Ind. Eng. Chem. Res. 2000, 39, 3521-3527
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Comparisons of Supercritical Properties from an Equation of State with a Hard-Sphere Repulsive Pressure Term and from the Peng-Robinson Equation of State† Don M. Zebolsky‡ Chemistry Department, Creighton University, Omaha, Nebraska 68178
An equation of state that adds a Percus-Yevick hard-sphere repulsive pressure term to a Morrison-McLinden term for attraction of two adjustable parameters correlates residual enthalpies and solubilities in supercritical carbon dioxide better than the Peng-Robinson cubic equation of state. Its use can be extended to mixtures of inorganic compounds in carbon dioxide. Volumes and enthalpies of mixing in general are correlated and predicted as well as or better by the equation with the hard-sphere repulsion, so there is no reason to continue to use cubic equations of state where the free-volume term is wrong. P ) RT/(V - b) - ac{1 + κ[1 - (T/Tc)1/2]}2/
Introduction Recent uses1-3 of van der Waals type equations of state have shown that the free-volume term can be replaced by a hard-sphere pressure to provide more conceptually satisfying equations. The Morrison and McLinden equation4 (MDS) has shown particular promise. Here it is compared with the Peng-Robinson equation of state5 (PR). First, the equations will be described. Second, their efficacy in correlating volumes and residual enthalpies of pure components will be shown. Third, mixture calculations of excess volumes, excess enthalpies, and solubilities will be discussed. Equations of State: P ) Po + Pa DeSantis et al.6 designed an attractive pressure term with one adjustable parameter to accompany the Carnahan-Starling hard-sphere repulsive pressure. Morrison and McLinden4 modified DeSantis’ term to include two adjustable parameters. Here for MDS the PercusYevick (PY) repulsive pressure of eq 1 is used with the Morrison-McLinden attractive term (eq 2):
Po ) RTV-1(1 + η + η2)/(1 - η)3
(1)
Pa ) -ac exp()/[V(V + b)]
(2)
(V2 + 2bV - b2) (3) where κ is 0.37464 + 1.54226ω - 0.26992ω2 and ω is Pitzer’s acentric factor. The values of ac and b at the critical point were derived from the critical point conditions of mechanical stability
(∂P/∂V)T ) (∂2P/∂V2)T ) 0
(4)
so that the critical temperatures and pressures agree with experiment. When the gas constant is used for R, critical volumes and compressibility factors are predicted to be higher than experiment. For MDS ac becomes 0.4610R2Tc2/Pc, bc is 0.1046RTc/Pc, and the compressibility factor, Zc, is 0.3149. For PR ac is 0.4572R2Tc2/Pc, b is 0.07780RTc/Pc, and Zc is 0.3074. In the limit of low density (large volume and small pressure), both equations reproduce the ideal gas. At high density (low volume and high pressure), the PR equation has a positive singularity at V ) b, whereas the PY term in the MDS equation has a positive singularity8 at V ) b/4. Therefore, at high pressures the PR equation predicts a fluid that is much less compressible than its hard-sphere replacement. The distinction becomes observable at densities above the critical point8 and is important in supercritical carbon dioxide. Molar Volumes and Residual Enthalpies
is1
where η is the molecular volume fraction, b/4V, temperature dependent and equal to A1(T - Tc) + A2(T2 - Tc2), and A1 and A2 are the two adjustable parameters. Morrison and McLinden used a quadratic expression in temperature with two added adjustable parameters for b, but here Fermeglia’s7 temperaturedependent term for b, bc{1.065655[1 - 0.12 exp(-2Tc/ 3T)]}3, that has no added adjustable parameters was used. The subscript c refers to the critical point. The PR equation5 is eq 3: † This article was presented in part at the 52nd Annual Calorimetry Conference, Pacific Grove, CA, Aug 4, 1997. ‡Present address: 2500 California Plaza, Rigge Science Building, Omaha, NE 68178-0104. Phone: 402-280-2814. Fax: 402-280-5737. E-mail:
[email protected].
Volumes were calculated from the equations of state by a combination of the half-interval, secant, and Newton methods. Analytical equations were derived to calculate residual enthalpies from eq 5:
H* ) ∫V∞[T(∂P/∂T)V - P] dV + PV - RT
(5)
Databases were prepared from single-phase liquid and gas residual enthalpies and molar volumes. For carbon dioxide9 444 data pairs were used from 220 to 700 K and from 40 to 200 bar. For toluene10 119 data pairs were used from 470 to 1000 K and from 10 to 100 bar. Recent data11 for carbon dioxide show that density variations average less than 0.1% from the IUPAC data9 used here. Residual enthalpies were unavailable for the
10.1021/ie000097y CCC: $19.00 © 2000 American Chemical Society Published on Web 08/26/2000
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Table 1. Parameters and Critical Volumes for Two EOS’s compound database range, reduced T, P CO2a 444 H*, V 0.72-2.3Tr, 0.54-2.7Pr C7H8b (toluene) 119 H*, V 0.79-1.7Tr, 0.24-2.4Pr SnCl4c 6V 0.53, 0.59Tr, 1.7-2.8Pr TiCl4d 6V 0.52-0.57Tr, 0.02-2.7Pr C10H8e (naphthalene) 72 φ, Lee-Kesler 0.4-1.5Tr, 1.2-7Pr
EOS MDS A1, ×10-3 A2, ×10-6
expt Tc, K Pc, bar Vc, cm3/mol
PR ω
-3.122 1.693 Vc ) 108 cm3/mol
0.225 105
304.2 73.8 94.0
-2.060 0.8813 Vc ) 378 cm3/mol
0.257 369
591.8 41.0 316
0.6489 -0.5328 Vc ) 413 cm3/mol
0.259 403
592 37.5 351
2.662 -2.348 Vc ) 358 cm3/mol
0.260 350
638 46.6 339
-1.572 0.4641 Vc ) 484 cm3/mol
0.302 472
748.4 40.5 410
C7H6O2e (benzoic acid) -1.827 0.2960 72 φ, Lee-Kesler Vc ) 432 cm3/mol
0.620 421
752 45.6 341
a
Reference 9. b Reference 10. c Reference 12. and 14. e Reference 15.
d
References 13
inorganic liquids tin(IV) and titanium(IV) chloride. Six single-phase liquid molar volumes12 of SnCl4 at 313 and 348 K and from 62.9 to 104 bar and six single-phase liquid molar volumes13,14 of TiCl4 at 333-363 K and from 1 to 125 bar were used. Lee-Kesler fugacity coefficients15 were chosen for naphthalene and benzoic acid. A simplex16 was added to the computer program that enabled the simultaneous variation of the adjustable parameters of the MDS equation. The adjustable parameters were varied to achieve the minimum standard deviation between calculated and experimental values of the database. They are shown in Table 1, together with the critical volumes predicted by the equations of state. The MDS values are always larger than the PR values but within 3% of those from PR. The best-fit standard deviations are displayed in Table 2. In Table 2 it can be seen that the MDS equation fits the residual enthalpies for carbon dioxide and toluene better but that the PR equation fits the volumes closer. Best-fit parameters are dependent upon the
Table 2. Relative Standard Deviations for Single-Phase, Pure-Component Molar Volumes and Residual Enthalpies Correlated with Two EOS’s molar volumes MDS PR
substance range of database
no. of data points
carbon dioxide 220-700 K, 40-200 bar Tc, Pc ) 304.2 K, 73.8 bar toluene 470-1000 K, 10-100 bar Tc, Pc ) 591.8 K, 41 bar tin(IV) tetrachloride 313, 348 K, 63-104 bar Tc, Pc ) 592 K, 37.5 bar titanium(IV) tetrachloride 333-363 K, 1-125 bar Tc, Pc ) 638 K, 46.6 bar
444a
0.043e
0.025e
119b
0.068e
0.049e
6c
0.009f
0.015
6d
0.040f
0.11
substance
no. of data points
carbon dioxide toluene
444a 121b
residual enthalpies MDS PR 0.026e 0.032e
a Reference 9. b Reference 10. c Reference 12. and 14. e Reference 1. f Reference 2.
0.093e 0.11e d
References 13
database. Only volumes were available for SnCl4 and TiCl4, and those were far below the critical temperature. Figure 1 shows the experimental and calculated residual enthalpies at 310 K, a supercritical temperature for CO2. The same comparisons are shown in Figure 2 at 330 K. Notice the large change in values from gaslike at low pressures to liquidlike at high pressures. In both figures the PR calculations are observed closer to experiment for the liquidlike values at all but the highest pressure. As the temperature rises, PR fits worsen. The worsening is seen at all higher reduced temperatures above 1.1. This pattern also has been observed for toluene and for n-hexane.1 Comparisons of molar volumes for carbon dioxide, in Figure 3, show the MDS and PR calculations are near identical at 310 K. Both improve as the temperature increases with MDS values even closer to experimental values than PR ones. However, volumes deviate increasingly from experiment for MDS as reduced temperatures decrease below 0.9, as much as 12% at 240 K. Excess Enthalpies and Volumes of Mixing Databases were prepared of 833 single-phase enthalpies1,17,18 of mixing, HmE, for toluene and carbon dioxide
Figure 1. Residual enthalpies against pressure for carbon dioxide at 310 K. Relative standard deviations: 0.056, MDS; 0.039, PR.
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Figure 2. Residual enthalpies against pressure for carbon dioxide at 330 K. Relative standard deviations: (330 K) 0.019, MDS; 0.047, PR; (380 K) 0.021, MDS; 0.082, PR.
Figure 3. Molar volumes against pressure for carbon dioxide at 310 K. Relative standard deviations: (320 K) 0.020, MDS; 0.032, PR; (310 K) 0.061, MDS; 0.065, PR; (240 K) 0.12, MDS; 0.040, PR. Table 3. Absolute Standard Deviations between Experimental and Calculated Single-Phase Enthalpies (HmE) of Mixing mixture database CO2 + C7H8a 298-573 K 75-125 bar 833 data CO2 + C7H8 + C6H14a 591 data CO2 + SnCl4b 313, 348 K 62.9-125 bar 149 data CO2 + TiCl4c 348 K 62.9, 125 bar 27 data a
HmE, J/mol maximum minimum 8060
-5660
standard deviations MDS PR 315 kij values 0.973
228 0.896
7250
-4300
291
367
110
-4760
347 kij values 0.995
290
228 kij values 0.855
286
150
-1660
0.872
0.827
References 1, 17, and 18. b Reference 12. c Reference 13.
from 298 to 573 K and from 75 to 175 bar, 149 singlephase HmE and volumes of mixing,12 VmE, for liquid SnCl4 and supercritical CO2 at 313 and 348 K and from 62.9 to 125 bar, and 27 single-phase13 HmE and VmE for liquid TiCl4 mixed with supercritical CO2 at 348 K and at 62.9 and 125 bar. The mixing rule used was A ) ∑∑xixjAij, where summations are over both components, i and j, and x is the mole fraction. The symbol A in the mixing rule represents b in both equations of state, exp() in the MDS equation, and the entire second term numerator
Table 4. Absolute Standard Deviations between Experimental and Calculated Single-Phase Volumes (VmE) of Mixing mixture database CO2 + SnCl4a 313, 348 K 62.9-125 bar 149 data CO2 + TiCl4b 348 K 62.9, 125 bar 27 data a
VmE, cm3/mol maximum minimum 3.7
8.0
-146
-104
standard deviations MDS PR 15.3 kij values 0.995
14.0
11.2 kij values 0.855
5.8
0.872
0.827
Reference 12. b Reference 13.
for PR. The combining rule, Bij ) (Bi + Bj)/2, was used for bij in both MDS and PR. The combining rule, Aij ) kij(AiAj)1/2, was used for exp()ij in the MDS equation (2) and for the second term numerator in the PR equation (3). The coefficient kij is the interaction coefficient and the one adjustable parameter used for the binary mixtures. The temperature derivatives of the mixing rules were used directly to calculate H* by eq 5 for the mixtures. In Table 3 can be seen standard deviations between experimental and calculated values of HmE for each of the mixtures with the corresponding equations of state. Some comparisons for VmE are presented in Table 4. The values of the interaction parameter, kij, are placed in the tables under each deviation. Excepting the enthalpies of titanium(IV) chloride with CO2, the binary mixture deviations are less for PR than for MDS. It is
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Figure 4. Enthalpies of mixing for carbon dioxide and tin(IV) chloride at 313 K and 62.9 bar. Absolute standard deviations: 313 J, MDS; 210 J, PR.
Figure 5. Enthalpies of mixing for carbon dioxide and tin(IV) chloride at 313 K and 125 bar. Absolute standard deviations: 153 J, MDS; 73 J, PR.
Figure 6. Enthalpies of mixing for carbon dioxide and titanium(IV) chloride at 348 K and 62.9 bar. Absolute standard deviations: 224 J, MDS; 488 J, PR.
surprising that the MDS equation, which correlates the residual enthalpies better, does not continue so consistently with the binary mixtures. The lower interaction parameter probably compensates. It is observed in Table 3 that the MDS equation predicts ternary mixture HmE values closer than the PR equation does. No ternary interaction parameters were used. Reasons why PR has lower standard deviations with experiment than MDS can be examined for selected temperatures in Figures 4-8. In Figure 4 is shown the large exothermic mixings for CO2 + SnCl4 that are characteristic of a temperature above the critical
temperature of CO2 but at a pressure below its critical pressure. The PR equation calculations are closer to experiment for the large negative enthalpies than the MDS equation, at CO2 mole fractions that are just less than 0.7 where the two-phase region begins. The display in Figure 5 shows the effect of an isothermal increase in pressure for CO2 + SnCl4 above the critical line to 125 bar. Enthalpies are less in absolute value than those in Figure 4, but at 125 bar and the same temperature, 313 K, enthalpies of mixing are both positive and negative depending on the composition. In Figure 5 it can be seen that the calculations from
Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000 3525
Figure 7. Volumes of mixing for carbon dioxide and tin(IV) chloride at 313 K and 62.9 bar. Absolute standard deviations: 8.2 cm3, MDS; 1.1 cm3, PR.
Figure 8. Volumes of mixing for carbon dioxide and tin(IV) chloride at 313 K and 83.6 bar. Absolute standard deviations: 13 cm3, MDS; 23 cm3, PR.
Figure 9. Mole fraction solubility of naphthalene in carbon dioxide against pressure at 333.5 K. Relative standard deviations, kij constant: 0.13, MDS; 0.40, PR.
the PR equation mimic in a small way this oscillation in the data, whereas the MDS equation predicts all negative values. Figure 6 contains enthalpies for CO2 + TiCl4 at a higher temperature. There the MDS equation better predicts the endothermic mixings that occur in mixtures rich in CO2. A similar pattern is found in the excess volumes shown in Figures 7 and 8. The large negative volumes are observed in Figure 7 to be calculated closer to experiment by the PR equation, yet when the pressure is raised as in Figure 8, the MDS equation is observed to calculate the volumes closer to experiment.
Solubilities Databases of the solubilities of two solids in supercritical CO2 were prepared. There are 51 mole fractions for naphthalene19,20 from 308 to 338 K and from 81 to 270 bar and 15 for benzoic acid19 from 318 to 338 K and from 120 to180 bar. To calculate the solubility from an equation of state, the solubility, y1 in eq 6, was equated19 to the product of three terms, the ideal solubility, the Poynting factor, and the reciprocal of the mixture fugacity coefficient
y1 ) (Pv/P) exp(Vs(P - Pv)/RT)(1/φ1)
(6)
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Table 5. Relative Standard Deviations between Experimental and Calculated Mole Fraction Solubilities for Two Solids in Supercritical Carbon Dioxide (Interaction Parameters, kij, Are Shown below Each Pair) solid solute database naphthalenea 308-338 K 81-270 bar 51 data
benzoic acidb 318-338 K 120-280 bar 15 data
range of mole fraction solubility × 100
standard deviations MDS
0.13-9.8
0.03-0.98
0.15 kij constants 0.9558
PR 0.35 0.8939
T, K 308 328 333 338 MDS 0.10 0.16 0.14 0.05 PR 0.12 0.27 0.40 0.25 kij ) f(T) MDS 0.9520s64s58s11 PR 0.9040s20s0.8939s0.896 0.22 0.23 kij constants 1.023 0.9882 T, K 318 328 MDS 0.27 0.12 PR 0.24 0.14 kij ) f(T) MDS 1.020s1.0277s1.0362 PR 0.990s1.0017s1.0124
a
338 0.05 0.08
Reference 20. b Reference 19.
where y1 is the solubility at pressure P and temperature T, Pv is the vapor pressure of the solid, Vs is the volume of the solid, and φ1 is the fugacity coefficient of the solid in the mixture. Vapor pressures for benzoic acid were from Kurnik et al.,19 and those for naphthalene were from Johnston et al.21 Volumes of the solid were calculated from densities and were assumed to be independent of temperature and pressure. When Vs was made a linear function of pressure using the isothermal solid compressibility, as recommended by Deiters,22 no significant differences were found in the calculated solubilities. Fugacity coefficients were calculated from the equation of state by means of eq 7.
ln φ1 )
(1/RT)∫∞V(∂P/∂ni
- RT/V) dV - ln(PV/RT) (7)
Equation 8 shows the results for the PR equation.
ln φ1 ) -ln[P(V - b)/RT] + b1/(V - b) - {a(b1/b) 2[y1a1 + (1 - y1)a12]/(2.8284bRT)} ln[(V - 0.4142b)/ (V + 2.4142b)] - aV(b1/b)/[RT(V2 + 2bV - b2)] (8) Equation 9 represents the fugacity coefficients for the MDS equation.
ln φ1 ) ln[1/(1 - η)] + 1.5[1/(1 - η)2] - 1.5 - ln (PV/RT) + (b1/b){3[1/(1 - η)3] - 3[1/(1 - η)2] + 1/ (1 - η) - 1} - a(b1/b){(1/b) ln[V(V + b)] + 1/(V + b)}/RT + 2[y1a1 + (1 - y1)a12] ln[V/(V + b)]/bRT, η ) b/4V (9) Because y1 appears in those equations, a value of zero initiated the calculations; iterations continued until successive ones differed by less than 0.0001. The results of the solubility calculations for the two equations of state are compared in Table 5. Naphthalene is more soluble than the polar benzoic acid. Calculations are seen in Table 5 to be not so close to the experimental
values but nevertheless provide useful estimates. The MDS equation is better than the PR equation for these estimates. The closeness of fit is sensitive to the interaction parameter, kij, and improves a little if it is made temperature dependent. Figure 9 holds comparisons with experimental data for the solute naphthalene for kij constant. McHugh and Paulaitis20 report an upper critical end point of 338 K and 240 atm. Above that, naphthalene liquefies.20 Calculated solubilities for naphthalene are better for MDS at all pressures. Conclusions Both the MDS and the PR equations can provide useful estimates of excess enthalpies and volumes for mixtures even when there is a paucity of enthalpy data for the pure components. Only volumes were used to calculate the parameters of the MDS equations for SnCl4 and TiCl4. Both equations have some predictive value in that equations with parameters determined from pure-component properties can be used to estimate mixture properties using only one adjustable interaction parameter. In general, the more conceptually satisfying replacement of the free-volume term with a hard-sphere repulsive pressure term as in the MDS equation provides improved estimates. The usefulness of the MDS equation can be extended to mixtures of spherically shaped inorganic compounds with supercritical carbon dioxide and to the solubilities of nonpolar solids in supercritical carbon dioxide. So, there is no need to continue to propagate the use of the incorrect freevolume term at pressures and temperatures that are supercritical. To do better than the MDS equation, one must pursue a more elegant incorporation of molecular properties into equations of state such as that by Deiters.22 Acknowledgment Gratitude is extended to Dr. Reed M. Izatt, Professor of Chemistry, Brigham Young University, who urged the use of the Peng-Robinson equation as a benchmark. His interest and encouragement are appreciated. Also I am grateful to Creighton University for support and facilities. Literature Cited (1) Zebolsky, D. M.; Renuncio, J. A. R. J. Supercrit. Fluids 1994, 7, 31. (2) Zebolsky, D. M. Thermochim. Acta 1997, 292, 51. (3) Christensen, J. J.; Izatt, R. M.; Zebolsky, D. M. Fluid Phase Equilib. 1987, 38, 163. (4) Morrison, G.; McLinden, M. NBS Technical Note 1226; NBS: Gaithersburg, MD, 1986. (5) Peng, D.-Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59. (6) DeSantis, R.; Gironi, F.; Marrelli, L. Ind. Eng. Chem. Fundam. 1976, 15, 183. (7) Skjold-Jorgensen, S. Fluid Phase Equilib. 1984, 16, 317. (8) Zebolsky. D. M. Thermochim. Acta 1989, 154, 107. (9) Carbon Dioxide, IUPAC Thermodynamic Tables of the Fluid State; Pergamon: Oxford, U.K., 1976. (10) TRC Thermodynamic Tables 23-2-(33.11002)-j, p2-4(pj3221-3), Thermodynamics Research Center, Texas A&M University: College Station, TX, Oct 31, 1979. (11) Span, R.; Wagner, W. J. Phys. Chem. Ref. Data 1996, 25, 1509. (12) Giles, N. F.; Oscarson, J. L.; Rowley, R. L.; Tolley, W. K.; Izatt, R. M. Fluid Phase Equilib. 1992, 73, 267.
Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000 3527 (13) Tolley, W. K.; Izatt, R. M.; Oscarson, J. L. Metall. Trans. B 1992, 23B, 65. (14) Gmelin’s Handbuch der Anorganischen Chemie, 8 Auflage, TITAN; Verlag-Chemie: Weinheim/Bergstrasse, Germany, 1951. (15) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. Properties of Liquids and Gases, 3rd ed.; McGraw-Hill: New York, 1977. (16) Cooper, J. W. Introduction to Pascal for Scientists; Wiley: New York, 1981. (17) Pando, C.; Renuncio, J. A. R.; Schofield, R. S.; Izatt, R. M.; Christensen, J. J. J. Chem. Thermodyn. 1983, 15, 747. (18) Cordray, D. R.; Christensen, J. J.; Izatt, R. M.; Oscarson, J. L. J. Chem. Thermodyn. 1988, 20, 877. (19) Kurnik, R. T.; Holla, S. J.; Reid, R. C. J. Chem. Eng. Data 1981, 26, 47.
(20) McHugh, M.; Paulaitis, M. E. J. Chem. Eng. Data 1980, 25, 326. (21) Johnston, K. P.; Ziger, D. H.; Eckert, C. A. Ind. Eng. Chem. Fundam. 1982, 21, 191. (22) Deiters, U. K. Equations of State: Theories and Applications; Chao, K. C., Robinson, R. L., Eds.; ACS Symposium Series 300; American Chemical Society: Washington, DC, 1986; p 371.
Received for review January 14, 2000 Revised manuscript received July 6, 2000 Accepted July 8, 2000 IE000097Y
