Kinetic parameter estimation of solvent-free reactions monitored by 13C NMR spectroscopy, a case study: Mono- and di-(hydroxy)ethylation of aniline with ethylene carbonate

Article abstract of DOI:10.1002/kin.20532

The kinetics of solvent-free reactions can be followed in situ by ¹³C nuclear magnetic resonance (NMR) spectroscopy, provided that the reaction mixture can be maintained liquid at the monitoring temperature. The pros and cons of the technique and the correct translation of the signal intensities into concentrations are discussed. A good model for this investigation is the reaction of ethylene carbonate (1) with aniline (2) at 140°C, two alkylation products of N-mono- and N, N-bis-(2-hydroxy)ethylation of aniline form (compounds 3 and 4, respectively). The overall reaction occurs with heavy volume shrinking, so that the physical as well as the chemical features evolve during the course of the process. The chemical evolution is described by the kinetic constants k₁ and k₂ of the two N-alkylation steps, the physical evolution by the time-dependent activity coefficients α(t). Two complementary procedures are utilized for the determination of these parameters.

Full text of DOI:10.1002/kin.20532

Kinetic Parameter
Estimation of Solvent-Free
Reactions Monitored by
13
C NMR Spectroscopy,
A Case Study: Mono- and
Di-(hydroxy)ethylation of
Aniline with Ethylene
Carbonate
`
VITTORIO LUCCHINI, MASSIMO FABRIS, MARCO NOE, ALVISE PEROSA, MAURIZIO SELVA
Dipartimento di Scienze Ambientali dell’Universit a` Ca’ Foscari, Calle Larga S. Marta 2137, 31023 Venezia, Italy
Received 15 June 2010; revised 26 August 2010; accepted 21 September 2010
DOI 10.1002/kin.20532
Published online 21 January 2011 in Wiley Online Library (wileyonlinelibrary.com).
ABSTRACT: The kinetics of solvent-free reactions can be followed in situ by 13C nuclear magnetic
resonance (NMR) spectroscopy, provided that the reaction mixture can be maintained liquid
at the monitoring temperature. The pros and cons of the technique and the correct translation
of the signal intensities into concentrations are discussed. A good model for this investigation
◦
is the reaction of ethylene carbonate (1) with aniline (2) at 140 C, two alkylation products of N-
mono- and N, N-bis-(2-hydroxy)ethylation of aniline form (compounds 3 and 4, respectively).
The overall reaction occurs with heavy volume shrinking, so that the physical as well as the
chemical features evolve during the course of the process. The chemical evolution is described
by the kinetic constants k1 and k2 of the two N-alkylation steps, the physical evolution by the
time-dependent activity coefficients α(t). Two complementary procedures are utilized for the
determination of these parameters. ꢀC 2011 Wiley Periodicals, Inc. Int J Chem Kinet 43: 154–160,
2011
INTRODUCTION
Correspondence to: Vittorio Lucchini; e-mail: lucchini@
unive.it.
Supporting Information is available in the online issue at
www.wileyonlinelibrary.com.
ꢀ
c 2011 Wiley Periodicals, Inc.
The greenest way to confront the problem of reaction
solvents is their elimination altogether. When one or
both reagents are liquid, or mutually soluble, at the re-
action and the monitoring temperature, the reaction

KINETIC PARAMETER ESTIMATION OF SOLVENT-FREE REACTIONS
155
is transferred to a screw-cap NMR tube. The tube
is then equipped with a sealed capillary, containing
pure [D6]DMSO, and blocked in a coaxial position by
means of a Teflon spacer, for locking and homogeneity
purposes. In our hands, and contrary to what stated [3],
the quantity of deuterated substance, in a capillary with
an inner diameter of 1.0 mm is sufficient for an accept-
able lock level and for easy homogeneity shimming.
Scheme 1 shows that CO2 develops in both reac-
tion steps, a situation that is evinced from the bubbling
observed in the NMR tubes when heated in the ther-
mostated bath and that obliges to a periodic release
of the tube pressure. As a consequence, and also be-
cause the reactions are of the condensation type, the
volume of the reaction mixture shrinks during the re-
action course.
Scheme 1
course can be adequately followed by 13C nuclear
magnetic resonance (NMR) spectroscopy. The advan-
tages are clear: The highest possible concentrations
ensure a swift acquisition of the signals, which are
also usually well separated. Also the disadvantages are
clear: quantitative estimation of the signal intensities
is not very precise (the integration of the NMR signals
is, among the NMR parameters, the one measured with
the lowest accuracy), and the correspondence between
the intensity of the signal and the relative number of
nuclei that originate the signal is not strictly propor-
tional (the rule of unitary molar extinction coefficient
for NMR spectroscopy is not rigorously observed).
To these problems, an exquisitely physical prob-
lem must be added: in reactions performed in solution,
the volume is buffered by the overwhelming presence
of the solvent, whereas in solvent-free reactions the
volume, and the associated properties, may drastically
vary. Thus both chemical and physical features evolve
during the reaction time. There follows the necessity
of discriminating between the two contributions and to
anchor the chemical parameters to a standard physical
state. This necessity has been already recognized [1].
We have used the alkylation of aniline 2 by ethylene
carbonate (EC) 1 (Scheme 1) as a model reaction to
address these problems. In this reaction, CO2 develops
in both steps, a fact that should lead to a heavy volume
variation. Contrary to a statement in the literature [2],
we report here that this reaction occurs, also in the
absence of solvent and of any catalyst, at an accessible
temperature and within a reasonable time, first to yield
N-2-hydroxyethylaniline 3 and then, with a 2:1 excess
of 1, N,N-bis(2-hydroxyethyl)aniline 4.
To evaluate the volume changes, an auxiliary screw-
cap NMR tube is charged with the same reacting mix-
ture, but without capillary; both tubes are then sub-
jected to the same reaction conditions.
◦
The NMR measurements are run at 25 C in a Varian
Unity 400-MHz spectrometer equipped with a reverse
13
insert, not optimized for C spectroscopy. Neverthe-
less, spectra with a good signal to noise ratio are col-
lected with 32 scans and a relaxation time of 16 s,
1
under the regime of an inverse-gated H decoupling,
to minimize nuclear Overhauser enhancement (NOE)
effects.
◦
The kinetics is run at 140 C in a thermostated sil-
icon oil bath, hosting both tubes. In the first run, a
volume expansion factor of 1.10 is observed on going
◦
from room temperature to 140 C, as directly measured
with a gauge in the auxiliary NMR tube, and the con-
centrations of EC and aniline are then corrected to 8.30
−1
and 4.19 mol L , respectively. The reaction times are
calculated from the moment when the tubes are dipped
into the bath to the moment when they are lifted for
the measurements, thus reasonably assuming that the
heating time and the cooling time are similar, and that
a negligible reaction advancement occurs at room tem-
perature. The capillary-equipped tube is subjected to
the NMR analysis; the volume change in the auxiliary
tube is measured with a gauge. The volume change is
shown later in Fig. 2.
The reaction is followed for 80 h of effec-
tive bath heating. The initial insurgence of N-2-
hydroxyethylaniline 3 is observed, which is then al-
most totally replaced by N,N-bis(2-hydroxyethyl)
aniline 4 (Scheme 1). The compounds are identified
by the GC-MS analysis of the final reaction mixture,
RESULTS AND DISCUSSION
Reaction Conditions
13
Quantities of EC and of aniline that reproduce the 2:1
molar ratio are weighed at room temperature in a vol-
umetric flask, resulting in EC and aniline concentra-
and the corresponding C signals easily attributed.
It should be noted that the additions are chemospe-
cific, as the nucleophilic attacks of aniline 2 or sec-
ondary aniline 3 occur exclusively at the β-carbon of
−
1
tions of 9.13 and 4.61 mol L . The correct volume
International Journal of Chemical Kinetics DOI 10.1002/kin

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56
LUCCHINI ET AL.
1
EC. Other noncatalyzed additions of aniline to EC or
to other dialkyl carbonates (dimethyl and dibenzyl car-
bonates, respectively) have been reported [2,4], where
the nucleophilic attacks at both the alkyl and carboxylic
carbons are observed.
ever, rare in H spectra (in our case, only one substance
is represented by an isolated singlet), where multiplets
and signal overlaps are the rule.
The comparison within the series of subsequent
spectra requires the individuation of an internal in-
tegration standard. This standard is recognized in the
sum of the integrated signals of a moiety, which is
present in a constant amount during the reaction course:
the ortho carbons of the aniline ring of 2, 3, and 4.
Thus the integration of every peak in the spectrum set
is multiplied by 4.19/sum, thereby returning the cor-
rect time-dependent peak intensities of reagents and
products.
The NOE perturbation is reduced by the technique
of inverse-gated decoupling, but not totally eliminated,
a fact that is recognized in the first control spectrum
(zero time spectrum), where the integration of the
ethylenic carbons of EC 1 is slightly overestimated
with respect to that of the ortho carbons of the aniline
2. Thus this signal of EC in the control and subsequent
spectra is correspondingly amended. The corrected in-
tegrated signals of the ortho carbons of 2, 3, and 4 and
that of the ethylenic carbons of 1, are then assumed to
faithfully represent the variations of concentration of
the involved species. The assumption proved correct,
as demonstrated by the fact that the weighted sum of a
different set of signals, those due to an ethylenic group
[1, 3, and 4 (twice)], remains constant for the whole
reaction time (as shown in the Supporting Informa-
tion). The time-dependent concentrations are graphi-
cally represented in Fig. 1.
The Quantitative Evaluation of the 13C
Resonances
As already stated, the 13C spectra are collected under
1
the regime of H inverse gated decoupling, with the aim
of minimizing NOE effects. To overcome the limits of
the numerical integration procedure of the spectrome-
ter, we adopted the strategy of fitting the peaks into a
combination of Gaussian and Lorentzian functions and
to evaluate the analytical integration of this combina-
tion. The procedure requires a correct definition of the
peaks, which is attained by registering the free induc-
tion decays (FIDs) with 32 K of memory, and trans-
forming them under zero filling to 262 K and multipli-
cation with the proper window function (the unshifted
Gaussian function); proper phase and baseline correc-
tions also must be performed. The fitting is computa-
tionally accomplished with the Levenberg–Marquardt
method [5]. This procedure allows the correct estima-
tion of the integrated value also of peaks, which are
scantily over the spectrum noise. One example of this
evaluation is shown in the Supporting Information.
The fit is most accurate when performed on a signal
represented by an isolated singlet. It is always possible
to detect isolated signals in 13C spectra; they are, how-
Figure 1 Time dependence of the molarities of EC 1 (ꢀ), aniline 2 (♦), N-2-hydroxyethylaniline 3 (◦), and N,N-bis(2-
hydroxyethyl)aniline 4 (ꢀ), normalized to the initial concentration of aniline 2. The best fit of the differential equation system
(1), obtained by the numerical Runge–Kutta integration and the Simplex search, is shown. [Color figure can be viewed in the
online issue, which is available at wileyonlinelibrary.com.]
International Journal of Chemical Kinetics DOI 10.1002/kin

KINETIC PARAMETER ESTIMATION OF SOLVENT-FREE REACTIONS
157
.
.
.
.
.
.
Figure 2 Symbols for compounds 1–4 as in Fig. 1 (left ordinate). The volume shrinking is represented by ꢁ (right ordinate).
The molarities are normalized according to the reaction volume. The best fit (obtained by the numerical Runge–Kutta integration
and the Simplex search) of the differential equation system (1), with concentrations prefixed by the volume factor (2), is shown.
[Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
This procedure assumes implicitly that the total tion is clearly not suited for our reaction system; there-
amount of a reference carbon (the ortho carbons of
the aniline rings) remains unchanged. On the other
hand, the volume change implies an increment of the
total amount, as detected by the measuring coil of the
instrument probe. Consideration of the volume shrink-
age brings about the time-dependent concentrations
shown in Fig. 2. As shown in the following paragraph,
this correction is not compatible with the system (1)
of differential equations describing the reactions of
Scheme 1.
fore, we resorted to a numerical method, the Runge–
Kutta integration of a system of ordinary differential
equations [7]. This method gives the calculated con-
centrations of reagents and products at any measure-
ment time, based on the equations and the parameters
in (1), the initial concentrations, and the kinetic con-
stants k1 and k2. The sum of the squares of the residuals
(the differences between experimental and calculated
concentrations) is minimized with the Nelder–Mead
Simplex search [8], by varying a selected set of param-
eters. In our case, the initial concentrations are pre-
cisely known and therefore kept unchanged, while the
variation is applied to the kinetic constants.
The Evaluation of the Kinetic Parameters
The system of consecutive reactions shown in Scheme
1
is kinetically described by the system of differential
The Fitting Trial of the Volume-Corrected Con-
centrations. Because of the volume shrinking, the
monitoring solenoid of the NMR probe is sensing an
increasing amount of matter (i.e., progressively higher
concentrations). This apparent “creation” of matter
cannot be accounted for by system (1) of differen-
tial equations that describes the chemistry. To include
the component due to volume shrinking, all concen-
tration terms in system (1) were divided by the factor
Vt/V0(V0 = 1). The Runge–Kutta procedure requires
the knowledge of the volume shrinking at the experi-
mental monitoring times but also at a number of inter-
mediate times. We utilized the following equation:
equations (1).
d[1]/dt = −k1[1][2] − k2[1][3]
d[2]/dt = −k1[1][2]
(1)
d[3]/dt = k1[1][2] − k2[1][3]
d[4]/dt = k2[1][3]
The analytical integration of this system is awkward.
Actually, although this set is described in an exhaustive
list of kinetic systems [6], no analytical integration is
offered, but rather a treatment based on the stationary
state approximation of 3 is described. This approxima-
Vt/V0 = 0.7088 + [1 − 0.7088 exp(−0.06711t)] (2)
International Journal of Chemical Kinetics DOI 10.1002/kin

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58
LUCCHINI ET AL.
obtained from fitting of the experimental volume
change represented by ꢁ in Fig. 2.
Thus, a time-dependent activity coefficient α must
be applied to the concentrations, and while the concen-
trations will change because of the reaction kinetics, α
will be a function of the reaction volume and of some
property thereof, and, in the last instance, of time. An
acceptable treatment would dictate that the coefficient
be the same for all species. The functional form of
this coefficient is unknown, but can, in any case, be
expanded as a McLaurin series:
No acceptable fit is achieved. We will show in the
next section (where the volume correction is neglected)
that the introduction of optimized time-dependent
activity coefficients allows to reach a good fit. In the
present case, this procedure is of no help. In fact, the
results, shown in Fig. 2, indicate that the Runge–Kutta
integration procedure refuses to calculate a cumulated
concentration of aniline 2 and substituted anilines 3
and 4 greater than the initial concentration of 2
ꢁ
ꢁꢁ
2
α(t) = α(0) + α (0)t + (α (0)/2!)t
ꢁꢁꢁ
3
+
(α (0)/3!)t + · · ·
(3)
The Fitting of the Volume Uncorrected Concentra-
tions. One alternative option is to calculate the con-
centrations with the Runge–Kutta integration proce-
dure on the basis of the unamended system (1) and
then correct the values by the measured volumes. A
second alternative is to perform no correction and to
assume no volume change (the concentration are nor-
malized to a constant value). The two choices are log-
ically and mathematically equivalent. We adopted the
second one.
The result of the fitting is presented in Fig. 1. The
inability of the best set of k1 and k2 to fit the ex-
perimental concentrations is evident. Because no other
reactions than those illustrated in Scheme 1 can be con-
ceived, we attribute the discrepancy to a heavy change
of the physical environment during the reaction course,
which depends on the volume change.
For the reasons outlined below, the zerotime coefficient
can be put to 1 (α (0) = 1), thus selecting the initial
state as the standard physical state.
The expansion can be inserted into system (1), uti-
lized in the Runge–Kutta numerical integration, and
ꢁ
ꢁꢁ
the parameters (k1, k2, α (0), α(0) , etc.) optimized
by the Simplex search. It turned out that, having put
ꢁ
α(0) = 1, only the first-order (α (0)) coefficient is nec-
essary and sufficient for a quite good fit (Fig. 3). The re-
−
2
−1 −1
h
sults are k1 = 1.80 × 10 L mol
, k2 = 5.22 ×
−
3
−1 −1
ꢁ
−2 −1
ꢁꢁ
10 L mol
h
, α (0) = 6.46 × 10
h
, α (0)/2,
and subsequent coefficients, negligible.
This result may, however, be misleading: covari-
ancy of the optimized parameters is not rigorously
ruled out, so that different combinations of them may
give equally acceptable fits. Thus no physical or chem-
ical meaning can be associated with the optimized
Figure 3 Symbols for compounds 1–4 as in Fig. 1. The numerical Runge–Kutta integration and the Simplex search are applied
to differential equation system (1), although with concentrations prefixed by the unique time-dependent activity coefficient α(t),
in the functional form (3). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
International Journal of Chemical Kinetics DOI 10.1002/kin

KINETIC PARAMETER ESTIMATION OF SOLVENT-FREE REACTIONS
159
parameters, and an independent method for the evalu-
ation of k1 and k2 must be looked for. It can be antici-
pated that one important finding of the Simplex search,
the uniqueness of the activity coefficient α(t), valid
for all involved species, will confer reliability to this
method.
the concentrations of 1 and 2 to an eighth-degree
−1
−1
h
polynomials gives −0.56 and −0.57 mol L
for the respective initial disappearance rates, whence
−
2
−1 −1
h . At the time when the
k1 = 1.62 × 10 L mol
concentration of 3 reaches a maximum, equation (5)
is still valid, referring to a situation where the vol-
ume shrinking is the same for all species. The value of
The k1 parameter can be derived from the evaluation
of the initial rates for the conversion of EC 1 and aniline
−3
−1 −1
h can be derived.
k2 = 3.56 × 10 L mol
2, which are equal:
CONCLUSIONS
d[1] / dt = d[2] / dt = − k1[1] [2]0
(4)
0
Under solvent-free conditions, both chemical and
physical reasons may affect the reaction outcome.
We have purposely chosen a reaction model where
the physical changes are most relevant. Quite clearly,
chemical parameters, as the kinetic constants, must re-
fer to a standard physical state. And quite reasonably,
the best choice for this state is the initial state.
Two different computational methodologies have
been adopted, to recover these standard state kinetic
constants. (a) The fitting of the experimental points
into the integrated expression (either analytical or, as
in our case, numerical) of the system of differential
equations (1). An acceptable fit requires the inclusion
of a time-dependent concentration coefficient, which
is adequately chosen as having a unitary value at the
standard state. The fact that the coefficient is unique
and applicable to all involved species will validate the
following alternative methodology. (b) The rate con-
stant k1, which is “operative” at the standard state, can
be deduced from the evaluation of the initial rate (by
differentiations of the correct polynomial fittings of the
reaction progress). The rate constant k2 is given from
the ratio k2/k1 measured at a singular state (when [3]
is highest) different from the standard state.
These rates are usually determined by the slope of the
straight line passing through the first experimental con-
centrations. In our case, an insufficient number of con-
centrations can be monitored, which are also biased
by the time-dependent activity coefficient; therefore,
we devised to take advantage of the procedure based
on the Newton polynomial interpolation of the exper-
imental data [9]. We selected an eighth-degree inter-
polation. The zero-order coefficient of the derivative is
the first-order coefficient of the interpolation and is the
only surviving coefficient at zero time, thus supplying
the estimation of the initial rate. The procedure gives
−1
−1
−
0.58 and −0.61 mol L
h for the initial disappear-
ance rates of 1 and 2. The degree of similarity of the
two values is a test for the reliability of the procedure.
The average value divided by the initial concentrations
−2
−1 −1
h . Be-
[
1]0 and [2]0 gives k1 = 1.71 × 10 L mol
cause of the covariance biasing the Simplex search, the
k1 values obtained with the two methods can agree as
for the order of magnitude only.
The evaluation of k2 by this procedure is impossible.
We can, however, evaluate the ratio k2/k1 at a singular
time: when the concentration of 3 reaches a maximum.
At this time
The comparison of the kinetic constants obtained
by the two methodologies is strongly biased by the co-
variance problem innate in the Simplex search. Thus
procedure (b) should be adopted when trustworthy ki-
netic parameters are required for meaningful compar-
isons within a coherent panel of solvent-free reactions
2
2
d[3] / dt = k1α(t) [1][2] − k2α(t) [1][3] = 0
and k2/k1 = [2] / [3]
(5)
This singular time is recognized by fitting the points of
in an eighth-degree polynomial and calculating the
[10].
3
concentrations of 3 from this polynomial, starting from
time zero and for small time increments, until the cal-
culated value starts to decrease. At this time (10.00 h)
k2/k1 = 0.25. From the above-reported value of k1,
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