A New Approach of Evaluating Bond Dissociation Energy from

The off-diagonal elements, representing bonding connections, are assigned values 1. Thus the constructed connection-matrix (CMij) (see eq 8) of bond R...
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J. Chem. Inf. Comput. Sci. 2003, 43, 600-608

A New Approach of Evaluating Bond Dissociation Energy from Eigenvalue of Bonding Orbital-Connection Matrix for C-C and C-H Bonds in Alkane Chenzhong Cao* and Hua Yuan Department of Chemistry, Xiangtan Normal University, Xiangtan, 411201, People’s Republic of China Received September 5, 2002

A new bonding orbital-connection matrix was constructed, in which the diagonal elements were assigned the chemical potentials E1 and E2 of two radicals Ri• and Rj• obtained by cutting the interested bond, and the off-diagonal elements, representing bonding connections, were assigned values 1. The eigenvalues X1CC and X1CH of the bonding orbital-connection matrix were obtained for C-C and C-H bonds, respectively. Also a steric effect parameter Sij was proposed for C-C bond. Using X1CC, X1CH, and Sij as bond descriptors, good correlations with the Bond Dissociation Energies (BDEs) of the C-C and C-H bonds were obtained for alkanes. The result shows that the eigenvalue of bonding orbital-connection matrix is a good descriptor for expressing the relative bond energies of C-C and C-H bonds in alkane. This work provides a new physical insight and a principally novel general approach to the evaluation of the bond dissociation energies of carbon-carbon (C-C) and carbon-hydrogen (C-H) bonds in alkane. Also it builds a simple bridge linking the adjacent matrix of radicals Ri• and Rj• with the BDE of Ri-Rj and Ri-H. Furthermore, the Heat of Atomization (HA) and Heat of Formation in Gas (HFG0) of alkane can be estimated well with the parameters X1CC, X1CH, and Sij. 1. INTRODUCTION

The bond dissociation energy (BDE) is a fundamental factor influencing chemical reactivity and is of paramount importance in radical processes. It has also been shown that the BDE can influence reactivity in polar processes. On the other hand, because BDE is related to the heat of formation of a compound, it is widely used for estimation of heats of formation for organic compounds. This method is the socalled bond additivity.1,2 For almost a century, the nature of chemical bonds has fascinated chemists, and the extensive work performed in this field has resulted in a number of methods for experimental determination or estimation of bond dissociation energies.1-17 Quantum chemical calculations have also proven to be a very useful tool for the determination of bond dissociation energies,18 even in relatively large molecules.19 Experimental determining of BDEs is tedious and time-consuming. Semiempirical quantum chemical calculation needs a time-consuming parametrization,18 and the application of high-level ab initio quantum chemical theory is very expensive.18 Thus, an estimation scheme is very valuable. A number of empirical methods for estimations of heats of formation of molecules and radicals (and thereby also for estimation of homolytic bond dissociation energies) have been put forward.1,15-17 The most famous of these methods is probably the empirical scheme by Benson and co-workers.1 Recently, some new estimation methods have also been investigated.15-17 In this work, we develop a novel semiempirical method for estimation of C-C and C-H bond dissociation energies (BDE) in alkanes. The approach is based on the eigenvalue of bonding orbital-connection matrix for carbon-carbon * Corresponding author phone: 00086-0732-8291336; fax: 00086-07328291001; e-mail: [email protected].

(C-C) and carbon-hydrogen (C-H) bonds in alkane and does not need complicated calculation. 2. METHOD

Pauling20,13

ever proposed that the relation between the heteropolar bond energy DAB and the homopolar terms DAA and DBB is

1 DAB ) (DAA + DBB) + k(∆χ)2 2

(1)

where k is an empirical constant (96.5 kJ/mol) and ∆χ is the difference in electronegativity between the atoms A and B. However, it has been shown13 that eq 2, with k ) 134 kJ/mol, gives better results than eq 1:

DAB ) (DAADBB)1/2 + k|∆χ|

(2)

Equation 2 implies if A and B are the same kind of atom, the DAB will be a constant. That is to say all C-C bonds in alkane have the same BDE and so do the C-H bonds. As we know, not all C-H or C-C bonds have the same BDE, and the BDE is actually related to its chemical environment. To resolve this problem, Smith suggested13 that the carboncarbon bond energy D(Ci-Cj) is variable according to eq 3

D(Ci-Cj) ) D°(C-C) - aqiqj

(3)

where i and j indicate adjacent singly bonded carbon atoms having fractional charges qi and qj, and a is an empirical constant, D°(C-C) is taken as 358.6 kJ/mol. Equation 3 shows that the carbon-carbon (Ci-Cj) bond-energy is enhanced by the fractional charges on themselves. From the principle of electronegativity equalization, Smith13 further changes eq 3 to eq 4

10.1021/ci020295v CCC: $25.00 © 2003 American Chemical Society Published on Web 03/05/2003

EVALUATING BOND DISSOCIATION ENERGY

D(Ci-Cj) ) D°(C-C) + k|∆χ|

J. Chem. Inf. Comput. Sci., Vol. 43, No. 2, 2003 601

(4)

where k is an empirical constant and ∆χ is the difference in electronegativity between the groups linked by the carboncarbon (Ci-Cj) bond. Equation 4 appears to be inconsistent with eq 2, but it is an extension of eq 2 and gives a new meaning of carbon-carbon bond-energy relating to its environment. It is known that the electronegativity χ of any chemical species (an atom or a molecule) is the negative of its chemical potential,21,22 µ, obtained from density functional theory as in eq 5

χ ) -µ ) -(∂E/∂N)V

Table 1. Ionization Potential for Some Alkyl Radials no.

radical

PEI(Ri)a

1 2 3 4 5 6 7 8 9 10

H3C• MeH2C• EtH2C• PrCH2• i-PrCH2• Me2CH• MeEtCH• Me3C• EtMe2C• Me3CH2C•

1 1.1405 1.1887 1.2122 1.2368 1.2810 1.3292 1.4215 1.4696 1.2849

9.83 (9.73) 8.4 (8.56) 8.1 (8.16) 7.5 (7.38)

method: photoionization

(5)

where E is the exact Hohenberg and Kohn energy functional for an N-electron system characterized by an external potential ν. Based on eqs 4 and 5, we think that the carboncarbon (Ci-Cj) bond-energy perhaps relates to the energies of “prebonded” atomic orbitals that form the Ci-Cj bond. Suppose the alkyl radical Ri• links with Rj• to form Ri-Rj (Ci-Cj) bond, its heat of reaction is equal to its dissociation energy. If the chemical potentials (or energy level) of orbitals

Ipexp (Ipcalc) (eV)b

Ipexp (Ipcalc) (eV)c

8.64 ( 0.05 (8.61) 8.35 ( 0.05 (8.47) 7.93 ( 0.05 (7.94) 7.42 ( 0.07 (7.42) 7.12 ( 0.1 (7.14) 8.33 ( 0.1 (8.20) method: electron impact

a PEI of radical, calculation based on ref 24. b The Ipexp is measured by photoionization, taken from ref 25. The Ipcalc is calculated by eq 7a. c The Ipexp is measured by electron impact, taken from ref 25. The Ipcalc is calculated by eq 7b.

and PrCH2• by semiempirical quantum chemistry method AM1(using MOPAC program packages in SYBL 6.7 of Tripos, Inc.) and got the values of -13.7962, -11.9448, -11.3913, and -11.2204 (eV), respectively. The obtained correlation between EHOMO and PEI(Ri) is

EHOMO(eV) ) -26.1568((0.6127) + 12.3914((0.5382)PEI(Ri) (7c) 1 and 2 in eq 6 are in low (large electronegativity), the formed carbon-carbon bond will be strong, in other words, its bond dissociation energy will be large. Otherwise, the bond will be weak and its bond dissociation energy will be small. The chemical potentials of orbitals 1 and 2 can be expressed by the ionization potentials of alkyl radical Ri• and Rj•, respectively. It should be noted that, up to now, only a few ionization potential values of alkyl radicals have been measured, so it is necessary to develop a method for estimating their ionization potentials. Our previous work23 has shown that the ionization potentials of monosubstituted alkanes have relation with the polarizabilities of their alkyl groups. Here, we also find a linear correlation between the ionization potentials of alkyl radicals and their polarizability effect index (PEI)24 when their ionization potentials are measured by a same method such as photoionization and electron impact (see eq 7 and Table 1). For examples, photoionization(the third column values of Table 1)25 of alkyl radicals has the correlation

Ip(eV) ) 18.092((0.911) - 8.358((0.787)PEI(Ri) (7a) r ) 0.9913, s ) 0.16, F ) 112.79, n ) 4 Electron impact ionization (the fourth column values of Table 1)25 of alkyl radicals has the correlation

Ip(eV) ) 15.554((0.535) - 5.724((0.403)PEI(Ri) (7b) r ) 0.9902, s ) 0.09, F ) 202.08, n ) 6 It can be seen from Table 1 that there are different ionization potential values in different measured condition for an alkyl radical. Further, we calculated the highest occupied molecular orbital(HOMO) energies(EHOMO) of H3C•, MeH2C•, EtH2C•,

r ) 0.9981, s ) 0.0886, F ) 529.93, n ) 4 Equations 7a, 7b, and 7c all show a same change rule that the chemical potentials (or energy level) of orbitals 1 and 2 in eq 6 can be expressed by E1 ) a + bPEI(Ri) and E2 ) a + bPEI(Rj), respectively. To relate the bond dissociation energies (BDEs) of carbon-carbon (Ci-Cj) bonds to the chemical potentials (or energy level) of orbitals 1 and 2, we construct an adjacency matrix for each carbon-carbon (CiCj) bond and assign the E1 and E2 as the diagonal elements of the matrix. The off-diagonal elements, representing bonding connections, are assigned values 1. Thus the constructed connection-matrix (CMij) (see eq 8) of bond RiRj (Ci-Cj) is similar to Hu¨ckel molecular orbital (HMO) method.26

CMij )

[ ][

E1 1 a + bPEI(Ri) 1 ) a + bPEI(Rj) 1 E2 1

]

(8)

The number 1 in CMij indicates the alkyl radical Ri• linked with Rj•. Here let the eigenvalues of connection-matrix (CMij) be OX1CC, OX2CC (and OX1CC < OX2CC which is similar to the HOMO and LUMO of Hu¨ckel molecular orbital method). It can be found (see the next section of this paper) that the bond energies D(Ci-Cj) (or BDE of bond Ci-Cj) have a linear correlation with the eigenvalues. For the calculation convenience, we further modify the connection-matrix (CMij) into another form, which is named as modified connection-matrix (CMij,m) (see eq 9), and assign its eigenvalues X1CC and X2CC(X1CC < X2CC).

CMij, m )

[

PEI(Ri) 1 PEI(Rj) 1

]

(9)

602 J. Chem. Inf. Comput. Sci., Vol. 43, No. 2, 2003

CAO

Table 2. Polarizability Effect Index Increment ∆PEI(l)24 of the lth Carbon Atom. lth carbon atom

∆PEI(l)

lth carbon atom

∆PEI(l)

1 2 3 4 5

1.0000 0.1405 0.0481 0.0235 0.0138

6 7 8 9 10

0.0090 0.0064 0.0047 0.0037 0.0022

X1CC )a1 + b1X1CC

(10a)

X2CC ) a2 + b2X2CC

(10b)

O

O

Thus this paper employs the eigenvalues X1CC and X2CC of modified connection-matrix (CMij,m), instead of OX1CC and O X2CC, to calculate the BDEs of carbon-carbon (Ci-Cj) bond in alkanes (all the eigenvalues in this paper are obtained by computer). Some calculation examples of eigenvalues X1CC and X2CC are listed in Table 3. The calculation procedure can be briefly divided into three steps. First Step. Cut an interested carbon-carbon (Ci-Cj) bond into two alkyl radicals Ri and Rj. Then calculate their polarizability effect index PEI(Ri) and PEI(Rj) according to the method in the literature,24 respectively. That is PEI(Ri) ) ∑∆PEI(l), in which ∆PEI(l) is the contribution of the lth carbon atom in alkyl radical to the PEI(Ri), for example, 2-methylbutyl radical (•CH2CH(CH3)CH2CH3), taking the radical carbon atom as the beginning one (first one), the next carbon atoms are the second one, and the next next carbon atoms are the third one, and so on (see Figure 1). Using the ∆PEI(l) values of Table 2, we can obtain its PEI(Ri):

PEI(Ri) ) ∆PEI(1) + ∆PEI(2) + 2∆PEI(3) + ∆PEI(4) ) 1.0000 + 0.1405 + 2(0.0481) + 0.0235 ) 1.2602 Second Step. Use the PEI(Ri) and PEI(Rj) values of radical Ri and Rj to construct the modified connection-matrix (see the CMij,m of Table 3). Third Step. Solve the modified connection-matrix CMij,m and get the eigenvalues (see the X1CC and X2CC of Table 3). For the carbon-hydrogen bond Ri-H (or Ci-H), we can use the chemical potentials of orbital 1 in alkyl radical Ri• and hydrogen atomic orbital, E1 and EH, to construct the connection-matrix (CMiH) (see eq 11).

CMiH )

[

] [

]

a + bPEI(Ri) 1 E1 1 ) 1 EH E0H + bPEI(H) 1 (11)

Because the electron in hydrogen atomic orbital connects with only a proton, its PEI(H) ) 0, that is EH ) E0H + 0 ) constant (the ionization potential of hydrogen atom) in eq 11. In accordance with eq 9, we obtain the modified connection-matrix (CMiH,m) of bond Ri-H (Ci-H) as follows.

CMiH, m )

[

PEI(Ri) 1 1 0

]

(12)

YUAN

If let the eigenvalues of connection-matrix (CMiH) be X1CH, OX2CH (here OX1CH < OX2CH) and the eigenvalues of modified connection-matrix (CMiH,m) be X1CH, X2CH, there are also linear correlations between X1CH, X2CH and OX1CH, O X2CH, respectively. That is O

X1CH ) a1H + b1HX1CH

(13a)

X2CH ) a2H + b2HOX2CH

(13b)

O O

Because a and b in expressions of E1 and E2 are constants, it can be demonstrated that there are linear correlation between the OX1CC or OX2CC of CMij and the X1CC or X2CC of CMij,m respectively. That is

AND

Thus the eigenvalues X1CH and X2CH of modified connection-matrix (CMiH,m), instead of OX1CH and OX2CH, are used to calculate the BDEs of carbon-hydrogen (Ci-H) bond in alkanes. Some calculation examples of eigenvalues are listed in Table 4. 3. CORRELATION BETWEEN EIGENVALUES AND BDE FOR Ci-Cj AND Ci-H BONDS

BDE for Ci-Cj Bond. Here we suppose that the BDE of Ci-Cj bond has a linear relation with OX1CC or OX2CC.

BDE(Ci-Cj) ) A + BOX1CC

(14a)

BDE(Ci-Cj) ) A + BOX2CC

(14b)

or

From eqs 10a and 10b, we can lead to eqs 15a and 15b.

BDE(Ci-Cj) ) A + B(a1 + b1X1CC) ) A1 + B1X1CC (15a) BDE(Ci-Cj) ) A + B(a2 + b2X1CC) ) A2 + B2X2CC (15b) where A, B, A1, B1, A2, and B2 all are coefficients of regression equations. Taking some measured BDEs12,16 of Ci-Cj as data set (listed in Table 5), we make a regression analysis with their eigenvalues X1CC and X2CC, respectively, and obtain the following correlation expressions.

BDE(Ci-Cj)(kJ/mol) ) 378.741((5.032) 210.237((15.169)X1CC (16) r ) 0.9872, s ) 5.22, F ) 192.08, n ) 7 BDE(Ci-Cj)(kJ/mol) ) 793.501((43.444) 207.206((18.766)X2CC (17) r ) 0.9801, s ) 6.50, F ) 121.92, n ) 7 Equations 16 and 17 show that there are indeed good linear correlations between the BDE and X1CC or X2CC. The correlation of eq 16 is better than that of eq 17, so we take eq 16 as the basic model to correlate the BDE of Ci-Cj bond. However the lager standard deviation of eq 16 also reminds us of the steric effect in the alkanes1,2,7,8 and the influence of steric effect on the BDE of Ci-Cj bond. Thus, for eq 16, we can expect an improved result if a steric term is added in. This paper proposes a simple steric effect parameter, Sij ) (ViVj)2, and employs it to scale the steric effect on the BDE of Ci-Cj bond (Vi and Vj are the vertex degrees of Ci and Cj, which is the number of carbon atoms linked to Ci

EVALUATING BOND DISSOCIATION ENERGY

J. Chem. Inf. Comput. Sci., Vol. 43, No. 2, 2003 603

Table 3. Some Calculation Examples of Eigenvalues X1CC and X2CC first step Ci-Cj bond

Ri

third step Rj

PEI(Ri) H3C• 1.0000

PEI(Rj) H3C• 1.0000

CH3-CH2CH3

H3C• 1.0000

H3CH2C• 1.1405

(CH3)2CH-CH2CH3

(CH3)2CH• 1.2810

H3CH2C• 1.1405

CH3-CH3

Figure 1. The 2-methylbutyl radical and carbon atom ordinal. Table 4. Some Calculation Examples of Eigenvalues X1CH and X2CH Ci-H bond

Ri

H

CH3-H

H3C•

H

CH3CH2-H

H3CH2C•

H

(CH3)2CH-H

(CH3)2CH•

H

[ [ [

CMiH,m

1.0000 1 1 1.0 1.1405 1 1 0 1.2810 1 1 0

] ]

]

X1CH

X2CH

-0.6180

1.6180

-0.5809

1.7214

-0.5470

1.8280

and Cj, respectively). That is to say the more the carbon atoms linking to the given carbon atom, the more crowded the space around the given one. For a pair of carbon atoms in a carbon-carbon bond, the more crowded space will lead to the more nonbonding interaction among the adjacent groups and raises the inner energy of the bond rapidly. For example, CH3-CH3, Vi ) Vj ) 1, Sij ) (1 × 1)2 ) 1; (CH3)2CH-CH2CH3, Vi ) 3, Vj ) 2, Sij ) (3 × 2)2 ) 36. Using the two variables X1CC and Sij, we obtained the following regression equation.

BDE(Ci-Cj)(kJ/mol) ) 376.160((1.777) 162.718((9.270)X1CC - 0.100((0.016)Sij (18) r ) 0.9988, s ) 1.79, F ) 833.73, n ) 7 Compared to eq 16, the correlation of eq 18 is much better and the standard deviation s is much less. The calculated BDEs of Ci-Cj bond by eq 18 are also listed in Table 5. BDE for Ci-H Bond. As did in the correlation of BDE for Ci-Cj bond, we carry out correlations between some measured BDEs15 and X1CH or X2CH for Ci-H bonds. The obtained regression equations are as follows:

BDE (Ci-H)(kJ/mol) ) 244.560((14.038) 311.043((25.365)X1CH (19) r ) 0.9806, s ) 2.43, F ) 150.38, n ) 8 BDE (Ci-H)(kJ/mol) ) 590.935((16.938) 96.072((9.306)X2CH (20) r ) 0.9730, s ) 2.86, F ) 106.58, n ) 8

second step CMij,m

[ [ [

1.0000 1 1 1.0000 1.0000 1 1 1.1405 1.2810 1 1 1.1405

] ] ]

X1CC

X2CC

0.0000

2.0000

0.0678

2.0727

0.2083

2.2132

Both eqs 19 and 20 show that there are good correlations between BDE of Ci-H and X1CH or X2CH. The correlation of eq 19 is better than that of eq 20, hence this paper employs eq 19 to evaluate the BDE of Ci-H bond. Here, the steric effect does not need to be considered because of the very small volume of hydrogen atom. The calculated BDEs of Ci-H bonds by eq 19 are listed in Table 6. The correlation between the eigenvalues of connectionmatrix and the bond dissociation energy may be understood as follows. Based on quantum chemistry, a bond combined by two lower energy atom orbitals will be in lower energy level26 or has higher BDE. In fact, it is true. For examples, the energy ordering of carbon atomic hybrid orbitals is sp < sp2 < sp3 (the ionization potential ordering is sp > sp2 > sp3), the carbon-carbon bond dissociation energy ordering is C(sp)-C(sp) >C(sp2)-C(sp2) > C(sp3)-C(sp3), and the carbon-hydrogen bond dissociation energy ordering is also C(sp)-H > C(sp2)-H > C(sp3)-H. As stated in the above section, the PEI-values for alkyl radicals are linearly correlated to the ionization potentials, so eqs 16, 17, 19, and 20 mean that the eigenvalues of a connection-matrix containing the ionization potentials would also be linearly related to the bond dissociation energies. According to the LCAOMO approximation, the molecular orbitals are linear combinations of the two atomic orbitals. That is a linear combination of the two atomic orbitals will lead to a secular determinant and result in two molecular orbitals,26 a bonding orbital with lower energy and an antibonding orbital with higher energy. The BDE is better related to the energy of its bonding orbital. Each connection-matrix (similar to a secular determinant) yields two eigenvalues, which are something corresponding to the energies of two molecular orbitals formed by two atomic orbitals. Perhaps, the smaller value (the first eigenvalue) is related to the energy of the bonding orbital and the larger value (the second eigenvalue) is related to the energy of the antibonding orbital. Therefore the first eigenvalue gives a somewhat better correlation with the BDE than the second eigenvalue. 4. DISCUSSION

Evaluation of BDE of C-H and C-C Bonds in Alkane. Equation 18 shows a good correlation between the BDE and parameters X1CC and Sij of carbon-carbon (Ci-Cj) bond. Equation 19 indicates a good correlation between the BDE and X1CH of carbon-hydrogen (Ci-H) bond. Therefore eqs 18 and 19 can be directly employed to evaluate the relative BDE of carbon-carbon (Ci-Cj) bond and carbon-hydrogen (Ci-H) bond for an assigned alkane molecule. Take 2,2,5trimethylheptane (Figure 2) for example. Cutting the C1-H

604 J. Chem. Inf. Comput. Sci., Vol. 43, No. 2, 2003

CAO

AND

YUAN

Table 5. Bond Dissociation Energies (BDEs) of the Ci-Cj Bonds in Alkane (in kJ/mol) Ri-Rj(Ci-Cj)

no. 1 2 3 4 5 6 7 a

X1CC

H3C-CH3 Me2HC-CMe3 Me3C-CMe3 MeH2C-CEtMe2 Me2HC-CEtMe2 EtMeHC-CHEtMe EtMeHC-CMe3

X2CC

0.0000 0.3488 0.4215 0.2916 0.3709 0.3292 0.3743

2.0000 2.3538 2.4215 2.3186 2.3798 2.3292 2.3764

376.0 ( 2.1 305.2b 282.4c 322.59b 298.32b 315.05b 302.5b

1 144 256 64 144 81 144

a

BDE(Ci-Cj)calcd 376.06 305.00 281.96 322.31 301.40 314.49 300.85

Taken from ref 12(a). b Taken from ref 16. c Taken from ref 12(b). d Calculated by eq 18.

Table 6. Bond Dissociation Energies (BDE) of the Ci-H Bonds in Alkanes (in kJ/mol) no.

Ri-H(Ci-H)

X1CH

X2CH

BDE(Ci-H)expa

BDE(Ci-H)calcb

1 2 3 4 5 6 7 8

CH3-H MeCH2-H MeCH2CH2-H Me2CH-H EtMeCH-H Me3C-H Me3CCH2_H EtMe2C_H

-0.6180 -0.5809 -0.5689 -0.5470 -0.5363 -0.5161 -0.5461 -0.5061

1.6180 1.7214 1.7576 1.8280 1.8653 1.9377 1.8310 1.9758

438.9 423 420 412.5 410.9 403.8 418 404

436.78 425.24 421.51 414.70 411.37 405.09 414.42 401.98

a

BDE(Ci-Cj)exp

Sij

Taken from ref 15. b Calculated by eq 19.

Table 7. Eigenvalues X1 for Ci-Cj and Ci-H and Sij Values for Ci-Cj in 2,2,5-Trimethylheptane bond

X1a

C1-H C3-H C4-H C5-H C6-H C7-H C10-H C1-C2 C2-C3 C3-C4 C4-C5 C5-C6 C6-C7 C5-C10

-0.5322 -0.4932 -0.4962 -0.4839 -0.5119 -0.5478 -0.5372 0.2304 0.3318 0.2725 0.2936 0.2593 0.1393 0.1970

Sij

BDEexpb (kJ/mol)

BDEcalcc (kJ/mol)

410.0 397.5

409.56 397.47 398.40 394.59 403.27 414.40 411.11 337.07 315.76 330.22 324.78 330.36 353.09 343.20

384.9

16 64 16 36 36 4 9

342.3 325.1 342.3 336.4 354.8 352.3

a For C -H bond, X ) X b i 1 1CH. For Ci-Cj bond, X1 ) X1CC. Taken from ref 17. c For Ci-H bond, calculated by eq 19. For Ci-Cj bond, calculated by eq 18.

Figure 2. Molecule 2,2,5-trimethylheptane (non-hydrogen atom).

bond, we can get 2,2,5-trimethylheptyl •CH2CMe2CH2CH2CHMeCH2CH3 (its PEI ) 1.3466) and a hydrogen atom H. Then construct its modified connection-matrix (CM1H,m) by means of eq 12. The modified connection-matrix (CM1H,m) of C1-H bond for 2,2,5-trimethylheptane is

CM1H, m )

[

1.3466 1 1 0

]

(21)

By solving the matrix CM1H,m (eq 21), the eigenvalue X1CH (-0.5322) is obtained. When cutting the C3-C4 bond, we can get 2,2-dimethylpropyl •CH2CMe2CH3 (its PEI ) 1.2849) and 2-methylbutyl •CH2CHMeCH2CH3 (its PEI ) 1.2603) and further construct its modified connection-matrix (CM34,m) by means of eq 9. The modified connection-matrix (CM34,m) of C3-C4 bond for 2,2,5-trimethylheptane is

CM34,m )

[

1.2849 1 1 1.2603

]

(22)

From eq 22, the eigenvalue X1CC, 0.2725 is calculated. Using the similar steps as above, we get different X1CH value for different Ci-H bond and different X1CC value for different Ci-Cj bond of 2,2,5-trimethylheptane molecule. These eigenvalues (X1) together with the steric effect parameter Sij values of Ci-Cj bonds all are listed in Table 7. Based on eqs 19 and 18, we evaluate BDE of each C-H and C-C bond for the 2,2,5-trimethylheptane molecule with

Figure 3. The plot of calculated BDE vs experimental BDE of C-H and C-C bond for 2,2,5-trimethylheptane.

their X1CH and X1CC values, respectively. The calculated results are listed in Table 7, and the plot of calculated BDE versus experimental BDE is shown by Figure 3, which illustrates that even small changes in the BDE of C-H and C-C bonds can be reproduced and the calculated BDE order is in good accordance with the experimental ones. We also noted that there are several discrepancies more than 10 kJ/ mol, which cannot be addressed better. Evaluation of Heats of Atomization for Alkanes. For an alkane molecule, its heat of atomization (HA) must be related to the bond dissociation energies of C-H and C-C bonds, which consist in the molecule. Of course it cannot be the simple sum of the bond dissociation energies of C-H

EVALUATING BOND DISSOCIATION ENERGY

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Table 8. Experimental and Estimated Heats of Atomization for 45 Alkanes no.

alkanesa

∑X1CC

∑X1CH

NC-C

NC-H

∑Sij

HAexpb (kJ/mol)

HAcalcc (kJ/mol)

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 33 34 35 36 37 38 39 40 41 42 43 44 45

1 3 4 2m3 5 2m4 22mm3 6 2m5 3m5 22mm4 23mm4 7 2m6 3m6 22mm5 23mm5 24mm5 33mm5 3e5 223mmm4 8 2m7 3m7 4m7 22mm6 23mm6 24mm6 25mm6 33mm6 34mm6 3e6 223mmm5 224mmm5 233mmm5 234mmm5 2m3e5 3m3e5 2233mmmm4 9 2233mmmm5 2234mmmm5 2244 mmmm 2334mmmm5 33ee5

0.0000 0.1356 0.3203 0.3921 0.5296 0.6219 0.7552 0.7533 0.8540 0.8753 1.0264 0.9654 0.9866 1.0914 1.1214 1.2794 1.2403 1.1998 1.3200 1.1517 1.4076 1.2266 1.3335 1.3679 1.3765 1.5296 1.499 1.4798 1.4418 1.5865 1.5284 1.4112 1.7023 1.6445 1.7215 1.6247 1.5374 1.6351 1.8843 1.4718 2.2165 2.1053 2.1076 2.1414 1.9716

-2.472 -4.5074 -5.5244 -5.5327 -6.5340 -6.5407 -6.5532 -7.5404 -7.5453 -7.5440 -7.5706 -7.5496 -8.5424 -8.5469 -8.5434 -8.5515 -8.5454 -8.5486 -8.5480 -8.5400 -8.5526 -9.5424 -9.5454 -9.5414 -9.5392 -9.5480 -9.5407 -9.5432 -9.5482 -9.5405 -9.5370 -9.535 -9.5556 -9.5487 -9.5399 -9.5409 -9.5357 -9.5358 -9.5454 -10.5392 -10.5260 -10.5301 -10.5398 -10.5252 -10.5152

0 2 3 3 4 4 4 5 5 5 5 5 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8

4 8 10 10 12 12 12 14 14 14 14 14 16 16 16 16 16 16 16 16 16 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 20 20 20 20 20 20

0 8 24 27 40 58 64 56 74 89 116 117 72 90 105 132 148 108 168 120 210 88 106 121 121 148 164 139 124 184 179 136 241 166 262 207 179 220 352 104 404 300 224 356 272

1663.1 4000.45 5176.18 5184.56 6349.91 6357.95 6369.46 7524.10 7531.22 7528.54 7542.48 7534.69 8698.16 8705.32 8702.68 8716.54 8709.63 8712.39 8711.94 8700.04 8715.20 9872.22 9879.26 9876.41 9875.87 9888.51 9877.71 9883.19 9886.42 9883.90 9876.77 9874.65 9883.90 9887.92 9880.22 9881.22 9874.99 9878.75 9889.68 11046.24 11054.45 11054.20 11059.18 11053.45 11049.17

1660.14 4002.43 5178.07 5184.04 6352.44 6357.78 6367.86 7526.45 7531.59 7531.09 7545.16 7535.22 8699.46 8704.66 8703.85 8712.14 8707.08 8709.11 8710.30 8703.09 8712.97 9872.13 9876.94 9876.17 9875.86 9884.14 9879.09 9880.82 9881.75 9881.76 9878.18 9875.04 9888.16 9888.57 9882.36 9882.05 9878.19 9879.92 9885.42 11043.99 11053.47 11057.07 11065.93 11053.10 11047.77

a m, methyl; e, ethyl; for example, 3m3e5 represents 3-methyl-3-ethylpentane. b All but for methane were taken from ref 27. The HA of methane is taken from ref 11. c Calculated by eq 26.

and C-C bonds calculated by the above approaches. For example, methane CH4 has four C-H bonds, and its process of atomization can be considered as the following: Step by step:

BDE of a real H-CH3 bond but is proportional to the BDE of a real H-CH3 bond. From the view of simultaneous process, alkane with formula CnH2n+2 can be considered as consisting of n-1 quasi-Ci-Cj and 2n+2 quasi-Ci-H bonds. Because that the BDE of Ci-Cj and Ci-H bond can be calculated by eqs 18 and 19, respectively, here we propose a model eq 25 to evaluate the HA of alkane.

HA ) d1NC-C + d2∑X1CC + d3∑Sij +

Or simultaneous:

CH4 f C + 4H

(24)

If eq 23 is employed to calculate its heat of atomization (HA), the HA will be the sum of the BDEs of the H-CH3, H-CH2, H-CH, and H-C bonds. In this case, the approach of this work cannot be used for calculating the HA. When eq 24 is employed to estimate the heat of atomization (HA), the HA may be considered as the sum of the BDEs of four quasi-H-CH3 bonds, whose BDE is not just equal to the

d4NC-H + d5∑X1CH (25)

Here di is the regression coefficients, NC-C and NC-H are the number of C-C and C-H bonds, which equal n-1 and 2n+2, respectively. In the right of eq 25, the first term together with the second term express the contribution of C-C bonds to the HA. The third term is the contribution of the steric effect to the HA, and the fourth term together with the fifth term indicate the contribution of C-H bonds to the HA. Using eq 25 and the experimental HA (see

606 J. Chem. Inf. Comput. Sci., Vol. 43, No. 2, 2003

CAO

AND

YUAN

Table 9. Estimated Enthalpy of Formation in Gas (HFG0) of Some Alkanes no.

alkanes

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 33 34 35 36 37 38 39 40 41 42 43 44 45 46

1 2 3 4 2m3 5 2m4 22mm3 6 2m5 3m5 22mm4 23mm4 7 2m6 3m6 22mm5 23mm5 24mm5 3 mm5 3e5 223mmm4 8 2m7 3m7 4m7 22mm6 23mm6 2 mm6 25mm6 33mm6 34mm6 3e6 223mmm5 224mmm5 233mmm5 234mmm5 2m3e5 3m3e5 2233mmmm4 9 2233mmmm5 2234mmmm5 2244mmmm5 2334mmmm5 33ee5

no. of no. of C H 1 2 3 4 4 5 5 5 6 6 6 6 6 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9

4 6 8 10 10 12 12 12 14 14 14 14 14 16 16 16 16 16 16 16 16 16 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 20 20 20 20 20 20

HFG0expa (kJ/mol) 74.85(74.4) 84.68(83.8) 103.85(104.7) 126.15(125.7) 134.52(134.2) 146.44(146.9) 154.47(153.6) 168.49(168.0) 167.19(166.9) 174.31(174.6) 171.63(171.9) 185.56(185.9) 177.78(178.1) 187.82 194.97 192.30 206.23 195.18 202.09 201.54 189.70 204.85 208.45 215.48 212.63 212.09 224.72 214.05 219.41 222.63 220.12 213.01 210.87 220.12 224.14 216.44 217.44 211.21 214.97 225.89 228.20 238.78 237.69 243.34 237.11 233.51

HFG0calcb HFG0calcc (kJ/mol) (kJ/mol) 71.47 85.48 108.40 131.37 137.34 153.07 158.40 168.49 174.40 179.53 179.03 193.11 183.16 194.73 199.93 199.12 207.41 202.35 204.38 205.57 198.35 208.24 214.72 219.53 218.76 218.45 226.73 221.68 223.41 224.34 224.36 220.78 217.63 230.75 231.16 224.95 224.64 220.78 222.51 228.02 233.91 243.39 246.99 255.85 243.02 237.69

72.25 84.61 106.20 128.02 134.13 148.73 154.20 164.58 169.16 174.49 173.99 188.14 178.21 188.67 194.08 193.33 201.91 196.70 198.78 200.05 192.62 202.73 207.89 212.93 212.24 211.98 220.54 215.37 217.15 217.97 218.23 214.52 211.24 224.54 225.25 218.85 218.52 214.57 216.40 221.91 226.35 236.84 240.60 249.78 236.49 231.16

a Taken from ref 27. The values enclosed in parentheses are taken from ref 13 and not be used for regression analysis in eq 29. b Calculated by eq 28. c Calculated by eq 29.

Table 8),27 we obtained the multiple regression equation as follows.

HA (kJ/mol) ) 410.338((7.966)Nc-c + 47.946((5.446)∑X1CC - 0.0729((0.0155)∑Sij +

210.575((20.198)NC-H - 330.841((32.251)∑X1CH (26) r ) 0.999999, s ) 2.64, F ) 5106213, n ) 45

There is an excellent correlation in eq 26, whose correlation coefficient r is 0.999999, and the standard deviation s is only 2.64(kJ/mol). The largest absolute error between the experimental and the calculated HA is 6.75 kJ/mol. S. S. Liu and co-workers27 estimated the HA of alkanes in Table 8 with molecular electronegative distance

Figure 4. The plot of calculated HFG0 by eq 28 vs experimental HFG0 for alkanes (HFG0exp ) 1.784 + 0.964HFG0calc r ) 0.9979, s ) 2.58, F ) 10476.65, n ) 46).

vector (MEDV) (10 variables) and got a result r ) 0.9998, s ) 29.98. The result of this paper is much better than that of the literature.27 Furthermore, eq 26 has less parameters and clearer physical meaning. In Liu’s work,27 all of -CH3 groups in alkane molecule are treated as the same, and their chemical environments are ignored. So are the groups >CH2, -CHC