Isoparametricity and related phenomena in reactions of trans-2,3-diaryloxiranes with arenesulfonic acids

This article is a continuation of a series of publications devoted to the systematic study of the still poorly investigated unique isoparametricity phenomenon in chemistry. Quantitative evidence for all variants of the second-order interaction structure-structure and structure-temperature types were obtained in the reactions between X-substituted trans -2,3-diaryloxiranes and Y-substituted arenesulfonic acids. Experimental evidence for the reality of isoparametric points with respect to the parameters of variable factors was obtained. Of particular importance is the proof of the physical reality of such an aspect of isoparametricity as the widely discussed enthalpy-entropy compensation effect. Important information was obtained about the intriguing properties of the three-parameter relationships with cross terms, concerning such attributes as the critical values of the parameters of the variable factors. At these critical values corresponding terms in three-parameter regressions disappear.


Introduction
Efficient control of chemical processes requires knowledge of the quantitative relations describing the effects of various internal and external factors (structure, solvent, temperature, pH of medium, pressure, etc.) on their kinetic, activation, thermodynamic, and other characteristics.This fundamental problem common to the whole of chemistry has been generally solved in terms of correlation analysis.][3][4][5][6][7] They demonstrate the surprising universality of the principle of linearity of free energies, which Palm has shown is a particular case of the polylinearity principle. 7Of special interest are polylinear equations (PLEs) with cross terms, taking into account non-additivity (interaction) of the effects of mutually varied factors.This makes them uniquely suitable for the analysis of chemical data.Predictive capabilities of PLEs are significantly improved compared to traditional one-parameter correlations.The presence of cross terms is implied by the general form of PLEs 7 and is interpreted as a change in the intensity of the effect of one factor under the influence of another factor.Non-additivity of the effects of two cross-varied factors determines such an intriguing property of PLEs as isoparametricity. 7,8This term means that the coefficient of sensitivity to the effect of one of factors in one-parameter correlations becomes zero for the special value of the parameter of another factor called isoparametric point (IPP).As a real phenomenon, isoparametricity is a challenge to traditional concepts in chemistry.Despite this, little attention is paid to its study.One of the reasons is that the IPPs in many processes fall within the area of distant extrapolation and are unattainable experimentally.In this context, the experimental evidence of isoparametricity is an actual phenomenological task.Our previous studies have shown that nucleophilic substitution reactions at carbonyl, benzyl, and benzhydryl carbon atoms, [9][10][11][12] along with nucleophilic reactions of oxirane ring opening, [13][14][15][16][17][18] are very promising for the experimental investigation of isoparametricity in all its aspects.The systematic study of multifactor effects on the kinetics of these reactions revealed a wide variety of IPPs in cross correlations of the structure-structure, structure-solvent, structure-temperature, and solvent-temperature types.Some of these points were passed through, which is a rare situation in chemical processes, and this was accompanied by reversal of the signs of the corresponding sensitivity coefficients because of reactivity reorientation (isoparametricity paradox).

Scheme 1. Reactions of oxiranes 1a-d with arenesulfonic acids 2а-e.
Since in the reaction series three factors were cross-varied (substituents X, Y, and temperature T), this allowed us to test the manifestation of various combinations of the second-order interactions of their effects, as well as the third-order interaction.A part of the results of cross correlation analysis of a kinetic experiment

Two-parameter correlations
To evaluate the effects of two-variable factors i and j at a fixed parameter of the third factor h, on the reaction rate the following PLE was used Here log k 00h is the value of log k ijh at randomly selected standard values of the parameters of the factors i and j, e.g., x i = 0 and x j = 0, q i 0h and q j 0h are coefficients of the sensitivity toward x i and x j under standard conditions (x j = 0, and x i = 0, respectively), q ij h is the cross-interaction coefficient, which reflects the perturbing (nonadditive) effects of factors i and j at a fixed factor h (in this and subsequent equations, the subscripts and superscripts relate respectively to the varied and fixed factors).The presence of a cross term q ij h in PLE 1 enables to calculate two IPPs: x i(j) h = -q j 0h (q ij h ) -1 and x j(i) h = -q i 0h (q ij h ) -1 (hereinafter, the subscript in parentheses refers to the second variable factor).At these IPPs the magnitude of log k ijh is the same, i. e. log k ijh = log k 00hq i 0h q j 0h (q ij h ) -1 , and remains constant when either the factor j at the IPP x i(j) h (q j ih = 0), or the factor i at the IPP x j(i) h (q i jh = 0), is varied.
The PLE 1 has the form of PLEs 2-4, describing the combined effects of substituents X and Y at a fixed temperature T, substituents Y and T with a fixed substituent X, and substituents X and T with a fixed substituent Y.
Here σ Y is the Hammett constant for substituent Y and τ X = log k 00Т -log k X0Т (Т = 265 K) is a quantitative characteristic of total effects of substituents Х (τ X for Х in oxiranes 1а, 1b, 1c, and 1d are equal to 0, 2.02, 3.47, and 4.38, respectively 15 ).The values of the coefficients of PLEs 2-4 calculated for various two-factor crossreaction series with the use of kinetic data 17 are given in Table 1.
Statistical significance of the cross-interaction coefficients q XY T , q YT X , and q XT Y show that all variants of the second-order interactions of the effects of the substituents X, Y and temperature T should be considered when assessing the reactivity of the studied reaction system.At the same time, the invariance of these coefficients when the parameters of fixed factors are changed indicates the absence of third-order interaction (q XYT = 0).
X , and Т (X) Y , only two points proved to be experimentally achievable.As can be seen from Table 1, in the cross-reaction series with variable substituents Y and temperature T at the fixed substituent X = 3-Br-5-NO 2 in oxirane 1d the IPP T (Y) X = 262 K is close to a temperature of 265 K in an experiment.In accordance with the regularities of isoparametric dependencies at this IPP substituents Y in the acidic reagent should not influence on the rate of these reactions (q Y ХT = 0).This is illustrated by a decrease in the sensitivity coefficient q Y XT to the effects of the substituents Y practically to zero in the reactions of oxirane 1d with a decrease in temperature: 17 q Y XT (T K) = 1.01 ± 0.09 (298 K), 0.50 ± 0.04 (281 K), 0.10 ± 0.05 (265 K).
The IPP τ X(Y) T = 4.73 was almost reached in the reaction series with variable substituents X and Y at a fixed temperature of 265 K, since the value τ X = 4.38 for the substituent Х in oxirane 1d is close to the value of this IPP.At this point the effects of the substituents Y should not appear, which is confirmed by a decrease in q Y XT (T = 265 K) = 1.57± 0.02, 1.00 ± 0.04, 1.50 ± 0.03, 0.10 ± 0.05 with an increase in the electron-withdrawing properties of the substituents X in oxiranes 1a, 1b, 1c, and 1d. 17n terms of activation parameters, the second-order interactions of the effects of structure and temperature can be described by a PLE: Here ΔG 00h ≠ is the free energy of activation under standard conditions (x i = x j = 0), Q i 0h and Q j 0h are parameters of the standard reactions at x j = 0 and x i = 0, respectively, Q ij h is the cross-interaction coefficient at a fixed factor h.The PLE 5 has the form of PLEs 6-8, describing the combined effects of substituents X and Y at a fixed temperature T, substituents Y and T with a fixed substituent X, and substituents X and T with a fixed substituent Y.
ARKAT USA, Inc ΔGXYT  = ΔGX00  + QY X0 σY + QT X0 T + QYT X σYT ( 7) The coefficients of PLEs 6-8 calculated for various two-factor cross-reaction series with the use of the ΔG XYT  values taken from our previous work 17 are presented in Table 2.The values of the cross-interaction coefficients Q XY T , Q YT X , and Q XT Y indicates the influence of all types of the second-order interactions of the effects of the cross-varied factors on the free activation energy ΔG XYT ≠ .At the same time, the invariance of these coefficients when the parameters of fixed factors are changed indicates the absence of third-order interaction (Q XYT = 0).Twenty IPPs (τ X(Y) , and , are given in Table 2.Only three of them turned out to be experimentally achievable.First of all it should be noted the IPP for the temperature T (Y) X(G) = 261 K in the reactions of oxirane 1d with acids 2a-2d, which is consistent with the above calculated value of the IPP T (Y) X = 262 K.At this IPP, the free activation energy ∆G XYT ≠IP should not depend on the effects of the substituents Y.The reason for the disappearance of the effects Y on G ХYT  at the IPP T (Y) X(G) is the EECE:  Y S XY  = 0 and ΔG XYT IP = constant (log k XYT = constant, q Y XT = 0).Because of the small difference between T (Y) X(G) = 261 K and temperature 265 K in the experiment, we have a rare opportunity to prove the physical reality of the enthalpy-entropy compensation phenomenon.
Due to the EECE in the reactions of oxirane 1d with acids 2a-d at 265 K, the substituents Y have no effect on the values of the free activation energy 18 : ΔG ХYT  (Y) = 93.3(4-OMe), 92.9 (4-Me), 93.0 (H), 93.1 (4-Cl) kJ mol -1 .Note that the IPP T (Y) X(G) , calculated from activation parameters (Equations 7, 8), is called the compensation temperature T comp , while the term isokinetic temperature T iso refers to the IPP T (Y) X , calculated from kinetic data (Equations 3, 4).These points, calculated in different ways, practically coincide in magnitude.Two IPPs for the constant of the substituent Y σ Y(T) X(G) = -Q T X0 (Q YT X ) -1 , namely, σ Y(T) X(G) = 0.52 (X = 4-NO 2 ) and σ Y(T) X(G) = 0.20 (X = 3-Br-5-NO 2 ), were realized in the reactions of oxiranes 1c and 1d with acids 2a-e at different temperatures. 16,18They fall in the experimental range of variation of σ Y = -0.27÷ 0.71.A remarkable feature of these IPPs is that the free activation energy ΔG XYT IP at these points does not depend on temperature, that is possible if in the expression ΔG XYT IP = ΔH XY  -TΔS XY  the activation entropy S ХY  = 0 and the free activation energy is determined only by the enthalpy term (ΔG XYT IP = ΔH XY  ).In these reactions we were able not only to demonstrate a rare case of realization of two IPPs for the structural parameter σ Y(T) , but also to realize unique transitions through these points, when the substituents Y were The reversal of the sign of the activation entropy S ХY  after passing through these IPPs (Figure 1) leads to the inversion of the temperature effect on the free activation energy ΔG XYT  .This unique situation is illustrated by Figure 2.

ARKAT USA, Inc
Calculated IPPs can be used as quantitative criteria of the reaction mechanism presented in Scheme 2. This mechanism provides for the formation of complex A with H-bond between the oxirane and acid НА (YC 6 H 4 SO 3 H) in the first equilibrium stage.In the second and rate-determining stage, the substrate activated in this way undergoes a nucleophilic attack by the second acid molecules with the formation of transition state (TS) B. This stage proceeds according to the mechanism А N D N with electrophilic assistance from the acid to the С-О bond opening in the oxirane ring.then reversed.This in turn causes an inversion of influence of the temperature effect on the free activation energy ΔG XYT ≠ (Figure 2).
According to the polylinearity principle the values of IPPs (Tables 1, 2) obtained by q ij h interaction of the effects of two cross-varied factors i and j proved to be dependent linearly on the effects of a fixed factor h: The values of the coefficients of Equations 9 and 10 are given in Tables 3 and 4. Using them, we can quantitatively predict the conditions for the implementation of any IPP.Of particular interest is prediction of conditions under which experimental realization of IPPs for temperature, connected with the EECE, will be possible.For example, it follows from the data in Table 3 that the accessible IPPs T (Y) X , e. g., T (Y) X = 283 K or 303 K, at which there is no influence of temperature on the effects of substituent Y (q Y XT = 0), can be implemented if τ X = 7.11 or 8.66, respectively.From coefficients of Equation 10 in Table 4 for the cross-reaction series in which the temperature T and substituents X are varied at a fixed substituent Y, it follows that easily accessible IPP τ X(T) Y(G) = 0 (X = H), at which there is no effect of temperature on the free activation energy (ΔS ХY ≠ = 0), is realized if σ Y = 0.80, for instance, in the case of acid 2 with a substituent Y such as 4-NO 2 (σ Y = 0.78).In this way we can predict at a quantitative level the value of any IPP x i(j) h or x i(j) h(G) in two-factor cross-reaction series for a given value of the parameter of a fixed factor h.However, it should be borne in mind that the change of the fixed substituents Y has practically no effect on the values of the IPPs T (X) Y (Tables 1, 3) and T (X) Y(G) (Tables 2, 4).

Figure 1 .
Figure 1.Sign inversion of the activation entropy S ХY  after crossing the IPP σ Y(T) X(G) = 0.52 in reactions of oxirane 1c with acids 2a-e; the values of S ХY  are taken from work. 17

Table 2 .
The coefficients of PLEs 6-8 (R  0.985) and IPPs for the two-factor cross-reaction series including

Table 3 .
The coefficients of Equation9for the two-factor cross-reaction series

Table 4 .
The coefficients of Equation10for the two-factor cross-reaction series