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Ind. Eng. Chem. Res. 2005, 44, 630-633
CORRELATIONS Aqueous Cross Second Virial Coefficients with the Hayden-O’Connell Correlation Kyle Bishop† and John P. O’Connell* Department of Chemical Engineering, University of Virginia, P.O. Box 400741, Charlottesville, Virginia 22904-4741
Using new data tabulations involving water, the predictive combining rule of the HaydenO’Connell correlation for cross second virial coefficients has been reformulated with a single binary parameter for pairs. Suggestions are made to estimate this parameter when data are not available. Using the binary parameters, new solvation parameter values have been found for substances that apparently solvate with water. In addition, an additive function has been formulated to improve the descriptions at high temperatures. With these modifications, much better agreement is found with experiment than with the original relations. Introduction The Hayden-O’Connell correlation for pure and cross second virial coefficients1-3 was developed using a corresponding-states formulation of the contributions of various intermolecular forces between pairs of molecules. The input properties were the substance critical temperature and pressure; the molecular radius of gyration and dipole moment; and if appropriate, an empirical parameter for association between like species or solvation between unlikes. Unlike most correlations that use empirical parameters, cross virial coefficients were estimated with a fixed combining rule of purecomponent properties. This provided generally satisfactory predictions, but the accuracy could be less than desired. Furthermore, the relations were formulated so that empirical adjustment for better results was inhibited. As a result, no revisions of the correlation seem to have been made. Recent treatments of volumetric, thermal, and phase behavior of high-temperature aqueous systems based on fluctuation solution theory4-6 use the second cross virial coefficient for solutes with water as an input limiting property. Such work requires more accurate results at extreme conditions than were available from any model.6 The purpose of this article is to demonstrate how the Hayden-O’Connell correlation can be modified to give better results for such systems and to suggest how similar strategies might be used for other mixed systems. Reformulation New Combining Rule. The original correlation obtained intermolecular potential parameters, and σ, for the nonpolar interactions between like species from * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: 434.924.3428. Fax: 434.982.2658. † Present address: Department of Chemical Engineering, Northwestern University, Evanston, IL 60208.
the input properties Tc, Pc, and RG, as well as µ and η if dipolar and associating interactions existed in the species. (Note that Stein and Miller7 pointed out that the particular relation originally chosen to obtain these parameters for associating species was not singlevalued; they suggested ways to treat the situation.) These parameters were used to compute values of purecomponent reduced second virial coefficients for the nonpolar contribution, B* ) BNP/NAσ3, at reduced temperatures T* ) kT/E. Then additional contributions for polarity and association were computed from other functions of reduced temperature and added to the nonpolar portion. For cross coefficients between unlike components i and j, combining rules without empirical parameters were chosen to provide predictive ability
(
ij ) 0.7(ij)1/2 + 0.6
1 1 + i j
σij ) (σiσj)1/2
)
(1) (2)
Recently, Plyasunov6 found inaccuracies with the Hayden-O’Connell correlation for aqueous systems, especially for T* > 2.5. Therefore, we propose new combining rules. Instead of eqs 1 and 2, it is now recommended that ij and σij be found from
ij ) (iEj)1/2(1 - kij)
(3)
σij3 ) 0.5(σi3 + σj3)
(4)
where kij is a small, positive, empirical parameter that depends on the substances involved. Equation 4 is unusual, but it seems more satisfactory for components of different sizes, such as are encountered in most aqueous systems, than other possibilities. Table 1 lists fitted values of kij for all substances that should not solvate with water (ηij ≡ 0). The results from the current method and the original are significantly different, with the original values being too negative in essentially all
10.1021/ie049267n CCC: $30.25 © 2005 American Chemical Society Published on Web 01/08/2005
Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 631 Table 1. Eq 3 Binary Parameters for Nonsolvating Compounds
a
compd
kij fitted
AAD Biw fitted
kij predicted, eq 5
AAD Biw predicted
AAD Biw original
ref(s)
argon butane CO cycloC6 ethane Hea heptane hexane methane N2 Nea O2 octane pentane propane
0.255 0.258 0.045 0.310 0.151 0.552 0.256 0.312 0.086 0.099 0.461 0.223 0.340 0.272 0.248
3.95 5.44 5.08 17.19 5.33 2.10 32.62 4.99 8.44 7.11 1.54 1.81 15.63 4.54 4.67
0.231 0.261 0.146 0.341 0.305 0.090 0.374 0.271 0.194
8.41 19.48 5.21 43.75 6.19 8.67 22.72 4.58 11.75
27.70 65.75 5.11 75.04 24.61 6.53 100.46 97.67 11.73 7.90 11.20 19.78 138.60 72.90 41.85
11, 16 11, 17 17 11, 17, 18 11, 17, 19-22 23 11, 17 11, 17, 24 17, 21, 24-27 17, 24, 26, 28, 29 16 11 8, 17, 24 17, 24 8, 17, 21
Pseudocritical constants from ref 30.
Figure 1. Second cross virial coefficients for ethane and heptane with water. Data references in Table 1.
cases. Figure 1 shows typical results for ethane and heptane. It should be noted that the accuracy of the correlation is essentially as good as or within a few cubic centimeters per mole of the fitted smoothing functions recommended by Dymond et al.8 The fitted kij values for all substances with Rg > 0.60, where Rg is the radius of gyration value for the substance in water, were correlated with the simple relation
kij ) 0.114RG - 0.0085Rg2
aqueous systems (5)
For solvating compounds, values of ηij need to be determined from the difference between experimentaldata and calculations with molecular parameters using the original function for Bchem,ij
[(
)]
650 2π - 4.27 × Bchem,ij ) - NAσ3ij exp ηij 3 ij/k + 300 1500ηij 1 - exp (6) T
(
[
)]
In this case, the required values of kij were predicted from eq 5, or if Rg < 0.60 (as for H2 and HCl), the cross coefficient data for the substance with argon were fitted to obtain a value of kij and multiplied by 2, a correlation that we found to be nearly as satisfactory as eq 5 for other substances. The objective function for fitting was
min
∑ k)data
[
]
2 [Bcalc - Bexp ij ij ]k
∆k2
(7)
Table 2 lists the results for solvating substances, comparing the new form with the original. Values of ηij here for the original form might differ from earlier tabulations because they might have been found using new data. The accuracy of the new form is similar to that of the old in nearly all cases. Note that there is a slight solvation needed for hydrogen, in agreement with the calculations of Hodges et al.9 As with the original correlation, values for specific group interactions were selected when data were available for more than one substance in a group. Table 3 reports the results. There is little change in the accuracy using group parameters compared to individually fitted values. High-Temperature Correction. In addition to the change in the combining rules, a new function was developed for high-temperature systems, given that, under these conditions, the original model gives values that are systematically too positive or not sufficiently negative, especially when eq 3 is used. Using accurate lower-temperature data for cross coefficients of water with Ar, CH4, C2H6, and C3H8, values of kij were found for each pair. Then, differences between the experimental data and calculations were obtained at higher reduced temperatures. The dependence for Ar was not the same as that for the polyatomic substances. We chose functions to correct for the high-temperature differences with a value of zero and a derivative of zero at the lowest temperature at which they are to be used. For Ar and other monatomics, the difference was significant for T* > 2.0, so it was modeled as
∆B/corr ) -[0.0873 ln(T*) - 0.0707 + 53.8 exp(-4.29T*)] monatomics with water for T* > 2.0 (8) For polyatomic substances, which include essentially all of the practical cases, the differences were significant for T* > 2.7, so we chose a function that closely modeled them
∆B/corr ) -[0.261 ln(T*) - 0.474 + 0.723 exp(-0.450T*)] polyatomics with water for T* > 2.7 (9) These functions are included in the fitting of kij values given in Table 1. Figure 2 shows the improvement in accuracy for the neon and propane systems within the
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Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005
Table 2. Eq 3 Binary Parameters and Eq 6 Solvation Parameters for Solvating Compounds compd
kij predicted, eq 5
ηij fitted
AAD Biw fitted
ηij group
AAD Biw group
ηij original
AAD Biw original
ref(s)
acetone benzene C2H4 C2H5Cl C3H6 C6F6 CH3Cl CHCl3 CO2 ethanol H2a H2S HCl methanol N2O NH3 SO2 toluene
0.249 0.266 0.155 0.217 0.212 0.343 0.147 0.276 0.105 0.213 0.040b 0.066 0.346b 0.155 0.124 0.091 0.167 0.292
1.647 0.409 0.092 0.187 0.093 0.398 0.079 0.644 0.240 1.660 0.026 0.181 1.232 1.602 0.238 1.462 1.106 0.506
18.93 8.83 11.64 7.06 1.91 6.62 1.95 8.29 12.31 19.31 1.40 6.98 30.45 32.57 12.11 22.18 9.48 8.43
0.43 0.09 0.15 0.09 0.43 0.15 0.24 1.63 1.63 0.24 0.43
10.25 11.59 9.11 1.92 6.72 17.20 12.31 18.98 32.42 12.37 10.94
1.81 0.13 0.01 0 0 0.02 0 0.17 0.11 1.73 0.04 0.12 1.29 1.65 0.17 1.42 0.92 0.13
18.56 4.66 11.81 23.67 12.32 7.60 19.54 12.49 12.06 11.64 1.31 6.88 28.27 29.15 12.95 17.22 4.33 9.78
12 8, 18, 24 8, 17, 29 17 8, 17 31, 32 17 31, 33 17, 22, 29, 34, 35 31 9 15 14, 36 31, 37 22 29, 38 13 31
a
Pseudocritical constants from ref 30. b From kiw ) 2kiAr.
This approach can be used for all pairs of substances for which the original model might be inadequate. Acknowledgment The authors are grateful to Dr. A. Plyasunov and Professor K. N. Marsh for sharing data from their compilations before publication. Notation
Figure 2. Second cross virial coefficients for neon and propane with water including high-temperature data. Data references in Table 1. Table 3. Group Solvation Parameters with Water group
new η12
original η12
alkene phenyl hydroxyl monochloro CO2/N2O
0.09 0.43 1.63 0.15 0.24
0.00 0.10 1.62 0.00 0.14
B ) second virial coefficient (cm3/mol) B* ) B/(2πΝΑσ3/3) ) reduced second virial coefficient ∆B/corr ) high-temperature correction to reduced second virial coefficient, eqs 5 and 6 k ) Boltzmann constant ) 1.3805 × 10-23 (J/molecule‚K) kij ) binary parameter in combining rule, eq 3 NA ) Avogadro’s number ) 6.0225 × 1023 (molecules/mol) Pc ) critical pressure (bar) R ) universal gas constant ) 83.143 (cm3‚bar/mol‚K) RG ) mean radius of gyration (Å) T ) absolute temperature (K) T* ) kT/ ) reduced temperature
experimental and computational chemistry ranges up to very high temperatures applicable to geochemical systems. It is expected that the binary parameter approach can be extended to other systems with similar improvements in accuracy for polar-nonpolar systems. A thorough tabulation for all systems will require examination of all data since 1974, such as in the compilations of Dymond et al.8,10,11 and the continuing work of Wormald 12-15 and of Hodges et al.9,16
Greek Symbols
Summary
Subscripts
A reformulation has been made of the HaydenO’Connell correlation for cross second virial coefficients to include an empirical binary parameter in the combining rule for energy parameters and an added function for Bij at very high temperatures. Fitted binary parameters are tabulated, and a reliable correlation is found as a function of the mean radius of gyration of nonsolvating compounds, where improvement is the greatest. For solvating compounds, solvation parameters are tabulated and group values determined where possible.
c ) critical property ij ) for interaction of species i with different species j chem ) for chemically bound pairs of molecules
∆ ) experimental uncertainty in Bij (cm3/mol) ) energy parameter (J/molecule) η ) association parameter for pure interactions; solvation parameter for unlike interactions µ ) molecular dipole moment (D) σ ) molecular size parameter (Å) Superscripts calc ) calculated from correlation exp ) experimental
Literature Cited (1) Hayden, J. G.; O’Connell, J. P. A Generalized Method for Predicting Second Virial Coefficients. Ind. Eng. Chem. Process Des. Dev. 1975, 14, 209. (2) Prausnitz, J. M.; Anderson, T. F.; Grens, E. A.; Eckert, C. A.; Hsieh, R.; O’Connell, J. P. Computer Calculations for Multi-
Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 633 component Vapor-Liquid and Liquid-Liquid Equilibria; PrenticeHall: Englewood Cliffs, NJ, 1980. (3) O’Connell, J. P. Letter to the Editor. AIChE J. 1985, 30, 1037. (4) Plyasunov, A. V.; O’Connell, J. P.; Wood, R. H. Infinite Dilution Partial Molar Properties of Aqueous Solutions of Nonelectrolytes. I. Equations for Partial Molar Volumes at Infinite Dilution and Standard Thermodynamic Functions of Hydration of Volatile Nonelectrolytes over Wide Ranges of Conditions. Geochim. Cosmochim. Acta 2000, 64, 495. Plyasunov, A.; O’Connell, J. P.; Wood, R. H. Infinite Dilution Partial Molar Properties of Aqueous Solutions of Nonelectrolytes. II. Equations for the Standard Thermodynamic Functions of Hydration of Volatile Nonelectrolytes over Wide Ranges of Conditions. Geochim. Cosmochim. Acta 2000, 64, 2779. (5) Plyasunov, A. V.; O’Connell, J. P.; Wood, R. H.; Shock, E. L. Semiempirical equation of state for the infinite dilution thermodynamic functions of hydration of nonelectrolytes over wide ranges of temperature and pressure. Fluid Phase Equilib. 2001, 183-4, 133. (6) Plyasunov, A. V. Arizona State University, Tempe, AZ, Personal communication, 2002. (7) Stein, F. P.; Miller, E. J. Extension of the HaydenO’Connell Correlation to the Second Virial Coefficients of Some Hydrogen-Bonding Mixtures. Ind. Eng. Chem. Process Des. Dev. 1980, 19, 123. (8) Dymond, J. H., Marsh, K. N., Wilhoit, R. C., Eds. Virial Coefficients of Pure Gases and Mixtures, B. Virial Coefficients of Mixtures; Landolt-Bo¨rnstein Numerical Data and Functional Relationships in Science and Technology; Springer-Verlag: New York, 2003; Vol. 21. (9) Hodges, M. P.; Wheatley, R. J.; Schenter, G. K.; Harvey, A. H. Intermolecular potential and second virial coefficient of the water-hydrogen complex. J. Chem. Phys. 2004, 120, 710. (10) Dymond, J. H., Marsh, K. N., Wilhoit, R. C., Eds. Virial Coefficients of Pure Gases and Mixtures. A. Pure Compoundss; Landolt-Bo¨rnstein Numerical Data and Functional Relationships in Science and Technology; Springer-Verlag: New York, 2003; Vol. 21. (11) Dymond, J. H.; Smith, E. B. The Virial Coefficients of Pure Gases and Mixtures: A Critical Compilation; Oxford University Press: Oxford, U.K., 1980. (12) Wormald, C. J. Water-acetone association. Second virial cross coefficients for water-acetone derived from gas-phase excess enthalpy measurements. J. Chem. Thermodyn. 2002, 34, 1639. (13) Wormald, C. J. Water-sulphur dioxide association. Second virial cross coefficients for water-sulphur dioxide derived from gas-phase excess enthalpy measurements. J. Chem. Thermodyn. 2003, 35, 91. (14) Wormald, C. J. Water-hydrogen chloride association. Second virial cross coefficients for water-hydrogen chloride derived from gas-phase excess enthalpy measurements. J. Chem. Thermodyn. 2003, 35, 417. (15) Wormald, C. J. Water-hydrogen sulphide association. Second virial cross coefficients for water-hydrogen sulphide derived from gas-phase excess enthalpy measurements. J. Chem. Thermodyn. 2003, 35, 1019. (16) Hodges, M. P.; Wheatley, R. J.; Harvey, A. H. Intermolecular potential and second virial coefficients of the waterneon and water-argon complexes. J. Chem. Phys. 2002, 117, 7169. (17) Wormald, C. J.; Lancaster, N. M. Excess enthalpies and cross-term second virial coefficients for mixtures containing water vapour. J. Chem. Soc., Faraday Trans. 1. 1988, 84, 3141. (18) Wormald, C. J.; Lancaster, N. M.; Sowden, C. J. Benzenewater association. Excess molar enthalpy and second virial crosscoefficients for (benzene-water)(g) and (cyclohexane-water)(g). J. Chem. Soc., Faraday Trans. 1 1997, 93, 1921. (19) Reamer, H. H.; Olds, R. H.; Sage, B. H.; Lacey, W. N. Composition of dew-point gas in ethane-water system. Ind. Eng. Chem. 1943, 35, 790. (20) Anthony, R. G.; McKetta, J. J. Phase equilibrium in the ethylene-ethane-water system. J. Chem. Eng. Data 1967, 12, 21.
(21) Lin, C.-L.; Wood, R. H. Prediction of the free energy of dilute aqueous methane, ethane, and propane at temperatures from 600 to 1200 °C and densities from 0 to 1 g cm-3 using molecular dynamics simulations. J. Phys. Chem. 1996, 100, 16399. (22) Coan, C. R.; King, A. D. Solubility of water in compressed carbon dioxide, nitrous oxide, and ethane. Evidence for hydration of carbon dioxide and nitrous oxide in the gas phase. J. Am. Chem. Soc. 1971, 93, 1857. (23) Hodges, M. P.; Wheatley, R. J.; Harvey, A. H. Intermolecular potential and second virial coefficient of the waterhelium complex. J. Chem. Phys. 2002, 116, 1397. (24) Abdulagatov, I. M.; Bazaev, A. R.; Gasanov, R. K.; Ramazanova, A. E. Measurements of the (p,F,T) properties and virial coefficients of pure water, methane, n-hexane, n-octane, benzene, and of their aqueous mixtures in the supercritical region. J. Chem. Thermodyn. 1996, 28, 1037. (25) Plyasunov, A. V.; Shock, E. L.; Wood, R. H. Second Cross Virial Coefficients for Interactions Involving Water: Correlations and Group Contribution Values. J. Chem. Eng. Data 2003, 48, 1463. (26) Rigby, M.; Prausnitz, J. M. Solubility of water in compressed nitrogen, argon, and methane. J. Phys. Chem. 1968, 72, 330. (27) Joffrion, L. L.; Eubank, P. T. P-V-T data and virial coefficients for gaseous methane-water mixtures with correction for adsorption effects. Fluid Phase Equilib. 1988. 43, 263. (28) Gallagher, J. S.; Levelt Sengers, J. M. H.; Abdulagatov, I. M.; Watson, J. T. R.; Fenghour, A. Thermodynamic properties of homogeneous mixtures of nitrogen and water from 440 to 1000 K, up to 100 MPa and 0.8 mole fraction N2; NIST Technical Note 1404; National Institute of Standards and Technology (NIST): Gaithersburg, MD, 1993; p 71. (29) Plyasunov, A. V.; Shock, E. L. Second Cross Virial Coefficients for Interactions Involving Water: Critical Data Compilation. J. Chem. Eng. Data 2003, 48, 808. (30) Poling, B. P.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids; McGraw-Hill: New York, 2001. (31) Nopper, R.; Ramsdorf, M.; Lichtenstein, W.; Opel, G. Die zweiten Virialkoeffizienten fur binare Gasmischungen von Methanol, Ethanol, Fluorbenzen and Toluen mit Wasser. Z. Phys. Chem. (Leipzig) 1981, 262, 1157. (32) Wormald, C. J.; Wurzberger, B. The water-hexafluorobenzene interaction. Second virial cross coefficients for waterhexafluorobenzene derived from gas-phase excess enthalpy measurements. Phys. Chem. Chem. Phys. 2000, 2, 5133. (33) Lancaster, N.; Wormald, C. J. The enthalpy of mixing of (water + trichloromethane) vapour. Z. Phys. Chem. 1981, 128, 43. (34) Vanderzee, C. E.; Haas, N. C. Second cross virial coefficients B12 for the gas mixture (carbon dioxide + water) from 300 to 1000 K. J. Chem. Thermodyn. 1981, 13, 203. (35) Patel, M. R.; Holste, J. C.; Hall, K. R.; Eubank, P. T. Thermophysical properties of gaseous carbon dioxide-water mixtures. Fluid Phase Equil. 1987, 36, 279. (36) Kindler, W.; Wuster, G. Equation of state for the vapour of concentrated and diluted hydrochloric acid. Ber.Bunsen-Ges. Phys. Chem. 1978, 82, 543. (37) Bich, E.; Hendl, H.; Vogel, E. A new evaluation of PFT measurements on the methanol vapor-steam mixture and its pure components with corrections for adsorption and impurity effects. Fluid Phase Equilib. 1997, 133, 129. (38) Wormald, C. J. Second virial cross coefficients for (ammonia + water) derived from gas-phase excess enthalpy measurements. J. Chem. Thermodyn. 2001, 33, 1193.
Received for review August 11, 2004 Revised manuscript received November 5, 2004 Accepted November 19, 2004 IE049267N
