Enthalpy-Entropy Compensation Effect in Chemical Kinetics and Experimental Errors: A Numerical Simulation Approach.

Many kinetic studies concerning homologous reaction series report the existence of an activation enthalpy-entropy linear correlation (compensation plot), its slope being the temperature at which all the members of the series have the same rate constant (isokinetic temperature). Unfortunately, it has been demonstrated by statistical methods that the experimental errors associated with the activation enthalpy and entropy are mutually interdependent. Therefore, the possibility that some of those correlations might be caused by accidental errors has been explored by numerical simulations. As a result of this study, a computer program has been developed to evaluate the probability that experimental errors might lead to a linear compensation plot parting from an initial randomly scattered set of activation parameters (p-test). Application of this program to kinetic data for 100 homologous reaction series extracted from bibliographic sources has allowed concluding that most of the reported compensation plots can hardly be explained by the accumulation of experimental errors, thus requiring the existence of a previously existing, physically meaningful correlation.


___________________________________________________________________________
ABSTRACT: Many kinetic studies concerning homologous reaction series report the existence of an activation enthalpy-entropy linear correlation (compensation plot), its slope being the temperature at which all the members of the series have the same rate constant (isokinetic temperature). Unfortunately, it has been demonstrated by statistical methods that the experimental errors associated with the activation enthalpy and entropy are mutually interdependent. Therefore, the possibility that some of those correlations might be caused by accidental errors has been explored by numerical simulations. As a result of this study, a computer program has been developed to evaluate the probability that experimental errors might lead to a linear compensation plot parting from an initial randomly scattered set of activation parameters (p-test). Application of this program to kinetic data for 100 homologous reaction series extracted from bibliographic sources has allowed concluding that most of the reported compensation plots can hardly be explained by the accumulation of experimental errors, thus requiring the existence of a previously existing, physically meaningful correlation.

INTRODUCTION
A common practice in chemical kinetics that provides useful information on the reaction mechanism is to study a series of closely related processes, differing among them either in the nature of an inert substituent in one of the reactant molecules 1,2 or in the solvent employed to perform the experiments. 3, 4 An intriguing result often found in this kind of kinetic studies is the observation of a linear correlation between the activation parameters [activation energy (E a ) and logarithm of the pre-exponential factor (A) or, alternatively, enthalpy ( o H   ) and entropy ( o S   ) of activation] corresponding to the members of the homologous series. 5 The existence of such linear relationship would be important at a theoretical level because it can be easily demonstrated that it would imply that at a certain temperature (called isokinetic temperature) all the members of the reaction series have the same rate constant. 6 Unfortunately, it has been shown by statistical methods that the experimental errors associated with the activation energy and to the logarithm of the pre-exponential factor are not mutually independent. In fact, they are linearly related, with a slope that depends on the mean experimental temperature. [7][8][9][10] The same can be stated with respect to the errors associated with the enthalpy and entropy of activation. As a consequence, when the divergences between the activation parameters for a series of related reactions are small in comparison with their respective experimental errors, a linear relationship tends to appear both in the E a vs. ln A and Because of this, some authors have considered that the existence of the so-called enthalpyentropy compensation effect is a mere artefact derived from the experimental errors. 11,12 However, other authors have argued that in certain cases, when the errors are quite small in comparison with the divergences between the activation parameters for the different members of the reaction series, the linear correlation must be considered as physically meaningful and caused by a mechanism other than the experimental errors committed in the determination of the activation parameters. Several theories have been proposed to explain the origin of this statistically meaningful compensation effect. A comprehensive review developing some of them has been published. 13 More recently, other theoretical interpretations of this phenomenon have been reported, such as the models of selective energy transfer (SET) 14 or multiexcitation entropy. 15 Nevertheless, although many different interpretations of the kinetic compensation effect have been propounded, a generally accepted theory is still missing.
An interesting point in this context is the fact that this kind of linear relationship tends to appear not only in the field of chemical kinetics, but also in other disciplines working with different processes ruled by mathematical laws of the type: where y  is the value of the magnitude y at T = T ∞ and x has the units of an absolute temperature. In the experimental studies involving this kind of equations it is often found that there exists a correlation between the values of x and y  , so that when one of them increases the other increases too. For instance, in chemical thermodynamics it has been found an enthalpy-entropy compensation effect for the temperature dependence of the equilibrium constants corresponding to many homologous reaction series (isoequilibrium relationship). 16 In addition, similar compensation effects have been reported for a wide range of phenomena including those such as heating-induced changes in food chemistry, 17 protein folding/unfolding and ligand binding/unbinding in biochemistry, 18 as well as polymer relaxation, 19 thermal electron emission from semiconductor traps 20 and electron conduction in chalcogenide glasses 21 in physics. Precisely, this ubiquitous character of the compensation effect makes it important as an objective of research. Calculations. The statistical method used to obtain the best fit to a linear relationship was that of least squares. The absolute errors associated to the intercept and slope were the corresponding standard deviations. 22 The hardware used in all the numerical simulations was either a Sony Vaio or a Toshiba personal computer, and the software employed for the calculations was the programming language BBC BASIC (version for Windows).

EFFECTS OF EXPERIMENTAL ERRORS
As reported in the bibliography, 7-10 when the temperature-rate-constant data are fitted according to an Arrhenius plot, the experimental errors associated with the pre-exponential factor and the activation energy are mutually interdependent. This result has been qualitatively illustrated in Figure 1. It can be observed that, when the laboratory uncertainties lead to a linear plot (green line) with a slope more negative than that corresponding to the real reaction, the intercept associated with 1/T = 0 is higher than the value that would be obtained in the absence of errors (blue line). On the contrary, when the experimental errors lead to a linear plot (red line) with a slope less negative than that corresponding to the real reaction, the intercept is lower than that to be obtained in the absence of errors. This means that a positive error in the activation energy is usually associated with a positive error in the pre-exponential factor, whereas a negative error in the activation energy is usually associated with a negative error in the pre-exponential factor. Moreover, big errors in the activation energy are associated with big errors in the pre-exponential factor, whereas small errors in the first parameter are associated with small errors in the second.
Given the relationships existing between the activation energy and enthalpy on the one hand, as well as between the pre-exponential factor and the activation entropy on the other, this result can be easily extrapolated from the E a vs. ln A plane to the Hence, it is clear that the ubiquity of experimental errors in all laboratory measurements should be considered as (at least) one of the possible explanations of the compensation effect: an increase of the activation energy (or the activation enthalpy) is usually associated with an increase of the pre-exponential factor (or the activation entropy), the former causing a decay of the reaction rate and the latter an enhancement, whereas a decrease of the first parameter (positive effect on the reaction rate) is very often associated with a decrease of the second (negative effect on the reaction rate), thus leading to the appearance of a compensation effect.

NUMERICAL SIMULATIONS
Parameters Involved. Some magnitudes had to be introduced at the beginning in order to perform the calculations. The first parameters required were the mean value (T m ) and the difference between the maximum and minimum values (∆T) of the experimental temperature. These equations have been used to obtain activation parameters either lower (eq 2) or higher (eq 3) than the mean values, respectively, and were applied in a parity manner, each being responsible for a half of the simulated activation parameters.
An additional parameter necessary to start the simulations was the experimental error  (5) This parameter quantifies the maximum uncertainty allowed to the laboratory determination of the rate constants, so that when Q = 1 the maximum relative accidental error is ± 10.0 %.
Again, eqs 4 and 5 were applied in a parity manner, each being responsible for a half of the simulated error-affected rate constants. The theoretical rate constants were obtained from the Eyring equation, using the activation parameters given by eqs 2 and 3, whereas their experimental counterparts were either lower (eq 4) or higher (eq 5) than them.
The final parameter required was the number of members of the homologous reaction series (N). When F = 0 (identical activation parameters for all the members of the reaction family), the plot is linear and with the slope almost identical to the mean experimental temperature. An increase of parameter F results in a decrease of both the apparent isokinetic temperature and the correlation coefficient (Table 1).

Results
The value of the dispersion factor has been systematically changed from zero to a certain maximum limit, obtaining the parameters associated with each enthalpy-entropy compensation straight line for a family of N = 1000 simulated homologous reactions and repeating the calculations at five different mean experimental temperatures in the range 100  500 K. The resulting error-driven compensation plots have been fitted to equations of the type: where the intercept (  (7) and the correlation coefficient ( Figure 6, bottom), approaching asymptotically the value corresponding to a perfect linear relationship: The next parameter systematically changed has been the experimental temperature range, that is, the difference between the maximum and minimum temperatures ( ∆T ) for the experimental interval chosen to perform the determination of the activation parameters. In this case, a decrease of ∆T leads to an increase of both the apparent isokinetic temperature ( Figure   7, top), reaching the limit: This is logical indeed, because it has been assumed as an initial hypothesis that there is no real correlation between the activation enthalpies and entropies of the members of the homologous reaction series (in Nature), so that the relationships found in the laboratory must necessarily come from the experimental errors. It can then be concluded that the confidence that is to be attributed to a compensation straight line observed in a certain kinetic research (in 13 the sense that it seems to be caused by a physically meaningful phenomenon) should improve when the experimental errors committed in the determination of the rate constants are as low as possible and when the experimental temperature range is as wide as possible.

p-Test
The main objective of the present work is that of developing a test designed to differentiate those isokinetic relationships with an experimental error origin from those caused by a physically meaningful phenomenon. Effectively, it has been observed in Figure 2 that an enthalpy-entropy compensation effect is found even when there is no real correlation between the activation parameters, provided that the accidental errors associated with the determination of the rate constants are high enough. Although some authors have published useful tools for this purpose, 4,23,24 the search for a different alternative, potentially providing further quantitative information, seems attractive enough. Thus, the following goal of this research will be the elaboration of a test capable of discerning whether a certain activation enthalpy-entropy linear relationship found in the laboratory can be explained by the occurrence of experimental errors or not.
Several couples of parameters can be used as the basic framework of the desired test, such as the intercept of the compensation straight line and its corresponding absolute error The values allowed for the dispersion factor will be in the range 0 ≤ F ≤ F max . The lower limit (F = 0) corresponds to a situation in which the activation parameters are identical for all the members of the homologous reaction series and the experimental dispersion is thus completely due to accidental errors (Q = F max ), whereas the upper limit (F = F max ) corresponds to a situation in which the real dispersion of the activation parameters is identical to the one experimentally observed due to the absence of errors ( Q = 0). Therefore, the values of parameter Q will obey the following equation: so that, as parameter F increases from 0 to F max , parameter Q decreases from F max to 0. The values of the other parameters will be taken as done for the maximum probability curve except the number of reactions involved in the simulated homologous series, being now equal to that of the experimental case (N = N exp ).
Given that the number of reactions studied in each homologous family is indeed much lower than that required to obtain the maximum probability curve (N = 10 5 ), when the numerical simulations are done with N = N exp and represented in a graphic the points will appear scattered above and below that curve. The objective of the test will be, precisely, to determine the probability of finding one of these simulation points in a region of the T ikr plane near the experimental point.
Although none of the simulations will be exactly coincident with the empirical situation, some of them will potentially lead to points placed close enough: the higher the distance between the experimental point and the maximum probability curve, the lower the probability of finding a simulation point in its vicinity. At this moment, there are two different ways to continue with the calculations. The simpler would be to accept as valid those simulations leading to results contained in certain ranges around (T ik ) exp and r exp . However, the limitation inherent to this procedure would be the choice of the range widths, since an increase of the region area would automatically result in an increase of the corresponding probability. In order to outline the second alternative, one must wonder why the experimental point is not exactly placed on the maximum probability curve. Only two alternative explanations are possible: i) the number of members in the reaction series is too low (N exp <<  ) and ii) the hypothesis based on which the numerical simulations were performed (no real correlation between the family activation parameters) is wrong.
Therefore, the chance of the compensation linear plot being originated by accidental errors will be lower when the experimental point is placed far from the maximum probability curve, either above or below it. However, as mentioned before, it is impossible to find a simulation exactly matching the empirical information obtained from the bibliography: Once assumed this, it is important to define which simulations will be taken as valid (capable of explaining the experimental isokinetic correlation) and which ones will not:

All the simulations leading to points located farther from the maximum probability curve (as concerning the apparent isokinetic temperature) than the experimental point should be considered as valid, provided that the associated correlation coefficient is high enough.
This condition derives from the fact that the probability of finding a particular (T ik , r) couple of values is inversely proportional to its distance with respect to the curve of maximum Hence, the test will count the number of simulation points in the T ikr plane fulfilling the following criteria: where the subscripts indicate whether the slope and the correlation coefficient of the enthalpyentropy compensation linear plots correspond to either the numerical simulation with N = N exp (sim), the curve of maximum probability with N =10 5 (cmp) or the experimental homologous series (exp). The value of (T ik ) cmp is calculated as that belonging to the curve with the abscissa r = r exp .
Because of the use of absolute values in eq 15, the only simulations regarded as capable of explaining the laboratory kinetic data will be those leading to isokinetic temperatures higher than or equal to the experimental value [(T ik ) sim ≥ (T ik ) exp ] when the point is placed above the maximum probability curve [(T ik ) exp > (T ik ) cmp ], and to isokinetic temperatures lower than or equal to the experimental value [(T ik ) sim ≤ (T ik ) exp ] when the point is placed below the maximum probability curve [(T ik ) exp < (T ik ) cmp ]. Moreover, in order for the numerical simulations to be considered as valid, it will be required that the corresponding correlation coefficient be higher than or equal to the experimental one (eq 16).  Table S1 (Supporting Information), and those corresponding to the reaction series with a notably high likelihood of presenting a physically meaningful compensation effect (p ≤ 0.000025) in Table 2. In four series the probability could not be obtained because of its extremely low value (p < 10 -6 ). Moreover, the condition of statistical significance (usually accepted as p < 0.05) was fulfilled for 63 homologous series, not being reached that level of significance in the other 37 cases. It can then be concluded that many of the isokinetic plots found in the bibliography (probably, more than 50%) are difficult to be explained just as a consequence of accidental errors, so that a real activation enthalpy-entropy correlation has to be necessarily postulated for those reaction families.
Although parameter p is a function of several variables (N exp , F max , T ik /T m and r ), the only one capable of predicting with some accuracy its value is the correlation coefficient of the experimental isokinetic plot, thus denoting a stronger influence than those of the other three variables. As shown in Figure 12 (top), an increase of the correlation coefficient results in a net decrease of the probability parameter. This means that the higher the linearity of the activation enthalpy-entropy compensation plot the lower the probability that it could be explained by accidental errors.
An attempt has been made to elucidate the combined effect of the four independent factors on the probability parameter. A reasonably linear relationship (Figure 12, bottom) where a = 0.0508, b = 0.0204 and c = 0.0738 quantify the statistical weight of each factor with respect to that of the correlation coefficient (arbitrarily taken as 1). These parameters were obtained by systematically changing their values in order to minimize the sum of the absolute errors of the individual points with respect to the linear relationship considered as the best fit. The fact that a, b, c << 1 confirms the outstanding influence of r in comparison with those of N exp , F max and T ik / T m , although the four of them have a positive effect on the value of 1  p. This indicates that a certain isokinetic plot should be regarded as more physically 22 meaningful (with a lower chance to be an artefact resulting from experimental errors) when the values of these four magnitudes are as high as possible.
Two interesting examples are shown in Figure 13, corresponding to the oxidations of substituted alkenes by permanganate ion in water 41 and by quaternary ammonium permanganate in dichloromethane. 1 In both cases the isokinetic temperature is higher than the mean experimental temperature (T ik / T m > 1) and both can be considered as statistically significant (p = 0.000197 and < 10 -6 , respectively).
The present work can be considered as a reductio ad absurdum argument: starting as an initial hypothesis that the activation enthalpy-entropy parameters for all the homologous series are randomly scattered in Nature, the final conclusion is that (at least for some reaction families) it cannot be true, since a pre-existing correlation must actually be postulated to explain the experimental data.
Several methods have been previously proposed to discriminate error-induced compensation plots from those with a real physical origin. 3,4,23,24,90,91 These methods can be regarded as complementary, often leading to coherent results. However, the one now presented (p-test) might be more intuitive, because it yields a parameter with a direct physical interpretation: the probability of an isokinetic relationship found for a particular homologous reaction series being explainable as a consequence of random experimental errors.

23
The Supporting Information is available free of charge on the ACS Publications website. .
Parameters involved in the application of the p-test to different experimental isokinetic plots extracted from the bibliography (Table S1). BASIC-language computer program for the determination of the probability of a particular enthalpy-entropy compensation plot being caused by accidental errors (p-test).

Notes
The authors declare no competing financial interest.

REFERENCES
(1) Perez-Benito, J. F. Substituent Effects on the Oxidation of Cinnamic Acid by