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Two empirical methods have been developed to estimate the activation energy (Ea) of hydrogen- abstraction reactions involving hydrocarbons by using ...
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Ind. Eng. Chem. Res. 2003, 42, 1151-1161

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Estimating the Activation Energy of Hydrogen-Abstraction Reactions Involving Hydrocarbons by Thermochemical Properties Xiaoliang Ma* and Harold H. Schobert The Energy Institute, The Pennsylvania State University, University Park, Pennsylvania 16802

Two empirical methods have been developed to estimate the activation energy (Ea) of hydrogenabstraction reactions involving hydrocarbons by using ground-state thermochemical properties: reaction enthalpy (∆H), broken bond energy (Db), and formed bond energy (Df). Ea ) 12.67 + 2.98Vc + 0.50∆H + 0.00604∆H2 kcal/mol (method I); Ea ) 54.47 + 4.21Vc - 1.00Df + 0.00581DbDf + 0.00681(Db - Df)2 kcal/mol (method II). The development of the methods is based on a fundamental understanding of the transition state structures (TSSs) and multiple regression analysis of a test set of 71 hydrogen-abstraction reactions involving the abstraction of alkane, allylic, and benzylic hydrogens. There is a significant effect of the π-conjugate TSS on the correlation, which is discussed in terms of quantum chemical understanding. The effect of the π-conjugate TSS is considered by adding an indicator variable Vc into the methods. The average absolute error and standard error are 0.47 and 0.64 kcal/mol for method I and 0.40 and 0.56 kcal/mol for method II for the test set. A comparison of the two proposed methods with previous empirical methods was performed statistically. The results show that the two developed methods significantly improve the estimation accuracy relative to the previous empirical methods. Introduction One of the most important elementary reactions in free-radical processes is hydrogen abstraction, which involves hydrogen atom transfer from a hydrogen donor (B-H) to a radical (A•).

+ B-H f [A‚‚‚H‚‚‚B] f A• H receptor H donor transition state A-H + B• (1) Obtaining the activation energy (Ea) of H-abstraction reactions is essential for fundamental understanding of the hydrogen-transfer mechanism and for kinetic modeling of the processes. However, the experimental determination of the activation energy for H-abstraction reactions is complicated and difficult because most radicals are very unstable and short-lived. Thus, the number of experimentally determined Ea values available in the literature is limited. As a consequence, many methods have been developed to estimate the Ea values of H-abstraction reactions on the basis of empirical, semiempirical, or theoretical studies. Molecular orbital theory based calculations have been used to examine the hydrogen-abstraction reactions and to predict the corresponding Ea. Litwinowicz et al.1 reported the estimation of Ea using ab initio methods at different levels. At the highest level in their work (PMP4/pVTZ//HF/DZP*), the average absolute error (AAE) from experimental Ea values was around 2 kcal/ mol for a test set of six reactions. The largest transition state structure (TSS) in their test set only contains three carbon atoms. Because of the high computational cost, using high-level ab initio methods to calculate a large * To whom correspondence should be addressed. Telephone: 814-863-8744.Fax: 814-863-8892.E-mail: [email protected].

set of TSSs, especially for the TSSs that contain more than eight carbon atoms, is currently still impractical. Many semiempirical molecular orbital methods, including MINDO/3,2 MNDO,3 AM1,4 and PM3,5,6 have been used to estimate Ea for H abstraction. The computational cost of these methods is much lower than the ab initio methods, but the calculation accuracy is unsatisfactory. For example, using the PM3 method to estimate Ea for a test set of 40 reactions involving hydrocarbons, the AAE is 5.7 kcal/mol.7 Therefore, additional scaling is usually necessary. Recently, we developed a semiempirical method,7,8 the PM3-FC method, to estimate Ea. This method combines semiempirical PM3 calculations of transition state enthalpies [∆Hf°(TS)] and family correlations between the PM3-calculated and experimental ∆Hf°(TS) values via regression analysis. The AAE between the PM3-FC-estimated and experimental Ea values is 0.26 kcal/mol for a test set of 40 reactions. Although molecular orbital calculations are readily available currently for estimating Ea of many H-abstraction reactions, chemists are still interested in the estimation of Ea by empirical or semiempirical methods because (1) empirical methods are relatively simple and computationally cheap, (2) with the Ea value for the systems related to those for which the empirical methods were well parametrized, the accuracy of the methods is comparable to, or even better than, that of many semiempirical and theoretical molecular orbital calculations, and (3) empirical approaches provide a better discrimination of the various factors that influence Ea. The Evans-Polanyi-Bell (EPB)9-11 method, an early empirical method, can be expressed by eq 2, in which Ea is a simple linear function of the reaction enthalpy

10.1021/ie020556q CCC: $25.00 © 2003 American Chemical Society Published on Web 02/19/2003

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(∆H). For the hydrogen-abstraction reaction shown by

Ea ) E0 + β∆H

(2)

eq 1, the reaction enthalpy would be given by

∆H ) Db - Df

(3)

where Db and Df are the bond dissociation energies of the broken bond (B-H) and the formed bond (A-H), respectively. Then eq 2 can be rewritten as eq 4, in which Ea is directly proportional to Db and -Df.

Ea ) E0 + βDb - βDf

(4)

In the last 60 years, numerous approaches have been made to improve and extend the EPB method or to establish other empirical or semiempirical relationships between Ea and various ground-state properties of the reactants and products. The early work in this area was reviewed by Berces and Marta.12 Later work was reviewed recently by Marston et al.13 and Denisov.14 According to the predictor variables, the previous relationships can be classified into three groups. In the first group, the response variable Ea is a function of the predictor variable ∆H [Ea ) f(∆H)]; in the second, Ea is a function of two predictor variables, Db and Df [Ea ) f(Db,Df)]; in the third, Ea is a function of predictor variables, which include more variables than ∆H, Db, and Df. In the first group, the EPB method9-11 (eq 2) is a typical method. Semenov15 gave the empirical parameters in eq 2 for exothermic reactions: E0 ) 11 kcal/ mol; β ) 0.25. Marcus proposed a second-order relationship between Ea and ∆H as follows:16,17

Ea ) E0(1 + ∆H/4E0)2

(5)

where E0 ) Ea at ∆H ) 0. Another empirical hyperbolic dependence of Ea on ∆H was reported by Rudakov and Volkova18 (RV method):

Ea(Ea - ∆H) ) E02

(6)

where E0 ) Ea at ∆H ) 0. The activation energy E0 was expressed by E0 ) pApB. pA and pB are empirical parameters of A and B atoms or fragments in eq 1. In the second group, Szabo´ proposed a linear dependence of Ea on Db and Df as follows:19

Ea ) Db + βDf

(7)

where β is an empirical parameter which depends on the transition state and the exo- or endothermic nature of the reactions. The recommended values of β are 0.83 and 0.96 for exo- and endothermic H-abstraction reactions, respectively. From the Morse function, Kagiya et al. derived an empirical nonlinear dependence of Ea on Db and Df:20

Ea )

Db[(1 - 2eβQ)Df + e2βQDb]2 (Df - e2βQDb)2

(8)

where Q is the reaction heat (Q ) -∆H ) Df - Db); β is an empirical constant equal to 0.0190. By approximate treatment of eq 8, Kagiya et al. further proposed a

simpler relationship between Ea and the bond dissociation energies:21

Ea ) Db - 0.0850Df(1 + 0.0944Db)

(9)

The estimated Ea values on the basis of eq 9 were reported to be in agreement with experimental Ea values within (2 kcal/mol. The third group includes the bond-energy-bond-order (BEBO) methods,22 Zavitsas’ method,23-25 the group contribution methods,26-28 the bond-strength-bondlength (BSBL) method,29 the extended EPB method,30,31 and the intersecting-state-model (ISM) method.32 Most of these estimation methods are based on correlations not only with the ∆H and bond dissociation energies but also with other molecular or transition state properties, for example, polarizability, electron affinity, electronegativity, bond order, or equilibrium bond lengths. Berces and Dombi compared BSBL, BEBO(JP),22 and BEBO(G)33 methods for estimating Ea for hydrogen abstraction.34 For the 17 hydrogen-abstraction reactions involving the hydrocarbons in their test set, the AAE from experimental Ea values is 1.0, 2.4, and 2.1 kcal/mol, respectively. Pivovarov and Stepukhovich reported the estimation of Ea using Moin’s method, which is based on the principle of additivity.27 The AAE is 0.53 kcal/ mol for 20 hydrogen-abstraction reactions involving the hydrocarbons in their test set.27 An extended EPB method, as expressed by eq 10, was proposed recently by Roberts and Steel,30,31

Ea ) E0 f + R∆H°(1 - d) + β∆χAB2 + γ(sA + sB) (10) where E0, R, β, and γ are four constants obtained from multiple regression analysis of the experimental data. The term f is defined by f ) D(B-H) D(A-H)/D(H-H)2. The term ∆χAB is the difference in electronegativities of radicals A• and B•. The structural parameters sA and sB are structural factors of the radicals A• and B•, respectively. For hydrocarbon radicals, the s factors are assumed to be the same. d is the delocalization term for the radical B•. Using this method to estimate Ea, the AAE is 0.70 kcal/mol for the nine hydrogen-abstraction reactions involving the hydrocarbons in their test set. This estimation accuracy is significantly higher than that of the previous empirical methods. However, the extended EPB method has a significant error for the reactions with a π-conjugate TSS when ∆H is equal to zero. For example, the Ea value estimated by eq 10 for the benzyl + benzyl-H reaction is 11.5 kcal/mol, whereas the experimental value is 15.8 kcal/mol.35 On the other hand, eq 10 is relatively complicated and contains eight predictor variables and four empirical parameters. The ISM method recently reported by Pais et al. on the basis of the bond order and the equilibrium bond length32 gives an AAE of 1.49 kcal/mol for estimating Ea for the 16 hydrogen-abstraction reactions involving hydrocarbons in their test set. In the present work, we attempted to determine whether there is a simpler empirical dependence of Ea on the ground-state thermochemical properties for H abstraction involving hydrocarbons, which could then be used to estimate Ea with a satisfactory accuracy. The attempts were made by a combination of the fundamental understanding of the TSS of H-abstraction reactions and the correlation between the Ea values and the thermochemical properties of ∆H, Db, and Df via multiple regression analysis for a test set of H-abstraction

Ind. Eng. Chem. Res., Vol. 42, No. 6, 2003 1153

reactions involving hydrocarbons. In the present approach, we did not consider the effect of hydrogen tunneling on Ea because (1) the effect of hydrogen tunneling is of subordinate magnitude in comparison with the error in the experimental data,1,36 except for hydrogen-abstraction reactions at very low temperature and (2) the estimation of Ea is based directly on the experimental data. Regression Method The multiple linear regression analysis on the basis of the least-squares method was used to correlate Ea with thermodynamic properties. The AAE and the standard error between the estimated and experimental Ea values were used as the indices to evaluate the various empirical methods. Because there are many possible correlation equations relating Ea to the groundstate thermodynamic properties, a backward elimination search procedure37,38 was conducted to select the best terms in the models according to the corresponding P value. The P value is a probability and a measure of the evidence against the null hypothesis that the corresponding regression coefficient is equal to zero. The smaller the P value, the more evidence against the hypothesis. The procedure begins with a model containing all potential terms and sequentially eliminates the least significant term from the model until all remaining terms have P values less than the predetermined level. The multiple regression analysis was performed using commercial software, Microsoft Excel in Microsoft Office 98. Sources of Experimental Data The experimental Ea values used in this study are from the available literature (Table 1). The reactions involve the abstraction of alkane, allylic, and benzylic hydrogens. The Ea values from different sources are somewhat inconsistent with each other for some reactions. Thus, we made the widest possible use of the selfconsistent and currently recommended experimental data, taken at a temperature of around 700 K. For the alkyl + alkyl-H reactions, we employed the Ea values recommended in the literature.39,40 For the reactions with a π-conjugated TSS, we used the Ea values adopted by Camaioni et al.35 In most cases, data totally inconsistent with an independent, consistent set of experimentally based values were rejected. When the Ea values from the cited references were different from each other, the arithmetic mean was used. Reassessment of the best experimental Ea values for these reactions is beyond the scope of the present paper. The bond dissociation energy [D(X-H)] values were calculated using the following equation:

D(X-H) ) ∆Hf°(X•) + ∆Hf°(H•) - ∆Hf°(X-H) (11) The enthalpies [∆Hf°(X•)] of radicals for calculation of D(X-H) are from experimental data in the literature, which were collected in one of our previous papers.41 For the enthalpies of some radicals that cannot be found in the literature, we used the PM3-FC method to estimate the values.41 Molecular Orbital Calculations Molecular orbital calculations of radicals and TSSs in this study were performed using a semiempirical

quantum chemistry method, the PM3 method, in CAChe MOPAC, version 94. The PM3 method determines both the optimum geometry and electronic properties of molecules by solving the Schro¨dinger equation using the PM3 semiempirical Hamiltonians developed by Stewart.5,6 The details were reported in our previous papers.8,41 Distribution of the single-occupied molecular orbital (SOMO) involving radicals and TSSs was described by using the electron density (CSOMO2) of the SOMO, where C is the coefficient of linear combination of atomic orbitals, obtained from the output of the PM3 calculation. Results and Discussion 1. Dependence of Ea on the Heat of Reaction. Figure 1 shows the dependence of Ea on ∆H. The Ea value increases with increasing ∆H, but the relationship is not linear in the range of ∆H from -25 to +25 kcal/ mol. If the EPB method (eq 2) is employed for the test set listed in Table 1, eq 12 is obtained, with an R2 value

Ea ) 0.500∆H + 15.04

(12)

of 0.855 and a standard error of 2.31 kcal/mol. Obviously, the fit is not good. After carefully examining the relationship between TSSs of the 71 H-abstraction reactions and their Ea-∆Η correlation, we found that the reactions can be divided into two families. The first family includes the reactions (1)-(32) listed in Table 1, where the corresponding TSSs do not have a π-conjugate character (in other words, either A• or B• radicals in eq 1 is stabilized by π-conjugate delocalization). The estimated values of this family in terms of eq 12 exhibit a negative deviation from the experimental values. The second family includes reactions (33)-(71) listed in Table 1. TSSs in this family are stabilized by π-conjugate delocalization. All reactions in the second family, except reactions (53) and (61), exhibit a positive deviation. This systematic deviation allows us to improve the accuracy of the EPB method by adding an indicator variable Vc into eq 2. Vc is a 1-0 variable, which is used to describe the character of the TSS. For the reactions with a π-conjugate TSS, the Vc value is taken to be 1, whereas for the reactions without a π-conjugate TSS, the Vc value is equal to zero. Another factor needing to be considered in the model is that the correlation between Ea and ∆H is curvilinear, especially for the reactions with π-conjugate TSS, as shown clearly in Figure 1. Consequently, it is necessary to build a nonlinear dependence of Ea on ∆H and Vc. To consider the probable forms of the variables, we first assigned a multiple regression model (eq 13a) to mimic the curvilinear relationship between Ea and ∆H. This model is a third-order polynomial function of the variables ∆H and Vc with the interacting terms of Vc∆H and Vc∆H2. The model can also be considered as a linear combination of six terms of the variables Vc, ∆H, ∆H2, ∆H3, Vc∆H, and Vc∆H2. β0, β1, β2, β22, β222, β12, and β122 are empirical

Ea ) β0 + β1Vc + β2∆H + β22∆H2 + β222∆H3 + β12Vc∆H + β122Vc∆H2 (13a) constants. To reduce the number of terms in eq 13a, the backward elimination search procedure was performed to eliminate stepwise the least significant terms. The

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Table 1. Experimental and Estimated Ea Values for H-Abstraction Reactions (kcal/mol) method Ii

reactants no.

radical A•

H donora B-H

Eab expt

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

C• C• C• C• C• C• C• C• C-C• C-C• C-C• C-C• C-C• C-C• C-C• C-C-C• C-C-C• C-C-C• C-C-C• C-C-C• C-C-C-C• C-C-C-C• C-C-C-C• C-C-C-C-C• C-C•-C C-C•-C C-C•-C C-C-C•-C C-C-C•-C C-C-C•-C CeC•(t-Bu) C3C•(t-Bu)

C C-C C-C-C C-C-C-C C-C-C-C-C C-C-C C-C-C-C C3C(t-Bu) C C-C C-C-C C-C-C-C C-C-C C-C-C-C C3C(t-Bu) C C-C C-C-C-C C-C-C C-C-C-C C C-C C-C-C C C C-C C-C-C C C-C C-C-C C C-C

14.6 11.6 11.5 11.4 10.9 10.0 10.0 8.3 15.4 13.4 12.3 12.3 11.4 10.4 10.0 15.4 12.3 12.3 10.9 10.4 16 13.1 13 15.7 15.9 13.0 12.9 16.1 12.7 12.6 17.5 15.4

33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

C• C• C• C• C• C-C• C-C• C-C• C-C-C• C-C-C• C-C•-C ph-C• ph-C• ph-C• ph-C• ph-C• ph-C• ph-C• ph-C• ph-C• ph-C• ph-C• ph-C•-C ph-C•(C)2 ph-C•-ph (ph)3-C• tetralin radical DHPhe radical fluorene radical DHAnt radical CdC-C• CdC-C• CdC-C• CdC-C• CdC-C• CdC-C•-C CdC-C•-C CdC-C•-C CdC-C•(C)2

CdC-C CdC-C-C CdC-C(C)-C ph-C ph-C(C)2 CdC-C CdC-C-C ph-C CdC-C CdC-C-C CdC-C C C-C CdC-C ph-C ph-C-C tetralin ph-C-ph (ph)3-C DHPhek fluorene DHAntl ph-C C ph-C ph-C ph-C ph-C ph-C ph-C C C-C C-C-C C-C-C ph-C C C-C C-C-C C

8.8 7.3 7.3 9.5 7.8 9.8 8.3 10.0 9.8 8.3 9.7 25.9 22.6 15.5 15.8 14.5 13.2 12.8 11.0 11.9 10.6 9.6 17.9 29.2 18.6 19.9 20.0 22.6 18.4 24.3 26.9 24.1 24.0 21.9 17.2 29.6 26.8 26.7 31.3

method IIj

Df

Vc

Ea estd

error

Ea estd

error

Without π-Conjugate TSS c 0.0 104.9 c, d -3.8 101.1 d -3.9 101.0 c, d -4.6 100.3 e -4.8 100.1 c-e -5.9 99.0 c, d -6.1 98.8 c, e -9.2 95.7 h 3.8 104.9 c, f 0.0 101.1 d -0.1 101.0 d -0.8 100.3 c, d -2.1 99.0 d -2.3 98.8 c -5.4 95.7 h 3.9 104.9 d 0.1 101.1 d -0.7 100.3 d, e -0.2 99.0 d -2.2 98.8 h 4.6 104.9 h 0.8 101.1 h 0.7 101.0 h 4.8 104.9 h 5.9 104.9 d, e 2.1 101.1 d, e 2.0 101.0 h 6.1 104.9 h 2.3 101.1 h 2.2 101.0 h 9.2 104.9 h 5.4 101.1

104.9 104.9 104.9 104.9 104.9 104.9 104.9 104.9 101.1 101.1 101.1 101.1 101.1 101.1 101.1 101.0 101.0 101.0 101.0 101.0 100.3 100.3 100.3 100.1 99.0 99.0 99.0 98.8 98.8 98.8 95.7 95.7

0 0 0 0 0 0 0 0 0 0 0 0. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12.7 10.8 10.8 10.5 10.4 9.9 9.8 8.6 14.6 12.7 12.6 12.3 11.6 11.5 10.1 14.7 12.7 12.3 11.7 11.6 15.1 13.1 13.0 15.2 15.8 13.7 13.7 15.9 13.8 13.8 17.8 15.5

-1.9 -0.8 -0.7 -0.9 -0.5 -0.1 -0.2 0.3 -0.8 -0.7 0.3 0.0 0.2 1.1 0.1 -0.7 0.4 0.0 0.8 1.2 -0.9 0.0 0.0 -0.5 -0.1 0.7 0.8 -0.2 1.1 1.2 0.3 0.1

13.4 11.2 11.1 10.7 10.6 10.0 9.9 8.4 15.0 12.7 12.6 12.2 11.4 11.3 9.7 15.0 12.7 12.2 11.5 11.4 15.3 13.0 12.9 15.4 15.9 13.6 13.5 16.0 13.6 13.6 17.6 15.1

-1.2 -0.4 -0.4 -0.7 -0.3 0.0 -0.1 0.1 -0.4 -0.7 0.3 -0.1 0.0 0.9 -0.3 -0.4 0.4 -0.1 0.6 1.0 -0.7 -0.1 -0.1 -0.3 0.0 0.6 0.6 -0.1 0.9 1.0 0.1 -0.3

With π-Conjugate TSS d, e -18.1 86.8 d, e -22.3 82.6 c, d -24.0 80.9 c, e -16.4 88.5 c -21.4 83.5 d -14.3 86.8 d -18.5 82.6 e, g -12.6 88.5 d -14.2 86.8 d -18.4 82.6 d -12.2 86.8 h 16.4 104.9 h 12.6 101.1 h -1.7 86.8 c 0.0 88.5 c -3.4 85.1 c -6.8 81.7 c -5.8 82.7 c -8.9 79.6 c -10.7 77.8 c -7.8 80.7 c -14.7 73.8 h 3.4 88.5 h 21.4 104.9 h 5.8 88.5 h 8.9 88.5 h 6.8 88.5 h 10.7 88.5 h 7.8 88.5 h 14.7 88.5 h 18.1 104.9 h 14.3 101.1 h 14.2 101.0 h 12.2 99.0 h 1.7 88.5 h 22.3 104.9 h 18.5 101.1 h 18.4 101.0 h 24.0 104.9

104.9 104.9 104.9 104.9 104.9 101.1 101.1 101.1 101.0 101.0 99.0 88.5 88.5 88.5 88.5 88.5 88.5 88.5 88.5 88.5 88.5 88.5 85.1 83.5 82.7 79.6 81.7 77.8 80.7 73.8 86.8 86.8 86.8 86.8 86.8 82.6 82.6 82.6 80.9

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

8.6 7.5 7.1 9.1 7.7 9.7 8.5 10.3 9.8 8.5 10.5 25.5 22.9 14.8 15.7 14.0 12.5 13.0 11.7 11.0 12.1 9.6 17.4 29.1 18.8 20.6 19.3 21.7 19.9 24.3 26.7 24.0 24.0 22.7 16.5 29.8 27.0 26.9 31.1

-0.2 0.2 -0.2 -0.4 -0.1 -0.1 0.2 0.3 0.0 0.2 0.8 -0.4 0.3 -0.7 -0.1 -0.5 -0.7 0.2 0.7 -0.9 1.5 0.0 -0.5 -0.1 0.2 0.7 -0.7 -0.9 1.5 0.0 -0.2 -0.1 0.0 0.8 -0.7 0.2 0.2 0.2 -0.2

8.8 7.4 6.9 9.4 7.7 9.9 8.3 10.5 9.9 8.4 10.5 25.9 23.2 14.7 15.6 13.9 12.4 12.8 11.6 10.9 12.0 9.5 17.3 29.1 18.7 20.5 19.2 21.6 19.8 24.2 26.9 24.2 24.1 22.7 16.4 29.7 26.8 26.8 30.9

0.0 0.1 -0.4 -0.1 -0.1 0.1 0.0 0.5 0.1 0.1 0.8 0.0 0.6 -0.8 -0.2 -0.6 -0.8 0.0 0.6 -1.0 1.4 -0.1 -0.6 -0.1 0.1 0.6 -0.8 -1.0 1.4 -0.1 0.0 0.1 0.1 0.8 -0.8 0.1 0.0 0.1 -0.4

ref

∆H

Db

For the Whole Test Set AAE standard error

0.47 0.64

0.40 0.56

a The italicized carbon is the carbon from which a hydrogen atom is abstracted. b Experimental E values are from the references a identified. When the values from different references are different, an arithmetic mean is used. c Reference 35. d Reference 39. e Reference f g h i 40. Reference 46. Reference 49. Derived from the Ea value of the corresponding reverse reactions. Ea ) (β0 + β1Vc) + β2∆H + β22∆H2. j E ) (β + β V ) + β D + β D D + β (D - D )2. k 9,10-Dihydrophenanthrene. l 9,10-Dihydroanthracene. a 0 1 c 4 f 34 b f 22 b f

Ind. Eng. Chem. Res., Vol. 42, No. 6, 2003 1155 Table 2. Backward Elimination of Ea on Six Predictors step 1 2 3 4

a

constant β0 β value P value β value P value β value P value β value P value SE of βa

12.63 3.7 × 10-67 12.63 3.5 × 10-68 12.63 3.3 × 10-69 12.67 1.4 × 10-77 0.11

Vcβ1 3.02 3.8 × 10-20 3.02 2.0 × 10-20 3.02 1.0 × 10-20 2.98 1.2 × 10-23 0.19

∆H2β22

∆Hβ2 0.498 1.6 × 10-26 0.498 5.8 × 10-27 0.500 1.8 × 10-64 0.500 2.2 × 10-65 0.0069

0.00818 0.140 0.00818 0.137 0.00818 0.134 0.00604 1.1 × 10-14 0.00061

∆H3β222 2.1 × 0.996

10-7

Vc∆Hβ12

Vc∆H2β122

R2

SE of Eab

0.0018 0.955 0.0019 0.947

-0.00127 0.695 -0.00217 0.693 -0.00271 0.6907

0.9891

0.659

0.9891

0.654

0.9891

0.649

0.9891

0.645

Standard error of the estimated β. b Standard error of the estimated Ea.

Figure 1. Dependence of Ea on ∆H.

Figure 2. Correlation of Ea with ∆H by eq 13c.

backward elimination steps and results are listed in Table 2. The P value for terms ∆H, ∆H2, Vc, and the constant term in step 4 is much less than 0.01, indicating strongly that these three variables and the constant term have significant influence on the Ea value and have to be included in the estimation model. Thus, the model can be written as follows:

Ea ) (β0 + β1Vc) + β2∆H + β22∆H2

for the backward reaction of eq 1 can be expressed by

Ea′ ) (β0 + β1Vc) - β2∆H + β22∆H2

(15)

where ∆H is the reaction enthalpy for the forward reaction (eq 1). Thus, from eqs 13b and 15

Ea′ ) Ea - 2β2∆H

(16)

(13b)

or

Ea ) 12.67 + 2.98Vc + 0.500∆H + 0.00604∆H2 kcal/mol (13c) The standard errors of the corresponding coefficients are listed in Table 2. The regression analysis shows that the effect of the π-conjugate TSS (Vc) on the response variable Ea is significant and additive, and the effects of ∆H3, Vc∆H, and Vc∆H2 are insignificant and negligible. The estimated Ea values from eq 13c are listed in Table 1, and the corresponding curves in comparison with the experimental data are shown in Figure 2, giving an R2 value of 0.9891 and a standard error of 0.64 kcal/mol. The results indicate that the fit of eq 13c is much better than that of the EPB method. It is worth pointing out that eq 13b is suitable for both exothermic (∆H < 0) and endothermic (∆H > 0) reactions, although the activation energy in the reverse reaction can be calculated directly using

Ea′(endothermic) ) Ea(exothermic) - ∆H (14) In fact, according to eq 13b, the activation energy (Ea′)

According to the regression analysis, the β2 value is 0.500, implying that the constructed model (eq 13b) with its parameters accords with the demand of eq 14. In fact, the parameter β2 (in eq 13b) equal to 1/2 is the requirement of eq 14. Moreover, when ∆H ) 0, eq 13b becomes

Ea ) β0 + β1Vc

(13d)

This indicates that β0 is the activation energy (E0) if the reaction without the π-conjugate TSS were to occur with zero reaction enthalpy. β1 is a contribution of the π-conjugate TSS to Ea, which can be denoted by Eπ. Thus, eq 13b can also be rewritten as

Ea ) E0 + Eπ + ∆H/2 + β22∆H2

(13e)

where β22 is a coefficient of the ∆H 2 term, indicating the effect of the ∆H 2 term on Ea. 2. Dependence of Ea on Db. The activation energy for H-abstraction reactions is dependent on the structures of both the hydrogen receptor (A•) and the hydrogen donor (B-H). For the reactions of the same radical with different hydrogen donors, the Ea value is dependent on the Db(B-H) of the donor. Thus, Ea becomes a function of Db. Figure 3 shows the dependence of Ea on Db. For identical radicals reacting with different donors, Ea increases as Db increases. However, such a correla-

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conjugate radicals + methane, conjugate radicals + ethane, and radicals + benzyl-H are similar to each other and that the position of the correlation lines is dependent on the nature of the hydrogen donors. The rates of change in Ea for the two families radicals + methane and radicals + ethane are also similar to each other. 4. Correlation of Ea with Db, Df, ∆H, and Vc. As mentioned above, the four variables Db, Df, ∆H, and Vc have significant influences on Ea. To determine the best model for estimating Ea on the basis of these four variables (in fact, there are only three independent variables due to eq 3), a multiple regression model (eq 17a), which contains 15 potential terms that are combinations of the four variables, was assigned:

Ea ) β0 + β1Vc + β2∆H + β3Db + β4Df + β22∆H2 + β33Db2 + β44Df 2 + β34DbDf + β34′Db/Df + β4′/Df + β12Vc∆H + β13VcDb + β14VcDf + β122Vc∆H 2 + β222∆H 3 (17a) Figure 3. Dependence of Ea on Db. The lines are the correlations of Ea with Db by eq 17e.

When eq 17a is treated by the backward elimination search procedure, 11 terms in the equation were sequentially eliminated. The remaining terms are Df, DbDf, ∆H 2, and Vc. As listed in Table 3, the P value for the Df, DbDf, ∆H 2, and Vc terms and the constant term is much less than 0.01, indicating strongly that these terms have a significant influence on the Ea value and have to be included in the model for efficiently estimating Ea. Therefore, the best model is given as

Ea ) (β0 + β1Vc) + β4Df + β34DbDf +β22∆H 2

(17b)

or

Ea ) 54.47 + 4.21Vc - 1.00Df + 0.00581DbDf + 0.00681(Db - Df)2 kcal/mol (17c)

Figure 4. Dependence of Ea on Df. The lines are the correlations of Ea with Df by eq 17f.

tion is discontinuous when the TSS corresponding to the reaction changes from a π-conjugate system to a nonπ-conjugate system, as shown in Figure 3. For example, methyl + alkyl-H reactions and methyl + alkenyl-HR/ aryalkyl-HR reactions display significantly different relationships. This result suggests that the contribution of Db to Ea is quite different in the two cases. 3. Dependence of Ea on Df. In the case of the reactions of different radicals with an identical hydrogen donor, the Ea value is dependent on the reactivity of the radicals (A•), which is related to the structure of the radical. The reactivity of the radical A• can be described by the corresponding Df(A-H) value.42 The effect of Df on Ea is shown in Figure 4. For reactions with and without the π-conjugate TSS, the Ea value decreases with increasing Df value. Similar to the dependence of Ea on Db, the linear correlation between Ea and Df is discontinuous when the TSS changes from a π-conjugate system to a non-π-conjugate system. It is found that the rates of change in Ea with Df for the three families

The standard errors of corresponding coefficients in eq 17b are listed in Table 3. The standard error of the Ea values estimated by eq 17c is 0.56 kcal/mol with an R2 value of 0.9919. This model indicates that Ea can be described by a linear combination of the Vc, Df, DbDf, and ∆H 2 terms. Because ∆H ) Db - Df, eq 17b can also be expressed by

Ea ) (β0 + β1Vc) + β4Df + β34DbDf +β22(Db - Df)2 (17d) This equation implies that Ea is a second-order function of Db and Df. The regression results also show that the effect of the π conjugation on Ea is additive and can be expressed by the β1Vc term. The effect of Db and Df on Ea is interactive, not simply additive. For the reactions of an identical radical with a different hydrogen donor, eq 17d can be rewritten as

Ea ) (β0 + β1Vc + β4Df + β22Df2) + [(β34 - 2β22)Df]Db + β22Db2 ) R0 + R1Db + β22Db2

(17e)

where R0 ()β0 + β1Vc + β4Df + β22Df2), R1 [)(β34 - 2β22)Df], and β22 are constant. Ea is a second-order function of Db. As ∂Ea/∂Db ) (β34 - 2β22)Df + 2β22Db > 0 in the range of Db and Df values from 60 to 120 kcal/mol, Ea increases with increasing Db and the change in Ea with

Ind. Eng. Chem. Res., Vol. 42, No. 6, 2003 1157 Table 3. Results from Backward Elimination of Ea on 15 Predictors step 12

a

β value P value SE of βa

constant β0

Vcβ1

Dfβ4

∆H2β22

DbDfβ34

R2

SE of Eab

54.47 8.89 × 10-48 1.37

4.21 4.9 × 10-17 0.37

-1.001 1.4 × 10-68 0.012

0.00681 3.7 × 10-16 0.0063

0.00581 6.3 × 10-49 0.00014

0.9919

0.561

Standard error of the estimated β. b Standard error of the estimated Ea.

a unit increase in Db is directly proportional to Db. Figure 3 shows that the dependence of Ea on Db according to eq 17e is in good agreement with the experimental data, except reactions (1), (53), and (61) with an absolute error greater than 1.0 kcal/mol. For the reactions of different radicals with an identical hydrogen donor, eq 17d can be rewritten as

Ea ) (β0 + β1Vc + β22Db2) + [β4 + (β34 - 2β22)Db]Df + β22Df2 ) γ0 + γ1Df + β22Df2

(17f)

where γ0 ()β0+ β1Vc + β22Db2), γ1 [)β4 + (β34 - 2β22)Db], and β22 are constant. Ea is a second-order function of Df. As ∂Ea/∂Df ) β4 + (β34 - 2β22)Db + 2β22Df < 0 in the range of Db and Df values, Ea decreases with increasing Df and the change in Ea with a unit increase in Df is a linear function of Db and Df. Figure 4 shows that the dependence of Ea on Df according to eq 17f is in good agreement with the experimental data. Similarly to eq 13b, eq 17b is also suitable for both exothermic (∆H < 0) and endothermic (∆H > 0) reactions. According to eq 17b, the activation energy (Ea′) for the backward reaction of eq 1 can be expressed by

Ea′ ) β0 + β1Vc + β4Db + β34DfDb + β22∆H 2

(18)

where Df, Db, and ∆H are the bond dissociation energies and the reaction enthalpy for the forward reaction (eq 1). Thus, from eqs 3, 17b, and 18, the following equation can be obtained.

Ea′ ) Ea - β4∆H

(19)

This indicates that the model (17b) is consistent with eq 14 because the β4 value is 1.00 according to the regression analysis. When ∆H ) 0 (Df ) Db), eq 17b becomes

Ea ) β0 + β1Vc + β4Df + β34Df2

(17g)

Ea ) β0 + β1Vc + β4Db + β34Db2

(17h)

or

This indicates that the parameter β0 is the activation energy if the reaction without the π-conjugate TSS were to occur with Df ) 0 (or Db ) 0). The parameter β1 indicates a contribution from the π-conjugate TSS to Ea. 5. Effect of the π Conjugation on Correlation. Regardless of whether Ea is correlated with ∆H or with the bond dissociation energies (Db and Df), there is a significant systematic error between the reactions with a π-conjugate TSS and the reactions without a π-conjugate TSS (see Figures 2-4). The ∆H, Db, and Df are concerned only with the ground-state thermochemical properties of the reactants and products, whereas according to the transition state theory,26,28,43-46 the Ea

value is concerned with the difference between the enthalpy of the transition state [∆Hf°(TS)] and the enthalpy of the reactants, which can be expressed by eq 20. If one assumes an identical hydrogen donor (B-

Ea ) ∆Hf°(TS) - [∆Hf°(A•) + ∆Hf°(B-H)] + 2RT (20) H), then according to eq 20 the change in Ea values is dependent on the change in ∆Hf°(TS) - ∆Hf°(A•) value. The regression analysis of Ea ) f(∆H) in the present study shows that the reactions with the π-conjugate TSS give higher Ea values than the reactions without the π-conjugate TSS by 2.98 kcal/mol (β1 ) 2.98 kcal/mol). This means that the difference between ∆Hf°(TS) and ∆Hf°(A•) for the reactions with the π-conjugate TSS is greater by 2.98 kcal/mol than that of the reactions without the π-conjugate TSS. In other words, the stabilization of the A• radical by an A substituent due to the π-conjugate delocalization appears to be stronger than the stabilization of the transition state by the A substituent. Because the ground-state thermochemical properties of reactants and products cannot reflect such a kinetic character, it is necessary to add the TSS indicator variable (Vc) to the estimating models. The different stabilizing effects of the π conjugation on the radical and TSS were also found by other researchers,30,47,48 but why the stabilization of TSS by the A substituent via the π conjugation is weaker than the stabilization of the radical by the same substituent is unclear in the previous studies. The SOMO electron densities (CSOMO2) of three radicalssethyl, allyl, and benzylsand three representative TSSs (C-C*‚‚‚H‚‚‚C*C, CdC-C*‚‚‚H‚‚‚C*-C, ph-C*‚‚‚H‚‚‚C*-C) are listed in Table 4 (C* are the C atoms that contact the abstracted H atom). The corresponding SOMOs are shown in Figure 5. For the ethyl radical (without π conjugation), the SOMO is predominantly located on the C* atom with a CSOMO2 value of 0.908. For the allyl and benzyl radicals, only about 50% of the SOMO is located on the C* atom (CSOMO2 ≈ 0.50) because of delocalization of the unpaired electron via π conjugation. For C-C*‚ ‚‚H‚‚‚C*-C, the SOMO is predominantly located on the two C* atoms with a CSOMO2 value of 0.904, similar to that for an ethyl radical. For CdC-C*‚‚‚H‚‚‚C*-C and ph-C*‚‚‚H‚‚‚C*-C, about 80% of the SOMO is located on the two C* atoms. Parts e and f of Figure 5 show clearly that there is significant delocalization of the unpaired electron by π conjugation. However, such delocalization is less than that in the allyl and benzyl radicals, as shown in Figure 5b,c. The SOMO analysis indicates that the greater stabilization by the A substituent via the π conjugation in radical A• relative to that in the corresponding TSS is due to the greater delocalization of the unpaired electron in the radical A• than in the TSS. 6. Comparison of Different Empirical Methods. On the basis of the present test set, the AAE and standard error of the Ea values estimated by various empirical methods are listed in Table 5. These methods

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Table 4. Electron Density of SOMO (CSOMO2) radical

transition state

no.a

C-C*•

CdC-C*•

ph-C*•

C-C*-H-C*-C

CdC-C*-H-C*-C

ph-C*-H-C*-C

C1 C2 C3 C4 C5 C6 C7 C8 C9 C*/∑C* ∑C

0.9075 0.0064

0.5000 0.0000 0.5000

0.4920 0.0000 0.1639 0.0000 0.1799 0.0000 0.1641

0.4529 0.0075 0.4510 0.0076

0.4050 0.0043 0.1162 0.4192 0.0071

0.9075 0.0064

0.5000 0.5000

0.4829 0.5080

0.9040 0.0151

0.8242 0.1275

0.4123 0.0043 0.0511 0.0001 0.0471 0.0001 0.0510 0.3825 0.0067 0.7948 0.1603

a

The carbon atoms are labeled as shown in Figure 5.

6) can be expressed by

Ea ) ∆H/2 + (∆H2 + 4E02)0.5/2

Figure 5. SOMO of radicals and TSSs.

include the EPB method (eq 2), Marcus’ method (eq 5), the RV method (eq 6), Szabo´’s method (eq 7), the Kagiya et al. methods (eqs 8 and 9), and our proposed methods (eqs 13b and 17c). A comparison of the experimental and estimated Ea values is also shown in Figure 6 for ease of reference. In the Ea ) f(∆H) methods, the EPB method gives AAE and standard error of 2.02 and 2.31 kcal/mol, respectively, with an R2 value of 0.855. As found from Figures 1 and 6a, there are two main sources of error in the EPB method: (1) The EPB method uses a simple linear function to mimic approximately the nonlinear relationship between Ea and ∆H. (2) The EPB method does not consider the systematic error caused by the π conjugation. Marcus’ method (eq 5) can also be expressed by

Ea ) E0 + ∆H/2 + ∆H2/16E0

(21)

where Ea is a second-order function of ∆H. The E0 value determined by the linear regression on the basis of the present test set is 15.10 kcal/mol. Using eq 21 to reproduce the Ea value, the AAE and standard error are 1.65 and 1.89 kcal/mol, respectively. The RV method (eq

(22)

The E0 value determined by linear regression on the basis of the present test set is 14.09 kcal/mol, with AAE and standard error of 1.30 and 1.46 kcal/mol, respectively. Because both Marcus’ method and the RV method treat the relationship of Ea and ∆H by a nonlinear function, both give better estimation accuracy than the EPB method. However, these two methods also do not consider the effect caused by the π-conjugate TSS. As shown in Figure 6b,c, there is a significant systematic deviation between the reactions with and without the π-conjugate TSS. This is the major source of the error in Marcus’ method and in the RV method. If we improve Marcus’ method by considering the effect of the π-conjugate, the AAE and standard error are 0.47 and 0.63 kcal/mol, respectively, for the test set. By comparison with eqs 21 and 22, the regression coefficient of the ∆H term in our method I (eq 13b) is the same as that in Marcus’ method and the RV method. The major difference is that we consider the effect of the π-conjugate TSS by adding the Vc variable to the estimation equation. Thus, as shown in Figure 6e, our method I gives a much better fit than the previous Ea ) f(∆H) methods. In the Ea ) f(Db,Df) methods, Szabo´’s method gives AAE and standard error of 3.32 and 4.28 kcal/mol, respectively. The fit is poor because an additive and linear function is used to treat the Ea ) f(Db,Df) function. Method I of Kagiya et al., with AAE and standard error of 1.21 and 1.46 kcal/mol, gives a better fit than their method II, with AAE and standard error of 2.52 and 3.02 kcal/mol, respectively. However, as shown in Figure 6d, a considerable scatter exists in method I of Kagiya et al., although no significant systematic error is evident between the reactions with the π-conjugate TSS and reactions without the π-conjugate TSS. Figure 6f shows that our proposed method II (eq 17c) gives the best accuracy among the Ea ) f(Db,Df) methods, with AAE and standard error of 0.40 and 0.56 kcal/mol, respectively. The expression of our method II is also simpler than method I of Kagiya et al. By comparison with other empirical methods, including BEBO, BSBL, the extended EPB, and ISM methods, our two methods show less error for estimating Ea of the hydrogen-abstraction reactions involving hydrocarbons. In comparison between our two methods, Ea in method I is a function of two independent variables, Vc and ∆H, while Ea in method II is a function of three variables, Vc, Df, and Db. The two methods give a similar standard error, as shown in Table 5, indicating

Ind. Eng. Chem. Res., Vol. 42, No. 6, 2003 1159 Table 5. Comparison of Different Empirical Methods on the Basis of the Test Set

method

EPB methoda (eq 2)

Marcus’ methoda (eq 5)

RV methoda (eq 6)

Szabo´’s methoda (eq 7)

R2 standard error, kcal/mol average absolute deviation, kcal/mol

0.855 2.31 2.02

1.89 1.65

1.46 1.30

4.28 3.32

a

Kagiya et al.’s method I method II (eq 8) (eq 9) 1.46 1.21

3.02 2.53

our methods method Ia method IIa (eq 13b) (eq 17b) 0.989 0.64 0.47

0.991 0.56 0.40

The empirical parameters in the expression are obtained by the linear regression on the basis of the present test set.

Figure 6. Plot of the experimental Ea vs the estimated Ea by different methods.

that Ea is dominantly dependent on variable ∆H, or Db - Df. Method II has one more adjustable parameter and, thus, has a lower error. On the other hand, because method II gives a formulated relationship between the

reactivation energy and bond dissociation energy, it is useful to examine quantitatively how the change in Db or Df results in the change in Ea, as shown in Figures 3 and 4.

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We should mention that the AAE or standard error used in the present study for the whole test set indicates an error of the estimated values from the experimental values but does not indicate an error of the estimated values from the true values. Similarly to all empirical and semiempirical methods, the success of our methods is also dependent on the accuracy of the experimental data used. The experimental data with higher accuracy will give more accurate regression parameters and then result in the estimated values with higher accuracy. In a sense, precision in model calculations (like ours) that is better than the precision in the experimental data has no particular meaning. However, as a minimum, this result shows that one can use our model to estimate Ea values that are at least of the same order of precision as the experimental results. Thus, others can use our methods to provide some guidance for experimenters; if the experimentally determined Ea is substantially outside that estimated from our methods, which derive from the largest known test set of reactions, then this may be a signal that there is a problem with the experimental result or that there is some unusual effect with this reaction. We also need to mention that, because the methods were built on the experimental data taken at a temperature of around 700 K, the Ea values estimated by the methods are also for a temperature of around 700 K. Finally, it is worth pointing out that our two methods not only have a relatively higher accuracy for estimating the activation energies of the hydrogen-abstraction reactions involving hydrocarbons but also are convenient and simple. Summary In the present study we proposed two empirical methods (eqs 13c and 17d) for estimating the activation energy of the hydrogen abstraction involving hydrocarbons on the basis of a fundamental understanding of the TSS of hydrogen- abstraction reactions and multiple regression analysis. In building the empirical models, we added a TSS indicator variable Vc to the expressions and selected the potential forms of the predictor variables by the backward elimination search procedure. The AAE and standard error for estimating Ea are 0.47 and 0.64 kcal/mol by method I and 0.40 and 0.56 kcal/ mol by method II on the basis of the 71 hydrogenabstraction reactions. The error analysis shows that these two methods improve significantly the estimation accuracy of the empirical methods. The error for the two methods is less than those in the previous methods. In addition, the physical insight of the empirical parameters in the models was discussed and why the reactions with and without the π-conjugate TSS show different correlations was explained in terms of the semiempirical quantum chemical calculations. Acknowledgment This work was supported by the Air Force Office of Scientific Research. Funding was provided by AFOSR under Contract F49620-99-1-0290. Literature Cited (1) Litwinowicz, J. A.; Ewing, D. W.; Jurisevic, S.; Manka, M. J. Transition-state Structures and Properties of Hydrogen-transfer Reactions of HydrocarbonssAb Initio Benchmark Calculations. J. Phys. Chem. 1995, 99, 9709-9716.

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Received for review July 29, 2002 Revised manuscript received December 11, 2002 Accepted December 21, 2002 IE020556Q