Group Additivity Determination for Enthalpies of Formation of

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Group Additivity Determination for Enthalpies of Formation of Carbenium Ions Kathryn Bjorkman, Chunyi Sung, Eric Mondor, Janine Chungyin Cheng, Deng-Yang Jan, and Linda J. Broadbelt Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie503348z • Publication Date (Web): 14 Nov 2014 Downloaded from http://pubs.acs.org on November 16, 2014

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Industrial & Engineering Chemistry Research

Group Additivity Determination for Enthalpies of Formation of Carbenium Ions Kathryn R. Bjorkman*,‡, Chun-Yi Sung, Eric Mondor, Janine C. Cheng, Deng-Yang Y. Jan‡ and Linda J. Broadbelt∗ *

Department of Chemical and Biological Engineering, Northwestern University, Evanston

Illinois 60208 ‡UOP LLC, A Honeywell Company, Des Plaines, Illinois 60017 Modeling of acid-catalyzed hydrocarbon conversion processes at the mechanistic level requires rate coefficients for a large number of reactions. The computational demand of finding activation energy barriers for each reaction is substantially reduced by employing structurereactivity correlations such as the Evans-Polanyi relationship that correlates activation energy with the enthalpy of reaction. However, there are many species for which enthalpies of formation are unknown. Therefore, group additivity methods to specify enthalpies of formation for each species involved in the reaction network are valuable. Quantum mechanical (QM) calculations and isodesmic reactions were used to calculate enthalpies of formation for a number of acyclic and cyclic carbenium ions, including allylic carbenium ions. These values compare

∗ Corresponding author e-mail: [email protected].

ACS Paragon Plus Environment

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favorably with experimental values, establishing Gaussian-4 as an accurate QM method for these calculations. Using these values, Benson-type group additivity values for enthalpies of formation were then derived through multiple linear regression. Enthalpies of formation values calculated from the group additivity scheme capture experimental and QM enthalpies of formation well and enhance the range of species that can be described by the group additivity approach. Introduction Alkylation chemistry is an important process in the petroleum industry that converts light hydrocarbons such as isobutane and butenes into alkylates, a high octane, low density fuel components with few aromatics, olefins, or sulfur compounds, and is a prototypical example of an acid-catalyzed hydrocarbon conversion process.1-3

Alkylate may be blended with other

gasoline components to produce fuels with an acceptable octane number according to increasingly stringent environmental regulations.4 While solid acid catalysts have the potential to avoid the toxicity and separation difficulties of liquid acids commonly used for alkylation chemistry (i.e. sulfuric or hydrofluoric acids), they are prone to rapid deactivation.1,3,4 Modeling of alkylation chemistry over solid acid catalysts has the potential to develop a better understanding of potential deactivation mechanisms and the overall process.2 An approach to developing detailed mechanistic models of alkylation chemistry is to use automated network generation software,5-7 but this approach produces large reaction networks that make it challenging to determine the rate coefficients necessary for each elementary step. Rate coefficients are often expressed in Arrhenius form (Equation 1):

k = Ae−E a / RT

(1)

where k is the reaction rate coefficient, A is the pre-exponential factor, Ea is the activation energy barrier, R is the ideal gas constant, and T is the temperature. For very complex reaction networks


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like those produced to model alkylation processes, it would be too computationally expensive to calculate rate coefficients directly using quantum mechanics and transition state theory for every step. Linear free energy relationships (LFER) may be used to relate a reaction rate constant to thermodynamic properties. The Evans-Polanyi relationship is one commonly used type of LFER, shown in Equation 2:8 (2) where Ea,i is the activation energy barrier for reaction i, ∆Hrxn,i is the enthalpy of reaction for reaction i, and E0 and α are fitted constants that are determined for each reaction family. The use of the Evans-Polanyi relationship reduces the computational demand considerably for these types of reaction networks, but enthalpies of reaction are still required for each individual reaction in the network. These values may be calculated from enthalpies of formation from their component species; however, enthalpies of formation may be challenging to locate for all possible species, especially for the allylic carbenium ions thought to be an integral part of alkylation chemistry. Group additivity is one approach that allows estimation of enthalpies of formation for large numbers of molecules based on the decomposition of a molecule into structural groups. Originally developed by Benson,9 this method has been shown effective for many different reaction systems.10-15 Other authors have derived group values for enthalpies of formation for simple carbenium ions16 and select allylic carbenium ions.17 However, additional groups are required to calculate enthalpies of formation for all potential allylic carbenium acyclic ions, as well as cyclic molecules. In this work, we performed QM calculations on a set of fifty-seven acyclic and cyclic carbenium ions, calculated enthalpies of formation for each using isodesmic reactions, and derived a set of Benson-like group values in order to allow for modeling of hydrocarbon


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conversion chemistries involving carbenium ions. Our results will be compared to experimental values where available, as well as to enthalpies of formation obtained from Dávila et al.’s recent set of group values.17

Methodology Quantum mechanical (QM) calculations were performed using four computational methods using the Gaussian 03 and Gaussian 09 software packages:

second-order Møller-Plesset

perturbation theory (MP2) with the 6-31G++(d,p) basis set,18 Gaussian-3 with geometries from B3LYP and the 6-31G(d) basis set (G3//B3LYP),19 a complete basis set model with geometries from B3LYP and the 6-31G† basis set (CBS-QB3),20 and Gaussian-4.21 Geometry optimizations were carried out to find a minimum energy structure, followed by frequency calculations to confirm that the structures were indeed minima, i.e. no imaginary frequencies, and to obtain thermodynamic data.

Dihedral scans were performed for larger molecules to ensure the

minimum energy geometry was found. The standard enthalpies of formation were calculated based on either atomization enthalpies or isodesmic reactions.

These methods are used because values obtained directly from QM

methods are absolute values and the enthalpies of all elements in their standard state cannot be determined with standard QM methods.22 The enthalpy of formation of an example carbocation, CxHy+, is calculated from atomization enthalpies in Equation 3: (3) where ∆Hf0(CxHy+) is the enthalpy of formation at standard state (298 K); ∆Hf0(H), ∆Hf0(C), and ∆Hf0(C+) are the enthalpies of formation of atomic hydrogen, carbon, and a positively charged carbon atom (+1), respectively (see Supplementary Information for values from the JANAF


4

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tables);23 and ∆Ha0(CxHy+) is the atomization enthalpy of CxHy+. The atomization enthalpy is defined as the enthalpy of decomposition of a molecule into its component atoms (Equation 4): (4) where H298(H), H298(C), H298(C+), and H298(CxHy+) are the enthalpies of atomic hydrogen, carbon, the plus-one carbon atom, and CxHy+ at 298 K, respectively. These enthalpies are calculated as the sum of contributions from QM calculations: electronic energies (Eel), zero-point energies (ZPE), and the vibrational (Evib298), translational (Etrans298), and rotational (Erot298) thermal corrections from 0 to 298 K based on standard formulae (Equation 5):24 (5) Isodesmic reactions are chemical reactions that conserve the number of like bonds from reactants to products. The unknown enthalpy of formation of a carbocation may be calculated by constructing an isodesmic reaction in which the enthalpies of formation of all other molecules in the reaction are known. These values are combined with values obtained from QM methods as in Equation 6:

(6) where H is the absolute enthalpy obtained from a QM calculation, ∆Hf0 is the experimentally determined enthalpy of formation at standard state, i is the number of other species in the isodesmic reaction, and Ai is a species in the reaction with stoichiometric coefficient, νi, following the usual convention of negative for reactants and positive for products. It is expected that the preservation of the same type of bonds on each side of the equation will effectively cancel any errors that may arise as a result of QM calculations. Isodesmic reactions have been


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shown effective in giving accurate estimates of enthalpies of formation when combined with QM methods13,25-30 and will therefore be used for enthalpy calculations where possible. Once the enthalpies of formation were determined, multiple linear regression was performed to obtain a set of fully significant group additivity values for carbocations, analogous to Benson’s group additivity scheme.9

Parameters in the model were deemed significant by standard

statistical tests, with p values less than α = 0.05. To accomplish this, values of certain groups were constrained, which will be further explained in the subsequent sections.

Results and Discussion Fifty-seven molecules (Figure 1) were used to calculate enthalpies of formation from QM results based on the methods described above; these values were then used to regress new Benson-type group additivity contributions for groups involving carbenium ion centers. The isodesmic reactions used to estimate enthalpies of formation for each molecule can be found in Supplementary Information.


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Figure 1. Molecules used to derive Benson-type group additivity values for enthalpies of formation. Their enthalpies of formation were calculated from G4 and isodesmic reactions.


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Figure 1. cont’d.

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Molecules used to derive Benson-type group additivity values for

enthalpies of formation. Their enthalpies of formation were calculated from G4 and isodesmic reactions.

Enthalpies of Formation from QM: Method Validation Four high-level QM methods were used to calculate enthalpies of formation for ten reference molecules that would be used in isodesmic reactions to calculate enthalpies of formation for the complete set of molecules: MP2/6-31G++(d,p), G3//B3LYP, CBS-QB3, and G4. Table 1 contains the results of the method validation for these ten reference molecules. The enthalpies of formation for the first five reference molecules (methane, ethane, ethylene and the methyl and ethyl cations) were calculated from atomization enthalpies. For this set of molecules, G4 was the most accurate method with a mean absolute deviation of 0.63 kcal mol-1, and minimum and maximum deviations of 0.08 and 1.44 kcal mol-1, respectively. G3/B3LYP was also reasonably


8

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accurate with a mean absolute deviation of 0.70 kcal mol-1, and minimum and maximum deviations of 0.01 and 1.86 kcal mol-1, respectively. MP2 was the least accurate of the four methods, with a mean absolute deviation of 25.31 kcal mol-1 for these molecules calculated with atomization enthalpies. The enthalpies of formation for the latter five reference molecules (propane, propylene, and the sec-propyl, propenyl, and tert-butyl cations) were calculated using the enthalpies of formation of the first five molecules in isodesmic reactions. For these molecules, G3//B3LYP was the most accurate method compared to experiment with a mean absolute deviation of 0.46 kcal mol-1, and minimum and maximum deviations of 0.03 and 0.74 kcal mol-1, respectively. CBS-QB3 was also very accurate with the a mean absolute deviation of 0.49 kcal mol-1, and minimum and maximum deviations of 0.07 and 0.72 kcal mol-1, respectively. MP2 was much more accurate for this set of molecules than it was for the first five species, with a mean absolute deviation of 0.73 kcal mol-1. G4 was reasonably close to the experimental values for neutral molecules, but had larger errors for the cations: 1.14 kcal mol-1 for propenyl, 1.39 kcal mol-1 for sec-propyl, 0.62 kcal mol-1 for tert-butyl.


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Table 1. Method validation results for enthalpies of formation for reference molecules. Enthalpies for the top five species were found using atomization enthalpies and for the bottom four species using isodesmic reactions. All values are in kcal/mol. MP2/6-31G++(d,p) Molecule

Exp.

Value

Dev.

CH4

-17.8[31]

3.87

C2H4

12.5[31]

C2H6

G3//B3LYP Value

Dev.

21.67

-17.90

-0.10

38.50

26.00

12.30

-20.1[31]

13.62

33.72

CH3+

261.3[32]

276.98

C2H5+

216.0[32]

245.47

CBS-QB3 Value

G4

Dev.

Value.

Dev.

-17.82

-0.02

-17.88

-0.08

-0.20

13.19

0.69

12.41

-0.09

-20.09

0.01

-19.77

0.33

-19.98

0.12

15.68

262.62

1.32

263.00

1.70

262.72

1.42

29.47

217.86

1.86

218.75

2.75

217.44

1.44

Mean Absolute Deviation

25.31

0.70

1.10

0.63

Minimum Absolute Deviation

15.68

0.01

0.02

0.08

Maximum Absolute Deviation

33.72

1.86

2.75

1.44

C3H6

4.88[31]

4.92

0.04

4.85

-0.03

4.81

-0.07

4.69

-0.19

C3H8

-24.82[33]

-25.14

-0.32

-25.11

-0.29

-25.10

-0.28

-25.19

-0.37

C3H5+

227.0[34]

226.14

-0.86

227.74

0.74

227.70

0.70

228.14

1.14

s-C3H7+

191.8[32]

192.47

0.67

192.48

0.68

192.48

0.68

193.19

1.39

t- C4H9+

170.0[[34]

168.26

-1.74

169.46

-0.54

169.28

-0.72

170.62

0.62


0.73

0.46

0.49

0.74


0.04

0.03

0.07

0.19


1.74

0.74

0.72

1.39

For the complete set of reference molecules, it was determined that the composite methods G3//B3LYP, CBS-QB3, and G4 were the three most accurate methods and deserving of additional scrutiny. Table 2 contains the enthalpies of formation calculated from G3//B3LYP, CBS-QB3, and G4 QM methods using isodesmic reactions for a subset of additional molecules for which experimental values were available. G4 was slightly more accurate compared to


10

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experiment with a mean absolute deviation of 0.76 kcal mol-1 compared to 0.79 and 0.84 kcal mol-1 for CBS-QB3 and G3//B3LYP, respectively. Based on these results, G4 was chosen as the QM method used to calculate enthalpies of formation for all molecules used in the group additivity training set via isodesmic reactions. Table 2. Enthalpies of formation calculated using G3//B3LYP, CBS-QB3, and G4 QM methods. The molecules corresponding to each number are shown in Figure 1. The values represent an average of values from isodesmic reactions (Table S.2). All values are in kcal/mol. G3//B3LYP Molecule

Exp.

(1)

184.12[31,35

CBS-QB3

G4

Value

Dev.

Value

Dev.

Value

Dev.

]

183.76

-0.36

183.91

-0.21

184.05

-0.07

(2)

161.5[36]

161.41

-0.09

161.35

-0.15

161.49

-0.01

(3)

153.2[37]

153.19

-0.01

153.26

0.06

153.29

0.09

(4)

155.6[37]

154.70

-0.90

154.79

-0.81

154.75

-0.85

(5)

155.3[37]

154.85

-0.45

155.01

-0.29

154.89

-0.41

(8)

211[38]

214.23

3.23

214.23

3.23

214.13

3.13


0.84

0.79

0.76


0.01

0.06

0.01


3.23

3.23

3.13

Enthalpies of Formation from QM: Comparison to Experiment With the methods established, enthalpies of formation for the complete set of fifty-seven molecules used for group additivity regression were calculated. The calculated enthalpies of formation based on the G4 results and the isodesmic reactions used to obtain them are summarized in Table S.2. The calculated values are compared to experimental values (where available) in Figure 2. The experimental values are calculated from proton affinity values35 and


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olefin enthalpies of formation,31 except for molecules where the carbocation enthalpy of formation was available from literature;36-40 values are tabulated in Table S.3. All G4 values fall within 3% error bounds of experimental values.

Figure 2. Comparison of calculated enthalpies of formation from G4 and isodesmic reactions versus experiment for available values from literature.

Group Additivity Value Regression and Discussion


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A total of sixteen groups were determined using multiple linear regression using the fifty-seven G4 values; these groups are listed in Table 3 along with their values and standard deviations. While the original formulation of the groups followed Benson’s prescription exactly, i.e. each polyvalent atom was assigned a group with its attached atoms explicitly delineated, it was determined through model discrimination that the effect of a positively charged carbon center attached to a carbon atom (e.g. C – C+ C H H) could not be deconvoluted from the charged atom as the center of a group. Therefore, the values of groups with pendant C+’s were fixed to analogous groups for uncharged carbon atoms as attached atoms found in literature (i.e. C – C+ C H H was fixed to the value for C – C C H H). The complete matrix of Benson contribution groups for the fifty-seven molecules can be found in Supplementary Information, along with the literature values used for fixed groups.


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Table 3.

List of sixteen regressed groups with values and standard error results.

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The

nomenclature follows standard Benson notation: CD is for a carbon atom comprising a double bond, and CB is an aromatic carbon atom. All values are in kcal/mol.

Group

Value

Standard deviation

C+ – C H H

211.40

4.45

C+ – C C H

205.10

1.65

C+ – C C C

194.64

1.32

C+ – CD H H

208.44

2.93

C+ – C CD H

187.61

1.78

C+ – C C CD

187.43

2.16

C+ – CD CD H

190.26

2.82

C+ – C CD CD

181.14

3.56

C+ – CD CD CD

181.10

4.87

C+ – CB H H

189.53

2.93

C+ – CB C H

183.16

3.13

C+ – CB C C

178.69

3.68

1.25

1.11

-7.02

1.63

-11.38

4.05

38.76

4.05

Allylic cis C+ – C – CD C+ cyclohexadiene (1,4) C+ cyclopentadiene

The majority of the groups found in Table 3 are traditional atom-centered groups; however, four other groups were introduced to account for special features of allylic carbocations. The allylic cis correction refers to the nonbonded, next-nearest neighbor interactions of groups on either side of a resonant C+ – CD bond; this is analogous to the cis correction described by Benson which is necessary for double bonds with more than two non-hydrogen groups.9


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Molecule (9) (Figure 1) is one example of a molecule requiring this correction factor. The C+ – C – CD correction refers to the physical deformation that occurs as a result of C+ – C – CD nextnearest neighbor interactions. The G4 optimized geometries of the molecules containing this group showed that the molecule would bend to bring the charged carbon in closer proximity to the double bonded carbon to stabilize the carbocation (Figure 3). Note that these corrections are not applied to 5-membered ring molecules with C+ – C – CD groups (i.e. molecule (38)) because the tight cyclic structure will prevent the molecule from buckling.

Figure 3. Molecule (12) with line drawing and optimized G4 geometry to illustrate physical bending of molecule to stabilize the charged carbon with the double bond. The central angle in the molecule is highlighted in both images to guide the eye. Special ring corrections are included for certain rings including charged carbon atoms: cyclohexadiene (1,4) (C+ cyclohexadiene (1,4)) and cyclopentadiene (C+ cyclopentadiene) rings. These corrections are added to the usual contributions of groups present in the molecule, as in


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Benson’s traditional formulation for ring corrections.9 The C+ cyclopentadiene interaction in particular is a very strong contribution to the enthalpy of formation, likely stemming from the anti-aromatic destabilization of this carbocation.41 Figure 4 contains the comparison of the enthalpies of formation used in the regression (derived from G4 and isodesmic reactions) to those fit by our regressed group values; values are listed in Table S.6. The group additivity values all fall within 4.8% of the G4 values and 89 % of the group additivity values fall within 3% of the G4 values, which is an excellent result.

Figure 4. Enthalpies of formation obtained from group additivity values compared to G4 values. Finally, the values obtained from our group additivity results were compared against the results from a similar approach recently published by Dávila et al.17 Table 4 compares the deviations from experimental enthalpies of formation for both group additivity schemes for a subset of molecules (molecules (1), (2), (3), (4), (5), (10), (11), (35), (36), and (44)) (values are listed in Supplementary Information). Both sets of values do reasonably well against experiment. Our


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results are slightly better overall compared to Dávila et al.’s results, with one fewer deviation greater than ± 2 kcal mol-1. While the mean absolute deviations are almost the same (2.66 and 3.0 kcal mol-1), the root mean square error based on our results (2.96 kcal mol-1) is smaller than that from Dávila et al.’s (3.39 kcal mol-1). The goodness of fit of our group additivity values compared to experimental values is particularly noteworthy, though, since unlike Dávila et al., we used G4 values in the regression rather than the experimental values directly as they did. Our approach resulted in an expanded set of group values that would not have been able to be obtained because of a lack of experimental values. Our results demonstrate that G4 calculations are a robust way to regress group additivity values for carbocations in the absence of sufficient experimental data. While Benson’s group additivity method is a widely accepted estimation technique for gas phase heats of formation of hydrocarbons, Lay et al.43 developed the hydrogen bond increment (HBI) method as an alternative for the estimation of the thermodynamic properties for hydrocarbon free radicals. The HBI method acquires an increment that corresponds to the enthalpy change upon the formation of a radical from its parent molecule via the loss of an H atom and the known heat of formation of the parent molecule. The increment can be added to the heat of formation of the parent molecule to get that of the radical of interest. Similarly, to obtain the heat of formation of the carbenium ion of interest, one could develop the increment from the ionization potential and add it to the heat of formation of the corresponding radical. This requires heats of formation of the radicals corresponding to the target carbenium ions. In addition, the accuracy of the QM method for estimating ionization potential should be validated first, and the choice of vertical or adiabatic ionization should be justified. While the increment method could have the advantage of better accounting for resonance effects that extend beyond the group region, it could also suffer from ambiguity in selecting the proper increment group, as


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indicated by Sabbe et al.

44

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The development and detailed assessment of the increment method

for the heat of formation of carbenium ions is beyond the scope of this work but deserves dedicated study in the future.

Table 4. Deviations from experimental values for enthalpies of formation calculated from group additivity schemes from this work and Dávila et al.17. All values are in kcal/mol.

Molecule

∆Hf° from Bjorkman et al. - ∆Hf° from Experiment

∆Hf° from Dávila et al. - ∆Hf° from Experiment

(1)

-4.42

-1.66

(2)

-2.51

1.43

(3)

-1.21

2.81

(4)

-1.51

2.40

(5)

-1.21

2.87

(10)

-4.83

-2.47

(11)

-2.17

-1.20

(35)

1.74

3.39

(36)

2.78

5.61

(44)

4.19

6.14


2.66

3.00

Root Mean Square

2.96

3.39

Conclusions


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In this work, QM methods and isodesmic reactions were used to calculate enthalpies of formation for fifty-seven acyclic and cyclic carbenium ions. Method validation suggested the choice of G4 to carry out these calculations with accuracy, and the G4 values compared well against the limited experimental values available.

These G4 enthalpies of formation were

subsequently used in multiple linear regression to determine group additivity values, particularly filling in gaps in existing literature values for allylic carbenium ions.

The enthalpies of

formation obtained using our group additivity values compared well versus experimental values and enthalpies obtained using a group additivity scheme in the literature. The goodness of fit of our group additivity values is noteworthy, given that G4 enthalpies of formation rather than experimental values were used in the regression.

Acknowledgments The authors are grateful for the financial support from UOP LLC and the Institute for Atomefficient Chemical Transformations (IACT), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, and Office of Basic Energy Sciences.

Supporting Information Enthalpy of formation values for atomic species, isodesmic reactions with enthalpies of formation calculated from G4 values, comparison of G4-derived enthalpies of formation to experimental values, complete matrix of Benson-type contribution groups, literature values used for fixed groups, comparison of G4-derived enthalpies of formation to those predicted with group additivity scheme, comparison of enthapies of formation calculated with group additivity values


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Group Additivity Determination for Enthalpies of Formation of

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