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Ind. Eng. Chem. Res. 2010, 49, 9394–9406
Group-Contribution Method for the Molecular Parameters of the PC-SAFT Equation of State Taking into Account the Proximity Effect. Application to Nonassociated Compounds Javier Vijande,* Manuel M. Pin˜eiro, and Jose´ L. Legido Departamento de Fı´sica Aplicada, Edificio de Ciencias Experimentais, UniVersidade de Vigo, E-36310 Vigo, Spain
David Bessie`res Laboratoire des Fluides Complexes, Groupe Haute Pression, UniVersite´ de Pau et des Pays de l’Adour, B.P. 1155, F-64013 Pau Cedex, France
A new group-contribution method to obtain the PC-SAFT molecular parameters of nonassociated compounds (linear alkanes, branched alkanes, linear monoethers, and esters) is developed on the basis of their regular trends against molecular mass. The method takes into account the proximity effect among all functional groups of a molecule in order to describe the deviations of shorter chain molecular parameters from the regular trend observed for the larger chain ones. The group-contribution scheme considers the molecular parameters as a linear combination of reference functional-group parameters and their mutual perturbations, which are responsible for the deviations from the reference ones. Both types of parameters (the reference functional-group parameters and the mutual perturbations) are optimized to the available molecular parameters published in the literature and obtained by direct fitting from experimental data of saturation pressures and saturated liquid densities. 1. Introduction Group-contribution models have demonstrated a reasonable capacity for predicting thermodynamic properties and phase equilibria of pure compounds and mixtures. This kind of model considers the molecules as constituted by several functional groups, usually defined by traditional carbon chemistry, and characterized by a few parameters representing their shape, size, or interaction energies. Their main advantage if compared with the molecular ones is that group-contribution models allow determining the properties of any compound of the same homologous series just by knowing the characteristic parameters of their constituent functional groups. However, considering functional groups as essential unities of molecules is only an aproximation and a criterion is necessary to define what set of atoms is considered as a functional group and if their properties can be considered as unalterable or not due to the presence of other groups in the molecule. The functional group assignment must be able to distinguish among several molecules containing the same functional groups located in different positions (structural isomers), for instance, diethyl ether (CH3-CH2-O-CH2-CH3) and methyl propyl ether (CH3-O-CH2-CH2-CH3). Concerning this, the Dortmund-UNIFAC model1,2 defines the sets CH3O- and -CH2Oas functional groups in order to characterize the molecules of ethers and it makes it possible to distinguish the molecules of the previous example: CH3-CH2O-CH2-CH3 and CH3O-CH2-CH2-CH3. However, another group assignment is possible for methyl propyl ether: CH3-OCH2-CH2-CH3, where -OCH2- is essentially the same group as -CH2O-. Other different functional group assignment criteria have been used as well in the development of group-contribution models. One of the most rigorous criteria is due to Carballo et al.3 These * To whom correspondence should be addressed. E-mail: jvijande@ uvigo.es.
authors developed an analytical quantum-mechanic method based in the “atoms-in-molecules theory” of Bader4 which allows a rigorous splitting of molecules into fragments called “quasi-atoms”. The method proposed by Carballo et al.3 allows identification of the set of quasi-atoms that contribute to the formation of a functional group on the basis of the size invariability in the molecule. On these grounds, the proximity effect between neighboring groups could be taken into account and described quantitatively. The authors applied this method first to redefine functional groups in n-alkane and ester molecules. The methyl and methylene group designations of n-alkanes were not modified and were kept in the traditional way, but on the other hand methyl and methylene groups in esters were redefined depending on their relative position with respect to the ester group. Therefore, the set of functional groups in ester molecules was composed of methanoate (HCOO-), methyl carboxylate (CH3COO-), ethyl carboxylate (-CH2COO-), methyl, and methylene in positions R and β (CH3R-, CH3β-, -CH2R- and -CH2β-) for positions adjacent to ester groups, and again traditional methyl and methylene, identical to the n-alkane groups, for other positions. The same procedure was applied by Gonzalez et al.5 to redefine the functional groups for primary alkanols, and the result was similar to that obtained for esters, as the hydroxyl functional group modifies the shape of the closer methyl and methylene groups. In both cases the functional group assignment criterion was applied to Nitta et al.’s6 group-contribution model and demonstrated that the main group (ester or hydroxyl) in the molecule modifies the structural characteristics of the nearest functional groups, methyl and methylene, with respect to those of reference groups (i.e., methyl and methylene groups in linear alkanes). This modification is extended only to a maximum distance of two functional groups. In past years, several group-contribution methods were applied to different SAFT versions derived from the original
10.1021/ie1002813 2010 American Chemical Society Published on Web 08/25/2010
Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010 7,8
SAFT equation of state due to Chapman et al. For instance, Lora et al.9 developed a pseudogroup-contribution method for the calculation of the segment number, m, and the segment volume, V°°, of pure polymer compounds for the original SAFT equation of state (EoS). However, this method could not be applied to determinate the attractive energy of a segment, u°/k, because it does not depend linearly on molecular structure. A similar pseudogroup-contribution scheme was developed in our previous work10 for the application of the PC-SAFT approach11 to hydrofluoroethers and n-alkanes. The method was defined on the basis of the linearity of coefficients m, mε, and mσ3 against the molecular mass of compounds, an event that was corroborated by several authors for different versions of the SAFT equation of state.11–13 The dependence of coefficients m, mε, and mσ3 on the molecular mass was described as a function of the functional-group number through eqs 1. The proximity effect among different functional groups was taken into account in a implicit manner, following Carballo et al.,3 by distinguishing different groups depending on their relative position with respect to the main group. This method consists of adjusting the group-contribution parameters involved to the molecular ones obtained by direct optimization of experimental data such as vapor pressures and saturated liquid densities or compressed liquid densities. Simultaneously, Tobaly and co-workers14–20 developed a different group-contribution method that was applied to original SAFT,7,8 PC-SAFT,11 and SAFT-VR21 including the association parameters, the molecular parameters to describe the multipolar term of residual Helmholtz free energy, and the binary interaction parameters of the combining rules for the mixtures. The dependence of the molecular parameters on the molecular mass was described as a function of the functional-group number using a different approach from that of Vijande et al.:10 the group parameters were obtained by direct optimization of the experimental data of a series of fluids. From a different perspective, Lymperiadis et al.22 developed a group-contribution SAFT equation of state, called SAFT-γ, as an extension of the SAFT-VR model.21 In SAFT-γ the traditional segments of the SAFT EoS are substituted by functional groups considered as heteronuclear spherical segments and characterized by a diameter, a well depth, and range parameters representing the dispersive interaction, and by a shape factor parameter which denotes the proportion of the spherical united-atom group that contributes to the properties of the molecule. In the association term, the bonding sites are located at the groups instead of segments and it is expressed in terms of the association parameters which are assigned to the groups. A subsequent work23 generalizes the SAFT-γ equation of state to treat functional groups which are represented by more than a single spherical segment, allowing a good description of properties of molecules built up of large functional groups such as carbonyl and carboxyl. Recently, another group-contribution method24 was developed on the basis of the linearity of coefficients m, mε, and mσ3 against the molecular mass of compounds. As in previous works,10,14–20 this method is used to determine the molecular parameters as a function of the functional-groups number present in the molecules. In this case, the simplified PC-SAFT25 is combined with the Constantinou-Gani group-contribution method26 to determine the three characteristic molecular PCSAFT parameters. This method includes two levels of contribution: in the first level, the molecular parameters are estimated as an addition of the contributions of first-order groups which are constituents of the molecules; the second level considers a
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set of simple and small second-order groups, which uses the first-order groups as building blocks and makes it possible to capture the proximity effects and distinguish among structural isomers. Finally, McCabe and co-workers27 have proposed a new group-contribution method, denoted as GC-SAFT-VR, based on the hetero-SAFT-VR equation of state28,29 that considers the molecules composed of tangentially bonded segments of different sizes and/or interaction energies. These different segments represent each type of functional group characterized by a diameter, a well depth, and a potential range. Thus, the different contributions to the free energy are expressed as a function of the number of each type of functional group for a pure compound, or for each compound in a mixture. The GC-SAFTVR approach accurately predicts the phase behavior of pure compounds and mixtures, as has been demonstrated in the original work27 for different mixtures of alkanes, ketones, alkylbenzenes, and esters and in a subsequent work30 where the phase behavior of polymer systems was studied. The linear dependence of coefficients m, mε, and mσ3, corroborated by several authors for different versions of the SAFT equation of state,11–13 is generally valid for hydrocarbons, but not for other compounds made up of functional groups which are structurally very different. This is due to the proximity effect between different functional groups. In normal alkanes, the functional groups methyl (CH3-) and methylene (-CH2-) are structurally similar and the effect of their mutual interactions is almost indistinguishable. However, in other compounds constituted by alkyl chains and another, different functional group, the main functional group modifies the properties of the nearest methyl and methylene groups. Consequently, the trends of m, mε, and mσ3 depart from linearity for compounds of shortest chains as shown by Lafitte et al.31 for alkanols, in the context of the SAFT-VR Mie32 equation of state, or by Vijande et al.10 for hydrofluoroethers, in the context of PC-SAFT. In our previous work,10 the proximity effect in hydrofluoroethers was partially taken into account by defining the groups adjacent to the main one as groups different from those of the same type located at other positions in the molecule. The problem of this option is that the number of group types necessary to constitute a given homologous series increases considerably. For this reason, the aim of the present work is to develop the original method in order to take into account the proximity effect without increasing the number of functional groups. Since the trends of m, mε, and mσ3 scarcely depart from linearity for normal alkanes, the proximity effect between methyl and methylene groups in these hydrocarbons is almost negligible. However, we have considered this effect in order to maintain the coherence in the formalism of the present group-contribution method. 2. Group-Contribution Method It is obvious that the segment number, m, increases with the molecular mass and this increment is linear if the molecular mass increases due to the same functional group. The parameter σ is the diameter of each segment of the molecule, so σ3 is related to the volume of this segment and mσ3 is related to the total volume of the molecule. Thus the total volume of the molecule linearly increases with the increment of a given functional group. The segment energy parameter, ε, is the depth of the segment potential well, and mε is the sum of the potential well depths of all segments that linearly increases with molecular mass for compounds of the same series. Both parameters σ and ε approach an asymptotic value when the chain length increases
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because the addition of new groups to longer chains does not essentially modify the structure of the molecule. In the original work,10 each of these coefficients was expressed as a function of the number (nR) of each functional group type (R) present in a molecule, and of their group contributions to the molecular parameters (mR, εR, σR): mGC ) mGCεGC ) mGCσGC3 )
∑n m R
R
(1)
R
∑n m ε R
R R
(1)
R
∑
πGC )
nRmRσR3
R
∑n π *
(3)
R R
R
and the perturbed coefficient can be expressed as
where the sums extend to all different types of groups (R) in the molecule. From now on, we will denote by π any of these coefficients m, mε, or mσ3, without subscript if it refers to molecular parameters obtained by direct optimization to experimental data, with the subscript GC if it refers to those obtained by the present group-contribution method, or with a subscript (R) that represents a kind of group if it refers to the group parameters. Equations 1 can be condensed in the following: πGC )
establish what the reference groups are, and how the functional groups modify mutually their properties. The reference groups must be defined for each compound series, and they will be treated later. The way that a given group alters the properties of others can be already defined. Let us denote as πR the coefficient π value for the reference group of type R, which we call the reference functional-group coefficient, and πR* is the value of the same coefficient perturbed by all the other groups present in the molecule that can affect it. Then, eq 2 can be rewritten as
∑n π
R R
(2)
R
where πR is the contribution of each group of kind R to the molecular coefficient π. However, the linearity of π coefficients is not perfect but some deviations are observed in compounds of shorter chain due to the perturbation that each group exercises over the others, an effect that, as said before, decreases for longer chain molecules. In the original group-contribution method, we considered that a functional group does not affect the properties of the others in an explicit way. Instead, we considered the proximity effect in an implicit way by the definition of different groups depending on their relative position respect to the main group. The definition of the groups in hydrofluoroethers and n-alkane molecules was made following the criteria exposed previously due to Carballo et al.,3 with the assumption that the ether group -O- modifies the properties of the methyl (CH3-), methylene (-CH2-), perfluoromethyl (CF3-), and perfluoromethylene (-CF2-) groups bonded directly to it (which was denoted in former works as the R position). Besides, the alteration of the properties on the groups in what was referred to as the β position (second neighbor) was considered negligible, because different attempts to define β groups did not bring any improvement in estimation results, but resulted in an undesirable, and in this case avoidable, increase of functional group number. The main problem created by this criterion of definition is that a large number of functional groups are necessary to describe different molecules, since the same atom group must be defined as two or more different functional groups depending on its position in the molecule, or the kind of molecule where it is located. Let us then reconsider the definition criterion of functional groups in order not to increase its number by extending the contribution method to any other molecule. Let us accept that a given atom set always constitutes the same functional group even in different molecules. Let us accept, too, that the properties of this group are affected by another functional group (of the same or another kind) in the molecule and, at the same time, accept the reciprocity of this effect. This criterion allows us to
πR* ) πR + ∆πR
(4)
where ∆πR is the total perturbation produced in the groups of R kind due to the other groups (of any kind). Thus, eq 3 can be split into two terms: πGC )
∑n π
∑ n ∆π
+
R R
R
R
(5)
R
R
In order to evaluate the perturbation caused in the groups of R kind by the others, we must take into account that each group occupies a different position and, therefore, its properties will be affected as a function of this position. That is, ∆πR has not a single value and it is not common to all groups of R kind, but it depends on the position of each group. Thus, it is necessary to evaluate each group of each kind separately. Let ∆πRi be the perturbation experienced by a single group Ri of R kind due to all other groups. The total perturbation will be
∑
nR
nR∆πR )
R
∑ ∑ ∆π R
(6)
Ri
i)1
A group Ri is perturbed by all and each of nβ groups of any kind β. Let ∆πRi-βj be the perturbation of group Ri due to one only group βj of kind β. Then, the total perturbation is
∑ n ∆π R
R
nR
R
)
∑ ∑ ∆π R
i)1
Ri
nR
)
nβ
∑ ∑ ∑ ∑ ∆π R
i)1
β
j)1 nR nβ
Ri-βj
∑ ∑ ∑ ∑ ∆π R
β
i)1 j)1
)
Ri-βj
(7)
It is obvious that the perturbation experienced by a group is in inverse relation to its relative position with respect to the causative group. That is, the closer both groups are, the bigger will be this effect. This is the well-known proximity effect studied in several group-contribution models,33,34 although not in a systematic way. The perturbation has a maximun value between two adjacent groups, and it tends to zero when the distance between them tends to infinity. Let pRi-βj be the relative position of group Ri with respect to group βj. Let us define this position as the number of bonds between both groups, i.e., 1 for neighbors, ... . The perturbation of the group Ri due to βj can be written as
∆πRi-βj )
µπ,R-β pRi-βj
(8)
Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010
where µπ,R-β is the perturbation caused in coefficient π of any group of kind R by any other of kind β adjacent to it. This new parameter, that we will call the mutual perturbatiVe parameter, is independent of the group position because it is defined as the perturbation produced between adjacent groups (that is, in a given position) and only depends on the type of interacting groups. Furthermore, this parameter is different for each coefficient π. Finally, taking into account eqs 5, 7, and 8, as well as the independence of µπ,R-β with the group position, the molecular coefficient π expressed as a linear combination of the contributions of each group is πGC )
∑n π
R R
+
R
nR
∑ ∑µ R
π,R-β
∑ ∑p1
nR
nβ
∑ ∑p
mR + mβ µε,R-β 2
µmσ3,R-β )
mR + mβ µσ,R-β3 2
(15)
so the mutual perturbative parameters of ε and σ are
(9)
µσ,R-β )
(
2µmσ3,R-β mR + mβ
(16)
)
1/3
where R is the perturbed group and β is the group which causes this perturbation. Note that the mutual perturbative parameters have the same units as the respective reference functional-group coefficients or parameters. In short, eq 11 can be expressed for each coefficient πGC as follows:
∑n m
mGC )
R
∑ ∑µ
+
m,R-βSR-β
R
∑n m ε R
+
R R
R
(10) mGCσGC3 )
R
R
mGCεGC )
1
2µmε,R-β mR + mβ
µε,R-β )
Ri-βj
i)1 j)1 Ri*βj
µmε,R-β )
Ri-βj
In eq 9, πR and µπ,R-β were determined by fitting to the molecular coefficients π obtained from the literature, as usual, by direct optimization of the experimental data of densities or the saturation curve using the PC-SAFT equation of state. The sums spread over all different groups (Ri * βj), including those of the same kind, since even though a group does not perturb itself, it does perturb another one of the same kind in the molecule; i.e., the coefficient µπ,R-R does not necessarily equal zero. Note that the sums of indices i and j in eq 9 only depend on the relative positions of the groups and they are a common factor to all coefficients π, i.e., to m, mε, and mσ3. They will be expressed by SR-β: SR-β )
perturbative parameters µε,R-β and µσ,R-β have not been defined yet. In order to establish a value for the perturbations of group contributions of the parameters ε and σ, keeping symmetry, we can define the mutual perturbative parameters of coefficients mε and mσ3 as follows:
nβ
i)1 j)1 Ri*βj
β
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∑n m σ R
β
∑ ∑µ
mε,R-βSR-β
R
3
R R
+
R
(17)
β
∑ ∑µ R
mσ3,R-βSR-β
β
so eq 9 can be summarized as πGC )
∑n π
R R
R
+
∑ ∑µ R
π,R-βSR-β
or else
(11)
Note, moreover, that the relative position of group Ri with respect to βj is equal to the relative position of βj with respect to Ri and, then, the sums mentioned before are symmetrical, i.e. SR-β ) Sβ-R
(12)
This means that these sums are a common factor to the perturbations of β over R (µπ,R-β) and R over β (µπ,β-R) and, therefore, only the sum of both parameters will be obtained in parameter optimization, but not each one of them separately. We have supposed that those nonzero perturbations (we insist on this condition of non-nullity) will be symmetric also since the meaningful result is not the value of each one separately, but the sum of both. Thus µπ,R-β ) µπ,β-R
(13)
which stands for µm,R-β ) µm,β-R µmε,R-β ) µmε,β-R
∑n m
mGC )
β
(14)
µmσ3,R-β ) µmσ3,β-R These equalities apply to the coefficients m, mε, and mσ3, but not to the parameters ε and σ necessarily because the mutual
mGCεGC )
R
R
+
R
∑n m ε R
R R
+
R
∑ ∑µ
m,R-βSR-β
R
1 2
β
∑ ∑ (m
R
R
+ mβ)µε,R-βSR-β
β
(18) mGCσGC3 )
∑n m σ R
R
3
R R
+
1 2
∑ ∑ (m
R
R
+ mβ)µσ,R-β3SR-β
β
3. Group-Contribution Parameters The process of determination of the reference functionalgroup parameters mR, εR, and σR, as well as the mutual perturbative ones µm,R-β, µε,R-β, and µσ,R-β consists of adjusting them to molecular parameters m, ε, and σ of the PC-SAFT EoS which are obtained by direct optimization from experimental data and are available in the literature. Gross and Sadowski11 determined the molecular parameters of n-alkanes, branched alkanes, linear monoethers, and esters using experimental data of the vapor pressures and saturated liquid densities of the pure compounds in a wide range of temperatures. The molecular parameters for other heavy n-alkanes were determined by Voutsas et al.,35 Agarwal et al.,36 and Tihic et al.37 Also, we have found parameters for heavy n-alkanes (from eicosane up to hexatriacontane) from von Solms et al.;25 however, they have not been fitted to experimental data, but have been extrapolated from the parameters of Gross and Sadowski.11 For branched alkanes, in addition to those of Gross and Sadowski,11 molecular
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parameters were obtained by Aparicio-Martinez and Hall38 and Tihic et al.37 and the molecular parameters of ethers and esters were optimized by Tihic et al.,37 extending the original matrix. Table SI-1 (see the Supporting Information) summarizes these molecular parameters from the authors cited. However, generally speaking, these parameter sets do not show any common and suitable trends against molecular weight, in contrast to what is found in the case of Gross and Sadowski,11 but in all cases they offer good predictive ability. Our main objective was to identify regular trends for the different parameters, keeping as much as possible the quantitative accuracy in the estimation of the molecular values, and for these reasons the Gross and Sadowski11 and Tihic et al.37 sets of molecular parameters were selected to correlate the group-contribution ones. The fitting process is carried out in several consecutive steps. First, the methane group is treated as an independent group and the reference group and perturbative parameters of methyl and methylene groups of n-alkanes are adjusted. Second, the reference group parameters of other groups and perturbative ones between methyl and methylene can be adjusted using molecular parameters of compounds containing methyl, methylene, and the group in question. Finally, the mutual perturbative parameters between two groups different from methyl and methylene can be adjusted with molecular parameters of compounds which contain both of them. In our case, this last step has not been attempted because the available molecular parameters correspond to compounds containing only one non methyl or methylene functional group. The molecular parameters obtained by direct fit from several authors and those obtained by this group-contribution scheme are listed in Table SI-1 (see the Supporting Information), and Figure 1 shows the dependence of coefficients m, mε, and mσ3 on molecular mass. On the other hand, Tables 1 and 2 show the reference functional-group parameters and the mutual perturbative ones respectively for the groups involved in the compound series cited before. Next, we explain in further detail for each compound series the reference groups and the development of eqs 17. 3.1. n-Alkanes. Methyl and Methylene Groups. We should consider methyl (CH3-) and methylene (-CH2-) as reference groups. Then, both methyl and methylene groups in n-alkanes would be perturbed by each other. However, neither methyl nor methylene can be located separately as a neutral molecule, but they will always be bonded to other functional groups. The optimal situation would be then having the main reference group bonded to a chain containing a maximum of only one different group. For example, in a n-alkane, the methyl group will be isolated from the other methyl when the chain length is long enough, but then it will not be isolated from a methylene group. Likewise, a methylene group located in the middle of the chain can be isolated from methyl groups for n-alkanes of a large enough chain, but then it cannot be isolated from other methylene groups. Then, we can consider that those are the reference situations: 1. The reference group methyl, CH3, is one bonded to a infinite chain of methylene groups: CH3-CH2-CH2-CH2-... 2. The reference group methylene, CH2, is one bonded to infinite chains of methylene groups: ...-CH2-CH2-CH2-CH2-CH2-...
Therefore, in an n-alkane, the group CH3 is not perturbed by the groups CH2, but by the other group CH3 located at the other end of the chain due to proximity effect. That is µπ,CH3-CH3 * 0 and µπ,CH3-CH2 ) 0 Likewise, the group CH2 is not perturbed by the others groups of his same kind, but by the two groups CH3 located at both ends of the chain. That is to say: µπ,CH2-CH3 * 0 and µπ,CH2-CH2 ) 0 Here, we can see a case where there is not symmetry of the mutual perturbative parameters between groups methyl and methylene, since the parameter µπ,CH3-CH2 is zero while µπ,CH2-CH3 is not. However, this is not an exception to the rule since the first parameter does not verify the condition of non-nullity, as stated before. n-Alkanes are made up by two methyl groups and nCH2 methylene groups, so eqs 17 can be summarized as follows: πGC ) 2πCH3 + nCH2πCH2 + µπ,CH3-CH3SCH3-CH3 + µπ,CH2-CH3SCH2-CH3
(19)
The additions SR-β are obtained by developing eq 10. A given n-alkane contains two groups CH3 at the ends of the chain, both of them separated by nCH2 groups of the kind CH2. Then, the relative mutual position is pCH3-CH3 ) 1 + nCH2 and, therefore 2 1 + nCH2
SCH3-CH3 )
(20)
However, the groups CH2 occupy all possible positions between the two groups CH3, and each of them is perturbed by both methyl groups. A CH2 group located at position p with respect to one of the methyl groups, will be at position nCH2 + 1 - p with respect to the other. Therefore, the corresponding addition, spread over all methylene groups and over the two methyl ones, is nCH2
SCH2-CH3
∑
1 + ) p p)1
nCH2
∑n
nCH2
p)1
CH2
∑
1 1 )2 +1-p p p)1
(21) In the correlation of eq 19 we have used only the molecular parameters optimized by Gross and Sadowski11 for n-alkanes from ethane to eicosane and we have exclude those of heavier alkanes obtained by other authors.35–37 The reason for this is that the latter parameters show a substantial dispersion with large deviations from the gradual trend of the molecular parameters obtained by the group-contribution method. 3.2. Branched Alkanes. Tertiary Aliphatic Carbon Group. The branched alkanes containing a tertiary aliphatic carbon (CH group) are constituted of three aliphatic chains of methylene groups bonded at CH, all ending in a methyl group. The reference group CH is then one bonded to three infinite chains of methylene groups. Therefore, the CH group changes the properties of both the methyl and methylene groups; each methyl group changes the properties of CH, methylene, and methyl groups; and methylene groups do not change the properties of each other. Thus πGC ) πCH + 3πCH3 + nCH2πCH2 + µπ,CH3-CH3SCH3-CH3 + µπ,CH3-CHSCH3-CH + µπ,CH2-CH3SCH2-CH3 + µπ,CH2-CHSCH2-CH + µπ,CH-CH3SCH-CH3 (22)
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Figure 1. Molecular coefficients, π, versus molecular mass obtained from literature parameters11,35–38 (points) and obtained by the group-contribution method (lines).
Since SCH3-CH ) SCH-CH3 and we have supposed that µπ,CH3-CH ) µπ,CH-CH3, then eq 22 is summed up in πGC ) πCH + 3πCH3 + nCH2πCH2 + µπ,CH3-CH3SCH3-CH3 + 2µπ,CH3-CHSCH3-CH + µπ,CH2-CH3SCH2-CH3 + µπ,CH2-CHSCH2-CH (23)
The CH group is bonded to three aliphatic chains that will be referred to as left, right, and up chains, with a number of methylene groups equal to nleft, nright, and nup, respectively, which add up to nCH2 groups of this kind. At the end of each chain is located a methyl group. The relative position of the methyl groups at, for example, the left and upper chains is 2 + nleft +
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Table 1. Reference Functional-Group Parameters R CH4 CH3-CH2>CH>C< -OHCOO-COO-
name
mR
methane methyl methylene tertiary aliphatic carbon quaternary aliphatic carbon ether formate carboxylate
(εR/k)/K
1.000 0 0.378 463 0.314 747 -1.597 00
150.03 160.224 273.852 -37.2512
SCH2-CH )
σR/Å 3.703 9 4.468 93 4.249 47 2.664 63
-1.236 40
-110.744
3.297 60
0.430 962 1.589 22 -1.140 69
132.160 332.148 -108.095
6.446 45 3.488 61 -3.561 24
∑ ∑ 2 + n1 + n i∈A j∈A j*i
i
i∈A
i∈A j∈A p)1 j*i
i
1 + nj - p
Most of the molecular parameters of methylalkanes were optimized by Gross and Sadowski,11 and those of 3-ethylpentane were optimized by Tihic et al.37 We have used both sets of parameters to correlate the group-contribution ones, obtaining good values for 3-ethylpentane, close to those obtained by Tihic et al. 3.3. Branched Alkanes. Quaternary Aliphatic Carbon Group. The C group (quaternary aliphatic carbon) is bonded to four aliphatic chains denoted by left, right, up, and down which contain nleft, nright, nup, and ndown methylene groups, respectively, and one methyl group at the end of every chain. As in the previous case, the reference group C is one bonded to four infinite aliphatic chains constituted by methylene groups. Taking into account the symmetry of the additions SR-β and mutual perturbative parameters µπ,R-β, the coefficients πGC are
(24) j
The rest of the additions can be obtained in a similar way:
∑ 1 +1 n
∑ ∑ ∑2 + n
(27)
where A is the set of aliphatic chains (left, right, up) in the molecule.
SCH3-CH )
(26)
i∈A p)1 ni
SCH2-CH3 ) SCH2-CH +
nup since the CH group, which separates both chains, occupies one position, too. In a similar way, the relative positions of the other methyl groups of the molecule can be obtained so SCH3-CH3 )
∑ ∑ p1
πGC ) πC + 4πCH3 + nCH2πCH2 + µπ,CH3-CH3SCH3-CH3 + 2µπ,CH3-CSCH3-C + µπ,CH2-CH3SCH2-CH3 + µπ,CH2-CSCH2-C (28)
(25) i
where Table 2. Mutual Perturbative Parameters µm,R-β perturbing group, β perturbed group, R
CH3
CH2
CH
C
O
HCOO
COO
CH3 CH2 CH C O HCOO COO
0.431 978 0.220 392 0.238 502 -0.066 208 0.171 809 0.355 359 0.777 015
0 0 0 0 0 0 0
0.238 502 0.068 613
-0.066 208 -0.179 736
0.171 809 0.001 950
0.355 359 0.029 420
0.777 015 0.531 642
(µε,R-β/k)/K perturbing group, β perturbed group, R
CH3
CH2
CH
C
O
HCOO
COO
CH3 CH2 CH C O HCOO COO
248.440 156.536 8.986 14 119.989 132.611 31.1860 -259.162
0 0 0 0 0 0 0
8.986 14 80.2345
119.989 134.743
132.611 4.964 83
31.1860 -50.3388
-259.162 -78.5781
µσ,R-β/Å perturbing group, β perturbed group, R
CH3
CH2
CH
C
O
HCOO
COO
CH3 CH2 CH C O HCOO COO
1.533 26 1.277 88 -2.317 11 -2.388 75 -4.017 47 -2.250 30 2.239 10
0 0 0 0 0 0 0
-2.317 11 -2.020 27
-2.388 75 -1.661 91
-4.017 47 -4.130 51
-2.250 30 -2.198 39
2.239 10 2.283 48
Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010
SCH3-CH3 )
∑ ∑ 2 + n1 + n i
i∈A j∈A j*i
SCH3-C )
∑ 1 +1 n i∈A
(29) j
(30) i
ni
∑ ∑ p1
SCH2-C )
(31)
i∈A p)1 ni
SCH2-CH3 ) SCH2-C +
∑ ∑ ∑2 + n i∈A j∈A p)1 j*i
i
1 + nj - p
(32) A is the set of aliphatic chains (left, right, up, down) in the molecule. The molecular parameters of 2,2-dimethylpropane and 2,2dimethylbutane have been optimized by Gross and Sadowski.11 For other compounds of the same series, Aparicio-Martinez and Hall38 have optimized the molecular parameters of 2,2-dimethylpentane and Tihic et al.37 have optimized those from 2,2dimethylhexane to 2,2-dimethyloctane. The molecular parameter sets that best fit to a common trend are those of Gross and Sadowski and Tihic et al.; on the other hand, these parameter sets were fitted from both vapor pressures and saturated liquid densities, whereas the Aparicio-Martinez and Hall set was optimized only from experimental vapor pressures. Therefore, we have used both sets of parameters (Gross and Sadowski and Tihic et al.) to correlate the group-contribution ones of the functional group C through eq 28. 3.4. Linear Monoethers. Ether Group. The reference ether group (-O-) is one bonded to two infinite aliphatic chains constituted by methylene groups, which do not change its properties. The linear monoethers contain nleft and nright methylene groups in the left and right aliphatic chains, respectively, and one methyl group at the end of every chain. The πGC coefficients are given by the expression πGC ) πO + 2πCH3 + nCH2πCH2 + µπ,CH3-CH3SCH3-CH3 + 2µπ,CH3-OSCH3-O + µπ,CH2-CH3SCH2-CH3 + µπ,CH2-OSCH2-O (33)
SCH3-O )
2 2 + nCH2
(34)
∑ 1 +1 n
(35)
i∈A
contribution parameters of the ether group and the perturbative ones because this way led to the best global results. 3.5. Esters. Formate Group. The alkyl methanoates are constituted by a formate or methanoate group (HCOO) located at an end of the aliphatic chain that contains nCH2 methylene groups and one methyl group located at the other end. The reference formate group is one bonded to a infinite aliphatic chain of methylene groups. Since there is a single methyl group in the chain, the addition SCH3-CH3 does not appear in the coefficient πGC expression. Moreover, since the methyl and formate groups are located at the ends of the aliphatic chain, the additions that relate these groups to methylene ones are the same. In short πGC ) πHCOO + πCH3 + nCH2πCH2 + 2µπ,CH3-HCOOSCH3-HCOO + µπ,CH2-CH3SCH2-CH3 + µπ,CH2-HCOOSCH2-HCOO where SCH3-HCOO )
SCH2-CH3 ) SCH2-HCOO )
i
∑ ∑ p1
∑ p1
(40)
p)1
πGC ) πCOO + 2πCH3 + nCH2πCH2 + µπ,CH3-CH3SCH3-CH3 + 2µπ,CH3-COOSCH3-COO + µπ,CH2-CH3SCH2-CH3 + µπ,CH2-COOSCH2-COO (41)
SCH3-CH3 )
2 2 + nCH2
SCH3-COO )
∑ 1 +1 n
(36)
∑ ∑ 2 + n1
CH2
-p
(37)
A is the set of aliphatic chains (left, right) in the molecule. The molecular parameters of some monoethers were optimized by Gross and Sadowski11 and Tihic et al.,37 and both parameter sets show approximately the same trends with molecular mass. However, only the Gross and Sadowski parameter set was used in this work to correlate the group-
i∈A
(42)
(43) i
ni
SCH2-COO )
i∈A p)1
i∈A p)1
(39)
In order to correlate the group-contribution parameters of group HCOO with eq 38, we have used only the molecular parameter set for alkyl methanoates found in the literature from Gross and Sadowski.11 3.6. Esters. Carboxylate Group. The alkyl alkanoates are constituted by a carboxylate or ester group (COO) bonded to two aliphatic chains ending in a methyl group each. The reference carboxylate group is one bonded to two infinite aliphatic chains, so it changes the methyl and methylene properties and the own are changed by the methyl but not by the methylene groups. The expression of πGC coefficients is
ni
SCH2-CH3 ) SCH2-O +
1 1 + nCH2 nCH2
ni
SCH2-O )
(38)
As in earlier cases, let nleft and nright be the number of methylene groups of the aliphatic chains located at left and right positions from the carboxylate group, respectively. The additions of eq 41 are
where the additions SR-β are SCH3-CH3 )
9401
∑ ∑ p1
(44)
i∈A p)1 ni
SCH2-CH3 ) SCH2-COO +
∑ ∑ 2 + n1 i∈A p)1
CH2
-p
(45)
A is the set of aliphatic chains (left, right) in the molecule. Even if the carboxylate is considered as a single group, due to its large size (it is constituted by a carbonyl group, >CdO, and one additional oxygen) it can be considered to occupy two positions instead of one. In this case, the position of one methyl
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Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010
Table 3. Average Percentage Deviations (APD%) and Absolute Average Deviations (AAD) of Vapor Pressures of Pure Compounds Obtained with Literature Parameters and Group-Contribution (GC) Parameters Psat literature parameters series
Np
linear alkanes branched alkanes linear monoethers esters
APD%
496 233 192 474
GC parameters
AAD/MPa
Np
-3
3 × 10 2 × 10-3 1 × 10-2 9 × 10-3
2.4 1.3 3.3 4.5
APD%
496 434 430 1202
AAD/MPa
source of exptl data
-3
3 × 10 1 × 10-2 1 × 10-2 1 × 10-2
3.6 6.0 6.2 12
40, 41 40–42 40–44 40–42,45–49
Table 4. Average Percentage Deviations (APD%) and Absolute Average Deviations (AAD) of Saturated Liquid (L) and Gas (G) Densities of Pure Compounds Obtained with Literature Parameters and Group-Contribution (GC) Parameters Fsat literature parameters series linear alkanes branched alkanes linear monoethers esters
Np L G L G L G L G
405 339 142 133 41 56 124 137
APD% 1.0 2.7 0.51 2.2 1.6 10 1.2 8.2
GC parameters -3
AAD/g · cm -3
5 × 10 1 × 10-3 3 × 10-3 9 × 10-4 9 × 10-3 2 × 10-2 1 × 10-2 5 × 10-5
with respect to the other is 3 + nCH2 instead of 2 + nCH2 as shown in eq 42. We must then replace the 2 with a 3 in eq 45 as well. The results of both considerations are very similar and no significant differences either in the parameter values of group contribution or in its deviations with respect to those obtained by direct optimization are observed. However, it might be interesting to consider this possibility in other molecules containing large or very asymmetric groups. Also, the asymmetry of the group COO led us to consider it as united to methyl or methylene groups (CH3COO- or -CH2COO-) in various functional group-contribution models1,3,39 in order to constitute the molecules of alkyl alkanoates: the left aliphatic chain is bonded to the carbon of COO grouping whereas the right one is bonded to an oxygen. Thus, the perturbation that this COO grouping causes should be different at both sides, so it would justify the consideration of the groupings CH3COO- and -CH2COO- as functional groups themselves instead of COO. However, the groupcontribution scheme presented in this work allows consideration of the COO grouping as a whole functional group because the mutual perturbations caused between two groups depends not only on the kind of functional groups, but also on the relative position of both groups. For this reason it is interesting to consider that the COO group occupies two positions instead of only one: one position would be occupied by the carbonyl group and another one by the additional oxygen. The molecular parameters of alkyl alkanoates were optimized by Gross and Sadowski11 and Tihic et al.,37 but both parameter sets do not fit well the same trend of variation against molecular mass. However, we have used both parameter sets to correlate the group-contribution ones with eq 41 in order to obtain the best results in the case of long-chain esters. 4. Results and Discussion The most important utility of a group-contribution method is the ability to interpolate and extrapolate the physical properties of homologous compounds series. Generally speaking, this task is accurately achieved by the present group-contribution scheme. Tables 3 and 4 show the average percent deviations (APD%) and the absolute average deviations (AAD) of predictions of
Np 405 339 142 133 41 56 161 185
APD%
AAD/g · cm-3
0.98 3.8 0.52 3.6 1.6 10 1.1 11
source of exptl data
-3
5 × 10 2 × 10-3 3 × 10-3 9 × 10-4 9 × 10-3 2 × 10-2 1 × 10-2 7 × 10-5
40, 50 40 40, 42 40, 42 42,43,51 42, 51 42 42
vapor pressures and saturated liquid and vapor densities respectively for homologous series of normal and branched alkanes, linear monoethers, and esters. More detailed results are provided in the Supporting Information (Tables SI-2 and SI-3). Additionally, Table SI-4, available as well as in the Supporting Information, shows the APD% and AAD of heats of vaporization predictions of these compounds series using the experimental data available in the literature. The average percent and absolute average deviations were obtained from the following equations: 100 APD% ) Np AAD )
1 Np
Np
∑ i)1
|
yexp - ycalc i i yexp i
|
(46)
Np
∑ |y
exp i
- ycalc i |
(47)
i)1
and ycalc where Np is the number of experimental data, and yexp i i are the experimental and PC-SAFT values of the property, respectively. The pure predictions with PC-SAFT, i.e., the property values of compounds obtained by using interpolated and extrapolated group-contribution parameters, are indicated in the tables available in the Supporting Information. We denoted as interpolated the parameters for (i) all molecules where no characteristic parameters were available within the molecular weight range of those used in the correlation and (ii) molecules in the same molecular weight range where such parameters existed but were not used in the group-contribution parameter correlation. We alternatively considered as extrapolated all those parameters determined by the group-contribution scheme outside the molecular weight range of the molecules used in the parameter correlation. Usually, the pure predictions are obtained for those compounds where the molecular parameters are not available. Actually, we have not used all parameter sets to optimize the group-contribution ones, so in some cases indicated in these tables the model is purely predictive even if a parameter set has been previously published. The PC-SAFT predictions of saturated pressures with the group-contribution parameters show reasonably good agreement
Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010
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Figure 2. Logarithm of saturated presure of n-alkanes, branched alkanes, ethers, and esters against inverse of temperature: experimental data40,41,48,49 (points) and PC-SAFT prediction with group-contribution parameters (lines).
pressures of these long-chain compounds. Figure 3 shows the vapor-liquid equilibria of n-alkanes in a Psat-F diagram. As usual with PC-SAFT predictions, in the vicinity of the critical point the saturated pressures are overestimated, but it offers very good predictions in the noncritical region, even in the liquid phase where the crossing point between the heptane and dodecane curves is captured. A T-F diagram of vapor-liquid
Figure 3. Vapor-liquid equilibria curve of normal alkanes: experimental data40 (points) and PC-SAFT prediction with group-contribution parameters (lines).
with experimental data, even for heavy compounds whose parameters were extrapolated by the group-contribution method, as can be seen in Figure 2. In this figure, it is advisible to emphasize the results obtained for heavier ethers and esters despite the apparently high values of APD% shown in Table 3 (see Table SI-2 in Supporting Information for more details), due to the very low values of the experimental saturated
Figure 4. Vapor-liquid equilibria curve of dodecane, 2,2-dimethylpropane, and dimethyl ether: experimental data40,51 (points) and PC-SAFT prediction with group-contribution parameters (lines).
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Figure 5. Heat of vaporization of dodecane, 2-methyl pentane, 2,2-dimethyl propane and ethyl propanoate versus reduced temperature: experimental data40,52 (points) and PC-SAFT prediction with group-contribution parameters (lines).
equilibria of dodecane, 2,2-dimethylpropane, and dimethyl ether is shown in Figure 4, where the good agreement with experimental data of liquid and vapor saturated densities against temperature must be emphasized. The prediction of thermal properties, which requires temperature derivatives of the Helmholtz free energy, represents a meaningful test of the predictive ability of an equation of state. Figure 5 compares the PC-SAFT predictions of the heat of vaporization using the group-contribution parameters of the experimental data for dodecane, 2-methylpentane, 2,2-dimethylpropane, and ethyl propanoate. The critical point, where the heat of vaporization is zero, is overestimated as usual with PCSAFT, but the prediction of this property shows good agreement in almost the whole temperature range. Finally, Figure 6 shows the molecular mass dependence of the heat of vaporization of n-alkanes (from ethane to eicosane) and methyl esters (from methyl methanoate to methyl octacosanoate) at 298.15 K, where the good agreement shown by the predictions with experimental data even for the heavier compounds, where the molecular parameters were extrapolated by using the present groupcontribution method, is to be emphasized. Summarizing, this group-contribution method is based on two observed facts: first, the linear dependence of the coefficients
m, mε, and mσ3 with molecular mass observed in homologous series; second, the deviations of these regular trends for shorter chain compounds of the same homologous series due to the proximity effect among the functional groups present in these molecules. The first step of the methodology proposed in this work was to define the reference functional groups in order to distinguish them from others of the same type but affected by the proximity effect due to neighboring groups. The functional groups are characterized by the reference functional-group parameters, and the deviations of the coefficients m, mε, and mσ3 from the linear trend are characterized by the mutual perturbatiVe parameters. Afterward, the coefficients m, mε, and mσ3 are expressed as a linear combination of both reference functional-group and mutual perturbative parameters taking into account the number of functional groups and their mutual relative positions in the molecule through the additions SR-β. Then, the reference group-contribution parameters of methyl and methylene groups and the mutual perturbative ones were determined by optimization from the molecular PC-SAFT parameters of linear alkanes obtained from the literature. Finally, the reference group-contribution parameters of the main functional groups of other molecules (branched alkanes, ethers, and esters) and the mutual perturbative ones between these groups and methyl and methylene groups were optimized using the same approach. The reference functional groups defined to develop this groupcontribution scheme introduce a limitation to the generality of the approach due to the importance that the methylene group acquires and, consequently, also introduces an asymmetry in the perturbative parameters. However, this scheme reduces considerably the number of fitting parameters, and although this imposes a limitation to the potential applicability of the method, most molecules involved in chemical engineering processes are constituted by aliphatic groups and one or just a few different functional groups. From this perspective, the method may be applied to most of the characteristic molecular SAFT parameters published so far in the literature. Acknowledgment The authors acknowledge Consellerı´a de Educacio´n e Ordenacio´n Universitaria (Xunta de Galicia) and Ministerio de Ciencia e Innovacio´n (Project No. FIS2009-07923), Spain, for financial support.
Figure 6. Heat of vaporization of n-alkanes (from ethane to eicosane) and methyl alkanoates (from methyl methanoate to methyl octacosanoate) at 298.15 K versus molecular mass: experimental data40,47–49,53,54 (points) and PC-SAFT prediction with group-contribution parameters (lines).
Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010
Supporting Information Available: The molecular parameters of linear and branched alkanes, linear monoethers, and esters, obtained by several authors via direct optimization to experimental data, that were used to adjust the groupcontribution ones in this work are listed in Table SI-1. This table also shows the molecular parameters obtained by the group-contribution method developed in this work, in order to compare with literature parameters. Tables SI-2 and SI-3 show the average percent deviations (APD%) and the absolute average deviations (AAD) of predictions of vapor pressures and saturated liquid and vapor densities respectively of normal and branched alkanes, linear monoethers, and esters. Furthermore, as a complement to the PC-SAFT results, obtained using the molecular parameters determined with the group-contribution method, Table SI-4 shows the average percent deviations (APD%) and the absolute average deviations (AAD) of heats of vaporization predictions of the compound series cited before using the experimental data available in the literature. This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited (1) Weidlich, U.; Gmehling, J. A Modified UNIFAC Model. 1. Prediction of VLE, hE, and γ∞. Ind. Eng. Chem. Res. 1987, 26, 1372. (2) Gmehling, J.; Li, J.; Schiller, M. A Modified UNIFAC Model. 2. Present Parameter Matrix and Results for Different Thermodynamic Properties. Ind. Eng. Chem. Res. 1993, 32, 178. (3) Carballo, E.; Mosquera, R. A.; Legido, J. L.; Romanı´, L. Quantum Mechanical Characterisation of Functional Groups for Molecular Solution Theories Using Bader Fragments. J. Chem. Soc., Faraday Trans. 1997, 93, 3437. (4) Bader, R. F. W. A Quantum Theory of Molecular Structure and Its Applications. Chem. ReV. 1991, 91, 893. (5) Gonza´lez, D.; Cerdeirin˜a, C. A.; Romanı´, L.; Carballo, E. Group Definition in Molecular Solution Theories by Quantum Mechanical Methods: Application to 1-Alkanol + n-Alkane Mixtures. J. Phys. Chem. B 2000, 104, 11275. (6) Nitta, T.; Turek, E. A.; Greenkorn, R. A.; Chao, K. C. A Group Contribution Molecular Model of Liquids and Solutions. AIChE J. 1977, 23, 144. (7) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. SAFT: Equation-of-State Solution Model for Associating Fluids. Fluid Phase Equilib. 1989, 52, 31. (8) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Liquids. Ind. Eng. Chem. Res. 1990, 29, 1709. (9) Lora, M.; Rindfleisch, F.; Mchugh, M. A. Influence of the Alkyl Tail on the Solubility of Poly(alkyl acrylates) in Ethylene and CO2 at High Pressures: Experiments and Modeling. J. Appl. Polym. Sci. 1999, 73, 1979. (10) Vijande, J.; Pin˜eiro, M. M.; Bessie`res, D.; Saint-Guirons, H.; Legido, J. L. Description of PVT Behaviour of Hydrofluoroetheres Using the PC-SAFT EOS. Phys. Chem. Chem. Phys. 2004, 6, 766. (11) Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40, 1244. (12) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284. (13) Pa`mies, J. C.; Vega, L. F. Vapor-Liquid Equilibria and Critical Behavior of Heavy n-Alkanes Using Transferable Parameters from the SoftSAFT Equation of State. Ind. Eng. Chem. Res. 2001, 40, 2532. (14) Tamouza, S.; Passarello, J. P.; Tobaly, P.; de Hemptinne, J. C. Group Contribution Method with SAFT EOS Applied to Vapor Liquid Equilibria of Various Hydrocarbon Series. Fluid Phase Equilib. 2004, 222223, 67. (15) Tamouza, S.; Passarello, J. P.; Tobaly, P.; de Hemptinne, J. C. Application to Binary Mixtures of a group Contribution SAFT EOS (GCSAFT). Fluid Phase Equilib. 2005, 228-229, 409. (16) Thi, T. X. N.; Tamouza, S.; Passarello, J. P.; Tobaly, P.; de Hemptinne, J. C. Application of Group Contribution SAFT Equation of State (GC-SAFT) to Model Phase Behaviour of Light and Heavy Esters. Fluid Phase Equilib. 2005, 238, 254.
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ReceiVed for reView February 4, 2010 ReVised manuscript receiVed June 30, 2010 Accepted July 26, 2010 IE1002813
