

104

 

 

N H m- Has-H NH .683 mHo NHmnmoVue
Ho 3+
I Q I
- - - . e
N + 2 on N Acorn V can 39.2 H: o :6 +N\+NH 938
sec mamN +
l I I o M %Q
HaHNV m + NN 3 m 2 H8. :3 o no +N\+mH 38
core mamN +
I I I. . m n o
HamHV m + NN 3 m 3H H8. :8 c on. +N\+mH on V8
H H no oo-m N3 mane mH + NHxnbVNHNeoVé-orneu
, To N\+m
H H no co-m car was: HHH + NercVNHNmoVas-nsu
To N\+m
. . on . N\
A a CV: owe .oo ownmm A>EV m Hm mumHouuoon oHonou
.oeoa

 

A.unooV .HH «Hana

105

Notes to Table 11.

aReversible half-wave potential in mV vs. ambient KSCE; determined
at 25°C by cyclic voltammetry using sweep rates of 50-500 mV 5-1
(see notes to Table 9 for more details). values in parentheses
are from literature sources and determined at the indicated ionic

strength u.

bReaction entrOpy of redox couple at 25°C in e.u. (see notes to
Table 9 for further details). Quantities in parentheses are
literature values determined at the indicated ionic strength.

No 11 value is given when the literature study and the present study
employed the same ionic strength.

cDetermined using HMDE.
dDetermined using platinum and glassy carbon electrodes.

een = ethylenediamine.

fphen - 1,10-phenanthroline.

8bpy 8 2,2'-bipyridine.

hsep - l,3,6,8,10,l3,16,l9-octaazabicyclo(6.6.6)eicosane. The trivial
name for this ligand is sepulchrate (see reference 88).

1Reference 119.

1Reference 120.

kReference 121.

1Reference 51.

mCalculated from isothermal cell data (A. Claus and V. Crescenzi,
quoted Table V of M. Chou, C. Creutz, and N. Sutin, J. Am. Chem.
Soc., 22, 5615 (1977).

nReference 88.

106

Relatively small values of As;c are also maintained for

cis- and trans-Ru(OH2)2(bpy)2+/2+ (Table ll). However, the measured
values of As;c for Co(phen)§+/2+ and Co(bpy)?"2+ are both substan-

tially different from zero (ca. +25 e.u., Table 11). It therefore
again appears that such Co(III)/(II) amine couples exhibit

"anomalously" large As;c values. On the other hand, As;c for the

"capped ethylenediamine" Co(sep)3+/2+ couple (88) is much smaller than
for Co(en)§+/2+ (Table 11).

E. Discussion

 

l. ILigand Effects on Reaction Entropies

It has been known for some time that the entropies of simple aquo
cations can be correlated with surprising success using empirical
relations involving the ionic charges and radii (41,45). Using the
Latimer-Powell relation (41) and known crystallographic radii (l8),
As;c for the first-row transition metal couples in Table 9 are cal-
culated to be ca. 45 e.u., in good agreement with the experimental

results. However, the smaller values observed for Ru(OHz)‘2+/2+ and

)3+/2+ 3+] 2+
2 6 n
and Yb(OH

V(0H (36 and 37 e.u.) and the larger values for Eu(OHZ)

) 3+] 2+
2 n

classical Born equation can only yield predicted values of As;c in

(48 e.u.) are not predicted by this relation. The

reasonable agreement with these results by inserting the radii of the
bare cations (45), rather than the radii including the coordinated
water molecules which intuitively seem more appropriate, especially
for transition metal cations. These large experimental values of

As;c are probably a consequence of the release of water molecules

107

surrounding the primary coordination sphere that are strongly
orientated ("frozen") in the tripositive oxidation state (47,48).
Indeed, such solvent structuring effects provide the most widely
accepted and intuitively reasonable explanation for the wide varia-
tions found in As;c (47-49,51).

In this connection, it is interesting to note that simple
M(III)/(II) ammine couples exhibit values of A3;c which are substan-

tially less than for the corresponding aquo couples (Tables 9 and 10).

3+] 2+ 3+] 2+
6 6 ’

and other hexaaquo couples, Astoc 2 36 e.u. Since the

Thus, for both Ru(NH3)

) 3+] 2+
2 6

size, shape, and electrostatic prOperties of ammonia and water ligands

and Os(NH3) AS° ~18 e.u., whereas
rc

for Ru(OH

are comparable (122), these differences suggest that rather specific
interactions between the coordinated and surrounding water molecules
are responsible for the large values of As;c observed for the aquo
systems. It seems reasonable that the large degree of solvent order-
ing around tripositive, compared to dipositive, aquo cations arises
partly from.the ability of the relatively acidic aquo protons to form
hydrogen bonds with surrounding water molecules (47,48). The weakly
acidic ammine protons presumably have a much lower tendency to aid
the central cationic charge in orienting solvating water molecules

in this manner. These results are not unexpected on the basis of the
empirical entropy correlation of George et al., which also indicates
that the entrOpies of ammine complexes decrease less with increasing
cationic charge compared with aquo systems (43,44). The much lower
"structure-making" ability of tripositive ammine, compared with aquo,

complexes is also borne out by the much smaller effective hydrated

108

radii for ion transport that are observed for the former complexes
(123). The differing magnitudes in the electrostatic double-layer
effects observed for the electroreduction of Cr(III) aquo and ammine
complexes suggest that such differences in the extent of hydration

also survive within the electrode-solution interfacial region (124).
47.1
re
where re is the effective radius of the ion (45), and the radius of
)3+/2+
3 6
couples AS]:c “l6 e.u. The close agreement between the Born prediction

3+] 2+ 3+]2+
6 6

(Table 10) indicates that there is no extensive solvent ordering

Since the Born relation predicts for +3l+2 couples that As;c - e.u.

M(NH is ca. 3 A (122), then this relation predicts for the

and the experimental results for Ru(NHB) and Os(NH3)
around these species even in the tripositive state, although dielec-
tric saturation effects (46) may complicate the application of this
simple model even to such substitutionally inert complexes. The

substitution of ammine by aquo ligands in the series of couples

3+]2+

O
Ru(NH3)6_x(OH2)x is accompanied by sizable increases in Asrc’
especially for x - 1 (Table 10), indicating that substantial solvent

structuring can occur even around an isolated aquo ligand. The

experimental values of As;c for both Eu(OH2):+/2+ (Table 9) and
Ru(NH3)g+/2'+ (Table 10) depend only slightly upon the total ionic

strength, confirming that this quantity is chiefly a consequence

of ion-solvent, rather than ion-ion interactions. The small decreases
in As;c at ionic strengths approaching unity are probably due in part

to ion-pairing in the tripositive oxidation state, so that the effec-

tive ionic charge of this state is less than +3.

109

The substitution of ammine or aquo ligands by simple anions in
Ru(III)/(II) couples consistently results in substantial decreases
in AS££ (Table 10). This effect could arise for at least two reasons.
Firstly, the reduction in the net ionic charges is always expected to
lower 63;; on the basis of the classical Born model since As;c is
predicted to depend on the difference of the squares of the ionic
charges on the two ions forming the redox couple. Secondly, the
replacement of hydrogen-bonding ligands by electronegative ligands
should decrease the extent of solvent structuring around these
cations. To varying degrees, these two factors are probably respon-
sible for the observed behavior of the chloro and isothiocyanato

complexes in Table 10. The more dramatic decrease in As;c for

2+]+ 3+] 2+
2

be attributed to the different hydrogen bonding characteristics of

Ru(NH3)SOH compared with Ru(NH3)SOH (Table 10) can reasonably

OH- and 0H2. Thus OH- can hydrogen bond to surrounding water mole-
cules via the oxygen atom, which should be favored on electrostatic
grounds by coordination to Ru(II) rather than Ru(III). In contrast,
hydrogen bonding involving aquo ligands will be electrostatically
favored by coordination to Ru(III) rather than Ru(II) since the
hydrogen atoms on the ligands will participate in the bonding.
Although the data for the aquo couples in Table 9 indicate that
some dependence of As;c upon the electronic structure of the central
metal cation is to be expected, part of the variations could arise

from differences in the number of aquo ligands that are bound to the

central metal ion. Thus n is undoubtedly greater than six for

110

3+]2+ 3+] 2+
2)n and Yb(0H2)n

hydrogen bonds with the surrounding solvent and may account for the

Eu(OH which can result in a greater number of
expecially large values of As;c that are observed for these couples.
The reduction of V(III) to V(II) could be accompanied by an increase
in the number of bound water molecules (including those held by
hydrogen bonding). Such a change is sterically reasonable in view of
the large ionic radius of v2+ (r “‘0.8 A (11)) and would result in a
smaller value of As;c, as is observed. Similarly, Cr(II) is highly
Jahn-Teller distorted (125), and the axial ligands in Cr(OH2)§+ are
very weakly held. As a consequence, the conversion of Cr(III) to
Cr(II) is accompanied by a decrease in the number of bound water
molecules. This could be responsible in part for the relatively

large value of As;c. Differences in the polarizing power of the
various cations may also be a factor. The As;c values for Ru(III)/(II)
couples containing simple unidentate ligands given in Table 10 could
provide workable estimates of As;c for other couples with the same
ligand constitution and charge type. However, it is apparent from the
data given in Table 11 that this assumption may not be reasonable for
complexes that contain chelating ligands. Thus the values of As;c

for Co(III)/(II) complexes containing ethylenediamine, o-phenanthro-
line, and bipyridine ligands are about 25 e.u. larger than for
corresponding complexes involving other metal cations. These results

are both surprising and somewhat puzzling. The behavioral difference

3+] 2+ 3+] 2+
3 3

iriligand conformation that are known for the trivalent complexes

and Co(en) might be explained by differences

'between Ru(en)

(2126) giving rise to differences in the surrounding water structure.

111

However, this explanation seems less plausible for the much larger

and structurally more open phenanthroline and bipyridine complexes.
The values of As;c close to zero that were previously found for other
phenanthroline and bipyridine couples have been explained by the
supposed ability of these aromatic ligands to "shield" the central
metal ion from the surrounding solvent (51,121). The contrasting
behavior of the corresponding Co(III)/(II) couples clouds this
explanation somewhat. Certainly a common feature of these Co(III)/(II)
couples is that the change in electronic structure

(t28)6-+ (t28)5(eg)2 that occurs upon reduction must result in
a greater degree of bond stretching and possibly stereochemical
change compared with Fe(III)/(II) and Ru(III)/(II) couples which
involve the conversion (t28)5-+ (t28)6. One of the consequences is
presumably the release of "bound" water molecules in going from
Co(III) to Co(II). The relatively small As;c (l9 e.u.) found for

the Co(sep)3+’/2+

couple (Table 11) could be a consequence of the
"tighter" structure of the macrobicyclic ligand effectively exclud-
ing specifically bound water in the higher oxidation state. Further-
more, the much smaller As;c values for the Ru(III)/(II) and
Fe(III)/(II) phenanthroline and bipyridine complexes may arise partly
from significant delocalization of the added t28 electron around the
aromatic rings. This delocalization can result in additional solvent
ordering around these ligands in the divalent state which will
counteract the decrease in charge density at the metal center. In

the Ru and Fe systems these effects apparently negate each other,

yielding As;c values close to zero.

112

It would be enlightening to ascertain the functional form of the
ligand effect on As;c, assuming, of course, that such a relation
exists. In other words, with respect to the reaction
MIIIL'L"+-e- +~ MIIL'L", it is useful to determine how AS° is

m n m n rc
affected by changes in,m and n. Such information will yield further
insight into the various ligand influences on solvent polarization
and will allow interpolation and extrapolation of As;c values
between related redox couples.

The simplest approach to the present question is to consider that
the observed value of As;c for a redox couple arises from independent,
additive contributions from each ligand in the complex. This partic-
ular model also forms the basis of the empirical entropy correlations
of George et al. (43,44). The dielectric-continuum Born model could
also be used to calculate As;c values, although one can expect some
problems since this model treats the nonsymmetrical complexes in this
study as uniformly charged spheres. Some of the results of this
investigation will be employed to judge the suitability of these
simple treatments for predicting the reaction entropies of mixed-
ligand complexes (i.e., those containing two different types of
ligands).

Table 12 summarizes the experimental values of As;c for octahedral
mixed ligand redox couples of the type MLéLg(III)/(II) for which
values are also available for the corresponding "pure ligand" couples
ML;(III)/(II) and ML;(III)/(II). Listed for each couple is the

Born

reaction entrOpy (as;c) that is calculated with the Born model

113

 

o

 

mH- om.Hs- -s\-mHzoVoe
mn- eN- oaN- -N\-.ao:zoVua
N N +N\+M 2%: we
m.m m.NH no +N\+MHseHVNaN=oVsm-otoxu
m .m m . NH no N 1M HRHVN HNmoVse-nso
A.sH on +N\+MHNmoVse
m.aH n.sN oN +N\+MHN=oVonszsa-oso
m.sH m.HN mN +N\+M=omHmszse
or: 3H +NH+MHH=zVsa
o.NH m.mH NH +NH+HHoHHHszne
as a: +NH+HcoaaaHHszVsa
n.0H m.NH m.NH +N\+mxnosHm=zVse
o.n N m.o +N\+MHenoVNHmszsa-wsc
on H +N 1” Ease
o .n.m .nuomnowmqv H .n.o .oumoaowmoV .n.o .m ammo oHonoo xoomm

 

aHHV\AHHHV nHan make on» no moHonoo xoomm mnOHum> How moHnouunm noHuommm .NH oHrme
:-

114

Notes to Table 12.

aExperimental value of as°rc taken from Tables 9-11 unless otherwise

noted.

bReaction entropy for mixed-ligand couples estimated from the experi-
mental values of the two appropriate pure-ligand couples by linear

interpolation (see text).
cReaction entropy calculated from the Born model by using equation (8).

dThese are estimated values for u a 0.1. Direct determinations at
this ionic strength are precluded by hydrolysis (see experimental
section). The tabulated values are estimated from those determined

in EE‘HpTS (Table 11) by assuming an ionic-strength dependence

3+/ 2+

2 (Table 11).

similar to that of cis-Ru(NH3)2(bpy)

eReferences 51, 114, and 121.

115

by using the relation (46):
(AS° )Born - 9 65 (22 IE - z2 I- ) (8)
, rc ' ox ox red ared

where 20x and zred are the net charges and 30x and fired are the
equivalent radii (81) of the oxidized and reduced species, respec-
tively. (The a value for a complex was taken to be equal to half of
the cube root of the product of the diameters along the three L-MrL
axes (81). Values of the equivalent radii were taken from reference 81
or estimated from ionic radii (11,18).) Also given for the mixed-

eat that were estimated by

ligand couples are reaction entropies (As;c)
linear interpolation from the experimental values of As;c for the
corresponding two pure ligand couples, i.e., by assuming that the two
types of ligands provide additive contributions to As;c that are
proportional to the number of each ligand type present in the mixed-

ligand couples.

For the Ru(III)/(II) couples containing ammine and bipyridine
) est

ligands, excellent agreement is seen between As;c and (As;c ,

whereas the Born model fails to predict the marked decreases in A8;c

that are observed as the number of bipyridine groups increases, par-

3+/ 2+ 3+/ 2+
2 3 °

results suggest that differences in local salvation of the individual

ticularly in going from c-Ru(NH3)2(bpy) to Ru(bpy) These
ligands rather than changes in the average distance of closest
approach of the solvent to the metal charge center are largely res-
ponsible for the observed large variations in As;c. This finding

is perhaps not surprising for complexes containing large ligands

such as bipyridine where a number of the solvating water molecules

116

will lie in the vicinity of the aromatic rings away from the metal
charge center and the other ligands. As mentioned previously, the
addition of an electron to such Ru(III) complexes to form Ru(II)
could well give rise to two competing effects upon the surrounding
solvent structure. The water molecules close to the ruthenium
center, including those surrounding any ammine ligands, will be less
polarized and therefore less "ordered" in the lower oxidation state,
giving rise to a positive contribution to 15;. However, the water
molecules adjacent to the bipyridine rings could well experience an
increase in polarization in going to the Ru(II) state since the

added t electron will be significantly delocalized around the

28
aromatic rings, acting to increase their net charge density. This

latter effect would yield a negative contribution to As;c which will
be roughly proportional to the number of pyridine rings. Therefore
the stepwise replacement of ammine by bipyridine or pyridine ligands

would result in an approximately linear decrease in Aszc, as observed.

3+/ 2+
3

As;c 3‘’0. The small (or slightly negative) values of As;c observed

For Ru(bpy) , these two effects presumably cancel, yielding

for Eu(III)/(II), Fe(III)/(II), and Os(III)/(II) polypyridine couples
have been previously ascribed to the efficient shielding of the metal
center from the solvent by these ligands (51). This last explanation
seems less reasonable since the polypyridine ligands will allow some
solvent molecules to approach close to the metal center along the Open
channels formed by the planar aromatic rings. Also, the markedly

larger values of As;c (22 e.u.) observed for both Co(bpy)§+/2+ and

3+ 2+
Co(phen)3 I (Table 11) can be more easily understood on the basis

117

of the present model. These Co(III)'+ (II) reactions involve the

electronic conversion t; +t5 e2 which should minimize the extent of

3 233
electron delocalization in the reduced state, therefore discouraging
any increase in solvent polarization in the vicinity of the pyridine
rings. This change in electronic configuration will also yield a
marked expansion of the cobalt center (127) and hence an especially
large decrease in the polarization of nearby water molecules.

We have obtained further evidence that favors the present inter-
pretation from the observation that the ferrocinium/ferrocene couple
has a distinctly negative value of as;c in aqueous media {-5 e.u.,
Table 11). Since this couple is of the charge type +1/0, the simple
dielectric-polarization model predicts a small positive value of As;c.
The negative As;c can be explained by noting that the electron added
in the reduction will be substantially delocalized around the cyclo-
pentadienyl rings in a manner similar to the polypyridine couples.
Furthermore, the "sandwich" structure of the ferrocinium/ferrocene
couple should largely prevent the close approach of solvent molecules
to the metal center. Consequently in this case the predominant con-
tribution to As;c is anticipated to be the increased polarization of
water molecules adjacent to the aromatic rings in the lower oxidation
state, in accordance with the observed negative value of As;c.
Negative values of As;c have also been observed for some blue copper
protein couples having the charge type +l/O and have been interpreted
in terms of hydrophobic effects in the vicinity of the copper redox

center (128).

118

As noted previously, the large values of As;c typically observed
for aquo couples have been attributed to field-assisted hydrogen
bonding between the aquo ligands and the surrounding water molecules.
This effect is apparent when it is noted that the substitution of a
single aquo ligand into Ru(NH3)2+/2+ yields a proportionately larger

increase in ASrc than those which result from subsequent aquo substi-

tutions (Tables 10 and 12). However, there is reasonable agreement

0 o estd 3+/ 2+
between ASrc and (Asrc) for o Ru(NH3)4(OHZ)2 . 0n the other
hand, substitution of one bipyridine in Ru(bpy)§+/z+ by two aquo

ligands yields somewhat smaller values of As;c compared to (Assc)estd

(Table 12). It is possible that the extent of hydrogen bonding involv-
ing aquo ligands is very sensitive to the electrostatic and steric
environment. Therefore, it may be generally difficult to achieve
accurate predictions of As;c values for mixed-ligand couples containing
aquo ligands.

Since a large number of complexes employed in redox kinetics
contain anionic ligands, it is of interest to examine the stepwise
changes in As;c that occur as anions are substituted into the coordina-
tion sphere. Substantial and even qualitative changes in As;c are
generally expected from the Born model as well as from local salvation
effects due to the variations in the overall charges of the complexes.
Unfortunately, suitable data are extremely sparse. Table 12 contains
data for three Fe(III)/(II) couples containing bipyridine and/or
cyanide ligands. Again it is seen that the estimated value

/2

for the mixed-ligand couple Fe(CN)4bpy- - is in reasonable agreement

119

with the experimental value of As;c, while the Born model fails to
provide adequate estimates of As;c for all three couples.

The assumption that the reaction entropies for mixed-ligand
couples arise from simple linear additive contributions from each
ligand would appear to provide a useful, though approximate, means
of estimating As;c under some circumstances. However, often one or
both As;c'values for the corresponding pure ligand couples are
unavailable. If it is assumed that values of As;c depend chiefly on
the ligands and the charge type of the redox couple, the required
reaction entropies could be inferred from those for other couples
containing the same ligands. George et al. (43,44) have used ionic
entrOpy data to obtain empirical parameters for various ligands which
allow estimates of As;c to be made for couples containing these
ligands. As was discussed earlier, this approach has serious limita-
tions since in some cases As;c can be significantly dependent on the
nature as well as the charges of the central metal ions. Nevertheless,
the method could be employed, albeit with caution, to estimate the
changes in As;c resulting from.minor alterations in ligand composition.

2. Correlations between Outer-Sphere Self-Exchange Rates and
Reaction Entropies

Aside from yielding information on the thermodynamics of ion
solvation, reaction entropies also provide insight into some factors
that influence redox reactivity. In order to recognize the origin and
the utility of the relationship between As;c and homogeneous electron-

transfer rates, one must recall that the free energy of activation

120

A631 for an outer-sphere self-exchange reaction consists of inner-shell
(AGZn) and outer-shell (Acgut) contributions arising from the struc-
tural reorganization of the reactants and the surrounding solvent,
respectively, that are required to yield the activated complex for
electron transfer. A correlation is expected between Acgut and As;c
since both reflect changes in the solvent structure which take place
in the course of a redox reaction. The outer-shell term stems from
the alterations in solvent polarization that transpire when the acti-
vated complex is formed from the reactants (or more precisely, from
the precursor complex). The reaction entropy is a measure of the
solvent reorganization which occurs around one of the reactants when
it is converted to a product. The Francerondon restrictions on
electron transfer tend to yield an activated complex having properties
intermediate to those of the reactants and products. Thus, in this
simple analysis, the entropy change associated with solvent reorganiza-
tion in the activation process is related to some fraction of As;c.

It must be emphasized that one cannot expect some sort of direct
correspondence (such as an equation) between As;c and Acgut’ After
all, the overall AS° for a self-exchange reaction is zero since no
net reaction occurs. Therefore, As;c is not linked to the electron-
transfer rates via an entropic driving-force effect. Furthermore,
Acgut refers to the activation of two reactants, while As;c is con-
cerned with just one redox center. Fortunately, the outer-sphere
reactions considered here are weakly adiabatic, and the absence of

strong coupling between the reactants in such processes allows each

redox center to be treated independently from a conceptual viewpoint.

121

(It remains to be seen whether the process of bringing two highly
charged species together will lead to effects that neither would
produce by itself, even if it is assumed that a weak mutual inter-
action occurs between these species. This point is discussed in
Chapter V.)

In addition to the possible correlation between As;c and Acgut’

one could expect some relation between reaction entropies and AGED.

As noted previously (e.g., in the discussion of the As;c values for
Co(III)/(II) couples), As;c becomes larger as the difference in the
effective radii A5 of the reduced and oxidized species increases.

Since large values of A; are also associated with large inner-shell

contributions to AGfl, some factors that tend to increase AGin are

expected to increase As;c as well. Thus, one generally cannot use

ASZe to separate the inner-shell and outer-shell components of AGfl,

but some correlation between As;c and A631 should be found.

The interpretation of AGfl is simplest for redox couples which
undergo weakly adiabatic electron transfer and for which the change
in metal-ligand bond distances is not large. For such systems the

inner-shell contribution AGin is small and calculable, and the

outer-sphere contribution A63“ can be obtained from AGf1 (81).

t
Although suitable systems are not abundant, Ru(III)/(II) couples con-

taining ammine and/or polypyridine ligands provide such a class (81).

3+/2+
at NH b
Estimates of AG I for various Rn( 3)2x( py)3_x

couples where
x - 0-3 have recently been made (81). These are listed in Table 13

along with the corresponding values of AG:ut (calcd). The latter

122

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.Az H.o u : cuwdouum 0H:0H um wocHounoV HH was 0H moHnma scum cQHMu .oHesoo xooou mo haouuoo doHuomom
.Hw mucouomou mo >H oHAMH scum coxmu “Amnv mucosa moons: scum wouoH=UHmo :OHuoaHuucoo HHoSquouso

.Hw mucouowou mo >H oHnmy Bonn ooxou moon

.muoouomou wouonmdom onu scum onmEoo oonHHHoo can show 0» wouHoomu Mao; UHuoumouuooHo osu mH MB woo

.oOHuanuucoo HHochuoccH onu mH dwoq .ucoumdoo «you co>uomno onu mH HHx .muamuomou mo uHoe HoHooHuuoe
.u sH H uoo

o3» yam hoooovouw :OHmHHHoo osu mH N muons 3 I «Uq I A H& aH I N dHVHm n auq scum ooonuno

.a0Huooou owoozoonMHom o£u Mom hmuodo ovum :OHuo>Huoo ass on :OHuonHuuooo Auso>Homv HHochuouoo

.Hw ouomuomou scum :oxou .aOHuoo waHuommu mo musmu unoHo>H=vm

0

o
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a

 

 

m
m.cH m.mH m.o m.n m.m oAmmzvsm
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o o o o o h“ .V m d.—
m 0H m NH H m H m o q +~\+mA Av A mzv m
N N m
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m
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o e w e o a A a o I

 

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omdo£OXMIMHom AHHv\AHHHV ESHGonuom meow you monuocm moum :OHuo>Huo< dooSqu oowwuoefioo .MH oHooH

123

quantities were calculated by using the Marcus model which treats
the surrounding solvent as a dielectric continuum (81). The values
of As;c for these ruthenium couples are given in Table 13 along with
the corresponding values of A8;c(calcd) calculated using the
dielectric-continuum Born model (see reference 81 and equation (8)
from the previous section). Table 13 also includes values of a, the
equivalent radius of the reacting cation. Because of the very
similar a values found for the trivalent and divalent Ru complexes
considered here (81), one can list a single value for both halves of
the redox couple.

It is seen that the increasing values of A63“t that are observed
as the smaller ammonia ligands replace bipyridine in the ruthenium
coordination sphere are also associated with markedly increasing
values of Asgc. This is not surprising since greater changes in
solvent polarization are required for electron transfer as the effec-
tive radii of the reactants decrease (79,81). The dielectric-
continuum estimates of AGgut(calcd) and As;c(calcd) are seen to be in
broad agreement with the corresponding experimental parameters.
However, it is interesting to note that there is substantially better
agreement between Acgut and AGgut(calcd) than between As;c and
As;c(calcd). This behavior may be related to the fact that
Acgut(calcd) is mainly determined by the optical dielectric constant
80p (see reference 79 and equation (12) from Chapter I), rather than
by the static dielectric constant as. The latter determines the value

of As;c(ca1cd) (46). It is known that eop is much less sensitive to

124

solvent structure than is as, so that any breakdown in the validity
of the dielectric-continuum model of ion-solvent interactions caused
by an alteration in the solvent structure in the vicinity of the
solute is likely to affect acgut to a much smaller extent than As;c.
Most other outer-sphere exchange reactions involve significant
changes in.metal—ligand bond distances so that the inner-shell
reorganization energy AG:n forms a sizable component of AGfl. This
is particularly true of M(OH2)2+/2+ reactions, and some reactions of
this type are listed in Table 14. This table also includes values
of the self-exchange rate constant kll’ AGfl, and As;c for the aquo
couples and a number of other 3+/2+ redox couples containing ammine,
ethylenediamine, or bipyridine ligands. (The tabulated values of the
activation free energy were corrected for the electrostatic work wr
required to form the collision complex by using the equation
AGEl - RT(ln Z - 1n kll) - wt where Z is the bimolecular collision
frequency (81)). The.ammine and ethylenediamine complexes, in par-
ticular, form an interesting comparison with the aquo complexes since
all three ligands are of comparable size so that the outer-shell terms
Acgut should be similar, at least on the basis of the dielectric-
continuum treatment of Marcus (79). The data from Tables 13 and 14
are also presented in Figure 4 as a plot of As;c versus AGfl.
Inspection of Table 14 and Figure 4 reveals that there is a general
tendency of Asgc to increase along with AGll’ although not surprisingly

there is a considerable scatter in the individual points. For the

aquo couples, the increasing values of As;c as well as of AG:1 are

125

Table 14. Comparison between Kinetic Parameters for Selected
Outer-Sphere Self-Exchange Reactions and Reaction Entropies for
Corresponding Redox Couples at 25°C.

 

 

k a * b c
Exchange Reaction/ ex A611 As;c
Redox Couple 'Muls-1 kcal.mol: e.u.
«(01196343 ~ 2 x 10'7(1.0)133 ~ 24 so
V(OH2)63+/2+ 1.5 x 10‘2(2.0)13S 17.4 37
Fe(OHz) 63442“ 4(0. 53136 13.8 43
Ru(OH2)63+/2+ ~ 80(1.0)132 ~ 12 36
Ru(NH3)63+/2+ 3 x 103(o.l)81 8.2 18.5
Ru(en)33+/2+ 3 x 103(o.1)81 8.9 13
1111(1)”):+ I 2+ 4 /x 108(o.1)137 ' 3.4 1
Co(en)33+/2+ 8 x 10"5(1.0)138 20.4 38
Co(sep)3+/2+ 5(o.2)88 14.4 19
Co(bpy) 33”“ ~ 20(o.1)139 13.4 22

 

8Rate constant for acid-independent self-exchange pathway; obtained
from indicated reference numbers. Ionic strengths are given in

parentheses.

bFree energy of activation for the self-exChange reaction, corrected

for the electrostatic work required to form the collision complex.

cReaction entropy of redox couple, obtained at (or extrapolated
to) ionic strength u I 0.1 M. Data from Tables 9-11.

126

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127

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128

broadly compatible with the conventional explanation which ascribes

the latter changes to increasing values of the inner-shell contribu-

tion AG* . Thus the Cr(OH2)3+/2+ exchange reaction involves transfer

in
of an antibonding e8 electron which is expected to yield substantially

3+/ 2+ 3+]2+

larger values of A; than the Ru(OH , Fe(OH and

3+/ 2+

2)6 2)6
exchange reactions which involve transfer of a nonbonding
3+/ 2+

V(OHZ)6

electron. Indeed, the Cr(OH reaction exhibits the largest

t23 2)6
values of both A611 and ASrc . However, substantial differences in

A611’ which do not correlate in any simple way with As;c, are

observed between the Ru(OH2)3+/2+, Fe(oaz)3+/2+ and V(OHZ)3+/2+
reactions. Of these aquo couples, A; has been directly determined
3+/ 2+

using crystallographic data only for Fe(OH2)6 ; a value of 0.14A

was obtained (129). This value of As leads to a calculated value of

k11 that is in good agreement with the measured value for the

3+/ 2+ 3+/ 2+

Fe(OH system (130). The approximation of A5 for V(OH

2)6 2)6
can be obtained by noting that the difference in ionic radii for six-

coordinate 173+”+ is 0.15 A (11). The corresponding difference for

Fe3+lz+ is 0.14 A (11) which agrees well with A5 for Fe(OH 3+]2+.

3+/ 2+

2)6
Thus, A; for V(OH is expected to be slightly larger than that

3+/ 2+

2)6

for Fe(OH2)6 , but this does not completely account for the sub-

stantially slower exchange rate of the vanadium.couple. Furthermore,
the A; values for these two couples would lead one to predict that

V(OH2)3+/2+ have a larger AS° cthan Fe(OH2 )3.”2+ which is the
opposite of what is actually found.

One factor which complicates the interpretation of the results

129

for the aquo redox couples is the lack of information concerning the
coordination numbers of many of the ions of interest. Thus, as

noted previously, the vanadium results can be explained by an increase
in the number of bound water molecules which could well accompany the
reduction of V(III) to V(II). This increase in the solvent struc-
turing around the V(II) ion will result in a smaller value of A8;c,

as is observed, and will yield an additional contribution to AGE1
which will act to decrease the self-exchange rate. Similar considera-
tions can be applied to Cr(II). The aquo ligands in the axial
positions of the hydrated ion are weakly held as a consequence of
Jahn-Teller distortion (as mentioned in the previous section). Thus,
the reduction of Cr(III) to Cr(II) will result in an effective

decrease in the number of bound water molecules. This change in the

coordination shell could be responsible in part for the large As;c and
3+/ 2+
6 O

The data in Table 14 and Figure 4 reveal-that-a dependence of A611

Ac;1 for Cr(OHz)

upon A3;c is also obtained for non—aqua redox couples. Thus both

AGI1 and As;c are markedly larger for Co(en)§+/2+ than for

Ru(en)§+/2+; similar differences are also seen between Co(bpy)3+/2+

3
3+]2+ .
and Ru(bpy)3 . It is likely that the larger values of ASrc for
the cobalt couples are related to their larger A5 values. (As an

example of the size of the values which are involved, As - 0.18 A

for Co(NH3):;+/2+ while the corresponding quantity for Ru(NH3):g+/2+
is 0.04 X (127).) The larger values of Acgn and hence of A611 for

the Co systems are at least partly a result of these larger A5

values (131).

130

The observed differences can also be discussed in terms of

electronic structure. The cobalt(III) complexes are low tgg spin
while the cobalt(II) complexes are high spin tgg e2, so a spin multi-

plicity change accompanies the electron transfer. The ruthenium(III)
and ruthenium(II) complexes are all low spin, and as a result no spin
multiplicity change accompanies the electron transfer. A consequence
of the spin multiplicity change in the cobalt system is nonadiabati-
city (that is, K, the probability of electron transfer within the
activated complex, is < 1; nonadiabaticity increases the apparent
value of AG* since AG*1 can be considered to include the term

11 1

-RT in K). It has been estimated (131) that K for the Co(NH3)3+/2+

6
exchange reaction is ~v10.4. If a similar factor applies to the

3+I2+ exchange reaction, AG;1 for the ethylenediamine couple

should be decreased by about 5 kcal mol-l. Although this correction

Co(en)

is appreciable (and reduces the scatter in Figure 4) substantial dif-
ferences between the cobalt and ruthenium systems remain even after
corrections for spin multiplicity.

Obviously other factors are also Operant. The different ligand
conformations in Co(en);+ and Ru(en)§+ (126) would have an effect on
the inner-shell reorganization of these complexes and hence on A031.
This difference could also be partly responsible for the observed
A8;c values since the availability of the metal ion to the solvent
will be affected by the conformations of the ligands around the cation.
Some of the variations between Ru(bpy):;+'/2+ and Co(bpy)?”2+ can be
explained in terms of the delocalization of the added electron in

Ru(bpy)§+ around the aromatic rings. As discussed previously, the

131

2+
3

this complex. In Ru(bpy)§+, the increased charge density of the

electronic structure of Co(bpy) precludes a similar effect within

ligands induces an increase in the degree of structuring in the

solvent immediately adjacent to the aromatic rings. Thus the reduc-

3+/ 2+
3

solvent ordering, and this provides some explanation for the small

tion of Ru(bPY) to Ru(bpy)§+ is accompanied by a small change in

As;c and A631 values of this couple. One can conclude that effects

other than A; and nonadiabaticity give rise to many of the differences
between the Co(III)/(II) and Ru(III)/(II) systems. Unfortunately,
it is more difficult to treat the ligand-conformation and electron—

delocalization factors in a quantitative manner.

The differences in As;c for Fe(OH2)2+/2+, Ru(OH2)g+/2+, and

Ru(NH3):+/2+ have recently been used to provide an estimate of

) 3+/ 2+
2 6

depends upon the extent to which As;c for the aquo couples differs

Aaa~0.l R for Ru(OH (132). The accuracy of this estimate

from similar ammine couples due to the hydrogen-bonding ability of

) 3+] 2+
2 6

the empirical entropy correlation of Powell and Latimer (41) predicts

the aquo ligands. Taking A; for the Fe(OH couple as 0.14 A,

that A8;c should only be about 10 e.u. lower for an analogous aquo

couple for which A5 - 0. However, A8;c for the Ru(NH3):;+/2+ couple
(A; ' 0°04 3 (127)) is 25 e.u. smaller than for Fe(OH2)2+/2+

(Table 14). Most likely, therefore, hydrogen—bonding effects do con-
tribute to As;c for the aquo couples.
Despite the complexities, there appears to be a consistent cor—

relation between AGfl and Asrc for the outer-sphere exchange reactions

132

considered here. The apparent simplicity of this correlation is
somewhat deceiving since some of the factors which contribute to A8;c
may not always affect AG:1 in a similar way. Nevertheless, the cor-
relation could be used in carefully selected systems as a means of
estimating A; and hence Asia. More importantly, the relationship

supplies further clues to the nature of the solvent-rearrangement

process required for electron transfer.

CHAPTER V
ACTIVATION PARAMETERS FOR HOMOGENEOUS OUTER-SPHERE ELECTRON-TRANSFER
REACTIONS: COMPARISONS BETWEEN SELF-EXCHANGE AND

CROSS REACTIONS USING THE MARCUS THEORY

133

A. Introduction

 

As noted in Chapter I, the adiabatic model of outer-sphere
electron transfer developed by Marcus and others predicts that there
should be a simple relationship between the kinetics of homogeneous
cross reactions and the corresponding self-exchange processes. This
relationship (the "Harcus cross-relation") has commonly been formu-

lated as equation (1):

1
/
k12 ' (kllkzlezflz) 2 (1)

log £12 - (logK12)2/4 log (kllkzzlzz) ' (1a)
where k11 and k22 are the rate constants for the two constituent self-
exchange (homonuclear) reactions, k12 and K12 are the rate and equi-
librium constants, respectively, for the corresponding cross (hetero-
nuclear) reaction, and Z is the bimolecular collision frequency in
solution. Nomerous tests of the applicability of eqn (1) to experi-
mental kinetic data have been made. It has frequently been observed
that the predictions of eqn (1) agree reasonably well with experimental
results (60,140). The observed values of klz, Rigs, are often within
an order of magnitude or so of the values, kiglc, that are calculated

from k11 and kfiz using eqn (1). However, it has recently become clear

that there are a disturbingly large number of reactions for which kggs
calc

and klz

are in substantial disagreement (132,134,141—145). For most

of these systems, it is found that Rigs <3 cgleand that the difference
between k;:8 and kiglc increases as the equilibrium.constant K12

increases (134,142-144).

134

135

Another puzzling feature of homogeneous outer-sphere redox
processes between like-charged ions is that the entropies of
activation AS* for self-exchange as well as cross reactions are
typically large and negative (ca. -20 to -35 e.u.) and insensitive
to variations in ionic strength (81,144b). Such values of A8* are
predicted by simple electrostatic theory at low ionic strengths, but

I

As for self-exchange reactions is theoretically predicted to increase

markedly with increasing ionic strengths to a value close to ~10 e.u.
(60,81,146).

In view of these striking discrepancies between theory and experi-
ment, it seems worthwhile to examine the capability of the Marcus
model to predict the relative values of the free energies, enthalpies,
and entropies of activation for corresponding self-exchange and cross
reactions. Such an investigation should help to pinpoint the factors
that are responsible for the observed breakdown of eqn (1). Few tests
have been made of the ability of the Marcus model to correlate activa-
tion parameters for corresponding self-exchange and cross reactions
(134,145) because many of the required enthalpies and entropies of
the cross reactions, AHiz-and A812, have been unavailable. However,the

standard (formal) potentials and reaction entropies tabulated in

Chapter IV can be used to obtain accurate AH;2 and A832 values at the
ionic strengths that are typically employed for kinetic measurements.
In the present work, these thermodynamic quantities are utilized to
compare the experimental activation parameters for a range of outer-

sphere cross reactions with those that are predicted by the Marcus

136

treatment from the kinetic parameters of the corresponding self-
exchange reactions.

This analysis will provide an additional insight into the limita-
tions of the Marcus and related models in describing the energetics

of homogeneous outer-sphere electron-transfer processes.

B. Kinetic Formulations and Results

1. Free Energies of Activation

 

For the present purposes, eqn (1) can be usefully rewritten in

terms of free energies of activation (134,147):

* a * * °
A612 0.5(AG11 + A622) + 0.5(1 + a) A612 (2)

3 ° * *
a A612/4(A611 + AGZ (2a)

2)

where AG;2 and A032 are the "standard" free energy driving force and
the activation free energy, respectively, for the cross reaction
(determined at the appropriate ionic strength), and AGfl'and A632 are
the free energies of activation for the self-exchange reactions. The
activation free energies appearing in eqn (2) should be corrected for
the work of forming the collision (precursor) complex from the separa-
ted reactants since they are actually reorganization energies within
such a binuclear assembly (79). Eqn (2) should therefore be written

for the experimentally accessible ("apparent") free energies of acti-

vation (AG*)app in the form (79):

 

w w w w
* a: * * .. _
(8612)app 0.5[(AGll)app + (AG22)app] + 0.5(AGlz+A621 6611 A622)
(6622 + Acgl - Ac¥2)2
+ O.SAG{2 + w w (3)
* * - _
8““19st + (AG22)app AG11 A“22]

137

The apparent free energies of activation are related to the

experimental second-order rate constants k°bs'by(74.79)

ksbs . z expl-(AG*)app/RT] (4)

where Z is the bimolecular collision frequency. In eqn (3), A312

and ACgl are the free energies required to form the precursor and
successor ("collision") complexes from the separated reactants and
products, respectively, and A621 and ACgZ are the corresponding work
terms for the constituent self-exchange reactions. For cross reactions
between like-charged ions of similar structure, it is expected that

W ‘W W W W

A012 ~ A021 ‘3 A611 *3 A622 - AG , so that eqn (3) can be simplified to

s . * + * + °
(A012) p 0.5[(AG11) p A022) p ] 0.5 (l+a)AG12 (5)
a o I _ W

In principle, the presence of positive values of ACW in eqn (5) can
explain the common observation that cation-cation reactions with large

driving forces have observed cross-reaction rate constants kggs are

calc

substantially smaller than the values k12 that are calculated using

eqn (1). For small values of Asz,

force term in eqn (5) will generally be small since aAsz is propor-

the influence of a on the driving-

tional to the square of Asz (eqn (5a)), so that eqn (5) will approxi-
mately reduce to eqn (2) under these conditions, irrespective of the
value of ACW. However, as the magnitude AG:2 increases, the effect

of 6 upon the driving-force term will increase so that the presence

of positive values of AC” will act to enlarge progressively the values

138

obs calc
of (A612)app calculated from eqn (5). This yields k12 <3k12 since

eqn (1) and its free energy analogue, eqn (2), do not include work
terms. The above argument is qualitatively reasonable, but the usual
procedure of calculating work terms from the Debye-Hflckel model is
inadequate in a quantitative sense because the values obtained,
(ACW)DH, are not large enough to explain the extent of the observed
discrepancies (134). Nevertheless, it seems quite plausible that
another, larger component of AC", ACE, could arise from the mutual
solvent ordering which accompanies convergence of the two cationic

reactants. Consequently eqns (5) and (5a) can conveniently be re-

written in terms of "Debye-Hflckel corrected" free energies of activa-

 

* 0
tion (AG )corr'
* u s * °
(AG12)corr 0.5[(Acn)corr + (AG22)corr] + 0.5(1+G)AG12 (6)
‘W
a: o * _
a AG12/4[(Acll)corr + (A622)corr ZAGs] (6a)
where (AG*) - (AG*) - (AGW) - AG* + AGW (7)
corr app . DH 5
w lezezN
and (81) (AG ) - T (8)
DH 0 ° Ib
. . esa(l+Bau )
where 21 and 22 are the charges of the two reactants, e is the elec-

tronic charge, N is Avogadro's number, as is the (static) dielectric
constant, B is the Debye-Hfickel parameter (82), u is the ionic strength,
and 3 is the distance between the centers of the reacting ions in the
collision complex.

Eqns (6) and (6a) will be employed as a convenient means of examin-

ing the relationship between the experimental activation free energies

139

of corresponding self-exchange and cross reactions in terms of the
Marcus model. These equations also provide the basis for exploring

the behavior of the individual enthalpic and entropic components.

Although the hypothesis embodied in the use of the term A0: in

eqn (6a) cannot be proven, it will be shown that the algebraic forms
of eqns (6) and (6a) are nicely consistent with.most of the available
experimental data.

Tables 15 and 16 summarize the rate parameters for the acid-
independent pathways of a number of homogeneous self-exchange reactions

between various cationic complexes. Table 15 also includes Ef and

A8;c values for the redox couples involved in these reactions. The

(or AG* ) were obtained in the followb

*
listed values of (A51 22)corr

l)corr
ing ways. For reactions for which the self-exchange rate constants

kg?“ have been determined, values of (AG*

11)corr were obtained by using

eqns (4), (7), and (8), assuming that Z - 6 x 1010M.1 sec.1 (152,153).
obs 3+/2+ 3+]2+ 3+]2+
The listed values of kll for Ruaq , Ru(NHB)6 3 ,

3+/2+ self-exchange are quoted from the literature (81, 1329

, Ru(en) and
Ru(NH3)5pY
145) and_were obtained from cross-reaction rate constants by using

eqn (1). These cross reactions had small driving forces and involved

the Ru complex of interest and a structurally similar coreactant. The
justification for this method of calculation is that eqn (1) has been

found to be consistently successful under these conditions (134).

values of the "Debye-Hfickel corrected" rate constants kiirr

are also
given in Table 15. These were obtained from kiis using the relation

corr obs W ob
lnk11 . lnk11 + (AG )DH/RT. No quantitative values of klls are

140

Table 15. Kinetic and Thermodynamic Parameters of Some Self-Exchange
Reactions at 25°C.

 

 

a b obs c corr

Er ASre k11 k11
Redox _1 _1 _1 _1
Couple mV. vs. a.c.e. e.u. M. sec. M. sec.
Caz/2+ 1680148 ~45e 3. 3(3) 151 8
Pei/2+ 500(o.2) 43 4(0.5)136 15
Ruiz/2+ -15(0. 3) 36 ~60(l)132 200
viz/2+ -475(o.2) 37 0.015(2)135 0.03
Buzz/2+ -625(0.l) 48 (4 x 10“)
Griz/2+ -660(1) 49 (2 x 10'6)
fizz/2+ -1425(o.l) 48 <~o.l)
114""3+ -880(0.5) 48 (0.5)

as

ling/3+ -80(1)149 52150 (~o. 02)
Ru(NH3)2+/2+ -180(0.2) 18 ~3 x 103(o.1)81 ~s x 10“
Ru(en)?/2+ -60(0.1) . 13 ~3 x 103(o.1)81 ~5 x 10"
Co(en)§+/2+ ~460(l) . 37 8 x 10'5(1)138 2.5 x10‘4
Co(phen)?/2+ 145(0.05) 22 ~40(o.1)139 ~150
Co(bpy)?/2+ 70(0.05) 22 ~ 20(o.1)139 ~80
Ru(bpy)?/2+ 1045(o.1) 1 2x 109(1)137 ~1 x 1010
Ru(N33)5py3+/2+ 75(0.l) 17 4.7 x 105(1)MS 1.5 x 106

 

141

Notes to Table 15.

 

aFormal potential of redox couple in mV vs. KSCE. Ionic strength u
given in parentheses. Data from Tables 9-ll except where otherwise

noted. For most systems, E becomes only ~5 - 10 mV more negative

f
‘with increasing u over the range u “v 0.1 to 1.0. Superscripts are

the reference numbers of the data sources.

bReaction entropy of redox couple. Data from Tables 9-ll except
where otherwise noted. Quoted values were obtained at the same

ionic strengths as E Superscripts are the reference numbers of

f.
the data sources.

cObserved second-order rate constant for the acid-independent pathway.
Corresponding ionic strength given in parentheses. Superscripts are

the reference numbers of the data sources.

dRate constant corrected for the Debye—Hfickel work (AGw)DH of forming

the collision complex from the separated reactants (eqn (8)). Quoted
corr obs W 9

values are obtained from 1n k11 ln k11 + (AG )DH/RT. The a

values used in eqn (8) were estimated from the sum of the radii of

the reactants. Estimated rate constants are given in parentheses

(see text).

eEstimated value.

142

 

 

Ru(NH3)Spy

Table 16. Kinetic Parameters of Some Self-Exchange Reactions at 25°C.
a b c d
corr * * *
kll (AGll)corr (AHll)corr -(Asll)corr
- Redox Couple Mtlsectl kcal.mol:1 kcal.mol-.-1 e.u.
c63+"2+ 8 13.5 10.2 11
as .
Fe3+l2+ 15 13.1 8.6 15
aq
Ru3+l2+ 200 11.6 ..7.08 —~153
aq
v34”+ 0.03 16.8 ~12.5 15
89
Buzz/2+ (4 x 10'4) 19.3e ~15.og .1158
Griz/2+ (2 x 10’6) 22.56 ..18.03 ~15g
szzlz+ («v0.1) ~15.5e ~11g '2153
U4+/3+ (0.5) 15.1e A.9.o .~2oh
aq
Npgz/3+‘ (1.0.02) «4.17.0f evllh .Nzoh
Ru(NH3)g+/2+ 7~5 x 104 8.3 3.2 17
Ru(en)3+/2+ ~5 x 104 8.3 3.2 17
Co(en)g+/2+ 2.5 x 10"4 19.6 13.4 22
Co(phen)§+/2+ ~.1so 11.7 4.5 24
Co(bpy)§+/2+ e-8o 12.1 7.0 17
Ru(bpy)§+/z+ tel x 1010 ~1 A.li .VOi
3+/2+ 1.5 x 106 6.3 2.4 13

 

143

Notes to Table 16.

8Rate constant corrected for (ACW)DH. values taken from Table 15.
For further details see notes to Table 15.

bFree energy of activation corrected for Debye-Hfickel work term.

values calculated from kiirr by using eqn (4) with
Z a’6 X 1010M 1s 1 (152). (Eqn (4) expresses a relationship

between k0 ”b and (AG*) p’ but the equation can be used with equal

validity for kF°rr and (AG*) .)
corr

cEnthalpy of activation corrected for Debye—Huckel work term. values
obtained from experimental activation enthalpies by using eqn (13)
‘W
and (AH11)corr-(AH11)aPp-(AG )DH (see text). Except where

otherwise noted, the experimental enthalpies were obtained from

the same data source as used for the corresponding kiisin

Table 15.

dEntropy of activation, obtained from experimental activation
entropies 68* by adding 10 e.u. (eqn (12)). Experimental values
are from the same data sources as the corresponding kiis values
listed in Table 15 (unless otherwise noted).

eEstimated from electrochemical exchange rate data (see text).

fEstimated from the kinetics of the Ru(en)§+ - szz reaction

(Table 19) by using eqn (7) (see text).

8(AS *1) assumed to be -15 e.u. (AH was calculated from

corr ll)corr
(AG

ll)corr on the basis of this assumption (see text).

h

(Asll)corr assumed to be -20 e.u. (AH was calculated from

ll)corr
(AG* 1corr1) on the basis of this assumption (see text).

1Estimated value (134).

144

apparently available for the aquo couples, Buzz/2+, Griz/2+ .Yb3z/2+
4+l3+ 4+/3+
an fld Np aq . However, self-consistent estimates of (A611)corr

for these reactions were obtained as follows. The electrochemical
exchange rates for these and other aquo redox couples have been deter-
mined at the mercury-aqueous interface and have been found to yield a
consistent correlation with the rates of corresponding homogeneous
self-exchange and cross reactions (94,154). This can be expressed as
(AG*):°rr - o.47(6c;*):‘or - o. 5 kcal (94), where (AG*):°rra and
(AG*)2orr are the corresponding electrochemical and homogeneous free

energies of activation, respectively, that have been corrected for the

apprOpriate electrostatic work terms. The listed values of (AG

3+/ 2+ Cr3+/2+ Yb3+]2+ and U4+/3+
aq aq aq 89

A justification for employing this method is that the resulting esti-

ll)corr

for Eua were obtained in this manner.

mates of (AC are consistently within ca. 0.5 kcal of the values

ll)corr

estimated (by use of eqn (6)) from the kinetics of cross reactions

having small driving forces (see below). The listed value of (Acll)corr
for Np2+l3+ was determined from the kinetics of the Ru(en)3+ 'NP3:

reaction (Tables 19 and 20) by using the latter method, since the
electrochemical exchange rate for this couple is not available. Dis-
cussion of the activation enthalpies and entrOpies will be reserved
for a more appropriate time (i.e., in section 2).

Tables 17 and 18 list the available kinetic parameters for the
acid-independent pathways of a number of cross reactions between the
various aquo couples listed in Tables 15 and 16. values of the

"Debye-Hflckel corrected" free energies of activation (AG were

12)corr

145

Table 17.
Between Aquo Cations at 25°C.

Driving Forces and Rate Constants for Cross Reactions

 

 

C C
a i 6:38 his"
Oxidant Reductant kcal mol-1 Mulsec"1 MZ-lsec-1
C03+ Fe2+ 27.2 50(1)155 150
c63+ v2+ 49.5 9 x 105(3)156 2 x 106
Cb3+ Cr2+ 53.9 1.3 3 104(3)156 2 x 104
C03+ 03+ 58.9 1.1 x 106(2)1448 3 x 106
Fe3+ Ru2+ 11.9 2.3 x 103(1)132 7 x 103
Fe3+ v2+ 22.5 1.8 x 104(1)144b 6 x 104
Fe3+ Eu2+ 25.9 7 x 103(1)”7 2 x 104
Fe3+ Cr2+ 26.7 2.3 x 103(1)158 7 x 103
Fe3+ U3+ 31.7 4 x 105(2)1448 8 x 105
Ru3+' v2+ 10.6 2.8 x 102(1)132 9 x 102
Np4+ v2+ 9.1 1.3(1)159 4
v3+ Eu2+ 3.5 9 x 10‘3(2)16o 2 x 10'
v3+ U3+’ 9.2 85(2)161 250
Eu3+ Cr2+ 0.8 .12 x 10‘5(0.5)160 8 x 10‘
cr3+ 03+ 5.0 6.2 x 10'2(2)144° 0.15

 

146

Notes to Table 17.

 

3Free energy driving force for cross reaction, determined from the

formal potentials listed in Table 15 by using AG;2 3 IKE:x - EEed)
where Box and Ered are the formal potentials of the couples under-

f f
going oxidation and reduction, respectively.

bObserved second-order rate constants for acid-independent pathway.
Corresponding ionic strength given in parentheses. Reference

numbers of the data sources appear as superscripts.

cRate constant corrected for the Debye-Hflckel work of forming the

collision complex from the separated reactants. See note (d) to
Table 15.

147

 

 

 

n.0I H.o w.nH o.m +m= +muu

N.OI o n.0NII w.o +~uo +m=m

o m.o e.HH N.o +mD +m>

A.o o o.~H m.m +N=m +m>

A.H m.o m.nH H.a +~> +~ez

m.H m.o A.OH c.0H +N> +mnm

m.o m.w m.e A.o A.Hm +m: +mom

¢.m H.m m.~ m.a A.o~ +~uo +mom

w.» o.m o.~ w.w ¢.m~ +Nom +mom

N.w m.c H.N N.w m.NN +~> +mom

H.m 5.0 m.a a.HH +~=m +mom

H.¢ H.HN N.mH ¢.m a.wm +m= +moo

0.5 o.AH H.0H 0.x a.mm +~uo +moo

m.s e.mH H.oH H.o m.me +N> +mou

N.¢H e.uH m.m A.HH ~.AN +~om +moo
HIHoa Hoax HIHoE Hoax HIHoa Hoax HIHoE Hoax HIHoa Hoax odouoswom udowao

u “be wNmusen.o u Nme<.em.o neweuANmosv stusu

.oenN us o:0Huoo osu< sooauom odOHuuoum ooouu now noouom de>Hua was doHuc>Huo< mo monuocm ovum .mH oHooe

148

Notes to Table 18.

3Free energy driving force for cross reaction, taken from Table 17.

bFree energy of activation, calculated from kigrr

.using eqn (4). See note (b) to Table 16.

(Table 17) by

coo determined from.eqn (6a) by using the (AG* values for the

ll)corr W
apprOpriate self-exchange reactions (Table 16) and ACS - 0.

(Table 16 contains (Acll)corr values but these can be used as
* * n II II II
(AG 11)corr or (A022)corr in eqn (6a) since the 11 and 22

subscripts only serve to distinguish one of the constituent self-

exchange reactions from the other.)

dCalculated from eqn (6) by inserting the appropriate values of
* * *
(AG ll)corr and (A622)corr (Table 16), and (A612)corr’ and

AGi2 (this table).
eApparent work required to form the collision complex from the
separated reactants; determined from the listed so and a values by
using eqn (9) (see text).

149

were again obtained from.the experimental rate constant kiss by using

eqns (4), (7), and (8). Tables 19 and 20 summarize the corresponding
data for cross reactions involving the non-aqua redox couples that
are listed in Tables 15 and 16. Also included in Tables 17-20 are

the free energies of reaction AG;2 for the cross reactions. These

latter quantities were determined from the difference in the formal

potentials E at 25°C that are given in Table 15 for the appropriate

f

pairs of redox couples. These values of Ef were obtained by means

of cyclic voltammetry (see Chapter IV) using suitably non-complexing
media at ionic strengths that are comparable to those employed for

the corresponding kinetic measurements. [Although some values of Ef

were determined at ionic strengths different from those employed for
the kinetic studies, this is of little consequence since the variation

of Ef with u is typically small (f;10 mV) in the range of ionic

strengths (u5'0.l-l) used for the kinetic measurements. Furthermore,

these differences will tend to cancel when the pair of Ef values used

to calculate A612 are determined at similar ionic strengths.]

The experimental values of (A612)corr for each cross reaction

given in Tables 18 and 20 were then compared with the predictions of

eqns (6) and (6a) using the following approach. The quantity 0.5010;2

was calculated for each cross reaction by inserting into eqn (6) the

(AG AG°

*
ll)corr’ (A622)corr’ and 12
are listed in Tables 16, 18, and 20. An analogous quantity, which

appropriate values of (Asz) that

corr’

o *
. will be designated 0.5oOA612, was determined from (Acll)corr’

(AG* , and AG;2 by using eqn (6a) with the assumption that

22)corr

150

 

 

Table 19. Driving Forces and Rate Constants for Cross Reactions
Involving Nan-Aqua Cations at 25°C.

a obs b . corr c

‘Aclz k12 1‘12

Oxidant Reductant kcal mol-1 M-J'sec_1 Mulsec"l
Ru(NH3):+ vi: 6.8 1.5 x 103(0.48)162 ~2 x 10"
Ru(NH3)2+ 3.1:: 10.3 2.3 x 103(1)163 7 x 103
Ru(NH3):+ or: 11.1 2 x 102(0.2)16“ 2 x 103
1:160:33)?“ vb: 28.7 4.5 x 107(1)163 1.5 x 108
Ru(NH3) 2+ Np2: -2 .3 0.3(1)119 ~ 2
Ru(NH3)2+ 0:: 16.0 1.5 x 105(1)165 9 x 105
Ru(en)§+ NPZ: 0.5 8.5(1)119 40
Ru(en)§+ (I: 18.8 8.4 x 105(1)165 4 x 106
Ru(NH3)5py3+ a: 12.7 3 x 105ml“ 1 x 106
Ru(NH3)5py3+ Hui: 16.1 5.4 x 104(1)134 1.5 X 105
Ru(NH3)5py3+ Ru(NH3)§+ 5.9 1.4 x 106mm“ 4 x 106
Fez: Ru(NH3)§+ 15.7 3.5 * 105(o.1)166 ~6 x 106
R1122 Ru(NH3)§+ 3.8 1.4 x 104(1)132 4 x 10"
Fez: Ru(en)§+ 12.9 1.4 x 105(0.1)166 ~3 x 106
Fez: M(NH3)5py2+ 9.8 5.8 x 104(1)”5 1.7 x 105
Co(phen)§+ Rummy? 7.5 1.5 x 10“(0.l)13" ~1 x 105
Co(bpy)g+ “(M3)? 5.8 1.1 x 10"(0.1)134 ~1 x 105
Co(phen)§+ Ru(NH3)5py2+ 1.6 2 x 103mm“ 6 x 103
Co(en);+ a: 0.3 7 x 10“'(1)167 2.5 x 10‘3
Co(en)3+ Eu: 3.8 4.5 x 10"3(1)168 2 x 10"2

 

151

 

 

Table 19. (cont.)
' a obs b corr c
‘A612 1‘12 k12
-1 -1 -1 -1 -1
Oxidant Reductant kcal mol M sec M sec
Co(en)§+ Griz 4.6 3 X 104(1)167 1 X 10-3
06(en)§+ a: 9.6 0.13(0.2)169 2.5
Co(en)§+ 1b: 22.2 4.5 x 102(0.18)170 ~1 x 10“
Co(phen)§+ a: 14.3 4 x 103ml“ ~1 x 10“
Co(bpy)§+ a: 12.6 1.1 x 103(2)171 2 x lo3
bumpy);+ Pei: 12.4 7.2 x 105(1)172 2 x 106

 

aFree energy driving force for cross reaction, determined from the E

ox red

values listed in Table 15 by using AG° - F(E - E
to Table 17.

12 f f

f
See note (a)

bObserved second-order rate constants for acid-independent pathway.

Corresponding ionic strength given in parentheses.

denote the reference numbers of the data sources.

Superscripts

cRate constant corrected for the Debye—Hfickel work of forming the

collision complex from the separated reactants.

Table 15.

See note (d) to

152

 

 

m.o N.o e.m m.m +wAmmzvam wmsm
m.m N.N e.A m.m A.nA +wAmmzvaa wmue
.3 me An an +..N.Ammzvsa +maamAmmzvam
6.6 m.N m.H b.A H.6A ”was +maamAm=zvsa
m.A m.o n.e A.NA WM> +mAanAmmzvam

on m
m.m e.m N.A A.m m.mA +m= +mAsuVam
H.o o m.NN m.o ”maz +MAsuvaa
an TN .3 es 0...: ”a +mAmmzvaa
o o m.eA m.NI Wmez +MAm=zvam
e.m o.s «.8 m.m A.mN Wms» +mAmmzvsa
m.o m.o N.¢A N.AA ”was +mAmmzvaa
as To ma 3: was +MANAA53A
a.c- 8.0 as An... ”MA, Mummies
HIHos Hoax HIHoE Hoax HIHoE Hoax HIHoa Houx HIHoE Hoax unmanned“ udoono

u “be eNmosem.o uNmu<.um.o swuouANmuav u Nmusu

 

.oomN on usofiuuo

doodlaoz maH>Ho>oH odOHuodom wmouo How wounom wdH>Hun was :OHuo>Huu< mo monuodm ovum .ow oHooH

153

 

 

mAAanvam
comm +m
+N m Assess
e.uH vo> +mA
. +N m nose
0 . so:
. H c «H oo> +mA
a A N.oA +N mAsmvou
- N.m . m.¢H one» +m
o.m n o .o +N mAcovoo
o.N . N N.NN vs +m
m o .m _+m: mAcmVoo
H o N o N 0 o $ UQHU +m
N.N . c.q swam +m
n.m m o m.wH +N mAsovoo
o.H .o m.m vo> +m
H.o H ¢.hH m.o +N MAao£avoo
m m 3a +
. I o .mH an A mzv mAhoavoo
m . +N
N o o o H oAmmzvsm +m
. m.m . +N mAconavou
N o o w n oAmmzvnm +m as m
0
Oh +N
.H m .n +m
. e.A . +N +m
o o q.o w a onovam
. .A + I so
A A s N.NA debug
w.N o o o.e ox uamuuswom
m
o.H Hos H
H-
w.H HOE Hoax NH <I
@ 0 Q . H031 Hagu— HI m 00
a... A... A- bases
as Anus A- NAu<.sm.o e
ans 6 NAossm.o u .
O
o o
3
o

 

ma
A.u:oov .oN oHn

154

NOtes to Table 20.

Free energy driving force for cross reaction, taken from Table 19.

corr

bFree energy of activation, calculated from k12 (Table 19) by

C

d

e

using eqn (4). See note (b) to Table 16.

s
so calculated by using eqn (6a) with the (AG11)corr values for the

appropriate self-exchange reactions (Table 16) and by assuming that

AG: - 0. See note (b) to Table 18.

Calculated from eqn (6) by inserting the appropriate values of

s * *
(A011)corr and (AGZZ)corr (from Table 16), and (AG1
A612 (from this table).

2)corr’ and

Apparent work required to form the collision complex from the
separated reactants; determined from the listed so and a values by

using eqn (9) (see text).

155

ACw
s

- 0. Regarded in terms of the constituent activation free
energies, 0. SaAGi2 contains contributions from self-exchange and cross
reactions while 0. So oAG12 contains only kinetic parameters from self-
exchange reactions. The resulting values of 0. 50IAG12 and 0.5010AGi2
are listed in adjacent columns in Tables 18 and 20. From eqns (6)

and (6a), it is seen that the difference between these two quantities,

O. 5AG° 2°(a—o ), equals the difference between the experimental values

)calc

12 that are calculated from
corr

of (A612)corr and the values of (AG*
the self—exchange and driving force parameters with eqns (6) and (6a)

and by assuming that AC:

- 0. For most cross reactions with small
enough driving forces so that the "quadratic term" 0. 5o oAG;2 is small
(5.40.5kcal mol-1), inspection of Tables 18 and 20 reveals that
0.50A8i2 ~0. 5a oAciz, at least within the expected accuracy of the
activation free energies used (probably i0.5-l kcal mol-l). In other
as (6632)

obeyed, in accordance with earlier conclusions (60,134,140).

calc

words, (AGIZ) corr

corr so that eqn (1) will be approximately
(Admittedly, for a few of these reactions, eqn (1) was employed to
obtain one or both of the self-exchange parameters used in Tables 15
and 16.) This agreement is expected since the relative values of

0. SaAGi2 and 0. So oAGi2 will be very insensitive to the magnitude of

 

ACw when the magnitude of A612 is small. However, for cross reactions
with larger absolute values of A612, it is seen that 0. So oAc12<0. SaAGi2
[i e (AG )calc < (AG* ) ] as ected if AG: >0 in e n (6a)
’ " 12 corr 12 corr ’ exp q
since
AG°
W 12 1. 1
AGs "' 8 (67"? (9)

156

Estimates of AC: obtained from eqn (9) for reactions with
suitably large values of the quadratic driving force term 0. 5a oAG12
(:>l kcal mol-1) are also given in Tables 18 and 20. For most

reactions involving two aquo reactants AG: ~v6-9 kcal mol 1

(Table 18). For reactions involving only one aquo reactant,

AC: ~v4-6 kcal mol“1 (Table 20).

2. Entropies and Enthalpies of Activation

 

In view of the marked differences between the experimental free
energies of activation and those calculated from the conventional
form of the Marcus cross-relation as expressed in eqns (1) or (2),
it is of particular interest to compare the behavior of the constit-
uent enthalpies and entropies of activation with the predictions of
the Marcus model.

Expressions similar to eqn (2) can be written for the corres-

ponding entropies and enthalpies of activation (134,147):

Asz 0.5(AS*1+ AS* 2)(1-40I2 ) + 0. 5(l+2a)ASh (10)

011:2 0.5(AH*1+ AH*2) (1-40:2 ) + 0. S(1+2<:I)AH12 (11)
where a is defined by eqn (2a), Asz and Ale are the activation
entropy and enthalpy for the cross reaction, A811 and A832 are the

activation entropies for the constituent self-exchange reactions,

AH* and AH;2 are the activation enthalpies for these self-exchange

11
reactions, and ASi2 and AH;2 are the entrapic and enthalpic driving

forces for the cross reaction. Since it is conventional to compute

157

entropies and enthalpies of activation from rate data by using the
pre-exponential factor kBT/h rather than 2, the quantities AS* and AH*
which appear in eqns (10) and (11) are related to the former quanti-

ties, AS* and AH*, by (134,147)

AS* 3 AS? - R ln(hZ/kBT) + 0.5R (12)

AH* - AH¥ + 0.5RT (l3)

Eqns (10) and (11) can be written in a form compatible with
eqn (6) by noting that AS* and AH* are reorganizational parameters
that may differ from the "Debye-Hfickel corrected" entropies and

* *
enthalpies of activation, (AS )corr and (AH )corr’

W W * . * W
A88 and Ana, respectively, such that (AS )corr A5 + A38 and

W' W W
* . * .
(AH )co AH + AHS. (Thus S8 and H8 are the entropic and

by the work terms

enthalpic components of S2, the solvent-related work term described
in the previous section.) From the preceding expressions, one can

write

W W 2
* - a * * - -
(Aslz)corr ASs o'5[(Asll)corr + (A822)corr 21583“1 4a )
+ 0.5(1+2a)ASi2 (14)
W W 2
* _ u * * -
(AH12)corr AHs 0'5 [(Anll)corr + (AH22)corr ZAHs](1.4a )
+ O.5(1+2a)AHi2 (15)

W

where a is given by eqn (6a) and -AGs - -AH: + TASZ. Since relatively

large values of AS: were required in order to fit the experimental

158

free energies of activation with eqn (6), it is of interest to compare
the predictions of eqns (14) and (15) with the experimental entropies

and enthalpies of activation, respectively, in order to ascertain if

the major contribution to AS: arises from.AS: or AHZ.

Tables 21 and 22 list the cross reactions considered in Tables

17-20 for which activation parameters have been determined. The

listed values of (AH were obtained from the experimental

12)corr

quantities AH* by using eqn (13) to yield (AHi'z)app and then by

assuming that (AHfz) -(AH (ASW)DH. This presumes that

corr 12)app

the coulombic work of forming the collision complexes from the sepa-
rated reactants is entirely of enthalpic origin at the high ionic
strengths encountered. The support for this assertion is that the
dependence of the rate constant upon ionic strength for a number of
outer-sphere redox reactions has been found to be due almost entirely
to the enthalpic term (81,144b). The listed values of (AS*)corr were
therefore obtained directly from the conventional experimental
quantities AS$ by using eqn (12) (which simply involved the addition
of 10 e.u. to AS*).

and (AS*1) for the self-

Activation parameters (A311)corr corr

exchange reactions of interest were given in Table 16. Since some of

these quantities have not been directly determined, it was necessary

to estimate them from the listed values of (AG11)corr° It was assumed
3+/ 2+ 3+/ 2+ 3+/ 2+ 3+/ 2+
that (Asll)corr- -15 e.u. for Rua aq Euaq Cra md Yba aq

since this value has been observed for both FeBZI/2+8 md Viz/2+ self-

4+/3+ 4+/3+

exchange. For Ua aq and Npa aq

exchange, (AS was taken as

11)corr

159

Table 21. Thermodynamic Parameters for Some Cross Reactions at 25°C.

 

 

o a o b o c
”AG12 ‘AH12 AS12
Oxidant Reductant kcal mol- kcal mo1- e.u.
3* Fe2+ 27.2 27 o
aq aq
3* c:2+ 53.9 55.5 -5
aq aq
3* + 58.9 60 -3
aq aq
rb3+ 2* 11.9 9.8 7
aq aq
Fe3* + 22.5 20.7 6
aq aq
Fe3+ Eu2+' 25.9 27.4 -5
aq aq
Fe3+ 3* 31.7 33.2 -5
aq 89
4+ +
Npaq aq 9.1 4.6 15
Eu2+ 3.5 6.8 -11
aq aq
+ 3+
3+ +
Ru(NH3)6 aq 6.8 12.5 -19
3+ 3+
Ru(NH3)6 Npaq -2.3 7.9 -34
3+ +
Ru(NH3)6 aq 16.0 25.0 -30
3+ 3+
Ru(en)aq Npaq 0.5 12.2 -39
3+ 4-
Ru(en)3 aq 18.8 29.3 -35
3+ +
Ru(NH3)5py aq 12.7 19.0 -21
Ru(NH ) pya+ Euz+ 16 1 25 7 -32
3 5 aq ' °

 

Table 21. (cont.)

160

 

 

 

o a o b o c
”A612 'AH12 AS12
Oxidant Reductant kcal mol-1 kcal mol-1 e.u.
3+ 2+
Feaq Ru(NH3)6 15.7 8.2 25
Fe3+ Ru(en)2+ 12.9 3.9 30
aq 3
3+ 2+
Feaq Ru(NH3)5py 9.8 1.7 27
3+ 2+
Co(phen) Ru(NH ) 7.5 6.3 4
3 3 6
3+ 2+
Co(phen)3 Ru(NH3)Spy 1.6 -0.2 6
Co(en)3+ Yb2+ 22.2 25.5 -11
3 aq
3+ +
Co(phen)3 aq 14.3 18.8 -15
3+ +
Co(bpy)3 aq 12.6 17.1 -15
3+ 2+
Ru(bpy)3 Feaq 12.4 25.3 -43
8Free energy driving force for the indicated cross reaction. Values

b

C

taken from Tables 17-20.

Entropy driving force for cross reaction, determined from the listed

AG° and AS°

O - O O
12 12 values by using Ale A612 + TAS

12'

Entropy driving force for cross reaction, determined from.the
reaction entropies for the appropriate redox couples (Table 15) by
. _ 0 red _ o ox . red 0 ox
using A812 (Asrc) (Asrc) , where (Asrc) and (Asrc)
the reaction entropies for the couples undergoing reduction and

are

oxidation, respect ively.

161

 

 

0N- N.N.. Ha NNI 9H- 36 Hz. cammzvé
+m= +N
vs 0 m
I o o I. o c an
mm o N N NH mm o N m 3H +m z +NH :55
. . . . S o N
«N N N .c a NN N o N N +N> +NA 56%
cm um
NN- H.m Ac.HH N I «a .H.HH in: +N>
vm cm
ON- «.3 NSH oN- o.HH oNH +N=m +N>
O O 0 O U“ V”
H. m N N NH .HHI o a o S +N> +3.5
MHI N. I . I . . was com
H N N NH . N H N 2.. +N in .H
«HI 0 . H . NH- o . N . Sam H5...
N m N N N +N 2H
HHI o.N .m oHI . N. vm> vmwm
N .c N N N in
I o o I. 0 cm dag HUGO
NH 0 N H N OH .H e n N +N .H
+ o .l o o a.“ V“
H N o o 0H N H N m +m= +N8
O l .0... o + .G . UNHU UQOU
N H H N N o N +N +N
a I .o .m I o. . H comm coco
N N NH N N H +N +N
.=.o IHoE Hoax HIHoa Hoax .=.o HIHoaN Hoax HIHoaN Hoax uamuosvom uauvon
on? NH uuoo uuoo uuoo
UHQUAN« mm<v «UHooAf Ev oHouAa «ocv 0 AN mm<v AN mm<v AN muqv

u

omonh nuH3 UomN um mocHahouoa m:0Huomom mmouo you muouoamumm :OHum>Huu< mo comHHomaoo

.huooza macaw: aoum voumHaono
.NN «Hana

162

 

 

| O ' O n' O 'I I %Q
NH N N N N NN N o H N +NmN +NH Nvam
an m
| o 0 II o o . ha
HN o N m m MN H m N CH +N> +mA avoo
co m
| o o | o o A“

«N m o o m mH c m N m +N> +mAamn you
- . . - . - . an N Nisms
oN m N N m Nm q o N m +Nn +m oo
.I. o o .l o o ham m m AH
NH N N N N N o N N N +N A szaa +NH=0N sou
II o 0 II o 0 Q m m Q
NH N H N N NH N N N N +NA szzm +NAam; Voo
I O o I o 0 kg“ m .0“

m m q e m 0H e c o n +N A mzvsm +mmm
I . . I . . mAcmv woo
w c e N m NH H N o o +N 2m +m m

0 o O o m a“
o I m N m ¢ NHI o N m m A mzvnm +mmm
vm n m
I 0 II o | o II 0 %Q
«N H H o 0 AN < o o A +N=m +m A mzvam
um n m
I . I . I . I . ma
HN H o H 0 MN ¢ 0 m o +N> +m A mzvam
oNI o.MI N.q HNI m.0I N.m 00: mAcovsm
+m +m
QMI o.H «.NH oeI m.o m.NH amnz mAcovam
+m +m
.=.o HIHoaNH Hoax HIHoaN Hoax .=.m HIHoEN Hoax HIHoEN Hoax acouoocom ucmuon
Huoo NH uuoo uuoo
HUHQUAN¥ Mm<v OUHGUA§ EV fiUHflUAN“ HUQV U A {WQV AN Hmdv AN HU<V

 

A.uaoov .NN «HNNN

163

Notes to Table 22.

a"Debye-Hiickel corrected" free energy of activation, taken from
Tables 18 and 20.

b"Debye-Hfickel corrected" enthalpy of activation, obtained from

experimental activation enthalpies AH* by using eqn (13) to obtain
‘W
* * I: * -
(A1112)app and then assuming (Afilz)corr (A1112)app (AG )DH
(see text). Data sources are the same as those given in
Tables 17 and 19.

cEntropy of activation, obtained for experimental activation entrOpies
by using eqn (12). See Tables 17 and 19 for data sources.

dFree energy of activation for cross reaction, calculated by using

* *
eqn (6) with the (Acll)corr and (AG22)corr values from Table 16
and the AG;2 values from Table 21, along with the value of a deter-

mined from eqn (6a) by assuming that AG: - O.

eEnthalpy of activation for cross reaction, determined from the
activation enthalpies of the corresponding self-exchange reactions
given in Table 16 by using eqn (15), the assumption that AH: - 0

(see text).

fEntrapy of activation for cross reaction, calculated from eqn (14)
by using the activation entropies for the corresponding self-exchange
reactions listed in Table 16, by assuming that ASw

s
culating a from eqn (6a) with the assumption that AC: 8 0 (see text).

- O, and by cal-

164

-20 e.u., since these values are more compatible with cross-reaction
data (see below). Although these estimates are admittedly somewhat
arbitrary, the actual, experimentally determined values of (Asfl)corr

for such small cationic reactants tend to fall within the narrow range

between ca. -15 to -20 e.u. (see Table 16). Such values are also

suggested by the experimental values of (A812)corr for cross reactions
3+] 2+ *
with small driving forces (see below). For Ru(bpy)3 , (Asll)corr

was assumed to be ~Oe.u. (134).
In order to compare the experimental parameters for cross reactions
with the predictions of eqns (14) and (15), values of the entropic and

enthalpic driving forces, AS° and AHiz, are required. The former can

12
red

be obtained from the expression: ASE2 - (As;c)

ox
) where

_ o
(Asrc

)red )ox

(As;c and (A3;c are the reaction entropies of the couples
undergoing reduction and oxidation, respectively. The A3;c values
were taken from Table 15 and were determined at ionic strengths com-

parable to those used for the kinetic measurements. The AS° values

12
are listed in Table 21, along with the corresponding values of AH;2
which were obtained from the relation: AH12 - AG;2 + TASi2 with

T - 298°K.
For each cross reaction in Table 22, the experimental activation

parameters (A612>corr’ (AH12)corr’ and (Asz) are listed alongside

)calc
2 w-o

C01“!

, (AHi2)ca1c, and (As: calc. The

*
the calculated parameters (AG1 w-o 2)w-o

calculated values were obtained by inserting the appropriate self-
exchange parameters from.Table 16 into eqns (6), (15), and (14),
setting As: and AH: equal to zero, and determining a from eqn (6a) by

assuming that AG: - O.

165

Inspection of Tables 21 and 22 reveals that when the cross
reactions have small values of A812 and A312’ good agreement occurs

*
between the experimental quantities, (A512)corr and (AHlZ)corr’ and

)calca )calc

the corresponding calculated parameters (AS* W2 nd (AH"‘1W_2

Naturally the small AH° and A3;2 values for these reactions yield

12

small A612 values, and the good agreement between the calculated and

experimental entropies and enthalpies dictates a similar relation

between (AG 12)corr and (AG*12)calc. As the magnitude of the free

energy driving force increases, the progressively smaller values of
calc *

(AG12)w-o that are seen relative to (Aclz)corr are typically

reflected in values of (A1112)calc that are markedly smaller than

calc
(Ale)corr and values of (A812)m that are less negative than
(A812)corr'

It remains to be seen if the difference between experimental and

calculated activation parameter can be traced to an enthalpic or

'W

entropic work term. The determination of AH8 and AS: is more dif-

ficult than that of AS: due to the greater complexity of eqns (14)
and (15) relative to eqn (6). However, it is possible to fit the
predictions of eqns (14) and (15) with the experimental quantities

(Asz) and (AHfz) by trying various values of A32, AHZ, and

(301'! 601'!

ASS. This procedure will be effective only in those cases where the
work terms have a significant effect on the values of the calculated
activation parameters. Inspection of eqns (l4) and (15) shows that

this will be true when AG°

12
significantly greater than zero (eqn (6a)). Table 23 lists the

is sufficiently large so that a is

 

166

 

QHI W o H U“ n m
NN- N.N N.N NN- N.N- N.N +N=N +NNN A mzvam N
m” m- E H..- . .. . z. m
N o N N +N: +NHcaN=N N
NH- NHN . . . Na N N
NN- H H- H N NN- N H- N N +N= +NH mzvam N
H I m.m . . as am
NN- o o N NH- N H N.N +N: +NmN N
OHI ¢.¢ vs vs
NH- N.N N.N NH- N.N N.N +NNN +NmN N
m I m.q vs vs
NH- N.N N.N NH- N.N N.N +N> +NNN N
NN- NHHN . . . NN Nmo
NN N N o HH NH N H N N +N= +N 0 N
NN- m- H .N . . Nmuo HUNo
H N N N N N +N +N 0 N
MHH Mum N.N NH- N. .HH Nam Nmoo
N N +N N +N H
.:.o HIHoE Hoax HIHoB Hoax .=.o HIHoa Hoax HIHoa Hoax ucmuosvmm quVon
NH uuoo uuoo uuoo
wOHMUA ¢m<v oonoANm=<v vonoANmu<v o ANMm<V a ANmm<v o ANMU<V

 

.m:0Hu0Hvoum mocha: co assay sacs oHnHmnusm com oHnouusm wchsHoaH
mo uoommm .moouom wcH>Hun swung wcH>om mGOHuomom mmouu maom pom unauoeuuom coHuo>Huo< .mN oHan

167

m m
m< n u< Haas

 

 

.3 3
waHabmmw scum muHsmou moo HQBOH unu «HHga .3m<aI I M0< :OHunasmmo was wsHms an woumHaono mH uHma cows «0
o=Hw> woman one .Amwv can AeHv macs scum voaHmuno .soHuommu muons you :oHuu>Huom mo hmouuco umuMHaonow
.Auxwu oomv Mm Imv< ou MOHuaasmmo osu wcchmno was maOHuoavo seam ocu wch:
.3 kumHaoHou who? mmon> HosoH on”. mHHHHa .3m<.H.II3u< uqsu mafia—mam .3 Aug cam AmHv 250 60.3 39330
mH moo Home: uSu .voumHH mosHm> mo uHmn sumo you .cOHuomou woouu now GOHum>Huom mo hmHmnucm vmumHsonuo
.Auxou oomv msoHuomou oovqusooao: How Hoa Hmux m I moq umnu was msoHuomou oavmloacm now
a HI NH 3. uuoo uuoo
HI Hoa Hoax o I Mo< uoau m:0Hu gamma on» was HN oHan Bonn oo< .eH uHama scum AN NU<V moo AH Ho<v
onmo has Awov was Aev scum aqu vwcHouno .GOHuomou macho now :OHum>Huuo mo Amused swam wouoHsonov
.ANN oHnma soumV :OHuo>Huom mo Naouucmo
.ANN oHawa aoumv 60Huo>Huoo mo hdesuoo :vouuouuoo HoxuwmIohnon:p
.AON can mH moHnoa soumv :OHuw>Huom mo hwuoco swam :wmuoouuoo Hoxummlomnmn:m
wHI m o m m MNI a CI H o +Nmm +mA avam MH
mHI N.m an m
HN- N.N N.N NN- N.N- N.N +NNN +NNamvoN NH
.V .- Q..v m. Um
' o 0 II o o 8 m
NH N N H N NH N N o N +NH Nam +N N HH
m I NIH o m Um
MHI m.H o.m NHI o.N m.m A mzvam +mmm OH
.:.m HIHoa Hmux H Hos Hoax .=.o HIHoa Hoax HIHoa Hoax acouoavum uammeo
NH NH NH uuoo NH uuoo NH uuoo NH
NonuH «NNN muHNoHE Ev coHuoN NNN u A may N N NNV A NNN

 

N.N:ouN .NN NHNNN

168

cross reactions from Tables 21 and 22 for which this condition holds.
Two sets of activation parameters are given for each system in

Table 23. The first set contains experimental values that are taken
from Table 22. The other activation parameters in Table 23, labelled
(AG*12)calc, (AH*12)calc, md (AS*12)calc, are determined from eqns (6),
(6a), (15), and (14) by including trial estimates of the apprOpriate
work terms, AGE, ABE, and A82. For the aquo-aquo reactions, AG: was
taken to be 6 kcal mol-1, whereas for the nonaquo-aquo reactions

AG: was chosen to be 5 kcal mol-1. (The larger AG: value for the

aquo—aquo reactions is in keeping with the trends found in Tables 18

and 20.) The estimates of (AG calc which result from the inclusion

12)
of these work terms are seen to agree much more closely with (Acf2)corr

calc calc

than did the (AGf2)F

calc

values listed in Table 22. Pairs of (AH*12)

and (Asz) values are given for each reaction in Table 23. The

W

upper values were determined by assuming that AG - -TA8:, and the

lower values by assuming that AG: - AHZ. These results are also given

in graphical form in Figures 5 and 6 which are plots of (AHlZ)corr
calc calc

versus (AHfz) and of (A812)corr , respectively.

Apart from the 003+ - Fe 2+ and Co3+— Cr2+ reactions, it is seen that
aq aq aq aq

versus (Asz)

considerably better agreement between the experimental and calculated
activation parameters is obtained using the former assumption

(i.e., the closed rather than open points in Figures 5 and 6).
Essentially the same results were obtained by choosing different values
of AG: within the range of the estimates given in Tables 18 and 20.

Reasonable concordance was also obtained by assigning both entrapic

169

Figure 5. Experimental (AH; for cross

2)corr
reactions corrected for Debye—Huckel work terms

plotted against (AHiz)calc

. * calc
Key. (AH12)

Data are taken from Table 23. The labels for

was calculated from eqn (15).

each point follow the nmmbering scheme in Table 23.

The calculated activation parameters were obtained

by assuming that AGZ=m-TASZ (closed circles) and

AG: - AH: (Open circles). Reactions 1-3 involving

Congz+ were omitted for clarity. The solid
straight line has unit slope and passes through

the origin.

170

C>O

 

 

 

13

6— 06
T 4
3
E
'5
C)
x
2. 2
o
9..
*9.
I
Q

0

-2 l ' l l J l I

 

‘2 O I 2
(AHE)corr , kcal mol"

Figure 5.

4

 

171

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onu

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:9: Tone .8. tSHNHN S
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172

 

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173

and enthalpic components to AGE, but only when the predominant

contributor was assumed to be ASE.

C. Discussion

 

It was noted in the introduction that increasingly marked discrep-
ancies between the experimental rate constants of outer-sphere cross
reactions and the predictions of eqn (1) are observed as the thermo-
dynamic driving force becomes extremely large. The foregoing results
indicate that such behavior is consistent with the presence of a
significant component of the overall free energy of activation which
does not respond to changes in the driving force. A simple, albeit
not the only, explanation of such behavior is that this part of the
activation free energy is associated with the workAGw of forming the
binuclear collision complex prior to the rate-determining electron-
transfer step. This consideration results in eqns (6), (6a), (14),
and (15). However, the derived values of AGN are typically much
larger than the Debye—Hfickel estimates (AGW)DH and appear to arise
chiefly from unexpectedly large and negative values of the entropic
component ASW (ca. -15 to -25 e.u.). It is tempting to speculate
that this entropy term has the same origin as the similarly large and
negative entrOpies of activation AS* that are typically observed for
self-exchange reactions (IN -15 to -20 e.u.). Since the reorganiza-
tion entropy within the binuclear complex is theoretically predicted
(60,146) and experimentally confirmed (173) to be close to zero for
self-exchange processes, the observed negative values of AS* in these

bimolecular reactions could also reflect the unfavorable entropic

174

work required to form the binuclear complex from the separated
reactants .

It remains to consider the possible origins of such large entropic
terms. A common feature of all the reactions considered here is that
they involve small multi-charged cationic reactants. Consequently,
it has been suggested that the negative activation entrOpies are
largely the result of concentrating charge in the dielectric
medium.(l73). Furthermore, the ionic entropy of the transition state
appears to be related to its net charge (l40a,l74). Thus the high
polarizing field within the vicinity of the collision complex could
result in a substantial local ordering of solvent molecules when this
complex is formed from.the separated reactants. Such a notion is
supported by the observation that the values of the reaction entropies
for the redox couples of interest here are frequently much larger than
the values predicted from the dielectric-continuum (Born) model (see
Chapter IV).

For a number of cross reactions with extremely large driving
forces, the derived estimates of AG: are close to the overall free

energy of activation (AG: r (Tables 18 and 20). This suggests

2)cor
that the rate-determining step under these conditions is the formation
of the collision complex rather than the electron-transfer step

itself (83). In this case estimates of AG: would refer to the forma-
tion of the transition state prior to the ground-state collision

complex, so that the actual values of AG: could be substantially

smaller than these estimates.

175

A.najor assumption that was made in obtaining eqns (5) and (6)
and also eqns (l4) and (15) from eqn (3) is that the work terms for
the corresponding self-exchange and cross reactions are the same.
Although this should be approximately the case for cross reactions
between structurally similar reactants such as the aquo-aquo reactions
considered here, this assumption is less likely to be valid for the
amine-aqua cross reactions since amine ligands appear to induce less
solvent ordering in the vicinity of multi-charged cations than aquo
ligands (see Chapter IV). However, from the form of eqn (3) it is
seen that if the work terms for the cross reactions, AG:2 and AGgl,
are both equal to the mean of the analogous quantities for the self-
exchange reactions, AG:

still only appear in the quadratic driving force terms in eqns (5) and

1 andAng, then work.terms of any sort will

(6). (It should be noted that if this cancellation of work terms does
not occur, eqn (1) will prove to be significantly in error. This
failure will be found even at low driving forces as long as the work
teams are sufficiently large.) The fact that the values of AGN for
aqua-amine reactions are typically smaller than for aquo-aquo reactions
may reflect a smaller degree of solvent ordering around the amine
redox center. Such findings are consistent with the markedly larger
values of As;c observed for aquo couples (Chapter IV). These As;c
values had suggested that the aquo complexes induce an unusual degree
of solvent ordering in the trivalent oxidation state. Substantial
differences have also been observed in the degree of solvent ordering
in the transition states for heterogeneous electron transfer for

otherwise similar aquo and ammine complexes (124,175).

176

These large and negative apparent entropies of activation are also
expected if nonadiabatic mechanisms are followed; i.e., if the
probability K of electron tunneling in the transition state is small
(74,80,140b). Such an effect would indeed appear as an apparent
negative entropic work term in eqn (14) if the values of K for self-
exchange and cross reactions are comparable. This possibility has
been discussed by Chou et al (134). Nonadiabatic pathways have been
suggested to explain the especially slow electron-transfer rates of
reactions involving lanthanide and actinide redox couples since the
transferred f electrons are shielded from the surrounding environment
and therefore are expected to have greater difficulties in ,.
tunneling (80). However, cross reactions between f- and d-electron
reactants do not exhibit any obviously larger discrepancies with
eqns (6), (l4), and (15) compared to reactions involving electron
transfer between d-acceptor and donor orbitals (Tables 17-23). This
provides one argument against the occurrence of strongly nonadiabatic
pathways and suggests that cross reactions involving f-electron
couples are adiabatic or not especially nonadiabatic. However, non-
adiabatic factors may be responsible for some of the large apparent
values of AG: observed for reactions involving Co(III)/(II)
couples (131).

An extensive comparison of experimental rate constants for outer-
sphere reactions with the predictions of eqn (1) and of some experi-
mental activation parameters with the predictions of eqns (10) and

(11) has recently been given by Chou, Creutz, and Sutin (134).

177

These workers found discrepancies between the experimental rate
parameters and the Marcus predictions which were similar to those
noted in the present work. A number of possible explanations for
these discrepancies were considered by Chou et al., including non-
cancellation of work terms and nonadiabatic factors. On the basis of
a comparison of the experimental activation parameters with the values
calculated from eqns (10) and (11) using the limited As;c data then
available, Chou et a1. tentatively concluded that the unexpectedly
small rate constants for the cross reactions originate primarily from
the enthalpic rather than the entropic component of the activation
free energies (134). This apparent disparity with the present con-
clusions arises partly from the use of different values of As;c in
ref. 134 and in the present work. The As;c data employed here are
considered to be more reliable as a consequence of the experimental
precautions described in Chapter IV. However, the chief reason for
the discord arises from the hypothesis embodied in eqns (l4) and (15)
that the discrepancies between the calculated and observed activation
parameters arise entirely from work terms that are independent of the
driving force. The observation that a major portion of the differences
between a set of observed activation parameters and the values calcu-
lated from eqns (l4) and (15) lies in the enthalpic (or entropic) terms
does not necessarily imply that the factor responsible for the dif-
ference is of enthalpic (or entropic) origin. (This connection was
made in ref. 134.) Such reasoning is fallacious because a contribu-
tion to G arising from either an entrapic or enthalpic contribution

to AC: (eqn (6a)) will affect both the activation entropies (A8352)calc

178

and enthalpies (AHfZ)calc calculated from eqns (l4) and (15).

calc

calc )
2 w=o

*
Although both (A312)w-o and (AS1

typically display pronounced
differences from the experimental quantities (AHf and (ASfZ)

(see Table 22), insertion of an entropic rather than enthalpic work

2)corr corr

term usually leads to a significantly better fit between the calcu-
lated and experimental activation parameters.

Because the Marcus model assumes the barriers to electron transfer
to be harmonic, it was reasonable for Chou et al. to consider the pos-
sibility that the observed differences between the experimental results
and the Marcus predictions are due to anharmonicity of the potential
energy barriers. However, it was concluded that the expected magnitude
of this effect was insufficient to account for the observed size of
the discrepancies (134). One piece of evidence which prompts further
consideration of anharmonicity factors is obtained from some recently
2:, Buzz, and

Vi: at the mercury—aqueous interface (176). It was found that all

observed kinetic data for the electrooxidations of Cr

three reactions are accelerated with increasing anodic overpotential
(i.e., thermodynamic driving force) to a strikingly smaller extent
than predicted by the harmonic Marcus model (176). In other words,
these systems exhibit the same type of driving-force—dependent devia-
tions from theory seen for homogeneous reactions. On the other hand,
discrepancies between experiment and theory that are substantially
smaller and in the opposite direction were observed for the electro-
3+ 3+

reduction rates of Craq and Euaq at cathodic overpotentials (176,177).

The observed anodic electrochemical behavior could be due to the

179
presence of strongly anharmonic free energy surfaces, apparently
arising from the need for additional orientation of solvent molecules
in the vicinity of the dipositive aquo reactant prior to electron
transfer (176). It is interesting to note that most of the homoge-
neous cross reactions listed in Tables 17-20 that have rate constants
markedly smaller than anticipated from the Marcus model also involve
the oxidation of aquo cations.

The present results therefore suggest that the conventional
dielectric-continuum models of electron transfer provide an incomplete
description of the role of the solvent in some outer-sphere reactions.
These models presume that at high ionic strengths essentially all of
the solvent reorganization that is required for electron transfer
occurs after formation of the precursor complex. In contrast, the
present findings suggest that a sizable solvent contribution to the
overall activation free energy arises from the formation of the pre-
cursor complex from the separated reactants, although the estimates
of AG: obtained from eqns (6) and (6a) will be markedly oversized if
nonadiabatic and anharmonic factors are indeed important. Whatever
the detailed reasons for the observed behavior, it seems plausible
that specific salvation factors can play an important role in deter-
mining the kinetics of cationic redox reactions in aqueous solution.
The further acquisition of kinetic and thermodynamic data, both in
solution and at electrode surfaces for redox reactions involving
systematic variations in ligand structure and thermodynamic driving
force should provide valuable additional information on this important

question.

CHAPTER VI

HOMDGENEOUS REDOX KINETICS OF THE EUROPIUM CRYPTATES

180

A. Introduction

The electrochemical results in Chapter III demonstrated that the

3+] 2+ a

heterogeneous charge-transfer rates of Eu(2.2.l) nd

Eu(2.2.2)3+/2+ were at least two orders of magnitude greater than
that for Eu::/2+. It would be of interest to learn if similar rate

enhancements take place in the homogeneous electron-transfer reactions
of these cryptates. For a wider perspective, one can attempt to
ascertain how and why the kinetic behavior of the Eu(III)/(II) couple
is altered by encapsulation within a cryptate cavity.

These objectives may be realized by determining the reaction
mechanisms and rate constants for redox reactions between the Eu
cryptates and other one-electron reductants or oxidants. The rate
constants for the outer-sphere cross reactions can then be used in
the Marcus cross-relation to obtain the self-exchange rate constants
for the Eu cryptates. These rate parameters will provide a good index

3+/ 2+ a

of the intrinsic redox reactivity of the Eu(2.2.l) nd

Eu(2.2.2)3+/2+

couples.

Other data from Chapter III have a bearing on this procedure.
The studies of the ligation kinetics indicate that the dissociation
of the Eu cryptates is a relatively slow process. Thus, these com-
plexes will retain their structural integrity for a considerable
period of time.‘ This substitutional inertness of the Eu cryptates
will play an important role in determining which electron-transfer

mechanisms are available to the cross reactions. The values of the

cross-reaction equilibrium constants employed in the Marcus

181

182

cross-relation can be calculated by using the thermodynamic data from
Chapters III and IV.
The literature does not appear to contain any studies on the

homogeneous redox kinetics of any cryptate couple. Some work has been

3+]2+

done on the Co(sep) couple (88). The sep (sepulchrate) ligand

is an octaaza analog of (2.2.2) except that the bridgehead nitrogens

in sep are joined to the other heteroatoms by methylene linkages.

3+] 2+

The self-exchange rate constant for Co(sep) has been determined

to be s‘gfl 3-1, although the interpretation of this result has not

been straightforward (88b).

B. Results

Rate constants at 25°C were determined for the following reactions:

2+ 3+ + 2+ 3+
aq, Eu(2.2.1) + viq, Eu(2.2.l) + Co(NH3)6 ,

Eu(2.2.2)3+ + Buzz, and Eu(2.2.2)3+ + Viz. In each case, the progress

Eu(2.2.1)3+ + Eu

of the reaction was followed by periodically recording the polaro-
graphic limiting current of the cryptate reactant. Pseudo first-order
conditions were employed with the noncryptate reactant present in at
least a ten-fold excess. Linear plots of 1n 1 vs. time (where i is
the polarographic limiting current corrected for background) were ob-
tained which indicate that a first-order rate law is operative under
these conditions. Since the cryptate concentration is directly pro-
portional to i (see eqn (6) in Chapter III), the slope of the

ln 1 vs. t plot will be the same as that for a 1n [cryptate] vs. time
plot. The slope of either plot yields an experimental rate constant

kexp which describes the overall rate of disappearance of the cryptate

183

reactant. Since this rate is the sum of the rates of cryptate con-
sumption via the electron-transfer and dissociation reactions, one
must sometimes subtract out the contribution of the latter process

in order to obtain the rate constant for the redox reaction. This

is not difficult since the dissociation rate constants of the cryptates
have already been determined under a wide range of conditions

(Chapter III). The dissociation process was found to be a minor but
significant contributor to the observed rates of consmmption of
Eu(2.2.2)3+ and Eu(2.2.1)2+. 0n the other hand, the dissociation rate
of Eu(2.2.1)3+ is quite slow and has a negligible (<112) influence on

the kexp values of the reactions which involve this cryptate.

The dissociation-corrected rate constant can be determined as a
function of the concentration of the noncryptate reactant. (This last
quantity can be calculated from the appropriate diffusion-limited
polarographic current.) A first-order relation is obtained, which
enables one to conclude that the overall rate law for these cryptate

redox reactions is:
Rate - ki2[oxidant][reductant] (1)

This rate law'was followed by all reactions except the one between

Eu(2.2.l)3+ and v“.

aq The rate of this reaction decayed more rapidly

than predicted by eqn (1), and one did not find the usual decrease in
the cryptate.current when this reaction was carried out in concen-
trated perchlorate media (a~O.l-0.2 g). A slight decrease followed

by an increase in current was observed instead. This last problem

184

was eliminated by the use of PF; electrolytes, but the unexpectedly
slow reaction rates were still encountered. A method of treating
the slow kinetics of the Eu(2.2.1)3+ - Vi: reaction was obtained by

considering the following scheme:

1".
‘ 12
Eu(2.2.1)3+ + v2+ 2 Eu(2.2.1)2+ + v3+ (2)
aq k! aq
-12

Because the equilibrium constant for this reaction is small (K97) the
back reaction will have a significant effect on the observed rate.
The current-time behavior was then found to conform to a modified form

of eqn (1) which includes a correction for the back reaction:

k.
_ . 3+ + _ 12 2+ +
Rate k12[Eu(2.2.l) ][v:q] K [Eu(2.2.1) ”“in (3)

where K is the equilibrium.constant for reaction (2). The other
redox reactions have large equilibrium constants or are chemically
irreversible, so no back-reaction corrections are needed in these
cases.

The source of the problems encountered in perchlorate media is
unclear. The observed current-time behavior indicates that V2: is

being produced by some means besides the cryptate reaction. While

4

appears to be too slow to explain the results obtained. However,

C10 slowly oxidizes Vi: to V2: (160,178), the rate of this reaction
these earlier studies were performed at significantly higher acid
concentrations ([H+] “‘0.5 g) than the present study, so an inverse-

acid catalysis may be producing the faster rates found in the latter.

185

Studies of the acid dependence of k12’ the experimental cross-

reaction rate constant, revealed a relation of the form

I a + 4
R12 R12 + kH/[H ] ( )
for the Eu(2.2.1)3+ - Buzz and Eu(2.2.2)3+ - Bug: reactions. The

other reactions did not show any significant [3+] dependence for 3+
concentrations in the 1-50 mg range. (Work involving Eu(2.2.2)3+ was
limited to 1-20 ml! 11+ in order to minimize the acid-catalyzed dis-
sociation of this cryptate.) Because of the low precision of the kiz

values for Eu(2.2.l.)3+ - Euiz reaction, one could only obtain estimates

of the quantities in eqn (4) (k12 as0.2 i 0.2 gfls'l and

2+
3Q
reaction also suffered from poor reproducibility, but it was possible

to determine that klz - 1.4 i 0.3 gfls’l and kn - (1.6 t 0.4) x 10'2 3’1.

(These problems in reproducibility do not appear to be specific to these

kH 5*‘0.4 i .02 8.1). The kiz values for the Eu(2.2.2)3+ - Eu

cryptate-Eu:: reactions. Similar results have been found for other
redox reactions which involve two lanthanide ions;
e.g., Eng: + Yb: (179).)

Reactions which show an inverse-acid dependence (and other redox
reactions as well) have been known to be accelerated by the presence
of anions such as 01-, Br-, and SCN- (59, 140a, 157). The reactions
in the present study did not display any discernible rate increases in
the presence of 01-, even though chloride concentrations of up to

50-100 mg were used.

After performing the various analyses described above, one can

186

obtain values of k12’ the acid-independent cross-reaction rate constant.

2+

These are given in Table 24. The k12 value for the Eu(2.2.l)3+ - Euaq

reaction was not included on account of its poor relative precision.
The 000133):+ - Eu(2.2.l)2+ reaction was conducted at an ionic strength
of u - 0.066 due to the limited solubility of the cobalt complex in
concentrated perchlorate media. The rate constants for this reaction
were corrected to u a 0.1 by using the following expression (165):

1la
log k a log k0 + A B (5)

1+B§u 2

2A Z Z n

where kc is the value of the rate constant at u = 0, ZA and 2B are the
charges of the reactants, 5 is the internuclear separation in the pre-
cursor complex, and A and B are Debye-Hfickel parameters with values of
.512 and 3.29 X 10'.9 cm-l, respectively, at 25°C (82,180). Approxima-
tions for 3 were obtained from the sum of the radii of the reactants.
An estimate for the radius of Eu(2.2.1)3+ (5.3 A) was calculated from
the sum of the ligand thickness and the cavity radius (3) while a value
of 3.5 A for Co(NH3)z+ was obtained by assuming the radius of this com-
plex to be closely similar to that given for Ru(NH3)2+ in Chapter IV.
The values of the self-exchange rate constants for the Eu

cryptates were then calculated from the k12 values in Table 24 by

means of the Marcus cross-relation:

l/ .
k12 " (knkz2K12f12) 2 (6)

where k11 and k22 are the self-exchange rate constants for the non-

cryptate and cryptate couples, respectively, is the equilibrium

K12

187

Table 24. Rate Constants and Equilibrium Constants

for Cross Reactions Involving Europium.Cryptates

 

 

 

 

k12

_1 _1 Ionic
Oxidant Reductant Q! s ) Strength K12
Eu(2.2.1)3+ V: 0.5 t 0.1 0.10 7.0
Co(NH3)2+ Eu(2.2.l)2+ .043 t .004 0.066 -—
00011113):+ Eu(2.2.1)2+ .055 t .005 0.10 --
Eu(2.2.2)3+ V: 1.5 t 0.2 0.10 3.7 x 104
Eu(2.2.2)3+ an: 1.4 i 0.3 0.10 1.3 x 107
constant for the cross reaction and £12 is given by

2
log f12 = (log K12) (7)

4 103(k11k22/ZZ)

where Z, the bimolecular collision frequency in solution, has a value

10 gel -1

of 6 X 10 s at 25°C (see Chapter V). The k11 values used for

these calculations are given in Table 25. These values are those for

Ribs from Table 15. The k1 for V3.”2+ was corrected to u=0-1 by
1 1 aq

3+/2+

using eqn (5) with 5 - 8 A. The Euaq value was obtained from the

COII

k11

term (see note d to Table 15). The formal potentials given in

in Table 15 by subtracting the appropriate Debye-Hfickel work

Chapters III and IV were used to calculate K12 by using
3 11_ red x
x12 eprRT(Ef - 12‘; >1 (8)

188

where Bred and on are the formal potentials of the species under-

f f
going reduction and oxidation, respectively. Table 24 summarizes

these K12 values.

Table 25. Self-Exchange Rate Constants
for Selected Noncryptate Couples

 

 

R11

-1 -l
Redox Couple Q! s )
V3-I-/2+ 2 x 10-3
aq
E 3+]2+ 1 x -5
uaq 10
Ru(NH3)2+/2+ 3 x 103

 

The analysis of the Co(NHs)‘:+ - Eu(2.2.l)2+ reaction 13 more
complicated. An accurate formal potential for the CO<NH3)Z+/2+
couple is not available due to the instability of the Co(NHa):+

3+]2+ self-

complex. Fortunately, the calculation of the Eu(2.2.l)
exchange rate constant is possible if the following series of

reactions is considered:

k
12
Co(NI-IB)‘:+ + Eu(2.2.l)2+ + Products (9)
A k13
Co(NH3):+ + Ru(NH3)§+ +‘ Products (10)

Ru(NH3)2+ + Eu(2.2.1)2+ : Ru(NH3)é+ + Eu(2.2.1)3+ (11)

189

One can then write the Marcus expressions for cross reactions (9) and

(10) and obtain:

1/2 (12)

1‘12 ' (k11k22K12f12)
1
/

k13 ' (k11k33K13f13) 2 (13)

where kll’ kzz, and k33 are the self-exchange rate constants of

)3+/ 2+ 3+/ 2+ )3+/ 2+
3 6 3 6

the ratio k§zlki3 is calculated, k11 and the E

Co(N‘H , Eu(2.2.1) , and Ru(NH , respectively. When

3+/ 2+

3)6 will

f for Co(NH

cancel , yielding

2 2
k12/“13 "' K231‘22/k333 (1“):

provided that f *‘f In eqn (14), K is the equilibrium constant

12 13° 23
for reaction (11) and can be calculated from the formal potentials of

3+/ 2+ 3+]2+

Eu(2.2.l) and Ru(NH3)6 by means of eqn (8). A value of

l

k13 . 0.01 M? s.1 (corrected to u - 0.1 by using 5 - 7 A in eqn (5))

was obtained from reference 164. Thus eqn (14) can be used to compute
a value of 1:22 for Eu(2.2.l)3+/2+.
By employing the procedures described above, values of the

cryptate self-exchange rate constants could be calculated. The

results are presented in Table 26.

190

Table 26. Self-Exchange Rate Constants for

 

 

Eu(2.2.1)3+/2+ and Eu(2.2.2)3+/2+

k22

_1 _1 Cross reaction used
Redox Couple (M s ) to calculate k22
Eu(2.2.l)3+/2+ 13 Eu(2.2.l)3+ + vi:
Eu(2.2.l)3+/2+ 6.1 00mg)?” + Eu(2.2.1)2+
Eu(2.2.2)3+/2+ 3.9 x 10‘2 1211(2.2.2)-3+ + V:
Eu(2.2.2)3+/2+ 4.2 x 10'2 1211(2.2.2)3+ + an:

 

C. Discussion

 

1. Reliability of the Self-Exchange Parameters of the Eu Cryptates
Some variation is observed in the k values in Table 26. The two

22
3+]2+ agree quite well, considering the precision

values for Eu(2.2.2)
of the cross-reaction rate constants used to calculate them. In
fact, this concurrence is probably fortuitous to a certain degree.

3”“ differ from

The self-exchange rate constants listed for Eu(2.2.l)
each other by a factor of two. This is of no great moment if one notes
the uncertainties involved in the calculations. The k11 value of
Viz/2+ was determined at u - 2 (135), and its correction to u - 0.1

is probably the most error-prone of all of the ionic-strength correc-
tions performed in this work. The validity of the k22 value determined
from the data of the Co(NH3)2+ - Eu(2.2.1)2+ reaction depends on the

soundness of the assumption that was used in deriving eqn (14).

Furthermore, the predictions of the Marcus cross-reaction are not

‘ 191

absolutely accurate. It is not uncommon to find that calculated and
observed cross-reaction rate constants are only within about an order
of magnitude of each other (see Chapter V and reference 60). There-
fore, it is not surprising that the same sort of small disparities
appear when self-exchange parameters are obtained from cross-reaction

values are 10 Mfl s.1 for

3+/ 2+

data. Thus, reasonable compromise k

22
3+]2+ -1

Eu(2.2.l) and 0.044;“1 s for Eu(2.2.2)

2. ‘Mechanistic Implications of the Inverse-Acid Dependence of kiz

 

The above self-exchange parameters were calculated with the assump-
tion that the acid-independent cross-reaction rate constants corres-
ponded to outer-sphere electron transfer. At the outset, one might
think that the encapsulation of the Eu(III) or Eu(II) ion within the
cryptate cavity would rule out inner-sphere reactions by preventing
the formation of ligand bridges. However, the [Ed-r1 dependences of
the experimental cross-reaction rate constant (such as those seen for
the Eu(2.2.1)3+ - Euiz’ and Eu(2.2.2)3+ - Bug: reactions) have
usually been ascribed to the presence of an inner-sphere mechanism
which proceeds via a hydroxide bridge. Contrary to initial impres-
sions, this type of mechanism is possible for the cryptate reactions.
As noted in Chapter III, Eu(2.2.1)3+ and Eu(2.2.2)3+ associate strongly
with 08-. Thus one can conceive of a reaction intermediate in which
a hydroxide ion is shared by the trivalent cryptate and Euiz. This
is not necessarily a classical ligand-bridged intermediate. A species

of this kind would require the hydroxide to penetrate between the

strands of the cryptate and to coordinate directly to the Eu(III) ion.

192

This arrangement may aid the conduction of the electron between the
redox centers. Such an effect would be especially important if one
believes electron transfer involving f orbitals to be nonadiabatic (80).
However, as pointed out in Chapter V, the degree of nonadiabaticity
for lanthanide reactions appears to be no greater than that
for other types of reactions. In addition, the inspection of molec—
ular models shows that the formation of a direct bridge will come at
the expense of a considerable’distortion of the ligand configuration.
'This alteration in the ligand conformation could result in an increase
in the inner-shell reorganizational barrier to electron transfer.
Therefore, a direct bridge is normally a reasonable explanation for
an inverse-acid pathway, but under the present circumstances, one
finds a number of structural arguments against the formation of such
species.

The interaction between the trivalent cryptate and OH- need not
be the direct complexation discussed above. It is possible that the
hydroxide ion could displace one of the water molecules immediately
adjacent to the cryptate. If this hydroxide were also to coordinate
with the Bug: coreactant, one would obtain an "indirect" bridge. The
formation of this species would seem more facile than that of the
direct bridge since no distortion of the cryptate would be necessary.
However, the indirect bridge does not supply unbroken communication
between the redox centers. Thus, it is more difficult to see how
this arrangement would significantly alter the probability of electron

transfer in the activated complex. It could be possible that the

193

indirect bridge catalyzes the redox reactions by other means. The
presence of the hydroxide ion reduces the electrostatic repulsions
which are associated with the convergence of the two cationic reac-
tants to form the precursor complex. Because the anion bridge

lowers the overall charge of the precursor complex, one could suspect
that the change in local solvent ordering that is required to reach
the activated complex will be less than that needed in the case where

the bridge is absent.

The factors discussed above could account for the observed

2+

aq’ but there is

inverse-acid catalyses of the reactions involving Eu
no obvious reason why this effect is not found when V3: is a co-
reactant. (Of course, no [11+]"l effect is expected for the
@0133):+ - Eu(2.2.1)2+ reaction since a hydroxide bridge cannot be
formed with the substitutionally inert cobalt complex). The rate
constants for some reactions between Vi: and certain aqua-cation
oxidants are acid independent (181-183), but these reactions are

5 Ed 3-1) . The rate of electron transfer is greater

rapid (k12 ~10
than that for ligand displacement on Viz, which is a relatively
substitution-inert species. Thus, the redox reaction takes place
before a potential bridging ligand has a chance to displace one of the
aquo ligands on Viz. However, when electron transfer is slower, some
hydroxide-bridged inner-sphere reactions should be found since there
will be an opportunity for ligand substitution to occur on Viz. For
example, the szz - Vi: reaction is relatively slow (k12 - 1.3 Mrl 3.1;
see Table 17) and has been found to have a [11+]“1 term in its rate

law (159). The rate constants for the cryptate-Vi: reactions are

194

comparable to that for Mpg: - V2+ and are certainly not fast enough

aq
to rule out OH- bridges on the basis of substitution-inertness argu—
ments.

In order to formulate explanations for the lack of a detectable
inner-sphere pathway in the Viz-cryptate reactions, it is worthwhile
to consider the formation of the bridge in greater detail. The usual
description of this process begins with the deprotonation of an aquo
ligand on one of the reactants which yields a hydroxo complex:

Ka
M2: 2 M011(“'1)+ + 11+ (15)
This hydroxide then displaces a ligand on the other reactant Ny+

while remaining complexed to the original reactant:

H
M011(“'1)+ + NZ: + M0N<n+7’1)+ + H20 (16)

Thus, the hydroxide bridge is formed. The deprotonation is cus-
tomarily assumed to occur on the more highly charged reactant since
this species tends to have the larger Ra value. By using the 81
value for Eu(2.2.1)3+ - 0H- from Chapter III in conjunction with

K; - KwBl’ one can calculate that pI<.a - 8.5 for Eu(2.2.1)3+. A value
of pK.a - 2.3 has been determined for Npiz (184,185). Thus the de-
protonation of an aquo ligand on Np:: is greatly favored over that of
a water molecule immediately adjacent to Eu(2.2.1)3+. In fact,

Vi: (pK.a - 6.5 (186)) has a stronger inclination to deprotonate than
Eu(2.2.l)3+. In any case, the less favorable deprotonation equili—

libria of the reactants in the Eu(2.2.1)3+ - Vi: reaction yield

195

relatively low concentrations of potential bridge-forming species.
Therefore, the concentrations of the intermediates can determine if
an effective inner-sphere pathway exists (as for the N92: - Vi:
reaction) or if electron transfer will be restricted to outer-sphere
processes (as in Eu(2.2.l)3+ - Viz). An analogous argument would use
the pK.a value for V2: (pKa *‘2.7 (187,188)) to account for the
presence of an inverse-acid dependence in the Viz/2+ self-exchange
rate (135).

Although the 81 value for Eu(2.2.2)3+-OH- has not been determined,
it should be comparable to that for Eu(2.2.1)3+¥OH-, if the similari-
ties of the 81 values of Eu(2.2.l)3+ - F- and Eu(2.2.2)3+ - F- (see
Table 2) are indicative of a general trend. Therefore, the lack of
a [Ha-l"-1 dependence for the Eu(2.2.2)3+ - Vi: reaction could be

explained in a manner similar to that used for Eu(2.2.l)3+ - V2+.

aq

The lack of any chloride catalysis of the cryptate reaction rates
can also be ascribed to an unfavorable preceding equilibrium. It was
noted in Chapter III that neither Eu(2.2.l)3+ nor Eu(2.2.2)3+ associ-
ates strongly with Cl-. Thus, the concentrations of species such as
Eu(2.2.l)3+ - Cl- and Eu(2.2.2)3+ - Cl- will be too low to have any
appreciable effect on the observed rates. (While the rate enhance-
ments afforded by the chloride-bridged pathways may be substantial,
it is assumed that they are not so phenomenally large that they over-
come the effects of the extremely low concentrations of the

Cl--bridged intermediates.)

The above discussion clarifies some of the observed rate behavior

196

but does not shed any light on the difference between the

cryptate - V2: and the cryptate - Hui: reactions. While no Ka value

for Hui: appears to be available, it should be less than that for

Viz, if one assumes that a smaller tendency towards hydrolysis follows

from the larger radius (18) and lower charge density of Ruiz. Thus

on the basis of the deprotonation equilibria, inverse-acid terms

would be expected in the rate laws of the cryptate - Vi: reactions.
One possible clue to the presence or absence of a hydroxide-

bridged pathway could reside in the disparate reaction entropies of

E023, 2+

for Viz/2+ is one of the smallest found for any aquo couple and is an

(AS° - 48 e.u.) and V3+l2+ (AS° - 38 e.u.). The AS° value
rc aq rc rc

indication that a relatively limited solvent reorganization is neces-
sary to reach the transition state for electron transfer. (This

rearrangement may be quite substantial on an absolute scale but it

3+/ 2+

should be considerably less than that for Euaq

.) As noted above,
one of the principal reasons for inverse-acid catalysis is that

the formation of the hydroxide bridge decreases the extent of solvent
reorganization required in the activation process (i.e., the outer-
shell component of the activation free energy is smaller for the
inner-sphere reaction). However, if this outer-shell term is rela-
tively small for the outer-sphere pathway in the cryptate - Vi:
reactions (as suggested by the as;c for Vizlz+), the hydroxide-
bridged mechanism may not offer a great advantage over the outer-

sphere route. 0n the assumption that such circumstances exist, the

inverse—acid catalysis may not be detected since the rate enhancement

197

associated with the hydroxide bridge is not enough to compensate for
the unfavorable position of the preceding deprotonation equilibrium
(reaction (15)) in acidic solution.

3. The Effect of Cryptate Formation on the Intrinsic Redox
Reactivity of the Eu(III)/(II) Couple

From the self-exchange rate constants for the Eu cryptates

(k22 - 10 571 s"1 for Eu(2.2.1)3+/2+ and R22 - 0.04 gfl s‘1 for
Eu(2.2.2)3+/2+), it is obvious that the intrinsic barriers to electron

transfer in these complexes are much smaller than the corresponding

3+]2+ ( 5 -1 -1
a

quantity for Eu q kll.- 10- .E. s ). Both outer-shell and inner-

shell factors contribute to the enhanced reactivity of the Eu cryptates.

The reaction entropies for the cryptate and aquo couples (As;c - 48,

3+/ 2+ 3+]2+ 3+/ 2+

27.5, and 21 e.u. for Eu
3Q

, Eu(2.2.l) , and Eu(2.2.2)

respectively) indicate that the outer-shell component, Acgut’ of the
activation free energy A052 should be smaller for the cryptate self-
exchange reactions than for the corresponding Eui§l2+

conclusion stems from the relationship between As;c and A63“t which

process. This

was described in Chapter IV. The As;c value of a transition metal
redox couple is determined by the interactions between its ligands
and the surrounding solvent. The encapsulation of Eu(III)/(II) in

the cryptate cavity largely precludes the formation of an extensive,

hydrogen-bonded network of "frozen" solvent such as that around Eu:+.

q
This substantial solvent structuring in the vicinity of Hui: yields
3+]2+

the large AS° value for Eu
rc aq

and also gives rise to the consid-

erable solvent-reorganization barrier to electron transfer.

198

In contrast, the smaller as;c values of the cryptates demonstrate that
the differences in solvent ordering around the trivalent and divalent
states are comparatively small, as are the solvent rearrangements
needed to reach the activated complexes.

The redox reactivities of the Eu aquo and cryptate couples can
also be discussed in terms of AGin,

activation free energy. In most cases, AGin is associated with the

changes in the metal-ligand bond distances which are needed to reach

the inner-shell component of the

the activated complex. As noted in Chapter III, the ionic radii of
Eu3+ and Eu2+ are 0.95 and 1.17 A, respectively, so sizable differences
are anticipated between the M-L bond lengths in Hui: and Ruiz.

implies that large changes in the bond distances are required to go

This

from the reactants to the transition state. On the other hand, the
relatively rigid structure of the cryptate should restrict the elonga-
tion or compression which the M—L bonds can undergo. While one cannot
expect the cryptands to be totally inflexible when complexed to
Eu(III) or Eu(II), they almost certainly represent a more constrained
ligand environment than the aquo ligands around Buzz/2+. The restraint
placed on the metal-ligand bonds in the cryptates should yield smaller
asgn values for these complexes. Thus, the faster self-exchange rates
for the Eu cryptates can be traced to these systems' smaller solvent-
and ligand-reorganizational barriers to electron transfer.

The preceding arguments did not make any special attempts to

3+/ 2+ 3+/ 2+

distinguish Eu(2.2.1) from Eu(2.2.2) since the major goal

was to point out the qualitative differences between the Eu aquo and

199

cryptate couples. However, it is clear that the redox reactivities

of the two cryptates are quite dissimilar. The k value of

22
3+/ 2+
Eu(2.2.l) is over two hundred times greater than that for
Eu(2.2.2)3+/2+. The former couple has a A8;c value of 27.5 e.u.

while as;c - 21 e.u. for the latter. Therefore it does not appear
that the outer-shell reorganization is less for the (2.2.1) complex.
It remains to consider the inner-shell barriers.

The crystal structure of Eu(2.2.2)3+ has been determined (5b).
A certain degree of ligand distortion was found, as evidenced by
the nonsymmetric arrangement of the donor atoms of the ligand. (This
deformation may be an attempt by the ligand to accommodate the highly
charged and relatively small Eu3+ ion by twisting its diether strands
in a manner which increases donor-acceptor interactions.) The crystal
structure for Eu(2.2.2)2+ is not available, although one may surmise
that it is similar to the corresponding structures of Ca(2.2.2)2+ (189)

and Pb(2-2-2)2+ (58) since the ionic radii of Ca2+ 2+

2+

and especially Pb
are comparable to that of Eu The ligand distortions in the Ca and
Pb cryptates are less severe and the distributions of the donor atoms
are closer to being symmetric. Thus, it seems reasonable that the
ligand conformation in Eu(2.2.2)3+ is quite different from that in
Eu(2.2.2)2+. Since the ligand configuration in the activated complex
is intermediate to those of the trivalent and divalent cryptates, a
sizable change in the ligand structure will be included in the acti-

vation process. This also means that AG* will contain a relatively

22
large inner-shell component. The ligand distortions in the
Eu(2.2.l)3+/2+ should be less. The smaller cavity of (2.2.1) is more

200

closely matched to the radius of Eu3+, and favorable interactions
between the cation and the donor atoms could be achieved without
large deformations in the ligand structure. Furthermore, the (2.2.1)
cryptand should be more rigid than (2.2.2) since the monoether strand
in the former reduces its flexibility. Thus, the very structure of

1m<2.2.1)3+’2+

may prevent this cryptate from undergoing the kind of
conformational changes assumed for Eu(2.2.2)3+. (Unfortunately, the
relevant crystallographic data for the (2.2.1) systems are not avail-

able, so the above statements are credible but unproven at present.)

These considerations indicate that the slower self-exchange in the

Eu(2.2.2)3+/2+ system is due to a larger AGin term which arises from
the ligand reorganization needed to reach the activated complex. This

inner-shell barrier for Bu(2.2.2)3+l2+

is sufficiently large to over-
come the effect of the smaller A63“t for this couple.

The Eu cryptate results do not conform to the correlation between
AS;c and AG* which was noted in Chapter IV. The couple with the

22
smaller As;c value (Eu(2.2.2)3+/2+

) does not have the larger self-
exchange rate. Some understanding of this apparent anomaly can be

gained by considering the factors which determine the As;c and k
3+/ 2+

22

values. The smaller As;c for Eu(2.2.2) can be ascribed to the
larger radii of the trivalent and divalent forms of the couple. The
charge density and the tendency to induce local solvent ordering will
be less for a Eu(2.2.2) cryptate than for the Eu(2.2.l) complex of
the same charge. It seems reasonable that this disparity in struc-

turing effects will be greater in the trivalent state, assuming that

the square of the charge is the relevant parameter in this situation.

201

Thus, the relatively lower degree of solvent order around Eu(2.2.2)3+
'is a major contributor to the smaller As;c for the (2.2.2) couple.

The correlation between reaction entropies and free energies of
activation was observed for systems which were significantly different
from the cryptates. The redox couples in Chapter IV had much simpler

ligand structures (except for Co(82p)3+/2+

). In these complexes, the
factors which determine the solvent-reorganizational (or outer-shell)
barrier to electron transfer also influence As;c. The inner-shell
barriers associated with compressing or stretching metal-ligand bonds
will also be reflected in the reaction entropies since the expansion
or contraction of an ionic species will affect the solvent structure
in its vicinity. However, inner-shell components to the activation
energy which arise from changes in ligand conformation are less likely
to be mirrored in As;c. It is plausible that the radii of Eu(2.2.2)3+
and Eu(2.2.2)2+ are quite similar. The alterations in ligand config-
uration which accompany the interconversion of the trivalent and
divalent forms of Eu(2.2.2) will modify some of the internal dimen-
sions of the ligand, but its overall size will remain relatively un-

3+/ 2+

changed. Thus, the As;c for Eu(2.2.2) only corresponds to the

outer-shell portion of the activation free energy. The reaction

entropy lacks an analog for the large change in ligand conformation

in Eu(2.2.2)3+/2+. Therefore, the As;c - k correlation does not

22
appear to be appropriate for the Eu cryptates.

CHAPTER VII

MISCELLANEOUS EXPERIMENTS

202

A. Electrochemistry of Other Transition Metal Cryptates

The electrochemistry of several other cryptates was studied. A
sample of Cu(2.2.1)(NO3)2 was obtained from Dr. Gansow‘s group, and
cyclic voltammograms of this cryptate in neutral 0.5.b_i.NaClO4 at 25°C
had sweep-rate—independent peak separations of 60 mV. This is quite
close to the 57 mV value which is expected for electrochemically
reversible, one-electron couples (92). Thus, the encapsulation of
Cu(I) in the cryptate cavity stabilizes this oxidation state to a
point where it is easily observed in aqueous solution. In contrast,
the cyclic voltammograms of Cu(N03)2 in 0.5.M1Na0104 had peak separa-

tions which were appropriate for a two-electron couple, indicating

that Gui: is reduced directly to Cu° and that the existence of a
Cu(I) state is at most transitory under these conditions.

2+]+

The formal potential of the Cu(2.2.l) couple (Ef - -0.103 V)

was obtained from the mean of the cathodic and anodic peak potentials.
By comparing this potential with Bf 8’--0.085 V for Cu::/+
concludes that the Cu(II) state is slightly stabilized relative to

(190), one

Cu(I) by the formation of the cryptate. According to equation (1)
from Chapter III, the difference between the free energies of complexa-
tion of Cu(2.l.l)+ and Cu(z.1.1)2+ is quite small (only about
0.4 kcal/mole).

Although no detailed studies were performed, the dissociation of
Cu(2.l.l)2+ appears to be quite slow. After one month in neutral
0.5 M NaClOa at ambient tanperature ( ~23°C), about 75% of the complex

7 -1

remains intact. This result allows a value of ka “’1 X 10- s to

be estimated for the dissociation rate constant.

203

204

The techniques described in Chapter IV were used to obtain a value

2+/"' in 0.5 _1_4_ NaClOl‘.
low reaction entropy is comparable to those found for other 2+/+

2+/+ a

of as;c - 12.5 e.u. for Cu(2.l.1) This rather

redox couples. For example, As;c - 15 e.u. for Ru(NH3)5NCS nd

A88 - 10 e.u. for Ru(NH3)SClZ+I+

rc (see Chapter IV). These last two

2+l+ since the Ru

systems may not be exact analogs to the Cu(2.1.l)
couples have significantly different ligand structures compared to
the cryptate. However, at present it appears that the overall charges
of these couples are the main influences on the reaction entropies.

A number of cryptate deposition (i.e., amalgam-formation)
reactions were also investigated in aqueous media. Solutions of
Cd(2.2.1)2+, Tl(2.2.l)+, Pb(2.2.l)2+, Cd(2.2.2)2+, and 1>h(2.2.2)2+
were made by combining a ten-fold excess of the cryptand with an
appropriate salt of the metal ion in 0.1 M_Et4NClO4. The cyclic
voltammograms of each of these cryptates were obtained using a HMDE.
All of the current-voltage curves had large, sweep-rate-dependent
peak separations. For example, the peak separations for Pb(2.2.2)2+
were 530 mV and 490 mV for scan rates of 200 and 50 mV/sec, respec-
tively. The smallest peak separations (250 mV at a sweep rate of
200 mV/sec) were found for Tl(2.2.l)+. Even with these large peak
separations, it was easily seen that the voltammetric waves of the
cryptates were shifted to more negative potentials than those of the
free metal ions. These results indicate that the metal ions are
stabilized by cryptate formation, although these effects cannot be

quantified since thermodynamic information cannot be extracted from

the irreversible cyclic voltammograms.

205

The electrochemical irreversibility of these cryptates stems
from the involvement of the metallic state in the electrode reactions.
Two mechanisms are available to the electroreduction of the cryptates.
The so-called EC mechanism consists of the electrochemical reduction
of the cryptate (e.g., Pb(2.2.2)2+ + 2e' + Pb(2.2.2)°) followed
by a chemical step in which the reduction product dissociates to
yield the free cryptand and a metal amalgam. The other possibility
is the CE mechanism in which the chemical step precedes the electro-
chemical one. Thus, the dissociation of the cryptate gives the free
ligand and metal ion, and this last species is then reduced to the
amalgam in a subsequent step.

An explanation of the observed voltammetric behavior in terms of
the EC mechanism could face some difficulties, especially if one
makes the assumption that the electroreduction of the cryptate yields
a metal atom inside the ligand cavity. Given that the atomic radii
of Pb, T1, and Cd are 1.80, 1.90, and 1.55 A, respectively (191), it
is unclear how the cavities of the (2.2.1) and (2.2.2) cryptands are
able to accommodate these atoms. The application of the EC mechanism
becomes more tenable if some modifications are made in the structure
which is assumed for the reduction product. It is conceivable that
the activation process for the electrode reaction involves a distor-
tion of the ligand to a point where the metal ion is no longer com-
pletely encapsulated. Such an arrangement would allow the metal ion
to be reduced without constraining the resulting metal atom to fit

inside the cryptate cavity. The reduction product in this case

206

would be some sort of loose association between the metal atom and
.ligand which would fall apart easily in the following chemical step.
The ligand reorganization in this mechanism would give rise to a sub-
stantial inner-shell component to the activation barrier to electron
transfer. The large activation energies for the electrode reactions
would then be reflected in slow heterogeneous charge-transfer kinetics
and electrochanically irreversible behavior.

The slow dissociation rates of the cryptates pose some problems
to the application of the CE mechanism to the present electrode
reactions. For example, the stability constant of Pb(2.2.2)2+ is
5 X 1012 (15). From Table 8, it is apparent that most divalent
(2.2.2) cryptates have formation rate constants of about

104 - lOs'Mfl s‘l. Thus, the dissociation rate constant for

Pb(2.2.2)2+ is in the neighborhood of 10.9 s-l. This seems to indi-
cate that the dissociation of the cryptate (i.e., the first step of
the CE mechanism) occurs too slowly to be seen on the cyclic
voltammetric time scale. However, it must be remembered that the
estimates made above apply only to bulk solution. The strong
electric fields in the interfacial region may bring about field-
assisted cryptate dissociation rates which are compatible with the
observed electrochemistry.

The slow cryptate dissociation rates and the CE mechanism can
also be reconciled if one assumes a somewhat different chemical

step. Instead of the complete dissociation of the cryptate, this

step would now consist of the partial removal of the metal ion from

207

the cryptate cavity. The rate of this step or the reduction rate of
the intermediate formed in this step would have to be slow enough to
account for the observed electrochemical irreversibility, but at the
same.time, these rates would be much faster than the extremely slow
dissociation rates of the cryptates. It should be noted that this
new CE mechanism is rather similar to the modified EC mechanism dis-
cussed earlier. The main difference is that the distortion of the
cryptate yields an activated complex in the latter mechanism while
it produces a reaction intermediate in the former. At the present
either mechanism appears to be adequate for explaining the observed
voltammetric behavior.

At the outset of this work, there had been some hope that the
formation of a cryptate would stabilize an uncommon oxidation state
of the encapsulated metal ion. This proved to be the case in the

2+f+ couple. On the other hand, no evidence for the exis-

Cu(2.l.l)
tence of Cd(2.2.l)+ or Cd(2.2.2)+ was found, although Cd(I) state has

been seen in the solid state (192). Evidently the stabilization which
Cd(I) would gain through complexation with a cryptand is not sufficient

to overcome the strong tendency of this oxidation state to dispro-

portionate to Cd(II) and Cd metal.

B. The Tl(2.2.2)+/T1(Hg) Couple as a Probe of Cryptate Thermodynamics

in anaqueous Solvents

 

2+ and Pb2+

While the amalgamrforming electroreductions of the Cd
cryptates were irreversible, the corresponding reaction for Tl(2.2.2)+

was found to be electrochemically reversible in a variety of solvents

208

(namely water, acetonitrile (MeCN), dimethylsulfoxide (DMSO),
N,N-dimethylformamide (DMF), and methanol). Cyclic voltammograms

of solutions containing Tl+ and at least a ten-fold excess of (2.2.2)
had sweep-rate-independent peak separations of about 60 mV, which is
quite close to the value expected for a reversible one-electron
process (92). Reversible voltammetric behavior was also obtained for
the Tl(I)/T1(Hg) couple in the absence of (2.2.2) in each of the five
solvents listed above. In all cases, the voltammetric half-wave
potential 355'“33 obtained from the mean of the cathodic and anodic
peak potentials. The dependence of Era upon the cryptand concentra-
tion was found to be in accordance with the Ligane equation, which
applies to the reversible amalgam-forming reduction of labile com-

plexes of high stability (193):

3112C - 211']: = -(RT/F) (1n x31 + m 1n [c]t) (1)

In eqn (1), E3: and Big/:0 are the half-wave potentials of the

Tl(I)/T1(Hg) couple in non-complexing media and in the presence of a

large excess of cryptand, respectively, [C]t is the total (analytical)

concentration of (2.2.2) cryptand, and'K:1 is the stability constant

of the thallium (I) cryptate, TlCm.

10

Data for the dependence of Hie on [C]t were analyzed using

eqn (1). It was found that m - l in all solvents; i.e., Tl+ and
(2.2.2) form a 1:1 complex, as expected. A value of log K:1 - 6.4
was obtained in aqueous media (u - 0.05) at 25°C. This agrees quite
well with an earlier value of 6.3 which was determined using a

potentiometric titration (6).

209

The reversible electroreduction of the Tl(2.2.2)+7Tl(Hg) couple
indicates that the exchange between Tl+ and Tl(2.2.2)+ is rapid on
the time scale of cyclic voltammetry. This finding is in harmony
with the rapid exchange rates obtained from NMR.measurements (3,19).
It is noteworthy that the previous electrochemical study of T1(2.2.2)+
found irreversible behavior for this complex in propylene
carbonate (34). Only a stoichiometric amount of the cryptand was
present in the experiments described in reference 34. These condi-
tions render the interpretation of the results difficult since the
conventional treatment of labile complex-forming systems requires
the ligand to be present in a large excess relative to the electro-
active metal ion (11). In the present investigation, the concentrae
tion of (2.2.2) was always at least ten times that of Tl+.

From the form of eqn (1), it can be seen that the Tl+/T1(Hg) and
Tl(2.2.2)+/T1(Hg) couples provide a direct means of monitoring the
activity (or concentration) of free cryptand. Systems of this kind
can be classified as electrodes of the second kind and have been
labeled "metal complex" electrodes (194,195). Such couples can be
combined with competitive complexation methods to allow the determina-
tion of the stability constants of the complexes of electroinactive
cations (194,195). A method of this type can be employed to determine
a wide range of stability constants by using the following procedure.
The half-wave potential, Eyé, for the T1(I)/T1(Hg) couple was deter-
mined in solutions containing 0.5‘mMLTl+ (E£;)’ and then after suc-

cessive additions of 5-10 m_M_ (2.2.2) cryptand (hf/:0) and 15-50 mg; of

210

the electroinactive metal cation M (Kaine). The ionic strength was

maintained at 0.05 or 0.1 M using tetraethylammonium perchlorate (TEAP).
Reversible cyclic voltammograms were obtained in each case, at least
when M was an alkali metal cation. In general, it was observed that
45?]: < -E¥/:MC < -E1i./:C. The proximity of 3‘12”" to FE]: or E323 was
dependent on how effectively M could compete with Tl+ for the cryptand.
In order to determine K2, the stability constant for the M(2.2.2)
complex, it is necessary to know the equilibrium concentrations of the
metal cryptate [MC], the metal ion [M], and the free cryptand in the
presence of M, [C]M. As outlined below, [C]M can be determined from
the electrochemical data. Knowing [C]M, the remaining terms [MC] and
[M] can be found directly since [MC] 8’ [Clt - [C]M and [M] - [Mlt - [MC],
where [Mlt is the analytical concentration of M. (Strictly speaking,
[MC] - [C]t - [C]M - [TIC], but the last term is usually quite small
relative to the others and its neglect represents an error of 3-52.

K31, and the analytical

If necessary, [TlC] can be calculated from [C]M,
concentration of Tl+, [T1+]t')

The determination of [C]M can be accomplished by noting that
Buck's equation for the effect of complexation upon reversible

voltammetric waves (196) can be written for the competition experiment

as

 

Tl
K [C] +1

TlMC T1 = _ RT Tl _ RT 3 M

1331/2 - E1,2 F ln(I(8 [C]M) F in T1 (2)
K3 [C]M

(The activity coefficient terms are omitted from eqns (1) and (2) and

211

elsewhere for simplicity.) Although Buck's equation normally only
applies to the case where the free ligand is present in a large
excess (197), it should apply in the present case even though the
free ligand concentration [C]M is typically less than [T1]t. This

is because the cryptand concentration will be "buffered" provided that
[C]t >’.[Tl]t and the equilibrium M + C ZIMC is maintained during the
cyclic voltammogram. (If this is not the case, then the shape of the
voltammogram will be severely distorted since the concentration of
cryptand increases during the cathodic part of the cycle.) Conse-
quently, by obtaining Kgl from eqn (1) in the absence of metal ion M

T1

and then inserting K8 into eqn (2), one can determine [C]M and hence

K12. In particular, it is interesting to note that when KEITC]M > 1
(i.e., when the stability of the thallium cryptate is sufficiently

greater than that for the competing metal cryptate so that

Tl TlMC
E - E 2
92 95

which allows eqn (1) to be subtracted from eqn (2) to yield

100 mV), the last term in eqn (2) can be neglected,

3],?“ - E12": - 531nm]: — ln[C]M) (3)

In this case, [01M and K? can be determined directly from eqn (3).
This procedure does not require a knowledge of K21. Therefore, under
these conditions, the Tl(I)/T1(Hg) couple responds only to the
decrease in free cryptand concentration which results when an excess
of metal ion M is added. It is noteworthy that K21 is not needed for
a determination of K: under these circumstances. In nonaqueous

solvents, it is possible that the small concentration of Tl+ (0.5 mg)

is preferentially complexed by water and other impurities, which could

212

lead to false values of Kzl. These incorrect Tl stability constants

will not affect the Kgrmeasurements unless the impurities vary in con-
centration upon the addition of M, or if they are capable of binding
significantly to the much higher (10-50 mg) concentrations of M that
were present in these experiments. It is also important to note that
[C]M can be calculated using eqn (2) or (3) even when [C]M < [C]t, and
that the method is generally capable of allowing K: to be obtained

‘ even when R: < K21. In fact, the practical lower limit of K1: that
can be determined using this method is simply that which corresponds
to appreciable cryptate formation at the concentrations of M and C
chosen. Since we find that K21 lies in the range 106 - 1010 in the
solvents studied to date, the method described here allows values of

K: in the range ca. 102 to 1010

to be accurately evaluated. (It is
difficult to obtain K: values when K: >‘K:1 since the T1+ ceases to
be an effective competitor. In practice, the difference between

T1 TlMC

Eyh and E55 becomes too small to be measured reliably.)

Table 27 lists some preliminary results for K: in DMSO and MeCN.
It is seen that good agreement is obtained between the present and
earlier determinations of K? for Cs(2.2.2)+ in DMSO (198). Potassium
and thallium cations were uniformly found to exhibit the largest
values of K: in each solvent, which is in harmony with the judicious
fit of these bare cations into the (2.2.2) cryptate cavity.

This simple electrochemical method therefore appears to provide
a powerful approach to the exploration of the thermodynamics of

cryptate and other macrocyclic complexation in a range of solvents.

213

Table 27. Formal Stability Constants (log Ks) for
Univalent (2.2.2) Cryptates in Dimethylsulfoxide
' and Acetonitrile at 25°C.

 

 

Cryptate DMSOa MeCNb
Li(2.2.2)+ < 1 7.3
Na(2.2.2)+ 5.3 10.6
1<(2.2.2)+ 6.0 11.1
Cs(2.2.2)+ 1.6 --

Tl(2.2.2)+ 6.2 12.3

 

aDetermined at u - 0.05.

bDetermined at u = 0.1.

While its requirement of electrochemically reversible systems may seem
restrictive, the T1 probe does offer an advantage over the principal
alternative, potentiometry. Because the T1 technique is not a zero-
current method, it is less likely to be affected by impurity currents.
Thus, the T1 probe can be considered an important complement to the

well-established potentiometric methods.

CHAPTER VIII

CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK

214

A. Conclusions

 

1. Transition Metal Cryptates

 

A recurrent theme in the studies of transition metal cryptates
is the important influence of ion hydration. The relatively low
stability constants of the trivalent Eu cryptates result from the
enthalpic destabilization associated with the extensive (or at least
partial) desolvation of the aquo cations upon their encapsulation
within the cryptate cavity. The surprising kinetic stabilities of
these trivalent cryptates stem from the large changes in solvent
structure which are needed to attain the transition states in the

dissociation reactions. One of the primary reasons for the enhanced

3+] 2+ is
aq
a smaller solvent-reorganization barrier to electron transfer in the

redox reactivities of the Eu cryptate couples relative to Eu

former species. The comparatively weak hydration of the trivalent
cryptates could well be the explanation for the tendencies of these
ions to associate strongly with F- and OH-.

The variations seen in the standard potentials of the

Eu(2 .2 .1)3+/2+ 3442+ 3+/ 2+

, Eu(ZB.2.l) , and Eu(2.2.2) couples indicate
that the structure of ligand can have a strong influence on the rela-
tive stabilities of the encapsulated Eu(III) and Eu(II) oxidation
states. Therefore, it would seem that the apprOpriate alterations in
ligand structure could yield a cryptate couple which has a specified
oxidizing or reducing ability. The cryptands could be modified by

replacing some of the ether oxygens with nitrogen or sulfur donor

215

216

atoms, by binding various functional groups to the ethylene bridges,
by lengthening or shortening these bridges, or by introducing substi-
tuents into the aromatic ring of (23.2.1). This last approach could
yield a series of structurally similar redox couples with varying
standard potentials. These cryptates could be useful in studies of
homogeneous electron-transfer kinetics since the intrinsic components
of the activation barriers of these complexes could well be reasonably
constant while the driving-force components obviously have to vary.
One could then investigate the driving-force dependence of homogeneous
cross—reaction rate constants. A.study of this kind could also be
undertaken with structurally dissimilar reactants, but some sort of
model would have to be employed in order to account for the effects

of the disparate intrinsic terms.

The rate data from Chapter VI appear to support the analysis of
Chapter V which suggested that electron transfer involving f orbitals
was not especially nonadiabatic. The cross-reaction rate constants
in Chapter VI conform to the Marcus cross-relation within the limits
of experimental uncertainty. Thus the electron-transfer reactions of

the Eu cryptates do not show any indications of nonadiabaticity. In

3+] 2+

fact, if the Eu
8Q

couple were nonadiabatic, one would expect the
cryptates to be even more so since the encapsulation of Eu(III) or
Eu(II) inside the cavity of a large organic ligand should signifi-
cantly diminish the already low probability for electron transfer
within the activated complex. (In other words, one would anticipate
that the transferred electron would have to tunnel over a greater

distance in the cryptate reactions.) On the basis of the present

evidence, the Eu(III)/(II) couples in this study are no less

217

adiabatic than couples which transfer electrons to and from d orbitals.

2. Reaction Entropies

The effects of hydration also pervade the reaction entropy study.
The large As;c values for M(III)/(II) aquo couples are due to the
extensive solvent structure which forms around the trivalent ion as
the result of field-assisted hydrogen bonding involving the protons
on the aquo ligands. The lower As;c values of the ammine couples
presumably result from.the weaker hydrogen bonding between the ammine
. protons and the surrounding solvent. Thus, interactions between the
ligands and the solvent play a major role in determining the Asgc
for a transition metal couple. Under certain circumstances, the
ligand effects predominate to such an extent that the reaction entrOpy
of a couple can be obtained simply by considering individual contri-
butions from each ligand in the complex. For example, the similar

As;c values of the Fe(bpy)3+/2+ )g+/2+

3)g+/2+ and Os(NH3):+/2+ couples can be explained in these terms.

and Ru(bpy couples or of the

Ru(NH
Perhaps the most graphic instance of the ligand influence is in the
3+/ 2+ 3+/ 2+ 3+/ 2+ 3+/2+

series, where the As;c values can be obtained by adding together

. Ru(NH3)2(bpy) . and Ru(bpy)

separate components for each bpy and NH ligand. However, the

3
reaction entropy is not always solely dependent on the ligands in the
redox couple. The different Asgc values for analogous Co and Ru
couples (see Table 11) indicate that the central metal ion can also

have a significant effect on the reaction entrOpy. It would appear

that two redox couples will have comparable reaction entrOpies when

218

they have the same ligand composition and when the electronic struc-
tures of the two central metal ions are similar. These conditions
are restrictive but some useful generalizations might be made. The
Ru(III)/(II) systems could serve as models for analogous couples in
which an electron is transferred to or from a nonbonding orbital, and
the appropriate Co(III)/(II) couples may be representative of those
systems where an antibonding orbital is involved in the electron
transfer. Unfortunately, only a limited number of metal ion couples
are suitable for studying the above assertions, so these will remain
largely untested at the present.

The reaction entropies provide a convenient means of calculating
the entropy of reaction for homogeneous redox processes. These quanti-
ties can also be obtained from the temperature dependence of the
appropriate equilibrium cell potentials. The potentiometric method
has a number of drawbacks. The cell potential of each reaction has
to be measured separately. In contrast, a relatively small number
of As;c values can be used to calculate the thermodynamic driving
forces of a host of homogeneous reactions. The potentiometric method
frequently requires the maintenance of stable concentrations of
reactants and products. This can be difficult considering that many
of the species in these reactions are unstable with respect to oxi-
dation or aquation, especially at high temperatures. In the As;c
method, these species only need to be present for the duration of a
cyclic voltammogram.(i.e., a few seconds). Finally, the uncertain-
ties in the os;c values which stem from extrathermodynamic assump-

tions will cancel when the difference between two reaction entrOpies

219

is taken to calculate the ASE2 value for a cross reaction. In
essence, these advantages of the As;c method enabled the study per-
formed in Chapter V to scrutinize a wide range of reactions. It
would have been impossible, or at best laborious, to obtain the
driving-force parameters by means of direct potentiometry.

The correlation between as;c and AGfl, the activation free energy
for self-exchange, proved to be effective in explaining certain
patterns in redox reactivity. However, this correlation cannot be
applied indiscriminately. It appears to retain general validity when
it is employed with relatively simple redox couples, but the problems
which are encountered when the correlation is applied to the Eu
cryptates may indicate that the correspondence between as;c and 06:1
is not as firm when more complex ligand structures are under considera-
tion. Large changes in ligand conformation can occur in the electron-
transfer activation process for such systems. These changes will be
reflected in As;c only if the alteration in ligand conformation has a
significant effect on the ability of the solvent to approach the redox
center. Therefore, it may well be that the correlation is only useful
to a limited degree for complex ligand systems such as those seen in
many biological redox couples. 0n the whole, the us;c - AGE1 correla-
tion is best used for interpreting the trends shown by a large number
of redox couples (Figure 4 is one example of this use.) The correla-
tion is not exact, so one cannot expect consistent success when

applying it to pairs or trios of redox couples unless these systems

are very similar to each other.

B. Suggestions for Further Work

 

Because the hydration of metal cations has emerged as a crucial
factor in the cryptate and reaction entrOpy studies, it will be of
interest to determine if similar investigations in nonaqueous
solvents yield the same sort of results. The different dielectric
constants and specific solvating abilities of these solvents should
produce a wide range of As;c values. These reaction entropies should
contain useful information on the solvent structure and the ligand-
solvent interactions in nonaqueous solution. One can also expect
some changes in the thermodynamics and kinetics of the Eu and Yb
cryptates when these complexes are studied in a nonaqueous solvent.
Some increases in the cryptate dissociation rates might be found in
weakly solvating media. The difference in the formal potentials of
the Eu cryptate and solvated Eu couples will be related to the extent
of solvation of Eu(III) relative to that of Eu(II).

A number of further investigations on the homogeneous redox
kinetics of cryptates could be performed. Information on the reac-

3+/ 2+

tivity of Yb(2.2.l) would be useful for comparison with the Eu

cryptates and also for its own sake. Because the Yb(2.2.l)3+/2+
couple has a very negative standard potential, the reactions of this
species with most commonly employed redox reagents will have large
driving forces. The relevant reactions in Chapter V all had sub-

stantial driving forces and at least one aquo reactant. It would be

interesting to determine if the reactions between the Yb cryptate

220

221

and some appropriate nonaquo coreactants contain the same solvent-
related work terms (06:) found in Chapter V. The presence of the

AC: term was ascribed to the solvent reorientation needed to form

the highly charged precursor complex. It seems likely that much of
the solvent structure in these complexes is induced by the aquo
ligands on one or both of the reactants. Because nonaquo complexes
are less strongly hydrated, it might be expected that a redox reaction
between two such species would have a lower AC: value. Unfortunately,

3+l2+ will cause

the extremely negative standard potential of Yb(2.2.l)
some difficulties in the determination of the self-exchange rate
constants for this couple. The Marcus cross-relation begins to break
down when the driving force of a cross reaction is large. This will
be the case for the reactions between the Yb cryptate and most well-

characterized coreactants. Therefore, one must employ redox couples

which also have particularly negative E° values; e.g., U22”+ or
Yb3+/ 2+.
aq

The homogeneous redox kinetics of the Cu(2.l.l)2+/+

couple would
be worthy of examination. The low As;c value of this couple and the
more rigid structure of the (2.1.1) ligand relative to (2.2.1) and

2”” should

(2.2.2) suggest that the self-exchange rate for Cu(2.l.l)
be rapid. Studies of the dissociation kinetics of the divalent and
monovalent Cu cryptates and the characterization of any anion effects
(such as those observed for the Eu(III) cryptates) would be useful
preliminaries for the investigation of the electron-transfer reactions

of Cu(2.l.l)2+/+.

222

The range of formal potentials shown by the Eu(2.2.l)3+/2+,

3+]2+ 3+/ 2+

Eu(ZB.2.l) , and Eu(2.2.2) couples demonstrates that the

Ef value of the Eu(III)/(II) couple can be altered considerably by
changes in the cryptand structure. Further insight into this kind
of ligand effect could be obtained by determining the formal poten-
tials for complexes formed between Eu(III)/(II) and the various
polyaza and polythia analogs of the (2.2.1), (2.2.1), and (2.1.1)
cryptands. The replacement of the oxygen atoms in the cryptands by
nitrogen and sulfur should produce large changes in the relative
stabilities of the Eu(III) and Eu(II) complexes. The nitrogen and
sulfur donor atoms could well stabilize the Eu(III) state via
covalent bonding and modify the ligation and electron-transfer
kinetics of the trivalent and divalent complexes.

The technique of determining stability constants for (2.2.2) com-
plexes by means of a competition with T1+ should prove to be applicable
to a large number of solvents. As noted in Chapter VII, this method can
be employed when the Tl+/T1(Hg) and Tl(2.2.2)+/T1(Hg) couple are
electrochemically reversible and when T1(2.2.2)+ and the other (2.2.2)
complexes under study are substitutionally labile. Since T1(2.2.2)+
is quite stable, the Tl+ competition method can be utilized to
determine stability constants which are too large to be obtained
from spectrophotometric techniques. This aspect of the Tl+ method
takes on special significance in those nonaqueous solvents where pH
titrations cannot be used to determine cryptate stability constants.
Thus, the Ks values for a number of (2.2.2) complexes can be obtained

in a range of solvents by means of this T1+ competition technique.

223

The enthalpies and entropies of complexation can then be calculated
from the temperature dependences of the stability constants. These
studies will furnish useful insights into the influences of the
solvent on cryptate stability. The Tl+ method should be easily
applicable to univalent cryptates, but some obstacles could be
encountered with divalent complexes. As noted in Chapters I and III,
a divalent cryptate has a larger stability constant and slower dis-
sociation rate constant than the corresponding univalent complex.
Therefore, the T1+ ion may not be capable of competing effectively
with alkaline earth ions for the (2.2.2) ligand. In other cases,
thermodynamic information will not be available since the voltammetric
waves will be distorted by the slow ligand exchange kinetics of the
divalent cryptate. Despite these possible limitations, the Tl+
competition method represents a simple and convenient means for

probing cryptate stabilities in a variety of solvents.

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224

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R. G. Linck in MI? International Review of Science, Inorganic
Chemistry, Series 1, Vol. 9, M. L. Tobe, editor, Park Press,
Baltimore, 1972.

R. J. Taylor and A. A. Humffrey, J. Electroanal. Chem., 42, 347
(1973).

D. J. Pruett, Michigan State University, Ph.D. Thesis (1978).

B. A. Moyer and T. J. Meyer, J. Am. Chem. Soc., 100, 3601 (1978).

87.

88.

89.
90.

91.

92.

93.
94.

95.

96.

97.

98.

99.

230

E. L. Yee and M. J. Weaver, Inorg. Chem., 22, 1077 (1980).

(a) I. I. Creaser, J. MacB. Harrowfield, A. J. Herlt,

A. M. Sargeson, J. Springborg, R. J. Geue, and M. R. Snow,
J. Am. Chem. Soc., 22, 3181 (1977); (b) A. M. Sargeson,
Chem. Br., 23 (1979).

F. Bottomley and S. B. Tong, Inorg. Chem., 22, 243 (1974).
J. M. Lehn and M. E. Stubbs, J. Am. Chem. Soc., 22, 4011 (1974).

M. J. Weaver and F. C. Anson, J. Electroanal. Chem., _6_§_, 711
(1975).

See, for example, (a) R. N. Adams, "Electrochemistry at Solid
Electrodes", Marcel Dekker, New York, 1969, pp. 145-149;

(b) D. T. Sawyer and J. L. Roberts, Jr., "Experimental
Electrochemistry for Chemists", Wiley, New York, 1974,
Chapter 7.

R. S. Nicholson, Anal. Chem., 21, 1351 (1965).
M. J. Weaver, J. Phys. Chem., g4, 568 (1980).

Strictly speaking, this procedure yields the so-called
"half-wave" potential E!” which will equal Ef only when the
diffusion coefficients of the oxidized and reduced forms of
the redox couple are identical. However, Ehh normally differs
from Ef by only 2-3 mV. See Chapter IV for a more detailed
discussion.

The (23.2.1) cryptand is similar to (2.2.1) except that one of
the central dioxyethylene groups is replaced by the analogous
catechol (3).

Although the determination of A8;c requires at least one
extrathermodynamic assumption to be made (see Chapter IV),
this uncertainty will cancel when differences in AS° are

rc
considered as in eqn (2).

It is important to note that the time required to attain true
equilibrium will be determined by the ligand exchange rate
(i.e., the first-order rate constant for dissociation of the
complex) irrespective of whether the complex is allowed to
aquate spontaneously or the free metal ion is allowed to
complex with added ligand.

See, for example, D. R. Crow, "Polarography of Metal Cbmplexes",
Academic Press, New York, 1969, pp. 73-77.

231

100. Equation (4) is a form of the well-known Lingane equation;
it should be carefully distinguished from equation (1) since
the former relation will apply when only the oxidized species
is complexed to form substitutionally labile complexes.

101. See, for example, K. J. Laidler, "Chemical Kinetics", Second
Edition, McCrawbHill, New York, 1965, pp. 88-90.

102. (a) L. Pauling, "The Nature of the Chemical Bond", Cornell
University Press, Ithaca, Third Edition, 1960, Chapter 13;
(b) S. W. Benson and C. S. Copeland, J. Phys. Chem., 21, 1194
(1963).

103. See, for example, J. P. Hunt, "Metal Ions in Aqueous Solution",
Benjamin, New York, 1963, Chapter 2.

104. J. C. Barnes and P. A. Bristow, Inorg. Nucl. Chem. Lett., 5,
565 (1969).

105. M. Diebler, M. Eigen, G. Ilgenfritz, G. Mesh, and R. Winkler,
Pure App. Chem., 22, 93 (1969).

106. The estimate, St *‘-10 to -15 e.u., arises from the conven-
tional use for bimolecular reactions of the transition-state
formalism with the pre-exponential factor ( T/h) exp (ASI/R)
rather than the simple collisional model wit the bimolecular
collision frequency Z. See, for example, reference 60 or
reference 101.

107. F. H. Spedding, J. A. Rard, and A. Habenschuss, J. Phys.
Chem., g;, 1069 (1977).

108. For further details, see Chapter 4 and reference 48.

109. For a general review, see J. N. Agar, Adv. Electrochem.
Electrochem. Eng.,.2, 31 (1963).

110. For example, see (a) E. D. Eastman, J. Am. Chem. Soc., 50,
292 (1928); (b) A. J. de Bethune, T. S. Licht, and "'
N. Swendeman, J. Electrochem. Soc., £22, 616 (1959);
(c) A. J. de Bethune, J. Electrochem. Soc., 197, 829 (1960);
(d) G. Milazzo and M. Sotto, Z. Phys. Chem. (Ffankfurt am
Main),_§2, 293 (1967); (e) G. Milazzo, M. Sotto, and
C. Devillez, Z. Phys. Chem., 22, 1 (1967).

111. The values for Hg and Pt are obtained by combining the
relative Thomsen coefficients (vs. Pb) quoted in
G. P. Harnwell, "Principles of Electricity and Magnetism",
MCGrawéHill, New York, 1939; p. 187, with the absolute
Thomsen coefficient for Pb given in reference 109.

112.

113.

114.

115.
116.
117.
118.

119.

120.

121.

122.

123.

124.

125.
126.
127.

128.

129.

232

R. S. Nicholson and I. Shain, Anal. Chem., _2, 706 (1964).

If the concentration of the electroactive species is known,
the diffusion coefficient can be calculated from equation (5)
or (6) from Chapter III.

C. I. H. Hanania, D. H. Irvine, W. A. Eaton, and P. George,
J. Phys. Chem., 1;, 2022 (1967).

Reference 39, p. 137.

Reference 92b, pp. 29-30.

Reference 82, pp. 42, 465.

R. R. Buckley and E. E. Mercer, J. Phys. Chem.,‘zg, 3103 (1966).

(a) D. K. Lavallee, C. Lavallee, J. C. Sullivan, and
E. Deutsch, Inorg. Chem., 22, 570 (1973); (b) C. Lavallee
and D. K. Lavallee, Inorg. Chem., 19: 2601 (1977).

J. J. Kim.and P. A. Rock, Inorg. Chem., 2, 563 (1969).

P. George, G. I. H. Hanania, and D. H. Irvine, J. Chem. Soc.,
2548 (1959).

See, for example, F. Basolo and R. G. Pearson, "Mechanisms
of Inorganic Reactions", Second Edition, Wiley, New York,
1967, pp. 62-64.

(a) E. R. Nightingale, J. Phys. Chem., 53, 1381 (1959);

(b) T. Takahashi and T. Kolso, Bull. Chem. Soc. Japan, 49,
2734 (1976) . ‘—

M. J. Weaver and T. L. Satterberg, J. Phys. Chem., 2;, 1772
(1977).

Reference 38, p. 592.

H. J. Peresie and J. A. Stanko, Chem. Commun., 1674 (1970).
H. C. Stynes and J. A. Ibers, Inorg. Chem., 19: 2304 (1971).
N. Sailasuta, F. C. Anson, and H. B. Gray, J. Am. Chem. Soc.,
101, 455 (1979). Itrshould be noted that the reaction
entropies tabulated in this paper are relative values based
on the assumption that As;c - 0 for the H+IH2 couple.

N. J. Hair and J. K. Beattie, Inorg. Chem., 12, 245 (1977).

130.

131.

132.

133.

134.

135.

136.

137.

138.

139.

140.

141.

142.

143.

233

N. Sutin, in "Tunneling in Biological Systems", B. Chance,
D. C. Devault, H. Frauenfelder, R. A. Marcus, J. R. Schreiffer,
and N. Sutin, editors, Academic Press, New York, 1979, p. 201.

E. Buhks, M. Bixon, J. Jortner, and G. Navon, Inorg. Chem., lg,
2014 (1979).

W. BBttcher, G. M. Brown, and N. Sutin, Inorg. Chem., l§,
1447 (1979).

Estimates by means of the Marcus cross-relation from the 3+/2+
rates of outer-sphere cross reactions which involve Cr(OHZ)6
and which have small driving forces (134).

M. Chou, C. Creutz, and N. Sutin, J. Am. Chem. Soc., 22, 5615
(1977).

K. V. Krishnamurty and A. C. Wahl, J. Am. Chem. Soc.,}§9,
5921 (1958).

J. Silverman and R. W. Dodson, J. Phys. Chem.,,éés 345 (1952)-

R. C. Young, F. R. Keene, and T. J. Meyer, J. Am. Chem. Soc.,fi,
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F. P. Dwyer and A. M. Sargeson, J. Phys. Chem., 22, 1892 (1961).

H. M. Neumann, quoted in R. Farina and R. G. Wilkins, Inorg.
Chem., 1, 514 (1968).

For recent reviews which deal with this point, see

(a) R. G. Linck in "Homogeneous Catalysis", G. N. Schrauzer,
editor, Marcel Dekker, New York, 1971, Chapter 7; (b) N. Sutin
in "Inorganic Biochemistry", G. L. Eichhorn, editor, Elsevier,
New York, 1973, Chapter 19; (c) R. G. Linck in "MTP Inter-
national Review of Science", Inorg. Chem. Series II, Vbl. 9,
Butterworths, London, 1974, Chapter 7; (d) R. G. Linck in
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(a) D. P. Rillema, J. F. Endicott, and R. C. Patel, J. Am.
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I. Bodek and G. Davies, Coord. Chem. Rev., 22, 269 (1974).

B. Falcinella, P. D. Felgate, and G. S. Lawrence, J. Chem.
Soc. Dalton, 1 (1975).

144.

145.

146.
147.

148.

149.

150.

151.

152.

153.

154 O

155.

156.

157.

158.

159.

234

(a) A. Ekstrom, A. B. MCLaren, and L. E. Smythe, Inorg.

Chem., 22, 2899 (1975); (b) A. Ekstrom, A. B. McLaren, and

L. E. Smythe, Inorg. Chem., 22, 2853 (1976); (c) A. Ekstrom,
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G. M. Brown, H. J. Krentzien, M. Abe, and H. Taube, Inorg.
Chem., 22, 3374 (1979).

Reference 74, pp. 123, 129.
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D. H. Huchital, N. Sutin, and B. Warnquist, Inorg. Chem.,_§,
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J. c. Hindman and E. s. Kritchevsky, J. Am. Chem. Soc., _7_2_,
953 (1950).

D. Cohen and J. C. Hindman, J. Am. Chem. Soc., 13, 4679 (1952).
The isothermal cell results in this reference were converted
to the absolute scale by using dsgc - 21 e.u. for the H+IHZ
couple.

H. S. Habib and J. P. Hunt, J. Am. Chem. Soc., 22, 1668 (1966).

The quoted 2 value _isll2 obtained from the expression (79, 81)

Z - r2N(8w k3 Tmr’I h°103where r is the distance between
the metal centers in) the collision complex and m1. is the
effective reduced mass of this complex. Typical values for
the present reactants (r- 7 X 10 8 cm and mi. - 100)
yie1d2~6><10 oM '-

Values of 3 used to calculate (AGW) from eqn (5) were
obtained from the sum of the radii o the reacting ions.

M. J. Weaver, Inorg. Chem., 22, 1733 (1976).

L. E. Bennett and J. L. Sheppard, J. Phys. Chem.,_§§, 1275
(1962).

M. R. Hyde, R. Davies, and A. G. Sykes, J. Chem. Soc., Dalton,
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D. W. Carlyle and J. H. Espenson, J. Am. Chem. Soc., 29
2272 (1968).

9

c. Dultz and N. Sutin, J. Am. Chem. Soc., 22, 829 (1964).

M. J. Burkhart and T. W. Newton, J. Phys. Chem., 73, 1741
(1969). '_—

235

160. A. Adin and A. G. Sykes, J. Chem. Soc. (A), 1230 (1966).

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162. C. A. Jacks and L. E. Bennett, Inorg. Chem., 22, 2035 (1974).
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164. J. F. Endicott and H. Taube, J. Am. Chem. Soc., 22, 1686
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165. C. Lavallee, D. K. Lavallee, and E. A. Deutsch, Inorg. Chem.
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166. T. J. Meyer and H. Taube, Inorg. Chem., 1, 2369 (1968),
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168. J. P. Candlin, J. Halpern, and D. L. Trimm, J. Am. Chem. Soc.
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169. R. T. Wang and J. H. Espenson, J. Am. Chem. Soc., 22, 380 (1971).

170. R. J. Christenson, J. H. Espenson, and A. B. Batcher, Inorg.
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171. R. Davies, M. Green, and A. G. sykes, J. Chem. Soc., Dalton,
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172. J. N. Braddock and T. J. Meyer, J. Am. Chem. Soc.,_2§, 3158
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173. H. Fischer, G. M. Tom, and H. Taube, J. Am. Chem. Soc., 22,
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174. T. W. Newton and F. E. Baker, Adv. Chem. Ser., 1;, 268 (1967).
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176. P. D. Tyma and M. J. Weaver, J. Electroanal. Chem., in press.
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180. Reference 82, p. 468.

181.

182.

183 O

184.

185.

186.

187.
188.
189.

190.

191.

192.

193.

194.

195.
196.
197.

198.

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J. H. Espenson, K. Shaw, and O. J. Parker, J. Am. Chem.
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J. C. Sullivan and J. C. Hindman, J. Phys. Chem., 22, 1332
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J. Podlaha and J. Podlahovfi, Coll. Czech. Chem. Commun., 22
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G. Jones and W. A. Ray, J. Am. Chem. Soc., 29, 1571 (1944).

L. Meites, J. Am. Chem. Soc., 12, 2476 (1953).

B. Metz, D. Moras, and R. Weiss, Acta Cryst, 329, 1377 (1973).
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M. C. Day and J. Selbin, "Theoretical Inorganic Chemistry",
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Reference 38, p. 507.

I. M. Kolthoff and J. J. Lingane, "Polarography", Second
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C. Luca, V. Magearu, and G. Popa, J. Electroanal. Chem., 22,
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G. Popa and V. Magearu, J. Electroanal. Chem., 22, 85 (1969).
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E. Mei, L. Liu, J. L. Dye, and A. I. Popov, J. Solution
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