Neutral Heteroleptic Lanthanide Complexes for Unravelling Host-Guest Assemblies in Organic Solvents: The Law of Mass Action Revisited

Article abstract of DOI:10.1021/acs.inorgchem.9b00755

The binding of lanthanide containers [Ln(β-diketonate)₃dig] [dig = 1-methoxy-2-(2-methoxyethoxy)ethane] to aromatic tridentate N-donor ligands (L) in dichloromethane produces neutral nine-coordinate heteroleptic [LLn(β-diketonate)₃] complexes, the equilibrium reaction quotients of which vary with the total concentrations of the reacting partners. This problematic drift prevents the determination of both reliable thermodynamic stability constants and intrinsic host-guest affinities. The classical solution theory assigns this behavior to changes in the activity coefficients of the various partners in nonideal solutions, and a phenomenological approach attempts to quantitatively attribute this effect to some partition of the solvent molecules between bulk-innocent and contact-noninnocent contributors to the chemical potential. This assumption eventually predicts an empirical linear dependence of the equilibrium reaction quotient on the concentration of the formed [LLn(β-diketonate)₃] complexes, a trend experimentally supported in this contribution for various ligands L differing in lipophilicity and nuclearity and for lanthanide containers grafted with diverse β-diketonate coligands. Even if the origin of the latter linear dependence is still the subject of debate, this work demonstrates that this approach can be exploited by experimentalists for extracting reliable thermodynamic constants suitable for analyzing and comparing host-guest affinities in organic solvents.

Full text of DOI:10.1021/acs.inorgchem.9b00755

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Cite This: Inorg. Chem. XXXX, XXX, XXX−XXX
Neutral Heteroleptic Lanthanide Complexes for Unravelling Host−
Guest Assemblies in Organic Solvents: The Law of Mass Action
Revisited
†
†
†
†
†
‡
́
Karine Baudet, Vishal Kale, Mohsen Mirzakhani, Lucille Babel, Soroush Naseri, Celine Besnard,
Homayoun Nozary,† and Claude Piguet*,†
‡Laboratory of Crystallography, University of Geneva, 24 quai E. Ansermet, CH-1211 Geneva 4, Switzerland
†Department of Inorganic and Analytical Chemistry, University of Geneva, 30 quai E. Ansermet, CH-1211 Geneva 4, Switzerland
S
* Supporting Information
ABSTRACT: The binding of lanthanide containers [Ln(β-diketonate)3dig] [dig
= 1-methoxy-2-(2-methoxyethoxy)ethane] to aromatic tridentate N-donor ligands
(L) in dichloromethane produces neutral nine-coordinate heteroleptic [LLn(β-
diketonate)3] complexes, the equilibrium reaction quotients of which vary with
the total concentrations of the reacting partners. This problematic drift prevents
the determination of both reliable thermodynamic stability constants and intrinsic
host−guest affinities. The classical solution theory assigns this behavior to changes
in the activity coefficients of the various partners in nonideal solutions, and a
phenomenological approach attempts to quantitatively attribute this effect to
some partition of the solvent molecules between bulk-innocent and contact-
noninnocent contributors to the chemical potential. This assumption eventually
predicts an empirical linear dependence of the equilibrium reaction quotient on
the concentration of the formed [LLn(β-diketonate)3] complexes, a trend
experimentally supported in this contribution for various ligands L differing in
lipophilicity and nuclearity and for lanthanide containers grafted with diverse β-diketonate coligands. Even if the origin of the
latter linear dependence is still the subject of debate, this work demonstrates that this approach can be exploited by
experimentalists for extracting reliable thermodynamic constants suitable for analyzing and comparing host−guest affinities in
organic solvents.
contribution),2 ensures an adequate procedure for estimating
INTRODUCTION
Basic Thermodynamics of Host−Guest Assemblies: A
Nontrivial Problem. Experimental coordination chemists
■
thermodynamic stability constants by using equilibrium
eq
reaction quotients QL1,,1M,eq = c /cLeqceMq defined by the speciation
LM
expressed in molar concentration units (right part of eq 2).
with (or without) interest in lanthanide complexation reactions
are determined to get reliable free-energy affinities ΔGasso
°
/cθ
eq
eq
γLM
γLγM
a
c
°
L,M
/RT)
LM
LM
accompanying the association between a host ligand (L) and a
guest metal container (M) occurring in a specific solvent
according to eq 1.
asso
β
= e−(ΔG
=
=
aLeqaMeq
1,1
(cLeq/cθ)(cMeq/cθ)
γLM
cθ
1,1,eq
L,M
=
Q
ΔGasso° = −RT ln(β L,M
)
γLγM
L + M V LM
(2)
(1)
1,1
The thermodynamic stability constant βL1,,1M, often unduly
Because coordination chemists are mainly focused on the
determination of the stability constants (and their associated
free-energy changes) by monitoring concentrations at equili-
brium, eq 2 provides considerable difficulties because the
activity coefficients are usually not accessible except for ideal
solutions (γi = 1), where all intermolecular interactions are
identical, a situation characterized by ξ = 0 within the frame of
the classical solution model (ξ is a dimensionless parameter
mixed by experimentalists with the equilibrium reaction
L,M
1,1,eq
quotient Q
expressed in concentration units, is defined
by the van’t Hoff isotherm as the ratio of the activities (aei q) of
the various partners at equilibrium (left part of eq 2). Except
for some rare cases where the activities can be monitored
directly by selective electrodes or by vapor-pressure measure-
ments, most experimental protocols consider the gathering of
concentrations at equilibrium for the estimation of βL1,,1M. The
introduction of activity coefficients γi, which transform mole
fractions (xi) or concentrations (ci) into activities according to
ai = γixi = γi(ci/cθ),1 together with the standard concentration
of the reference state cθ (cθ = 1 M is used for the rest of this
Special Issue: Innovative f-Element Chelating Strategies
Received: March 15, 2019
© XXXX American Chemical Society
A
DOI: 10.1021/acs.inorgchem.9b00755
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Inorganic Chemistry
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that estimates the energy of solute−solvent interactions
relative to that of solvent−solvent and solute−solute
interactions).1 For chemical reactions involving ionic partners
in solution, the Debye−Huckel limiting law provides a
̈
satisfying key because the activity coefficients of the ions
only depend on their intrinsic charge and size and on the ionic
strength of the mixture, a parameter that can be fixed by the
addition of a large excess of dissociated nonreacting electro-
lytes. Even though the ratio γLM/γLγM is unknown for charged
partners at equilibrium, it corresponds to a constant for a fixed
eq
ionic strength. The equilibrium reaction quotient QL1,,1M,eq = c
/
LM
ceLqceMq estimated in concentration units (we take for granted the
original choice of cθ = 1 M) is therefore proportional to the
L,M
thermodynamic stability constant β
(eq 2). For these
1,1
reasons, numerous equilibrium reaction quotients have been
reported for the formation of charged coordination complexes
in polar solvents, for which specific ionic strengths are duly
mentioned.3 The situation becomes critical when association
reactions are conducted (i) in a polar solvent in the absence of
added polyelectrolytes or in a nonpolar (usually organic)
solvent where the ionic strength cannot be fixed and (ii) when
neutral partners are considered. A very rough, but instructive
approach considers the classical solution theory for mastering
the free-energy changes accompanying the mixing of two
nonideal chemical components A and B, with the latter one
being taken as the solvent (B, mole fraction xB), while A (mole
fraction xA = xM + xL + xLM) is assigned to the sum of the
solute particles controlled by equilibrium (1). According to
this hypothesis, the free energy of mixing nRT[xA ln(aA) + xB
ln(aB)] deviates from that of an ideal solution nRT[xA ln(xA) +
xB ln(xB)] by an excess enthalpic contribution nξRTxAxB,
which develops for n moles of the mixture when the solute−
solvent (A−B) interactions differ from the solvent−solvent
(B−B) and solute−solute (A−A) interactions (ξ < 0 means
exothermic mixing and the dominance of favorable solute−
solvent interactions, while the reverse situation characterizes ξ
> 0).1 Because aA = γAxA, the activity coefficient of the solute in
a nonideal binary mixture finds a mathematical expression
ln(γA) = ξ(xB)2 = ξ(1 − xA)2 (Figure 1a), often referred to as
Figure 1. Plots of (a) activity coefficients γsolute and (b) activities asolute
for a solute dispersed in a solvent in a binary mixture according to the
Margules equation.4 ξ is the dimensionless parameter measuring the
energy of the solute−solvent interactions relative to that of the
solute−solute and solvent−solvent interactions.1
°
β
= e−(ΔG
L,M
/RT)
asso
1,1
Ä
Å
Å
Å
É
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
2
2
cLtot + cMtot − c
c
eq
eq
i
j
j
y
z
z
z
z
z
Å
Å
Å
eq cθ
LM
LM
= exp −ξ 1 −
j
j
j
Å
Å
Å
Å
Å
eq
cB
c c
L
M
k
{
Ñ
Ñ
Ö
Å
Ç
Ä
Å
Å
Å
Å
Å
Å
É
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
cLtot + cMtot − c
eq
LM
i
j
j
y
z
z
z
z
z
L,M
Q
1,1,eqcθ
= exp −ξ 1 −
j
j
j
Å
Å
Å
Å
Å
cB
k
{
(5)
Ñ
Ö
Å
Ç
Because (i) the sum cLtot + ctMot − c varies during titration of
eq
LM
the receptor L with an M guest and (ii) βL1,,1M is a constant, eq 5
predicts that the reaction quotients recorded at equilibrium
L,M
eq
Q
= c /ceq ceq will indeed change for various total
1,1,eq
LM LM LM
concentrations of host and guest. It is therefore not so
surprising that titrations of ligands L1−L3 with [La(hfa)3dig]
(eq 6), where hfa is the hexafluoroacetylacetonate anion (see
Figure 2a), in dichloromethane systematically exhibit signifi-
cant variations of the reaction quotients at equilibrium (Figure
2b),5 a trend previously noted for titrations of EDTA4−
(EDTA = ethylenediaminetetraacetic acid) with Ca2+ con-
ducted in aqueous buffered solutions.6
the Margules equation,4 which finally leads to the solute
2
activity aA = γAxA = eξ(1−x ) xA (Figure 1b).
A
Lk + [La(hfa)3dig] V [LkLa(hfa)3] + dig
Using concentration units for diluted solutions (xB ≥ 0.9
ΔGexch° = −RT ln(β Lk,La
)
eq
and cB ≫ ceLq + cMeq + c ), the mole fraction of the solute xA in
(6)
1,1,exch
LM
our “binary” mixture at equilibrium is given in eq 3 and its
activity coefficient in eq 4.
Because (i) the stepwise appearance of free diglyme in
solution exactly matches (stoichiometry 1:1) the complexation
of Lk during the NMR titrations and (ii) [LkLa(hfa)3] is the
only complex characterized in the solid state and in solution,7
there is no ambiguity concerning the exclusive formation of 1:1
complexes in eq 6. In this situation, how may coordination
chemists decide which one of the various reaction quotients
collected in Figure 2b is pertinent for applying the van’t Hoff
eq
eq
LM
cLeq + cMeq + c
cLtot + cMtot − c
LM
xA =
≃
cLeq + cMeq + c + cB
cB
eq
(3)
LM
Ä
Å
Å
Å
Å
Å
Å
É
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
2
eq
LM
cLtot + cMtot − c
i
j
j
y
z
z
z
z
z
γA = γ
= exp ξ 1 −
j
j
j
Å
Å
Å
Å
solute
equation, leading to the searched free-energy change ΔGexch
°
cB
k
{
Å
Ñ
(4)
Å
Ç
Ñ
Ö
associated with equilibrium (6)? According to the classical
solution theory, one expects γsolute → 1 for large concentrations
of reacting partners (Figure 1), a situation impossible to reach
in a diluted solution. On the other hand, working at infinite
dilution should also fix the activity coefficients to constant
values, but their considerable sensitivities to minor changes in
L,M
1,1,eq
Introducing eq 4 to eq 2 yields eq 5, where Q
experimentally accessible reaction quotient, which is (very)
often mistaken by coordination chemists for the thermody-
is the
L,M
namic stability constant β
.
1,1
B
DOI: 10.1021/acs.inorgchem.9b00755
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Figure 2. (a) Graphical illustration of the host−guest association involving the exchange of diglyme (dig) with tridentate ligands L1−L3 around
[La(hfa)3] according to eq 6). Color code: C, gray; O, red; N, blue; F, light blue; La, orange. H atoms are omitted for clarity. Reaction quotients
Q1L,k1,,Lexach,eq = (|LkLa(hfa)3|eq|dig|eq)/(|La(hfa)3dig|eq|Lk|eq) estimated during 1H NMR titrations performed at total 10 mM host concentrations in the
(b) absence and (c) presence of an excess (0.14 M) of diglyme. Color code: L1, red; L2, green; L3, blue.5
the composition of the mixture also make this approach
precarious (Figure 1). We conclude that the systematic
measurements of concentrations instead of activities during
titration processes is a severe handicap, which finds a single
acceptable key for the assembly of ionic partners in polar
solvents where the ionic strength is fixed.
titration is reduced by 1 order of magnitude (Figure 2c).5
For a fixed concentration of diglyme, eq 6 transforms into the
conditional association process summarized in eq 7, which is
reminiscent of equilibrium (1).
Lk + [La(hfa)3] V [LkLa(hfa)3]
Lk,La
β
= β1L,1k,,eLxach/|dig|tot
(7)
1,1,asso
RESULTS AND DISCUSSION
■
Some Practical Attempts To Unravel the Thermody-
namics of Host−Guest Assemblies in Solution. Whereas
host−guest assemblies involving charged species can be
satisfyingly modeled and understood thanks to the theory of
the ionic atmosphere,1 we are not aware of any comparable
rationalization for related reactions conducted between neutral
partners in poorly polar solvents. However, a simple look at the
Margules equation plotted in Figure 1a suggests that γsolute
varies less when at least one member of the chemical entities
under equilibrium can be fixed at high concentration, thus
leveling out the change in the composition accompanying the
complexation reaction. With this in mind, the titrations
summarized in eq 6) were repeated, still using a 10 mM
concentration of ligand receptors L1−L3 but in the presence
of a constant total concentration of 0.14 M diglyme, which is
La
The plots of the occupancy factors θ = |La(hfa)3|ebqound
/
Lk
|Lk|tot as a function of the free concentration of the [La(hfa)3]
guest (left part of eq 8) for the titrations of L1−L3 conducted
in CD2Cl2 + 0.14 M diglyme (diamonds in Figure 3a) indeed
highlight only some minor shifts with respect to the theoretical
binding isotherms (right part of eq 8) built by using a single
Lk,La
1,1,asso
thermodynamic constant β
= (1/N) ∑iN=1 (Q Lk,La
)
i
1,1,asso,eq
taken as the average of the N equilibrium reaction quotients
determined along the titrations (dotted green traces in Figure
3a).5
|La(hfa)3|ebqound
|La|tot − |La(hfa)3|eq
|Lk|tot
La
θ
=
=
Lk
|Lk|tot
Lk,La
β
|La(hfa)3|eq
one of the products of the reaction. As anticipated, the
1,1,asso
Lk,La
1,1,asso,eq
=
variation of the novel equilibrium reaction quotients Q
Lk,La
1 + β
|La(hfa)3|eq
(8)
1,1,asso
= (|LkLa(hfa)3|eq|dig|tot)/(|La(hfa)3dig|eq|Lk|eq) during the
C
DOI: 10.1021/acs.inorgchem.9b00755
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Figure 3. (a) Experimental (diamonds) and fitted (dotted green traces using eq 8; dashed red traces using eq 11) binding isotherms for the
Lk,La
titration of Lk with [La(hfa)3dig] in CD2Cl2 + 0.14 M diglyme at 298 K. (b) Dependences of the equilibrium reaction quotients −RT ln(Q
)
1,1,asso,eq
according to eq 10.5
Lk,La
Lk,La,S
on the progress of the association reactions highlighting ΔG
and ΔG
1,1,asso
1,1,asso
Lk,La
Table 1. Average Free Energies −RT ln(βL1,k1,La) (eqs 8 and 12, Ideal Solutions) and Thermodynamic Parameters ΔG
and
1,1
Lk,La,S
1,1
ΔG
(eq 10, Nonideal Solutions), Determined for the Titrations of Lk with [La(hfa)3dig] in CD2Cl2 + 0.14 M Diglyme (eq
a
7, Columns 2−4) or in Pure CD2Cl2 (eq 6, columns 5−7)
)/kJ·mol−1
ΔGL1,k1,,Lasaso/kJ·mol−1
ΔGL1,k1,,Laass,So/kJ·mol−1
−RT ln(β
)/kJ·mol−1
Lk,La
1,1,exch
ΔGL1,k1,,Lexach/kJ·mol−1
ΔGL1,k1,,Leaxc,Sh/kJ·mol−1
Lk,La
host
−RT ln(β
1,1,asso
L1
L2
L3
L7
−13.2(6)
−9.8(6)
−9.5(8)
−11.5(2)
−12.2(1)
−8.8(1)
−9.0(1)
−11.6(1)
−305(22)
−254(24)
−285(36)
16(9)
−4.6(2.1)
−3.4(2.7)
−3.9(2.4)
−7.8(2.1)
5.9(3)
5.6(3)
4.0(3)
−1.7(2)
−1615(50)
−1503(71)
−2020(92)
−729(27)
a
The uncertainties are those obtained by least-squares fits using eqs 8 and 10.
Interestingly, Castellano and Eggers6 attempted to assign the
measured by the equilibrium concentration of the formed final
eq
change in the activity coefficients occurring during the
titrations to some desolvation processes involving the solvent
molecules in contact with the reactants and products. Their
simple derivation postulates that a p subset of solvent
molecules in second-sphere contact with the partners of the
reactions (Scontact) possesses a specific chemical potential,
which is different from that of the bulk solvent (Sbulk). In these
conditions, the conditional association reaction (eq 7)
transforms into equilibrium (9).
[LkLa(hfa)3] complex (c ). The factor of proportionality is
LM
L,M,S
1,1,asso
written as a free-energy change ΔG
assigned to some
solvation effects accompanying the association reaction, which
is not taken into account by the standard chemical potentials of
the various species at equilibrium (for a detailed derivation, see
Appendix 1 and ref 6).
L,M
1,1,asso,eq
L,M
eq
L,M,S
−RT ln(Q
) = ΔG
+ (c /cθ)ΔG
1,1,asso
LM
1,1,asso
(10)
Empirical plots of −RT ln(QL1,k1,,Lasaso,eq) as a function of
|LkLa(hfa)3|eq for titrations of L1−L3 in the presence of a
constant excess of diglyme (0.14 M) indeed support
approximate linear dependences predicted by eq 10 (Figure
3b), from which the thermodynamics free-energy changes
Lk + [La(hfa)3] + pScontact F [LkLa(hfa)3] + pSbulk
(9)
This approach has been rebutted recently with the support
of sophisticated and probably irrefutable statistical thermody-
namic arguments.8 In order to avoid any controversy, it is fair
to mention here that an intuitive and simple balance of
chemical potentials pertinent to equilibrium (9) led Castellano
and Eggers to propose eq 10,6 which catches the variation of
the activity coefficients, with the progress of the reaction
Lk,La
Lk,La,S
ΔG
and correction factors ΔG
can be extracted
1,1,asso
1,1,asso
(Table 1, columns 2−4).5 The rebuilt binding isotherms using
the latter two fitted parameters in eq 11 slightly improve the
matching with the experimental data (Figure 3a, dashed red
traces).
D
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Figure 4. (a) Dependences of the equilibrium quotient reactions −RT ln(QL1,k1,,Leaxch,eq) on the progress of the association reactions highlighting
ΔGL1,k1,,Leaxch and ΔGLk,La,S
1,1,exch according to eq 10. (b) Experimental (diamonds) and fitted (dashed green traces using eq 12; dashed red traces using eq
13) pseudobinding isotherms for the titration of Lk with [La(hfa)3dig] in CD2Cl2 at 298 K.5
Lk,La
|La(hfa)3|ebqound
|Lk|tot
On the contrary, Castellano−Eggers’ approach summarized
Q
|La(hfa)3|eq
|La(hfa)3|eq
1,1,asso,eq
La
Lk,La
Lk,La
in eq 10, where β
is now replaced with β
, provides
θ
=
=
1,1,asso
1,1,exch
and ΔG
Lk
Lk,La
1,1,asso,eq
Lk,La
1,1,exch
Lk,La,S
1,1,exch
1 + Q
thermodynamic free-energy changes ΔG
(Table 1, columns 6 and 7) adapted to the satisfying
reproduction of the experimental pseudobinding isotherms
using eq 13 (Figure 4b, red traces).
Lk,La
Lk,La,S
= exp{−[(ΔG
+ |LkLa(hfa)3|eqΔG
)
1,1,asso
1,1,asso
/RT]}|La(hfa)3|eq
Ä
Å
Å
Lk,La
Lk,La
Lk,La,S
Q
|La(hfa)3dig|eq
|La(hfa)3dig|eq
Å
/ 1 + exp{−[(ΔG
+ |LkLa(hfa)3|eqΔG
)
1,1,exch,eq
Å
1,1,asso
1,1.asso
La
Lk
Å
Ç
θ
=
É
Ñ
Ñ
Ñ
Ñ
Lk,La
1 + Q
1,1,exch,eq
/RT]}|La(hfa)3|eq
(11)
Ñ
Ö
Lk,La
Lk,La,S
= exp −[(ΔG
+ |LkLa(hfa)3|eqΔG
)
Related linear correlations between −RT ln(QL1,k1,,Lexach,eq) and
|LkLa(hfa)3|eq were observed for the ligand-exchange reaction
(eq 6) conducted in pure deuterated dichloromethane in the
absence of a fixed concentration of diglyme (Figure 4a).
Because the activity coefficients vary much more dramatically
during the exchange process (compare Figure 2b for the
exchange reaction with Figure 2c for the association reaction),
the pseudobinding isotherms rebuilt with eq 12 and using a
{
1,1,exch
1,1,exch
/RT] (|La(hfa) dig| /|dig| )
}
3
eq
eq
Ä
Å
Lk,La
Å
Å
/ 1 + exp −[(ΔG
Å
{
1,1,exch
Å
Å
Ç
Lk,La,S
+ |LkLa(hfa)3|eqΔG
)
1,1,exch
É
Ñ
Ñ
Ñ
Ñ
Ñ
/RT] (|La(hfa) dig| /|dig| )
}
3
eq
eq
(13)
Ñ
Ö
s i n g l e a v e r a g e t h e r m o d y n a m i c c o n s t a n t
Lk,La
1,1,exch
La
β
= (1/N) ∑iN=1 (Q Lk,La ) (Table 1, column 5) are
Note that the occupancy factors θ associated with the
exchange reactions are plotted as a function of the ratio
i
1,1,exch,eq
Lk
clearly not adapted (green traces in Figure 4b).
|La(hfa)3dig|eq/|dig|eq in eqs 12 and 13 because the
concentration of diglyme is no longer constant.5
|La|tot − |La(hfa)3dig|eq
La
Lk
θ
=
=
Lk,La
1,1,exch
The positive free-energy changes ΔG
column 6) imply logically that the replacement of a tridentate
O-donor ligand (diglyme) with a less electronegative tridentate
> 0 (Table 1,
|Lk|tot
β1L,1k,,eLxach(|La(hfa)3dig|eq /|dig|eq)
N-donor ligand Lk around oxophilic trivalent lanthanides is
1 + β Lk,La
1,1,exch(|La(hfa)3dig|eq/|dig|eq)
thermodynamically unfavorable.9 The trend ΔG
>
L1,La
1,1,exch
(12)
E
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Figure 5. (a) 1H NMR titration of L1 with [La(tta)3dig] in CD2Cl2 at 298 K with a numbering scheme (8.3 × 10−3 ≤ |L1|tot ≤ 1.1 × 10−2 M and
4.3 × 10−4 M ≤ |La|tot ≤ 2.7 × 10−2 M). (b) Experimental (diamonds) and fitted (dotted green traces using eq 12; dashed red traces using eq 13)
pseudobinding isotherms. (c) Dependence of the equilibrium reaction quotients −RT ln(QL1,La(tta)
1,1,exch,eq ) on the progress of the association reaction
3
L1,La(tta)3
L1,La(tta)3,S
highlighting ΔG
and ΔG
according to eq 10.
1,1,exch
1,1,exch
ΔGL1,21,,Lexach > ΔG1L,31,,Lexach points to some minor, but specific, effects
of the peripheral lipophilic chains on the energetic balance
pertinent to the ligand-exchange process. In the presence of a
constant and large excess of the leaving diglyme ligands, the
energetic profiles ΔGL1,k1,,Lasaso < 0 (Table 1, column 3) reflect the
solvation energies produced by dipole molecules immersed in a
10
dielectric (1−30 kJ·mol−1). We conclude that ΔG
Lk,La,S
or
1,1,exch
Lk,La,S
ΔG
cannot be interpreted in terms of simple contact
solvation but that they mainly reflect the changes in the activity
coefficients, a phenomenon that becomes crucial at high
1,1,asso
LM
eq
L,M,S
1,1
affinity of the neutral lanthanide carrier [La(hfa)3] for the
concentration because the (c /cθ)ΔG
terms in eqn 10
L1,La
entering tridentate Lk ligand. The associated trend ΔG
<
become dominant and drive the quantitative replacement of
diglyme with Lk at decimolar concentrations. Attempts to
separate some pertinent enthalpic and entropic contributions
using low-temperature NMR proved to be very delicate
because of the technical limitations of our setup, which
required a return to room temperature for opening of the
NMR tube and the addition of one more crop of metal prior to
restoration of the working temperature. The resulting scattered
data are too imprecise to be exploited for the exchange process
(equilibrium 6; see Figure S1 and Table S1). For the
conditional association reaction (eq 7), the trends are more
1,1,asso
L2,La
L3,La
ΔG
≈ ΔG
is surprisingly reversed compared with
1,1,asso
1,1,asso
that found for ΔGL1,k1,,Lexach, which suggests some delicate balances
in solvation processes affecting the lipophilic N-donor ligand
upon changes in the nature of the solvent (pure dichloro-
Lk,La
methane for ΔG
and dichloromethane with 0.14 M
1,1,exch
diglyme for ΔGL1,k1,,Lasaso). Castellano−Eggers’ approach (eq 10)
Lk,La,S
provides a second parameter ΔG
or ΔGLk,La,S
1,1,asso, which is
1,1,exch
supposed to estimate the effect of the change in contact
solvation. The data collected in Table 1 (columns 4 and 7)
show huge negative values with no relationship with standard
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Figure 6. (a) 1H NMR titration of L1 with [La(pbta)3dig] in CD2Cl2 at 298 K with a numbering scheme (7.7 × 10−3 ≤ |L1|tot ≤ 1.1 × 10−2 M and
1.8 × 10−4 M ≤ |La|tot ≤ 2.7 × 10−2 M). (b) Experimental (diamonds) and fitted (dotted green traces using eq 12; dashed red traces using eq 13)
L1,La(pbta)3)
1,1,exch,eq
pseudobinding isotherms. (c) Dependence of the equilibrium reaction quotients −RT ln(Q
on the progress of the association reaction
L1,La(pbta)3
L1,La(pbta)3,S
highlighting ΔG
and ΔG
according to eq 10.
1,1,exch
1,1,exch
Lk,La
precise and probably reliable for ΔG
(see Figure S2 and
nonsymmetrical [La(tta)3dig] (Htta = 2-thenoyltrifluoroace-
tone; Figure 5a) and [La(pbta)3dig] (Hpbta = perfluorophe-
nyltrifluoroacetone; Figure 6a) complexes, with the structure
of the latter being isostructural with that recently reported for
[Eu(pbta)dig] (see Figure S3 and Tables S3−S7).12 Again, a
simple look at the experimental binding isotherms extracted
1,1,asso
Table S2), which are characterized by slightly unfavorable
Lk,La
enthalpy changes (ΔH
≥ 0). The opposite entropic
≪ 0) are responsible for the
1,1,asso
Lk,La
contributions (−TΔS
1,1,asso
driving force leading to nonnegligible lanthanide-ligand
association (ΔGL1,k1,,Lasaso < 0) at room temperature, a mechanism
in line with the compensation model popularized during the
early 1980s by Choppin for rationalizing lanthanide complex-
ation in aqueous solution.11
Applying the “Contact-Solvent Correction” Patch to
the Binding of Nonsymmetrical [La(β-diketonate)3]
Guests to Tridentate Receptors L1−L3. After exploration
of the systematic lipophilic changes in the ligands L1−L3, the
nature of the lanthanide containers was varied by using the
1
from the H NMR titrations of L1−L3 with these nonsym-
metrical lanthanide containers in pure dichloromethane
[equilibrium (6); Figures 5b, 6b, and S4 and S5] or in
dichloromethane containing 0.14 M diglyme [equilibrium (7);
Figure S6] unambiguously shows that they obey eq 10, thus
leading to approximately linear dependences of the logarithms
of the equilibrium reaction quotients on the concentrations of
the [LkLaX3] adducts (X = tta, pbta; Figures 5c and 6c). The
G
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) (eqs 8 and 12) and Thermodynamic Parameters ΔGL1,k1,LaX3 and ΔG1L,k1,LaX ,S (eq
Lk,LaX3
3
Table 2. Average Free Energies −RT ln(β
1,1
10), Determined for the Titrations of Lk with [La(X)3dig] (X = tta, pbta) in CD2Cl2 + 0.14 M Diglyme
host/guest
−RT ln(β1,1,asso
Lk,La) /kJ·mol−1
ΔGL1,k1,,Lasaso/kJ·mol−1
ΔGL1,k1,,Lasas,So/kJ·mol−1
−RTln(β
)/kJ·mol−1
Lk,La
1,1,exch
ΔGL1,k1,,Leaxch/kJ·mol−1
ΔGL1,k1,,Leaxc,Sh/kJ·mol−1
L1/tta
L2/tta
L3/tta
L1/pbta
L2/pbta
L3/pbta
−10.4(6)
b
b
−12.8(2)
−9.3(9)
b
−8.8(1)
b
b
−12.3(1)
−8.4(2)
b
−463(16)
b
b
−100(24)
−524(69)
b
−2.5(1)
−0.2(2)
3.0(1)
0.6(2)
−0.3(2)
−1.8(2)
3.0(2)
4.6(1)
7.0(3)
6.2(1)
6.0(1)
4.6(2)
−848(30)
−973(15)
−1510(122)
−1197(24)
−1368(18)
−2080(80)
a
b
The uncertainties are those obtained by least-squares fits of eqs 8 and 10. Too small affinities for being analyzed by NMR at 10 mM
concentrations.
Lk,LaX3
ligands (1.5 ≤ μ ≤ 3.4 D),10,12
Δ
solvG0[LkLaX ] (−10 to −20 kJ·
associated thermodynamic free-energy changes ΔG
and
1,1,exch
3
Lk,LaX3,S
mol−1) dominates ΔsolvGL0k (−1 to −2 kJ·mol−1) by 1 order of
ΔG
for the exchange reaction (CH2Cl2 + 0 M diglyme)
1,1,exch
Lk,LaX3
Lk,LaX3,S
1,1,asso
and ΔG
and ΔG
for the conditional association
1,1,asso
magnitude. With this in mind, eq 15 predicts that the changes
in ΔsolvG[0LkLaX ] control those of ΔG1L,k1,,LexacXh3,sol along the ligand
reactions (CH2Cl2 + 0.14 M diglyme) are gathered in Table 2
and can be directly compared with those previously reported
for the symmetrical [Ln(hfa)3dig] container (Table 1).
While the magnitude of the contact-solvent corrections,
which take into account the change in the activity coefficients,
is difficult to rationalize (column 7 in Tables 1 and 2), the
exchange free energies extrapolated at infinite dilutions
3
series for a given lanthanide container. Consequently, the only
minor changes computed recently10 for the total pseudospher-
ical volumes and electric dipoles accompanying specific ligand
substitution on going from L1 to L3 fully justify the limited
Lk,LaX3
variation of ΔG
observed in solution (column 6 in
1,1,exch,sol
Tables 1 and 2).
(column 6 in Tables 1 and 2) are all positive and do not
Lk,LaX3
1,1,exch
Considering the alternative situation, for which different
lanthanide containers are connected to the same tridentate
ligand Lk, the situation is more complicated because the
Born−Haber cycle in eq 14 translates into eq 16, which
contains three variable contributions.
largely vary (3.0 ≤ ΔG
≤ 7.0 kJ·mol−1) whatever the
choice of ligands or lanthanum containers. The Born−Haber
thermodynamic cycle built in Figure 7 and summarized in eq
14 is well-suited for clarifying this observation.
Lk,LaX3
1,1,exch,sol
Lk,LaX3
Lk,LaX3
1,1,exch,sol
Lk,LaX3
+ ΔsolvG[0LkLaX ]
ΔG
= ΔG
+ ΔsolvGd0ig − ΔsolvG[0LaX dig]
ΔG
= ΔG
1,1,exch,gas
3
1,1,exch,gas
3
+ ΔsolvG[0LkLaX ] − ΔsolvGL0k
− ΔsolvG[0LaX dig] + constant
(14)
3
(16)
3
The dipole moments of the lanthanide containers [La-
(hfa)3dig], [La(pbta)3dig], and [La(tta)3dig] decrease in the
order hfa (7.7 D) > pbta (6.7 D) > tta (4.6 D), whereas those
of [L1La(hfa)3], [L1La(pbta)3], and [L1La(tta)3] are more
scattered (hfa, 13.8 D; pbta, 13.1 D; tta, 14.9 D).12 With these
numbers in hand, the pertinent difference ΔsolvG[0LkLaX ]
−
3
Δ
solvG0[LaX dig] found in eq 16 is systematically negative and
3
maximizes its favorable contribution according to the order tta
≪ hfa ≈ pbta if we reasonably assume similar sizes for
[L1La(X)3]. We therefore expect more favorable free energies
of exchange for the [La(tta)3] container, a trend roughly
obeyed in Tables 1 and 2 (column 6).
Figure 7. Thermodynamic cycle for ligand exchanges around
[La(X)3] containers (X = hfa, tta, pbta).
For a given lanthanide container [LaX3], the gas-phase
Lk,LaX3
Extending the “Contact-Solvent Correction” Patch to
Successive Host−Guest Assemblies in Solution. The
detailed thermodynamic investigation of the association of
monotridentate ligands L1−L3 with the [La(X3)dig] container
contribution ΔG
is not expected to vary significantly
1,1,exch,gas
because peripheral substitution on going from L1 to L3 has
little, if any, inductive effect on the basicity of the N-donor
atoms of the entering tridentate ligands. With the contribution
of ΔsolvG0dig − ΔsolvG0[LaX dig] being independent of the nature of
in dichloromethane demonstrated that the equilibrium
3
L,M
quotient reaction Q
may be an ambiguous reporter of
1,1,asso,eq
the ligands, the Born−Haber cycle summarized in eq 14 can be
abridged to give eq 15.
L,M
the thermodynamic stability constants β
. Irrespective of
1,1,asso
the chemical origin of the dependence of the equilibrium
reaction quotients on the advance of the association reactions
(change in the activity coefficients according to the classical
solution theory and/or consequences of contact-solvent
molecules on the chemical potential of the solvent), the
Lk,LaX3
1,1,exch,sol
ΔG
= ΔsolvG[0LkLaX ] − ΔsolvGL0k + constant
(15)
3
The variable solvation energies of the neutral dipolar
molecules Lk and [LkLaX3] can be estimated with the help
of the Onsager equation, which predicts a μ2/RH3 dependence
(μ is the dipole moment of the particle, and RH is the radius of
a spherical cavity cut from the dielectric when a spherical
solute is immersed into the solvent).13 Because of the much
larger dipole moments calculated for the [LkLaX3] complexes
(13.1 ≤ μ ≤ 14.9 D) compared with those of the free Lk
L,M
LM
empirical eq 10, −RT ln(QL1,,1M,asso,eq) = −RT ln(β
) + (c /
1,1,asso
eq
cθ)ΔGL,M,S
1,1,asso, proposed by Castellano and Eggers6 satisfyingly
correlates these two parameters, thus leading to a specific free-
L,M
L,M
energy change at infinite dilution ΔG
= −RT ln(β
).
1,1,asso
1,1,asso
However, in coordination and supramolecular chemistry, the
consideration of a single equilibrium is rare and eq 10 leads to
H
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Figure 8. Host−guest association for the connection of ditridentate ligands L4−L6 to [La(hfa)3] containers.
Lk,La
major inconsistencies when, for instance, the ditridentate
ΔG
= −RT ln(β Lk,La
)
1,1,asso
1,1,asso
ligands L4−L6 successively fix two [La(hfa)3] guests in
CH2Cl2 + 0.14 M diglyme to give [LkLa(hfa)3] or [Lk(La-
(hfa)3)2] (eqs 17 and 18 and Figure 8).10
Lk,La
1,1,asso,eq
= −RT ln(Q
)
− (|LkLa(hfa)3|eq /cθ)ΔG
Lk,La,S
1,1,asso
(19)
Lk,La
Lk + [La(hfa)3] F [LkLa(hfa)3]
ΔG
= −RT ln(β Lk,La
)
1,2,asso
1,2,asso
Lk,La
Lk,La
Q
and β
Lk,La
1,1,asso,eq
1,1,asso
(17)
= −RT ln(Q
)
1,2,asso,eq
Lk,La,S
− (|Lk(La(hfa)3)2|eq /cθ)ΔG
1,2,asso
(20)
Lk + 2[La(hfa)3] F [Lk(La(hfa)3)2]
Because G is a state function, standard thermodynamics
Lk,La
Lk,La
Q
and β
1,2,asso
LkLa,La
1,2,asso,eq
(18)
require that the free-energy change ΔG
= −RT
1,1,asso
ln(KLkLa,La
1,1,asso ) associated with the fixation of the second
Application of eq 10 to equilibria (17) and (18) provides
[La(hfa)3] container to [LkLa(hfa)3] (eq 21) is given by the
I
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LkLa,La
1,1,asso
difference between the two cumulative processes ΔG
=
Lk,La
Lk,La
ΔG
− ΔG
.
1,1,asso
1,2,asso
[LkLa(hfa)3] + [ln(hfa)3] F [Lk(La(hfa)3)2]
LkLa,La
1,1,asso,eq
LkLa,La
Q
and K
1,1,asso
(21)
In ideal solutions, the equilibrium reaction quotients strictly
mirror the thermodynamic stability constants, and eq 22 holds.
LkLa,La
LkLa,La
ΔG
= −RT ln(K
) = −RT ln(Q LkLa,La
)
1,1,asso
1,1,asso
1,1,asso,eq
Lk,La
= −RT ln(Q Lk,La
1,2,asso,eq/Q
)
1,1,asso,eq
(22)
For nonideal solutions, the use of eqs 19 and 20 gives eq 23
LkLa,La
LkLa,La
ΔG
= −RT ln(K
)
1,1,asso
1,1,asso
= − RT ln(Q Lk,La
1,2,asso,eq/Q
)
1,1,asso,eq
Lk,La
Lk,La,S
− (|Lk(La(hfa)3)2|eq /cθ)ΔG
1,2,asso
Lk,La,S
+ (|LkLa(hfa)3|eq /cθ)ΔG
1,1,asso
(23)
which is close, but not equal, to the application of Castellano−
Eggers’ approach (eq 10) to equilibrium (21) summarized in
eq 24.
LkLa,La
LkLa,La
ΔG
= −RT ln(K
)
1,1,asso
1,1,asso
Lk,La
= − RT ln(Q Lk,La
1,2,asso,eq/Q
)
1,1,asso,eq
− (|Lk(La(hfa)3)2|eq /cθ)ΔG
LkLa,La,S
1,1,asso
(24)
The unacceptable discrepancy between eqs 23 and 24
originates from the contribution of contact solvation, which
considers different corrections for the formation of [Lk(La-
(hfa)3)2] depending on the “mechanism” of its formation,
either from Lk and two [La(hfa)3] containers [equilibrium
(18) and eq 20] or from [LkLa(hfa)3] and one [La(hfa)3]
container [equilibrium (21) and eq 24]. This limitation can be
overcome by considering the sum of the two cumulative
complexation reactions in equilibrium (25) because the
relative fractions of each formed complex [LkLa(hfa)3] and
[Lk(La(hfa)3)2] during the titration procedure are now
associated with their specific contact solvation correction in
Figure 9. Plots of (a) −RT ln(Q L4,La
2,3,asso,eq) as a function of
|L4La(hfa)3|eq and |L4(La(hfa)3)2|eq according to eq 26). (b)
Projections onto the |L4(La(hfa)3)2|eq = constant plane (left) and
|L4La(hfa)3|eq = constant plane (right). (c) Projection roughly
orthogonal to the best least-squares plane (shown in blue) for the
titration of L4 with [La(hfa)3dig)] in CH2Cl2 + 0.14 M diglyme (298
K).
Lk,La
Lk,La
eq 26. Now, the Born−Haber cycle ΔG
= ΔG
+
2,3,asso
1,1,asso
Lk,La
ΔG
is obeyed (see eqs 19, 20, and 26).
1,2,asso
and “solvation” corrections ΔGLk,La,S
1,1,asso and ΔGLk,La,S
1,2,asso gathered in
2Lk + 3[La(hfa)3] F [LkLa(hfa)3] + [LkLa(hfa)3]
Table 3.
Lk,La
Lk,La
Q
and β
1,1,asso
The fitted binding isotherms rebuilt using eq 27 are shown
as red traces in Figure 10 and do not display major
improvements compared with the previously reported fits10
taking the constant activity coefficients for granted (green
dotted traces in Figure 10).
1,1,asso,eq
(25)
Lk,La
ΔG
= −RT ln(βLk,La
)
2,3,asso
2,3,asso
Lk,La
2,3,asso,eq
= − RT ln(Q
)
Lk,La,S
− (|LkLa(hfa)3|eq /cθ)ΔG
Lk,La,0
Table 3. Free-Energy Changes ΔG
and Associated
1,1,asso
2,3,asso
Contact Solvation Variations ΔGLk,La,S
1,1,asso and ΔGLk,La,S
1,2,asso (eq 26)
Lk,La,S
− (|Lk(La(hfa)3)2|eq /cθ)ΔG
2,1,asso
(26)
for the Titration of Lk with [La(hfa)3dig] in CH2Cl2 + 0.14
M Diglyme (298 K)
Mathematically speaking, eq 26 corresponds to the equation
of a plane in a three-dimensional ⟨|LkLa(hfa)3|eq; |Lk(La-
(hfa)3)2|eq; −RT ln(QL2,k3,,Lasaso,eq)⟩ Cartesian frame (Figures 9 and
S7 and S8). Bilinear least-squares fits of the titration data
collected10 for the titrations of L4−L6 with [La(hfa)3dig)] in
CH2Cl2 + 0.14 M diglyme provide total free-energy changes
L4
L5
L6
ΔGL2,k3,,Lasas,0o/kJ·mol−1
ΔGL1,k1,,Lasas,So/kJ·mol−1
ΔGL1,k2,,Lasas,So/kJ·mol−1
−36.1(8)
−514(351)
525(144)
−32.2(7)
552(282)
−378(100)
−43.3(8)
1190(357)
559(158)
J
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Deciphering Lipophilicity as a Major Issue for
Programming Host−Guest Assemblies in Solution.
Having now in hand a safe procedure for analyzing the
intrinsic affinities of lanthanide containers for tridentate
binding sites, it is worth questioning the unsolved stepwise
decrease of the latter parameter with the length of the
oligomeric host (Figure 11).10,14 The logarithmic dependence
Figure 10. Experimental (diamonds) and fitted (dashed red traces
using eq 27) binding isotherms for the titrations of (a) L4, (b) L5,
and (c) L6 with [La(hfa)3dig] in CH2Cl2 + 0.14 M diglyme (298 K).
The green dotted traces correspond to previous fits assuming fixed
activity coefficients as reported in ref 10.
Figure 11. (a) Chemical structures and (b) Intrinsic association free
Lk,La
energies −RT ln(f ) for equilibrium (7) using monomeric ligand
asso
L1 (N = 1), dimer L4 (N = 2), and polymers PN (N = 10, 12, 20, and
31) with [La(hfa)3] (N is the number of available tridentate binding
sites; CD2Cl2 + 0.14 M diglyme, 298 K).10,14
|LkLa(hfa)3|eq + 2 |Lk(La(hfa)3)2|eq
La
Lk
θ
=
2 |Lk|tot
(27)
Lk,La
Lk,La
of −RT ln(f
) on the increasing size (N) of the
When standard thermodynamics is followed (i.e., ΔG
1,2,asso
=
asso
2,3,asso
Lk,La
multitridentate oligomers (Figure 11b) was tentatively
assigned in ref 10 to the balance of the solvation energies,
which is controlled by the systematic freezing of rotational
degrees of freedom accompanying complexation of the
entering metals to the receptors to give the target metal-
lopolymers {PN[La(hfa)3]n} (1 ≤ n ≤ N). However, a simple
look at PN shows that the amount of lipophilic chains also
linearly grows with the number of binding sites N and may
therefore offer an alternative explanation, which is supported
by the decrease of the affinities of the tridentate binding site
for [La(hfa)3] on going from L1 [no lipophilic hexyloxy
ΔG
+ ΔGL1,k1,,Lasaso), the introduction of contact-solvent
corrections for multiple successive complexation processes
(eqs 17 and 18) performed in excess of diglyme does not
improve significantly the thermodynamic analysis (Figure 10).
Consequently, the cooperativity factors for the successive
binding of two [La(hfa)3] to L4 [ΔELa−La = 0.1(2) kJ·mol−1],
L5 [ΔELa−La = 1.15(5) kJ·mol−1], and L6 [ΔELa−La = −0.8(2)
kJ·mol−1]10 deduced from the simple site-binding model
applied in the absence of contact-solvent corrections (eqs 28
Lk,La
and 29) are reliable (f
is the intrinsic affinity, and ΔELa−La
asso
is the closest-neighbor intersite interaction).10
= −RT ln(f ) = −12.2(1) kJ·mol−1] to L3
L1,La
L1,La
chains; ΔG
asso
asso
L3,La
asso
L3,La
asso
[four lipophilic hexyloxy chains; ΔG
− RT ln(f
=
=
Lk + [La(hfa)3] F [LkLa(hfa)3]
−9.0(1) kJ·mol−1]. Interestingly, the intrinsic affinity ΔG
L4,La
asso
Lk,La
L4,La
asso
ΔG
= −RT ln(β Lk,La
)
−RT ln(f ) = −11.6(1) kJ·mol−1 of the dimeric ligand L4,
1,1,asso
1,1,asso
Lk,La
which possesses two binding sites separated by a phenyl spacer
bearing two hexyloxy chains, lies between those of L1 and L3.
The nonsymmetrical monomeric ligand L7 (Figure 12a and
Scheme 1) with two lipophilic hexyloxy chains thus represents
the missing link for assigning the thermodynamic trend
depicted in Figure 11 either to the larger size of the molecular
receptor or to the larger amount of lipophilic hexyloxy chains
brought by the disubstituted p-phenylene spacers as N
= −RT ln(f
) − RT ln(2)
(28)
(29)
asso
Lk + 2[La(hfa)3] F [Lk(La(hfa)3)2]
Lk,La
ΔG
= −RT ln(β Lk,La
)
1,2,asso
1,2,asso
Lk,La
= −2RT ln(f
asso
) + ΔELa−La
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Inorganic Chemistry
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Figure 12. (a) Association reaction between the nonsymmetrical ligand L7 and [La(hfa)3dig]. (b) Experimental (diamonds) and fitted (dotted
green traces using eq 8; dashed red traces using eq 11) binding isotherms for the titration of L7 with [La(hfa)3dig] in CD2Cl2 + 0.14 M diglyme at
L7,La
298 K. (c) Dependence of the equilibrium reaction quotients −RT ln(QL1,71,,Lasaso,eq) on the progress of the association reaction highlighting ΔG
1,1,asso
L7,La,S
and ΔG
according to eq 10.
1,1,asso
Scheme 1. Intrinsic Binding Affinities of Monomeric
Ligands L1, L3, and L7 for La(hfa)3 in CD2Cl2 + 0.14 M
Diglyme at 298 K
We conclude that the surprising logarithmic dependence
observed for the intrinsic affinity on the length of the polymer
(Figure 11b) does not originate from the increasing number of
connected binding sites as originally thought,10 but it is the
result of the reduced intrinsic affinities of the tridentate
binding units for La(hfa)3 when peripheral lipophilic 1,4-
dihexyloxyphenyl substituents are connected to the distal
benzimidazole rings (Scheme 1). In other words, the lipophilic
ligand L3 may be considered as a valuable model of the
binding sites in the bulk of the linear polymers PN, whereas L7
is well-suited for modeling the terminal sites of these polymers.
However, ligand L1 is not pertinent for modeling any of the
binding sites in polymer PN.
CONCLUSIONS
■
The host−guest assemblies between the neutral lanthanum
carriers [La(β-diketonate)3] and the tridentate N-donor
binding sites found in ligands L1−L7 unambiguously rely on
nonideal solute−solution mixtures in dichloromethane. The
additional nonelusive enthalpic contribution to the mixing
entropy modeled by the classical solution theory predicts that
the equilibrium reaction quotients Qeq deduced from molar
speciation in solution (i) significantly deviate from the
thermodynamic equilibrium constants β and (ii) depend on
the total concentrations of the hosts and guests (eq 5). These
previsions were experimentally confirmed by systematic NMR
titrations (Figure 2). For an analytical chemist, this behavior is
usually assigned to some changes in the activity coefficients γi,
and only setting them to a fixed value during the complete
titrations is acceptable for restoring standard binding isotherms
reminiscent of those observed in ideal solutions (γi = 1). This
situation is common for reactions involving ionic partners in
increases. The tricky synthesis of L7 is reported in Appendix 2.
1H NMR titrations of L7 with [La(hfa)3dig] conducted in the
absence of diglyme (Figure S9) and in the presence of an
excess (0.14 M) of diglyme (Figures 12 and S10) indeed gave
= −RT ln(f ) = −11.5(2) kJ·mol−1 (Table 1), a
L7,La
L7,La
ΔG
asso
asso
value intermediate between those found for the related
monomeric ligand without a lipophilic chain (L1) and that
L7,La
with four chains (L3). Moreover, the ΔG
found for the
polar solvents, and the Debye−Huckel approach demonstrates
̈
asso
L4,La
nonsymmetrical monomeric ligand L7 exactly fits ΔG
=
that a large and constant ionic strength ensures constant
activity coefficients. For host−guest associations involving
neutral partners, rough consideration of the classical solution
asso
−11.6(1) kJ·mol−1, where two related tridentate sites are
connected in the dimeric ligand L4.
L
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Inorganic Chemistry
Forum Article
theory limited to binary mixtures combined with the Margules
equation (summarized in eq 5) suggests that setting one of the
partners of the reaction at a large and constant concentration
should restore proportionality between the thermodynamic
stability constant β and equilibrium reaction quotients Qeq. We
therefore decided to set the concentration of diglyme (dig) at
0.14 M for our association reactions summarized in eq 6, which
corresponds to sufficient excess for considering the concen-
tration of this partner of the reaction as dominant and constant
during the NMR titrations of ligands L1−L7 with [La(β-
diketonate)3dig] containers. In these conditions, only minor
drift could be detected between the ideal binding isotherms
(dashed green traces in Figures 3a, 10, and 12) and the
experimental occupancy factors (black diamonds in these
figures). We therefore recommend this approach for the easy
collection of conditional stability constants pertinent to
association processes involving neutral partners in nonideal
solutions. When working with pure solvents with no excess of a
given partner of the reaction in solution, the fluctuation of the
activity coefficients with the evolution of the nonideal mixtures
prevents extraction of the pertinent stability constants from the
various equilibrium reaction quotients (see, for instance, the
green traces in Figures 4b, 5b, and 6b). On the basis of some
interesting and chemically intuitive considerations of the
solvent reorganization processes, which accompany the host−
guest assembly depicted in eq 9, Castellano and Eggers6
proposed that deviation from the ideality estimated by RTξ[1
solvent has changed. However, there is no reason for inducing
any significant reduction of the fluctuation of the activity
coefficients during the titrations. This statement was confirmed
1
by repeating the H NMR titrations of L1−L3 in dichloro-
methane by adding 0.2 M of either benzene (Figure S11) or
NBu4PF6 (Figure S12), which indeed showed no smoothing in
the change of the activity coefficients of the reacting partners
(Tables S8 and S9).
ASSOCIATED CONTENT
* Supporting Information
The Supporting Information is available free of charge on the
ACS Publications website at DOI: 10.1021/acs.inorg-
chem.9b00755.
■
S
Derivation of eq 10) (Appendix 1), synthesis of ligand
L7 (Appendix 2), tables of crystal data, geometric
parameters, and thermodynamic data, and figures
1
showing molecular structures, H NMR titrations, and
thermodynamic binding isotherms (PDF)
Accession Codes
CCDC 1902650 contains the supplementary crystallographic
data for this paper. These data can be obtained free of charge
via www.ccdc.cam.ac.uk/data_request/cif, or by emailing
data_request@ccdc.cam.ac.uk, or by contacting The Cam-
bridge Crystallographic Data Centre, 12 Union Road,
Cambridge CB2 1EZ, UK; fax: +44 1223 336033.
− (cLtot + ctMot − c /cB)]2 in eq 5 in terms of free energy could
eq
LM
AUTHOR INFORMATION
Corresponding Author
*E-mail: Claude.Piguet@unige.ch.
be modeled using a simple linear correction ΔGScLeqM in eq 10
■
(Appendix 1). ΔGS thus stands for some additional changes in
the solvent−solute contact interactions, which are not taken
into account by the chemical potential of the pure solvent and
partners. The empirical application of eq 10 to the NMR
ORCID
́
Celine Besnard: 0000-0001-5699-9675
titrations of monomeric ligands L1−L3 and L7 indeed shows
Claude Piguet: 0000-0001-7064-8548
eq
LM
roughly linear plots between −RT ln(Qeq) and c (Figures 4a,
Notes
5c, 6c, and 12), a behavior paralleled in two dimensions when
using ditridentate ligands L4−L6 with two available binding
sites (Figure 9). This gives access to the free-energy change
The authors declare no competing financial interest.
ACKNOWLEDGMENTS
■
L,M
eq
ΔG
at infinite dilution (c → 0), which is related to a
1,1
LM
Financial support from the Swiss National Science Foundation
is gratefully acknowledged. We thank Prof. Tatjana Parac-Vogt
(Katholieke Universiteit Leuven) and Prof. Eric Bakker
(University of Geneva) for fruitful discussions.
L,M
L,M/RT
1,1
thermodynamic equilibrium constant β
= e−ΔG
,
1,1
together with a second parameter ΔGS, which measures the
sensitivity of the activity coefficients to the exact composition
of the nonideal mixture. The often huge experimental values
estimated for ΔGS (hundreds to thousands of kilojoules per
mole in Tables 1 and 2) prevent its interpretation as a
straigthforward balance of solvation energies brought by the
replacement of the reactants with products during the chemical
reaction because the Onsager equation returns a maximum of a
few tens of kilojoules per mole for the solvation energies of
these complexes and ligands in dichloromethane.10 Whatever
its theoretical justification, ΔGS can be considered by
experimental coordination chemists as a constant specific to
a given solvent and a specific reaction, which transforms the
equilibrium reaction quotients QL1,,1M,eq obtained by speciation at
equilibrium into a single thermodynamic constant at infinite
dilution β1L,,1M. This approach restores some pertinent
comparisons between the intrinsic affinities recorded for
various ligands (Scheme 1) and different lanthanide containers
(Table 2) in a given solvent. It is probably worth mentioning
here that the addition of external chemical species, not
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