Thermodynamics and kinetics of indole oligomerization in 0.5molL -1 aqueous sulfuric acid: Evaluation of some temperature-dependant parameters

Article abstract of DOI:10.1002/poc.3319

Indole is present in a wide variety of natural compounds with physiological activities, as well as it is a very important intermediate in medicinal and industrial chemistry. For this reason, the evaluation of indole protonation, oligomerization equilibria and the study of the kinetics of dimer and trimer formation in diluted sulfuric acid at various temperatures are of paramount importance because of their practical and scientific implications. Here, we calculate the protonation equilibria by using the literature titration data together with a quantum chemical computational approach, in order to obtain reliable pK_a value of indole from 288 to 313K. Starting from these calculations, we are able to measure the oligomerization equilibrium constants of indole, their kinetic constants, whose values are dependent of indole pK _a, at different temperatures. Enthalpy and entropy of the reactions are calculated by means of Van't Hoff equation, while the activation parameters of Eyring-Evans-Polanyi equation are evaluated for the whole kinetic constants. Copyright

Full text of DOI:10.1002/poc.3319

Research Article
Received: 6 February 2014,
Revised: 14 March 2014,
Accepted: 1 May 2014,
Published online in Wiley Online Library: 9 June 2014
(wileyonlinelibrary.com) DOI: 10.1002/poc.3319
Thermodynamics and kinetics of indole
ꢀ1
oligomerization in 0.5 mol L aqueous sulfuric
acid: evaluation of some temperature-
dependant parameters
Giuseppe Quartaronea, Andrea Pietropolli Charmeta, Lucio Ronchina*,
Claudio Tortatoa and Andrea Vavasoria
Indole is present in a wide variety of natural compounds with physiological activities, as well as it is a very important intermediate
in medicinal and industrial chemistry. For this reason, the evaluation of indole protonation, oligomerization equilibria and the
study of the kinetics of dimer and trimer formation in diluted sulfuric acid at various temperatures are of paramount importance
because of their practical and scientific implications. Here, we calculate the protonation equilibria by using the literature titration
data together with a quantum chemical computational approach, in order to obtain reliable pKa value of indole from 288 to 313 K.
Starting from these calculations, we are able to measure the oligomerization equilibrium constants of indole, their kinetic con-
stants, whose values are dependent of indole pKa, at different temperatures. Enthalpy and entropy of the reactions are calculated
by means of Van’t Hoff equation, while the activation parameters of Eyring–Evans–Polanyi equation are evaluated for the whole
kinetic constants. Copyright © 2014 John Wiley & Sons, Ltd.
Keywords: activation parameters; Indole protonation; oligomerization equilibria; oligomerization kinetics
of several indoles observed in aqueous acid solution.[11,12] As a
matter of fact, indole protonation equilibrium deviates from those
INTRODUCTION
of other indoles, and this is connected to the involvement of the
molecule in other equilibria, such as that of dimerization. The
evaluation of the protonation equilibrium by spectrophotometric
measurements is, for these reasons, cumbersome and uncertain,
and the values of the pKa are not reliable especially at temperature
higher than room temperature. For these reasons, a theoretical
evaluation of the pKa in conjunction with the more recent reinter-
pretation of the classical spectrophotometric measurements
seems to be a better way for obtaining reliable values of equilib-
rium and kinetic constants of indole oligomerization.[10–12]
The formation of dimer and trimer of indole, as well as the re-
versible nature of these molecules, is known from long time,[13,14]
but there are no data on the equilibrium constants of these reac-
tions except those reported in our preliminary studies.[10] Early
studies on dimerization and trimerization of indoles pointed out
that the oligomer formation is related both to indole and acid con-
centration, as well as to the nature of the acid itself.[15,16]
The heterocyclic ring system of indole is present in a wide variety
of natural compounds (for instance, tryptophan, serotonin, etc.);
many of which have important physiological activities.[1,2]
Indole derivatives play a very important role in the synthesis
of several compounds with pharmacological activity such as di-
indolemethane, which is a well-known anticancer and antiviral
agent.[1,2] Indoles and heterocyclic compounds are, however,
important chemicals in intermediates chemistry, on material sci-
ences and, in general, in theoretical investigation.[3–5] Besides,
another important field of employment of indoles is the inhibi-
tion of corrosion, for instance, they are used as corrosion inhibi-
tor for ferrous and nonferrous alloy, but the nature of the
protective phenomenon is still not completely understood, and
the role of the indole polymers is only faintly explored.[6–9]
The behaviour of indoles in aqueous acid solution is strictly
connected to their chemical nature, and the substituents play a
crucial role in the equilibria of formation of the respective dimer
and trimer.
Several studies on indole and derivatives, both experimental
and computational, can be found in the literature.[3–5] In a previous
paper, we studied some aspects of indole dimerization and
trimerization by evaluating some kinetic and thermodynamic
parameters; however, this research has been intended as a prelim-
inary step for more detailed evaluation of the whole process, which
is also comprehensive of the protonation equilibria.[10] Indole pKa
value is characterized by a large uncertainty because of the pres-
ence of these oligomerization equilibria, which cause large devia-
tion on the activity coefficient function. This is clearly evidenced
by the deviation of the linearity of the substituent effect on pKa
The synthesis of indole polymers and of their reactions in
different media has been carefully studied for long time, and
several strategies have been reviewed.[17,18] The early paper of
* Correspondence to: L. Ronchin, Department of Molecular Science and
Nanosystems, Università Ca’ Foscari Venezia, Dorsoduro 2137, 30123, Venice, Italy.
E-mail: ronchin@unive.it
a G. Quartarone, A. Pietropolli Charmet, L. Ronchin, C. Tortato, A. Vavasori
Department of Molecular Science and Nanosystems, Università Ca’ Foscari
Venezia, Dorsoduro 2137, 30123, Venice, Italy
J. Phys. Org. Chem. 2014, 27 680–689
Copyright © 2014 John Wiley & Sons, Ltd.

INDOLE OLIGOMERIZATION
sum (Eqn (1)) was achieved by a step-descent method, and the conver-
gence was verified by reducing the step of two orders of magnitude
and obtaining constant values of the square residuals sum.[25,26] The
regression variables were the direct kinetic constants kD and kT, while
equilibrium constants were independently evaluated from an average
of the final values of the kinetics.
Schmitz-Dumont reports some data relating the repartition of
indole, dimer and trimer in biphasic condition starting from indole
dispersed in aqueous solution of acids at different concentrations:
Anyway, no thermodynamics and/or kinetics data were given.[16]
The reactivity of indoles as nucleophile has been carefully investi-
gated towards several electrophiles.[19–22] The general behaviour
of these reactions resembles the majority of aromatic substitutions
in which the rate-determining step is the formation of a Wheland-
like intermediate.[19–21]
i¼n
X
2
S ¼
½Yi ꢀ fðXi; kD; kT Þꢁ
(1)
i¼0
In our previous paper, we report some preliminary results on the
thermodynamics and kinetics of indole oligomerization at 298 K,
but because of the lack of information on indole protonation
equilibria at various temperatures, the complete investigation on
equilibria and kinetics has been delayed until now.[10] Recently,[23]
high-level quantum chemical calculations suggested a revision of
the experimental indole gas-phase basicity value, thus confirming
the work of da Silva et al.[24] The computed values of pKa were
found in good agreement with the most recent data.[10] In this
paper, together with the thermodynamics and kinetics behaviours
of dimerization and trimerization of indole in aqueous sulfuric acid
at different temperatures ranging from 288 to 318 K, we propose,
in the same range of temperature, an evaluation of the protonation
equilibrium of indole by using the same theoretical methods.
Errors and correlation matrix of the kinetic constants have been eval-
uated using a first-order approximation of the expectation surface at the
convergence point and the variance (σ) approximated to the normalized
residual mean square [σ2 ≅ s2 = S/(n ꢀ 2)].[25] Figure 1 shows a typical plot
of measured versus calculated concentrations of the species involved in
the oligomerization reaction. It is noteworthy that the experimental point
is well behaved around the identity line suggesting a good reliability of
the model.
Quantum chemical calculation and theoretical determina-
tion of indole pKa value
Recently,[23] a thorough ab initio study on the indole molecule was
carried out combining the information derived from different quantum
chemical calculations to derive accurate predictions of its aqueous pKa
value. Besides the vibrational analysis performed within the framework
of generalized second-order vibrational perturbation theory,[28] gas-
phase basicity and proton affinity values were computed by different
approaches relying on both several model chemistries and high-level
single-point energy corrections computed at coupled cluster level of the-
ory with single, double and perturbative triple excitations (CCSD(T));[29]
the results obtained were then employed to benchmark the correspond-
ing predictions yielded by different density functional theory methods.
Aqueous pKa values of indole were predicted at T = 298.15 K with remark-
able accuracy (i.e. well within 0.3 pKa unit of the recent experimental
data[10]) when the solvation free energy is computed by the SM8[30]
implicit solvation model.
EXPERIMENTAL
Materials
Solvents and reagents were purchased from Aldrich and used after the
usual purification. Aqueous sulfuric acid solutions were prepared by
diluting the concentrated acid, and their composition was determined by
automatic potentiometric titrations against standard solutions of NaOH.
Indole dimer hydrochloride (which was used instead of indole dimer
sulfate) was prepared following the literature:[13–16] A chilled solution of
indole in 1,2-dichloroethane was saturated with dry hydrogen chloride, a
white precipitate was obtained, filtered, washed with 1,2-dichloroethane
and dried under vacuum. NMR, UV–Vis and IR spectra of the precipitate
coincide with literature data.[13] Indole trimer sulfate was prepared by direct
synthesis from aqueous solution of sulfuric acid following the procedure
described in literature;[13–16] NMR, UV–Vis and IR spectra of the com-
pound match those already published thus confirming the goodness
of the synthesis.[13,14,22]
In the present work, the geometries of indole and its protonated form
were optimized in the gas phase employing the Becke, three-parameter,
Lee–Yang–Parr (B3LYP)[31] functional in combination with the correlation
consistent basis set[32] of triple-ζ quality augmented by diffuse functions
(aug-cc-pVTZ). To provide the high-level corrections on the electron
Reaction kinetics
The kinetic runs were performed in a well-stirred thermostated reactor at
298 K and at atmospheric pressure, containing weighed samples of sulfu-
ric acid (volume, 0.1 Lꢀ1) at 0.5 mol Lꢀ1. The desired concentration of sub-
strate (1–6 mmol Lꢀ1) was obtained by dilution of an ethanol solution of
the substrate. All the operations were carried out under nitrogen in order
to avoid indole oxidation. Small amounts of the solution were drawn at
different times, and the samples were analysed by HPLC, using a Perkin
Elmer 250 pump equipped with a Phenomenex (C-8 5 μm) column and
a diode array Perkin Elmer LC 235 UV detector. From a practical point
of view, the interval of concentrations used for studying the reaction is
compelled by the necessity of balancing the solubility of reagent and
products with the equilibrium conversion. As a matter of fact, conversion
of indole at concentrations lower than 1 mmol Lꢀ1 is too low for kinetics
purpose because of the sensibility limits of the analytical method.
Nonlinear regression analysis
The regression of the data was carried out on simultaneous algebraic
differential equations numerically evaluated at each experimental point
(Xi, Yi). Mass balance is the regression function (f) that accounts for the
constraints of the model.[25–27] The minimization of the square residual
Figure 1. Experimental versus calculated concentration of indole oligo-
merization. Run conditions: indole concentration initial time 0.005 molLꢀ1
,
T 288 K and H2SO4 0.5mol Lꢀ1
J. Phys. Org. Chem. 2014, 27 680–689
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G. QUARTARONE ET AL.
Table 1. Protonation equilibrium of indole in 0.5 mol Lꢀ1
aqueous H2SO4: KI calculated by using different activity coef-
ficient functions from pKIH
correlation energy that are necessary for reliably computing the gas-
phase basicity for indole, single-point energy calculations were carried
out at CCSD(T) level of theory and using the aug-cc-pVTZ basis set on
these optimized geometries. Solvation effects in the temperature that
range 288–318 K were investigated by means of the SM8T[33] implicit
model; to our knowledge, this framework is the only one developed
and specifically parameterized to study the temperature dependence
of solvation energy of both neutral and charged compounds, as, for
example, recently used by Gupta et al. in their investigation on a set of dif-
ferent amines.[34] Gaussian 09[35] and GAMESS softwares[36] were used for
geometry optimizations and calculations of the gas-phase basicity, while
the SM8T computations were carried out by using the GAMESSPLUS pack-
age.[37] Vibrational analysis performed at B3LYP/aug-cc-pVTZ level of theory
on the previous optimized geometries led to the calculation of zero-point
energy, entropy, and thermal corrections to enthalpy and Gibbs free energy
for both indole and its protonated form.
a
Activity coefficient
pKIH
103 KIa
T (K)
Reference
function
[37]
[11]
[36]
[10]
H0
HI
X
ꢀ2.8
ꢀ3.6
ꢀ2.2
ꢀ2.4
1.59
0.32
6.33
4.0
298
298
298
298
Mc
ꢀ1
aKI = K
.
IH
energy, ΔG*(aq), by combining the corresponding gas-phase
basicity of indole, ΔG°(g)(Ind), to the free energies of solvation.
The geometries of indole and its protonated form were first
optimized at B3LYP level of theory using the aug-cc-pVTZ basis
set; then, as previously reported in the Experimental section,
for computing the ΔG°(g)(Ind) with the smallest error, high-level
corrections on the electron correlation energy were computed
by performing single-point calculations at CCSD(T) level of theory
on the B3LYP/aug-cc-pVTZ optimized geometries. Similar
approaches were proven to be very effective (refer to Ervin and
De Turo[41] and Pickard et al.[42] for examples and references
therein) in predicting gas-phase free energies for a variety of
compounds. By means of the CCSD(T)/aug-cc-pVTZ//B3LYP/aug-
cc-pVTZ free energy data, the corresponding gas-phase basicity
for indole, ΔG°(g)(Ind), was calculated, according to Eqn (3):
RESULTS AND DISCUSSION
Protonation equilibrium of indole
In Scheme 1, the protonation of indole (1) in aqueous acid solu-
tion is reported. Indole is a weak base that is slightly protonated
at the carbon in three positions. Equation (2) is the relationships
of the equilibrium constant of indole protonation. Equation (2)
derives from the definition of pKIH of the conjugated acid of in-
dole, the latter being a fairly weak base.
c2
γ2
KI ¼
(2)
c1cHþ γ2γHþ
In Table 1, several values of pKIH and of equilibrium constant
KI are reported. Despite of the large number of studies on the
argument, there is no general agreement on the value of the
equilibrium constant of indole protonation. As a matter of fact,
at 298 K, the most accepted values of pK is ꢀ3.6, but recent
studies suggest values much higher close to ꢀ2.5 in agreement
to the earlier studies of Berti, which suggested a value of ꢀ2.8
calculated by H0 acidity function. Similar values have been pro-
posed by Andonowsky and Stojkovich by using X excess acidity
function. More recently in our preliminary paper, relating the
kinetics of indole oligomerization, we used the Mc activity
coefficient function to evaluate the value of the pKIH obtaining
ꢀ
ꢁ
ΔG°ðgÞðIndÞ ¼ G°ðgÞðIndÞ þ G°ðgÞðHþÞ ꢀ G°
IndHþ
(3)
ðgÞ
where G°(g)(Ind) and G°(g)(IndH+) refer to gas-phase Gibbs free
energy of indole and its protonated form, respectively. For the pro-
ton in the gas phase, the free energy was computed by employing
the standard equations of thermodynamics and of Sackur–Tetrode
equation.[43]
After having obtained the ΔG°(g) (indole gas-phase basicity)
at a given temperature, following the thermodynamic cycle used
in the previous work,[22] the corresponding overall free energy
change in solution, ΔG*(aq), was computed by adding to the
former the change in free energy of solvation. The solvation free
energies for both the indole and its protonated form were calcu-
lated by carrying out single-point calculations with the SM8T
implicit model. For the latter calculations, we employed the
M06-2X functional[44] and the 6-31G(d) basis set, given the good
results provided by this combination of functional and basis
set as reported in literature for computing the solvation free en-
ergy (refer to Chamberlin et al.[45] for example). The results of
ΔGsolv(Ind) (indole solvation free energy) and of ΔGsolv(IndH+)
(indole-protonated form free energy of solvation) are reported
in Table 2. As expected, there are a decreasing trend of the Gibbs
free solvation energy as temperature rises; the much large values
observed (above 10 times) for the protonated indole are
ascribed to the stabilization effect of the carbonium ion solva-
tion, which is more effective on the charged molecules than
the uncharged ones.
a
value close to that measured by Andonowsky and
Stojkovich.[10,38–40]
Evaluation of the temperature dependence of pKIH by
quantum chemical calculations
The difficulties encountered in a reliable evaluation of the indole
protonation constants especially at temperature higher than
298 K induced us to employ quantum mechanical calculation
for evaluating its value at various temperatures. Following the
methodology of our previous study,[23] which led to an excellent
agreement between experimental and predicted data, to
compute the aqueous pKa value, we employed a thermodynamic
cycle that determines the overall solution phase reaction free
The term of the free energy change related to the conversion
between the different standard states used for the gas phase and
the aqueous solution was taken into account, and the solvation
free energies of indole and of its protonated form were computed.
Finally, at each considered temperature, by employing the
Scheme 1. Indole protonation equilibrium
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Copyright © 2014 John Wiley & Sons, Ltd.
J. Phys. Org. Chem. 2014, 27 680–689

INDOLE OLIGOMERIZATION
Evaluation of activity coefficients of indole, indole dimer and
trimer in solution
Table 2. Solvation free energies of ΔGsolv(Ind) and its pro-
tonated form, ΔGsolv(IndH+), obtained by SM8T implicit
model at different temperatures by employing the M06-2X/
6-31G(d) level of theory
Equation (5) is the thermodynamic definition of pKIH of indole
rewritten in the Mc activity function notation.[46] The nbb Mc
term represents the activity coefficients of reagents and prod-
ucts where Mc is the activity coefficient function of the primary
nitroanilines in H2SO4, while nbb (nbb = 1.152) is the specific coef-
ficient of indole in sulfuric acid with respect to Mc function.[46]
T (K)
ΔGsolv(Ind) (kJ molꢀ1
)
ΔGsolv(IndH+) (kJ molꢀ1
)
288
298
308
318
ꢀ23.7
ꢀ22.6
ꢀ21.6
ꢀ21.1
ꢀ252.8
ꢀ250.8
ꢀ249.6
ꢀ249.0
!
c2
γ2
c2
c1cH
pKIH ¼ Log
¼ Log
þ þ nbbMc
(5)
c1cHþ γ1γHþ
The uncertainty on the protonation equilibria of indoles re-
flects the experimental evidence that oligomerization reactions
may take place during equilibrium measurements, causing a
poor reliability of the titration.[10] Recently, we have investi-
gated[23] the indole protonation equilibrium by quantum chem-
ical calculations obtaining computed values very close with the
most recent experimental data. The calculated pKIH value at
298 K is ꢀ2.54 being in close agreement with our previous
evaluation of pKIH by using Mc acidity function.[10] In addition,
a temperature dependence of the equilibrium constants has
been evaluated. Equations (2) and (3) allow the calculation of
the concentration of (2), which is the electrophile to be taken
into account in kinetic calculation. Furthermore, for the Mc func-
tion, it has been quantified a temperature dependence by an
empirical relationship with respect to the Mc(298) (Eqn (6), which
is the activity function measured at 298 K.[46]
corresponding free energy change in solution, the aqueous pKa
was derived by using Eqn (4):
ΔGꢂðaqÞ
pKa ¼
(4)
RT Lnð10Þ
where R is the universal gas constant and T is the temperature (K).
In Table 3, the results of these calculations at the different temper-
atures considered in the present work are reported.
Figure 2 shows the Van’t Hoff plot of protonation equilibrium
constants, the reaction is slightly exothermic, and the entropy is
negative because of the reduction of mole number (pKIH is calcu-
lated considering that the conjugated acid of indole as the acid,
which dissociates to indole and H+). As a matter of fact, indole is
a weak base, and only a small fraction is in the protonated form
under the condition investigated.
!
200
T
McðTÞ ¼
The term (γ2/(γ1 γH
þ 0:03292 Mcð298Þ
(6)
Table 3. Protonation equilibrium of indole in 0.5 mol Lꢀ1
aqueous H2SO4 obtained by quantum mechanical calcula-
tions performer at different temperatures
+
)), which represents the deviation from ide-
ality of the solution, can be reduced to (1/γ1), if we consider that
charged ions have comparable activity coefficients. This is ac-
ceptable in a range of sulfuric acid concentration where activity
coefficients of the charged species are the function of the ionic
strength (typically lower than 0.5 mol Lꢀ1).[47] From a molecular
point of view, this means that each ion has similar interaction
both with solvent molecules and with the other ions.[46] For in-
stance, in this range of concentrations, it is possible to reliably
estimate the activity coefficient of ionic species in solution by
correlation models for aqueous electrolytes, which are the func-
tion of the ionic strength irrespective of the type of ion.[47]
As pointed out by several authors,[48,49] the activity coefficient of
the uncharged species in diluted aqueous solution of sulfuric acid
shows a typical salting-out effect, thus following the Sechenov re-
lationship (Eqn (7)), where b is the Sechenov parameter character-
istic of the indole and CA is the concentration of sulfuric acid.[49]
a
T (K)
ΔG* (kJ molꢀ1
)
pKIH
103 KIa
Reference
288
298
308
318
ꢀ13.5
ꢀ14.5
ꢀ15.4
ꢀ16.3
ꢀ2.45
ꢀ2.54
ꢀ2.61
ꢀ2.68
3.55
2.88
2.46
2.09
[this work]
[22]
[this work]
[this work]
ꢀ1
aKI = K
.
IH
Log γ1 ¼ b CA
(7)
Typically, Sechenov parameter is calculated by solubility data,
but because of difficulty on the evaluation of the solubility of (1)
in sulfuric acid because of the parallel oligomerization reaction,
the evaluation of the Sechenov parameter for indole can be
obtained by using the methods proposed by Yates and co-workers,
who derived the values of the parameter (b = 1.5) by comparing H0
and HI acidity functions.[49] The comparison of the activity coeffi-
cients calculated by literature of Sechenov parameters at 298 K
with those obtained from Mc activity function is reported in
Table 4.[49] The quite good agreement of the two methods (e.g.
compare entries 1, 2 and 5) suggests that, at the concentration of
Figure 2. Protonation equilibria: Van’t Hoff plot of protonation
equilibrium constant in the range 288–318 K: ΔH = ꢀ13 kJ molꢀ1 and
ΔS = ꢀ93 J molꢀ1 K
J. Phys. Org. Chem. 2014, 27 680–689
Copyright © 2014 John Wiley & Sons, Ltd.
wileyonlinelibrary.com/journal/poc

G. QUARTARONE ET AL.
indole to the trimer (3,3′-(-2-aminophenethylidene) di-indole (4);
Table 4. Activity coefficient of uncharged molecules in
diluted electrolite
1
Scheme 3 and Eqn (9). Besides, in H2SO4 0.5 mol Lꢀ , this com-
pound is a fully protonated ortho-substituted alkyl-aniline, which
a
does not react further with indole under the condition investigated.
Entry
T (K)
Activity model
γ1
Reference
[46]
1
2
3
4
5
298
288
298
308
318
Sechenov
Mc
Mc
Mc
Mc
0.75
c3 γ3
c1c2 γ1γ2
KD
KT
¼
¼
(8)
(9)
0.676
0.682
0.687
0.693
This work
This work
This work
This work
c4 γ4
c3c1 γ3γ1
aγ2 / (γ1 γH+) ≈ 1 / γ1.
Equations (8) and (9) show the thermodynamic constant of (3)
and (4) formation, respectively. As pointed out in the previous
section, the values of the activity coefficients of (1) in both equi-
libria of dimerization and trimerization (Eqns (8) and (9)) are con-
sidered equal to that of indole protonation equilibrium (Eqn (5))
and calculated by Eqn (10).
sulfuric acid and indole employed in the present study, a reliable
evaluation of the activity coefficient of (1) can be obtained starting
from the values of the Mc activity function. Furthermore, the Mc
activity function has been calculated by titration of nitro-anilines
measured at different temperatures;[46] in this way, the tempera-
ture variation of the activity coefficients can be estimated by using
Eqn (4). This method allows to obtain reliable behaviour of the
activity coefficients for uncharged molecules of various nature if
γ2
γ3
γ4
1
bbMc
¼
¼
¼
¼ 10n
(10)
γ1γHþ γ1γ2 γ1γ2 γ1
applied in a concentration range of sulfuric (0–1 molLꢀ1 [47]
acid
)
Such an evaluation of the activity coefficients holds because
the value of the activity coefficient of the individual ionic species
is approximately the function of the mere ionic strength and not
of the type of ion.[47] Starting from this hypothesis, the non ionic
species (1) is the same in all equilibria; for this reason, the value
of the activity coefficient term in each equilibrium relationship
(Eqns (5), (8) and (9)) is constant, and it could be calculated from
the Mc activity function value of sulfuric acid at the concentra-
tion of 0.5 mol Lꢀ1, (Eqn (10)) and taking into account the specific
nbb values of indole (Table 4).
where different charged species have practically the same activity
coefficient. Large deviation can be encountered at higher acid
concentration where the activity coefficients of charged species
differ noticeably because of the lacking in the pure electrostatic na-
ture of their interactions (between them and with the solvent).[47]
As a matter of fact, also the activity coefficients of the equilib-
ria of dimerization and trimerization can be calculated by using
this method because the ionic strength of the media is the same
and activity coefficient of the ionic species does not depend on
the nature of the ions.
It is clear that the formation of the dimer depends on the
protonation of (1) (Scheme 1, Eqn (5)); then, the value of KD is
the function of pKIH, (1) and H+ concentrations, as well as to
the activity coefficient term that could be considered equal to
that of the equilibrium of indole protonation. In this way, we
can rewrite Eqn (8) as Eqn (11), which is composed of all known
terms, being c1 and c3, the measured concentration of (1) and (3)
at equilibrium and cH+ is the H+ concentration, which derives
from dissociations of sulfuric acid reported in literature,[50,51]
while the activity coefficient term is calculated from Mc and
nbb values.[12,46] In the same way, the calculus of KT is carried
out by rewriting Eqn (9) as Eqn (12), where c1, c3 and c4 are the
measured concentration of (1), (3) and (4), respectively, and
the activity coefficients ratio, for the reasons explained in the
previous section, is the same of Eqn (11).
Dimerization and trimerization equilibria of indole
It has been pointed out that the oligomerization of (1) is revers-
ible in aqueous acid at 298 K.[10] This is quite singular features of
this molecule because the starting indole is a weak base, while
the oligomers (3) and (4) are fully protonated aliphatic amines,
which are stabilized by solvation in acid aqueous solution.
Schemes 2 and 3 show dimerization and trimerization of (1). The
resulting carbonium ion (2) is an electrophile that is able to attack
the nonprotonated indole giving (3): the corresponding 3
substituted indole dimer (3-(indolin-2yl) indole (3); Scheme 2 and
Eqn (8)). The latter is in protonated form, and it reacts further with
KIc3 γ3 γ2
¼
c21cHþ γ1γ2 γ1γH c21cH
þ
KIc3
bbMc
KD
¼
¼ 102n
(11)
(12)
þ
c4 γ4
c4
bbMc
KT
¼
¼
¼ 10n
c3c1 γ3γ1 c3c1
Scheme 2. Indole dimerization equilibrium
In Tables 5–8, the calculated values of the equilibrium constants
of dimerization and trimerization at different temperatures,
starting from indole, from dimer and from the trimer, are reported.
The results of Table 5–7 show that the KD and KT, measured
starting from (1), (3) or (4), have a good reliability also by varying
the temperature. This suggests that the variation of the starting
substrate does not influence the final concentrations of (1), (3)
and (4) because they reach the equilibrium without noticeable
Scheme 3. Indole trimerization equilibrium
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INDOLE OLIGOMERIZATION
Table 5. Equilibria of (1) oligomerization in 0.5 mol Lꢀ1 aqueous H2SO4 in the range 288–318 K
T (K)
(1)
(3)
(4)
γꢀ1
105KD (L molꢀ1
)
102KT (L molꢀ1
)
(103 mol Lꢀ1
)
(103 mol L-1)
(103 mol Lꢀ1
)
t0
tend
tend
tend
288
298
308
318
5.0
6.3
5.0
5.0
1.1
1.62
1.8
0.57
0.77
0.53
0.58
0.89
1.03
0.62
0.45
1.479
1.466
1.454
1.443
5.42
3.99
2.74
2.19
23.7
11.8
9.01
5.06
2.3
t0, initial time; tend, final time; the final time concentration is obtained from the average of five independent measurements.
Table 6. Equilibria of (3) in 0.5 mol Lꢀ1 aqueous H2SO4 in the range 288–308 K
T (K)
(1)
(3)
(4)
γꢀ1
105KD (L molꢀ1
)
102KT (L molꢀ1
)
(103 mol Lꢀ1
)
(103 mol Lꢀ1
)
(103 mol Lꢀ1
)
tend
t0
tend
288
298
308
0.82
0.98
1.2
1.02
1.01
1.04
0.32
0.26
0.2
1.48
1.47
1.45
1.479
1.466
1.454
5.39
3.88
2.66
22.9
11.6
9.12
t0, initial time; tend, final time.
Table 7. Equilibria of (4) in 0.5 mol Lꢀ1 aqueous H2SO4 in the range 288–308 K
T (K)
(1)
(3)
(4)
γꢀ1
105KD (L molꢀ1
)
102KT (L molꢀ1
)
(103 mol Lꢀ1
)
(103 mol Lꢀ1
)
(103 mol Lꢀ1
)
tend
tend
t0
tend
288
298
308
0.81
1.12
1.14
0.20
0.32
0.29
0.99
0.94
1.0
0.41
0.33
0.21
1.479
1.466
1.454
5.31
3.67
2.65
22.9
11.1
8.91
t0, initial time; tend, final time.
Figure 3 shows the Van’t Hoff plot of dimerization and
trimerization equilibrium constants in the range 288–318 K. It is
interesting to observe that ΔH of reaction of dimer and trimer
formation (ꢀ24 and ꢀ37 kJ mol-1, respectively) is compatible
with literature data.[52] On the contrary, the large difference of
ΔS (28 and ꢀ65 J molꢀ1 K, respectively) between dimer and tri-
mer is not easily explainable especially because dimerization
shows a positive ΔS, although there is a reduction of mole num-
bers, which is, from thermodynamic point of view, not a com-
mon behaviour. As a matter of fact, this positive value in the
variation of entropy observed for the dimerization equilibrium
can be ascribed to the large variation of entropy observed in
the protonation equilibrium of indole, which is responsible for
the large decrease of the global entropy of the reaction. In other
words, the change of nature of the ion may explain this behav-
iour because (2) is a carbonium ion, while (3) is an ammonium
salt. It is likely that the stabilization of the carbonium ion (2) re-
quires a number of water of solvation larger than those needing
for (3). On the contrary, the negative variation of entropy of (4)
from (3) is in agreement with the overall mole reduction because
of the addition of (3) to (1). In this case, it is likely that the
Table 8. Indole oligomerization averaged equilibrium con-
stant at different temperatures
T (K)
105 KD (L molꢀ1
)
105 SD 102 KT (L molꢀ1
)
102 SD
288
298
308
318
5.37
3.85
2.68
2.19
0.18
0.18
0.19
0.36
23.2
11.5
9.01
5.06
2.1
1.2
1.7
1.1
The intervals of confidence of KD and KT at 318 K are obtained
taking into account only the measurements for indole.
parasitic reactions. To say the truth, a perceptible colouration of
the solution is observed at the higher temperature after quite
long time (after 20 h), but the concentration of the dyed
substances is below the experimental error and for the sake of
simplicity neglected. Table 8 reports the averaged values of the
equilibrium constants of dimerization and trimerization
employed in the nonlinear regression of the kinetics.
J. Phys. Org. Chem. 2014, 27 680–689
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G. QUARTARONE ET AL.
Figure 5. Time versus concentration profiles of (3) reactivity in H2SO4
0.5 mol Lꢀ1. Run conditions: T 308 K and H2SO4 0.5 mol Lꢀ1. Δ = (1),
○ = (3), □ = (4)
Figure 3. Dimerization and trimerization equilibria: Van’t Hoff plot of
dimerization and trimerization equilibrium constants in the range
288–318 K. Dimerization: ΔH = ꢀ24 kJ molꢀ1 and ΔS = 28 J molꢀ1 K;
trimerization: ΔH = ꢀ37 kJ molꢀ1 and ΔS = ꢀ65 J molꢀ1 K
solvation grade of (3) and (4), which are both protonated
amines, slightly influences the overall entropy variation.
Kinetics of indole oligomerization
Figure 4 shows the time versus concentration profile at 288 K of
indole oligomerization. The formation of (4) is much slower than
that of (3); however, the latter cannot be considered at stationary
state. As a matter of fact, the reactivity of the dimer shows mea-
surable reaction rate in reaching equilibrium as can be observed
in Fig. 5 where the reactivity of the dimer at 308 K is reported.
The reactivity of (4) at 288 K is reported in Fig. 6. It is evident that
(4) reacts to give (1) and (3) at measurable reaction rate. This
suggests that a kinetic scheme that take into account of all the
stage of the reaction needs to give a reliable fitting of the exper-
imental data.
Figure 6. Time versus concentration profiles of (4) in H2SO4 0.5 mol Lꢀ1
Run conditions: T 288 K and H2SO4 0.5 mol Lꢀ1. Δ = (1), ○ = (3), □ = (4)
.
These considerations, however, were already present in the
preliminary study, and the kinetic model proposed in that pre-
liminary paper is now tested at different temperatures.[10] The ki-
netic model is a straight formulation of the equilibria in terms of
reaction rate by considering that the reverse kinetic constants
are obtained from the direct one divided by the respective
equilibrium constant measured at infinite time of reaction,
that is, kDR = kD/KD and kTR = kT/KT. Equations (11)–(13) are the
mathematical formulation of the kinetic model, and the results of
the fitting of these simultaneous equations with experimental
data are reported in Table 9.
dc1
ꢀ
¼ kDc1c2 ꢀ kDRc3 þ kT c1c3 ꢀ kTRc4
(13)
dt
dc4
¼ kT c3c1 ꢀ kTRc4
(14)
(15)
dt
c01 þ 2c30 þ 3c40 ¼ c1 þ c2 þ 2c3 þ 3c4
The measured values of the kinetic constants kD and kT at
different initial concentrations of indole, as well as
those obtained starting from (3) and (4), are quite in good
agreement (at 298 K, kD kT are in the range 380–510 and
0.094–0.098 L molꢀ1 sꢀ1, respectively). Similar results are ob-
served at 288, 308 and 318 K, thus suggesting that the kinetic
constants are barely the function of the temperature and not
to the starting substrate, almost in the limit of the experimental
error. As a matter of fact, the kinetic constant calculated by
fitting the simultaneous Eqns (11)–(13) with time versus con-
centration profile of the reactivity of (1), (3) and (4) gave similar
values suggesting a good reliability of the model (Table 9). In
Figure 4. Time versus concentration profiles of indole oligomerization.
Run conditions: T 288 K and H2SO4 0.5 mol Lꢀ1. Δ = (1), ○ = (3), □ = (4)
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J. Phys. Org. Chem. 2014, 27 680–689

INDOLE OLIGOMERIZATION
fact, despite of the complex environment, where oxidation of
(1) is possible, the constancy of the parameters suggests that
this parasitic reaction is negligible.
Activation parameters
In Figs 7 and 8, the Eyring plot of kD, kT, kDR and kTR is reported.
The corresponding activation parameters are evaluated by fitting
the linearized Eyring–Evans–Polanyi equation (Eqn (16)).
k
1
T
kB
RLn ¼ ꢀΔH‡ þ RLn ΔS‡
(16)
T
h
As expected, the enthalpy of activation of the direct reactions
is lower than that of the reverse ones because of the enthalpy of
reaction of dimerization and trimerization that is summed to that
of the activated state. An interesting feature is, however, the
values of the entropy of activation that are negative both for di-
rect and reverse reactions. This suggests that the formation of (3)
and (4), which both cause a reduction of the overall number of
moles, is anyway accompanied by a negative variation of the en-
tropy of the respective activated complexes because of their sol-
vation. It is likely that the activated complexes are in highly
solvated state; in this way, it is reasonable to account a negative
ΔS‡ also in the reverse reaction because of the large number of
Figure 7. Eyring plot of dimerization and trimerization direct kinetic
constant: ΔH‡kD = 33 kJ molꢀ1, ΔS‡kD = ꢀ83 J molꢀ1 Kꢀ1, ΔHk‡T = 29 kJ molꢀ1
and ΔS‡kT = ꢀ169 J molꢀ1 Kꢀ1. □ = RLn kD, ○ = RLn kT
Figure 8. Eyring plot of dimerization and trimerization reverse kinetic
constant: ΔH‡kDR = 57 kJ molꢀ1, ΔSk‡DR = ꢀ108 J molꢀ1 Kꢀ1, ΔHk‡TR = 65 kJ
molꢀ1 and ΔS‡kTR = ꢀ109 J molꢀ1 Kꢀ1. □ = RLn kDR, ○ = RLn kTR
J. Phys. Org. Chem. 2014, 27 680–689
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G. QUARTARONE ET AL.
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of both reagent and products because of the much ordered
structured of the highly solvated transition state. This is con-
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(Fig. 8). In fact, it is to be emphasized that both products ((3)
and (4)) and transition states should have similar molecular
structure, but they should differ for the respective solvation
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CONCLUSIONS
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The main thermodynamic and kinetic parameters of indole
oligomerization in 0.5 mol Lꢀ1 H2SO4 have been measured here.
Calculation of both equilibria and kinetics of oligomerization are
based on the values of the protonation equilibria of indole
evaluated at different temperatures by using a computational
approach, by which a reliable value of pKIH has been calculated
taking into account the experimental one at 298 K. In this way,
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and transition plays a role of paramount importance in the over-
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formation of (3) is positive despite of the reaction formerly pro-
ceeds with reduction of mole number, thus suggesting a strong
influence of solvation between reagent and product. On the con-
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expected behaviour. Furthermore, the important role of solva-
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evidenced by the Eyring–Polanyi equation relative to the kinetic
constant of dimerization and trimerization (kD and kT, direct
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the formation of a highly solvated activated complexes whose
complex solvation environment in a highly ordered state,
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variety of biologically active molecules; for this reason, a
detailed study of the thermodynamic and kinetic properties
of their reactions may be of aid for the comprehension of
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