GENERAL RESEARCH

Group Contribution Prediction of Vapor Pressure with Statistical AssociatingFluid Theory, Perturbed-Chain Statistical Associating Fluid Theory, andElliott-Suresh-Donohue Equations of State

Fateme Sadat Emami,† Amir Vahid,† J. Richard Elliott, Jr.,†,* and Farzaneh Feyzi‡

Department of Chemical and Biomolecular Engineering, The UniVersity of Akron, Akron, Ohio 44325-3906,and Chemical Engineering Department, Iran UniVersity of Science and Technology, Tehran, 16846-13114, Iran

The group contribution methodology developed by Elliott and Natarajan has been extended to the statisticalassociating fluid theory (SAFT) and perturbed-chain statistical associating fluid theory (PC-SAFT) equationsof state (EOS). Thermodynamic properties were correlated and predicted for a database of 878 compounds,including associating compounds. Association contributions were treated with Wertheim’s theory. The databasecovers 19 chemical families including hydrocarbons, cyclic hydrocarbons, aromatic hydrocarbons, alcohols,amines, nitriles, thiols, sulfides, aldehydes, ketones, esters, ethers, halocarbons, and silicones. The presentgroup contribution (GC) method was developed in two stages. Initially, pure component parameters of eachEOS were obtained by matching their boiling temperatures at 10 or 760 mmHg and available GC estimatesof solubility parameter and liquid density, while applying standard hydrogen-bonding parameters. Then, groupcontributions were regressed for the shape factor parameters of each EOS. Group contributions are presentedfor 84 first-order functional groups (FOG). Given the GC shape factors, the same GC estimates of solubilityparameter and liquid density can be applied to estimate all EOS parameters on a GC basis. The resultingcorrelation enables three-parameter corresponding states predictions without any experimental data. A byproductof the correlation for equation of state parameters is the capability to predict vapor pressure only on the basisof chemical structure. This capability was evaluated by computing the vapor pressures at 10, 100, and 760mmHg. On the basis of the present work, vapor pressure average absolute percent deviations (P AAD%)were 36% for Elliott-Suresh-Donohue (ESD) EOS, 65% for SAFT, 32% PC-SAFT. For comparison, thefirst- and second-order groups (FOG and SOG) provided by Tihic et al. (for simplified PC-SAFT) have beenapplied to ∼650 nonassociating compounds. The resulting P AAD% were 53% for Tihic FOG and 42% forTihic SOG. An alternative characterization of accuracy is the average absolute deviation (|∆T|) betweenexperimental and calculated saturated temperature. These were 8, 12, 8, 10, and 9 K for ESD, SAFT, PC-SAFT, Tihic FOG, and Tihic SOG equations, respectively.

Introduction

Prospective applications and new compounds often requireknowledge of properties and phase behavior that have not beenmeasured. For example, polymeric materials are an ever-increasing segment of the chemical industry, both as endproducts and intermediates. Representation of polymer mixturesby equations of state for this task is a fairly mature area.1 Thephase equilibria of mixtures of polymers in organic liquidsolvents and in supercritical fluids has been a subject of someresearch.2-4 Nevertheless, existing methods do not generallyinclude explicit treatment of hydrogen-bonding contributionsand there is a need for predictive rather than correlative models.As another example, alternative energy and pharmaceuticalapplications often encounter compounds that have not beenextensively characterized. Experimental values for the meltingtemperature and the density may be all that is available. In thesecases, the components must be characterized entirely in termsof their chemical structure. With this background, the presentresearch describes a group contribution method for solvents from

several families that extends naturally to polymer species andtheir solution phase behavior.

Our method is based on EOS models that adapt Wertheim’stheory to characterize hydrogen bonding as an explicit contribu-tion to the free energy. We consider two variants of statisticalassociating fluid theory (SAFT)5,6 and we update the Elliott-Suresh-Donohue (ESD) treatment considered by Elliott andNatarajan.7,8 Wertheim’s theory is based on a rigorous analysisof cluster diagram expansions.9 As such, it includes both theentropic and enthalpic contributions to hydrogen bonding subjectto the assumptions of the potential model. The potential modelis composed of square-well attractive sites strategically locatedto model steric hindrance and short-range, linear bonding.

In optimizing the group contributions, vapor pressure wasselected as a sensitive and valuable property for characterizingthe shape factor. Consequently, the results provide a basis forvapor pressure prediction that is evaluated crudely by computingthe vapor pressures in the database that result from applyingthe group contribution correlation developed here.

Previous work has addressed group contribution predictionsof polymer properties and solvent vapor pressures to varyingextents. Benzaghou et al.10 correlated EOS parameters in termsof various functional groups and extrapolated them to thepolymer limit, but did not report on vapor pressure accuracy.

* To whom correspondence should be addressed. E-mail: jelliott@uakron.edu. Tel.: (330) 972-7253. Fax: (330) 972-5856.

† The University of Akron.‡ Iran University of Science and Technology.

Ind. Eng. Chem. Res. 2008, 47, 8401–8411 8401

10.1021/ie800329r CCC: $40.75 2008 American Chemical SocietyPublished on Web 09/25/2008

Jansen et al.11 and later Yair and co-workers12 developed aUNIFAC group contribution method for predicting vaporpressures of pure compounds. They claimed a percent averageabsolute deviation (AAD) of roughly 4% for vapor pressure.On the other hand, Asher et al.13 recently updated the UNIFACmethod for 76 typical aerosol compounds, but they reportedaccuracy within a factor of 2 (i.e., -50 to 100% AAD). Theyalso reported deviations for the Lee-Kesler method and severalothers to be roughly within a factor of 3 (-70 to 200% AAD).A follow-up study by Asher and Pankow13,14 showed similaraccuracy for an additional 208 hydrocarbons. Similarly, Bureauet al.15 reported deviations of 16-285% AAD for seven estersusing the UNIFAC method.

In work closely related to the present effort, Elliott andNatarajan8 developed a correlation for polymer parameters andevaluated accuracy for vapor-liquid equilibrium in polymersolutions. They presented a group-contribution approach for theESD equation,14 an equation similar to SAFT and PC-SAFT,to estimate the shape parameter. They did not evaluate theaccuracy of vapor pressures computed from the correlated groupcontributions, however. Group contribution correlations forboiling temperature and heat of vaporization like those ofConstantinou et al.16 and Kolska et al.17 can also be used forvapor pressure prediction. The accuracy of this approachgenerally yields 50-100% AAD. This approach does notprovide a consistent basis for extension to solution properties,however, placing it outside the scope of the current study. Weplan to evaluate this approach in greater detail separately.

Recently, Tamouza et al.,18,19 Nguyen Thi,20 and Huynh etal.21 have developed group contributions for the SAFT approachapplied to pure compounds and binary mixtures of esters andaromatic compounds. Although they have reported vaporpressure accuracy of 10-50% AAD for 35 esters, 25 hydro-carbons, and 9 alcohols, their analysis focuses on limitedvariability in molecular structure (e.g., only linear alkyl for-mates) and applies a very different approach from the onedeveloped in the present work.

The most recent work on GC-SAFT families has been doneby Tihic et al.22 with application to polymer solution phaseequilibria using first-order and second-order groups with 400hydrocarbons of low molecular weight. Unfortunately, they havenot reported the details of the chemical families in their database.Furthermore, their training compounds do not include associat-ing compounds. Finally, they have not reported the vaporpressure accuracy for their method.

In this paper, the method developed by Elliott and Natarajan8

is adapted to the statistical associating fluid theory (SAFT)model5 and perturbed-chain statistical associating fluid theory(PC-SAFT)6 model. In addition, we evaluate the resultingmethod for vapor pressure prediction as well as boilingtemperature of a database of 871 organic compounds. Themethod implemented in this work is basically the same as Elliottand Natarajan.8 The primary difference is that alcohols withmultiple hydroxyl segments have been omitted. We observedthat properties varied too strongly when second or third hydroxylsegments were located at different positions in the molecule,possibly related to intramolecular hydrogen bonding. Therefore,considering alcohols with multiple hydroxyls must be the subjectof a future study. Excluding these alcohols has reduced the PAAD% from ∼81% to ∼40%, compared to using the previouslyreported shape factors.8 The updated group contribution shapefactors are reported for the ESD8 model along with new SAFT5

and PC-SAFT6 groups in Table 1. Our database includes 181associating compounds. For comparison, the reported FOG and

SOG by Tihic et al. have been used for prediction of vaporpressure of ∼650 nonassociating compounds.

Group Contribution Form of SAFT and PC-SAFT Equa-tions. Generally, the SAFT and PC-SAFT models characterizeeach pure component in terms of three parameters. Theparameters for the SAFT EOS are the temperature-independentsegment volume, υ°°, the shape parameter, m, and the temper-ature-independent dispersion energy of interaction betweensegments, u°/k. For PC-SAFT the parameters are the temper-ature-independent segment diameter, σ, the depth of the potentialwhich is related to Barker-Henderson23 approach, ϵ/k, and thenumber of segments per chain, m. ESD also has three param-eters, a molecular volume, b, a shape parameter, c, and a vander Waals attractive energy, ϵ/κ. To establish a correspondencebetween the three EOSs we have focused on b, molecularvolume. For SAFT models, b ) πmσ3/6.

Meanwhile, all three models apply three additional parametersfor hydrogen-bonding species: the number of hydrogen-bondingsegments per molecule, Nd; the bonding volume, KAD; and thehydrogen-bonding energy, ϵHB. Following Elliott and Natarajan,the present work computes ϵHB/k from the hydrogen-bondinggroup contribution method effectively as an average overbonding sites: 4 kcal/mol for hydroxyl groups and 1.25 kcal/mol for amine, amide, nitrile, and aldehydes groups. Thesevalues were derived from previous studies of hydrogen-bondingenergies for a wide range of components.24 The number ofadjustable parameters can be further reduced by noting othertrends pertinent to hydrogen bonding. Nd is obvious from themolecular structure, and we can correlate the bonding volume,KAD, in terms of b and m.

KAD ) 0.035b ⁄ m (1)Elliott and Natarajan8 used KAD ) 0.025 b/c as a similar

equality for the ESD model. We have assigned two associationsites to all associating components (often referred to as the 2Bmodel25). Consequently, we only need three parameters, υ°°,m, and u°/k for the SAFT EOS and σ, m, and ϵ for the PC-SAFT EOS, and b, c and ϵ for ESD EOS to be characterizedfor each component.

At this point, the task is to calculate three parameters of anyEOS by satisfying three criteria. First, the estimated compress-ibility factors must be consistent, using molar volumes presentedby Hoy26 et al. as extended by Elliott and Natarajan. Thiseffectively fixes the molecular volume, b:

ZL298 )

0.1VL298

298R(2)

where VL298 is the liquid molar volume at 298 K, and R is the

universal gas constant. Note that this constraint is slightlydifferent from ZL

298 ) 0, as applied by Elliott and Natarajan.The extended Hoy26 correlation has the form

VL298 ) 12.1+∑ νi∆Vi (3)

where ∆Vi is the UNIFAC group contribution for liquid molarvolumes as cited in table 2.

Second, the internal energy departure functions should beconsistent with the internal energy estimated from liquid molarvolume and heat of vaporization provided by the Constantinouand Gani16 GC method. We use the relationship of the solubilityparameter, δ ) [(Hvap

298 - 298R)/VL298]1/2, in terms the internal

energy departure function for each EOS, neglecting the departurefunction of the vapor.

8402 Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008

δ2VL298

298R)- U

298R(4)

We can readily derive the internal energy departure functionfrom the Helmholtz energy. Detailed expressions for the internal

energy and the solubility parameter in terms of the SAFT andPC-SAFT models are given in Appendix A.

Third, the shape factor is determined in one of two ways,depending on whether an optimal value is being determined

Table 1. Group Contribution for Estimating the Shape Parameter, the Liquid Molar Volume, and the Heat of Vaporizationa

group PC-SAFT ∆m SAFT ∆m ESD ∆c ∆V298L ∆H298

vap group PC-SAFT ∆m SAFT ∆m ESD ∆c ∆V298L ∆H298

vap

CH3- 0.483 0.995 0.228 21.6 4.116 CCL3 5.475 8.850 2.269 62.1 33.400CH2< 0.390 0.462 0.207 15.6 4.650 ACCL 0.523 0.448 0.345 26.9 11.883>CH- -0.187 -0.331 0.019 9.6 2.771 CH2NO2 1.935 3.312 1.151 50.2 30.644>C< -0.515 -0.994 -0.090 3.6 1.284 CHNO2 1.225 1.749 0.805 46.3 26.277CH2dCH 0.180 0.457 0.135 32.4 6.714 ACNO2 0.044 -0.213 0.194 31.4 19.700CHdCH -0.118 -0.091 0.040 26.4 7.370 CH2SH 0.293 0.474 0.292 46.7 14.931CH2dC 0.054 -0.122 0.112 26.4 6.797 I -0.048 -0.105 0.105 42.6 14.364CHdC 0.134 0.470 0.162 20.4 8.178 Br 0.353 0.975 0.324 25.3 11.423CdC 0.117 -0.386 -0.022 14.4 9.342 CHtC 0.167 0.723 0.168 40.2 7.751ACH 0.150 0.363 0.109 13.4 4.098 CtC 0.696 2.778 0.755 28.8 11.549AC- 1.824 2.287 0.917 7.4 12.552 CL(CdC) 0.631 1.389 0.376 19.5 7.000ACCH3 0.681 1.115 0.416 29.0 9.776 ACF 0.467 0.815 0.246 18.6 4.877ACCH2 0.640 -0.133 0.358 23.0 10.185 CF3 4.878 6.629 1.546 37.2 8.901ACCH 0.496 0.345 0.345 17.0 8.834 CF2 2.426 4.316 0.759 26.0 1.860OH 0.636 1.853 0.378 12.5 24.529 CF 2.487 -4.458 0.805 14.8 8.901ACOH 3.613 6.028 1.765 19.9 40.246 COO 0.401 1.008 0.498 25.7 13.4CH3CO 0.965 2.244 0.664 38.9 18.999 SiH3 0.408 0.329 0.215 21.6 3.4CH2CO 1.169 2.289 0.787 32.9 20.041 SiH2 0.408 0.329 0.215 58.4 3.4CHO -0.042 0.922 0.243 23.3 12.909 SiH 0.408 0.329 0.215 53.7 3.4CH3COO 2.502 4.290 1.241 43.0 22.709 Si 0.408 0.329 0.215 50.3 3.4CH2COO 0.847 0.891 0.639 37.0 17.759 SiH2O 1.819 2.537 2.537 33.8 6.8HCOO 0.785 1.047 0.556 43.3 14.500 SiHO 1.819 2.537 2.537 33.8 6.8CH3O 1.383 2.650 0.688 28.0 10.919 SiO 1.676 2.672 0.796 33.8 6.8CH2O -0.230 0.471 0.129 22.0 7.478 TERT-N -1.129 0.467 -0.337 12.6 4.190CH-O -0.083 -0.322 0.057 16.0 5.708 CCL2F 0.647 0.976 0.413 53.8 13.322FCH2O -0.018 1.452 0.188 33.2 11.227 CCLF2 0.993 2.213 0.382 45.5 8.301CH2NH2 0.133 1.190 0.296 32.6 14.599 CONH2 0.707 2.875 0.695 34.3 41.9CHNH2 0.031 -0.199 0.144 26.6 11.876 CONHCH3 8.837 14.293 14.293 49.9 38.5CH3NH 0.776 1.734 0.444 32.6 14.452 CONHCH2 8.722 13.854 13.854 43.9 51.787CH2NH 0.329 1.419 0.510 26.6 14.481 CON(CH3)2 9.340 15.368 15.368 78.9 38.9CHNH 2.430 3.885 1.092 20.6 14.000 CONCH3CH2 9.225 14.929 14.929 72.9 39.1CH3-RN 0.222 0.478 0.104 28.2 6.947 CON(CH2)2 9.110 14.490 14.490 66.9 39.3CH2-RN 0.222 0.478 0.104 28.2 6.918 C2H5O2 0.791 5.054 5.054 50.0 36.657ACNH2 2.322 3.690 1.310 24.4 28.453 C2H4O2 0.210 0.273 0.273 44.0 14.956C5H4N 1.277 2.037 0.885 75.7 31.523 CH3S 0.884 1.464 0.498 39.6 16.921C5H3N 1.639 2.367 2.367 69.7 31.005 CH2S 1.167 1.763 0.654 33.6 17.117CH2CN 0.032 0.663 0.233 38.7 23.340 CHS 0.038 -0.027 -0.027 27.6 13.265CH2CL 0.604 1.004 0.376 35.1 13.780 C4H3S 0.127 -0.063 -0.063 65.7 27.966CHCL 0.533 1.008 0.392 29.1 11.985 C4H2S 0.068 -0.124 -0.124 59.7 28CCL 0.342 0.619 0.295 23.1 9.818 RCH2< 0.294 0.467 0.180 15.6 4.650CHCL2 1.036 2.249 0.655 48.6 19.208 >RCH- -0.060 -0.148 -0.013 9.6 2.771CCL2 1.841 3.061 0.923 42.6 17.574 >RC< -0.275 -1.031 -0.124 3.6 1.284

a Bold-italic values are Elliott and Natarajan additions to Hoy’s volumetric parameters.

Table 2. Coverage of Compounds in the Database and Percent Error in the Correlation for the Shape Parameters of Different Equations ofStates and Liquid Molar Volume

no. in database rms% error in ∆m PC-SAFT rms% error in ∆m SAFT rms% error in ∆c ESD rms% error in V

alcohols 77 14.85 30.76 10.03 7.14amines 55 19.89 34.16 11.64 8.93nitriles 10 20.68 39.81 14.75 6.95silanes 10 16.84 17.62 13.86 21.02imines 7 9.80 14.01 7.45 9.81tert-Ns 6 26.71 16.23 17.95 7.60nitrates 19 19.33 35.62 11.04 9.11thiols 27 14.16 17.87 8.44 8.39aldehydes 23 22.81 24.53 13.39 6.23epoxides 13 24.23 12.14 18.89 8.69ethers 38 17.77 40.71 8.01 8.43formates 16 13.04 25.22 7.52 10.33esters 92 12.84 19.00 7.18 5.44ketones 28 13.68 18.66 7.31 3.96alcohol-ethers 18 26.54 55.97 17.35 5.27aliphatic hydrocarbons 157 12.47 19.56 9.40 9.58cyclic hydrocarbons 41 12.74 15.92 7.29 5.88aromatic hydrocarbons 137 10.82 26.55 6.55 6.61halocarbons 103 24.04 54.82 11.52 10.12overall 878 16.34 30.97 9.80 8.27

Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008 8403

for inclusion in the database or whether one seeks the result ofthe group contribution correlation. For an optimal value, thelast constraint to be satisfied is the equality of fugacity of vaporand liquid phases at the temperature of interest (boilingtemperature at 760 or 10 mmHg, whichever is nearer to 400K).

ln φL ) ln φV (5)

ln φ) &µres(T, K)- ln Z (6)

µres(T, K)kT

) Ares + (Z- 1) (7)

Ares ) Ares

NkT(8)

where Ares is the reduced Helmholtz free energy which theirrelations are presented in the Appendix A for SAFT and PC-SAFT EOSs.

In the calculation of vapor phase fugacity, a truncated virialexpansion after its second coefficient is used as follows

ZVsat )Psatb ⁄ ηVRT) 1+B2η

v (9)

The relations for the second viral coefficient are given inAppendix A. The saturated liquid and vapor densities at thiscondition are computed based on the experimental vaporpressure.

When the group contribution result is sought, shape factorsare computed from

m) 1+ (∑ υi∆mi) (10)

where ∆mi is the group contribution and 1 is the numberassigned to a spherical molecule.

The optimal shape parameters for all compounds weretabulated with the ones in terms of the group contributioncorrelations. This table was formulated as an Excel worksheetusing the “sumproduct” operator to compute the GC values andthe Excel solver was applied to minimize %RMSm betweenmopt and mGC by changing the functional group factors. Therms error plays the role of an objective function and has theform of below:

%RMSm) 100"∑ (mopt -mGC

mopt )2

NDP(11)

where NDP is the number of data points.

Results and Discussions

Optimized group contribution values are listed in Table 1for all three EOS models, along with the contributions for the

Table 3. Average Boiling Temperature Deviations at 760, 100, and10 mmHg for GC-PC-SAFT, GC-SAFT, GC-ESD, and Tihic et al.22

Approach

no. ofoccurrence

TAAD%

Tbias%

100 ln(Tcalcd/Texptl) ∆T |∆T|

SAFT 878 3.45 -1.76 -1.87 -5.41 12.48PCSAFT 878 2.20 0.52 0.46 1.43 8.12ESD 878 2.21 -0.05 -0.11 -1.00 8.13Tihic FOG 643 2.79 -0.43 -0.53 -0.16 9.91Tihic SOG 666 2.67 -0.48 -0.57 -0.40 9.52

Table 4. Average Vapor Pressure Deviations at 760, 100, and 10 mmHg for Different Families of Organic Compounds

family EOSno. of

occurrenceP

AAD%P

bias%ln(Pcalcd/

Pexptl) family EOSno. of

occurrenceP

AAD%P

bias%ln(Pcalcd/

Pexptl)

alcohols C3-C24 SAFT 77 73.57 63.07 35.48 thiols C2-C12 SAFT 27 53.25 41.51 24.80PC-SAFT 32.62 8.22 -0.78 PC-SAFT 25.14 2.18 -4.03ESD 39.44 23.10 11.79 ESD 24.67 7.50 0.86

amines C2-C14 SAFT 55 99.61 86.09 35.33 aldehydes C2-C13 SAFT 23 95.68 89.09 45.61PC-SAFT 48.55 7.02 -13.42 PC-SAFT 34.56 29.86 18.02ESD 59.21 25.78 -0.06 ESD 34.96 21.33 10.59

nitriles C3-C8 SAFT 10 150.58 149.91 70.25 epoxides C3-C7 SAFT 13 176.53 176.49 73.25PC-SAFT 43.23 9.68 -3.93 PC-SAFT 56.89 26.59 7.88ESD 39.93 -6.56 -18.84 ESD 82.95 60.13 22.87

silanes C-C12 SAFT 10 63.29 63.29 46.84 ethers C2-C16 SAFT 38 77.23 71.40 42.34PC-SAFT 29.40 0.80 -5.85 PC-SAFT 31.30 -14.19 -24.24ESD 33.86 14.20 6.72 ESD 27.67 6.78 -0.14

imines C5-C13(e.g., acridine)

SAFT 7 191.90 191.59 81.91 formates C2-C11 SAFT 16 103.80 93.43 41.04

PC-SAFT 34.57 24.57 16.99 PC-SAFT 18.77 -0.94 -4.32ESD 55.60 46.92 28.72 ESD 19.46 8.17 4.04

tert-Ns C3-C15(e.g. trimethylamine)

SAFT 6 139.69 53.34 62.76 esters C3-C28 SAFT 92 45.66 18.99 2.66

PC-SAFT 37.76 -11.14 -15.06 PC-SAFT 26.56 -11.41 -18.67ESD 37.50 -10.69 -14.88 ESD 27.53 8.46 1.98

nitrates C2-C18 SAFT 19 78.12 51.10 13.92 ketones C3-C12 SAFT 28 80.42 76.05 38.70PC-SAFT 47.52 -0.37 -23.85 PC-SAFT 52.41 11.77 -13.30ESD 47.25 -8.57 -34.12 ESD 59.77 28.65 -3.74

alcohol-ethers C3-C10(e.g., carbitol)

SAFT 18 110.75 95.17 44.64 hydrocarbons C3-C36 SAFT 157 51.09 38.15 22.97

PC-SAFT 53.71 -19.33 -47.64 PC-SAFT 18.83 6.02 1.84ESD 62.96 17.39 -10.03 ESD 22.31 8.88 3.61

cyclic hydrocarbons C3-C16 SAFT 41 41.28 39.01 30.15 halocarbons C3-C10 SAFT 103 77.19 51.60 4.01PC-SAFT 14.28 1.59 -0.73 PC-SAFT 48.50 5.87 -36.68ESD 12.67 -4.31 -6.08 ESD 37.76 -5.40 -37.98

aromatic hydrocarbonsC4a-C24

SAFT 138 54.00 35.46 17.54 total SAFT 878 69.30 54.07 24.18

PC-SAFT 35.33 19.84 7.02 PC-SAFT 32.21 4.63 -8.74ESD 53.33 39.69 16.19 ESD 38.26 15.81 -0.42

a Conjugated double bonds (e.g., 1,3-butadiene) behave like aromatic hydrocarbons.

8404 Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008

VL298 and Hvap. Note that Table 1 does not report any value for

carboxylic acid groups owing to the peculiar hydrogen bondingbehavior in these complexes. A revised association model forcarboxylic acids is the subject of an upcoming study. Thecoverage of families of compounds is given in Table 2 wherethe deviations correspond to the deviations of the correlatedshape parameters from the optimized values. The correlationerrors are generally around 17% for PC-SAFT, 31% for SAFT,and 10% for ESD. The fairly large deviations are due to thesensitivity of the shape parameters to the accuracy of thesolubility parameter estimation. As before, we recommend thatthe group contribution estimation of the shape parametercorrelation should only be applied when reliable vapor pressuresor critical constants are not available.

Note that most of the engineering methods for predictingvapor pressure based on molecular structure are somewhatlimited and based on predicting the boiling temperature.27

Hence, we consider the boiling temperature deviations first inassessing the quality of vapor pressure predictions resulting fromthe shape factor correlation. Table 3 summarizes the boilingpoint temperature deviations for all organic families consideredhere. The deviations in Table 3 correspond to the boiling pointtemperatures of pure compounds for several different familiesat pressures of 760, 100, and 10 mmHg. Keep in mind that ourobjective function minimized %rmsm, not P AAD% or TAAD%. As demonstrated in Table 3, the average value of ∆Tis around 10 K for all EOS models, demonstrating the feasibilityof this approach when no experimental data are available.

Table 4 shows a detailed analysis of vapor pressure deviationsfor various families of organic compounds as an average at thethree boiling temperatures: 760,, 100, and 10 mmHg. A detailedtable of vapor pressure deviations is available in the SupportingInformation. Note that the best results are obtained at the normalboiling temperature. Also, PC-SAFT and ESD generally haveclose prediction of P AAD% and close to 35%. This is because

the ESD model is effectively a simplified form of SAFTequation of state with an alternative approximation for the radialdistribution function. PC-SAFT and ESD may have better resultsbecause their disperse attraction contributions were characterizedwith direct consideration of nonspherical molecules, whereasthe SAFT model is based on adaptation of a disperse attractionterm that has been optimized for spherical molecules.23,28-30

Among these organic compounds, a few families have distinctlylarge deviations: epoxides (176 P AAD% for SAFT, 57 PAAD% for PC-SAFT, and 83 P AAD% for ESD), alcohol-ether(110 P AAD% for SAFT, 54 P AAD% for PC-SAFT, and 63P AAD% for ESD). These large deviations are attributed tothe peculiar chemical structures of the epoxides (i.e., 1,2-epoxy-2-methylpropane) and the dual functionality of alcohol-ethercompounds (e.g., carbitol, acetovanillone, ethyl lactate). Thebehavior of alcohol-ether compounds may reflect interactionbetween the functional groups that undermines the groupadditivity. On the other hand, hydrocarbons have the best resultswith vapor pressure deviations of 51% P AAD for SAFT, 19%P AAD for PC-SAFT, and 22% P AAD for ESD. These smalldeviations reflect effective group additivity. Overall, maximumdeviations were 1578%, 1184%, and 1581% for SAFT, PC-SAFT, and ESD, and 90% of compounds had less than 210%,80%, and 88% deviations.

Tables 5 and 6 provide a comparison between our currentapproach with the FOG and SOG used in the simplified PC-SAFT EOS by Tihic and co-workers.22 Note that the simplifiedPC-SAFT EOS is the same as PC-SAFT EOS for purecompounds. According to Tables 5 and 6, the FOG and SOGof Tihic et al.22 have larger vapor pressure deviations, 53% PAAD for Tihic FOG and 42% P AAD for Tihic SOG at 10,100, and 760 mmHg boiling point temperatures compared to∼28% with the present work for the same compounds. Readersare referred to the Supporting Information for detailed vaporpressure deviations for each EOS. The perfluoro groups (i.e.,

Table 5. A Comparison between the Current Study and FOG of Tihic et al.22 in Terms of Vapor Pressure Errors

groupno. of

occurrence PAAD%P

bias%ln(Pcalcd/

Pexptl) groupno. of

occurrenceP

AAD%P

bias%ln(Pcalcd/

Pexptl)

nitriles Tihic FOG 18 35.58 -23.81 -41.05 ketones Tihic FOG 28 108.27 72.48 2.11this work 46.98 2.79 -20.44 this work 52.56 11.77 -13.30

thiols Tihic FOG 27 122.34 111.04 53.28 hydrocarbons Tihic FOG 157 22.87 5.58 0.35this work 25.14 2.18 -4.03 this work 18.83 6.02 1.84

epoxides Tihic FOG 13 124.15 107.23 39.62 cyclic hydrocarbons Tihic FOG 41 96.06 95.56 55.44this work 56.89 26.59 7.88 this work 14.28 1.59 -0.73

ethers Tihic FOG 38 33.08 -12.55 -24.90 aromatics Tihic FOG 138 24.54 -6.38 -14.05this work 31.30 -14.19 -24.24 this work 35.33 19.84 7.02

formates Tihic FOG 16 47.88 31.59 14.96 halocarbons Tihic FOG 71 123.01 94.72 13.97this work 18.77 -0.94 -4.32 this work 48.25 13.94 -29.49

esters Tihic FOG 92 33.16 17.09 1.35 total Tihic FOG 643 53.71 28.53 3.24this work 26.56 -11.41 -18.67 this work 28.50 4.02 -7.43

Table 6. A Comparison Between the Current Study and SOG of Tihic et al.22 in Terms of Vapor Pressure Errors

groupno. of

occurrenceP

AAD%P

bias%ln(Pcalcd/

Pexptl) groupno. of

occurrenceP

AAD%P

bias%ln(Pcalcd/

Pexptl)

nitriles Tihic SOG 18 35.58 -23.81 -41.05 esters Tihic SOG 92 54.36 35.27 14.44this work 46.98 2.79 -20.44 this work 26.56 -11.41 -18.67

thiols Tihic SOG 27 109.14 100.64 52.45 ketones Tihic SOG 27 37.02 3.35 -9.09this work 25.14 2.18 -4.03 this work 28.72 -13.42 -21.05

aldehydes Tihic SOG 23 35.04 15.07 4.87 hydrocarbons Tihic SOG 157 24.97 13.71 7.82this work 34.56 29.86 18.02 this work 18.83 6.02 1.84

epoxides Tihic SOG 13 121.54 109.83 43.11 cyclic hydrocarbons Tihic SOG 41 50.48 17.27 0.37this work 56.89 26.59 7.88 this work 14.28 1.59 -0.73

ethers Tihic SOG 38 31.15 -10.62 -21.51 aromatics Tihic SOG 137 24.32 -4.81 -12.68this work 31.30 -14.19 -24.24 this work 35.33 19.84 7.02

formates Tihic SOG 16 47.07 32.39 16.06 halocarbons Tihic SOG 71 123.06 96.39 15.79this work 18.77 -0.94 -4.32 this work 34.71 0.40 -31.54

total Tihic SOG 666 42.35 32.29 16.21this work 27.70 4.15 -6.68

Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008 8405

CF3 and CF2) represent a particular case of interest. Forexample, a deviation of 3000% was obtained for 1,1,1-trifluoroethane with the Tihic correlation, compared to 70% bythe present work. Generally, the high vapor pressure deviationsfor perfluoro compounds relate to low heat of vaporization andhigh vapor pressure relative to compounds with similar molec-ular volume. Sans and Elliott31 explained this peculiar behaviorusing a soft shoulder in their transferable potential. Theyreported 8% PAAD and 3% in liquid density for thesecompounds. Finally, Tihic et al. reported that the densitydeviations improve when using SOG. A similar observationapplies to the vapor pressure deviations, and it is more obviousat lower pressures.

It is notable that a few compounds have been excluded fromour database because they are very small molecules or havepeculiar molecular structures. For instance, compounds like 1,1-difluoroethane, propyleneimine, 1,3-propyleneoxide, and 1,2-propyleneoxide are too small to exhibit group additivity. Also,if the structure of a molecule is too complicated then some otherinteractions between the structural groups would come intoeffect. Trioxane and dimethylmaleate are examples. Moreover,as mentioned before, some fluorinated species are problematicsuch as dichlorofluoromethane, 1,2-dibromotetrafluoroethane,and octafluorocyclobutane. Some other compounds like ethylfluoride and furan would have a smaller shape factor than 1when we want to calculate their parameters by this method,which is disallowed. Shape factor is representative for thesphericity of the molecule and the most spherical compound ismethane. Its assigned shape factor is 1, therefore, all othercompounds should have greater shape factors than one. Anotherissue to be considered is the acceptance of the experimentaldata. The Othmer-Yu and Riedel references in the DIPPRdatabase are not based on experimental data, although they oftenform the basis of DIPPR’s “accepted” values. 1,2-Epoxy butanewas excluded for this reason. Overall, these 12 compounds outof 890 have been deleted from our database in addition to thealcohols with multiple hydrodroxyl groups.

The pressure-temperature diagram is presented in Figure 1for several alcohols. The predicted pressure by the presentedmethod is plotted using GC-PC-SAFT (red dashed lines). Asshown in the plot, the GC-PC-SAFT overestimates the vapor

pressure of small alcohols (e.g., ethanol); also molecules likemethanol cannot be treated by this method at all, which is notplotted because its curves overlap with ethanol and decreasethe clarity of the plot. However, the fact is that the groupcontribution method is not intended for small molecules, sincethe experimental data are available. On the other hand, accurateagreement is observed between the experimental and estimatedvapor pressures for larger alcohols (e.g., n-pentanol and n-octanol). Furthermore, our method can be applied simply andrapidly. For example, TraPPE predictions for n-decanol are notavailable, while the present method has been applied simply,with very small deviations from the experimental data. Also itis noticeable that this method has been used at low temperatures(e.g., less than 0.45 reduced temperatures) while the TraPPEmethod has not generally been applied below reduced temper-atures of 0.6. According to Table 4 the lowest vapor pressuredeviations for alcohols can be achieved by using PC-SAFT asthe basis, and slightly larger deviations are achieved with ESDand SAFT.

Figure 2 shows the predictions for olefins by GC-ESD andGC-SAFT compared with experimental and TraPPE data. Asbefore, small molecules like ethylene cannot be treated withthis approach. Also for small compounds the deviation betweenthe GC-ESD and experimental data are large. Again, our modelcan successfully predict the vapor pressure at low temperatureswhere TraPPE has not been applied. PC-SAFT results in slightlylower deviations than SAFT or ESD. Figure 3 provides a similarillustration for n-alkanes with GC-SAFT alone.

Conclusions

We have proposed a generalized group contribution approachbased on the principle of three-parameter corresponding statesfor 16 different families of compounds. In this method, SAFTand PC-SAFT models were used to demonstrate the groupcontribution approach, along with an updated version of theElliott and Natarajan correlation for the ESD model. The resultsare more accurate than many previous group contributionmethods for predicting thermodynamic properties such as vaporpressure and boiling temperature. A distinctive part of this workis proposing groups for varied associating compounds with the

Figure 1. The plot of the saturated vapor pressure versus the inverse of thetemperature for alcohols. Experimental data32 are shown with points. Thesolid lines and the dashed lines represent the TraPPE33 and GC-PC-SAFT,respectively.

Figure 2. The plot of the saturated vapor pressure versus the inverse of thetemperature for olefins. Experimental data32 are shown with points. Thesolid lines represent the TraPPE34 model, the dashed lines represent GC-ESD, and the dashed-dot lines represent the GC-SAFT results.

8406 Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008

SAFT methodology. The PC-SAFT and ESD models providedaccuracy of roughly 40% P AAD. This is comparable to theaccuracy achieved by the TraPPE35 method involving molecularsimulation, but less accurate than the AUA35,36 or SPEADMD27

molecular simulation methods (10-20% P AAD). Goingforward, we expect this group contribution method to set anupper bound on % P AAD to be accepted from vapor pressurepredictions of more detailed molecular models. In futureresearch, this method could be readily extended to mixturesincluding polymer solutions as demonstrated by Elliott andNatarajan. Associating blends and their compatibilization wouldbe of particular interest. Also, proposing a method for treatingcarboxylic acids and hydrogen fluoride is challenging. We hopeto address the carboxylic acids in the very near future.Furthermore, predicting other thermodynamic properties suchas entropy, speed of sound, and heat capacity can be the subjectof future studies. Basically, once we obtain the EOS parametersby the present approach we can calculate many other quantitiesthrough the equation of state.

Acknowledgment

The authors are grateful to Professor Joachim Gross fromTechnical University of Delft, Netherlands, for his eruditediscussion and distributing the PC-SAFT code. This researchwas supported in part by ChemStations Inc., Houston, TX.

Supporting Information Available: Detailed anaysis ofvapor pressure deviations. This material is available free ofcharge via the Internet at http://pubs.acs.org.

Appendix

A. The GC-SAFT and PC-SAFT Equations of State

A.1. The SAFT Equation of State. A.1.1. Compress-ibility Factor. The SAFT equation of state for pure fluids isspecified by

Z) 1+ Zseg + Zchain + Zassoc (A.1.1.1)

Zseg )m[4η- 2η2

(1- η)3+∑

i∑

j

jDij[&u]i[ητ ]j] (A.1.1.2)

where m is a shape factor equal to the number of sphericalsegments per molecule, η ) bF is the reduced fluid density(segment packing fraction), b ) τmυ° is the molar volumeoccupied by molecules themselves, τ ) 0.74048, υ° is thesegment molar volume in a closed-packed arrangement, i.e., thevolume occupied by NAv closely packed segments, in millilitersper mole of segments. By considering the definition of thepacking fraction we can express υ° as

υo)πNAv

6τd3 (A.1.1.3)

Since υ° is implicitly temperature dependent (d is temperaturedependent) Huang and Radosz16 decided to include a corre-sponding, temperature-independent segment molar volume atzero temperature, T ) 0, which would be denoted υ°° and wouldbe referred to as the segment volume:

υoo)πNAv

6τσ3 (A.1.1.4)

where σ is a temperature-independent segment diameter. Tem-perature dependence of the segment diameter d is based on theBarker-Henderson approach.27 Specific equations for d and υ°

are given below.

d) σ[1-C exp(-3&u0)] (A.1.1.5)

υo) υoo[1-C exp(-3&u0)]3 (A.1.1.6)

Therefore, we can rewrite the packing fraction as below:

η)πNAv

6Fmd3 (A.1.1.7)

Zchain ) (1-m)[ 52

η- η2

(1- η)(1- η2)] (A.1.1.8)

Zassoc ) -F2(-η2 + 2η+ 2)(1- η)(2- η)

(A.1.1.9)

F) 2NdR1⁄2 ⁄ [1+ (1+ 4NdR)1⁄2] (A.1.1.10)

where Nd is the number of segments per molecule possessinghydrogen bonding sites.

R)FKADYHB(1- η

2)(1- η)3

(A.1.1.11)

YHB ) exp(&εHB)- 1 (A.1.1.12)

KAD is the hydrogen bonding volume, and εHB is the energy ofhydrogen bonding.

A.1.2. Helmholtz Free Energy.

Ares )m[4η- 3η2

(1- η)2+∑

i∑

j

Dij(&u)i(ητ )j]+

(1-m) ln1- η

2

(1- η)3+ 2Nd ln(1-F2 ⁄ Nd)+F2 (A.1.2.1)

A.1.3. Solubility Parameter. For the association contribu-tion, we use the simplified form of Wertheim’s theory developedby Elliott.37

Figure 3. plot of the saturated vapor pressure versus the inverse of thetemperature for n-alkanes. Experimental data32 are shown with points, solidlines show the TraPPE35 results, and blue dashed lines represent the GC-SAFT.

Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008 8407

δ2VL298

298R)- U

298R(A.1.3.1)

URT

) & ∂ (A ⁄ RT)∂&

(A.1.3.2)

AASSOC

RT) 2Nd ln(XA)+Nd(1-XA)+ 2Nd ln(1-F2 ⁄ Nd)+F2

(A.1.3.3)

Uassoc

RT) & ∂ (Aassoc ⁄ RT)

∂&) [1- 2Nd

Nd -F2]2F&∂F∂&

(A.1.3.4)

∂F∂&

) εHB√R2 [ Nd -F2

1+ 2F√R]ΥHB + 1

ΥHB(A.1.3.5)

F√R ≡ ( 1

XA- 1); R ≡ g(σ)KAD [exp(ε ⁄ kT)- 1]

(A.1.3.6)

The expressions for the solubility parameter, δ, and the molarvolume, VL

298, in terms of the SAFT EOS is as follows:

δ2VL298

RT) 9ηmC&uo exp(-3&uo)(2η- 4)

(1- η)3[1-C exp(-3&uo)]

- m(1+ 2&e)1+ &e ∑

i∑

j

iDij[&u]i[ητ ]j

-

9mC&uo exp(-3&uo)

1-C exp(-3&uo)∑

i∑

j

jDij[&u]i[ητ ]j

+

9ηC&uo(η- 52)(1-m) exp(-3&uo)[1-C exp(-3&uo)]

(1- η)(1- η2)

+

&εHBF√R(ΥHB + 1)

ΥHB

Nd +F2

1+ 2F√R(A.1.3.7)

where & ) 1/kT, T ) 298 K, and R is the universal gas constant.A.1.4. Vapor Packing Fraction. We can write the vapor

phase compressibility factor as below:

ZVsat )Psatb ⁄ ηVRT) 1+B2η

V (A.1.4.1)

B) ηV +B2(ηV)2 ) Pb

RT(A.1.4.2)

or

ηV )-1+ √1+ 4BB2

2B2(A.1.4.3)

To calculate the second virial coefficient, we have

B2 ) limηf0(Z- 1

η )) (dZdη)η)0

(A.1.4.4)

By setting Z ) 1 + B2η, we can define B2 for each EOS:

B2 )B2seg +B2

chain +B2assoc (A.1.4.5)

where the derived second Virial coefficient according to abovedefinition is

B2seg ) 4m (A.1.4.6)

B2chain ) 2.5 (A.1.4.7)

B2assoc is the same for both SAFT and PC-SAFT equations of

state and is derived as follows:

B2assoc )-Nd

2[0.035m

ΥHB])-Nd2[KADΥHB

b ] (A.1.4.8)

Another alternative to calculate the ηV is from isofugacitycriteria which leads the relationship

2B2ηV - ln( Pb

ηVRT)) Ares + ZL - 1- ln(ZL) (A.1.4.9)

or

ηV ) ηL exp[µLres - µV

res] (A.1.4.10)

where µres ) &µres/N is the reduced chemical potential and theresidual chemical potential.

A.2. The PC-SAFT Equation of State. A.2.1. Compress-ibility Factor. The PC-SAFT equation of state was derived anddescribed in detail by Gross and Sadowski.19 In terms of thecompressibility factor Z, the equation of state is given as thesum of the ideal gas contribution (Zid = 1), the hard-chain term(hc), the dispersive part (disp), and the contribution due toassociation (assoc) according to

Z) 1+ Zhc + Zdisp + Zassoc (A.2.1.1)

Note that the effect of multiple interactions (such as dipole-dipoleand quadrupole-quadrupole forces) is not separately taken intoaccount in eqs 1 and 12. This contribution will be the subjectof further investigations.

Zhc )mZhs - (m- 1)(ghs)-1F∂ghs

∂F (A.2.1.2)

where Zhs is the residual contribution of the hard-sphere fluid,F is the total number density of molecule given by Gross andSadowski,19 and ghs is the radial distribution function of thehard-sphere fluid; that is,

ghs ) 1(1- *3)

+3d*2

2(1- *3)2+

(*2d)2

2(1- *3)3

(A.2.1.3)

with *n defined as

*n )π6Fmdnn∈ {0, 1, 2, 3} (A.2.1.4)

The temperature-dependent segment diameter d is given by

d) σ[1- 0.12 exp(-3&ε)] (A.2.1.5)

where ε is the disperse attraction energy.The dispersion contribution to the compressibility factor of

the PC-SAFT EOS can be written as

Zdisp )-2πF∂(ηI1)

∂ηm2&εσ3 -πFm[C1

∂(ηI2)

∂η+C2ηI2](m&ε)2σ3

(A.2.1.6)

where

∂(ηI1)

∂η)∑

j)0

6

aj(m)(j+ 1)ηj (A.2.1.7)

∂(ηI2)

∂η)∑

i)0

6

bi(m)(j+ 1)ηi (A.2.1.8)

where aj(m) and bi(m) are two functions of m that were definedby Gross and Sadowski.19

C1 ) (1+ Zhc +F∂Zhc

∂F )-1

(A.2.1.9)

8408 Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008

C2 )∂C1

∂η(A.2.1.10)

As in the ESD and SAFT equations we can introduce themolecular volume in PC-SAFT as follows:

b) π10

md3 (A.2.1.11)

The definition of the association contribution parameters incompressibility factor for PC-SAFT EOS is the same as theSAFT equation.

A.2.2. Helmholtz Free Energy.

Ares ) m*0[ 3*1*2

(1- *3)+

*23

*3(1- *3)2+ (*2

3

*32- *0) ln (1- *3)]-

(m- 1) ln ghs(σ)- 2πFI1(η, m)m2&εσ3 -

πFI2(η, m)(&ε)2(mσ)3 + 2Nd ln(1-F2 ⁄ Nd)+F2 (A.2.2.1)

A.2.3. Solubility Parameter. The expression for the solubil-ity parameter, δ, and the molar volume in terms of the SAFTEOS is as follows:

δ2VL298

RT) T ×

[m*0(3(*1,T*2,T + *1*2,T)

(1- *3)+

3*1*2*3,T

(1- *3)2+

3*22*2,T

*3(1- *3)+

*23*3,T(3*3 - 1)

*32(1- *3)

+

(3*22*2,T*3 - 2*2

3*3,T

*33 ) ln(1- *3)+ (*0 -

*23

*32) *3,T

(1- *3))-

(m- 1)(ghs)-1( *3,T

(1- *3)2+ (dT

2 ) 3*2

(1- *3)+

(d2)( 3*2,T

(1- *3)2+

6*2*3,T

(1- *3)3)+ (ddT

2 ) 2*22

(1- *3)3+

(d2)( 4*2*2,T

(1- *3)3+

6*22*3,T

(1- *3)4) )-

2πF(∂I1

∂T-

I1

T)m2&εσ3 -πFm(∂C1

∂TΙ2 +C1

∂I2

∂T- 2C1

I2

T)(m&ε)2σ3

]+&εHBF√R(ΥHB + 1)

ΥHB

Nd +F2

1+ 2F√R(A.2.3.1)

where VL298is the liquid molar volume at 298 K, and R is the

universal gas constant.Note that the association term is the same as SAFT EOS.

*n,T )∂*n

∂T) π

6FmndTd n-1n∈ {1, 2, 3} (A.2.3.2)

dT )∂d∂T

) σ(3&εT )[-0.12 exp(-3&ε)] (A.2.3.3)

where T ) 298 K and the number density of molecules, F, iscalculated from η through

F) 6π

η(md3)-1 (A.2.3.4)

∂I1

∂T)∑

i)0

6

ai(m)i*3,Tηi-1 (A.2.3.5)

∂I2

∂T)∑

i)0

6

bi(m)i*3,Tηi-1 (A.2.3.6)

∂C1

∂T) *3,TC2 (A.2.3.7)

A.2.4. Vapor Packing Fraction.

B2 )B2hc +B2

disp +B2assoc (A.2.4.1)

We can readily derive the second viral coefficient for PC-SAFTEOS according to the definition presented in the previous sectionas below:

B2hc ) 3

2m+ 5

2(A.2.4.2)

and

B2disp )

-12a0

md3m2&εσ3 -

6b0

d3(m&ε)2σ3 (A.2.4.3)

and the association contribution is the same as SAFT EOS.

B. Sample Calculations

Here,we show the detailed calculation of three compoundsto elaborate the method more clearly. The first example is1-hexadecanol whose step by step calculations are shown byGC-SAFT method. The second example illustrates applicationto polymers with polymethyl-methacrylate, calculated by theESD EOS. The third example illustrates application to a cyclicether, tetrahydrofuran, using the PC-SAFT EOS.

Example B.1. (1-Hexadecanol, GC-SAFT). 1-Hexadecanolis composed of 1CH3, 15CH2 and 1OH and has a molecularweight of 242.446 kg/mol. Using Hoy values for liquid molarvolume, we obtain

V298L ) 12.1+ (21.6+ 15 × 15.6+ 12.5)) 279.35 cc ⁄ mol

(B.1.1)

By governing Gani’s approach for heat of vaporization, we get:

H298vap ) 6.829+ (4.116+ 15 × 4.65+ 24.529)

) 105.224 kJ ⁄ mol (B.1.2)

Therefore, solubility parameter can be readily calculated as

δ [(J ⁄ cc)0.5]) [Hvap298 [J ⁄ mol]- 298 [K] R [J ⁄ mol . K]

VL298 [cc ⁄ mol] ]1⁄2

) [105.224 × 1000- 298 × 8.314279.35 ]

) 19.18(J ⁄ cc)0.5

) 9.38(cal ⁄ cc)0.5 (B.1.3)

The group contribution shape factor for SAFT EOS is equal to

mGC-SAFT ) 1+ (0.995+ 15 × 0.462+ 1.853)) 10.772(B.1.4)

At this point, we have one parameter (shape factor) of SAFTEOS, and we have to calculate the other two (the temperature-independent segment volume, υ°° temperature-independentdispersion energy of interaction between segments, u°/k). Forthis reason we can use the following equations (B.1.5 and B.2.6):

ZL,298calcd )

0.1 [MPa] ×VL298 [cc ⁄ mol]

298 [K] ×R [J ⁄ mol K](B.1.5)

Using Hoy values for liquid molar volume at 298 K and 1 barwe obtain

0.1VL298

298R) 0.1 × 279.35

298 × 8.314) 0.011275

This value should be equal to the calculated compressibilityfactor calculated by SAFT EOS. This is our first constraint.

Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008 8409

-Ucalcd

298R)

δ2 [J ⁄ cc] VL298 [cc ⁄ mol]

298 [K] R [J ⁄ mol K](B.1.6)

δ2VL298

298R) 19.178252 × 279.35

298 × 8.314) 41.4706

This equation is our second constraint. By satisfying these twoequations we can readily calculate the remaining parameters.

υoo) 14.00 [mL ⁄ mol] and υo

k) 240.02 [K]

Using this approach to estimate the vapor pressures at 760,100 and 10 mmHg one will get ∼29.3% error as an average.

Example B.2. Poly Methyl Methacrylate, Mn ) 20 024,GC-ESD. The repeat unit of this polymer is methyl methacrylatewhich consists of 2 CH3, 1 CH2, 1 >C<, and 1 COO. Themolecular weight of the repeat cell is equal to 100.12, thereforeto get the Mw of the polymer we almost need 200 of the repeatcell units which means we now have 400 CH3, 200 CH2, 200>C< and 200 COO. At this time we can calculate theparameters as the same way explained in the previous example:

V298L ) 12.1+ (400 × 21.55+ 200 × 15.55+ 200 × 3.56+

200 × 25.7)) 17594.1 cc ⁄ mol (B.2.1)

H298vap ) 5520.03 kJ ⁄ mol (B.2.2)

δ [(J ⁄ cc)0.5]) 8.66 (cal ⁄ cc)0.5 (B.2.3)

cGC-ESD ) 1+ (400 × 0.228+ 200 × 0.207+ 200 × (-0.090)+200 × 0.498)) 215.21 (B.2.4)

Then by satisfying equations B.1.5 and B.1.6 we can calculatethe size parameter, b, and the depth of potential, ε. The resultsare as follows:

b) 6851.65 cc ⁄ mol, εk) 340.36 K (B.2.5)

Example B.3. Tetrahydrofuran, GC-PC-SAFT. Tetrahy-drofuran consists of three RCH2 and one CH2O, and itsmolecular weight is 72.1057 kg/mol. By the same way asexplained in the previous example we can calculate the liquidmolar volume, heat of vaporization, and solubility parameteras below:

VL298 ) 80.75 [cc ⁄ mol], Hvap

298 ) 28.257 [KJ ⁄ mol], and

δ) 17.87 [(J ⁄ cc)0.5] (B.3.1)

The group contribution shape factor would be obtained as

mGC-PCSAFT ) 1+ 3 × 0.294+ 1 × (-0.229)) 1.653(B.3.2)

After solving two constraints stated in Example B.1 we get

σ) 4.07 [Å] and εk) 340.36 [K] (B.3.3)

which would result in ∼85% error in vapor pressure as anaverage. Again, we have to stress that this method is suggestedonly when the experimental data are unavailable.

Readers are referred to http://gozips.uakron.edu/∼elliott1/ foravailable spreadsheets to facilitate their calculations.

Nomenclature

m ) shape parameter for SAFT andPC-SAFT EOSs∆m ) UNIFAC group contribution for shape parameter of SAFT

and PC-SAFT EOSsυ° ) temperature-dependent segment volume for SAFT, mL/mol

υ°° ) temperature-independent segment volume for SAFT, mL/mol

u° ) temperature-independent dispersion energy of interactionbetween segments for SAFT, J

τ ) 0.74048σ ) temperature-independent segment diameter, Åε ) the depth of the potential for, Jc ) shape parameter for ESD∆c ) UNIFAC group contribution for shape parameter of ESDb ) characteristic size parameter for ESD EOS, Cm3/molNd ) number of hydrogen-bonding segments per moleculeKAD ) hydrogen-bonding volume, (Å)3

εHB/k ) hydrogen-bonding energy, k cal/molk ) Boltzman factor ≈ 1.381 × 10-23 J/KVL

298 ) liquid molar volume of the liquid at 298 K, cc/mol∆Vi ) UNIFAC group contribution for liquid molar volumes at

298K, cc/mol∆H298

Vap) heat of vaporization at 298 K, kJ/molδ ) the solubility parameter, (J/cc)1/2

& ) 1/kT, J-1

B2 ) second virial coefficientXA ) mole fraction of molecules NOT bonded at site Aghs ) the radial distribution function of the hard-sphere fluid*n ) π/6 Fmdn, n ∈{0,1,2,3}η ) *3 ) πNAv/6 Fmd3

F ) number density (number of molecules in unit volume)µres ) &µres/N ) reduced chemical potential

Literature Cited

(1) Sandler, S. I. Models for Thermodynamic and Phase EquilibriaCalculations; Marcel Dekker: New York, 1987.

(2) Folie, B.; Radosz, M. Phase Equilibria in High Pressure PolyethyleneTechnology. Ind. Eng. Chem. Res. 1995, 34, 1501–1516.

(3) Harismiadis, V. I.; Kontogeorgis, G. M.; Fredenslund, A.; Tassios,D. P. Application of the van der Waals Equation of State to Polymers II:Prediction. Fluid Phase Equilib. 1994, 96, 93–117.

(4) Xiong, Y.; Kiran, E. Comparison of Sanchez-Lacombe and SAFTModel in Predicting Solubility of Polyethylene in High-Pressure Fluids.J. Appl. Polym. Sci. 1995, 55, 1805–1818.

(5) Huang, S.; Radosz, M. Equation of State for Small, Large, Poly-disperse, and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284.

(6) Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An Equation of StateBased on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res.2001, 40, 1244–1260.

(7) Elliott, J. R.; Suresh, S. J.; Donohue, M. D. A Simple Equation ofState for Nonspherical and Associating Molecules. Ind. Eng. Chem. Res.1990, 29, 1476.

(8) Elliott, J. R.; Natarajan, R. N. Extension of the ESD Equation toPolymer Solutions. Ind. Chem. Eng. Res. 2002, 41, 1043.

(9) Wertheim, M. S. Fluids with Highly Directional Attractive Forces.I. Statistical Thermodynamics. J. Stat. Phys. 1984, 35, 19.

(10) Benzaghou, S.; Passarello, J. P.; Tobaly, P. Predictive Use Of ASAFT EOS For Phase Equilibria Of Some Hydrocarbons And Their BinaryMixtures. Fluid Phase Equilib. 2001, 180, 1.

(11) Jensen, T.; Fredenslund, A.; Rasmussen, R. Pure-Component VaporPressures Using UNIFAC Group Contribution. Ind. Eng. Chem. Fundam.1981, 20, 239–246.

(12) Yair, O. B.; Fredenslund, A. Extension of the UNIFAC Group-Contribution Method for the Prediction of Pure-Component Vapor Pressures.Ind. Eng. Chem. Process Des. DeV. 1983, 22, 433–436.

(13) Asher, W. E.; Pankow, J. F.; Erdakos, G. B.; Seinfeld, J. H.Estimating the Vapor Pressures of Multi-functional Oxygen-ContainingOrganic Compounds Using Group Contribution Methods. Atmos. EnViron.2002, 36, 1483–1498.

(14) Asher, W. E.; Pankow, J. F. Vapor Pressure Prediction for Alkanoicand Aromatic Organic Compounds by a UNIFAC-Based Group ContributionMethod. Atmos. EnViron. 2006, 40, 3588–3600.

(15) Bureau, N.; Defiolle, D.; de Hemptinne, J.-C. Phase Equilibria of(Methane-Long Chain Ester Cuts) Systems in Drilling Conditions. FluidPhase Equilib. 2002, 194-197, 831–846.

8410 Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008

(16) Constantinou, L.; Gani, R. New Group Contribution Method ofEstimating Properties of Pure Compounds. AIChE J. 1994, 40, 1697–1710.

(17) Kolska, Z. R.; V.; Gani, R. Estimation of the Enthalpy ofVaporization and the Entropy of Vaporization for Pure Organic Compoundsat 298.15 K and at Normal Boiling Temperature by a Group ContributionMethod. Ind. Eng. Chem. Res. 2005, 44, 8436–8454.

(18) Tamouza, S. P., J.-P.; Tobaly, P.; de Hemptinne, J.-C. Applicationto binary mixtures of a group contribution SAFT EoS (GCSAFT). FluidPhase Equilib. 2005, 228-229, 409–419.

(19) Tamouza, S. P., J.-P.; Tobaly, P.; de Hemptinne, J.-C. Groupcontribution method with SAFT EoS applied to vapor liquid equilibria ofvarious hydrocarbon series. Fluid Phase Equilib. 2004, 222-223, 67–76.

(20) Nguyen Thi, T. X. T., S.; Passarello, J.-P.; Tobaly, P.; deHemptinne, J.-C. Application of Group Contribution SAFT Equation ofState (GC-SAFT) to Model Phase Behaviour of Light and Heavy Esters.Fluid Phase Equilib. 2005, 238, 254–261.

(21) Huynh, D. N. B., M.; Passarello; J.-P.; Tobaly, P.; de, Hemptinne.Application of GC-SAFT EOS to polycyclic aromatic hydrocarbons. FluidPhase Equilib. 2007, 254, 60–66.

(22) Tihic, A.; Kontogeorgis, G. M.; von Solms, N.; Michelsen, M. L.;Constantinou, L. A Predictive Group-Contribution Simplified PC-SAFTEquation of State: Application to Polymer Systems. Ind. Eng. Chem. Res.2007,

(23) Barker, J. A.; Henderson, D. Perturbation Theory and Equation ofState for Fluids: The Square-Well Potential. J. Chem. Phys. 1967, 47, 2856.

(24) Coleman, M. M.; Painter, P. C. Hydrogen-Bonded Polymer Blends.Prog. Polym. Sci. 1995, 20, 1.

(25) Suresh, S. J.; Elliott, J. R., Jr. Multiphase Equilibrium Analysisvia a Generalized Equation of State. Ind. Eng. Chem. Res. 1992, 31, 2783.

(26) Hoy, K. L. Tables of Solubility Parameters. J. Coated Fabrics 1989,19, 53.

(27) Elliott, J. R.; Gerek, Z. N.; Gray, N. H. Combining MolecularDynamics and Chemical Process Simulation: The SPEAD Model. Asia-Pacific J. Chem. Eng. 2007, 2, 257–271.

(28) Barker, J. A.; Henderson, D. Perturbation Theory and Equation ofState for Fluids. II. Successful Theory of Liquids. J. Chem. Phys. 1967,47, 4714.

(29) Henderson, D. Perturbation Theory for a Mixture of Hard Sphereand Squre-Well Molecules. J. Chem. Phys. 1974, 61, 926.

(30) Henderson, D.; Barker, J. A. Perturbation Theory and the Equationof State of Mixtures of Hard Spheres. J. Chem. Phys. 1968, 49, 3377.

(31) Elliott, J. R.; Vahid, A.; Sans, A. D. Transferable Potentials forMixed Alcohol-Amine Interactions. Fluid Phase Equilib. 2007, 256, 4–13.

(32) DIADEM 2004; Design Institute for Physical Properties (DIPPR),2004.

(33) Chen, B.; Potoff, J. J.; Siepmann, J. I. Monte Carlo Calculationsfor Alcohols and Their Mixtures with Alkanes. Transferable Potentials forPhase Equilibria. 5. United-Atom Description of Primary, Secondary, andTertiary Alcohols. J. Phys. Chem. B 2001, 105, 3093–3104.

(34) Wick, C. D.; Martin, M. G.; Siepmann, J. I. Transferable Potentialsfor Phase Equilibria. 4. United-Atom Description of Linear and BranchedAlkenes and Alkylbenzenes. J. Phys. Chem. B 2000, 104, 8008–8016.

(35) Martin, M. G.; Siepmann, J. I. Transferable Potentials for PhaseEquilibria. 1. United-Atom Description of n-Alkanes. J. Phys. Chem. B1998, 102, 2569–2577.

(36) Bourasseau, E.; Haboudou, M.; Boutin, A.; Fuchs, A. H.; Ungerer,P. New Optimization Method for Intermolecular Potentials: Optimizationof a New Anisotropic United Atoms Potential for Olefins: Prediction ofEquilibrium Properties. J. Chem. Phys. 2003, 118, 3020–3034.

(37) Elliott, J. R. Efficient Implementation of Wertheim’s Theory forMulticomponent Mixtures of Polysegmented Species. Ind. Eng. Chem. Res.1996, 35, 1624.

ReceiVed for reView February 27, 2008ReVised manuscript receiVed July 16, 2008

Accepted August 4, 2008

IE800329R

Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008 8411