Development of a new topological index for the prediction of normal boiling point temperatures of hydrocarbons: The Fi index

Journal of Molecular Liquids 165 (2012) 125–132

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Development of a new topological index for the prediction of normal boiling point temperatures of hydrocarbons: The Fi index Fernando C.G. Manso a, b, d, Hélio Scatena Júnior a, Roy E. Bruns c, Adley F. Rubira d, Edvani C. Muniz d,⁎ a

Universidade Federal do Rio Grande do Norte, Departamento de Química, UFRN, Natal-RN, Brazil Universidade Tecnológica Federal do Paraná, UTFPR, Campo Mourão-PR, Brazil Universidade Estadual de Campinas, Instituto de Química, Campinas-SP, Brazil d Grupo de Materiais Poliméricos e Compósitos – GMPC, Departamento de Química, Universidade Estadual de Maringá, UEM, 87020-900, Maringá-PR, Brazil b c

a r t i c l e

i n f o

Article history: Received 3 January 2011 Received in revised form 27 October 2011 Accepted 28 October 2011 Available online 12 November 2011 Keywords: Topology Molecular descriptors Normal boiling temperature Topological indexes

a b s t r a c t The goal of this paper is proposing a simple molecular descriptor, based on the molecular structures, for predicting the normal boiling temperature (B.T.) of hydrocarbons. To this end, the topological index Fi was developed and used to correlate the topology of alkanes, alkenes, alkynes and cycloalkanes possessing normal or branched chains to their B.T. The robustness of predictor Fi was evaluated by comparison with the most cited predictors in the literature: Weiner, Hosoya and Randić. The quadratic model developed in this work predicts very well the B.T. of hydrocarbons. In the first moment, the developed model was tested for predicting the B.T. of alkanes. After, it was applied with success for predicting the B.T. of other compounds (alkenes, alkynes and cycloalkanes). The topological index Fi proved to be rather effective and produced small deviations, as compared to other topological indexes used for comparison. Based on the topological index Fi, other properties of interest can also be further explored and the concepts developed in this work can be easily adapted to other families of compounds, mainly in liquid phase. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The use of molecular descriptors in quantitative structure–activity relationship analysis (QSAR) and in quantitative structure–property relationship (QSPR) targets the estimation of specific characteristics based on the structures of the compounds under study. Thus the main goal is simply to correlate, as accurately and unequivocally as possible, some important functionality to the molecular structure. To this end, topological indexes or molecular descriptors were proposed in QSPR studies based on Wiener's pioneering studies in the late 1940s [1]. Since then, some hundreds of molecular descriptors have been proposed for predicting the values of several physical properties, as well as the properties related to biological activities through QSAR (quantitative structure–activity relationships). Graph theory is used for building these topologies and new concepts are developed to transform such graphs, which are non-numerical mathematical objects, into related numbers that are topological indexes or molecular descriptors. Since the pioneering work of Harry Wiener [1] in the late 1940s, of Huruo Hosoya [2] and of Millan Randić [3] in the 1970s, and of many others, the study of quantitative correlations of molecular structure and properties has experienced many advances. Topological indexes have been successfully used for estimating several physical properties

⁎ Corresponding author. Tel.: + 55 44 3011 3664; fax: + 55 44 3011 4125. E-mail address: [email protected] (E.C. Muniz). 0167-7322/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2011.10.019

[4,5], including normal boiling temperature (B.T.) [6]. These indexes have been successfully used for estimating data related to biological activities as well. First, molecular descriptors (or topological indexes) that numerically represent the structural characteristics of molecules are obtained by the use of graph theoretical concepts applied to molecular structure. Next, these descriptors are correlated to the properties of interest for obtaining quantitative structure–property correlations. According to Mihalic and Trinajstic [7], a fundamental chemistry concept is that the structural characteristics of a molecule are responsible for its properties. Thus, it is possible to link the molecular structure to the molecular properties, and to quantify them through structure– property correlations, the so-called quantitative structure–property relationships (QSPR) have been used. The term topological (from topos in Greek, meaning place) originates from topology, a branch of pure mathematics that deals only with the properties of a figure X that hold for every figure into which X can be transformed with a one-to-one correspondence, that is continuous in both directions [8]. A topological property remains constant when an object is submitted to deformations without undergoing rupture. This is what happens if molecular structures are subjected to relaxation points of bonded atoms as a function of several vibrational movements. Thus, several molecular properties are, in fact, topological properties and, therefore, they are functions of their structures. Theoretical tools have played a fundamental role in the development of science over the years, and this is the function of molecular

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descriptors: to establish mathematical models capable of theoretically making good predictions of compound properties. Thus, in chemistry, which is an essentially experimental science, theoretical results provided by molecular descriptors can be used for predicting properties and/or planning and optimizing experiments, thus saving time and money. This is a key strategy in developing new materials, as well as those with properties related to biological activities. These arguments make the development of molecular descriptors, as done in this work, highly justified. The normal boiling temperature (B.T.) of a compound is dependent on intermolecular interactions existing in the liquid phase and on the difference in internal molecular partition functions between gaseous and liquid phases at its boiling temperature [9]. Therefore, the B.T. can be related to the chemical structure of the molecule. Several methods have been developed to estimate the boiling temperature at normal pressure (1 atm) of compounds based on their structures [10,11]. In 2009, Zenkevich [12] published an interesting paper in which a given nonlinear function of physical-chemistry parameter (A) was linearly modeled just by the use of a recurrent equation A (n + 1) = a A(n) + b, where n is the number of carbons of a compound in a homologous series; and a, b and A are constants that can be determined for a given homologous series. Using this approach, Zenkevich [12] obtained a linear plot of B.T. for a series of perfluoro-n-alkanes CnF2n + 2 and several other properties related to n parameter. Ramjugernath and co-workers [13] used group contributions and group interactions for predicting the normal boiling points and vapor pressure of non-electrolyte organic compounds. The method was developed with the aid of the Dortmund Data Bank [14], which contains over 180,000 data points for both solid and liquid vapor pressures. Group parameters were regressed to a training set of nearly 114,000 data points for more than 2330 compounds. As in the case of the method of Nannoolal et al. [15], the model proposed by Ramjugernath and co-workers [13] only requires knowledge of the molecular structure and a single vapor pressure point in order to generate the vapor pressure curves. The method of Nannoolal et al. [15], based on group contribution, is an excellent contribution to the field and was applied, with success, to 2850 components stored in the Dortmund Data Bank. Other properties can be calculated from inherent chemical parameters. For instance, values of the solute descriptors for 3-nitrobenzoic acid can be calculated from experimental solubility data for organic solvents using the Abraham general solvation model, and the obtained descriptors reproduced the experimental data with a standard deviation of only 0.082 log units [16]. The objective of present work is to develop a molecular descriptor based on simple concepts that are easily applied to hydrocarbons for estimating their normal boiling temperature (B.T.). To target this goal, the methodology for calculating the proposed topological index Fi is explained, the results obtained for predicting the B.T. of hundred alkanes through the model is analyzed and compared to other predictors, and finally applied to predicting B.T. of alkenes, alkynes and cycloalkanes. In spite of the existence of other similar studies [1– 3,7,13–16], the results obtained in this work are convincing for the use of Fi descriptor for predicting properties based on molecular structure of a wider collection of compounds.

index is a invariant graph, that is, the same graph cannot generate more than one numerical value. The graphs used to represent the chemical structures are called structural graphs, chemical graphs, molecular graphs, or constitutional graphs [7]. This paper uses connected structural graphs, which are graphs that have all pairs of points (vertexes) connected by a path, that is, by a sequence of adjacent points. The Methyl butane was used for exemplifying how the several molecular descriptors are determined. For simplifying the correlation of molecular descriptors to the Methyl butane structure, all hydrogen atoms of such molecule are suppressed in the representation of Methyl butane, being each carbon atom represented by a small circle (○) called graph vertex. Adjacent carbons (vertexes) are connected with a line, thus giving, for instance, the following molecular graph for Methyl butane

The characterization of a molecule with a numerical index results in a considerable loss of information. In this process, threedimensional objects, or molecules, are described by one-dimensional objects, the topological indexes. However, surprisingly, relevant structural information can be retained from the topological indexes. Thus, topological descriptors have been widely and successfully used in QSPR studies for estimating a large number of physical, chemical, and biological properties. According to Katritzky and Gordeeva [17], topological indexes developed by Wiener [1] (W), Hosoya [2] (Z), and Randić [3] ( 1χ) are among the best known and most used ones. To better compare the topological index Fi, proposed in this work, each of the above-mentioned indexes is defined and its application is briefly explained, taking Methyl butene as a molecule model. 1.1.1. The Wiener topological index (W) The Wiener topological index, W, is determined through equation W¼

. 1

dij 2∑ i;j

ð1Þ

where dij is an element of the atomic distance matrix of the molecular graph, quantified as the number of chemical bonds between atoms i and j. For Methyl butane: 2

0 61 6 D¼6 62 43 2

1 0 1 2 1

2 1 0 1 2

3 2 1 0 3

1

3 2 17 7 27 7 35 0

2

3

4

1.1. Molecular descriptors The essence of QSPR methods is to transform the chemical structure of a compound into numerical descriptors using graph theory elements and then establish quantitative relations using some mathematical operator, including these descriptors and the property to be optimized [17–21]. Molecular descriptors, such as topological descriptors, are some of these [1–3,22]. Mihalic and Trinajstic [7] defined topological descriptor as index term that characterizes a structure of a molecule by a single number, which means that a topological

5 therefore, W = ½ (36) = 18.0. So, 18.0 is the value of Wiener topological index (W) for the Methyl butane.

F.C.G. Manso et al. / Journal of Molecular Liquids 165 (2012) 125–132

1.1.2. The Hosoya topological index (Z) The Hosoya topological index, Z, is determined through equation Z¼

n=2 X

mðG; K Þ

ð2Þ

K¼0

where n is the number of vertexes, K the number of k2 (k2 being the graph with two independent adjacent vertexes), m is the number of paired K, and G stands for graph: by definition, m(G,0) = 1. For Methyl butane:

1

2

3

4

127

is invariably equal to 3, i.e., i + j = 3. Thus, j is a function of i as expressed by j ¼ 3–i:

ð4Þ

In the cases evolving alkanes, i corresponds to the number of hydrogens bonded to a given carbon. The following relationships can be obtained: Primary carbon: (Cp) → i = 3, j = 0, Secondary carbon: (Cs) → i = 2, j = 1, Tertiary carbon: (Ct) → i = 1, j = 2, Quaternary carbon: (Cq) → i = 0, j = 3, For methane (isolated carbon): i = 4, j = −1. Therefore, for the Methyl butane, the graph vertexes were associated with the following ordered pairs:

1

5

3,0

m(G,0) = 1, m(G,1) = 4, which are the (k2) graphs between: C1–C2, C2–C3, C3–C4, C2–C5, m(G,2) = 2, which are the (k2) graphs between C1–C2–C5, C2–C3–C4. Therefore, Z = 1.0 + 4.0 + 2.0 = 7. In this way, 7.0 is the value of Hosoya topological index (Z) for the Methyl butane. 1.1.3. The Randić topological index ( 1χ) The Randić topological index, 1χ, is determined through equation 1

χ ¼∑

!1 , =2 1 ′  n m′

ð3Þ

where m′ and n′ are the valences of the bound carbons. For Methyl butane:

1

2

3

4

2 1,2

3

4

2,1

3,0

3,0 5 For the vertexes 1, 4, and 5 the ordered pair (i,j) is (3,0); for vertex 3, the ordered pair (i,j) is (2,1); and for vortex 2, the ordered pair (i,j) is (1,2). These vertexes represent primary, secondary, and tertiary carbons, respectively. The topological index Fi is mathematically defined as: Fi ¼ ∑Oi þ ∑T j

ð5Þ

where Oi is the value of i for a given ordered pair (i, j) and Tj is the value for (jh × jh′) 1/2, being h and h′ any adjacent vertexes. Therefore, for Methyl butane, the following values are obtained: ∑Oi ¼ 3 þ 1 þ 2 þ 3 þ 3 ¼ 12:

5

Note that Σ Oi is simply the total number of hydrogen atoms present in the structure of Methyl butane, C5H12 (rule applied for all alkanes on general formula CnH2n + 2). 1=2

and for all the bonds , !1 =2 1 ¼ 0:57735 1×3 , !1 =2 ¼ 0:40825 C2−C3 1 3×2 , !1 =2 ¼ 0:57735 C2−C5 1 1×3 , !1 =2 ¼ 0:70711 C3−C4 1 2×1 C1 −C2

∑T j ¼ ð0  2Þ ¼ 1:4142:

þ ð2  1Þ

1=2

1=2

þ ð1  0Þ

1=2

þ ð2  0Þ

1=2

¼ ð2Þ

So, Fi = 12 + 1.4142 = 13.4142. Thus, the value of “Fi” topological index for Methyl butane is 13.4142. For the first two compounds of the alkane series, CH4 and C2H6, which have, respectively, isolated carbon and primary carbons, the degree of connectivity,, is equal to −1 for methane (a special case where the Tj value is the proper value of j) and null, for the ethane because Tj = (0 × 0) 1/2. Thus, for these compounds, the values of Fi are 3.0, for methane, and 6.0 for ethane. 2. Analysis and discussion of results

1

χ ¼ 0:57735 þ 0:40825 þ 0:57735 þ 0:70711 ¼ 2:2701:

Therefore, 2.2701 is the value of Randić topological index ( 1χ) for the Methyl butane. 1.1.4. The topological index “Fi” The topological index Fi, object of this work, associates each vertex of the molecular graph to an ordered pair i,j whose sum i + j

The topological index Fi proposed in this paper was applied to a group of alkane molecules (CnH2n + 2) being the n changed from 3 to 11 carbons in normal and branched chain compounds. Initially, a training set containing 25 compounds was analyzed. Table 1 describes the values of W, Z, 1χ and Fi topological indexes. Fig. 1 presents the plot of normal boiling temperatures (B.T.), from literature [17], for the compounds of training set against (Fi) 1/2.

F.C.G. Manso et al. / Journal of Molecular Liquids 165 (2012) 125–132

Alkane

log W

log Z

(1χ)½

Fi½

Propane Butane Pentane Methyl butane 2-Methyl pentane 2-Methyl hexane 2-Methyl heptane 2-Methyl octane 2-Methyl nonane 3-Methyl pentane 3-Methyl hexane 3-Methyl heptane 3-Methyl octane 3-Methyl nonane 2.3-Dimethyl pentane 2.2.3-Trimethyl butane 2.3.3-Trimethyl pentane 2.3.4-Trimethyl pentane 2-Methyl decane Methyl propane 2.3-Dimethyl butane 2.3-Dimethyl hexane 2.3-Dimethyl-3-ethyl pentane 2.3.3.4-Tetramethyl hexane 2-Methyl-3-isopropil hexane

0.6021 1.0000 1.3010 1.2553 1.5051 1.7160 1.8976 2.0569 2.1987 1.4914 1.6990 1.8808 2.0414 2.1790 1.6628 1.6232 1.7924 1.8129 2.3263 0.9542 1.4624 1.8451 1.9345 2.0607 2.0934

0.4771 0.6990 0.9031 0.8451 1.0414 1.2553 1.4624 1.6721 1.8808 1.0792 1.2788 1.4914 1.6902 1.8865 1.2304 1.1139 1.3617 1.3802 2.0569 0.6021 1.0000 1.4314 1.5798 1.7482 1.7853

1.1892 1.3835 1.5538 1.5067 1.6644 1.8083 1.9417 2.0664 2.1841 1.6757 1.8188 1.9514 2.0756 2.1927 1.7835 1.7156 1.8719 1.8851 2.2957 1.3161 1.6348 1.9185 2.0161 2.1035 2.1428

2.8284 3.3166 3.7417 3.6625 4.0514 4.4062 4.7344 5.0412 5.3305 4.1022 4.4529 4.7779 5.0822 5.3692 4.4062 4.2953 4.7097 4.6904 5.6048 3.1623 4.0000 4.7344 5.0905 5.3210 5.3305

The values of topological indexes of Wiener [1] (log W), Hosoya [2] (log Z), and Randić [3] ( 1χ) 1/2 for the 25 compounds of training set are also described in Table 1. Figs. 2, 3 and 4 show, respectively, the plots of B.T. values from literature [18] for the compounds of training set against the log (W), log (Z) and ( 1χ) 1/2. Linear dependence is obtained in each plot of Figs. 2 to 4. The coefficient of determination (R 2) for the linear regression is shown in respective figure. The equations obtained for linear regression of curves presented in Figs. 1 to 4 are, respectively, 1=2

B:T:predict ¼ −273:61 þ 82:41ðFiÞ

ð6Þ

B:T:predict ¼ −46:04 þ 142:97ð log W Þ

ð7Þ

B:T:predict ¼ −92:84 þ 143:56ð log Z Þ

ð8Þ

B:T:predict ¼ −289:84 þ 211:30



1

χ

§

:

ð9Þ

Table 2 shows the variance and highest (positive and negative) differences between experimental values of B.T. (from literature

150 100 50 0 -50 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Log (W ) Fig. 2. Plot of normal boiling temperature (B.T.) from literature [18] versus log(W) (Wiener index) for the compounds of training set described in Table 1.

200 2

R = 0.9915

150 100 50 0 -50 0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Log (Z) Fig. 3. Plot of normal boiling temperature (B.T.) from literature [18] versus log(Z) (Hosoya index) for the compounds of training set described in Table 1.

[18]) and the values obtained using W, Z, 1χ, and Fi for the 25compounds of training set (Table 1). The coefficient of determination (R 2) for applying Eq. (6) to the experimental data (Table 1) is 0.9999 and it is higher than the respective parameter obtained for the others predictors. Thus, according to this result, the topological Fi can be used for predicting the normal boiling temperature (B.T.) of alkanes. But after applying Eq. (6) to others 75 alkanes of validation set (Table 3) we observed a significant deviation at low and higher temperatures. The 25 compounds used in training set were included in the validation set and are labeled by superscript letter a in

2

R = 0.9999

B.T. from Literature (OC)

B.T. from Literature (oC)

2

R = 0.9923

200

200 150 100 50 0

R2 = 0.9969 150 100 50 0 -50

-50 2.5

200

B.T. from Literature (OC)

Table 1 Compounds and their respective values of topological indexes log W (Wiener), log Z (Hosoya), (1χ)½ (Randić) and Fi½.

B.T. from Literature (OC)

128

3.0

3.5

4.0

4.5

5.0

5.5

6.0

1/2

(Fi)

Fig. 1. Plot of normal boiling temperature (B.T.) from literature [18] versus (Fi)1/2 for the compounds of training set described in Table 1.

1.2

1.4

1.6

1.8

2.0

2.2

2.4

(1χ )1/2 Fig. 4. Plot of normal boiling temperature (B.T.) from literature [18] versus (1χ)1/2 (Randić index) for the compounds of training set described in Table 1.

F.C.G. Manso et al. / Journal of Molecular Liquids 165 (2012) 125–132 Table 2 Variance, highest (positive and negative) differences between experimental values of B.T. (from literature [18]) and the values obtained using W, Z, 1χ, and Fi for the 25compounds of training set (Table 1). Predictor

Linear model

Variance

Fi Weiner Hosoya Randić

B.T.pred = 82.41 (Fi)1/2–273.61 B.T.pred = 142.97 (log W)–146.04 B.T.pred = 143.56 (log Z)–92.84 B.T.pred = 211.30 (1χ)½–289.84

1.090 57.382 63.256 22.840

Highest differences from literature value Positive/negative + 1.9 °C/−1.6 °C + 17.9 °C/−9.3 °C + 13.7 °C/−17.8 °C + 10.0 °C/−5.9 °C

Table 3. Fig. 5 shows the plot of normal boiling temperatures (B.T.), from literature, for the compounds of Table 3 against (Fi) 1/2. The coefficient of determination (R 2) of plot of Fig. 5 is not so bad, it is 0.9967. Besides the prediction of B.T. for alkanes with (Fi) 1/2 from 2.8 to 7 does not produces serious deviation as compared to literature values [18], the linear model falls for predicting the B.T. for alkanes with (Fi) 1/2 lower than 2.8 or higher than 7. The plot of B.T. values form literature [18] against the predicted B.T. values through the use of topological index Fi for the alkanes listed in Table 3 using Eq. (6) is presented in Fig. 6. The straight line in Fig. 6 refers to the data produced by an “ideal” model (y = x). Negative and low deviations were observed in the middle B.T.-scale while high and positive deviations were observed in both ends B.T.-scale. For instance, the ability of model (Eq. (6)) to predict B.T. of compounds situated in the ends of B.T.-scale of Fig. 6, viz. Methane (B.T. = − 161.5 °C) and Pentatriacontane (B.T. = 490 °C) falls a lot: the deviation at B.T = − 165.5 is close to 38 °C and the deviation at B.T. = 490 °C is ca. 34 °C. Aiming to increase the robustness for predicting the B.T. through the Fi topological index, we proceeded a second-order polynomial regression in the curve of Fig. 5 that produced the following quadratic model 1=2

B:T:predict ¼ −329:82 þ 105:10 ðFiÞ

–2:36Fi

ð10Þ

with coefficient of determination (R 2) of 0.9986. The new regression is shown in Fig. 7. As can be seen in Fig. 7, the quadratic model fit very well the curve obtained by plotting the B.T. from literature [18] against (Fi) 1/2. The robustness of the as-obtained quadratic model is increased as compared to the linear model. The following equation 1=2

B:T:predict ¼ −276:34 þ 83:73 ðFiÞ

−0:153 Fi

ð11Þ

is obtained when only the data of training set (the 25 alkanes listed in Table 1) are used for built a quadratic model. This model is also powerful in predicting the B.T. of alkanes mainly in the middle-Fi scale, but it falls for predicting B.T. for alkanes possessing high and low values of Fi. For instance, the predicted value of B.T. for Pentatriacontane, the alkane in Table 3 with highest Fi value [(Fi) 1/2 = 10.1980], is 561.63 °C as Eq. (11) is used. But, using Eq. (10) the obtained value for such compound is 496.6 °C. The value of B.T. from literature [18] for this compound is 490.0 °C. In the other end of Fi scale, the predicted value of B.T. for Methane, the alkane in Table 3 with lowest Fi value [(Fi) 1/2 = 1.7321], is −131.8 °C as Eq. (11) is used. The value for B.T. obtained for Methane through Eq. (10) is − 159.5 °C. Comparing with the reference value (B.T. from literature is −161.5 °C). So, it is clear that the quadratic model (Eq. (10)) based on compounds of Table 3 presents higher robustness than the one based only in training set (Eq. (11)). The main reason for that is the higher degree of freedom of quadratic model described by Eq. (10) that was obtained based on hundred compounds and in a wider (Fi) 1/2 scale (from 1.8

129

Table 3 Compounds, boiling temperatures [18] and values of the square root of descriptor Fi [(Fi)½] after the use of the fitting factor. Alkane

Boiling temp. (°C)

Index (Fi)½

Methane Ethane Propanea Butanea Pentanea Hexane Heptane Octane Nonane Decane Undecane Dodecane Tridecane Tetradecane Pentadecane Hexadecaneb Heptadecane Octadecaneb Eicosane Uneicosane Docosane Tricosane Tetracosane Pentacosane Hexacosane Octacosane Nonacosane Triacontane Entriacontane Dotriacontane Methyl butanea 2-Methyl pentanea 2-Methyl hexanea 2-Methyl heptanea 2-Methyl octanea 2-Methyl nonanea 3-Methyl pentanea 3-Methyl hexanea 3-Methyl heptanea 3-Methyl octanea 3-Methyl nonanea 4-Methyl heptane 4-Methyl octane 4-Methyl nonane 3-Ethyl pentane 4-Ethyl heptane 4-Ethyl octane 2,3-Dimethyl pentanea 2,4-Dimethyl pentane 2,4-Dimethyl hexane 2.5-Dimethyl hexane 3.4-Dimethyl hexane 2.2.3-Trimethyl butanea 2.2-Dimethyl pentane 3.3-Dimethyl pentane 3.3 Dimethyl hexane 3-Methyl-3-ethyl pentane 2.3.3-Trimethyl pentanea 2.3.4-Trimethyl pentanea 3.3-Dimethyl heptane 61-3.5-Dimethyl heptane 2.3.5-Trimethyl hexane 2.2.4.4-Tetramethyl pentane 3.5-Dimethyl octane 2.4.5-Trimethyl heptane 3.3.4-Trimethyl heptane 2.2-Dimethyl butane 2.4.4-Trimethyl hexane Dimethyl-4-ethyl hexane 2-Methyl decanea 3.3-Dimethyl octane 4-Methyl decane Pentatriacontane

− 161.5 (7459) − 88.6 (5448) − 42.1 (9834) − 0.5 (3229) 36.0 (8709) 68.7 (6731) 98.5 (6355) 125.6 (8355) 150.8 (8120) 174.1 (5333) 195.9 (11,938) 216.3 (5292) 235.4 (11,871) 253.5 (11,613) 270.6 (8642) 286.8 (6332) 302.0 (6306) 316.3 (8234) 343.0 (5350) 356.5 (6294) 368.6 (5268) 380.0 (11,855) 391.3 (11,603) 401.9 (8637) 412.2 (6622) 431.6 (8220) 440.8 (8099) 449.7 (11,819) 458.0 (6299) 467.0 (5348) 27.8 (3397) 60.2 (8823) 90.0 (6834) 117.6 (6422) 143.2 (8408) 167.1 (8140) 63.2 (8824) 92.0 (6835) 118.9 (6423) 144.2 (8409) 167.9 (8141) 117.7 (6424) 142.4 (8410) 165.7 (8142) 93.5 (8809) 141.2 (6408) 163.7 (8404) 89.7 (8761) 80.4 (8762) 109.5 (6773) 109.1 (6774) 117.7 (6776) 80.8 (3457) 79.2 (8760) 86.0 (8763) 111.9 (6775) 118.2 (8814) 114.8 (8849) 113.5 (8850) 137.3 (6388) 136.0 (6390) 131.4 (6864) 122.2 (8841) 159.4 (8381) 156.4 (6446) 161.9 (6449) 49.7 (3289) 130.7 (6865) 162.9 (6824) 189.3 (5054) 161.2 (8379) 187.0 (5056) 490.0 (8999)

1.7321 2.4495 2.8284 3.3166 3.7417 4.1231 4.4721 4.7958 5.0990 5.3852 5.6568 5.9161 6.1644 6.4031 6.6332 6.8557 7.0711 7.2801 7.6811 7.8740 8.0623 8.2462 8.4261 8.6023 8.7750 9.1104 9.2736 9.4340 9.5917 9.7468 3.6625 4.0514 4.4061 4.7344 5.0412 5.3305 4.1022 4.4529 4.7779 5.0822 5.3692 4.7779 5.0822 5.3692 4.4992 5.1228 5.4076 4.4062 4.3392 4.7162 4.6721 4.7779 4.2953 4.3281 4.4118 4.7396 4.8162 4.7097 4.6904 5.0462 5.0653 4.9828 4.8440 5.3532 5.3144 5.3475 3.9664 4.9878 5.3861 5.6048 5.3352 5.6417 10.1980 (continued on next page)

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600 Boiling temp. (°C)

Index (Fi)

132.9 (6385) 136.0 (6386) − 11.7 (10,073) 57.9 (3290) 118.6 (6815) 106.8 (6771) 115.6 (6772) 143.0 (6407) 35.2 (6387) 140.6 (6389) 140.5 (6866) 144.7 (8811) 165.1 (8143) 166.5 (8403) 157.5 (6434) 155.0 (8373) 156.0 (8375) 162.1 (8384) 165.0 (6411) 329.9 311.0 (6310) 177.8 (6433) 164.6 (6850) 158.5 (8376) 160.8 (8382) 165.0 (6414) 165.0 (6839)

5.0242 5.0242 3.1623 4.0000 4.8211 4.6618 4.7344 5.1228 4.9828 5.0822 5.0592 5.0905 5.3692 5.4076 5.4076 5.2661 5.3144 5.3692 5.4076 7.4833 7.2398 5.6826 5.3210 5.3144 5.3532 5.4076 5.3305

No superscript: the datum was obtained from ref. [18]. The number in parenthesis refers to the code of the substance on Refs. [18]. a Compounds of training set (Table 1). b The B.T. obtained from ref. [23].

to ca. 10) while the quadratic model described by Eq. (11) was based on 25 compounds in which the (Fi) 1/2 scale spreads from 2.7 to ca. 5.7. Fig. 8 shows the plot of normal boiling temperatures (B.T.) predicted by quadratic (Eq. (10)) against the experimental data for the compounds of Table 3. The dashed line inserted in the plot obeys the equation y = x, i. e. an “ideal” model without residues. The analysis of Fig. 8 is convincing that the B.T. can be predicted by Eq. (10) for alkanes within a wide range of normal boiling temperatures and with low deviation. Table 4 shows the values of (Fi) 1/2 and B.T. from literature [18] for some alkenes, alkynes and cycloalkanes. The quadratic model of Eq. (10) was also used for predicting the normal boiling temperature (B.T.) for the hydrocarbons listed in Table 4. As expected, for the compounds constant in Table 4, the values of B.T. predicted by quadratic model (Eq. (10)) are closer to those published by literature [18] as compared to predicted by the linear model (Eq. (6)). This shows that the topologic index Fi can be used not only for predicting the

B.T. from Literature (oC)

500 400 300 200

Linear model described by equation (6)

100

R2 = 0.9967

0 -100 -200 2

4

6

8

10

(Fi)1/2 Fig. 5. Plot of normal boiling temperature (B.T.) versus (Fi)1/2 for the hundred compounds described in Table 3.

500 400 300 200 100 0

y=x

-100 -200 -200 -100

0

100

200

300

400

500

600

B.T. from literature (oC) Fig. 6. Plot of normal boiling temperature (B.T.) predicted by linear model based on (Fi)1/2 against the B.T. from literature for compounds described in Table 3.

500

B.T. from Literature (oC)

2.4-Dimethyl heptane 2.5-Dimethyl heptane Methyl propanea 2.3-Dimethyl butanea 3-Ethyl hexane 2.2-Dimethyl hexane 2.3-Dimethyl hexanea 3-Ethyl heptane 2.6-Dimethyl heptane 3.4-Dimethyl heptane 3.3.4-Trimethyl hexane 2.3-Dimethyl-3-ethyl pentanea 5-Methyl nonane 3-Ethyl octane 4-n-Propyl heptane 2.2-Dimethyl octane 2.4-Dimethyl octane 4.5-Dimethyl octane 3-Ethyl-4-methyl heptane Nonadecane 2-Methyl heptadecane 2.2.4.6.6-Pentamethyl heptane 2.3.3.4-Tetramethyl hexanea 97-2.5-Dimethyl octane 3,6-Dimethyl octane 3-Methyl-4-ethyl heptane 2-Methyl-3-isopropyl hexanea

400

B.T.predict = -329.82 + 105.10 Fi1/2 - 2.36 Fi

300 200 100

R2 = 0.9986

0 -100 -200 2

4

6

8

10

(Fi )1/2 Fig. 7. Plot of normal boiling temperature (B.T.) from literature [18] versus (Fi)1/2 for the hundred compounds described in Table 3. The solid line refers to data produced by quadratic model.

normal boiling temperatures of alkanes but also for other types of hydrocarbons. The values of Fi described in Table 4 were calculated using the same procedures as done for alkanes, as earlier described. The plot of experimental values of B.T. against the predicted values through the use of topological index Fi (Eq. (10)) for the compounds listed in Table 4 is presented in Fig. 9. A straight line is obtained showing the robustness of Eq. (10) for predicting the B.T. of hydrocarbons others than alkanes.

B.T. predict by quadratic model (oC)

Alkane

½

B.T. predict by linear model (oC)

Table 3 (continued)

600 500 400 300 200 100 0

y=x

-100 -200 -200 -100

0

100

200

300

400

500

600

B.T. from literature (oC) Fig. 8. Plot of normal boiling temperature (B.T.) predicted by quadratic model based on (Fi)1/2 against the B.T. from literature [18] for compounds described in Table 3.

B.T. predict by quadratic model (oC)

F.C.G. Manso et al. / Journal of Molecular Liquids 165 (2012) 125–132 Table 4 Values of Fi and B.T. fom literature [18,23,24] and the values of B.T. predict by use of linear (Eq. (6)) and quadratic model (Eq. (10)) for alkenes, akynes and cycloalkanes. Compound

(Fi)

1-Butene 1-Decene 1-Dodecene 1-Heptene 1-Hexadecene 1-Octadecene 1-Octene 1-Tetradecenea 2,3-Dimethyl-1-butenea 2,Ethyl-1-butenea cis 1,2-dimethylcyclo-hexanea cis 1,3-dimethylcyclo-hexane cis 1,4-dimethylcyclo-hexane Cyclohexane Cyclopentane Ethylcyclo-pentane Methylcyclo-hexane Methylcyclo-pentane n-Propyl-cyclohexanea 1,3-Butadiene Ethyne 3-Methyl-1-butine 1-Hexineb 1-Heptine 1-Eicosine

1/2

B.T. (°C) from literature

3.3166 5.3852 5.9161 4.4721 6.8557 7.2801 4.7958 6.4031 4.0000 4.1022 4.8814 4.8638 4.8638 4.2426 3.8730 4.6090 4.5638 4.2224 5.2194 3.3166 2.4495 3.6625 4.1231 4.4721 7.6811

− 6.3 170.5 213.8 93.6 284.9 314.9 121.3 251.2 55.7 64.7 129.8 120.1 124.4 80.8 49.3 103.5 101.0 71.9 156.8 − 4.4 − 84.7 26.3 71.0 100.0 342

B.T. (°C) predicted by linear model, Eq. (6)

B.T. (°C) predicted by quadratic model, Eq. (10).

− 0.5 166.8 209.7 92.9 285.7 320.0 119.1 249.1 54.8 63.1 126.1 124.6 124.6 74.4 44.5 104.0 100.4 72.8 153.4 − 0.5 − 70.6 27.5 64.7 93.0 352.4

− 4.4 170.4 212.4 95.4 283.8 314.8 122.4 249.8 55.3 64.1 129.5 128.0 128.0 76.1 44.4 106.9 103.1 74.3 157.1 − 4.4 − 83.0 26.1 65.9 95.5 343.4

500 75 compounds [Tab.3] 25*compounds [Tab.1] 25 compounds [Tab.4]

400 300 200 100

y=x 0 -100 -200 -200

-100

0

100

200

300

400

500

B.T. from literature (oC) Fig. 10. Plot of normal boiling temperature (B.T.) predicted by quadratic model based on (Fi)1/2 against the B.T. from literature [18] for the alkanes of training set (Table 1) (cicle); alkanes of validation set (Table 3) (square); alkenes, alkynes and cycloalkanes (Table 4) (triangle).

3. Conclusions

400

The main goal of this paper was to propose a new and simple molecular descriptor based on the structure molecular of hydrocarbons for predicting the normal boiling temperature (B.T.) of this type of compounds. To this end, the topological index “Fi” was proposed. This index associates an ordered pair (i,j) with each vertex of the structural graph and introduces a new concept in the construction of the topological indexes. A training set constituted by 25 compounds, all alkanes, was used to built a linear model. The robustness of such model was evaluated by comparison with the most cited predictors in the literature: Weiner, Hosoya and Randić. Although, compared to other predictors, the linear model based on index Fi was robustness for predicting the B.T. of the alkanes containing in the training set, it falls for predicting B.T for alkanes with wider range of Fi values. Then a quadratic model was built based on Fi indexes of hundred alkanes (Table 3). The as-obtained quadratic model predicts very well the B.T. of alkanes and also of other compounds (alkenes, alkynes and cycloalkanes). The topological index Fi proved to be rather effective and produced small deviations, as compared to the topological indexes cited in the literature. Based on the topological index Fi, other properties of interest can also be explored and the concepts developed in this work can be easily adapted to other families of compounds, mainly in the liquid phase. In conclusion, jointly with the descriptors cited in the literature, the molecular descriptor “Fi” is a contribution for researchers interested in studying the relation between the structure and property of chemical substances, which is essential for developing the physics and chemistry of molecular materials.

300

Acknowledgments

No superscript: the B.T. was obtained from ref. [18]. a The B.T. was obtained from ref. [23]. b The B.T. was obtained from ref. [24].

Fig. 10 shows the plot of normal boiling temperature (B.T.) predicted by quadratic model based on (Fi) 1/2 against the B.T. from literature [18] for the akanes of training set (Table 1), alkanes of validation set (Table 3); alkenes, alkynes and cycloalkanes (Table 4). It can be noticed that no significant deviation was observed. These findings reinforce the ability of topological index Fi in predicting the normal boiling points of hydrocarbons. It can be speculated that the topologic index Fi may also be used for predicting other properties such as vapor pressure, surface tension, refraction index, etc. and such results can be applied in both academic and technological fields. Further studies in this direction should be going on for developing predictors based on simple topological index Fi.

B.T. predict by quadratic model (oC)

131

The authors thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil) and CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Brazil) for their financial support.

200 100 0

y=x References

-100 -100

0

100

200

300

400

B.T. from literature (oC) Fig. 9. Plot of normal boiling temperature (B.T.) predicted by quadratic model based on (Fi)1/2 against the B.T. from literature [18] for the alkenes, alkynes and cycloalkanes listed in Table 4.

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