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Original article
10 (
4
); 439-447
doi:
10.1016/j.arabjc.2013.11.016

Quantitative structure–property relationship study of standard formation enthalpies of acyclic alkanes using atom-type-based AI topological indices

,
a Department of Chemistry, Rasht Branch, Islamic Azad University, Rasht, Iran
b Department of Chemistry, Faculty of Science, University of Guilan, P.O. Box 1914, Rasht, Iran

⁎Corresponding author. Tel.: +98 131 4224080; fax: +98 131 4223621. Safa@iaurasht.ac.ir (Fariba Safa) Safa_f@yahoo.com (Fariba Safa)

Received: , Accepted: ,
© The Author(s) 2013
Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

A quantitative structure–property relationship (QSPR) study was performed for prediction of enthalpies of 134 acyclic alkanes using modified Xu (mXu) index and atom-type-based AI topological indices. At first, a simple linear regression model was developed using mXu index alone and the statistics were R2 = 0.947, F = 2335 and standard error of 1.00. The results showed that combination of the atom-type-based AI topological indices and mXu index can produce significant improvement in the statistical quality of the model, especially the decrease in the standard error was 33% relative to the simple linear model. The final model was validated to be statistically significant and reliable using external validation technique. External validation was performed by dividing the entire data set into three subsets and predicting enthalpy values for each subset from the other two as training sets. Average standard error of calibration of 0.66 and average standard error of prediction of 0.68 demonstrated the validity and good efficiency of the topological indices in modeling enthalpies of alkanes. The obtained results showed that the enthalpy for acyclic alkanes is dominated by molecular size and the atomic groups are also important although their contributions are much smaller than that of the molecular size.

Keywords

Quantitative structure–property relationship
Standard formation enthalpy
Acyclic alkanes
Atom-type-based topological indices
Multiple linear regression
1

1 Introduction

Graph-theoretical approach represents simple and efficient means to QSPR studies. Graph-theoretical topological indices are high potential descriptors for modeling and predicting physicochemical properties of chemical compounds. Different types of topological indices have been proposed since the pioneering work of Wiener (1947). Conventional topological indices such as the well-known molecular connectivity index (Kier and Hall, 1976) characterize a molecule as a whole and do not take into account the separate contributions of individual molecular fragment and atomic groups to properties. This causes some difficulties in developing high quality QSPR models. Another type of topological indices is atomic level topological indices that are particularly of interest because they code the structural environment of each atom type in a molecule and describe the structural information of a molecule at the atomic level. Kier et al. (1991) first introduced the concept of atom-type-based topological indices, i.e. the so-called electrotopological state index. Ren (1999) proposed a different set of atomic-based AI topological indices that along with modified Xu index (Ren et al., 1999) were successfully used for QSPR studies (Ren, 2002a–d, 2003). New set of atomic level topological indices proposed by Lu et al. (2006a) were also used for modeling different physicochemical properties (Lu et al., 2006b,c).

The enthalpy of organic compounds as a physicochemical parameter is of great importance in chemical engineering and chemical reactions. In past years, different models for prediction of alkane enthalpies have been established using multiple linear regression (MLR) (Toropov et al., 2004) and neural networks (Gakh et al., 1994; Ivanciuc, 1999; Yao et al., 2001; Zhang, 2009). However, the use of atomic level topological indices in modeling enthalpy has been largely neglected.

The main aim of this study is to illustrate the usefulness of atom-type based AI topological indices in QSPR study of the enthalpy of alkanes. The effects of structural features on enthalpies of the model compounds are also illustrated. As far as we are aware, this is the first QSPR study for prediction of alkanes enthalpies using atom-type-based topological indices.

2

2 Materials and methods

2.1

2.1 Data set

The experimental data of the standard formation enthalpies for acyclic alkanes at 300 K were taken from the literature (Ivanciuc, 1999). The data set contains 134 alkanes including C6–C10 linear and branched alkanes (Table 1). The enthalpy values of the model compounds fall in the range of 24.77–45.31 kJ/mol.

Table 1 Experimental and calculated values of standard formation enthalpies of acyclic alkanes (ΔHf) at 300 K.
No. Compound Experimental ΔHf (kJ/mol) Calculated ΔHf (kJ/mol) Relative error (%)
1 3-Methylpentane 26.32 25.99 −1.25
2 2,2-Dimethylbutane 25.40 24.95 −1.77
3 2,3-Dimethylbutane 24.77 24.80 0.12
4 3-Methylhexane 30.71 30.99 0.91
5 3-Ethylpentane 31.71 30.68 −3.25
6 2,2-Dimethylpentane 29.50 29.65 0.51
7 2,3-Dimethylpentane 28.62 29.41 2.76
8 2,4-Dimethylpentane 29.58 29.46 −0.41
9 3,3-Dimethylpentane 29.33 29.41 0.27
10 2,2,3-Trimethylbutane 28.28 28.07 −0.74
11 n-Octane 38.12 37.54 −1.52
12 2-Methylheptane 35.82 35.72 −0.28
13 3-Methylheptane 35.31 35.67 1.02
14 2,4-Dimethylhexane 33.76 33.91 0.44
15 2,5-Dimethylhexane 33.39 33.91 1.56
16 3,3-Dimethylhexane 33.43 34.03 1.79
17 3,4-Dimethylhexane 32.47 33.89 4.37
18 3-Ethyl-2-methylpentane 34.31 33.68 −1.84
19 3-Ethyl-3-methylpentane 33.26 33.68 1.26
20 2,2,3-Trimethylpentane 32.13 32.52 1.21
21 2,2,4-Trimethylpentane 32.55 32.49 −0.18
22 2,3,3-Trimethylpentane 32.17 32.43 0.81
23 2,3,4-Trimethylpentane 32.55 32.36 −0.58
24 2,2,3,3-Tetramethylbutane 31.84 31.15 −2.17
25 2-Methyloctane 40.42 40.18 −0.59
26 3-Methyloctane 39.92 40.21 0.73
27 3-Ethylheptane 40.71 39.85 −2.11
28 4-Ethylheptane 40.50 39.73 −1.90
29 2,2-Dimethylheptane 38.83 38.64 −0.49
30 2,3-Dimethylheptane 37.82 38.48 1.75
31 2,4-Dimethylheptane 38.16 38.35 0.50
32 2,5-Dimethylheptane 37.53 38.25 1.92
33 2,6-Dimethylheptane 37.99 38.18 0.50
34 3,3-Dimethylheptane 38.20 38.55 0.92
35 3,4-Dimethylheptane 37.02 38.39 3.70
36 3,5-Dimethylheptane 38.07 38.30 0.60
37 3-Ethyl-3-methylhexane 37.36 38.13 2.06
38 4-Ethyl-2-methylhexane 39.25 38.01 −3.16
39 2,2,4-Trimethylhexane 36.61 36.78 0.46
40 2,2,5-Trimethylhexane 36.36 36.69 0.91
41 2,3,3-Trimethylhexane 36.28 36.88 1.65
42 2,3,4-Trimethylhexane 36.86 36.67 −0.52
43 2,3,5-Trimethylhexane 36.02 36.60 1.61
44 2,4,4-Trimethylhexane 36.44 36.70 0.71
45 3,3,4-Trimethylhexane 35.98 36.75 2.14
46 3,3-Diethylpentane 38.37 37.72 −1.69
47 3-Ethyl-2,2-dimethylpentane 36.15 36.62 1.30
48 3-Ethyl-2,3-dimethylpentane 36.61 36.51 −0.27
49 2,2,3,3-Tetramethylpentane 35.86 35.36 −1.39
50 2,2,3,4-Tetramethylpentane 35.06 35.23 0.48
51 2,3,3,4-Tetramethylpentane 36.23 35.22 −2.79
52 3-Ethyloctane 45.31 44.31 −2.21
53 4-Ethyloctane 45.10 44.16 −2.08
54 2,2-Dimethyloctane 43.43 42.85 −1.34
55 2,4-Dimethyloctane 42.76 42.68 −0.19
56 2,5-Dimethyloctane 41.92 42.56 1.53
57 3,4-Dimethyloctane 41.80 42.78 2.34
58 3,5-Dimethyloctane 42.47 42.67 0.47
59 3,6-Dimethyloctane 41.63 42.55 2.21
60 3,3-Dimethyloctane 42.30 42.90 1.42
61 4,5-Dimethyloctane 41.51 42.77 3.04
62 4-n-Propylheptane 44.85 43.99 −1.92
63 4-Isopropylheptane 43.10 42.34 −1.76
64 2-Methyl-3-ethylheptane 43.30 42.46 −1.94
65 2-Methyl-4-ethylheptane 43.64 42.23 −3.23
66 3-Methyl-4-ethylheptane 42.47 42.29 −0.42
67 3-Methyl-5-ethylheptane 43.35 42.31 −2.40
68 2,2,3-Trimethylheptane 41.30 41.26 −0.10
69 2,2,4-Trimethylheptane 41.05 41.04 −0.02
70 2,2,5-Trimethylheptane 40.50 40.84 0.84
71 2,2,6-Trimethylheptane 40.96 40.68 −0.68
72 2,3,3-Trimethylheptane 41.00 41.23 0.56
73 2,3,4-Trimethylheptane 40.96 40.98 0.05
74 2,3,5-Trimethylheptane 40.12 40.81 1.72
75 2,3,6-Trimethylheptane 39.75 40.67 2.31
76 2,4,4-Trimethylheptane 40.54 40.99 1.11
77 2,4,5-Trimethylheptane 39.98 40.78 2.00
78 2,4,6-Trimethylheptane 41.13 40.66 −1.14
79 2,5,5-Trimethylheptane 40.54 40.81 0.67
80 3,3,5-Trimethylheptane 40.46 40.94 1.19
81 3,4,4-Trimethylheptane 40.08 41.09 2.52
82 3,4,5-Trimethylheptane 41.14 40.89 −0.61
83 2-Methyl-3-isopropylhexane 40.46 40.68 0.54
84 3,3-Diethylhexane 42.43 42.01 −0.99
85 3,4-Diethylhexane 43.68 42.01 −3.82
86 2,2-Dimethyl-3-ethylhexane 40.50 40.86 0.89
87 2,2-Dimethyl-4-ethylhexane 41.92 40.70 −2.91
88 2,3-Dimethyl-3-ethylhexane 40.71 40.79 0.20
89 2,3-Dimethyl-4-ethylhexane 42.43 40.62 −4.27
90 2,4-Dimethyl-4-ethylhexane 40.29 40.63 0.84
91 3,3-Dimethyl-4-ethylhexane 39.92 40.73 2.03
92 3,4-Dimethyl-4-ethylhexane 40.42 40.71 0.72
93 2,2,3,3-Tetramethylhexane 40.00 39.67 −0.83
94 2,2,3,4-Tetramethylhexane 39.25 39.39 0.36
95 2,2,3,5-Tetramethylhexane 39.54 39.26 −0.71
96 2,2,4,5-Tetramethylhexane 38.83 39.17 0.88
97 2,2,5,5-Tetramethylhexane 39.37 39.14 −0.58
98 2,3,3,4-Tetramethylhexane 40.04 39.40 −1.60
99 2,3,3,5-Tetramethylhexane 39.16 39.28 0.31
100 2,3,4,4-Tetramethylhexane 38.87 39.31 1.13
101 2,3,4,5-Tetramethylhexane 40.71 39.16 −3.81
102 3,3,4,4-Tetramethylhexane 39.87 39.46 −1.03
103 2,4-Dimethyl-3-isopropylpentane 39.25 39.01 −0.61
104 2-Methyl-3,3-diethylpentane 39.25 40.39 2.90
105 2,2,3-Trimethyl-3-ethylpentane 38.41 39.29 2.29
106 2,2,4-Trimethyl-3-ethylpentane 38.87 39.16 0.75
107 2,3,4-Trimethyl-3-ethylpentane 38.58 39.14 1.45
108 2,2,3,3,4-Pentamethylpentane 38.62 37.95 −1.73
109 2,2,3,4,4-pentamethylpentane 38.81 37.88 −2.40
110 2-Methylpentane 26.61 26.32 −1.09
111 n-Heptane 33.56 32.78 −2.32
112 2-Methylhexane 31.21 31.01 −0.64
113 4-Methylheptane 35.06 35.66 1.71
114 3-Ethylhexane 36.07 35.29 −2.16
115 2,2-Dimethylhexane 34.23 34.25 0.06
116 2,3-Dimethylhexane 33.05 34.02 2.93
117 4-Methyloctane 39.71 40.22 1.28
118 4,4-Dimethylheptane 37.53 38.51 2.61
119 3-Ethyl-2-methylhexane 38.70 38.09 −1.58
120 3-Ethyl-4-methylhexane 38.07 38.01 −0.16
121 2,2,3-Trimethylhexane 36.61 36.97 0.98
122 3-Ethyl-2,4-dimethylpentane 36.07 36.44 1.03
123 2,2,4,4-Tetramethylpentane 36.44 35.27 −3.21
124 2,3-Dimethyloctane 42.43 42.79 0.85
125 2,6-Dimethyloctane 42.09 42.42 0.78
126 2,7-Dimethyloctane 42.58 42.29 −0.68
127 3,3-Dimethyloctane 42.80 42.90 0.23
128 2-Methyl-5-ethylheptane 42.93 42.21 −1.68
129 3-Methyl-3-ethylheptane 42.13 42.55 1.00
130 4-Methyl-3-ethylheptane 42.51 42.41 −0.24
131 4-Methyl-4-ethylheptane 41.46 42.42 2.32
132 3,3,4-Trimethylheptane 40.46 41.11 1.61
133 2,5-Dimethyl-3-ethylhexane 41.34 40.51 −2.01
134 2,2,4,4-Tetramethylhexane 40.29 39.34 −2.36

2.2

2.2 Descriptor generation

As known, each molecule may be represented by a topological graph G = {V,E} where V(G) and E(G) are the vertex and edge sets, respectively. Vertices correspond to individual atoms in the graph and the edges correspond to chemical bonds between them. The vertex–adjacency matrix, A = [aij]n × n, is a square symmetric matrix whose elements are 1 if vertices i and j are adjacent and 0 otherwise. The distance matrix, D = [dij]n × n, is also a square symmetric matrix whose entries are the length of the shortest path between the vertices i and j in the molecular graph. The sum over row i or column j of matrix A yields local vertex-degree vi and the sum for matrix D yields distance sums si. For a simple molecular graph, Xu index is defined as follows (Ren, 1999):

(1)
Xu = n 1 / 2 log ∑ i = 1 n v i s i 2 ∑ i = 1 n v i s i where the sum is over all i vertices in graph G and n is the number of vertices.

For any atom i that belongs to jth atom type in the graph, corresponding topological index is defined as (Ren, 2002a),

(2)
AI i ( j ) = 1 + ϕ i ( j )
(3)
ϕ i ( j ) = v i ( j ) s i 2 ( j ) ∑ i = 1 n v i s i where the parameter ϕi(j) is considered as a perturbing term reflecting the effect of the structural environment of the ith atom on its topological index value, and the sum is over all i vertices in G. According to this definition, for jth atom-type in a graph, the corresponding atom-type topological index, AI(j), is the sum of all AIi(j) values of the same atom type in the molecular graph G: AI ( j ) = ∑ i = 1 m AI i ( j ) = m + ∑ i = 1 l ϕ i ( j )
(4)
= m + ∑ i = 1 m v i ( j ) s i 2 ( j ) ∑ i = 1 n v i s i where m is the count of atoms of the same type. Clearly, the AI(j) value is equal to the count of jth atom-type plus total perturbation terms and is related to its structural environment. For differentiation of the heteroatoms and multiple bonds in calculating AI indices, new degree of vertex, vm (Ren, 2002a) based on the valence connectivity of Kier–Hall (Kier and Hall, 1986) can be successfully used.

2.3

2.3 Regression analysis and model validation

In the present work, linear regression analyses were performed using SPSS/PC software package (version 17.0). Criteria for selection of the best multiple linear regression model were the statistics: squared multiple correlation coefficient (R2), adjusted correlation coefficient (R2adj), Fisher-ratio (F) and standard error of estimate (SE). The validity and stability of the model obtained was tested using external validation technique by a procedure in which the entire data set was divided into three subsets and the enthalpy values for each subset as prediction set were predicted by the other two subsets as a training set. Then, standard error of calibration (SEC) and standard error of prediction (SEP) were used for evaluating quality of the MLR model (Kramer, 1998).

3

3 Results and discussion

3.1

3.1 Linear regression models for enthalpy of alkanes

After calculation of mXu and atom-type based AI indices (Table 2), structure–property relationship models were generated. At first, a simple linear model was developed using mXu index alone. Specification of the model found for the entire data set along with the statistical parameters is given as follows:

(5)
Δ H f = 7.002 ( ± 0.649 ) + 7.793 ( ± 0.161 ) m Xu n = 134 R 2 = 0.947 R adj 2 = 0.946 SE = 1.00 F = 2335 It can be seen that the mXu index shows high correlation with alkane enthalpies. However, the index cannot give very accurate correlation with ΔHf data judging from the standard error. Therefore, to obtain accurate model and to account the contribution of atom types or groups to the enthalpy data of the model compound, multiple linear regression analysis using mXu and all the atom-type AI indices of the molecules was performed to choose a high quality subset of indices for development of the model. Table 3 shows specification of the best MLR model found for alkanes enthalpies. It may be easily found that combination of mXu and AI indices significantly improves quality of the model. The statistics show that the regression explained by the model is significant at 99% confidence level. The model explains more than 97% of the variances in the experimental values of alkanes enthalpies with a standard error of only 0.67. As shown, the standard error is reduced by 33% relative to the simple linear model with mXu index and the error is 1.76% of the mean enthalpy which is approaching the experimental error in determining enthalpy. Values of relative error for predicted enthalpies (Table 1) show that maximum relative error was 4.37% for 3,4-dimethylhexane. Graphical indication of the quality of the MLR model can be seen in Figs. 1 and 2. The regression line in Fig. 1. shows good agreement of the calculated slope and intercept with the ideal values of 1 and 0, respectively. The plot of the residuals versus experimental values of enthalpy (Fig. 2.) falls within a horizontal band centered around zero showing absence of systematic error in development of the proposed model. The results evidently demonstrate the efficiency of AI indices in structure–enthalpy correlation of alkanes.
Table 2 Values of mXu and atom-type-based AI indices employed in multiple linear regression model of alkanes enthalpies.
No. mXu AI(–CH3) AI(>CH–) AI(>C<)
1 2.4381 7.0980 2.3130 0
2 2.3052 9.6280 0 2.7560
3 2.3429 9.6280 5.4180 0
4 3.0167 8.7600 2.8990 0
5 2.9613 8.7810 2.6200 0
6 2.9015 10.4220 0 3.2820
7 2.8963 10.5850 5.8240 0
8 2.9472 10.8280 6.8400 0
9 2.8423 10.3420 0 2.9100
10 2.7652 12.1350 2.9290 3.0320
11 3.7280 7.6000 0 0
12 3.6351 9.8740 4.3350 0
13 3.5852 9.6620 3.3710 0
14 3.4783 11.7710 7.1850 0
15 3.5321 12.0680 8.4000 0
16 3.4079 11.3010 0 3.2830
17 3.4227 11.4820 6.0000 0
18 3.3992 11.5660 6.1040 0
19 3.3395 11.2800 0 3.0000
20 3.3071 13.1460 2.8520 3.4690
21 3.3668 13.5200 3.8270 3.7690
22 3.2848 13.1250 3.2500 3.0830
23 3.3464 13.4970 9.7490 0
24 3.1762 14.7300 0 6.5460
25 4.1884 10.8780 5.1330 0
26 4.1395 10.5890 3.9430 0
27 4.0586 10.5350 3.2330 0
28 4.0315 10.5280 3.0090 0
29 4.0532 12.6170 0 4.7670
30 4.0259 12.6460 7.5090 0
31 4.0230 12.7740 7.8570 0
32 4.0451 12.9910 8.6720 0
33 4.0977 13.3440 10.0660 0
34 3.9713 12.3070 0 3.8130
35 3.9682 12.4290 6.5090 0
36 3.9906 12.6400 7.2860 0
37 3.8764 12.2220 0 3.2840
38 3.9589 12.7820 7.4930 0
39 3.8910 14.5390 3.5260 4.3680
40 3.9495 14.9560 4.7500 4.6130
41 3.8382 14.1740 3.6670 3.3470
42 3.8620 14.4800 10.1950 0
43 3.9213 14.8910 11.7490 0
44 3.8620 14.4860 4.2840 3.6490
45 3.8005 14.0790 3.1000 3.4140
46 3.8045 12.2280 0 3.0570
47 3.7965 14.1890 2.8110 3.8000
48 3.7637 14.1210 3.4820 3.1180
49 3.6841 15.7590 0 6.8910
50 3.7508 16.2240 6.6880 3.8820
51 3.7189 16.0660 7.1140 3.1820
52 4.5988 11.4310 3.7000 0
53 4.5590 11.3940 3.2850 0
54 4.6034 13.7900 0 5.6640
55 4.5635 13.8230 8.7530 0
56 4.5700 13.9830 9.2820 0
57 4.5116 13.4240 7.2540 0
58 4.5178 13.5760 7.7400 0
59 4.5439 13.8300 8.6960 0
60 4.5231 13.3590 0 4.4630
61 4.4911 13.3240 6.8140 0
62 4.5257 11.4150 3.1040 0.
63 4.4398 13.5220 7.0460 0
64 4.4745 13.5700 7.4350 0
65 4.4698 13.7940 8.0530 0
66 4.4145 13.3990 6.5770 0
67 4.4550 13.6020 7.4430 0
68 4.4265 15.3990 3.2600 4.7210
69 4.4308 15.6480 3.4950 5.0650
70 4.4574 15.9780 4.2560 5.3410
71 4.5128 16.4440 5.6930 5.5410
72 4.3931 15.2890 4.1650 3.7660
73 4.3984 15.5410 11.0470 0
74 4.4257 15.8690 12.2000 0
75 4.4820 16.3310 13.9160 0
76 4.3877 15.5190 4.7970 3.7660
77 4.4163 15.8450 12.0700 0
78 4.4724 16.3000 13.7410 0
79 4.4267 15.8870 5.3200 4.3270
80 4.3685 15.4250 3.8990 4.1300
81 4.3291 15.0700 3.4420 3.5730
82 4.3582 15.3920 10.3970 0
83 4.3459 15.5710 10.8040 0
84 4.3164 13.1690 0 3.2840
85 4.3604 13.3760 6.2300 0
86 4.3232 15.2600 2.9300 4.2560
87 4.3630 15.6280 3.3480 4.8650
88 4.2868 15.1480 3.8060 3.3320
89 4.3291 15.5210 10.6050 0
90 4.3184 15.4910 4.6480 3.5730
91 4.2684 15.0840 3.0100 3.6810
92 4.2566 15.0440 3.2940 3.3810
93 4.2298 16.8700 0 7.5560
94 4.2594 17.2690 6.5550 4.3930
95 4.3221 17.7760 8.1050 4.5920
96 4.3298 17.8920 7.9700 4.9410
97 4.3627 18.0720 0 10.4400
98 4.2230 17.0860 7.2700 3.4320
99 4.2867 17.5900 8.8000 3.6260
100 4.2330 17.2180 7.2620 3.7380
101 4.2958 17.7120 15.1180 0
102 4.1679 16.7280 0 7.0840
103 4.2421 17.5740 14.3810 0
104 4.2120 15.1200 3.6560 3.1420
105 4.1506 16.8010 0 7.1660
106 4.2161 17.2800 6.7510 4.1240
107 4.1779 17.0940 7.4300 3.1900
108 4.1118 18.8360 3.8420 7.2860
109 4.1498 19.0900 2.9070 8.5320
110 2.4752 7.9380 2.9590 0
111 3.1691 6.8460 0 0
112 3.0645 8.6220 3.6020 0
113 3.5699 9.6020 3.0780 0
114 3.5098 9.6550 2.8620 0
115 3.4859 11.4900 0 3.9650
116 3.4647 11.5890 6.5540 0
117 4.1175 10.4690 3.4080 0
118 3.9442 12.2180 0 3.5130
119 3.9344 12.5550 6.6640 0
120 3.9014 12.4380 6.1540 0
121 3.8692 14.2360 2.9860 4.0410
122 3.8267 14.5180 10.0940 0
123 3.7802 16.3740 0 8.4280
124 4.5741 13.7510 8.6350 0
125 4.5956 14.2420 10.2590 0
126 4.6457 14.6520 11.8180 0
127 4.5231 13.3590 0 4.4630
128 4.5098 14.0080 8.9870 0
129 4.4207 13.1930 0 3.7140
130 4.4276 13.3480 6.5200 0
131 4.3849 13.1490 0 3.4730
132 4.3408 15.1030 3.1570 3.8760
133 4.3769 15.9210 12.0360 0
134 4.2679 17.3950 0 8.7360
Table 3 Specification of the best MLR model for alkanes enthalpies along with the relative and fraction contributions of the topological indices.
Descriptor Coefficient Standard error Ψr Ψf (%)
Constant 4.819 ±0.600
mXu 8.011 ±0.184 31.95 76.21
AI(–CH3) 0.376 ±0.109 5.14 12.26
AI(>CH–) −0.445 ±0.064 −2.26 5.40
AI(>C<) −0.711 ±0.114 −1.56 3.73
Statistics
R2 0.976
R2adj 0.976
F 1326
SE 0.67
n 134
Plot of the predicted versus experimental enthalpies for 134 alkanes based on the developed MLR model.
Figure 1 Plot of the predicted versus experimental enthalpies for 134 alkanes based on the developed MLR model.
Plot of the residuals versus experimental enthalpies for 134 alkanes based on the developed MLR model.
Figure 2 Plot of the residuals versus experimental enthalpies for 134 alkanes based on the developed MLR model.

In order to verify the statistical validity of the MLR model and to demonstrate their utility to predict the enthalpy values for compounds outside the training set, external validation technique using the method used by Amboni et al. (2002) was employed. The procedure was conducted by dividing the entire data set into three subsets, leaving out one subset as prediction set, and regenerating the model coefficients for the training set composed of the other two subsets. Then the developed model was used for calculating the enthalpy values for the prediction set and SEC and SEP values were obtained. The procedure was repeated until the other subsets were used as prediction set. The results obtained are illustrated in Table 4. As shown, average values of SEC and SEP for different subsets were 0.66 and 0.68, respectively. Good agreement between the calculated and experimental enthalpies for the three prediction sets is also shown in Fig. 3. The results of external validation suggest that the MLR model can be used for predictive analysis of new observations for this class of compounds.

Table 4 Specification of the developed MLR models based on the external validation along with their statistical parameters.
Coefficients
Training subsets
1 and 2 1 and 3 2 and 3
Constant 5.453 4.741 4.457
mXu 8.002 7.956 8.108
AI (CH3) 0.304 0.412 0.373
AI (>CH–) −0.405 −0.468 −0.440
AI (>C<) −0.620 −0.746 −0.727
Statistics
R2 0.973 0.980 0.976
R2adj 0.972 0.979 0.975
F 770 1062 853
SEC 0.69 0.60 0.69
SEP 0.62 0.79 0.62
nta 89 89 90
npa 45 45 44
a nt and np are the number of molecules in the training and prediction sets, respectively.
Plot of the calculated versus experimental enthalpies for different prediction sets.
Figure 3 Plot of the calculated versus experimental enthalpies for different prediction sets.

3.2

3.2 Structural interpretation of alkanes enthalpies

In order to obtain insights into the role and importance of different structural features of alkanes in determining their enthalpies, the relative contribution (Ψr) and fraction contribution (Ψf) of the topological indices to ΔHf were calculated using the following equations (Needham et al., 1988):

(6)
Ψ r ( i ) = c i TI ‾ i
(7)
Ψ F ( i ) = R 2 × | Ψ r ( i ) | / ∑ i | Ψ r ( i ) | × 100 % where ci and TI ‾ i are the coefficient and the average value of the ith topological index entered in the model. The sum is over all indices in the model.

Table 3 shows the relative and fraction contributions of the individual topological indices to alkanes enthalpies. As may be easily seen, mXu index with the Ψf value of 76.21% is the most important descriptor appeared in the model. This finding shows bulkiness or size of alkane molecule plays a dominant role in determining its enthalpy because mXu index characterizes the molecular size (Ren et al., 1999). Positive Ψr value for this descriptor indicates that the larger the size of the molecule, the greater is the ΔHf value.

On the other hand, the obtained results indicate that alkanes enthalpies depend not only on the molecular size but also on various parts of the molecules. The AI indices included in the model had smaller contributions to the enthalpy data than mXu index and the AI indices decrease in the order of AI(–CH3) > AI(>CH–) > AI(>C<). Relatively large Ψf value of 12.26% for AI(–CH3) shows that the peripheral methyl groups make a larger contribution to enthalpy than the inside groups (>CH– and >C<) indicating that the branching is an important factor influencing alkane enthalpies, because AI(–CH3) index is clearly related to the number of terminal methyl groups which is a crude measure of branching (Needham et al., 1988). As expected, Ψr value for the descriptor is positive that indicates larger the number of the branches, the greater is the ΔHf value. The inside groups >CH– and >C< had minor contributions to the enthalpy values. Fraction contributions of the AI(>CH–) and AI(>C<) were 5.40% and 3.73%, respectively. Presence of these descriptors in the model indicates that the position of branching in the molecular structures or steric factor is also important to alkane enthalpies. Based on the results, the descriptors included in the developed model provided useful information about structural features important in determining alkane enthalpies.

4

4 Conclusion

In the present study, a good QSPR model for enthalpy of a group of acyclic alkanes was developed using a combination of mXu index and atom-type-based AI topological indices. The results showed that molecular size plays major role in determining alkane enthalpies and among the atomic groups, methyl group makes greater contribution to enthalpy than the inside groups (>CH– and >C<). Based on the obtained results, modified Xu and atom-type-base AI topological indices demonstrate high efficiency in model development for prediction of the enthalpies for alkanes. Good prediction quality of the proposed model allows the estimation of enthalpies for similar compounds using only the knowledge of two dimensional structures of the molecules.

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