Making Equation of State Models Predictive−Part 3 ... - ACS Publications

J. Phys. Chem. C 2007, 111, 15533-15543

15533

Making Equation of State Models Predictive-Part 3: Improved Treatment of Multipolar Interactions in a PC-SAFT Based Equation of State† Kai Leonhard, Nguyen Van Nhu, and Klaus Lucas* Lehrstuhl fu¨r Technische Thermodynamik, RWTH Aachen, 52056 Aachen, Germany ReceiVed: April 3, 2007; In Final Form: July 17, 2007

Recent investigations have demonstrated the merit of quantum mechanically determined molecular parameters and new combination rules in equation of state (EOS) models. In this paper we study several perturbation theory models for application in a predictive EOS. Two models have been compared, namely, the perturbation theory with a spherical reference and an approach where the free energy has been fitted directly to molecular simulation results of certain polar fluids. We apply both methods without adjustable binary parameters. A comparison with experimental phase equilibirum data reveals that both models have strengths and weaknesses. This insight leads to an improved perturbation theory based EOS model that performes better than the two preceding ones and is likely to lend itself to further improvements in the future.

1. Introduction

2. Perturbation Theories for Electrostatic Interactions

Various different approaches to a predictive equation of state (EOS) have been pursued. One frequently used procedure starts from existing models designed to predict mixture properties, such as group contribution methods (e.g., (mod)UNIFAC1), and surface charge interaction models, such as COSMO-RS2, that were originally constructed for the dense liquid state only. They can be made density dependent by combination with an empirical EOS.3 Such a combination extends the range of application for the computation of vapor-liquid equilibria (VLE) to higher pressures, but this ad-hoc extension of contact interactions over a large density range usually yields only limited accuracy. Also, the basic restrictions of insufficient knowledge of group interaction parameters, and the fact that the partitioning into groups does not always work, limit the general applicability, as does the limited accuracy of the quite recent COSMO-RS model, especially for small molecules. Another path relies on physically based EOS models. In such an approach, an analytical EOS model is usually used for repulsive interactions, and then dispersion interactions, electrostatic interactions, and possibly others, are added by a perturbation treatment. Recently, we used quantum mecanically (QM) computed molecular properties, namely, dipole and quadrupole moments, as well as static and dynamic polarizabilities, in combination with the PCP-SAFT EOS,4-6 to investigate the improved predictive performance of that EOS when QM information is used.7,8 It turned out that this combination of an EOS and a QM-based combination rule (CR) is successful only for a limited class of systems because the perturbation theory (PT) employed in PCP-SAFT has several important restrictions. In this work, we combine PC-SAFT4 with an earlier PT based on a spherical reference system and compare it to experimental data. We classify the results in terms of types of intermolecular interactions and indicate the direction of further improvements to be made.

2.1. PT with a Spherical Reference. In a PT with a spherical reference, the assumption is made that the spherical pair correlation function (PCF) of the reference fluid is adequate to incorporate multipole moments and other force types even when the shape of the molecules changes. Under these conditions, the internal energy of the system can be computed by an integration of the product of the molecular interaction energy and the PCF of the spherical reference system over the configuration space. We obtain the change in free energy by a suitable Boltzmann averaging of the internal energy, which involves the computation of the logarithm of an averaged exponential function and is therefore expanded in a series. Usually, the expressions are worked out up to third-order, and the limit of the expansion is approximated by a Pade´ approximant. A more detailed description of this approach can be found in refs 9 and 10. Some features of this theory are important for our purpose. To determine the PCF of the reference fluid, for which a 1-center Lennard-Jones (1CLJ) fluid has been chosen, Monte Carlo (MC) simulations have been performed at various densities and temperatures. On the basis of these results, the required integrations have been performed, and correlation expressions have been fitted to reproduce the integrations of the intermolecular distances that have been numerically obtained from the MC simulation results. The integrations over the orientation have been performed analytically. Hence, the advantage of this method is that analytical results for any desired type of interaction can be obtained based on the same series of MC simulations of a single reference fluid. So, this approach presents a systematic correction of conformal solution theory. The disadvantage is that the accuracy of this method decreases when shape effects are too strong to keep the strucural properties of the reference fluid unchanged. The equations for the perturbation theory with spherical reference are given in the Appendix (Section 7). Established EOS models on this basis are the GT,12 the SLM,13-15 and the BACKPF16,17 EOS models. 2.2. PT Used in PCP-SAFT. Another branch of PT models has evolved for the extension of the PC-SAFT model to

†

Part of the “Keith E. Gubbins Festschrift”. * To whom correspondence should be addressed. E-mail: lucas@ltt. rwth-aachen.de; phone: (+49)241-8098350; fax: (+49)241-8092-255.

10.1021/jp0726081 CCC: $37.00 © 2007 American Chemical Society Published on Web 10/13/2007

15534 J. Phys. Chem. C, Vol. 111, No. 43, 2007 quadrupole-quadrupole5 and dipole-dipole6 interactions. Monte Carlo simulations have been performed with a symmetric 2-center LJ (2CLJ) fluid to obtain macroscopic data such as vapor pressures, saturated liquid and vapor densities, critical point data, and virial coefficients. However, the perturbation formalism has not been worked out explicitly on the basis of the pair correlation function of the underlying 2CLJ fluid. Instead, the third-order, two-body integral has been set to zero (J3,ij ) 0), and the two remaing integrals (second-order, two body (J2,ij) and third-order, three-body (J3,ijk)) are treated as functions of density, temperature, and bond length with coefficients adjusted to make the result of the PT fit the results of MC simulations for the above-mentioned macroscopic properties. The advantage of this approach is that no structure of a reference fluid is used explicitly, which may spoil the results in the case of a strong perturbation. Instead, the macroscopic properties used for fitting are recovered with high accuracy from the PT. However, this advantage comes with several disadvantages. First, there is no guarantee that a property not used for fitting is described equally well. Second, in principle, for each kind of interaction and each shape of the molecule a new timeconsuming series of MC simulations covering a sufficiently large region of density and temperature has to be performed. In practice, only a limited number of simulations with linear symmetric molecules has been selected as the basis of the theory, with inaccuracies resulting in the application to most real systems with nonlinear molecules. As for quadrupoles, the restrictions for PCP-SAFT are specifically that, first, only squared scalar quadrupole moments Q2i are included in the model, and that, second, the simulations have been performed for prolate linear molecules with one of the quadrupole principal axes aligned with the symmetry axis of the molecule only. More generally and in contrast to PCPSAFT, an attractive effect of mixing molecules with the same absolute quadrupole moment but with different signs may influence the macroscopic behavior. Also, the use of scalar quadrupole moments instead of tensors is appropriate for highly symmetric molecules only (here “highly symmetric” means “at least C3V symmetry”, e.g. linear molecules) because all other molecules have two independent elements of the quadrupole tensor in the quadrupole principal axis frame. Also, quadrupolar oblate molecules may behave differenty. Concerning dipoles, the interactions have been incorporated into the EOS in an analogous manner. Monte Carlo simulations, with molecules of varying elongation and the dipole vectors always aligned with the symmetry axis of the molecules, have been performed. Then, the results have been used to adjust coefficients in the J integrals. It is questionable as to whether more general dipole vectors, along with arbitrary shapes, are incorporated adequately by this approach. A dipole-quadrupole contribution for PCP-SAFT has been developed but not yet published18 and was therefore not considered in this study. 2.3. Combined PT. Because of the good performance of the PC-SAFT EOS in combination with the London CR that was described in Part 28 of this series, we have abandoned the earlier versions of the spherical-reference PT EOS models. Instead, we have combined the PT with spherical reference with PCSAFT, which we call PC-SAFTP1, for PC-SAFT polar, version 1. During the course of our investigations it turned out that a combination of PC-SAFT with the dipole-dipole term of PCPSAFT and all other (dipole-quadrupole and quadrupolequadrupole) terms from the PT with spherical reference performed particularly well. Therefore, we implemented that

Leonhard et al. combination as well, and we denote it as PC-SAFTP2, for PCSAFT polar, version 2. The details of the two new EOS models are described in the Appendix (Section 7). For models without a dipole moment the two models are identical, and we may just write PC-SAFTP without further specifying the version. 3. Combination Rules Whereas the mixing rules follow directly from the PT treatment in the EOS models, combination rules (CR) are necessary for the adjustable pure compound PC-SAFT energy parameter ii and the size parameter σii. In the case of hard spheres, the exact combination rule for the diameter (i.e., the contact distance) is obvious (eq 1).

σij )

σii + σjj 2

(1)

For the parameter  we use a modification of London’s combination rule for molecules consisting of segments (eqs 2 and 3) instead of the original geometric CR.19

ij ≈

6 6 2 iiσiijjσjjRi,sRj,s 2 2 σ6ij Rj,8 iiσ6ii + Ri,s jjσ6jj

(2)

with8

Ri,s )

Ri mi

(3)

Here, Ri is the static polarizability of molecules i, and mi is the number of segments in molecule i. The polarizabilitites have previously been computed quantum mechanically.7 The advantage of London’s CR over the originally used geometric combination rule is that it does not require an adjustable binary parameter but uses QM obtained information on the dispersion interaction instead. When the EOS model is able to account for all other interactions properly, London’s CR provides an accurate estimate of the nonideality due to dispersion interactions. 4. Comparison to Experiment 4.1. Adjustment of Pure Compound Parameters. The three pure compound parameters that cannot be directly determined by quantum chemistry have been adjusted to experimental vapor pressure data, to saturated liquid volume data, and to critical data for each compound. The resulting parameters and the resulting average absolute deviations, in percent, of the vapor N (|p pressure (%AAD ) 100/N ∑i)1 calc - pexp|/pexp)) and of the liquid’s molar volume (computed analogously) are shown in Tables 1, 2, and 3. The averages of the vapor pressure deviations over all systems are 0.8%, 1.0%, and 0.8%, and for the liquid volume they are 1.0%, 1.1%, and 0.8% for the PC-SAFTP1, the PC-SAFTP2, and the PCP-SAFT models, respectively. So, all studied EOS models allow a satisfactory correlation of pure compound VLE data using only three adjustable parameters. This indicates that the most important physical interaction effects are essentially captured equally well by the models. The two PC-SAFTP models are slightly less accurate than PCP-SAFT. This is because of the larger error for linear molecules with a strong dipole moment (e.g., acetonitrile) for which PCP-SAFT was specifically designed. Pure compound paramters for argon and krypton are only given in Table 1 because they are identical for all three models because they coincide with the PC-SAFT EOS for nonpolar molecules.

Making Equation of State Models Predictive

J. Phys. Chem. C, Vol. 111, No. 43, 2007 15535

TABLE 1: Pure Compound Parameters and Average Absolute Deviations of Vapor Pressure (∆p/pexp) and Molar Volume of the Liquid Phase (∆W/Wexp) in % for the PC-SAFTP1 EOSa average absolute deviation (%) compound

m

σ (Å)

 (kB/K)

vapor pressure

liquid volume

temp. range (K)

number of points

data source (ref)

argon krypton ethane propane butane pentane octane decane neopentane isobutane cyclohexane benzene C6F6 CO2 CS2 R22 R23 R32 R41 R125 R134a R227ea DME DMF acetonitrile acetone

1.0000 1.0000 1.6076 1.9857 2.2908 2.6872 3.8725 4.6438 2.4098 2.3046 2.4919 2.2524 3.6330 1.2578 1.5947 1.8887 1.9230 1.2747 1.3423 2.6175 2.1808 3.3879 1.8625 2.2961 1.7696 2.2690

3.3768 3.6010 3.5251 3.6263 3.7308 3.7599 3.8265 3.8560 3.9302 3.7281 3.8513 3.7701 3.4502 3.4182 3.7036 3.4666 3.1849 3.6606 3.3398 3.3338 3.4949 3.3621 3.5176 3.6743 3.4893 3.4911

117.825 164.167 191.266 209.148 225.135 231.471 240.666 244.369 222.611 214.272 280.973 292.179 222.336 179.986 345.053 196.405 132.763 147.321 157.722 154.649 161.132 162.546 224.140 259.664 121.217 222.952

0.9 1.0 0.3 0.4 0.6 0.4 0.7 0.5 0.4 1.1 0.5 0.9 0.4 0.3 2.0 0.3 0.5 0.9 1.2 0.2 0.5 0.8 1.7 2.1 2.7 0.4

2.1 1.7 0.5 0.5 0.6 0.3 1.2 1.3 1.1 0.4 0.6 0.5 0.8 0.7 0.8 0.4 0.7 1.4 0.9 0.8 1.1 0.3 1.1 2.1 4.2 1.2

86-150 120-208 95-305 100-360 150-420 160-444 230-560 270-610 256.6-430 160-400 270-550 250-555 280-510 220-304 200-540 130-365 120-295 149-349 130-309 186-338 170-370 170-370 150-400 270-640 250-530 180-500

33 23 43 27 28 30 35 35 19 25 29 15 24 22 35 48 36 50 13 40 41 21 26 38 29 33

20 20 21 20 20 22, 23 20 20 20 20 20 24 20 25 20 26 27 28 29 30 31 20 20 20 20 20

0.8

1.0

average a

The subscript “exp” denotes either true experimental data or pseudo experimental data created with a reference EOS. Multipole moment elements have been given elsewhere7 as have polarizabilities and dispersion coefficients needed for the CRs.

TABLE 2: Pure Compound Parameters and Average Absolute Deviations of Vapor Pressure (∆p/pexp) and Molar Volume of the Liquid Phase (∆W/Wexp, in %) for the PC-SAFTP2 EOS compound

m

σ (Å)

 (kB/K)

ethane propane butane isobutane pentane neopentane octane decane cyclohexane benzene C6F6 CO2 CS2 R22 R23 R32 R41 R125 R134a R227ea DME DMF acetonitrile acetone

1.6076 1.9857 2.2908 2.3046 2.6872 2.4098 3.8725 4.6438 2.4919 2.2524 3.6330 1.2578 1.5947 2.1349 2.3089 1.7002 2.0370 2.7891 2.5135 3.4766 2.0052 2.6853 2.2745 2.6889

3.5251 3.6263 3.7308 3.7281 3.7599 3.9302 3.8265 3.8560 3.8513 3.7701 3.4502 3.4182 3.7036 3.3148 2.9760 3.2395 2.8939 3.2559 3.3081 3.3298 3.5165 3.4969 3.2125 3.3026

191.266 209.148 225.135 214.271 231.471 222.611 240.666 244.369 280.973 292.179 222.336 179.986 345.053 187.429 130.208 140.248 151.256 152.897 158.782 161.634 217.559 275.705 205.293 225.835

average

average absolute deviation % vapor pressure liquid volume 0.3 0.4 0.6 1.1 0.4 0.4 0.7 0.5 0.5 0.9 0.4 0.3 2.0 0.3 1.6 2.2 2.5 0.3 0.8 0.8 1.3 2.4 2.8 0.5

0.5 0.5 0.6 0.4 0.3 1.1 1.2 1.3 0.6 0.5 0.8 0.7 0.8 0.5 1.5 2.2 1.2 0.7 2.0 0.4 0.9 2.5 3.0 0.9

1.0

1.1

4.2. Prediction of Mixture Behavior. To compare the predictive performance of the three perturbation theories of PCSAFTP1, PC-SAFTP2, and PCP-SAFT for mixtures, we consider VLE, hE, and liquid-liquid equilibria (LLE) data. Although these data are not simultaneously available for many systems, we can present them for some characteristic examples.

We classify the studied systems into groups depending on the polarity of the molecules they contain. 4.2.1. Systems with Weak and Simple Electrostatic Interactions. This class contains systems that consist of two nonpolar compounds or of one nonpolar and one quadrupolar component with high symmetry. Hence, the results of the PC-SAFTP1 and the PC-SAFTP2 EOSs are identical. The predictions for the argon-krypton mixture are equally good for all three models, because they are identical for this system, and show that the underlying PC-SAFT model can handle the size effects in this mixture and that the London combination rule can describe the dispersion nonideality. In the case of alkane-alkane mixtures, the three PTs perform equally well because the influence of the multipolar interactions is practically negligible; see Table 4. These mixtures demonstrate the ability of the PC-SAFT model to predict chain-lengths effects. Concerning mixtures of carbon dioxide with an alkane, the performance of the three models is similar but no longer identical (only the average pressure deviations over all systems of PCP-SAFT and PC-SAFTP are identical by coincidence). The PCP-SAFT EOS with the geometric CR always underestimates the nonideality of the systems, and London’s CR improves the prediction significantly except for CO2-neo-pentane, where the accuracy of both CRs is approximately the same. The PC-SAFTP EOS produces results slightly more nonideal. This leads to a slight overestimation of the nonideality for some systems with the London CR. Because the vapor pressures of CO2 and ethane are relatively similar, the system exhibits an azeotrope and is the most sensitive one of this group to inaccuracies of the model. We show the detailed VLE and HE diagrams for this system in Figures 1, 2 and 3. From these diagrams, we can see that the performance for both models is similar for VLE and hE in the sense that a deviation in one property leads to a deviation in the other one in the same direction, but the deviations in hE

15536 J. Phys. Chem. C, Vol. 111, No. 43, 2007

Leonhard et al.

TABLE 3: Pure Compound Parameters and Average Absolute Deviations of Vapor Pressure (∆p/pexp) and Molar Volume of the Liquid Phase (∆W/Wexp, in %) for the PCP-SAFT EOSa compound

m

σ (Å)

ethane propane butane pentane octane decane isobutane neopentane cyclohexane benzene C6F6 CO2 CS2 R22 R23 R32 R41 R125 R134a R227ea DME DMF acetonitrile acetone

1.6075 1.9856 2.2904 2.6860 3.8701 4.6411 2.3049 2.4098 2.4918 2.0259 3.3720 1.5747 1.5927 1.9909 1.8984 1.7881 2.0501 2.5275 2.2709 3.3530 2.0024 2.3879 2.2552 2.5135

3.5252 3.6264 3.7310 3.7606 3.8276 3.8570 3.7278 3.9302 3.8513 3.9429 3.5660 3.1383 3.7064 3.4128 3.2306 3.1952 2.8896 3.3943 3.4751 3.3828 3.4289 3.6688 3.2150 3.3853

average absolute deviation %

 (kB/K)

µ (D)

Θ (DÅ)

vapor pressure

liquid volume

191.272 209.140 225.151 231.517 240.712 244.408 214.237 222.611 280.974 302.587 226.148 162.190 345.089 197.780 149.750 168.785 151.260 161.531 176.508 164.584 220.005 301.211 209.072 237.618

0.000 0.087 0.036 0.000 0.000 0.000 0.133 0.000 0.000 0.000 0.000 0.000 0.000 1.473 1.667 2.010 1.875 1.562 2.078 1.410 1.331 3.840 3.938 2.928

0.752 0.781 0.968 1.486 2.844 3.579 0.689 0.000 0.729 7.908 8.177 4.271 3.194 4.131 3.807 4.047 0.420 5.114 5.990 4.926 3.495 7.692 2.400 4.414

0.3 0.4 0.6 0.4 0.7 0.5 1.1 0.4 0.5 0.9 0.8 0.3 2.0 0.4 0.4 0.7 2.3 0.4 0.4 0.8 1.5 1.6 1.5 0.5

0.5 0.4 0.6 0.3 1.2 1.3 0.4 1.1 0.6 1.1 1.2 0.9 0.7 0.4 0.3 0.7 0.6 0.5 0.2 0.4 0.6 2.8 1.4 1.2

0.8

0.8

average a

The scalar quadrupole moments transformed as described in ref 8 and the scalar dipole moments obtained quantum chemically are shown as well. Conversion of the multipole moment units used to SI units: 1 D ) 3.336 × 10-30 C m and 1 B ) 3.336 × 10-40 C m2.

TABLE 4: Average Absolute Deviations of Vapor Pressure (∆p/pexp) and Gas-Phase Composition (∆y, in %) Computed with the PCP-SAFT and PC-SAFTP EOSs with the London Combination Rule for Systems with Weak and Simple Electrostatic Interactionsa system argon-krypton CO2-ethane CO2-propane CO2-n-butane CO2-iso-butane CO2-neo-pentane CO2-octane CO2-cyclohexane ethane-pentane ethane-n-decane benzenecyclohexane average

source of PCP-SAFT PC-SAFTP1 PC-SAFTP2 exp. data ∆p/pexp ∆y ∆p/pexp ∆y ∆p/pexp ∆y (ref) 0.7 3.1 2.8 2.8 5.0 6.4 6.9 3.3 2.2 4.3 4.2

0.3 0.8 0.6 0.6 1.7 1.2 0.4 0.4 0.7 1.1 1.6

0.7 1.6 2.2 3.5 7.5 7.2 7.2 2.5 2.2 4.3 3.0

0.3 0.7 0.5 0.6 1.8 1.3 0.4 0.5 0.7 1.1 1.1

0.7 1.6 2.2 3.5 7.5 7.2 7.2 2.5 2.2 4.3 3.0

0.3 0.7 0.5 0.6 1.8 1.3 0.4 0.5 0.7 1.1 1.1

3.8

0.9

3.8

0.8

3.8

0.8

32 33 34 35 35 36 37, 38 39 40 41 42

a

The static polarizabilities necessary for London’s CRs have been given in ref 7.

are more pronounced. For this system, the accuracy of the PCSAFTP model is slightly better than that of PCP-SAFT. In general, we can conclude that both EOS models show an excellent predictive performance for this kind of systems. When we consider the mixture of prolate and oblate quadrupolar molecules, the accuracy of PCP-SAFT, in which parameters have been adjusted to simulations of prolate molecules, is lower than that of PC-SAFTP with a spherical reference, as can be demonstrated with the cyclohexane-benzene system (Figure 4). 4.2.2. Systems with Quadrupolar-Quadrupolar Interactions. The benzene-perfluorobenzene system is an extremely difficult system. The two compounds have similar sizes and similar absolute quadrupole moments but with different signs. It is known that this leads to an attractive contribution at high

Figure 1. Vapor-liquid equilibrium of the system ethane(1)-carbon dioxide(2). Experimental data are taken from ref 33.

densities.19 Further, alkanes and perfluorinated alkanes separate into two phases because of the unfavorable dispersion interaction between fluorine and hydrogen. The effect is similar for aromatics and is seen when the QM based CRs, which model the dispersion nonideality relatively accurately, are employed. In addition to that, the boiling points of both components are almost identical. Together with the two oppositely acting previous effects, this leads to a VLE with two azeotropic points, compare Figure 5. Because the PCP-SAFT EOS does not consider the sign of the quadrupole moment, only the positive deviation from ideal behavior, because of the small difference between the quadrupole moments and the dispersion nonideality, is found. The last contribution leads to a positive excessenthalpy, whereas experimental data show that the attractive effect of the quadrupole moments dominates (Figure 6). The PC-SAFTP

Making Equation of State Models Predictive

Figure 2. Excess enthalpy (hE) of the system ethane(1)-carbon dioxide(2). Experimental data are taken from ref 43.

Figure 3. Excess enthalpy (hE) of the system ethane(1)-carbon dioxide(2). Experimental data are taken from ref 43.

Figure 4. Vapor-liquid equilibrium of the system cyclohexane(1)benzene(2). Experimental data are taken from ref 42.

model, in principle, contains the attractive quadrupole-quadrupole interaction, but it is too weak ascompared to other interaction contributions. Therefore, hE is negative, at least at the lower temperature, but is not negative enough. Also, the

J. Phys. Chem. C, Vol. 111, No. 43, 2007 15537

Figure 5. Vapor-liquid equilibrium of the system benzene(1)perfluorbenzene(2). Experimental data are taken from ref 44.

Figure 6. Excess enthalpy (hE) of the system benzene(1)-perfluorbenzene(2). Experimental data are taken from ref 45.

azeotrope is weaker with the PC-SAFTP EOS, but the second azeotrope, because of attractive interactions, is still not found. An analysis of the carbon disulfide-carbon dioxide system is more complicated. In this system there is a strongly nonideal dispersion, and there are quadrupoles of different signs and different absolute magnitude. Beyond that, the density is lower, and vapor pressures differ by more. The geometric mean CR underestimates the nonideality of the system. With the new combination rule, the agreement of the PCP-SAFT results with experimental VLE data is reasonable, but at 280 K the experimentally found LLE is not predicted (Figure 7). At 360 K, the Henry coefficient is predicted reasonably well, but the critical point is too high (Figure 7). The PC-SAFTP EOS predicts the LLE at 280 K reasonably well, and at 360 K, the Henry coefficient is predicted slightly more accurately, but the critical point is even higher. The overall prediction of the phase diagram of this difficult system is very satisfying. In this class of systems the average absolute deviation of the vapor pressure is almost twice as large as that in the class of simple systems (see Table 5) for the PCP-SAFT EOS, but it is similar in both cases for the PC-SAFTP EOS. Perhaps more interesting, the inclusion of a more physical combination rule for dispersion interactions does not always lead to an improvement of the predictions of PCP-SAFT. We attribute this to the fact that in some systems (e.g., benzene-perfluorobenzene) the

15538 J. Phys. Chem. C, Vol. 111, No. 43, 2007

Leonhard et al. TABLE 5: Average Absolute Deviations of Vapor Pressure (∆p/pexp) and Gas-Phase Composition (∆y, in %) Computed with the PCP-SAFT and PC-SAFTP EOSs with the London Combination Rule for Systems with QuadrupolarQuadrupolar Interactions system CO2-CS2 benzene-C6F6 CO2-benzene

PCP-SAFT ∆p/pexp ∆y

PC-SAFTP1 ∆p/pexp ∆y

PC-SAFTP2 ∆p/pexp ∆y

12.0 3.2 7.3

0.9 1.7 0.4

7.4 2.1 2.7

1.1 1.2 0.3

7.4 2.1 2.7

1.1 1.2 0.3

7.5

1.0

4.1

0.9

4.1

0.9

average

source of exp. data (ref) 46 44 39

a

The static polarizabilities necessary for London’s CRs have been given in ref 7.

TABLE 6: Average Absolute Deviations of Vapor Pressure (∆p/pexp) and Gas-Phase Composition (∆y, in %) Computed with the PCP-SAFT and PC-SAFTP EOSs with the London Combination Rule for Systems with Nonpolar-Moderately Dipolar Interactionsa Figure 7. Vapor-liquid and liquid-liquid equilibria of the system carbon dioxide(1)-carbon disulfide(2). Experimental data are taken from ref 46.

system

source of PCP-SAFT PC-SAFTP1 PC-SAFTP2 exp. data ∆p/pexp ∆y ∆p/pexp ∆y ∆p/pexp ∆y (ref)

R125-propane R134a-propane propane-R227ea

13.6 9.6 15.9

5.4 3.6 6.0

11.4 2.1 15.4

4.5 1.0 5.8

12.7 4.6 15.8

5.0 1.5 6.0

average

13.0

5.0

9.7

3.8

11.0

4.2

48 29 49

a

The static polarizabilities necessary for London’s CRs have been given in ref 7.

TABLE 7: Average Absolute Deviations of Vapor Pressure (∆p/pexp) and Gas-Phase Composition (∆y, in %) Computed with the PCP-SAFT and PC-SAFTP EOSs with the London Combination Rule for Systems with Nonpolar-Strongly Dipolar Interactionsa system

source of PCP-SAFT PC-SAFTP1 PC-SAFTP2 exp. data ∆p/pexp ∆y ∆p/pexp ∆y ∆p/pexp ∆y (ref)

R32-propane DMF-cyclohexane

2.8 2.1

2.3 0.7

27.7 5.8

8.6 5.8

13.5 2.2

3.6 1.0

average

2.5

1.5

16.7

7.2

7.9

2.3

29 50, 51

a The static polarizabilities necessary for London’s CRs have been given in ref 7.

Figure 8. Vapor-liquid equilibrium of the system propane(1)-R134a(2). Experimental data are taken from ref 47.

non-inclusion of repulsive dispersion and attractive oppositesign quadrupole effects cancel each other. Also, the oblate shape of benzene and perfluorobenzene is not taken into account by any of the studied EOS models. 4.2.3. Systems with Nonpolar-Moderately Dipolar Interactions. We present in Figure 8 the VLE diagram of a mixture of moderately-dipolar molecules (R134a) with a reduced dipole moment (see appendix) µ/2 ≈ 2 and essentially nonpolar propane molecules. All three models correctly predict an azeotrope, but the one predicted by PCP-SAFT is not strong enough. The azeotrope predicted by PC-SAFTP2 is closer to the experiment but is still not strong enough. The PC-SAFTP1 model shows the best prediction but slightly overestimates the azeotrope. One reason for the relatively weak behavior of PCP-SAFT is the missing dipole-quadrupole interaction in PCP-SAFT, which leads to a parameter  that incorporates not only dispersion but also dipole-quadrupole interactions for pure R134a. The true dipole-quadrupole interaction creates some nonideality in the studied mixture with propane, but the nonideality of the dispersion term that is used instead is much weaker. Also, the dipole in R134a is not aligned with the symmetry axis of R134a, which probably contributes to some inaccuracies in PCP-SAFT that are also present in PC-SAFTP2.

Taking the other systems in this class into account, we find a satisfactory performance for all three EOSs with average vapor pressure deviations of 13.0%, 9.7%, and 11.0% for the PCPSAFT, the PC-SAFTP1, and the PC-SAFTP2 EOS with London’s CR, respectively (Table 6). 4.2.4. Systems with Nonpolar-Strongly Dipolar Interactions. When we turn to a more strongly polar refrigerant (R32) with a reduced squared dipole moment of about 3, the PCP-SAFT EOS with the geometric mean CR tends to underpredict the strength of the azeotrope in a mixture with propane.8 With London’s CR, the nonideality is described much better, compare Table 7 . The EOS predicts a hetero-azeotrope at temperatures below 243 K. Experimentally, we find the hetero-azeotrope at temperatures up to 310 K. Note that the dipole vector of R32 is aligned with the C2V symmetry axis, but R32 is not a linear molecule. In contrast to PCP-SAFT, PC-SAFTP1 shows a too strong nonideality with a too large LL-immiscibility region with London’s CR for this aspherical and strongly dipolar molecule, compare Figure 9. The upper limit of the hetero-azeotrope is 294 K. PC-SAFTP2 yields a too strong azeotrope but gives a reasonable width of the LLE at 280 K with a high-temperature limit of the hetero-azeotrope at 283 K. Another interesting system in this class is the DMFcyclohexane system, Figure 10. The experimental upper critical LLE temperature (UCT, not shown in the diagram) is about

Making Equation of State Models Predictive

J. Phys. Chem. C, Vol. 111, No. 43, 2007 15539

TABLE 8: Average Absolute Deviations of Vapor Pressure (∆p/pexp) and Gas-Phase Composition (∆y, in %) Computed with the PCP-SAFT and PC-SAFTP EOSs with the London Combination Rule for Systems with Quadrupolar-Dipolar Interactionsa system

PCP-SAFT ∆p/pexp ∆y

PC-SAFTP1 ∆p/pexp ∆y

PC-SAFTP2 ∆p/pexp ∆y

CO2-R22 R22-CS2 CO2-acetone R41-CO2 R23-CO2 R23-CS2 benzene-DMF

6.8 6.4 36.8 57.9 11.8 31.0 37.6

1.9 2.5 0.8 6.7 3.0 2.0 6.5

1.8 5.5 23.6 11.1 5.4 28.3 35.2

0.7 2.2 0.8 1.5 5.4 2.2 5.2

1.6 7.6 13.6 2.5 6.6 33.5 14.7

0.9 2.6 0.7 0.6 2.2 2.5 3.0

average

26.9

3.4

15.8

2.0

11.5

2.3

source of exp. data (ref) [55] [54] [56] [29] [54] [54] [51]

a The static polarizabilities necessary for London’s CRs have been given in ref 7.

TABLE 9: Average Absolute Deviations of Vapor Pressure (∆p/pexp) and Gas-phase Composition (∆y) in % Computed with the PCP-SAFT and PC-SAFTP EOSs with the London Combination Rule for Systems with Quadrupolar-strongly Dipolar Interactionsa system benzeneacetonitrile CO2-acetonitrile average

PCP-SAFT ∆p/pexp ∆y 102.2

PC-SAFTP1 ∆p/pexp ∆y

33.6

93.1

191.8

2.3

147.0

17.9

Figure 10. Vapor-liquid equilibrium of the system DMF(1)cyclohexane(2). Experimental data are taken from refs 50 and 51.

source of PC-SAFTP2 exp. data ∆p/pexp ∆y (ref)

31.7

4.0

1.5

[58]

275.2

1.2

6.8

0.4

[57]

184.1

16.4

5.4

1.0

a The static polarizabilities necessary for London’s CRs have been given in ref 7.

Figure 11. Vapor-liquid equilibrium of the system R41(1)-carbon dioxide(2). Experimental data are taken from ref 29.

Figure 9. Vapor-liquid equilibrium of the system propane(1)-R32(2). Experimental data are taken from ref 29.

320 K, depending on the data source.52,53 For this system, the PCP-SAFT EOS gives a good prediction of the VLE with the geometric CR, and the UCT is predicted slightly (13 K) too high.8 London’s CR yields an improved agreement for the VLE, but the UCT is increased to more than 355 K. As already observed in the last example, PC-SAFTP1 exhibits the greatest nonideality, and PC-SAFTP2 lies somewhere in between the other two. None of the EOS models studied here is able to predict the VLE and, at the same time, the UCT of the LLE accurately. For the strongly non-spherical and dipolar molecules in this small class, the PCP-SAFT EOS shows a much better performance than PCP-SAFTP1. The PCP-SAFT PT describes the

coupling between the shape of the repulsive forces and a strong dipole moment well, but the PT employed in PC-SAFTP1 fails when this coupling is strong, presumably because of the dipoledipole term that is based on a spherical reference and tends to overpredict the nonideality when the dipole moment is very large. PC-SAFTP2 lies somewhere in between the other two EOS models. 4.2.5. Systems with Quadrupolar-Dipolar Interactions. It is known that dipoles and quadrupoles interact attractively.19 This reduces the nonideality caused by the different multipole moments. However, the PCP-SAFT EOS used here5,6 was not explicitly designed for such systems. Therefore, it does not contain any dipole-quadrupole interaction, and it predicts a too strong positive deviation from ideal behavior for the VLE and a too positive excess enthalpy for such systems, as can be seen in Figure 11 for the R41-carbon dioxide system. Both PC-SAFTP EOSs perform much better for this kind of systems because they contain the dipole-quadrupole interaction. For the carbon disulfide-R23 system, all three models yield very accurate predictions of the VLE at 323.15 K; see Figure 12. At 423.15 K, all models predict a binary critical point that is much too high and leads to the large average-% deviations observed for this system. Because all three PT models behave

15540 J. Phys. Chem. C, Vol. 111, No. 43, 2007

Leonhard et al. 5. Conclusion

Figure 12. Vapor-liquid equilibrium of the system carbon disulfide(1)-R23(2). Experimental data are taken from ref 54.

Figure 13. Vapor-liquid equilibrium of the system acetonitrile(1)carbon dioxide(2). Experimental data are taken from ref 57.

In two recent publications7,8 we have shown that multipole moments and dispersion nonidealities of small to medium-sized molecules can be computed with reasonable accuracies with standard quantum chemistry models. For systems with relatively simple multipole interactions, the use of consistently computed quantum chemical multipole moments and the new CRs presented in that work yield a considerable improvement of predictive performance of the PCP-SAFT EOS. In the present paper we have combined the PC-SAFT EOS with two other PT models; the combination of PC-SAFT with a spherical reference PT for multipolar species is called PC-SAFTP1, and the combination of the dipole-dipole term from PCP-SAFT and all other terms from the PT with spherical reference is called PC-SAFTP2. We have compared the predictive performance of all three PT models in the PC-SAFT EOS for a variety of systems and found that the PC-SAFTP2 EOS, based on a combination of both methods, shows the best overall performance. Still some questions remain open: (1) The coupling between dipole and quadrupole orientation and shape anisotropy (The shape-dependent quadrupole term of PCP-SAFT compares better with molecular simulations of prolate quadrupolar molecules than the PT with spherical reference, but it performs poorer for experimental systems with oblate molecules, e.g., benzene-cyclohexane). Also, PCP-SAFT works well for linear molecules (e.g., CO2 or acetonitrile) but is less accurate for nonlinear molecules where the dipole is not aligned with the longest principal axis (e.g., R227ea). (2) The coupling between dipole and quadrupole orientation is contained in the PT with spherical reference and not in the PCP-SAFT version used here.5,6 For the special case of linear molecules, the coupling is included in the most recent PCP-SAFT version,18 but the importance of this effect has not yet been investigated systematically. (3) Structural information (the PCF) of multipolar and anisotropic fluids has not been used to investigate the limits of the PT with spherical reference nor to parametrize nor check the PCP-SAFT PT. These questions require further research, and we think that further progress toward a generally applicable predictive EOS can be made on the basis of this work. 7. Appendix: The Equation of State Models

similarly, the reason for this behavior probably lies in the PCSAFT EOS itself and not in any of the PT theories. An overview of results for systems with quadrupolar-dipolar interactions is presented in Table 8. 4.2.6. Systems with Quadrupolar-Strongly Dipolar Interactions. In the CO2-acetonitrile system, CO2 has a large quadrupole moment, acetonitrile has a very strong dipole moment, and both have a rather anisotropic linear shape. The PCP-SAFT and the PC-SAFTP1 models strongly overestimate the nonideality in this system; see Figure 13 and Table 9. In the PCPSAFT model, the reason is the missing dipole-quadrupole interaction. Using the PC-SAFTP1 model, the agreement with the experimental data is even poorer, and this time the reason is the strongly overestimated effect of the dipole on the anisotropic acetonitrile. PC-SAFTP2 predicts the phase diagram with satisfying accuracy. For the second system (benzene-acetonitrile) the PCP-SAFT and the PC-SAFTP1 models predict a hetero-azeotrope instead of an azeotrope at 1.033 bar (diagram not shown), but PCSAFTP2 correctly predicts an azeotrope at the pressures studied. The effects leading to this behavior are the same as those for the previous system.

Here we present a detailed description of the implementation of the PCP-SAFT EOS, the equations used for the PT with spherical reference, the new PC-SAFTP1, and the new PCSAFTP2 EOS. We implemented the EOS models in the ThermoC package,59 which was used to perform all EOS calculations. In this package, the residual free energy and the pressure as functions of temperature (T) in Kelvin, molar volume (V) in cm3/mol, and mole fraction (x1) are used to compute phase equilibria. The fugacity is not used explicitly. 7.1. Implementation of the PT with Spherical Reference. Here, we present the formulae for electrostatic interactions of the PT with spherical reference used in our new EOS models, because not all previous publications11-17 of these expressions have been completely free of missprints. The contribution for multipole interactions to the free energy, ares,att,aniso is evaluated in second (aλλ) and third (aλλλ) order PT and a Pade´ approximation is used (eq 4), ares,att,aniso RT with

)

aλλ 1 - aλλλ/aλλ

(4)

Making Equation of State Models Predictive aλλ

)-

πn10-38

∑∑

(TkB)2

RT

]

R

β

J. Phys. Chem. C, Vol. 111, No. 43, 2007 15541

[

2 xRxβ (µ2Rx + µ2Ry + µR2 z)(µβ2 x + µβ2 y + 3

(8) (6) JRβ JRβ 4 µβ2 z) + (µ2Rx + µ2Ry + µR2 z)(θβ2 xx + θβ2 yy + θβ2 zz) + 3 5 3 σRβ σRβ (10) JRβ 56 2 (θRxx + θ2Ryy + θR2 zz)(θβ2 xx + θβ2 yy + θβ2 zz) (5) 45 σ7 Rβ

Rλλλ ) Rλλλ,A + Rλλλ,B aλλλ,A

)-

RT

πn10-57 (TkB)

µR2 zθRzz)(µβ2 xθβxx µ2RyθRyy

[

8 2 (µR θR + µ2RyθRyy + 5 x xx

∑ ∑x x

×

+

µβ2 zθβzz)

R β

3

R

+

β

µβ2 yθβyy

+

µR2 zθRzz)(θβ3 xx

+

(6)

(11) JRβ 8 σRβ

+

θβ3 yy

Ares ) Ahs + Achain + Aassoc + Adisp + AQQ + ADD

192 2 (µR θR + + 105 x xx (13) JRβ

θβ3 zz)

+

10 σRβ

]

(15) JRβ 256 3 3 3 3 3 3 (θR + θRyy + θRzz)(θβxx + θβyy + θβzz) (7) 245 xx σ12 Rβ

aλλλ,B

)

π3n210-57 3

RT

(TkB)

∑ ∑ ∑x x x

R β γ×

R

β

γ

32 135

µR2 z)(µβ2 x + µβ2 y + µβ2 z)(µγ2 x + µγ2 y + µγ2 z) × 128 315

x3π(µ2R

x

vanish19) in Buckingham (1 B ) 1 DÅ), and the diameters of the molecules in Ångstro¨m (1 Å ) 10-10 m) and from using the relation 1 D2 ) 10-49 J m3. These expressions lead to the relation 1.1126 C2 ) 1 J m, which may look strange at first glance. However, from Coulomb’s law we find that 1 J ) 1 C2/(4π0m). With 0 ) 8.854 × 10-12 C/(V m), one can see that the unit conversions are correct. The expressions necessary to compute the pressure can be derived from the given ones by differentiation (p ) -(∂A/∂V)T). 7.2. Implementation of the PCP-SAFT EOS. The PCPSAFT EOS is based on a perturbation expansion in which the residual Helmholtz energy is written as a sum of several contributions (eq 9.)

x

14π 2 (µRx + µ2Ry + 5

KRβγ(222;333)

+

σRβσRγσβγ

Chapman et al.61 applied Wertheim’s thermodynamic PT62-64 (TPT) to a system of C different flexible chains of mi hard spheres with a diameter di that are tangentially connected (SAFT, statistical associating fluid theory). This model for the repulsive interactions was first combined with an isotropic dispersion interaction. Later, Gross et al.65 applied a Barker-Henderson perturbation theory to the hard-chain as a reference fluid. This led to a segment specific dispersion interaction model Adisp with model constants that have been adjusted to experimental data of pure n-alkanes (PC-SAFT, perturbed-chain SAFT). The original expression for interchain association (Aassoc) is not used in the present study.

θγ2 zz) ×

( )

[

]

ζ32 ζ32 Ahs 1 3ζ1ζ2 )m j + + - ζ0 ln(1 - ζ3) NkBT ζ0 1 - ζ3 ζ (1 - ζ )2 ζ23 3 3 (10) Achain

+ µ2Ry + µR2 z)(µβ2 x + µβ2 y + µβ2 z)(θγ2 xx + θγ2 yy + KRβγ(233;344)

(9)

)-

NkBT

∑x (m - 1)ln g i

hs ii (σii)

i

(11)

i

-

2 2 σRβσRγ σβγ

x

22π 2 (µRx + µ2Ry + µR2 z)(θβ2 xx + θβ2 yy + θβ2 zz)(θγ2 xx + θγ2 yy + 405 7 KRβγ(334;445) 256 x2002π(θ2Rxx + θ2Ryy + θR2 zz) + θγ2 zz) × 2 2 3 54675 σRβ σRγ σβγ KRβγ(444;555) (θβ2 xx + θβ2 yy + θβ2 zz)(θγ2 xx + θγ2 yy + θγ2 zz) × (8) 3 3 3 σRβ σRγ σβγ 128

where R is the gas constant, T the temperature, n is the number density, kB is Boltzmann’s constant, µRi are the components of the dipole vector of molecule R, and θRii are the diagonal elements of the quadrupole tensor in its principal axes frame. The required J(n)(T*, n/0) and K(l, l′, l′′; n, n′, n′′; T*, n/0) integrals are taken from ref 60. The σRβ term is a size parameter whose definition depends on the EOS model used in combination with the PT and will therefore be defined later, as will be the reduced temperature T* and the reduced density n/0. The terms n and l, l′, l′′; n, n′, n′′ are specifiers for the type of J and K integrals, respectively. The numerical factors given in the above equations result from using the elements of the dipole vector in Debye (1 D ) 10-18 esu ) 3,336 × 10-30 C m), the diagonal elements of the quadrupole tensor (which must be expressed in the quadrupole tensor principal axis frame, therefore all non-diagonal elements

Adisp ) -2πFI1(η, m j )m2σ3 - πFm j C1I2(η, m j )m22σ3 NkBT With m2σ3 )

∑ ∑x x m m i j

i

m22σ3 )

i

j

∑∑ i

kBT ij

xixjmimj

j

π ζn ) F 6

( ) ( ) ij

j

σ3ij

(12)

(13)

2

kBT

σ3ij

∑x m d

n i i

i

i

[

( )]

ii dii ) σii 1 - 0.12 exp -3 kBT 6

η

π

F)

∑x m d

3 i i

i

i

η ) ζ3 m j)

∑x m i

i

i

(14)

15542 J. Phys. Chem. C, Vol. 111, No. 43, 2007

Leonhard et al.

where ii, σii, and mi are the pure compound segment energy parameter, segment diameter, and segment number, respectively. They are fitted to pure compound VLE and critical data. The terms Ahs, Achain, and Adisp are the contributions of hard spheres, chain formation, and dispersion to the residual free energy, respectively. The N term is the number of particles, kB is Boltzmann’s constant, and T is the temperature. The mathematical form of the perturbation integrals I1 and I2, their coefficients, the pair correlation function at contact, ghs ij (σii), and C1 are given in ref 65. The original combination rule for the size parameter is σij ) (σii + σjj/2), and for the interaction parameter it is ij ) xiijj(1 - kij), where kij is a parameter adjusted to mixture data. In this work, we use the predictive combination rule without an adjustable parameter as described in section 3. Note that i and j denote molecules in PCP-SAFT, but R and β have been used in the PT with a spherical reference. Subsequently, the model has been extended to include quadrupole-quadrupole,5 dipole-dipole,6 and dipole-polarizable molecule66 interactions by use of PT and cluster renormalisation theory. The resulting free energy for the quadrupolequadrupole interaction is given in the Pade´ approximation (eqs 15-17), A2/NkBT AQQ ) NkBT 1 - A3/A2 A2

) -π

NkBT

)

2

F

xi xj

4

NkBT A3

( ) ∑∑ 3

i

j

ii jj σ5ii σ5jj /2 nQ,inQ,jQ/2 i Qj J2,ij kBT kBT σ7 ij (16)

() ( ) ( ) ( ) ∑∑∑ π 3

3

∑ ∑x x

F

3 4

i j

i

j 3/2σ5/2 ii

jj

kBT

3

4

3

F

2

i

j

(15)

ii

A2

) -πF

NkBT A3

∑∑

4π2 2 F ) 3 NkBT

i

xi xj

j

ii jj kk σ3ii σ3jj σ3kk

∑ ∑ ∑x x x k T k T k T σ σ σ i j k

i

j

k

B

T* )

σ15/2 4π2 jj /3 Q J + nQ,inQ,jQ/3 i j 3,ij 3 σ1ij2 5 5 5 ii jj kk σii σjj σkk xi xj xk × kBT kBT kBT σ3 σ3 σ3 k

n)F)

ij ik jk /2 /2 nµ,inµ,jnu,k µ/2 i µj µk

T miii /kB

where nQ,i is the number of quadrupole moments in molecule i. It is always unity in our applications, but may take larger values (e.g., for a polymer with quadrupolar building blocks). The term ) Q2i /(miiiσ5ii) is the reduced quadrupole moment of Q/2 i molecule i; hence, the sign and the tensor information are lost. The mathematical form of the perturbation integrals J3,ij((ij/T), η, m) and J3,ijk(η, m) and their coefficients are given in ref 5. Dipolar interactions have been incorporated into the EOS in a manner analogous to the one employed for quadrupolar interactions. The functional form of the dipolar free energy is derived from PT, and one J integral is omitted and the other ones are fitted to macroscopic pure compound simulation data of 2CLJ molecules with a point dipole aligned to the molecular axis. The maximum considered elongation in the fitting procedure is equivalent to m ) 2, as in the case of quadrupolar molecules. The resulting equations are eqs 18-20, (18)

DD J3,ijk (20)

NA 1024V

(21) (22)

(V in cm3)

σeff ) 3xmid0

J3,ijk (17)

×

B

n 0/ ) Fσ3eff

kBT

A2/NkBT ADD ) NkBT 1 - A3/A2

B

(19)

where nµ,i is the number of dipole moments in molecule i, and ) µ2i /(miiiσ3ii) is the reduced quadrupole moment of µ/2 i molcule i. The original authors also applied the model to nonlinear molecules, where the dipole moment is not aligned to the symmetry axis or where such an axis does not exist (e.g., ketones), although the statistical mechanics is then no longer correct. Still, here we also apply the QM determined scalar dipole moment without a restriction on molecular shape. 7.3. The PC-SAFTP1 EOS. On the one hand, PC-SAFT provides a good description of size and shape effects and allows the use of physical combination rules for the dispersion interactions. On the other hand, the λ PT based on a spherical reference system is able to capture more types of multipolemultipole interactions. Hence, we combine both methods. We take without change the hard sphere, the chain, and the dispersion expression from the PC-SAFT EOS. The perturbation expressions for electrostatic interactions are taken from eqs 4-8. The following definitions of reduced quantities hold (eqs 2124).

3/2

ij ik jk /2 /2 nQ,inQ,jnQ,kQ/2 i Qj Qk

ii jj σ3ii σ3jj /2 DD nµ,inµ,j µ/2 i µj J2,ij kBT kBT σ3 ij

(23) (24)

For mixtures, some parameters (e.g., ) that are used to reduce multipole moments in the λ perturbation theory are not needed, and therefore they are not defined in the PC-SAFT theory. Therefore, we have introduced the following definitions (eqs 25-28). T* )

T m/kB

(25)

∑ ∑x x m m  σ i j

)

i

3 j ij ij

i

j

(26)

m2σ3 with

∑x m ) ∑ ∑x x σ m)

σeff

i

i

3 i j ij,eff

i

(27) (28)

j

The definition of the one-fluid  is based on eq 13. 7.4. The PC-SAFTP2 EOS. Based on the experience that the dipole-quadrupole interaction is missing in PCP-SAFT and that the PT in PC-SAFTP1 does not converge properly when a molecules has a high dipole moment and a significant shape

Making Equation of State Models Predictive anisotropy, we combined contributions of both models. To do so, we use the PT with a spherical reference without the purely (6) and KRβγ(222;333) to zero) and dipolar terms (i. e., we set JRβ apply the Pade´ approximation for the remaining contributions. Then, we compute PCP-SAFT dipolar perturbation contribution, eqs (18-20). Finally, we add both PT results to the free energy of our system. We call this approach PC-SAFTP2. Acknowledgment. We acknowledge financial support from the German research council (Deutsche Forschungs Gemeinschaft, DFG) under the priority program SPP 1155. References and Notes (1) Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. AIChE J. 1975, 21, 1086-1099. (2) Klamt, A. J. Phys. Chem. 1995, 99, 2224-2235. (3) Holderbaum, T.; Gmehling, J. Fluid Phase Equilib. 1991, 70, 251265. (4) Gross, J.; Sadowski, G. Ind. Eng. Chem. Res. 2001, 40, 12441260. (5) Gross, J. AIChE J. 2005, 51, 2556-2568. (6) Gross, J.; Vrabec, J. AIChE J. 2006, 52, 1194-1201. (7) Singh, M.; Leonhard, K.; Lucas, K.; Fluid Phase Equil. 2007, 258, 12-28. (8) Leonhard, K.; Van Nhu N.; Lucas, K.; Fluid Phase Equil. 2007, 258, 41-50. (9) Gray, C. G.; Gubbins, K. E. Theory of Molecular Fluids; Clarendon Press: Oxford, 1984. (10) Lucas, K. Molecular Models for Fluids; Cambridge University Press: Cambridge, 2007. (11) Gubbins, K. E.; Twu, C. H. Chem. End. Sci. 1978, 33, 863-878. (12) Gubbins, K. E.; Twu, C. H. Chem. End. Sci. 1978, 33, 879-887. (13) Shukla, K. P.; Lucas, K.; Moser, B. Fluid Phase Equilib. 1983, 15, 125- 172. (14) Shukla, K. P.; Lucas, K.; Moser, B. Fluid Phase Equilib. 1984, 17, 19-55. (15) Shukla, K. P.; Lucas, K.; Moser, B. Fluid Phase Equilib. 1984, 17, 153- 186. (16) Siddiqi, M. A.; Lucas, K. Fluid Phase Equilib. 1989, 51, 237244. (17) Siddiqi, M. A.; Lucas, K. Fluid Phase Equilib. 1990, 60, 1-9. (18) Vrabec, J.; Gross, J. J. Phys. Chem. B 2007, in press. (19) Lucas, K. Applied Statistical Thermodynamics; Springer-Verlag: Berlin, 1991. (20) Daubert, T. E.; Danner, R. P. Physical and Thermodynamic Properties of Pure Chemicals; Taylor & Francis: London, 1998. (21) Friend, D.; Ingham, H.; Ely, J. J. Phys. Chem. Ref. Data 1991, 20, 275-347. (22) Ambrose, D.; J., W. Pure & Appl. Chem. 1989, 61, 1395-1403. (23) Cibulka, I. Fluid Phase Equilib. 1993, 89, 1-18. (24) Vargaftik, N. B. Handbook of Physical Properties of Liquids and Gases; Springer-Verlag: Berlin, 1975. (25) Span, R.; Wagner, W. J. Phys. Chem. Ref. Data. 1996, 25, 15091596. (26) Kamei, A.; Beyerlein, S.; Jacobsen, R. Int. J. Thermophys. 1995, 16, 1155-1164. (27) Penoncello, S.; Lemmon, E.; Shan, Z.; Jacobsen, R. J. Phys. Chem. Ref. Data 2003, 32, 1473-1499. (28) Tillner-Roth, R.; Yokozeki, A. J. Phys. Chem. Ref. Data 1997, 26, 1273-1297. (29) Holcomb, C. D.; Magee, J. W.; Scott, J. L.; Outcalt, S. L.; Haynes, W. M. Selected Thermodynamic Properties for Mixtures of R32 (Difluoromethane), R125 (Pentafluoroethane), R134a (1,1,1,2-Tetrafluoroethane), R143a (1,1,1,Trifluoroethane), R41 (Flu- oromethane), R290 (Propane), and R744 (carbon dioxide); Technical Note 1397, National Institute of Standards and Technology, 1997.

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