Ind. Eng. Chem. Res. 1993,32, 1734-1746
1734
Estimation of Thermodynamic and Physical Properties of Acyclic Hydrocarbons Using the ABC Approach and Conjugation Operators Leonidas Constantinou, Suzanne E. Prickett, and Michael L. Mavrovouniotis' Department of Chemical Engineering and Institute for Systems Research, A. V . Williams Building, University of Maryland, College Park, Maryland 20742
Fast and reliable estimation of thermodynamic and physical properties of organic compounds is essential for the analysis and design of chemical processing systems. The ABC approach is based on the contributions of Atoms and Bonds in the properties of Conjugate forms of a molecular structure. In this article, the principles of this approach are applied to the estimation of thermodynamic and physical properties of hydrocarbons. Standard enthalpy of formation, standard entropy, normal boiling point, standard enthalpy of vaporization, critical pressure, critical temperature, and critical volume are estimated. Compared t o a group-contribution approach, the ABC technique is more accurate, and it involves the same or substantially smaller number of parameters. 1. Introduction
A variety of physical and chemical properties of pure compoundsand mixtures, under specified conditions, must be used in the modeling, design, and analysis of chemical process systems. When experimental values for the necessary parameters are not available,the properties must be estimated from informationon the molecular structures of the compounds involved. If several systems or alternatives must be evaluated or designed quickly and approximately, the initiation of complex and time-consuming quantum-mechanical computations or experimental measurements cannot be justified. Consider, for example, the selection of a compound based on a set of specifications. In order to narrow the field of candidates and focus on sets of promising compounds for detailed study, one must have very fast evaluation of computergenerated candidate compounds. One must also be able to determine trends which allow the elimination of whole sets of compounds; only simple estimation methods can make such trends apparent. Thus, simple and efficient methods for the approximateestimation of thermodynamic and physical properties from the molecular structure of substances are essential for the preliminary modeling, analysis, and design of chemical processes. In additive group contribution methods (Joback and Reid, 1987; Benson, 1968; Domalski and Hearing, 1988; Mavrovouniotis, 1990b, 1991;Reid et al.,1987),a property is estimated as a summation of the contributions of the groups which occur in the structure of the molecule. The advantage of these methods is that they can give quick estimates without requiring substantial computational resources. They also readily provide trends within sets of compounds. For example, the variation of a property with elongation of a side chain is apparent in the contribution of the elongating group; substitutions that would have a specific effect on a property are apparent in the sign and magnitude of the relevant contributions, permitting systematic generation of molecules that meet property specifications (Joback and Stephanopoulos, 1989). However, the simplified view of molecular structures inherent in group-contribution methods leads to significant limitations in their scope and accuracy. Group-contribution methods largely ignore concepts such as delocalized bonds, resonance, and other electronic interactions among groups
(Mavrovouniotis,1990a). For example, the properties of ions are strongly dependent on charge dispersion which cannot be readily modeled by group contributions. In the ABC method, which is based on conjugation (also known as resonance), compounds are represented as hybrids of many conjugate forms. Each conjugate form is an idealized structure with integer-order localized bonds and integer charges on atoms. The real compound is a hybrid of a number of conjugate forms, and as a result it may contain fractional charges and bonds which are delocalized and stronger or weaker than ideal integerorder bonds. To estimate the properties of a compound in ABC, one must determine and combine formal properties of its conjugate forms. The properties of each conjugate are estimated from the contributions of individual atoms and individual bonds through conjugation operators, rather than larger functional groups. The intramolecular interactions among atoms and groups are thus not captured through a large variety of groups (which would require a large number of parameters), but through a variety of computationally-generatedconjugate forms, whose properties can in fact be captured with just a small number of contributions from atoms and bonds; such contributions are expressed through conjugation operators. This article focuses on the estimation of properties of acyclic hydrocarbons (saturated and unsaturated). The properties covered here are the standard enthalpy of formation, standard entropy, normal boiling point, standard enthalpy of vaporization, critical pressure, critical temperature, and critical volume. The ABC approach is under continued development, aimed at the accurate and general estimation of a variety of properties, and thus more efficient analysis and design of products and processes. The method is computer-based because the generation and analysis of a fairly large number of conjugates cannot be carried out manually. The generation of the conjugates is a nontrivial computational aspect of the method. With the widespread availability of powerful symbolic computing environments,however, the requisite manipulation of molecular structures is greatly facilitated, and the portability of the method among different computer systems is ensured. The computational aspects of the technique, discussed in detail by Prickett et al. (1993), will be addressed only briefly in this paper, which will place more emphasis on the conceptualand mathematical aspects. 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993 1735 In section 2, an analysis of the quantitative use of conjugation is provided; some aspects are covered only briefly, because they are explained more comprehensively elsewhere (Mavrovouniotis, 1990a). In section 3, the ABC framework is applied to the estimation of the enthalpies of formation of acyclic hydrocarbons; the participation of conjugation operators is also explained. The estimation of other thermodynamic and physical properties is presented in section 4; the results are compared with those of widely-used methods. The discussion in section 5 underlines the significance of the approach and its contribution to the field of property estimation.
Table I. Analysis of Effect of CCCH Conjugates on Enthalpies of Formation of Alkanes.
compound n-pentane 2-methylbutane 2,2-dimethylpropane
-
y-c-c-x
...
y'+ c=c ...XIwhere each of X and Y can be either a carbon or a hydrogen. Since carbon is less electronegative than hydrogen, the most important conjugates are the ones in which X is a hydrogen (receiving the negative charge in the alternative conjugate) and Y is a carbon (receivingthe positive charge), as can be inferred from the fact that hydrogen generally bears a fractional negative charge in hydrocarbons (Fliszar, 1983). These will be called CCCH conjugates or C3H conjugates. Each distinct subchain of the form C-C-C-H in a compound gives a distinct conjugate form. There are a number of convenient procedures for counting these chains manually. For large, highly branched alkanes, it is safer to determine the number of CCCH subchains, NCCCH, using two of the procedures described in Appendix A, to obtain a more reliable count. As an illustration, examination of the CsH12 isomers (Table I) shows that there are 14 CCCH conjugates for n-pentane, 20 for 2-methylbutane, and 36 for 2,2-dic,
enthalpy of formation,* kJ/mol
no. of CCCH conjugates
-146.9 -154.2 -168.5
14 20 36
ca12
ca1z
,
CaH12
,
a The number of CCCH conjugates explains the trends in the enthalpies of formation within seta of isomeric alkanes. In this set, CCCH conjugates capture differences in the degree of branching. b Cox and Pilcher, 1970.
1
2. Quantitative Use of Conjugation
Conjugate forms are alternative formal arrangements of valence electrons in a molecule: The purely covalent form is the dominant conjugate and ionic forms are recessive conjugates (Mavrovouniotis, 1990a). A real compound can be considered the hybrid of all ita conjugates. Many empirical rules used in organic chemistry are based on conjugation. Consider first the widely-used rule stating that branched alkanes have enthalpies lower than their straight-chain isomers (Morrison and Boyd, 1973). Such differences cannot be due to steric hindrance among side chains, because such steric interactions should lead to the reverse of the observed order of enthalpies. The basis of the rule lies with an electronic conjugation effect. Those isomers which have more conjugate forms, or forms that are more similar energetically, are expected to be more stable. This is a general rule about conjugation: The presence of alternative conjugates stabilizes a compound, i.e., lowers its enthalpy (although we will see in section 4 that the effect of conjugation on other properties is not monotonic). Westart from the standard form of the compound, which is essentially its most dominant conjugate form, and we proceed to consider alternative conjugates. One would expect that lower-energy conjugates (that exert the strongest influence on the enthalpy of a compound) are those that affect as few bonds as possible. One-bond conjugates cannot be used in the comparison of isomeric hydrocarbons, however, because the number of carboncarbon bonds and the number of carbon-hydrogen bonds does not vary within a set of isomeric alkanes. The next set of candidates would involve three bonds and their difference from the dominant conjugate would be described by the form
molecular formula
~ n=6arbonmtom
.
..
.
.
l
nr7 ~ urbon atom
ii
..
. f
.I
8 I
E
- 1 -
-12-
10
20
30
40
20
30
40
50
Number o( CCCH conjugan
numbn ol CCCH conjugnu
Figure 1. Enthalpyof formation (Cox and Pilcher, 1970)of isomeric alkanes, as a function of the number of CCCH conjugates. Each graph represents alkanes with a fixed number of carbon atoms (n). These observations indicate the potential of a quantitative framework for estimating enthalpies of formation from the number of conjugates.
methylpropane. In agreement with the stated rule and the experimental enthalpies of formation, the higher number of conjugates leads to lower enthalpy. Even though other conjugates which correspond to the form Y-C-C-X also have an appreciable contribution, our observations are based only on the CCCH conjugates because this type of conjugate has higher a priori significance originating from conjugation. A similar empirical rule which states that the enthalpies of alkenes increase with their degree of substitution, Le., the number of side chains attached on the double-bonded carbons, can also be explained by conjugation (Mavrovouniotis, 1990a). Figure 1presents further evidence that conjugation bears a quantitative relationship to the enthalpy of formation. Plots of the enthalpy of formation as a function of the number of CCCH conjugates, for alkanes with a fixed number of carbons, show a strong linear correlation. To generalize over the whole class of alkanes, one can obtain, quantitatively, those enthalpy differences that are due to additive bond energies, by performing a linear regression of the form
+
AHf=a bn where the parameters a and b are linear combinations of the carbon-carbon bond energy E(C-C) and the carbonhydrogen bond energy E(C-H): u = 2E(C-H) - E(C-C)
b = 2E(C-H)
+ E(C-C)
The results of this linear fit, with experimental data from Cox and Pilcher (19701, are a = -52.3 kJ/mol; b = -20.42 kJ/mol The standard deviation is 6.1 kJ/mol and the average absolute value of the errors is 5.1 kJ/mol. To test the
1736 Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993
-
n Q)
4,
I
0
e
7
8
2:
'
Figure 3. Conjugation operators (indexed 1-5) involving only one bond.
0.
C 1
I
-2.
Q
c
4-1
P
+
-64
a
0
W
I 2
4
6
N/n
8
Figure 2. Correlation of the errors from a bond-energy fit to the number of CCCH conjugates ( M .
hypothesis that conjugation should explain the portion of the enthalpy that is not due to bond energies, a graph is prepared of the error of the fit (defined as the estimated value minus the experimental value), for each compound, as a function of the number of CCCH conjugates; both axes are normalized by the number of carbon atoms n (Figure 2). The graph confirms that deviations from bondenergy additivity are correlated to the number of CCCH conjugates.
3. Conjugation Operators and Enthalpies of Formation The ideal-gas enthalpy of formation a t 298 K is the first property of interest here; in the rest of this article, it will be referred to as the energy E. In subsequent sections, the energy is used as a weighting factor for conjugate forms in the estimation of other thermodynamic and physical properties. However, accurate estimation of the enthalpy of formation is valuable in itself, and even modest differences may be important in applications. For example, in parallel chemicalreactions, the selectivity is often correlated to the enthalpies of the alternative products, which may be similar isomeric compounds. A small difference in the enthalpies of two compounds or reaction intermediates can have a substantial impact on the selectivity of a process. These differences are more pronounced in compounds that are affected significantly by conjugation, making our approach, which handles conjugation in a systematic way, all the more valuable. In Appendix B, we focus our attention on the quantitative combination of the energies of conjugate forms to determine the energy of the hybrid compound. Equation B24 is the outcome of our analysis:
E = Eo- c N i e i I20
where Eo is the energy of the dominant conjugate, Ni is the number of conjugates characterized by a particular energy Ei, and ei is related to the difference between a conjugate of energy Ei and the dominant conjugate. The last parameter is given by the relation ei = A exp[A-'(E,
- Ei)l
(I3251 where A is a positive parameter which has molar energy units (kJ/mol). To derive a concrete model from the general quantitative analysis which leads to (B24) and (B25), one must identify the types of conjugates that will be used and relate Eo and ei to specificatom and bond contributions. For a restricted class of compounds, one must be especially careful to include only those adjustable parameters (contributions) which are mathematically independent.
For acyclic hydrocarbons, the energy of the dominant conjugate is simply taken to be a linear function in the number of bonds of different types in the dominant conjugate:
Eo = X,(C-C) n(C-C)
+
+ Xd(c=c)
Xd(c=c)
n(C=C) n(C=C) + X,(C-H) n(C-H) (1)
where, for each type of bond, n(.) represents the number of bonds and Xd(.) the contribution. Contributions from atoms cannot be included because the number of carbon (or hydrogen) atoms is linearly dependent on the numbers of bonds. For the same reason, a constant term (usually present in semiempiricallinear correlations) is not included in the energy. A recessive conjugate refers toa specificalternative form of a particular compound. A whole class of conjugates can be described by a conjugation mode or conjugation operator which is a recipe that yields a recessive conjugate when applied to the dominant form of any compound from a given class. The description and computational identification of conjugation operators is discussed by Prickett et ai. (1993). Here, conjugation operators that affect either one bond or three bonds will be used. Each of the five potential one-bond operators (Figure 3) modifies, by a constant amount, the energy of any dominant conjugate on which it acts. Thus, each operator corresponds to a fixed Eo- Ei and a fixed ei given by (B25). The parameters el to e5 will be treated as adjustable parameters. For the estimation of the enthalpy of formation, it turns out that the contribution of the basic operator 5 cannot be distinguished, in the regression, from the other adjustable parameters-especially the contribution of operator 4. Thus, operator 5 is dropped. Each Ni in (B25) represents the number of ways in which the operator can be applied to the dominant form of the compound. The Ni's are related to the numbers of bonds:
= 2n(C=C); N4 = N5= n(C-H) (2) As explained in section 2, conjugation operators involving three bonds are particularly important for distinguishing among isomers. There are, theoretically, 25 three-bond operators, and their occurrences (Nil must be counted from the molecular structure, manually (Appendix A) or computationally (Prickett et al., 1993). Their corresponding contributions (ei) are actually not extra adjustable parameters, because thev can be derived from those of the one-bond operators. Cbnsider, as an example, the operator C3H or CCCH: Nl = 2n(C-C); N2 = 2n(C=C);
N3
C-C-C-H c+ C'+ ...C =C ...H1(3) Using ei (1Ii I5) to denote the contribution of a basic operator i, the effect of C3H on the dominant conjugate can be decomposed as follows: C-C-C-H
el .-+
C'+ ...C'--C-H
e4
+ e*-'
...
C'+ ...C1--C1+...H'-+ C'+ C=C ...HI- ( 4 ) with el and e4 denoting application of the corresponding operators from Figure 3, and e2-l application of operator
Ind. Eng. Chem. Res., Vol. 32, No. 8,1993 1737 Table 11. Contributions of Conjugation Modes to Estimation of the Enthalpy of Formation decornpnskbn lnm basic operaton:
C = r
;
C
C-!
&l*;
t
t
l + M l -
C=C
e
e-l
;
C&+
t
; t
l+C..,Cl- C& e1
conjugation modes H-C C-H C-C
C-H P t
l+C,..Hl-
e3-l
e4
r-elm:
t4-C C ............ C C ......H1Figure 4. Decomposition of the operator given by form 6 into basic one-bond operators. The one-bond Operators have been arranged in the sequence that corresponds to the bonds in the dominant and recessive formsofthelong operator. Thecontributionofthisoperator, is derived from the product of the contributionsof the incident onebond operators (7).
2 in the reverse direction. The effect of the operator C3H on the energy is merely the combination of the effects of the operators in the decomposition (B15); since operator 2 is used in the reverse direction, its energy must be subtracted rather than added. I t can be shown from this principle and (B11) that multiplication of the corresponding ei)s (or their inverses for a reversed operator) yields the ei of the composite operator. Here
e(CCCH) = ele4e;'
(5)
A similar decomposition can be carried out for each of the 25 three-bond operators. Longer operators can also be decomposed in this way. For example, the operator
can be decomposed as shown in Figure 4. The contribution for each occurrence of this mode of conjugation would be: e [ ( 6 ) ]= e2eC1e1e;'e4 = e1ec1e4
(7)
Thus, regardless of the length of the operator chains used, the set of adjustable parameters includes only the bond contributions Xd(.) and the contributions of the basic operators, ei for i = 1-4 (since operator 5 has been dropped). The contribution of any other (longer) conjugation operator for hydrocarbons can be expressed as a function of the basic operators, without any new adjustable parameters. A regression was carried out using experimental data for the enthalpy of formation of 117 acyclic hydrocarbons, as given by Cox and Pilcher (1970). The optimization algorithm was a modified Levenberg-Marquardt method used to solve nonlinear least squares problems (Dennis and Schnabel, 1983). This method was used to carry out all regressions presented in this article. The standard deviation of the regression was 4.2 kJ/mol. The values derived for the adjustable parameters are e, = 2.035; e2 = 14.205; e3 = 28.848; e4 = 3.530 (in kJ/mol) (8)
xd(c-c)= 20.574; xd(c=c) = 144.045;xd(c=c) = 317.154; xd(C-H) = -13.692 (in kJ/mol) (9) Using the values of el to e4, we estimate the ei contributions of the three-bond conjugation modes by decomposingthem into the basic operators (Table 11). For comparison, a group-contribution fit on the same set of compounds, using a total of nine groups yielded the following contributions:
c=c c=c
H-C-C-H H-C-C-C H-C-C=C H-C-CEC H-CeC-H H-C=C-C C-C-C-H C-C-C-C
ei
(kJ/mol) a 3.530 2.035 14.205 28.848 a a a a
a a 0.506 0.292
conjugation modes c-C-C=c C-C-CEC C-C=C-H c-C=c-C C-C-C-H c-C-C-C c=c-C=c c=c-C=c CXX-H C5C-C-C c=c-C=c CIC-CIC
ej (kJ/mol)
2.035 4.133 0.249 0.144 3.530 2.035 14.205 28.848 7.168 4.133 28.848 58.581
a The contribution of these conjugation modes is zero because operator 5 (H-C) was neglected.
k(-CH,) = -42.781; L(>CHZ) = -20.569; k(>CH-) = -2.340; k(>C
