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Ind. Eng. Chem. Res. 2000, 39, 4828-4830
Krichevskii Parameters and the Solubility of Heavy n-Alkanes in Supercritical Carbon Dioxide Takeshi Furuya and Amyn S. Teja* School of Chemical Engineering, Georgia Institute of Technology, 778 Atlantic Drive, Atlanta, Georgia 30332-0100
Krichevskii parameters were calculated from measurements of the solubility of heavy n-alkanes in supercritical carbon dioxide and also from critical point measurements for lower n-alkane + carbon dioxide systems. The results show smooth behavior of the Krichevskii parameter with carbon number of the n-alkane. This behavior can be used to validate the consistency of data sets obtained in different types of experiments. They also demonstrate the potential utility of the Krischevskii parameter in interpolating/extrapolating limited data sets. Introduction ∞,c T,v,
is defined as The Krichevskii parameter, (∂P/∂x) the change in pressure that occurs when a solute is added to a solvent at its critical point, with temperature and volume being kept constant. It is related to several important thermodynamic properties of dilute solutions1-5 and therefore plays an important role in supercritical fluid systems.1,6 The parameter can be calculated from the limiting slopes of critical lines in binary systems7,8 and also from solid solubilities in the supercritical fluids.9,10 As a result, an independent consistency check can be developed by comparing the value obtained from the critical line in a binary mixture with that obtained from solid solubility data for the same binary system.10 Unfortunately, complete sets of critical point and solid solubility data do not exist for many systems10 and are nonexistent for systems of interest in supercritical fluid technology. Consequently, the practical utility of such a consistency test is very limited. In the present work, we demonstrate that the Krichevskii parameter can be used to validate solubility data for a homologous series of substances in a common supercritical solvent and can be used to identify gross inconsistencies in the data. This is shown for the homologous series of n-alkanes in supercritical CO2. Knowledge of heavy n-alkane solubilities in supercritical CO2 is important in the design of hydrocarbon processing systems. However, literature data on these systems are scarce and often unreliable. A method that allows the user to identify consistent data sets and to interpolate/extrapolate limited data is therefore of great practical interest.
T ln E ) A + BF
The Krichevskii parameter is related to the limiting slopes of the critical locus in a binary system via1,7,8
(1)
where (dP/dT) cσ is the slope of the vapor pressure curve of the solvent at its critical point and (dPc/dx)∞ and (dTc/dx)∞ are the initial slopes of the critical locus * Author to whom correspondence should be addressed. E-mail:
[email protected]. Fax: 1-404-894-2866.
(2)
where T is the temperature, A and B are temperatureindependent parameters, and F is the density of the supercritical solvent. More recently, Me´ndez-Santiago and Teja9 have compiled available solid-supercritical fluid data and shown that eq 2 is valid over a considerable range of temperatures and pressures, including solvent reduced pressures up to 2.5. In eq 2, E is the enhancement factor given by
E ) y2P/P2sub
(3)
where P is the pressure and y2 and P2sub are the solubility and sublimation pressure, respectively, of the solute at temperature T. The slope B in eq 2 is related to the Krichevskii parameter via10 ∞,c /(RFc12) B ) - (∂P/∂x) T,v
Calculation of the Krichevskii Parameter
∞,c (∂P/∂x) T,v ) (dPc/dx)∞ - (dP/dT) cσ (dTc/dx)∞
of a binary mixture at infinite dilution. Thus, a value for this parameter for a binary solute-supercritical fluid system can be obtained if mixture critical locus data and solvent vapor pressure data near the critical point of the solvent are available. Because of the regularity in the behavior of a homologous series of solutes in a common solvent, it is expected that the parameter can be correlated with some property (e.g., size) of the solute.7,8 Krichevskii parameters can also be calculated from the solubilities of solids in supercritical fluids. Using the theory of dilute solutions, Harvey4 has shown that a simple linear relationship can be used to describe the solubility of solids in supercritical fluids as follows:
(4)
where R and Fc1 are the ideal gas constant and the critical density of the solvent, respectively. If critical loci and solid solubility data in the dilute concentration range are available for a binary mixture, then the two values can be compared as a check of consistency of the data. In the more common event that the two types of data do not exist for the same mixture, then the regularity in behavior of homologous series of components can be utilized to estimate the Krichevskii parameter and/or interpolate data. This is demonstrated in the following section.
10.1021/ie000324w CCC: $19.00 © 2000 American Chemical Society Published on Web 11/10/2000
Ind. Eng. Chem. Res., Vol. 39, No. 12, 2000 4829
Figure 2. Kirchevskii parameters as a function of solute carbon number in CO2 + n-alkane systems.
Figure 1. (a) and (b) Critical temperatures and pressures of dilute CO2 + n-alkane mixtures as a function of the n-alkane composition. Data from refs 11 and 12: (O) methane, (4) ethane, (0) propane, (]) n-butane, and (2) n-pentane. Table 1. Calculated Slopes of Critical Lines and Values of the Krichevskii Parameter for CO2 + n-Alkane Systems system
(dTc/dx)∞ (K)
(dPc/dx)∞ (MPa)
∞,c (∂P/∂x) T,v (MPa)
CO2 (1) + methane (2) CO2 (1) + ethane (2) CO2 (1) + propane (2) CO2 (1) + butane (2) CO2 (1) + pentane (2)
-98.4 -46.1 9.4 128.2 208.6
3.9 -4.0 -5.6 2.9 10.5
20.7 3.8 -7.2 -18.9 -25.7
Results and Discussion The critical temperatures and pressures of CO2 + methane, ethane, propane, butane, and pentane mixtures were obtained from the literature11,12 and used to calculate the limiting slopes (dPc/dx)∞ and (dTc/dx)∞, as shown in Figure 1. The limiting slope of the vapor pressure curve of CO2, (dP/dT) cσ, was obtained using the vapor pressure correlation of Span and Wagner,13 resulting in a value of 0.1712 MPa/K for this slope. Substitution into eq 1 resulted in the Krichevskii parameters presented in Table 1. These values are
plotted as a function of carbon number of the n-alkane in Figure 2. As expected, Krichevskii parameters decrease monotonically from small positive values for methane- and ethane-containing mixtures to increasingly negative values as the carbon number of the solute increases. As discussed by Gude and Teja,7 systems in which the solute is more volatile than the solvent generally exhibit Krichevskii parameter values that are positive (except when the components exhibit strong interactions). By contrast, systems in which the solute is less volatile than the solvent and the two components exhibit weak interactions exhibit Krichevskii parameter values that are negative. Methane and ethane are more volatile than carbon dioxide and thus exhibit positive ∞,c . The higher n-alkanes are less values of (∂P/∂x) T,v volatile than carbon dioxide and therefore exhibit nega∞,c . This type of behavior also tive values of (∂P/∂x) T,v indicates that there are no strong interactions between carbon dioxide and the n-alkanes. Solid solubility data for heavy n-alkanes with 24-36 carbon atoms in supercritical CO2 were also obtained from the literature. Table 2 lists the temperature and pressure ranges of these data and provides data references. The solubility data were correlated using eq 2, and best-fit values of the slope A and intercept B are presented in Table 2. The sublimation pressure needed to calculate the enhancement factor E (eq 3) was calculated using the method of Pouillot et al.14 The density of CO2 was calculated using the equation of state proposed by Span and Wagner,13 using the critical properties of CO2 reported by Span and Wagner.13 Figure 3 shows the correlation for a typical data set of octacosane (C28) in CO2. As can be seen in this figure, all data sets measured by different researchers at different temperatures collapse to a single line. The selfconsistency of different data sets can also be checked by selecting all data that follow this linear trend9 and disregarding outliers.
Table 2. Constants of Equation 2 and Calculated Krichevskii Parameters for CO2 + Heavy n-Alkane Systems solute
T range (K)
P range (MPa)
A
B
∞,c (∂P/∂x) T,v (MPa)
data reference
tetracosane pentacosane octacosane nonacosane triacontane dotriacontane tritriacontane hexatriacontane
308-320 308-313 308-325 308-318 308-318 318-338 308-333 308-338
5.0-26.1 10.4-20.7 8.0-48.9 7.6-21.5 9.0-25.0 4.7-20.6 10.0-24.0 3.5-20.7
4494 3788 4581 4351 4418 4279 5324 4829
84.59 140.46 132.78 165.86 181.28 227.30 256.67 274.59
-79.4 -131.8 -124.6 -155.7 -170.2 -213.4 -240.9 -257.7
15, 16, 17, 18 15, 17, 19 15, 20, 21, 22 15, 19 20 15, 16 15 15, 16
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Ind. Eng. Chem. Res., Vol. 39, No. 12, 2000 Superscripts c ) critical ∞ ) infinite dilution Subscripts c ) critical σ ) saturation
Literature Cited
Figure 3. Solubility of n-octacosne in supercritical CO2. Data from ref 20 (O), ref 21 (4), ref 22 (0), and ref 15 (]).
The Krichevskii parameters calculated from the intercept B using eq 4 are shown in Table 2 and plotted as a function of the carbon number in Figure 2. We note that Krichevskii parameters of heavy n-alkanes in CO2 are much smaller than those for light n-alkanes. Also, these values decrease rapidly with increasing carbon number, indicating that interactions between heavy n-alkanes and CO2 decrease very rapidly as the size of the n-alkane increases. Conclusions Krichevskii parameters of carbon dioxide + n-alkanes were calculated using critical point data of carbon dioxide + light n-alkane systems and solubility data of carbon dioxide + heavy n-alkane systems. It was shown that these parameters exhibit regular behavior with carbon number, irrespective of the type of data used in their calculation. As a result, inconsistencies in the two types of solubility data can be detected. Because of errors (which we estimate to be on the order of 20%)7 ∞,c associated with the calculation of (∂P/∂x) T,v from different types of data, it is only possible to detect gross inconsistencies in the data sets. However, judicious interpolation/extrapolation can be carried out to obtain values for systems containing heavy hydrocarbons that are difficult to measure. Krichevskii parameters also provide a qualitative understanding of solute-solvent interactions in carbon dioxide + n-alkane mixtures. Acknowledgment T.F. gratefully acknowledges financial support from the Japan Science and Technology Corporation in the form of an Overseas Research Fellowship. Nomenclature A, B ) constants in eq 2 P ) pressure, MPa P2sub ) sublimation pressure of component 2 R ) ideal gas constant T ) temperature, K x ) solute mole fraction in liquid y2 ) mole fraction solubility of component 2 F ) density of carbon dioxide, mol/L Fc1 ) critical density of carbon dioxide, mol/L
(1) Krichevskii, I. R. Thermodynamics of critical phenomena in infinitely dilute binary solutions. Russ. J. Phys. Chem. 1967, 41, 1332-1338. (2) Sengers, J. M. H. L. Solubility near the solvent’s critical point. J. Supercritical Fluids 1991, 4, 215-222. (3) Harvey, A. H.; Sengers, J. M. H. L.; Tnger, J. C., IV. Unified description of infinite-dilution thermodynamic properties for aqueous solutes. J. Phys. Chem. 1991, 95, 932-937. (4) Harvey, A. H. Supercritical solubility of solids from nearcritical dilute-mixture theory. J. Phys. Chem. 1990, 94, 84038406. (5) Harvey, A. H.; Sengers, J. M. H. L. Correlation of aqueous Henry’s constants from 0C to the critical point. AIChE J. 1990, 36, 539-546. (6) Harvey, A. H.; Crovetto, R.; Sengers, J. M. H. L. Limiting vs apparent critical behavior of Henry’s constants and K factors. AIChE J. 1990, 36, 1901-1904. (7) Gude, M. T.; Teja, A. S. The critical properties of dilute n-alkane mixtures. Fluid Phase Equilib. 1993, 83, 139-148. (8) Gude, M. T.; Teja, A. S. Near-critical phase-behavior of dilute mixtures. Mol. Phys. 1994, 81, 599-607. (9) Me´ndez-Santiago, J.; Teja, A. S. The solubility of solids in supercritical fluids. Fluid Phase Equilib. 1999, 160, 501-510. (10) Me´ndez-Santiago, J. Extensions of the theory of dilute solutions. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, 1999. (11) Hicks, C. P.; Young, C. L. The gas-liquid critical properties of binary mixtures. Chem. Rev. 1975, 75, 119-174. (12) Sadus, R. J. High-pressure phase behaviour of multicomponent fluid mixtures; Elsevier: New York, 1992. (13) Span, R.; Wagner, W. A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressures up to 800 MPa. J. Phys. Chem. Ref. Data 1996, 25, 1509-1596. (14) Pouillot, F. L. L.; Chandler, K.; Eckert, C. A. Sublimation pressures of n-alkanes from C20H42 to C35H72 in the temperature range 308-348 K. Ind. Eng. Chem. Res. 1996, 35, 2408-2413. (15) Chandler, K.; Pouillot, F. L. L.; Eckert, C. A. Phase equilibria of alkanes in natural gas systems. 3. Alkanes in carbon dioxide. J. Chem. Eng. Data 1996, 41, 6-10. (16) Yau, J. S.; Tsai, F. N. Solubilities of Heavy n-Paraffins in Subcritical and Supercritical Carbon Dioxide. J. Chem. Eng. Data 1993, 38, 171-174. (17) Smith, V. S.; Campbell, P. O.; Vandana, V.; Teja, A. S. Solubilities of long-chain hydrocarbons in carbon dioxide. Int. J. Thermophys. 1996, 17, 23-33. (18) Schmitt, W. J.; Reid, R. C. The solubility of parafinic hydrocarbons and their derivatives in supercritical carbon dioxide. Chem. Eng. Commun. 1988, 64, 155-176. (19) Smith, V. S. Solid-fluid equilibria in natural gas systems. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, 1995. (20) Reverchon, E.; Russo, P.; Stassi, A. Solubilities of Solid Octacosane and Triacontane in Supercritical Carbon Dioxide. J. Chem. Eng. Data 1993, 38, 458-460. (21) Swaid, I.; Nickel, D.; Schneider, G. M. NIR-spectroscopic investigations on phase behavior of low-volatile organic substances in supercritical carbon dioxide . Fluid Phase Equilib. 1985, 21, 95-112. (22) McHugh, M. A.; Seckner, A. J.; Yogan, T. J. High-pressure phase behavior of binary mixtures of octacosane and carbon dioxide. Ind. Eng. Chem. Fundam. 1984, 23, 493-499.
Received for review March 16, 2000 Revised manuscript received September 22, 2000 Accepted September 25, 2000 IE000324W
