A polylinear regression model taking into account the cross effects of the structure and temperature in the reactions of phenyloxirane with benzoic acids, catalyzed by pyridines: Evidence of the isoparametricity phenomenon

Article abstract of DOI:10.1134/S107042721308017X

The joint effect of the structure and temperature on the rate of the reactions of phenyloxirane with Y-substituted benzoic acids in acetonitrile, catalyzed by X-substituted pyridines, was studied. A polylinear regression model that adequately describes the effects of cross-varied factors was calculated. Owing to intensive interaction of the effects of the substituents X and temperature, experimental evidence for the isoparametricity phenomenon was obtained. Pleiades Publishing, Ltd., 2013.

Full text of DOI:10.1134/S107042721308017X

ISSN 1070-4272, Russian Journal of Applied Chemistry, 2013, Vol. 86, No. 8, pp. 1256−1261. © Pleiades Publishing, Ltd., 2013.
Original Russian Text © I.V. Shpan’ko, I.V. Sadovaya, 2013, published in Zhurnal Prikladnoi Khimii, 2013, Vol. 86, No. 8, pp. 1282−1287.
CATALYSIS
A Polylinear Regression Model Taking into Account the Cross
Effects of the Structure and Temperature in the Reactions
of Phenyloxirane with Benzoic Acids, Catalyzed by Pyridines:
Evidence of the Isoparametricity Phenomenon
I. V. Shpan’ko and I. V. Sadovaya
Donetsk National University, Donetsk, Ukraine
e-mail: shpanko09@rambler.ru
Received May 18, 2013
Abstract—The joint effect of the structure and temperature on the rate of the reactions of phenyloxirane with Y-
substituted benzoic acids in acetonitrile, catalyzed by X-substituted pyridines, was studied.Apolylinear regression
model that adequately describes the effects of cross-varied factors was calculated. Owing to intensive interaction
of the effects of the substituents X and temperature, experimental evidence for the isoparametricity phenomenon
was obtained.
DOI: 10.1134/S107042721308017X
Efficient control of chemical processes requires
knowledge of the quantitative relationships that take into
account the joint effect of various internal and external
factors (structure, solvent, temperature, pH of the me-
dium, pressure, etc.) on their kinetic and thermodynamic
characteristics. Representation of such relationships in
the form of multiparameter correlations based on the
polylinearity principle (PLP) is the most promising [1].
Polylinear equations (PLEs) contain cross terms taking
into account nonadditivity (interaction) of the effects of
mutually varied factors, whereby their predictive capa-
bilities are significantly improved compared to traditional
one-parameter correlations. Over the recent decades,
PLEs have been successfully used for obtaining quanti-
tative estimates of the rates of nucleophilic substitution
reactions at various electrophilic reaction sites under
multifactor conditions, as well as for interpreting their
mechanisms [2–7]. Owing to interaction of the effects
of the structural factors in those processes, experimental
evidence for the unique isoparametricity phenomenon
was obtained [1, 3, 5–7].
in empirical one-parameter correlations, e.g., α (β) in
the Brönsted equation, or ρ in the Hammett equation,
or the coefficient proportional to the activation energy
in the Arrhenius equation, at the isoparametric value of
the parameter of another factor, called the isoparametric
point (IPP). In other words, at the IPP for the param-
eter of one of the factors no influence is exerted on the
correlated value of another factor. On passing through
the IPP, the inversion of the signs of the corresponding
sensitivity coefficients takes place (isoparametricity
paradox).
Though broadening and deepening the knowledge
of previously unknown properties of physicochemical
systems that are in conflict with conventional chemical
theory, the isoparametricity studies have received little
attention so far. This is due, on the one hand, to high la-
boriousness of multifactor experiments, and on the other
hand, to the virtual nature of the IPPs in many reaction
series (RSs), because they fall into a far extrapolation
region and are experimentally inaccessible. Further prog-
ress in study of the isoparametricity phenomenon may be
achieved with RSs for which cross interactions involve
other, along with the structural, factors. In this context,
Isoparametricity is manifested as the zero value of the
sensitivity coefficient for the effects of one of the factors
`
1256

A POLYLINEAR REGRESSION MODEL
1257
temperature deserves special attention as a universal fac-
tor characterizing the degree of hotness of the system.
and natural origin [11, 12]. We showed previously [13]
that, in the reactions of aryloxiranes with arenesulfonic
acids, intensive interaction of the effects of the structure
of the acid reactant and the temperature is manifested,
which allowed the first experimental observation of the
isoparametricity phenomenon in the oxirane chemistry.
For study of the joint effects of the structure and
temperature we chose oxirane ring opening reactions
which are of interest from both theoretical and practical
viewpoints. The specific organization of the atoms in the
oxirane ring, combined with its strain and unsaturated
nature, are responsible for a wide variety of practically
important reactions of oxirane substrates with compounds
from different classes [8–10]. These reactions are widely
used in organic synthesis, as well as in production of poly-
mers, pharmaceuticals, and pesticides, and also underlie a
number of biochemical processes, in particular, the physi-
ological process of detoxification of metabolites of exog-
enous substances. The oxirane ring is a structural element
of many biologically active substances of both synthetic
Here, we studied the joint effect of the structure and
temperature on the rate of the reactions of phenyloxirane
with Y-substituted arenecarboxylic (benzoic) acids (Y =
H, 3-Br, 3-NO2), catalyzed by X-substituted pyridines
X-Py (X = 4-OMe, 4-Et, H, 3-COOEt, 3-CN), in ace-
tonitrile at 279, 295, 308, 323 and 343 K, examined the
isoparametric properties of the cross reaction series, and
calculated a polylinear regression model of the system
reactivity on the basis of the results of a multifactor ki-
netic experiment.
OH
X-Py
+
YC6H4COOH
(1)
O
OC(O)C6H4Y
EXPERIMENTAL
k3m (correlation coefficient r ≥ 0.998), whose extrapola-
tion to the origin indicates the absence of a noncatalytic
process. The accuracy of the kinetic and correlation pa-
rameters was estimated in terms of the standard deviation
SD, which was derived by the statistical method from N
experimental data points. Statistical data processing was
carried out using Origin 6.1 package for the confidence
level of 0.95.
As solvent we used analytically pure-grade acetonitrile,
which was dried and distilled successively from P2O5
and CaH2. Phenyloxirane (Merck, content of the main
substance no less than 98%) was vacuum-distilled.
Pure-grade benzoic acids were recrystallized twice from
aqueous ethanol (1 : 1). Pure- and chemically pure-
grade pyridines were vacuum-distilled. The products of
reactions (1) are primary alcohols, 2-(Y-benzoyloxy)-
2-phenylethanols [14]. The rate of the reactions was
determined from the loss of the acid reactant HA, as we
described previously [14]. In the kinetic experiments we
used the oxirane substrate S in no less than tenfold excess
with respect to HA: [S]0 >> [HA]0 = 0.062–0.090 M. The
concentration of the X-Py catalyst m was varied within
0.00953–0.0473 M. Under these conditions, the reaction
rate is described by the equation d[HA]/dt = k1[HA] =
k2[S]0[HA] = k3[S]0[HA]m. In all the kinetic experiments
the observed pseudo-first-order rate constants k1, s–1,
remained unchanged during the process up to 70–80%
conversions of the acid reactant. The k1 constants were
determined accurately to within 5%. The second-order
rate constants k2, L mol–1 s–1, were obtained using the re-
lationship k2 = k1/[S]0. The apparent third-order catalytic
rate constants k3, L2 mol–2 s–1, were calculated for four or
more concentrations m from the linear dependences k2 =
The rate constants k3 listed in the table for all the re-
actions studied reflect the joint effect of the cross-varied
factors (substituents X in the pyridines, substituents Y
in the benzoic acids, temperature T) on the rate of the
catalytic process (1). It is noteworthy that, at 295 K, in all
the RSs with fixed substituentsY, the substituents X have
negligible effect on k3.At a lower temperature (279 K) k3
tends to increase with weakening of the electron-donating
and strengthening of the electron-accepting power of the
substituents X. By contrast, at temperatures above 295 K,
the identical change in the power of the substituents X
causes k3 to decrease, more intensively at higher tempera-
tures. These changes in k3 are indicative of interaction
(nonadditivity) of the effects of the substituents X and
temperature in cross RS (1). For quantitative estimation
of this interaction the kinetic data were subjected to cor-
relation analysis.
The effect of the substituents X on the rate of reactions
(1) is well described by the Hammett equation
RUSSIAN JOURNAL OF APPLIED CHEMISTRY Vol. 86 No. 8 2013

1258
SHPAN’KO, SADOVAYA
Rate constants k3 and ρXYT coefficients in the Hammett equation for the reactions of phenyloxirane with Y-substituted benzoic
acids, catalyzed by X-substituted pyridines, in acetonitrile at different temperatures
k3 × 105, L2 mol–2 s–1, at indicated temperature, K
X (σX)
279
295
308
323
343
Y (σY) = 3-NO2 (0.71)
4-OMe (–0.27)
4-Et (–0.15)
H (0)
2.26 ± 0.01
2.88 ± 0.03
3.7 ± 0.1
7.0 ± 0.1
–
16.1 ± 0.1
15.8 ± 0.2
63.7 ± 0.9
48 ± 2
219 ± 2
133 ± 3
1260 ± 20
701 ± 4
287 ± 7
–
15.6 ± 0.1
34.8 ± 0.1
19.2 ± 0.5
14.9 ± 0.7
–0.75 ± 0.04
78.8 ± 0.3
–
3-COOEt (0.37)
3-CN (0. 56)
ρXYT
14.4 ± 0.2
13.7 ± 0.2
16.9 ± 0.2
–1.30 ± 0.07
–
0.761 ± 0.009
–0.083 ± 0.003
–2.4 ± 0.1
Y (σY) = 3-Br (0.39)
4-OMe
4-Et
–
7.40 ± 0.09
7.20 ± 0.03
6.99 ± 0.03
6.7 ± 0.2
32 ± 2
120 ± 2
80 ± 1
650 ± 10
302 ± 6
141 ± 2
–
1.39 ± 0.03
2.00 ± 0.01
3.7 ± 0.1
–
26.2 ± 0.2
18.1 ± 0.1
10.69 ± 0.02
7.0 ± 0.3
H
45.1 ± 0.1
–
3-COOEt
3-CN
6.26 ± 0.07
–0.077 ± 0.009
7.7 ± 0.1
–1.74 ± 0.04
–
YT
ρX
0.81 ± 0.08
–0.77 ± 0.04
–2.4 ± 0.1
Y (σY) = H (0)
4-OMe
4-Et
–
3.4 ± 0.1
3.28 ± 0.07
3.22 ± 0.05
3.00 ± 0.06
–0.078 ± 0.002
14.9 ± 0.2
11.5 ± 0.5
8.2 ± 0.1
51.9 ± 0.8
32 ± 1
292 ± 1
136 ± 6
64 ± 3
–
0.692 ± 0.006
0.894 ± 0.006
1.71 ± 0.01
0.751 ± 0.005
H
18.9 ± 0.2
–
3-COOEt
4.7 ± 0.1
YT
ρX
–0.77 ± 0.06
–1.62 ± 0.06
–2.4 ± 0.2
logk3 = logk3st + ρХYTσX,
(2)
calculated using the linear dependence ρXYT = ρXYT = ∞
+
qXYT ×103/T, which is represented in the series of the fixed
substituents Y = H, 3-Br, 3-NO2 by the equations (r ≥
where k3st is the rate constant at temperature T with the
substituent Y under standard conditions (X = H, σX = 0),
and ρXYT, coefficient of sensitivity to the effects of the
substituents X (from here on, the sub- and superscripts
refer to the varied and fixed factors, respectively).
The ρXYT coefficients (see table) for partial RSs
depend on the temperature which affects not only the
magnitude but also the sign of ρXYT. The reason is the
above-mentioned interaction of the effects of the X
substituents and temperature in reaction (1). According
to the PLP, the cross-interaction coefficient qXYT can be
0.991) ρXYT = (–16.6 ± 0.4) + (4.9 ± 0.1) × 103/T, ρXYT
=
(–17.1 ± 0.7) + (5.0 ± 0.2) × 103/T, and ρXYT = (16.0 ±
0.7) + (4.7 ± 0.2) × 103/T, respectively.As seen from these
equations, qXYT is unaffected by the substituentsY, which
fact suggests the lack of pair interactions of the effects of
X and Y (ρXYT = 0), on the one hand, and of the effects of
Y and temperature, on the other (qXYT = 0), as well as of
the ternary interaction of the effects of the substituents
X and Y and temperature T (qXYT = 0). The lack of these
interactions is responsible for a narrow range of variation
RUSSIAN JOURNAL OF APPLIED CHEMISTRY Vol. 86 No. 8 2013

A POLYLINEAR REGRESSION MODEL
R = 0.998, SD = 0.0362, N = 58, F = 6226.
1259
(6)
of the parameters of sensitivity ρYXT to the effects of the
substituents Y, calculated using the tabulated data, in the
equation log k3 = log k3st + ρYXTσY for partial reactions
in the case of different fixed substituents X and different
fixed temperatures T (ρYXT = 0.86–0.96, r ≥ 0.992).
The statistical characteristics of regression (6) are
indicative of high reliability of all its coefficients. The
adequacy of the rate constants log k3calc, calculated us-
ing this regression, to the experimental log k3exp data is
demonstrated by Eq. (7) in which the slope is virtually
equal to the expected unity:
In the general case, according to the PLP, the joint
effect of the substituents X and Y and temperature T on
the rate of reactions (1), taking into account the cross-
interaction of the effects of all the three varied factors,
should be estimated using the PLE:
log k3calc = (0.029 ± 0.026) + (1.004 ± 0.007)k3exp
r = 0.999, SD = 0.0379, N = 58.
,
(7)
logk3 = logk3st + ρXstσX + ρYstσY + BTst × 103/T + ρXsYt σXσY
Elimination from Eq. (7) of the statistically insignifi-
cant intercept term gives logk3calc = (1.004 ± 0.007)k3exp
+ q σX × 103/T + qYsTtσY × 103/T + qXYTσXσY × 103/T, (3)
s t
XT
,
which relationship demonstrates an excellent agreement
between the calculated and experimental log k3 values.
where k3st is the rate constant under standard conditions
st
st
st
(σX = σY = 0, T = ∞ K); ρX , ρY , and B , parameters of
the standard reactions at σY = 0 and T =T∞ K, σX = 0 and
Owing to the statistically significant coefficient at
the cross-member (qXT = –2.85 ± 0.02), regression
(6) is isoparametric. It is characterized by two IPPs,
namely, that for the constant of the substituent X,
σXIP = –BTst/qXT = 0.63, and that for the reciprocal of the
s t
st
T = ∞ K, and σX = σY = 0, respectively; ρXY, qXT, and
qYsTt , pair interaction coefficients in the standard reaction
series (at T = ∞ K, σY = 0, and σX = 0, respectively); and
qXYT, triple cross-interaction coefficient.
temperature 103/TIP= –ρXst/qXT = 3.42, TIP = 292 K, as well
By processing the results of the multifactor kinetic
experiment using Eq. (3), a polylinear regression was
calculated (R is the multiple correlation coefficient, and
F, Fisher’s exact test):
st
as by isoparametric value log k3IP = log k3st – ρX BTst/qXT
=
–4.57, identical for the both IPPs. It should be noted that
st
log k3IP is part of the apparent log k3 = log k IP + ρY σY =
–4.57 + (0.91 ± 0.02)σY value, and only at 3the standard
value σY = 0 (Y = H) log k3 = log k3IP.
logk3 = (5.1 ± 0.2) + (–15.6 ± 0.7)σХ + (1.2 ± 0.3)σY
+ (–2.83 ± 0.05) × 103/T + (0.4 ± 1.4)σХσY
+ (4.6 ± 0.2)σX × 103/T + (–0.08 ± 0.09)σY × 103/T
+ (–0.1 ± 0.4)σXσY × 103/T,
At the IPP σXIP = 0.63 the rate of the process (logk3)
should be temperature-independent. The figure illustrates
a decrease in sensitivity to the effect of temperature in the
reaction involving 3-bromobenzoic acid, with the IPPσXIP
being approached on changing from electron-donating to
electron-accepting substituent X in pyridine. The reac-
tions involving 3-CN-pyridine, for which the constant
σX = 0.56 in the case of the substituent X = 3-CN is little
different from the isoparametric value σXIP = 0.63, exhibit
low sensitivity to the effect of the temperature. These
reactions are characterized by close to zero slopes BTXY
in the Arrhenius equation log k3 = A X=Y∞ + BTXY×103/T
(r ≥ 0.995): Y = 3-Br, log k3 = (–3.17T± 0.06) + (–0.30 ±
0.02) × 103/T;Y= 3-NO2, log k3 = (–2.8 ± 0.1) + (–0.31 ±
0.03) × 103/T. As a consequence, the apparent activation
energies EaXY = –2.303R × BTXY×103 (R is gas constant)
are low: 5.7 and 5.9 kJ mol–1, respectively.
R = 0.998, SD = 0.0367, N = 58, F = 3260.
(4)
As might be expected, in regression (4) the already
discussed ρXY, qYT, andqXYT coefficients are statistically
insignificant.After exclusion of the cross-terms with these
coefficients, PLE (3) is simplified to the form
log k3 = log k3st + ρXstσX + ρYstσY + BTst × 103/T
+ qXTσX ×103/T.
(5)
Processing the kinetic data by Eq. (5) gave
log k3 = (5.18 ± 0.08) + (–15.4 ± 0.3)σХ
+ (0.91 ± 0.02)σY + (–2.85 ± 0.02) × 103/T
+ (4.5 ± 0.1)σX × 103/T,
As to the IPP for temperature TIP= 292 K, its realiza-
tion in experiment is clearly demonstrated by the figure,
in which it appears as the intersection point of the family
of the correlation straight lines in the Arrhenius equa-
RUSSIAN JOURNAL OF APPLIED CHEMISTRY Vol. 86 No. 8 2013

1260
log k3
SHPAN’KO, SADOVAYA
enabled calculation of a polylinear regression model that
adequately describes the effect of cross-varied factors
(structure, temperature) on the rate of the reactions of
phenyloxirane with benzoic acids, catalyzed by pyridines.
(2) The interaction of the effects of the substituents
X in pyridine and of the temperature was revealed,
whereby experimental evidence for the isoparametricity
phenomenon was provided: At the isoparametric point
for temperature TIP (isokinetic temperature) the rate of
the reactions studied is independent of the structure of
the substituents X, and on passing through this point the
order of the effect of X on the catalytic activity of the
pyridines is reversed (isoparametricity paradox). At the
isoparametric point σXIP for the constant of the substitu-
ent X the temperature has a negligible effect on the rate
of the process.
Variation of log k3 (see table) with 103/T (T, K) for the reaction
of phenyloxirane with 3-bromobenzoic acid in acetonitrile,
catalyzed by X-substituted pyridines: X = (1) 4-OMe, (2) H,
and (3) 3-CN. The log k3 values for the reaction involving
3-CN-pyridine at 279 and 343 K were calculated by Eq. (6).
(3) It was shown that the multifactor experiments and
cross-correlation analysis of their results enable identi-
fication of previously unknown fundamental properties
of cross reaction series, such as nonadditivity of the
joint effects of the varied factors, the isoparametricity
phenomenon, and the isoparametricity paradox. The
knowledge of these properties allows the use of quan-
titative approaches to designing optimal conditions for
organic reactions, in particular, for oxirane ring opening
reactions. Using the polylinear model of the reactivity of
the phenyloxirane-benzoic acid-pyridine catalyst systems
at different temperatures, it is possible to predict with high
reliability the rate of the catalytic process for any given
set of parameters of the varied factors.
tion coordinates for partial reaction series with fixed
substituents X in pyridine. At this IPP, substituents X
have no effect on the rate of the process (ρYT = 0). We
came very close to this IPP in the experiment at a close
to isoparametric temperature of 295 K. For all the fixed
substituents Y, the ρXYT values for the reactions at this
temperature are slightly different from zero (see table).
It should be emphasized that, in the case of cross RS
(1), a passage through IPP TIP with varied temperature
was realized, which is an extremely rare situation in
chemical processes. The observed inversion of the sign
of the parameter of sensitivity ρXYT to the effects of the
substituents X demonstrates the isoparametricity para-
dox, namely, reversal of the order of the effect of the
substituents X on the catalytic activity of the pyridines
on passing through the TIP.
REFERENCES
1. Pal’m, V.A., Osnovy kolichestvennoi teorii organicheskikh
reaktsii (Fundamentals of the Quantitative Theory of Or-
ganic Reactions), Leningrad: Khimiya, 1977
Thus, the joint effects of the structure and temperature
in cross RS (1) are adequately described by polylinear
regression model (6) which takes into account both the
additive contribution from the effects of the cross-varied
factors to the rate of the process and the nonadditive ef-
fects of the temperature and substituents X in pyridine.
The use of this polyilinear model, taking into account
its isoparametric properties, allows very accurate pre-
diction of the reactivity of the phenyloxirane-benzoic
acid-pyridine catalyst systems for any set of parameters
of the varied factors.
2. Lee, I., Chem. Soc. Rev., 1990, vol. 19, no. 3, pp. 317–333.
3. Shpan’ko, I.V., Mendeleev Commun., 1991, no. 4, pp. 119–
120.
4. Lee, I., Adv. Phys. Org. Chem., 1991, vol. 27, pp. 57–117.
5. Shpanko, I.V., Kim, S.I., Koh, H.J., and Lee, I., Bull. Ko-
rean Chem. Soc., 1995, vol. 16, no. 6, pp. 533–538.
6. Shpan’ko, I.V., Teor. Eksp. Khim., 1999, vol. 35, no. 2,
pp. 67–78.
7. Shpan’ko, I.V., Teor. Eksp. Khim., 2001, vol. 37, no. 5,
CONCLUSIONS
p. 265–279.
8. Paquin, A.M., Epoxidverbindungen und Epoxidharze,
(1) Processing the multifactor kinetic experiment data
RUSSIAN JOURNAL OF APPLIED CHEMISTRY Vol. 86 No. 8 2013

A POLYLINEAR REGRESSION MODEL
1261
Berlin: Springer, 1958.
Synthesis and BiologicalActivity), Moscow: Bogorodskii
Pechatnik, 1999.
9. Kas’yan, L.I., Kas’yan, A.O., Okovityi, S.I., and
Tarabara, I.N., Alitsiklicheskie epoksidnye soedineniya.
Reaktsionnaya sposobnost’ (Alicyclic Epoxy Compounds:
Reactivity), Dnepropetrovsk: Dnipropetrovsk. Univ., 2003.
12. Kas’yan, O.A., Golodaeva, E.A., Tsigankov, A.V., and
Kas’yan, L.I., Zh. Org. Khim., 2002, vol. 38, no. 7,
pp. 1661–1670.
10. Aziridines and Epoxides in Organic Synthesis, Yudin,A.K.,
13. Shpan’ko, I.V. and Sadovaya, I.V., Kinet. Katal., 2011,
Ed., Weinheim: Wiley, 2006.
vol. 52, no. 5, pp. 662–669.
11. Dryuk, V.G., Kartsev, V.G., and Voitsekhovskaya, M.A.,
14. Shpan’ko, I.V., Sadovaya, I.V., and Kitaigorodskii, A.M.,
Ukr. Khim. Zh., 2003, vol. 69, no. 6, pp. 111–115.
Oksirany – sintez i biologicheskaya aktivnost’ (Oxiranes:
RUSSIAN JOURNAL OF APPLIED CHEMISTRY Vol. 86 No. 8 2013