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change thus refers to the chosen reference state, that is,
c
u
¼
1M, and intramolecular bind-
ing is favoured (
EM
1M). In this context, a deep analysis of
elementary processes in the chelate effect was recently presented by Ercolani [11,12].
The chelate effect efficiency depends on the concentration of
L
and on the magnitude of
EM
. However, the chelating donor atoms in common ligands are localized in the topologi-
cal proximity and form preorganized binding sites, where the reaction with a neighbour-
ing donor atom is strongly favoured because of a much higher effective molarity
(
EM
>
1M) or disfavoured (
EM
<
1M) compared with the donor atoms belonging to other ligand molecules. The
favourable short-distance chelate effect thus occurs in a large domain of ligand concentra-
tions. However, an optimal degree of freedom must be programmed in polydentate ligands
that are often incorporated in segmental ligands for preparing helicates. The optimal ring
size for a maximum energetic gain is five or six atoms including the coordinated metal ion
[2]. The choice of donor atoms and ring sizes depends on the overall system design.
Considering the kinetic point of view, the binding of the second donor atom proceeds
much more rapidly compared to the second monodentate ligand [2]. The
x
-dentate ligand
connects to the receptor in a single connection point and in one reaction step, and such
binding events can be globally viewed as an intermolecular connection. Moreover, a
knowledge of the overall energetic effect associated with different multidentate binding
sites is more suitable for a quantitative comparison between systems, without evaluating
the contributions of individual donor ions. The total number of such intermolecular con-
nections between metallic receptors and identical
x
-dentate ligands will be equal to
CN
/
x
.
The above simplification of the thermodynamic description found a practical utility in the
interpretation and mechanistic descriptions of self-assembly processes [14] and in the ther-
modynamic modelling of stability constants [15]. The global free energy changes related to
this complexation process are given as D
G
inter
¼
0[13,15],where
k
i
is the
microscopic affinity constant related to the intermolecular connection of one
x
-dentate
coordination site
i
to a metallic receptor. In the case of the bipyridyl ligand containing two
nitrogen donor atoms, the affinity constant can be calculated as
k
N
2
¼
k
i
RT
ln
ð
Þ <
N
k
N
is
the microscopic affinity of the pyridyl nitrogen donor atom to the metal ion. D
G
inter
,which
is associated with intermolecular connections, is negative and represents the main driving
force responsible for the completion of thermodynamic assemblies.
2
k
:
EM
,where
3.2.1.2 Statistical Factors
The formation of self-assembled edifices may occur through different reaction pathways.
The number of these possibilities is referred to as the degeneracy of the microscopic state
[16]. This purely entropic driving force corresponds to the changes in rotational degener-
acy related to the transformation of reactants into products. The statistical factors
v
M;L
m
;
n
for
the general complexation process (Equation 3.1) are calculated as
r
[10,13],
where
l
is the number of microspecies [M
m
L
n
] that can be formed, if all the identical
atoms are labelled and
r
corresponds to the same definition for the reverse reaction [13].
v
M;L
m
;
n
¼
l
=
l
r
M;L
m
;
n
½
M
ð
solv
Þ
x
þ
n
L
½
M
m
L
n
þ
x
solv
b
ð
3
:
1
Þ
m
Ð
These factors can be conveniently calculated by using: (i) the direct counting method
[10] or (ii) the symmetry number method [18], which provide equivalent results, as
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