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THE
JOURNAL
OF
CHEMICAL
PHYSICS
VOLUME
24,
NUMBER
5
MAY,
1956
On the Theory of Oxidation-Reduction Reactions Involving Electron Transfer.
I*
R.
A.
MARCUS
Department
of
Chemistry, Polytechnic Institute
of
Brooklyn, Brooklyn, New York
(Received
July
28,
1955)
A mechanism for electron transfer reactions is described,
in
which there is very little spatial overlap of the electronic orbitals
of
the
two reacting molecules in
the
activated complex. Assuming
such a mechanism, a quantitative theory of the rates
of
oxidation-
reduction reactions involving electron transfer in solution is
presented.
The
assumption of "slight-overlap" is shown
to
lead
to
a reaction
path
which involves an intermediate
state
X*
in
which
the
electrical polarization
of
the solvent does not have
the
usual value appropriate for
the
given ionic charges (i.e.,
it
does
not have an equilibrium value). Using an equation developed else-
where for
the
electrostatic free energy of nonequilibrium states,
the free energy
of
all possible intermediate states is calculated.
The
characteristics of
the
most probable
state
are
then
deter-
mined with
the
aid of the calculus of variations
by
minimizing
its
free energy subject
to
certain restraints. A simple expression for
INTRODUCTION
D
URING
recent years oxidation-reduction reac-
tions involving the transfer of an electron between
the reactants have been the subject
of
many kinetic
studies.
1
Several generalizations may be drawn from
this data. For example,
it
was found
that
isotopic ex-
change reactions between
ions~differing
only in their
valency are generally slow if simple cations are involved
and fast
if
the ions are relatively large, such as com-
plex ions.
This behavior has been qualitatively explained
by
Libby
2
on the basis of related ideas
of
Franck, applying
the Franck-Condon principle. The degree
of
orientation
of the solvent molecules toward
an
ion greatly depends
on the charge of
that
ion. For a given ion, it will there-
fore be different before and after this ion undergoes an
electron transfer. Libby observed
that
the solvent
molecules near the reacting ions cannot adjust them-
selves immediately to the change in ionic charges result-
ing from an almost instantaneous electronic jump. A
state of high energy, he suggested,
is
therefore produced.
Such a barrier to reaction would be greater for small
ions, since they are more highly solvated than large ones.
This conclusions
is
in agreement with the fact
that
in
most cases the smaller ions react more slowly in these
isotopic exchange redox reactions.
Another observation which can be drawn from a
*This
research was supported in
part
by
the
Office
of Naval
Research under Contract No. Nonr839(09). Reproduction in
whole or in
part
is permitted for
any
purpose of
the
U.
S.
Govern-
ment.
1
See
review articles: Zwolinski, Marcus (Rudolph
J.),
and
Ey-
ring, Chern. Revs. 55,
157
(1955);
C.
B. Amphlett, Quart. Revs. 8,
219 (1954);
0.
E. Myers and
R.
J.
Prestwood, Radioactivity Applied
to
Chemistry, edited
by
Wahl
and
Bonner (John Wiley and Sons,
Inc., New York, 1951), Chap. 1 ; Betts, Collinson, Dainton, and I vin,
Ann. Repts. on Progr. Chern. (Chern.
Soc.
London) 49,
42
(1952);
R. R. Edwards, Ann. Revs. Nuclear Sci. 1,
301
(1952); M.
Haissinsky,
J.
chim. phys. 47, 957 (1950); and recent reviews in
Ann.Rev.Phys.Chem.
2
W.
F.
Libby,
J.
Phys. Chern. 56,
863
(1952).
the
electrostatic contribution
to
the
free energy of formation of
the
intermediate
state
from the reactants,
!J.F*,
is thereby obtained
in terms
of
known quantities, such as ionic radii, charges,
and
the
standard free energy of reaction.
This intermediate
state
X* can either disappear
to
reform
the
reactants, or
by
an
electronic jump mechanism
to
form a
state
X
in which
the
ions are characteristic of the products. When
the
latter
process is more probable
than
the
former, the over-all
reaction
rate
is shown to be simply
the
rate
of formation of the
intermediate state, namely the collision number
in
solution multi-
plied by
exp(-!J.F*/kT).
Evidence in favor of this is cited.
In
a
detailed quantitative comparison, given elsewhere,
with
the
kinetic data, no
arbitrary
parameters are needed to obtain reason-
able agreement
of
calculated
and
experimental results.
summarf
of
data
on isotopic exchange reactions having
simple mechanisms
is
that
the entropy of activation of
such reactions
is
large and negative.
It
is
of interest
that
all these reactions were between ions of like sign.
It
was assumed
3
that
a reorganization of the solvation
atmospheres about the reacting ions occurred prior to
reaction,
but
it
was believed
that
this would con-
tribute a positive term to the entropy
of
activation.
It
was suggested
that
the reorganization would involve
a partial "melting" of the solvent attached to the ions,
and
that
this would involve an increase in entropy.
To
explain the observed entropy
of
activation there
would have to be a larger, negative term.
It
was sug-
gested
that
this term was due to the
low
probability of
an
electron tunnelling
4
through a solvation barrier,
from one reactant to the other in the intermediate state.
However, several aspects of this interesting treatment
are open to question.
5
In
fact, using the values given
a Marcus
(Rudolph].),
Zwolinski, and Eyring,
J.
Phys. Chern. 58,
432 (1954). These authors summarize some
of
these
data
in their
Table
I.
In
Table
II,
reactions are given having apparent positive
entropies of activation. However, in
at
least all
but
one
of
the
reactions in Table
II
the
mechanism is complex and
the
concen-
trations
of
the actual reactants are unknown. Accordingly,
the
so-called entropies
of
activation
of
such reactions have no im-
mediate theoretical significance.
The
lone possible exception,
incidentally, does not involve reacting ions of like sign.
4
J.
Weiss, Proc. Roy.
Soc.
(London) A222,
128
(1954), has also
discussed the electronic jump process. Unlike reference 3 the
necessity for the reorganization
of
the solvent occurring prior
to
the
electronic transition was not considered there.
6
The
mechanism used there was incomplete in
that
only one
fate
of
the
intermediate
state
in
the
reaction was considered.
It
was tacitly assumed
that
this
state
involving the reorganized
solvent could only produce products,
but
not reform
the
reactants.
(The former would occur by
an
electron
jump
process, the
latter
by
a disorganizing motion
of
the
solvent.)
It
is shown later
that
this omission can significantly affect the role played
by
the
electronic jump process.
The
number of times per second
that
the electron in one
of
the
reactants struck
the
barrier was not included in
the
over-all calcu-
lation. Effectively, this made electron tunnelling appear about one
thousand-fold less frequent
than
would otherwise have been
estimated.
966
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OXIDATION-REDUCTION
THEORY
967
there for the probability
of
electron tunnelling
and
using the detailed
treatment
given in the present paper,
a different conclusion will be drawn about the origin
of
the observed entropy of activation.
An object
of
the present paper
is
to devise a method
of calculating the free energy of reorganization
of
the
solvent molecules about the reactants prior to the
electronic jump process, and from this to develop a
quantitative theory
of
electron transfer reactions.
THEORETICAL
General
In
most bimolecular reactions, appreciable changes in
various interatomic distances within each molecule
generally occur during the course of a collision.
The
potential energy of this system, arising from the stretch-
ing and compression
of
various chemical bonds, usually
passes through a maximum in the collision. The con-
figuration of the atoms
at
the maximum
is
the well-
known activated complex,
and
a detailed knowledge
of
it
permits
an
a priori
calculation
of
the reaction rate.
In
most reactions there usually
is
a transfer of atoms
or groups of atoms between the reactants,
and
a rear-
rangement
of
atoms within each reactant.
In
order for
this to occur, there presumably must be a strong inter-
action of the electronic structures
of
the two reactants
in the activated complex.
That
is, there would be a
considerable spatial overlap of the electronic orbitals
of
the two reacting molecules in this complex.
In
contrast to such reactions, some reactions may
merely involve the transfer
of
an
electron between the
reacting molecules. For such reactions to occur, only
a slight overlap of the electronic orbitals is perhaps
necessary.
Only
a slight electronic interaction may be
sufficient to electronically couple the two molecules
and
permit the electron transfer to occur.
If
this is indeed
the case, then its consequences are far-reaching.
In
the present paper a quantitative theory for electron
transfer reactions will be developed on the basis
of
the
assumption
that
there is little overlap
of
the electronic
orbitals of the two reacting particles in the activated
complex. The final formula
of
this paper
is
therefore not
applicable to
any
electron transfer reaction having a
large-overlap activated complex.
Electronic Configuration of the Activated Complex
Just
before a collision the electronic configuration
of
the reacting pair
of
molecules is the same as
that
of
reactants.
Just
after a successful collision, their elec-
tronic configuration is the same as
that
of
the products.
The electronic configuration
of
the intermediate stage
in the reaction, i.e.,
of
the activated complex,
is
pre-
sumably
of
an
intermediate nature. We may readily
determine
it
for activated complexes in which there is
but
slight overlap
of
the electronic orbitals
of
the two
reacting particles.
One
may write down Schrodinger's wave equation,
describing the wave function
cf>
of the electrons of the
reacting particles in the activated complex, taking into
account their interactions with each other and with all
the solvent molecules.
Let
us consider first
any
given
configuration of all the atoms in the system, i.e.,
of
the atoms of the two reacting particles and
of
the sol-
vent.
If
there were no overlap of the electronic orbitals
of
the two reacting particles there would be no elec-
tronic interaction of the two molecules. Therefore
an
exact solution
of
the wave equation would then simply
be
that
wave function which characterizes the electronic
configuration
of
the two reactants when they are far
apart
in the solvent. For the given atomic configura-
tion
of
the reacting particles, let us denote this wave
function by
cf>x*·
Again,
an
equally valid solution to
the wave equation would be
that
which characterizes
the electronic configuration
of
the two products when
they are far
apart
in the solvent. For the given atomic
configuration
of
the reacting particles, let this function
be
cf>x·
In
the case
of
weakly interacting electronic
orbitals
of
the two reacting particles the linear com-
bination
(cf>x+ccf>x•),
where
cis
a constant, would be the
appropriate wave function for the activated complex,
but
not
cf>x
or
cf>x•
alone.
It
can be shown
6
that
this is the
appropriate solution for weakly interacting orbitals
only if the
total
energy
of
the system is the same for each
of
electronic configurations
cf>x
and
cf>x*
in
any
given
atomic configuration.
Presumably, in our activated complex the two elec-
tronic configurations,
cf>x
and
cf>x•,
make equal contribu-
tions to the total wave function. The important thing,
however,
is
that
for every atomic configuration of the
activated complex the
total
energy
of
a hypothetical
system having the electronic configuration of the
reactants
(cf>x•)
must be the same as
that
of a hypo-
thetical system having the electronic configuration of
the products
(cf>x).
Since this is a thermodynamic sys-
tem, there will be many atomic configurations
of
all the
solvent molecules
and
of the reacting pair
of
molecules
in the activated complex which
will
conform to this
energy restriction. Thus, the energy in a thermo-
dynamic sense, which is the average
of
the energies
of
all the
suitable
atomic configurations, must be the
same for both electronic configurations. These two
hypothetical thermodynamic states
of
the system
will
be called the intermediate states,
X*
and
X.
6
The Schrodinger equation can be written as
H¢=E</>;
E
is the
energy of an atomic configuration.
The
Hamiltonian operator
H
includes terms expressing
the
interaction
of
the
electrons
and
nuclei
of
the reacting particles with each other
and
with the solvent
molecules.
In
the
case
of
no overlap,
cf>,
and
cf>,•
were shown
to
be
solutions
to
this wave equation.
Let
their corresponding energies
beE,
and
E,•,
respectively, so
that
we have:
llcf>x=Excf>x
and
llcf>x•=E,•cf>x*·
If
c
is
any
constant, a linear combination of
cf>x
and
cf>,•
is
(cf>x+ccf>,•).
When introduced into
the
wave equation
this yields:
H(<Px+ccf>x•)=Exc!>x+E,•ccf>,•.
Only when
Ex
equals
Ex•
is
the
right-hand side equal
to
E,(cf>,+ccf>,•).
That
is, only
under these conditions does
(cf>,+ccf>x*)
satisfy
the
equation
llcf>=E¢.
It
is also seen
that
for such a linear combination,
the
total energy
E
equals
E,
and
therefore
Ex•.
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968
R.
A.
MARCUS
These considerations of the energy restriction are
amplified later in an application of the uncertainty
principle to a discussion
of
the validity of assuming a
small-overlap activated complex.
The
total
energy condition can readily be shown to
place a severe restraint upon the solvation of the
activated complex. The degree of orientation
of
the
solvent molecules about
any
ion will strongly depend
on its charge. Accordingly, the
equilibrium set of con-
figurations of all the atoms of the solvent in the neigh-
borhood of the reacting particles
will
greatly depend on
whether these particles have the ionic charges of the
reactants or of the products. Now the average configura-
tion of the solvent was seen to be the same in the two
states,
X* and X. These states differ in the charges of
the reacting particles. Therefore, the average configura-
tion of the solvent in the activated complex cannot be
an equilibrium one.
(In
this respect
it
differs from the
large-overlap complex, as discussed in a later section.)
The
average configuration of the solvent in the activated
complex must also be such as to satisfy the energy
restriction noted earlier.
That
is,
in the activated com-
plex the solvent configuration must be such
that
the
total
energy of the system, solvent plus reacting par-
ticles, must be the same, regardless of whether these
particles are the reactants or the products.
It
is
of
interest
that
the foregoing discussion can be
rephrased in terms of the Franck-Condon principle:
When one electron configuration
is
formed from the
other
by
an electronic transition, the electronic motion
is
so
rapid
that
the solvent molecules do not have time
to move during the electronic jump.
That
is,
the reac-
tion proceeds
by
way of two successive intermediate
states,
X*
and
X,
which have the same atomic con-
figurations
but
different electronic configurations. Con-
servation
of
energy leads to the requirement
that
the
total energy of these two states must be the same.
The electronic wave function of the activated com-
plex derived previously, a linear combination
of
cf>x•
and
cf>x,
admits
of
a simple interpretation. The function
is
a function
of
the position coordinates of all the elec-
trons of the two reacting particles.
It
can be plotted in a
many-dimensional space as a function
of
all these
co-
ordinates.
In
such a plot
cf>x•
will be large in certain
regions
of
this many-dimensional coordinate space,
and
cf>x
will be large in other regions. The function
cf>x•
will be large when the coordinates of all the electrons
are such
that
the number of electrons in the vicinity
of each of the reacting particles is the same as when these
particles are reactants. Since the electrons are indis-
tinguishable there will be a number
of
such regions in
the many-dimensional space. Similarly,
cf>.,
will be large
when the number
of
electrons in the vicinity
of
each
of
the reacting particles
is
the same as when these
particles are products. Again there will be a number
of
such regions. The wave function for the activated
complex, being a linear combination
of
these two wave
functions,
is
large in all these regions.
The
reaction
ultimately involves going from the regions character-
istic
of
cf>x•
to those characteristic of
cf>x·
Since the wave function
is
the sum
of
two wave func-
tions, each corresponding to a different electronic con-
figuration,
we
can also interpret the wave function as
representing a quantum-mechanical resonance of two
electron configurations, one being the electronic con-
figuration of the reactants, the other
that
of the
products.
Inasmuch as there will be some overlap
of
the elec-
tronic orbitals
of
the two reactants, the description of
the activated complex given in this section
is
but
a first
approximation, which
is
the better the less the overlap.
Reaction Scheme
The occurrence
of
a small overlap in the activated
complex introduces another consideration which
is
normally not present in the usual large-overlap acti-
vated complexes. Since the electronic interaction be-
tween the reacting particles in a small-overlap complex
is
weak, the rate
at
which this electronic interaction can
effect any change
of
electronic configuration may be-
come a slow step in the over-all process. We can envisage
the over-all reaction as occurring in the following way.
As
the two reactants approach each other there
is
a
certain probability
that
a suitable fluctuation of the
solvent molecules which satisfies the restriction de-
scribed in the previous section will occur, such
that
an activated complex
could
be formed. An electronic
interaction
of
the reacting particles could then result
in the correct electronic configuration of the activated
complex. A theoretical treatment
of
this aspect of the
problem could involve the use
of
several quantum-
mechanical methods including the use of time-dependent
perturbation theory
2
•
6
a or electron tunneling formulas.
3
•
4
We shall return to this later. We can suppose, then,
that
when the reactants are near each other a suitable
solvent fluctuation can result in the formation
of
the
state,
X*, whose atomic configuration of the reacting
pair and
of
the solvent
is
that
of the activated complex,
and whose electronic configuration
is
that
of the re-
actants. This state
X* can either reform the reactants
by
disorganization of some of the oriented solvent
molecules, or it can form the state
X
by
an
electronic
transition, this new state having an atomic configura-
tion which
is
the same as
that
of
X*
but
having an
electronic configuration which
is
that
of the products.
The state
X can either reform
X*
by an electronic
transition, or alternatively, the products in this
state
can merely move apart, say.
The pair
of
states
X*
and X constitute the activated
complex.
If
the electronic interaction between them
were large, the formation of one from the other would
be very rapid and one need then not speak
of
them
••
L. Pauling and E. B. Wilson, Introduction
to
Quantum Me-
chanics
(McGraw-Hill Book Company, Inc., New York, 1935).
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OXIDATION-REDUCTION
THEORY
969
separately. An analogous situation also arises in very
different reactions, such as some cis-trans-isomerizations
in which spin-conservation requirements can cause the
effective electronic interaction to be very weak.
7
It
may
also be remarked
that
the term activated com-
plex was defined earlier in the usual way as the atomic
configuration
at
the potential energy maximum along
the reaction coordinate. This does
not
mean, however,
that
the reaction rate can be calculated in the usual
way simply
by
calculating the free energy
of
formation
of the activated complex from the reactants and intro-
ducing this into the well-known absolute reaction rate
theory formula
7
for the
rate
constant. Instead, the
present reaction has been shown to consist of several
elementary steps, several
of
which may be slow.
In
such cases the rate constants of all the elementary steps
must be evaluated individually, and for this purpose,
too, the absolute rate theory formulas will not be used
as such.
The reaction scheme described above can be written
as the sequence Eqs.
(1) to (3).
In
this treatment
it
is
not necessary
that
all
of
the reactants or products have
charges.
In
this reaction sequence A and B will denote
the reactants involved in the electronic transition.
kl
A+B~X*
(1)
k-1
t2
X*~
X
(2)
k-2
k3
X
~products.
(3)
The reverse step
of
(3)
does not have to be consid-
ered, even though
it
may occur when the concentration
of
products
is
appreciable, since
we
are only interested
here in calculating the rate constant of the over-all
forward reaction. The rate constant for the over-all
backward reaction could then be calculated from this
with the aid
of
the equilibrium constant for the over-all
reaction.
The sequence
(1) to (3) will in many cases represent
the complete reaction.
In
more complex systems, how-
ever,
A
and
B may not be the actual compounds intro-
duced into the reaction system,
but
would be the active
entities formed from them. The over-all rate
of
this
reaction sequence will be written as
kb;CaCb where
c's denote concentrations
and
kb;
is
the observed rate
constant
of
this reaction sequence. According to Eq.
(3), the rate
is
also given
by
k3cx.
We may therefore
write
(4)
7 Glasstone, Laidler,
and
Eyring, The Theory
of
Rate Processes
(McGraw-Hill Book Company, Inc., New York, 1941).
The steady-state equations for the concentrations
of
X* and
of
X,
Cx•
and
c,,
are given
by
Eqs. (5) and (6).
Introducing into Eq.
(4)
the value obtained for c,
by solving these simultaneous equations,
we
find
The various rate constants appearing in this expres-
sion for the over-all rate constant,
kb;,
will be esti-
mated in the present paper.
It
is
shown later
that
when the forward step in reaction
(2)
is
more probable,
or about as probable, as the reverse step in reaction
(1),
Eq.
(7)
reduces to a particularly simple form (neglect-
ing factor
of
about two, which
is
of
minor importance):
(8)
Otherwise, Eq.
(7)
would be used. Equation
(8)
will
be used extensively in correlating observed and calcu-
lated rates of oxidation-reduction reactions.
We proceed now to estimate the properties of the
intermediate states X* and
X,
in order to be able to
calculate their rate
of
formation.
Solvation of Activated Complexes
As
noted earlier, in the activated complex all the
solvent molecules are oriented in some nonequilibrium
configuration. This
is
in marked contrast to what
is
usually assumed for large-overlap activated complexes.
In
the latter, the solvent configuration is assumed to be
in equilibrium with the ionic charges
of
the activated
complex. For example,
it
is
generally assumed
that
the
electrical polarization
of
the solvent
at
any
point can
be calculated from the dielectric constant and the ionic
charge and radius
of
the complex, by standard electro-
static procedures.
It
is
usually assumed, for example,
for purposes of calculating the free energy of solvation
of
the complex,
that
the complex can be treated as a
sphere having a charge equal to the sum
of
the charges
of
the reactants.
7
This theory has proved very useful
in interpreting the effect
of
dielectric constant on the
reaction rate. However,
we
have seen
that
such a
description would be quite inapplicable to electron
transfer reactions in which the overlap
of
the electronic
orbitals
of
the two reacting particles is small in the
activated complex.
In
order to calculate the thermodynamic properties,
such as the energy,
of
the intermediate states X* and X
it
is necessary to use expressions which do not assume
that
the solvent molecules are oriented toward the ions
in
an
equilibrium manner. More explicitly, the electrical
polarization
of
the solvent
at
each point is not in elec-
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970
R.
A.
MARCUS
trostatic equilibrium with the electrical field produced
by
ionic charges.
That
is,
it
cannot be predicted from
the known ionic charge distribution
by
standard meth-
ods. Recently, however, a method for calculating the
thermodynamic function of such systems was devised
8
and
will be used to calculate the free energy
of
forma-
tion
of
the intermediate states
X*
and X from the
reactants.
There are
an
infinite number of pairs
of
(thermo-
dynamic) intermediate states,
X*
and
X,
just as there
are an infinite number of thermodynamic states
of
any
system, each pair satisfying the energy restriction
described earlier. Actually
it
is the most probable pair
of
intermediate states which constitutes the activated
complex. The most probable pair of intermediate states
can be determined with the aid of the calculus
of
varia-
tions
by
minimizing the free energy of formation
of
X*
from the reactants subject to the energy restriction
found earlier,
that
is,
subject to the restriction
that
X
and
X*
have the same total energy. This mini-
mization procedure serves to determine the electrical
polarization of the solvent
at
each point of the system
in the intermediate state. This can be used to calcu-
late the free energy of formation
and
rate
of
formation
of
the intermediate
state
from the isolated reactants
in the medium.
A Model for the Reactants
The model which will be used for the structure
of
the
reactants will be closely akin to
that
which is generally
employed in the treatment of ionic interactions.
It
will
be assumed
that
each reactant
may
be treated as a
sphere, which in
turn
may
be surrounded
by
a concen-
tric spherical region of saturated dielectric,
9
outside
of
which the medium
is
dielectrically unsaturated.
We let the sphere bounding the saturated region
have a radius
a.
The radii,
a1
and
a2,
for the two reac-
tants
could change somewhat when the two ions
approach each other though this effect
is
invariably
ignored in the treatment
of
ionic interactions and will
be ignored here. For a given element
of
the Periodic
Table
it
will also depend to some extent on the valence
of the ion.
In
the case
of
monatomic ions, however, a
is
generally assumed to equal the sum
of
the crystal-
lographic radius
and
the diameter of a solvent molecule,
since only the innermost layer
of
solvent molecules
is
usually assumed to be saturated.
9
However, since the
crystallographic radius varies relatively little with the
valence
of
the ion,
10
a would be expected to
vary
but
little with the ion' valence. A refinement
of
the present
s R.
A.
Marcus,
J.
Chern. Phys.
24,
979 (1956).
9 Numerous theoretical
treatments
of
the
free energy of solva-
tion which
have
assumed this model include: (a)
J.D.
Bernal and
R. H. Fowler,
J.
Chern. Phys.
1,
515 (1933); (b) D. D. Eley and
M. G. Evans, Trans.
Faraday
Soc. 34, 1093 (1938);
(c)
E.
J.
W.
Verwey, Rec.
trav.
chim.
61,
127
(1942); (d) R. W. Attree,
Dissertation Abstr.
13,
481
(1953).
1
0
This is especially
true
when
the
valence of
the
ion before
and
after
the
reaction differs
by
only one unit. This will be shown
to
be
the
case of greatest interest, in later applications of this paper.
treatment would take this variation into consideration.
In
general when a
is
slightly different before
and
after
the electron transfer reaction a mean value for
it
will
be adopted. To sum up
we
shall suppose
that
the region
inside a sphere of radius
a about a reactant
is
rigid,
all groups within
a being fully oriented (saturated
dielectric). A refinement of this assumption
of
constant
a will be described in a later paper of this series.
The usual treatment
of
ionic interactions assumes
that
the free energy
of
interaction
of
two ions of
charges
q
1
and
q2
a distance R
apart
in a medium of
dielectric constant
D
is
q
1
q
2
/
DR. This implies several
assumptions
11
•
8
and
we
shall make analogous ones in
the present treatment. We shall
treat
an
ion plus its
rigid, saturated dielectric region as a conducting sphere
of
radius
a.
Now the free energy of the entire system
is
the sum
of
several contributions; one
is
the free energy
of
interaction of all the atoms within one sphere with
each other
and
with the central ionic charge in
that
sphere. A second
is
the free energy of interaction
of
all
the atoms within the sphere about the second reactant
with each other and with the central ionic charge
of
the
second reactant. A third
is
the free energy
of
interaction
of
all the molecules outside of the two spheres with
each other
and
with the charges
of
the spheres. A fourth
is the interaction
of
the two ionic spheres with each
other.
As
in the treatment
of
ionic reactions which
employs the
q1q2/DR
law,
we
observe
that
if,
as as-
sumed, the atoms in the spheres are not to change their
average positions during the mutual approach
of
the
ions, the first two contributions to the free energy will
remain fixed and, therefore,
not
contribute to the free
energy
of
formation of the state X* from the reactants,
and similarly will not contribute to the free energy
of
formation of the products from the state X. The re-
maining two contributions to the free energy are calcu-
lated, as previously observed,
by
treating each ion plus
saturated sphere as a conducting sphere
of
radius
a.
We proceed to consider the properties
of
the dielectric,
assumed unsaturated,
9
outside
of
these saturated spheres.
Electrostatic Characteristics of the Activated
Complex
As
noted previously, each of the intermediate states
X*
and
X,
can be treated as a macroscopic system
having a definite value of the electrical polarization
of
the medium
at
each point of the system. The primary
problem then becomes one
of
determining this polariza-
tion function in these two intermediate states, in the
volume
outside of
that
occupied
by
the two reactants
plus saturated spheres.
The polarization
of
any dielectric medium is generally
regarded as consisting
of
electronic, atomic,
and
orienta-
tion contributions.
As
observed previously, the two
intermediate states
X* and X have similar configura-
tions
of
all atomic nuclei in the system. Since the atomic
11
SeeR.
Platzman
and
J.
Franck,
Z.
Physik
138,
411
(1954).