82
J. Chem. Inf. Comput. Sci. 2002, 42, 82-86
Spectral Moments of Polycyclic Aromatic Hydrocarbons. Solution of a Kinetic Problem Svetlana Markovic´,*,† Zoran Markovic´,† Johan P. Engelbrecht,‡ and Robert I. McCrindle† Department of Chemistry and Physics and Department of Environmental Engineering, Technikon Pretoria, P.O. Box 56208, Arcadia 0007, Republic of South Africa Received July 2, 2001
The relationship between the rate of supercritical fluid extraction of polycyclic aromatic hydrocarbons (PAHs) from coal-tar pitch and some topological invariants are examined. The aim is to explain the appearance of a minimum value on the activation energy/molar mass curve of the PAHs. Rate constants are approximated using spectral moment expansion. It is shown that the size of the PAHs determines more than 95% of the extraction rate. Activation energy for the extraction of the PAHs was found to increase with increasing molar mass. The appearance of a minimum value on the activation energy/molar mass curve is the consequence of experimental difficulties resulting from the tendency of lower members of the PAHs to sublime. 1. INTRODUCTION
Supercritical fluid extraction (SFE) has become an accepted alternative to traditional solvent extraction methods, due mainly to its speed and the reduced amount of liquid solvent required.1,2 In general, the rate determining step in the extraction of solutes from a matrix is either the desorption or the solubility/elution step.2-6 The “hot-ball model” was put forward in order to describe desorption as the diffusion of homogeneous spherical particles into a medium, where their concentration is infinitely dilute.3 This model was further developed, so that the effect of solubility of the analytes in the solvent was also included.4 For many dynamic SFE experiments, the rate of overall extraction is limited by the rate of diffusion or a similar process or by that of transfer out of the matrix. In an extraction of polycyclic aromatic hydrocarbons (PAHs) from soil, the effect of flow rate on SFE rate was used in order to elucidate the mechanism of the extraction process.5 In recent literature,6,7 extraction of coal-tar pitch with supercritical carbon dioxide was examined. Some of the PAHs were identified in appreciable amounts in the extracts (see Table 1). In addition, some minor PAHs were also detected in the extracts (Table 4). Experimental concentration-time data were utilized to determine the rate constants, and the temperature dependence of the extraction rates was used to calculate the activation energies for the extraction of the components given in Table 1.6 Naphthalene and 2-methylnaphthalene were identified in the extracts at significant concentrations, but their rate constants and activation energies were not determined, because the loss of these compounds, due to their sublimation, was unavoidable.6,8 It was found that the activation energies of PAHs slightly decreased with increasing molar mass up to molecules of mediate molar masses and then increased with further increase in molar mass. Such behavior of the molecules of lower molar masses was unexpected, and it * Corresponding author’s permanent address: Faculty of Science, University of Kragujevac, P.O. Box 60, Yu-34000 Kragujevac, Yugoslavia. fax: +381 34 335040; e-mail:
[email protected]. † Department of Chemistry and Physics. ‡ Department of Environmental Engineering.
Table 1. Molecular and Line Graphs, Rate Constants, and Activation Energies of the Major Components of Coal-Tar Pitcha
a 1 - naphthalene, 2 - 2-methylnaphthalene, 3 - acenaphthene, 4 - fluorene, 5 - phenanthrene, 6 - anthracene, 7 - fluoranthene, 8 - pyrene, 9 - benzo(a)anthracene, and 10 - chrysene. * Rate constants and activation energies approximated by eq 1, p ) 2.
was not explained in the ref 6. In this work we attempt to clarify this problem by means of theoretical methods.
10.1021/ci0100604 CCC: $22.00 © 2002 American Chemical Society Published on Web 01/28/2002
SPECTRAL MOMENTS
OF
POLYCYCLIC AROMATIC HYDROCARBONS
J. Chem. Inf. Comput. Sci., Vol. 42, No. 1, 2002 83
Table 2. Data Showing the Quality of the Approximations 1-3a eq 1 p/q
eq 2
0
2
4
6
8
10
0
2
3
4
5
6
eq 3
70 °C
R ARE f
0.994 3.70
0.997 2.54
0.997 2.54
0.997 2.37
0.9990 1.55
0.9998 0.64
0.997 2.60
0.997 2.54
0.997 2.54
0.997 2.83
0.998 1.73
0.9990 1.41
0.98 7.03
90 °C
R ARE f
0.98 5.20
0.993 4.01
0.988 4.94
0.995 3.29
0.97 7.41
125 °C
R ARE f
0.985 4.35
0.998 1.31
0.991 3.24
0.9987 1.50
0.98 6.46
a
0.26
0.14 0.988 4.97
0.19
0.18 0.988 4.97
0.14 0.992 3.32
0.25
0.18 0.992 3.32
0.14
0.38 0.990 4.71
0.75 0.992 3.99
0.20 0.998 1.20
0.57
0.14
0.15 0.998 1.29
0.16
0.14 0.988 4.97
0.14
0.15
0.17 0.988 4.97
0.14 0.992 3.32
0.15
0.19 0.992 3.32
0.14
0.26 0.991 3.40
0.22 0.995 3.38
0.27 0.992 3.32
0.14
0.16 0.998 2.70
0.78
0.17
R - correlation coefficients, ARE - average relative errors, and f - results of the F-test.
Table 3. Fitting Parameters for the Eq 1, p ) 2 70 °C 90 °C 125 °C
a0
a2
a4
-0.05976 0.02742 0.11784
-0.17081 -0.18488 -0.21607
9.36963 8.89717 8.81020
Table 4. Molecular Graphs, Rate Constants, and Activation Energies of the Minor Components of Coal-Tar Pitcha
bonds), whereas line graphs reflect the adjacency of these bonds. The graphs G and L can be represented by their adjacency matrices A and E, respectively. It is well-known that the physicochemical behavior of PAHs is highly dependent on molecular topology,9-12 and it can be successfully described by topological invariants such as spectral moments and topological indices. Spectral moments of G and L, Mk, and µk, are equal to the numbers of self-returning walks of length k contained in molecular and line graphs, respectively. They are defined as
Mk ) Tr[Ak] and
µk ) Tr[Ek] where Tr denotes the traces of the matrices A and E. Spectral moments of molecular graphs have been employed in the physical chemistry of solid state13-15 and in the theoretical chemistry of conjugate hydrocarbons.16-26 Spectral moments of line graphs have found remarkable applications in quantitative structure-property (QSPR) and structureactivity (QSAR) relationship studies and in the theoretical chemistry of aromatic compounds.27-32 An important structural descriptor of organic molecules is the Wiener index (Wiener number), that is defined as the sum of the distances between all pairs of vertices of the molecular graph G
W(G) ) ∑ d(Vi,Vj) 1 - indene, 2 - 1-methylnaphthalene, 3 - ethylnaphthalene, 4 dimethylnaphthalene, 5 - methylbiphenyl, 6 - methylfluorene, 7 methylphenanthrene, and 8 - phenylnaphthalene. Rate constants and activation energies were approximated by means of eq 1, p ) 2. a
Chemical graphs are the basic tools used in applying the techniques of mathematical graph theory to the specific problems of chemistry.9-11 The molecular graph of a conjugated hydrocarbon is a graph representing the carbon atom skeleton of the corresponding molecule. Consider a molecular graph G with n vertices and m edges. The line graph of G (L) can be constructed in the following manner: a vertex of the line graph is associated to each edge of G. The edges of L are now obtained by connecting those vertices of L which represent adjacent edges in G. Molecular graphs represent the adjacency of carbon atoms (existing of chemical
i
