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:
a
3
for
ligand and the derived expression for the effective concentration depends on
c
eff
a long and flexible chain of optimized length
d
opt
.
The present relations can be advantageously applied for estimating the entropy depen-
dence of both processes affected with effective molarity
EM
: (i) chelate effect, which
depends on the distance
d
separating binding sites in a ligand, (ii) macrocyclic intra-
molecular connection, where the receptor coordination sites are separated by the distance
a
. The enthalpic part of
EM
, which usually translates into ring strains in semirigid and
rigid ligands, is more difficult to access theoretically. Therefore, related thermodynamic
data extracted rigorously from real systems are required for developing satisfactory
models.
3.2.2.3 Number of Metal-Ligand Connections
Supramolecular systems tend to form a maximum of connections in order to mini-
mize D
G
. It usually happens in agreement with the principle of maximum occu-
pancy, although that can be disobeyed in some cases (e.g., lanthanide helicates in
[21]). Although the number of intra- and intermolecular connections can be accessed
via direct counting, it can often be convenient to use a simple mathematical calculus.
We will limit ourselves to discrete binary complexes [M
m
L
n
], which are formed with
m
metal ions and
n
ligands, that is,
N
n
components. In addition, we assume
that metal ions can form
CN
/
x
connections and each ligand has
yx
-dentate binding
sites. To access correctly the total number of connections, two situations must be
considered. First, if all ligand binding sites are occupied by metal ions (ligand satura-
tion), the total number of connections in this assembly is equal to
B
¼
m
þ
y
. This case is
met more often and accounts for the majority of helicates. Second, if all metal binding
sites are occupied (metal saturation), the total number of connections is equal to
B
¼
n
¼
m
CN
/
x
. Among the total number of connections, we distinguish intermolecular (inter
¼
N
inter) connections. The validity of the
formula is illustrated for 1D, 2D and 3D helicates in Figure 3.5. The above relations can
find applications in the quantitative description of self-assembly, and in thermodynamic
modelling (see below).
1
¼
m
þ
n
1) and intramolecular (intra
¼
B
3.2.2.4 Statistical Factors in Polynuclear Systems
The number of possibilities to arrange microspecies in polynuclear complexes signifi-
cantly increases and the statistical factors become an important entropic contribution to
the overall free energy. Fortunately, the symmetry numbers method introduced in Section
3.2.1 is particularly convenient and robust for calculating statistical factors. The applica-
tion of Equation 3.2 is then straightforward. Since its introduction to supramolecular
chemistry by Ercolani
et al.
[10], this method is now systematically applied in the ther-
modynamic modelling of supramolecular assemblies. Probably the most difficult task
consists in determining the point groups of all compounds in equilibrium. An example of
calculating for model triple-stranded helicates with Ga(III) [24] is given in Figure 3.6a
[13]. For comparison, the same assembly is analysed with the direct counting method in
Figure 3.6b. It is evident that applying the latter approach requires much more effort than
using the first method.
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