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Ind. Eng. Chem. Res. 2002, 41, 5534-5537
CORRELATIONS Group Vector Space Method for Estimating Melting and Boiling Points of Organic Compounds Xu Wen* and Yang Qiang School of Chemical Engineering, Tianjin University, Tianjin 300072, China
The specific position of a group in the molecule has been considered and a group vector space method for estimating boiling and melting points of organic compounds developed. Expressions for melting point Tm and boiling point Tb have been proposed, with the numerical values of relative group parameters presented. The average percent deviations of estimation of Tm and Tb are 6.80 and 1.40, respectively. Introduction As fundamental physical properties of organic compounds, melting and boiling points have found wide use in the domains of process simulation and product design. However, it is not always possible to find reliable experimental values of melting and boiling points for the compounds of interest, nor is it practical to measure the properties as the need arises. So, estimation of the melting and boiling points is required. For estimation of the melting and boiling points of pure compounds, the simple group technique developed by Joback and Reid (1987)1 may be useful as a guide in obtaining approximate values of them. This method provides the important advantage of quick estimates without requiring substantial computational resources. However, it is of questionable accuracy and utility. To overcome this limitation, complex group contribution methods have been reported in the literature. The two-level group contribution method proposed by Constantinou and Gani (1994)2 performed the estimation of the melting and boiling points, and the group-interaction contribution (GIC) method of Marrero-Morejon and Pardillo-Fontdevila (1999)3performed the estimation of the boiling points only. For the Constantinou and Gani method, the estimation is performed at tow level: the basic level uses the contribution from first-order groups, while the second level increases the consideration of the second-order groups. Their 78 first-order groups are not sufficient to describe the molecules of some common compounds, and under certain circumstances, the same molecule may be described in different ways because of overcomplication of this method. For the GIC method, the number of parameters in the property correlation is 223, which has no very large difference to the number of substances in some property regressions. As we know, only if the number of substances in linear data regression is much more than that of parameters in the model does the group method have a function of extrapolation predicting. In this paper, the specific position of the group in the molecule will be taken into consideration, and a new method for estimating physical properties to limit the * To whom correspondence should be addressed. E-mail:
[email protected].
number of model parameters with higher accuracy will be proposed. A Group Vector Space (GVS) for Organic Compounds In this work, we select 40 simple groups to describe organic compounds. These groups are the same as those used by Joback and Reid.1 The molecule is considered to be in a given space, and every group in the molecule is only a point in the space. For convenience, graphs with different numbers of points are all expressed as graphs with five points. Thus, the organic molecule can be expressed as the following seven topologic graphs:
Considering the chain graph first, the dimension number of the space is equal to the number of end points on the chain, and one end point has determined a dimension of the space. The coordinate of an end point in the dimension determined by it is zero, while the
10.1021/ie010989d CCC: $22.00 © 2002 American Chemical Society Published on Web 10/04/2002
Ind. Eng. Chem. Res., Vol. 41, No. 22, 2002 5535
coordinate of another point in this dimension is the distance from that point to the end point. For the cyclic graph, one ring represents a dimension. In that dimension the coordinate of the ring point equals the number of points on the ring, and the coordinate of the nonring point equals the sum of the distance from the point to ring and the number of points on the ring. If the route from the ring point to the end point is nonunique, the shortest route should be selected. So, the dimension number m of the space for a graph is equal to the sum of the number ke of end points and the number kr of rings in the graph. Every point in the graph has m coordinates in the m-dimensional space. The graph may be described by a space matrix, where the number of rows in the matrix equals the number of points in the graph and the number of columns equals the dimension number of the space. The space matrices of the above seven topologic graphs are as follows:
f)a+
∑i ni∆fi
(1)
where subscript i represents the group type, ∆fi is the contribution value of the i-type group, ni is the number of i-type groups in the molecule, and a is the correlation constant. In this study, the group contribution was divided into two parts: the position contribution of the group and the independent contribution of the group. The physical property f is expressed as follows: ni
f)a+
νj∆fPi + ni∆fIi + ∆f0i) ∑i ∑ j)1 (
(2)
where ∆fPi is the position contribution of the i-type group, ∆fIi is the independent contribution of the i-type ni group, f0i is the constant of the i-type group, and ∑j)1 νj is the module index sum of the i-type groups. To improve the property estimation accuracy, the power index “1” of property f in eq 2 is substituted by a variable, the optimum value of which can be obtained by trial computation. Upon doing this, the expressions of Tm and Tb may be written as follows: ni
0.6
Tm
) 11.488 +
νj∆TmPi + ni∆TmIi + ∆Tm0i) ∑i (∑ j)1 (3) ni
Tb1.5 ) 1304.13 +
νj∆TbPi + ni∆TbIi + ∆Tb0i) ∑i ∑ j)1 (
(4)
The matrices show that the space position of point i in the graph can be represented by an m-dimensional vector (bi1, bi2, ..., bim). Then the module Ri of point i vector is m
bij ) ∑ j)1
2 1/2
Ri ) (
(i ) 1-5)
The average square root of the module of some point i in the graph is defined as the module index vi of this point vector. That is, 5
νi ) Ri/(
Rj2)1/2 ∑ j)1
(i ) 1-5)
The quantity vi is used to describe the point i position in the space. In the analogy of this, the module index vi of group i in the molecule is taken to characterize the position of that group in the molecular space. Thus, every simple group, except halogen groups, has its own independent module index. For the four halogen groups, their module indexes were determined to be the same as those of the hydrocarbon groups with which they were connected. Correlation and Group Parameters The expression of physical property f for the simple group method is
where Tm and Tb are the melting point and boiling point, respectively, and the units employed are all kelvin. A great deal of experimental data in the literature4-6 have been used to optimize the values of the group parameters in eqs 3 and 4, which are shown in Table 1. Estimation Results and Method Comparison Melting points of 529 compounds and boiling points of 669 compounds are divided into six groups; the average deviations between the calculated value by this method and the experimental data are listed in Table 2. The Joback and Reid1 method, the Marrero-Morejon and Pardillo-Fontdevila3 method, and the Constantinou and Gani2 method have been chosen as the comparative methods to examine this work. The results are listed in Table 3. The proposed method gives, as shown in Table 3, more accurate predictions for Tm and Tb than the compared methods. For the Constantinou and Gani method, the accuracy of the Tb estimation is rather higher, but the number of their estimated data is too small to illustrate the real situations. Moreover, the Constantinou and Gani method is of questionable utility. Illustrative Examples of Estimation The procedure for estimating the physical property of an organic compound can be readily performed. First, the structural formula and corresponding topologic space matrix for the molecule were drawn. Then, module ai of group i in the molecule and corresponding
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Table 1. Values of Group Parameters boiling points (K) group serial no.
groupa
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
-CH3 >CH2 >CH>C< dCH2 dCHdC< dCd tCH tC(>CH2)R (>CH-)R (>CCdO)R OdCH-COOH -COO- (ester) dCdO -NH2 >NH (>NH)R >N-Nd (-Nd)R -CN -NO2 -SH -S(-S-)R
a
∆TbPi ×
10-2
7.969 17.987 1.483 -24.914 1.735 5.345 -21.584 -0.014 -10.168 20.065 6.810 2.768 -18.975 20.438 7.160 24.254 28.204 39.657 14.303 62.320 14.291 30.764 60.126 43.431 -226.513 23.522 38.802 35.221 22.905 -4.422 57.454 140.594 69.051 -0.058 22.232 38.685 71.876 33.316 44.943 -0.648
∆TbIi ×
melting points (K)
10-2
∆Tb0i ×
2.209 3.547 11.312 23.773 4.352 7.409 20.609 0.013 17.203 5.973 7.384 10.577 20.145 2.235 11.122 -9.025 0.448 2.396 23.294 3.786 -34.092 -0.304 -8.527 4.715 77.280 13.001 7.283 12.876 0.193 25.453 -26.956 -42.190 -7.143 0.226 -0.604 13.393 -0.252 1.857 5.244 1.524
10-2
0.224 -1.204 -0.037 -4.304 0.008 -0.037 -1.782 12.964 -7.123 -3.498 -6.829 -0.027 -2.623 -1.784 0.142 -8.934 -1.311 -3.448 -0.307 -8.833 0.492 -2.610 -5.581 2.704 46.456 -44.672 21.483 -1.234 0.159 -2.998 -0.621 1.551 -0.181 19.615 6.516 0.293 -0.274 -0.018 1.654 -0.183
∆TmPi
∆TmIi
∆Tm0i
4.261 4.673 7.184 -3.084 -0.919 4.611 -13.270 14.986 3.970 13.256 4.443 -7.634 -34.126 6.084 4.060 3.214 8.202 -11.770 -7.539 5.485 39.203 11.210 13.886 -15.807 75.110 14.259 10.591 19.929 7.525 12.454 9.947 -0.163 37.934 -0.174 38.652 16.956 18.454 25.924 18.098 -0.135
0.554 0.102 -1.739 2.888 2.971 -0.082 6.088 -3.526 1.438 0.317 0.438 3.033 14.096 0.110 1.402 -0.922 0.343 -0.942 11.378 2.502 -7.478 -0.414 1.336 6.551 -10.565 -1.256 6.578 -2.346 0.110 0.886 1.519 5.658 -10.424 -0.059 -10.148 -1.960 0.023 -12.074 -5.132 5.112
-1.252 -1.822 -0.787 -2.079 -0.143 -1.080 -1.567 0.066 2.288 -4.948 -0.349 -1.367 -1.682 -2.412 -1.455 2.624 -0.172 14.110 0.119 1.491 0.512 -1.887 -4.773 5.989 -15.452 3.146 0.495 -0.323 0.148 -0.052 -0.014 0.171 0.067 -0.137 -0.194 -0.087 0.019 0.078 2.272 -0.775
R represents the ring group.
module index νi may be calculated. Finally, substituting ni each ni and each ∑j)1 νj along with the values of the corresponding group parameters found in Table 1 into a relative correlation, the estimation value of the physical property can be obtained. Two examples are shown as follows: Example 1. Estimate Tm of 2,2,3-trimethylpentane. (a) Structural formula and corresponding topologic space matrix:
The results of computation are as follows: group no. 1
2
3
4
5
6
7
8
33 33 33 36 57 32 17 16 Ri ΣRi2 257 0.3583 0.3583 0.3583 0.3743 0.4709 0.3529 0.2572 0.2495 νi 2
(c) Values of group parameters and corresponding ni νj and ni: ∑j)1
νi ) [Ri2/
ni
∆TmP
∆TmI
∆Tm0
1.9201 0.3529 0.2572 0.2495
5 1 1 1
4.261 4.673 7.184 -3.084
0.554 0.102 -1.739 2.888
-1.252 -1.822 -0.787 -2.079
and
(i ) 1, 2, ..., 8)
Example 2. Estimate Tb of isopropylcyclohexane. Its experimental value is 427.7 K.
5 bij2 ∑j)1
8 Rj2]1/2 ∑j)1
Σνj
-CH3 >CH2 >CH>C
CH(-CH2-)R (>CH-)R
0.6964 0.3015 1.7085 0.2801
2 1 5 1
7.969 1.483 6.810 2.768
2.209 11.312 7.384 10.577
0.224 -0.037 -6.829 -0.027
Using the same procedure as that of example 1, the Tb value estimated with eq 4 is 426.3 K. The relative deviation is -0.33%.
Tm ) melting point, K Tb ) boiling point, K vi ) module index of the i-type group ∆fPi ) position function parameter of the i-type group ∆fIi ) independent function parameter of the i-type group ∆f0i ) constant of the i-type group
Literature Cited (1) Joback, K. G.; Reid, R. C. Estimation of Pure-Component Properties from Group-Contributions. Chem. Eng. Commun. 1987, 57, 233. (2) Constantinou, L.; Gani, R. New Group Contribution Method for Estimating Properties of Pure Compounds. AIChE J. 1994, 40, 1697. (3) Marrero-Morejon, J.; Pardillo-Fontdevila, E. Estimation of Pure Compound Properties Using Group-Interaction Contributions. AIChE J. 1999, 45, 615. (4) Hall, K. R.; Beach, L. B. Selected Values of Properties of Hydrocarbons and Related Compounds; Thermodynamics Research Center, Texas A & M University: College Station, TX, 1980. (5) Reid, R. C.; Prrausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill Book Co.: New York, 1987. (6) West, R. C.; Astle, M. J. CRC Handbook on Organic Compounds; CRC Press, Inc.: Boca Raton, FL, 1985.
Received for review December 6, 2001 Revised manuscript received July 22, 2002 Accepted August 9, 2002 IE010989D
