ARTICLE pubs.acs.org/IECR
Graph-Based Method for the Automated Calculation of Thermochemical Properties of Components and Intermediate Species in the Hydroprocessing of Light Oil Fractions Ytalo O. Davila G.* and Jorge M. Martinis C.† PDVSA INTEVEP, P.O. Box 76343, Caracas 1070A, Venezuela ABSTRACT: A method based on graph theory has been formulated to estimate some key thermochemical properties commonly utilized in modeling acid-catalyzed transformations of hydrocarbons at the elementary step level. Presently, modeling techniques for these transformations frequently convey the computer-aided generation of reaction networks involving a very large number of molecules and intermediate species such as carbenium ions. Although quantum mechanical calculations have been widely used to estimate thermodynamic properties of carbenium ions accurately, the current performance of such techniques is yet insufficient for very large sets of species. The proposed method handles this problem by simply analyzing the connectivity of the carbon atoms within a molecule, which is less demanding from a computational point of view. Thus, it is possible to obtain information on the contributions from different structural groups and their indistinguishability, making property estimation straightforward. This work presents a set of rules for calculating the symmetry numbers of hydrocarbons with up to 2 rings, and Benson-like structural contribution groups for estimating the enthalpy of carbenium ions.
1. INTRODUCTION Traditional modeling of acid-catalyzed reactions (such as those occurring during hydrocarbon hydroprocessing), based on the lumping of components according to their family or carbon number,14 is useful for optimizing the operating conditions of existing units and evaluating renovations or changes in capacity. However, this technique fails in predicting the effects of changes in feed composition. This is because properties such as enthalpy, as well as the reactivity, vary significantly, depending on the type, number, and position of substituents. As a result, in the past several years, much effort has been placed on the development of more-robust models, based on fundamental detailed analysis at a molecular level.510 The present work describes a methodology developed for the prediction of thermochemical properties of light oil components, as part of a larger work for the development of detailed kinetic models for hydrocarbon hydroprocessing based on single events.7,8 This approach implies the computer-based generation of reaction networks, which could lead to thousands of species. This, in turn, calls for an automated technique for calculating thermochemical properties, avoiding the use of costly quantum mechanical calculations, which would be impractical for such a large number of species. The proposed methodology makes use of the graph theory,5,6,11 which allows for information to be obtained rapidly from the molecular structure of a compound, with rather limited computer requirements. A graph is used to store structural information from molecules, following the approach of Martinis12 and Martinis and Froment.13 In this way, molecular structures are manipulated according to the rules of specific elementary steps, in order to generate reaction networks starting from feedstock species.1218 An algorithm has been developed to identify Joback contribution groups19 and Benson contribution groups20 from the r 2011 American Chemical Society
carboncarbon connectivity information stored in graphs, allowing further estimation of physical and thermodynamic properties.21 A similar approach was used previously to calculate the properties of molecules and radicals,18,22,23 but not carbenium ions, for which only some specific values have been reported in the literature.2428 Therefore, new Benson-like structural contribution groups are proposed to estimate the enthalpy of these species. In addition, specific rules are defined for the topological analysis of the molecules commonly found in naphthas, in order to estimate their symmetry numbers. This property is required to calculate the number of single events8 for elementary reaction steps.
2. METHODOLOGY 2.1. Identification of Structural Contribution Groups. Both Joback and Benson group contribution methods where used to estimate thermodynamic and physical properties of molecules. The structural groups of each molecule were identified starting from their molecular structures stored in graphs. A graph is an ndimensional square Boolean matrix, where n is the number of carbon atoms in the molecule. Each element ai,j of a graph is defined according to eq 1. Figure 1 shows examples of graphs for pentane and benzene. 8 > < 1 if atoms i and j share a bond ð1Þ ai, j ¼ 0 if atoms i and j do not share a bond > : 1 if i ¼ j and atom i is sp2 hybridized Received: August 3, 2010 Accepted: September 21, 2011 Revised: August 29, 2011 Published: September 29, 2011 12774
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Industrial & Engineering Chemistry Research The algorithm used to identify Joback contribution groups is presented in Figure 2, and a step-by-step example is shown in Figure 3 for 3-methyl cyclopentene. The first step in both figures consists of building the graph representation of the molecule i, according to the rules proposed in eq 1. The second step consists on the construction of the connectivity matrix and the hybridization vector. The connectivity matrix is the molecule’s graph without the main diagonal, and it keeps only information about
Figure 1. Graph examples: (a) pentane and (b) benzene.
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carboncarbon connections. The hybridization vector is built from the information in the main diagonal of the graph. In the example of 3-methyl cyclopentene (see Figure 3), it can be seen that elements 1 and 4 are nonzero, corresponding to the olefinic carbons in the molecule. Step 3 identifies, for each and every carbon in the molecule, whether it is cyclic or not, and in the case of cyclic carbons, it distinguishes aromatic from nonaromatic. For example, in Figure 3, elements 15 of the cyclic vector are nonzero, which corresponds to the five carbon atoms that conform the ring in 3-methyl cyclopentene. In the fourth step, the corresponding Joback structural group is assigned to each carbon atom k, and finally, in the fifth step the information is stored in a database. Similar to the previous case, Figure 4 presents the algorithm used to identify Benson contribution groups, and Figure 5 shows an example of its application to 3-methyl cyclopentene. Step 1 gets information of the previously executed Joback groups’ identification procedure. Step 2 identifies the type of carbon, for every carbon j. Step 3 identifies number and type of every
Figure 2. Algorithm use to identify Joback contribution groups in a molecule from its graph. Numbers shown in boxes represent the steps to be taken. 12775
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Figure 3. Example of the application of the algorithm used to identify Joback contribution groups for 3-methyl cyclopentene. Numbers shown in boxes represent the steps to be taken.
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adjacent group k and assigns a Benson structural contribution group to carbon j. In the case of carbenium ions, it is identified whether carbon j bears a positive charge, and a Benson structural contribution is assigned to it, depending on the nature and number of subtituents. The proposed Bensonlike structural contributions for carbenium ions are explained in more detail in section 3.1. Once structural group contributions have been assigned to each and every carbon, Step 4 assigns correction factors (if any) to the entire structure. In the case of protonated monoaromatic and naphthenic species, correction factors Bz+ and Naph+, respectively, are proposed, which are also explained in more detail in the next section. They do work just as the conventional Benson correction factors for cyclic molecules. Finally, Step 5 saves the information in a database. The result in either case is a row vector, where each element corresponds to a Joback (or Benson) contribution group. Thus, every element contains a number that indicates the number of times this group is present in the molecule. The properties then can be calculated by simple multiplication of this vector by a matrix containing the contribution factors. 2.2. Symmetry Number Calculation. The entropy of a molecule depends on how many equivalent configurations it can adopt. These configurations also have an effect on the reactivity of
Figure 4. Algorithm used to identify Benson contribution groups in a molecule from its graph. Numbers shown in boxes represent the steps to be taken. 12776
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Figure 5. Example of the application of the algorithm used to identify Benson contribution groups for 3-methyl cyclopentene. Numbers shown in boxes represent the steps to be taken.
a molecule through a specific reaction path. The symmetry number is a measure of this property, and it is divided into internal and external contributions. The external symmetry is defined as the total number of equivalent positions a molecule can adopt, by rotations along axes, considering it as a rigid body.29,30 On the other hand, the internal symmetry is related to rotations along bonds between atoms. The total symmetry number is defined according to eq 2.12 σT ¼
σEXT σINT σ EXT σINT ¼ Noi 2nq
ð2Þ
where σEXT is the external symmetry, σINT the internal symmetry, Noi the number of optical isomers, and nq the number of chiral centers. In this work, Noi is calculated as 2nq. The external symmetry number can be obtained through group theory,27,31 using computational chemistry packages. However, this approach is time-consuming and impractical for handling hundreds of species, apart from the fact that it does not report the total symmetry (internal plus external). As an alternative, the symmetry number can be obtained by doing the mental exercise of identifying all rotational axes in a molecule, which is practical only if calculations are required for a handful of molecules, in addition to being subject to human error. In this work, a methodology similar to that originally developed by Muller et al.,22,32 and then elaborated by Walters and Yalkowsky,33 was used. Once the structure of a molecule is stored in a graph, all of the information about atom connectivity and hybridization is available. In this way, the symmetry number can be estimated from the molecules’ topology, which is faster and less demanding, from a computational point of view. The main disadvantage of this approach is that graphs do not distinguish cis and trans configurations; therefore, in this case, the isomer with the largest symmetry number is considered. The steps followed for the calculation of the symmetry number are described below.
Figure 6. Examples of center identification for (a) 2,2-dimethyl pentane, (b) 3-methyl hexane, and (c) isopropyl cyclopentane.
2.2.1. External Symmetry. 2.2.1.1. Center Identification. The external symmetry number depends, first, on the nature of the center, which can be an atom, a bond, or a ring, and, second, on the number, nature, and position of the branches that grow from that center. The method developed by Walters and Yalkowsky33 apparently considers only the equality of branches for linear branches, but does not consider the symmetry of the branches in the external symmetry of the molecules. However, the nature of the branch, mainly whether it is planar or tetrahedral, has an influence on the external symmetry number. The center is identified by “peeling” the molecule by layers, using a procedure similar to that shown in Figure 2 to identify cyclic carbons. Examples are given in Figure 6 for three cases in which the center is an atom, a bond, and a ring. Each layer is conformed by all primary carbons (marked with a red circle) that are identified in the structure; once an outer layer is removed, a new structure is generated and the algorithm removes all primary carbons that conform the next inner layer. This procedure is repeated until the generated structure has no primary carbons, thus, the remaining carbons conform the center of the molecule or carbenium ion. 2.2.1.2. Branch Building. To enable comparison, branches are rebuilt inversely, with respect to how the center was identified, that is, from inside out. Therefore, the layers in a branch are numbered in an inverse order, compared to the example shown in Figure 6 for the center identification, so the most inner layer will be numbered as 1, with n being the most external. First, it is determined how many carbons are linked to the center, by identifying those that belong to the most inner layer. A branch then is built, starting from each carbon, using a recursive method. The branches are stored as m 3 vectors, in which each row represents a carbon atom, with m 12777
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Industrial & Engineering Chemistry Research being the total number of carbon atoms that conform a specific branch. The order (or layer) of each new carbon (from 1 to n, with n being the total number of layers for a specific branch), its type (1 for primary, 2 for secondary, and 3 for tertiary), and its hybridization (1 for sp2, 0 for sp3) are stored in columns 1, 2, and 3, respectively. This information is sufficient to establish the uniqueness of each branch. As an example, 2,3-dimethyl pentene has a tertiary carbon as its center; therefore, three branches grow from it; Figure 7 shows the vectors generated for each branch. 2.2.1.3. Branch Comparison. The vectors of each branch generated in the previous step are compared, and, according to the criteria shown in Table 3 (presented later in this paper),
Figure 7. (a) Identified branches for 2,3-dimethyl pentene and (bd) vectors corresponding to branches 13, respectively.
Figure 8. Carbenium ion resulting from the protonation of 2,3-dimethyl pentene.
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the external symmetry number is determined. In Figure 7, it can be seen that, for 2,3-dimethyl pentene, all branches are different, and therefore, its external symmetry number is 1. 2.2.2. Internal Symmetry. The internal symmetry is related to rotations of CC bonds that generate equivalent structures. They are primarily associated with methyl and tert-butyl groups, which have a symmetry of 3, and isopropyl groups with a positive charge on carbon 2 (in the case of carbenium ions), as well as aromatic rings and cyclopentanes, which have a planar structure and therefore have a symmetry number of 2. Consequently, the program identifies the presence of each of these groups in the molecule and, in the case of rings, determines whether their rotation contributes to the internal symmetry. The internal symmetry is then defined as the product of all independent internal symmetries. Going back to the example of 2,3-dimethyl pentene (see Figure 7), its internal symmetry number is dependent only on methyl terminal groups. There are three methyl groups, so the internal symmetry number is 3 3 3 = 27. It is important to notice that the carbenium ion coming from the protonation of this olefin has a higher internal symmetry (see Figure 8). By protonation of the double bond, a protonated isopropyl is generated, which is planar, with symmetry 2 (by rotation of the bond between carbons 2 and 3). In addition, a methyl group is formed. Therefore, the resulting internal symmetry number is 3 3 3 3 2 = 162. 2.2.3. Chirality. Chirality arises when overlapping the spectacle image of a molecule with the original structure is not possible. In this work, chiral carbon atoms are identified as those tetrasubstituted whose substituents are all different.34 The program analyzes each tri- and tetra-substituted sp3 carbon, builds branches starting from them similarly to how they are built from the center for external symmetry determination, and then
Table 1. Estimated Benson Structural Contribution Groups for Carbenium Ions hof (kJ/mol) group
adjacent
1
CH2+
C
2
CH+
C, C
3 4
C+
6
950.45 ( 15.76
950.45 ( 17.19 868.61 ( 9.93
824.38 ( 11.14
824.38 ( 12.16
C, C, C
829.64 ( 5.57
829.33 ( 8.57
C, C, dC
799.85 ( 9.10
C C, C
747.24 ( 7.05 740.65 ( 7.05
868.81 ( 10.92 829.94 ( 7.72 785.96 ( 15.44
748.50 ( 9.93 730.96 ( 9.93
9
C, C, C
693.86 ( 11.14
693.86 ( 12.16
10
C, C, C, C
633.86 ( 15.76
633.86 ( 17.19
745.36 ( 10.92 755.18 ( 10.92
11
C, C, C, C, C
685.66 ( 15.76
685.66 ( 15.44
12
C, C, C, C, C, C
675.76 ( 15.76
675.76 ( 15.44
13
dC
690.40 ( 9.10
14
dC, C
674.29 ( 5.57
674.24 ( 8.60
dC, C, C
662.06 ( 15.76 733.46 ( 15.76
662.06 ( 17.19 733.46 ( 17.19
C
703.01 ( 11.14
15 16
Naph+
17
Fitting root-mean-square error (RMSE)
15.76
Cross-validation RMSEa a
dataset 2
799.85 ( 8.92
785.96 ( 15.76
Bz+
7 8
dataset 1
868.68 ( 7.04
C, dC
5
complete data
690.40 ( 8.92 674.34 ( 7.72
703.01 ( 10.92 17.19
15.44
17.33
15.13
Reported RMSE corresponds to the evaluation of dataset 1 with contributions fitted from dataset 2 and vice versa. 12778
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Table 2. Recurrence of Proposed Benson Contribution Groups on the Carbenium Ion Database groupa
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
SETb
toluene
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
ethylene
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2-butene
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
benzene
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
2
propene
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
cyclopentene
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
1,3-butadiene
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
cyclohexene ethyl benzene
0 0
1 0
0 0
0 0
0 0
0 0
0 1
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
2 1
propyl benzene
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
2
isopropenyl benzene
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
butyl benzene
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
2
p-xylene
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
1,2-dimethyl benzene
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
2
2-methyl propene
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
naphthalene hexene
0 0
0 1
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
1 0
0 0
1 1
2-methyl 2-butene
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2-methyl 2-pentene
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2,4-dimethyl 2-pentene
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
2
1,3-dimethyl benzene
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
3-methyl 2-pentene
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2,3-dimethyl 2-butene
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1-methyl cyclopentene 1,2-dimethyl cyclopentene
0 0
0 0
0 0
1 1
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
2 1
2-methyl, 1,3-butadiene
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
2
2-methyl naphthalene
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
propenyl benzene
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
2
1-methyl naphthalene
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
2,3-dimethyl 1,3-butadiene
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
2
1,3,5-trimethyl benzene
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
styrene 1,2,3,5-tetramethyl benzene
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 1
0 0
0 0
1 0
0 0
0 0
0 0
0 0
2 1
1,3,5-triterbutyl benzene
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
indene
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
1-ethenyl,3-methyl benzene
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
2
pentamethyl benzene
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
2
1-ethenyl, 2-methyl benzene
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
2
1-methyl, 2-isopropenyl benzene
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
1-isopropenyl, 4 methyl benzene tetralin
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 1
0 0
0 0
0 0
0 0
0 0
1 0
0 0
0 0
0 0
2 1
alphamethyl styrene
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
2
1,3-pentadiene
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
hexamethyl benzene
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
2
1-ethenyl, 4-methyl benzene
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
2
1-methyl, 3-isopropenyl benzene
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
3-methyl,1,3-pentadiene
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
2
isopropenyl, 2,2-dimethyl benzene 1-(2,2-dimethyl-1-methylenepropyl), 3,5-dimethyl benzene 1-(2,2-dimethyl-1-methylenepropyl),
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
1 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 1
0 1 0
0 0 0
0 0 0
2 1 1
4-methyl benzene recurrence
1
5
2
8
3
1
5
5
2
1
1
1
3
8
1
1
2
-
Group numbering according to Table 1. b Dataset to which each carbenium ion was assigned, for cross-validation.
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compares them. In the example of 2,3-dimethyl pentene, the carbon 3 is chiral.
3. RESULTS 3.1. Structural Group Contributions for Estimating the Enthalpy of Carbenium Ions. The kinetic parameters of acid-
catalyzed reactions such as cracking and isomerization are related to the thermodynamic properties of the corresponding carbenium ions. The prediction of enthalpies of formation is particularly important, because the activation energies of elementary steps are calculated from the heats of reaction through the EvansPolanyi relationship.8 However, there are few references in the literature for these enthalpies.25,26 Therefore, the Benson groups were extended in this work in order to automate the calculation of the enthalpy of formation for each and every carbenium ion created during reaction network generation. Proton affinities were used to calculate standard heats of formation in the gas phase. Then, the enthalpy contributions of the groups listed in Table 1 were obtained by minimization of the resulting overspecified equation system, with the generic structure shown in eq 3. 2 3 2 3 2 3 hof1 ΔHf1 gc1, 1 gc1, 2 3 3 3 gc1, n 6 7 6 7 6 7 6 gc2, 1 6 7 6 7 ⋱ l 7 6 7 6 hof2 7 ¼ 6 ΔHf2 7 ð3Þ 6 l 7 6 7 6 7 ⋱ 4 5 4 l 5 4 l 5 gcm, n hofn ΔHfm gcm, 1 3 3 3 where gc is an m n matrix with m > n, each gci,j element indicates the number of times a structural contribution group j is present in molecule i, hof is an n-dimensional vector that contains the structural contributions for enthalpy calculations, and ΔHf is the vector of literature values of the enthalpies of formation. As can be seen from Table 1, 17 structural contributions were fitted; therefore n = 17. For the estimation of these contributions, 50 species were used: 16 aromatics, 11 aromatic olefins, 10 acyclic olefins, 4 cyclic olefins, 5 diolefins, 3 diaromatics, and indene;25 therefore m = 50. Those species are listed in Table 2, where the recurrence of each structural contribution group in the database is also shown. Simplifications were made to reduce the number of proposed structural contribution groups, according to the limited available data. For instance, the effect of adjacent groups on the charged carbon is taken into account, but not the effect of the latest on the previous ones. Nearest neighbors can stabilize the positive charged carbon, depending on whether they are hybridized sp2 or sp3. Meanwhile, the effect of a positively charged carbon as a nearest neighbor is taken as if it was an olefin, in the case of primary, secondary, and tertiary carbenium ions. On the other hand, for protonated aromatic species, ring correction factors Bz+ are proposed. These correction factors are added at the end, just as the conventional correction factors for cyclic molecules in the original Benson method. Therefore, the contributions of the groups linked to the protonated aromatic ring are taken as if they were connected to an aromatic molecule. These correction factors vary, depending on the number and nature of the substituents on the aromatic ring. However, they do not take into account the relative position effects of the substituents. Similarly, Naph+ correction factors were defined for naphthene-like protonated structures.
Figure 9. Comparison of calculated standard enthalpy of formation of carbenium ions with literature values,25 for 16 aromatics, 11 aromatic olefins, 10 linear and branched olefins, 4 cyclic olefins, 5 diolefins, 3 diaromatics and indene, with 5% error bands.
Furthermore, no entropy or specific heat corrections were considered in this study, and heteroatoms and the dCd group were not taken into account. In order to validate the estimates, the data was divided into two subsets for independent fitting. Only those groups that have a recurrence higher than three (3) were considered for the cross-validation. In the last column of Table 2, the subset of data to which each component was assigned is shown. The root-mean-square error (RMSE) is evaluated for the fittings themselves, and also for the cross-validation, where dataset 1 is evaluated on the fit obtained from dataset 2 and vice versa. In Table 1, it can be seen that the structural enthalpy contributions obtained for groups 2, 4, 7, 8, and 14 are quite similar, and that the RMSE error does not increase significantly when the fitting of the entire set of data is compared to those of subsets 1 and 2. Both subsets of data cover approximately the same region, since all contribution groups are independent from each other, and data points of the above-mentioned groups were evenly distributed to ensure roughly the same recurrence. However, for the rest of the groups, a higher uncertainty is expected, because of the smaller number of data points available for the fitting. It is important to notice that, although the Benson method only takes into account next-nearest neighbors, it is considered the most accurate among similar methods that have been used to compute formation properties.21 In addition, the use of second-order methods does not improve the predictions significantly and generally gives additional complexity that is not justified in the present work. Up to two fused aromatic or naphthenic rings are considered; therefore, no additional corrections are introduced for polyaromatic species, such as those proposed by Yu et al.,35,36 where resonance is expected to have a high influence. Only paraffins, olefins, naphthenes, and mono-aromatic species were considered in this work, while heteroatoms were not included. Figure 9 compares the enthalpy estimated from the expanded Benson groups with that calculated directly from the proton affinities of the 50 carbenium ions shown in Table 2. It can be 12780
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Table 3. Rules for External Symmetry Number Estimation Depending on the Nature of the Center and the Number and Nature of the Branchesa
a
For cases in which the center is a ring, only those configurations with a symmetry higher than 1 are indicated. The identifications (IDs) AD provide a generic way to denote whether the branches are equal or different. σR is used to indicate the symmetry number of the branch. Column p indicates the position of the substituent according to IUPAC rules. The assumptions are made in order to obtain the highest possible symmetry number for that configuration.
seen that all data points fall between the 5% error bands. Therefore, the methodology proposed is good enough to establish reactivity levels when the EvansPolanyi relation is applied. 3.2. Symmetry Number Calculation. Table 3 summarizes the rules that must be applied to calculate the external symmetry number of a species, once its center as well as the branches linked to it have been identified. The third column specifies how many branches are attached to the center of the molecule. Columns R1R4 report the characteristics of each branch. The identification (ID) of a branch can be A, B, C, or D and is used simply to indicate whether two or more branches are equal or different. σR reports the rotation symmetry contribution of a branch, and column p indicates the position of that branch according to IUPAC rules, which is relevant only for cyclic species.
Even though the identification of the number of chiral centers might be trivial in most cases, this is not the case for cyclic species. Because of the intrinsic symmetry of a ring, two or more chiral centers may cancel each other, depending on their relative position. Table 4 gives some basic rules to compute the number of chiral centers, depending on the type of cyclic species, and the type and number of branches linked to it, up to three branches. The nomenclature used is similar to that in Table 3. On the other hand, references in the literature of symmetry numbers are scarce. Table 5 compares the symmetry numbers computed for a few species with those reported in the literature. Muller and co-workers32 developed a method similar to that presented here, but it does not include carbenium ions. Moreover, no similar set of rules for symmetry calculations based on the nature of the center of the molecule, and the nature and 12781
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Table 4. Rules for Chiralily Identification in Cyclic Speciesa
The proposed extension of Benson groups to compute the enthalpies of formation of carbenium ions gave results with errors of
