386
J. Phys. Chem. 1991, 95, 386-399
6. The slope fitted by linear least squares yielded values of k&. Similarly, the coefficient k i e was determined from the plots of (y - k3 - k& [CS,]) vs [He], as illustrated in Figure 7. The values k& and k& are also listed in Table VI. At 298 K, k& = 5.79 X cm3 molecule-’ s-l and kze = 7.36 X cm3 molecule-’ s-I. The rate coefficients determined by fitting the double-exponential decay plot are not as accurate as those determined from a simple exponential decay. The uncertainties in the measurements of k & and kg! are greater than those of kEe and k;: due to the limited range ok [CS,] that can be used to produce good plots of [OH] decay. We estimate the 95% confidence limits to be f25% for k!;, and k&, and f20% for kFf! and kEe. The values of kFfL, k&, kze, and k& are shown in an Arrhenius plot, Figure 8. These rate coefficients were fitted to lead to activation energies E, = -1 3.4 f 0.3, 34.6 f 4.1, -34.7 f 3.2, and 6.7 f 0.8 kJ mol-’ for kf.ff kze, k&, and k&, respectively; the uncertainties represent one standard deviation. The 95% confidence limits for the measurements of E are estimated to be -f25%. The temperature dependence of kg[, and k&, yield AHo
for the title reaction, -49.7 f 10.4 kJ mol-], in agreement with that determined from the van’t Hoff plot. The AHo derived from ktfb and k i e is slightly greater because kib and k i e were determined at only three temperatures; and the data at 249 K gave a relatively large value of K,. The -E, value for kg! is approximately 20 kJ mol-’ greater than that for k;:, presumafoly because the deactivation by the polyatomic CS2molecules is relatively more efficient at low temperatures. In summary, the reaction between OH and CS2has been studied by means of the laser-photolysis/laser-induced-fluorescence technique. The equilibrium constants were determined to be smaller than those from previous studies, whereas the value of AHo agrees well. The rate coefficients for the forward and reverse reaction (for M = He and CS2) have been reported for the first time; their dependence on temperature has also been studied. Acknowledgment. This research is supported by the National Science Council of the Republic of China. We are grateful to Dr. A. R. Ravishankara for communicating results to us before publication.
Influence of Solute-Solvent Asymmetry upon the Behavior of Dilute Supercritical Mixtures Irena B. Petsche and Pablo G. Debenedetti* Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544-5263 (Received: May 3, 1990)
Infinitely dilute supercritical mixtures may be classified into three categories: attractive, weakly attractive, and repulsive, according to the sign of the solute’s partial molar properties, and of the excess number of solvent molecules surrounding a given solute molecule. These quantities are arbitrarily large near the solvent’s critical point. Their sign is determined by differences in size, energy, and shape between solute and solvent (molecular asymmetry). The attractive or repulsive character of a van der Waals mixture is determined by the ratio of solute to solvent specific energies, referred to the respective molecular volumes. For a lattice-gas mixture with constant cell size, the boundaries between the three regimes are a function of chain length ratio, segment energy ratio, and solvent length. A simplified perturbed hard chain model predicts van der Waals-like behavior in the limit of unit chain length, and lattice-like behavior in the limit of constant segment size. The classification into attractive and repulsive behavior can also be expressed in terms of slopes of critical lines. Attractive behavior near the soloent’s critical point is a necessary condition for supercritical solubility enhancement; repulsive behavior near the less volatile component’s critical point is closely related to gas-gas immiscibility.
Introduction The partial molar properties (volume, enthalpy, excess entropy with respect to an infinitely dilute ideal gas mixture at the same density and temperature) of an infinitely dilute solute diverge at the solvent’s critical point. The solute acts as a local density perturbation whose effect is propagated over a length scale given by the correlation length. Its partial molar properties scale as the solvent’s compressibility, and their divergence is a critical phenomenon, indicative of long-ranged correlations in density fluctuations. Although this divergence is common to every infinitely dilute near-critical system, its sign is not. For a given system, the solute’s partial molar volume, enthalpy, and excess entropy diverge with the same sign, and can therefore tend either to +m or to -a. Short-ranged interactions occurring over distances of a few angstroms determine the sign of the diverging solute partial molar properties. In this paper we investigate the effects of differences in size, shape, and interaction energies between solute and solvent molecules upon the sign of the solute’s diverging partial molar properties. Understanding the relationship between solute-solvent differences in molecular size. architecture, and energetics, on the one hand, and bulk thermodynamic behavior, on the other, is especially 0022-3654/91/2095-0386%02.50/0
important in the case of dilute mixtures of nonvolatile solutes in supercritical solvents. In this class of system, negative solute partial molar volumes and enthalpies near the mixture’s lower critical end point give rise to a solubility increase with pressure (at constant temperature) and to a decrease in solubility with temperature (at constant pressure), respectively. For a nonvolatile solute, the mixture’s lower critical end point is very close to the solvent’s critical point. Since, furthermore, the mixture is dilute, the large and negative solute partial molar volume and enthalpy are a consequence of proximity to the solvent’s critical point. Thus, the question naturally arises as to which aspects of molecular asymmetry between solute and solvent will give rise to negatively diverging partial molar properties. Our theoretical study reflects the current widespread interest in the topic of molecular interactions in dilute, supercritical systems.’,* In recent years, experimental studies have probed the behavior of dilute supercritical mixtures over a range of length scales. Long-ranged correlations and cooperative behavior have been investigated via solute partial molar property measure( I ) Brennecke, J. F.; Eckert, C. A. AIChE J . 1989, 35, 1409. (2) Johnston, K . P.; Peck, D. G.; Kim,S.Ind. Eng. Chem. Res. 1989, 28, 1115.
0 199 1 American Chemical Society
Behavior of Dilute Supercritical Mixtures ments3s4 Short-ranged interactions, on the other hand, have been studied by solvatochromic and by fluorescence spectroscopy.8-'0 Theoretical treatments include the use of fluctuation t h e ~ r y , ~ . " -integral '~ equations,leI7 and computer simulations.I8 A convenient scheme for the classification of near-critical behavior according to the sign of diverging, long-ranged quantities has been recently proposed by Debenedetti and Mohamed,I3 who introduced the notion of repulsive, weakly attractive, and attractive behavior. In an attractive binary, long-ranged solvent enrichment around the solute occurs in the vicinity of the solvent's critical point, resulting in large and negative solute partial molar volumes. In a repulsive binary, long-ranged solvent depletion occurs around the solute in the vicinity of the solvent's critical point, which results in the solute's partial molar volume diverging to +m. In the weakly attractive case, the long-ranged enrichment is insufficient to overcome the purely osmotic effect associated with the introduction of the solute into the solvent, and thus the solute's partial molar volume diverges to +,.I3 In every case, the solute's partial molar enthalpy and excess entropy diverge with the same sign as that of the partial molar volume, and scale as the compressibility too.I2 Applications of supercritical fluids to processes such as polymer fracti~nation,'~ production of fine powders,20enzymatic catalysis in nonaqueous recovery of toxic wastes from contaminated soils,22 regeneration of activated carbon,23 etc. involve attractive near-critical behavior. Thus, large, negative solute partial molar properties at infinite dilution, near the soluent's critical point, and solvent enrichment around the solute (attractive behavior) are necessary (but not sufficient) conditions for supercritical solubility. I n this paper, we address two questions: how do differences in size, shape, and interaction energy between solute and solvent determine the attractive, weakly attractive, or repulsive character of a binary system, and over what range of densities and temperatures will each type of behavior occur? Regarding the second question, it is important to understand that the distinction between the above three types of behavior becomes significant in the near-critical region, where the partial molar quantities diverge.
(3) Eckert, C. A.; Ziger, D. H.; Johnston, K. P.; Ellison, T. K. Fluid Phase Equilib. 1983, 14, 167. (4) Eckert, C. A.; Ziger, D. H.; Johnston, K. P.; Kim, S. J . Phys. Chem. 1986, 90, 2738. ( 5 ) Kim, S . ; Johnston, K . P. AIChE J . 1987, 33, 1603. (6) Kim, S.; Johnston, K. P. Ind. Eng. Chem. Res. 1987, 26, 1206. (7) Johnston, K. P.; Kim, S.; Combes, J. In Supercritical Fluid Science and Technology; ACS Symposium Series 406; Johnston, K. P., Penninger, J. M. L., Eds.; American Chemical Society: Washington, DC, 1989; Chapter 5. (8) Brennecke, J. F.; Eckert, C. A. Proc. Inr. Symp. Supercrir. Fluids (I) Nice, Fr. 1988, 263. (9) Brennecke, J. F.; Eckert, C. A. In Supercrirical Fluid Science and Technology; ACS Symposium Series 406; Johnston, K. P., Penninger, J. M. L., Eds.; American Chemical Society: Washington, DC, 1989; Chapter 2. (10) Brennecke, J. F.; Tomasko, D. L.; Peshkin, J.; Eckert, C. A. Ind. Eng. Chem. Res. 1990, 29, 1682. ( I I ) Debenedetti, P. G . Chem. Eng. Sei. 1987, 42, 2203. (12) Debenedetti, P. G.; Kumar, S. K. AIChE J . 1988, 34, 645. ( I 3) Debenedetti, P. G.; Mohamed, R. S. J . Chem. Phys. 1989,90,4528. (14) Cochran, H . D.; Pfund, D. M.; Lee, L. L. Proc. Inr. Symp. Supercrir. Fluids (I),Nice, Fr. 1988, 245. (15) Cochran. H. D.; Lee, L. L. In Supercritical Fluid Science and Technology; ACS Symposium Series 406; Johnston, K. P., Penninger, J. M. L., Eds.; American Chemical Society: Washington, DC 1989; Chapter 3. (16) Wu, R. S.: Lee, L. L.; Cochran, H. D. Ind. Eng. Chem. Res. 1990, 29, 977. ( I 7) McGuigan, D. B.; Monson, P. A. Fluid Phase Equilib. 1990,57,227. (18) Petsche, 1. B.: Debenedetti, P. G.J . Chem. Phys. 1989, 91, 7075. (19) Kumar, S. K.; Chabria, S. P.; Reid, R. C.; Suter, U. W. Macromolecules 1987, 20, 2550. (20) Matson, D. W.; Fulton, J. L.; Petersen, R. C.; Smith, R. D. Ind. Eng. Chem. Res. 1987, 26, 2298. (21) Randolph, T. W.; Blanch, H . W.; Prausnitz, J. M. AIChE J . 1988, 34, 1354. (22) Roop, R. K.; Hess, R. K.; Akgerman, A. In Supercritical Fluid Science and Technology; ACS Symposium Series 406; Johnston, K. P., Penninger, J . M. L., Eds.; American Chemical Society: Washington, DC, 1989; Chapter 29. (23) de Fillipi, R . P.; Krukonis, V. J.; Modell, M. Environmental Protection Agency, Rep. No. EPA-600/2-80-054, 1980.
The Journal of Physical Chemistry, Vol. 95, NO. 1 , 1991 387
In the case of attractive near-critical mixtures, the large negative solute partial molar volumes give rise to an increase in solubility with pressure. Once this initial increase in solubility occurs, in the near-critical region, it persists as long as the solute's partial molar volume is less than the molar volume of the pure solute. When these two quantities become equal, the solubility reaches a maximum at constant temperature. Under these conditions, the mixture is normally repulsive, but distinctions between the three types of behavior are meaningless away from criticality. It is the attractive behavior of a system in the vicinity of the solvent's critical point that causes the initial solubility increase. In mapping the boundaries between attractive and repulsive behavior, accordingly, we are ultimately interested in determining what features of molecular asymmetry cause the near-critical region to belong to one or the other regime. We begin by reformulating the classification scheme of Debenedetti and Mohamed in terms of the solute-solvent direct correlation function integral. We show that the attractive, weakly attractive, or repulsive character of a particular binary system depends on the infinite dilution limit of this quantity. We then investigate how differences in size and interaction energy between solute and solvent determine the attractive or repulsive character of a given near-critical van der Waals binary. We subsequently study the effects of temperature and density upon the location of the boundaries between the different regimes for given van der Waals binaries. There result necessary criteria for supercritical solubility in terms of ratios of pure component critical parameters. Furthermore, we show that, in a van der Waals mixture, attractive behavior is always enhanced (i.e., its boundaries enlarged) by an increase in the strength of solutesolute interactions, whereas increasing the solute diameter relative to the solvent's will eventually result in loss of attractive behavior. The relevant parameter in determining a van der Waals system's attractive or repulsive character is the ratio of solute to solvent specific energies, where the specific energy is based upon molecular volume. The lattice-gas model of Panayiotou and Vera24also predicts unbounded enhancement of attractive behavior due to energy asymmetries. Geometric effects, on the other hand, give rise to different behavior than that predicted by the van der Waals model in that attractive behavior never disappears by making the solute longer than the solvent. The simplified perturbed hard chain model of Kim et al.25yields predictions which agree with the van der Waals model in the limit of single-bead molecules, and with the lattice gas model of Panayiotou and Vera in the limit of similar segment size for both solute and solvent. A comparison of the predictions of the van der Waals, lattice-gas, and perturbed hard chain models yields useful insights into the importance of molecular shape in determining global phase behavior. Finally, we discuss the macroscopic (thermodynamic) implications of the attractive, weakly attractive, and repulsive regimes with respect to global phase behavior and slopes of critical loci. There results an interesting connection between repulsive behavior near the critical point of the less volatile component and fluid-fluid immiscibility. It is important to note that the direct correlation function integral is short-ranged and does not exhibit critical anomalies. In all of the calculations reported here, the limit of infinite dilution was taken first, and the behavior of the sign of the direct correlation function integral was then studied. In all cases, these sign changes occur away from the solvent's critical point. Classification Scheme In order to show the relationship between the sign of diverging solute partial molar properties and short-ranged quantities, we reformulate Debenedetti and Mohamed's classification scheme for near-critical behavior in terms of the direct correlation function. To this end, we first write p v l m=
Kd
(1)
(24) Panayiotou, C.; Vera, J. H. Polym. J . 1982, 14, 681. (25) Kim, C. H.; Vimalchand, P.; Donohue, M. D.; Sandler, S.1. AIChE J . 1986, 32, 1726.
388 The Journal of Physical Chemistry, Vol. 95, No. 1, 1991 where VImis the solute's partial molar volume at infinite dilution, p is the solvent's number density, K7 is its isothermal compressibility, and 6 denotes the infinite dilution limit of the rate of change of pressure upon solute addition at constant temperature, volume, and solvent mass
Petsche and Debenedetti \
2.0, 1.9
-
1 .8
-
1 .7
-
WEAKLY ATTRACTIVE REPULSIVE
1
where subscript 1 denotes solute, 2 denotes solvent, x is a mole fraction, and N = N , N 2 . Equation 1 is simply the chain rule, written for an infinitely dilute mixture. Next, we write VIm= kTKdI - C12-) (3)
+
where
C,Zm= p
1
clZm dr
(4)
Comparison with eq 1 shows that the sign of the solute's partial molar volume is determined by the value of the direct correlation function integral. The direct correlation function is short-ranged, finite, and well behaved at the solvent's critical point. Equation 3 can also be written in terms of a long-ranged fluctuation integral6*I1 p V l m = pkTK, -
r =P
1
r
(glzm- 1) dr
(6)
r = pkTKKI2" (8) Equation 8 shows that the excess number of solvent molecules surrounding the infinitely dilute solute will diverge at the solvent's critical point, the sign of this divergence being that of the direct correlation function integral. The latter quantity thus determines three possible types of behavior
-
VIm +a; VIm+
+OD;
r -,-m r -,+m r++m
1 3
0
.2
.4
.6
.8
1.0
1.2
1.4
1.6
1.8
2.0
Pr Figure 1. Boundaries between the three regimes for a van der Waals representation of the naphthalene-C02 system. Dashed lines represent boundaries based upon the experimental data of Eckert et aL4 Axes are solvent reduced quantities.
or repulsive behavior. We consider in the first place the classical continuum model of van der Waals for which P = pkT(1 - pb)-l - ap2. For the mixture parameters a and b we use the common mixing rules originally proposed by van der Waalsz8 b = Cxibi and a = CCxix,qj,where the Berthelot rule, or geometric mean, is used for aij [ai, = ( u , u , ) ~ / For ~ ] . this model, the infinite dilution limit of the direct correlation function integral (see Appendix) is given by
where p is the solvent's number density. The mixture exhibits attractive behavior when CIzm > 1, which from eq 12 is true when
where a = a l / a 2 y, b l / b 2 ,p, P I P c , and T, = TITc [ p c and T, are critical properties of the solvent, and we have used the fact that a2 = 27kTcb2/8and b2 = l / ( 3 p c ) ] . The left-hand side of eq 13 is the upper limit of T , for which attractive behavior is possible at a given density. Repulsive behavior is exhibited when c12- < 0:
(9) (IO)
CI2-> I : VIm-,- a ', (11) Equations 9-1 1 define the repulsive, weakly attractive, and attractive near-critical regimes. The former is characterized by large and positive solute partial molar properties, and longranged solvent depletion around the solute and the latter, by large and negative solute partial molar properties and long-ranged solvent enrichment around the solute. In a weakly attractive binary, the long-ranged enrichment is insufficient to overcome the purely osmotic effect associated with the introduction of solute molecules into the s01vent.I~
The van der Waals Mixture We now investigate how differences in size and interaction energies between solute and solvent determine a system's attractive (26) Kirkwood, J. G.;Buff, F. P. J . Chem. Phys. 1951, 19, 774. (27) OConnell, J. P. Mol. Phys. 1971, 20, 27.
1
(7)
The quantity r is therefore the statistical excess number of solvent molecules surrounding an infinitely dilute solute molecule with respect to a uniform distribution at bulk conditions. In the above equation, g12denotes the unlike pair correlation function, and the superscript denotes infinite dilution conditions. Combining eqs 3 and 6 , we obtain
CI2-< 0: 0 5 ClzmI1:
1
1
In eq 4, c12denotes the solutesolvent direct correlation function, Clz the direct correlation function integral, and the superscript denotes infinite dilution conditions (limit of vanishing solutesolvent interactions). Equation 3 is an identity4 and follows from reformulating the Kirkwood-Buff fluctuation theory of solutionsz6 in terms of direct correlation f~nctions.~'It follows from eqs 1 and 3 that
where
FL 1
< CIzm< 1: 9 a q 3-pry < T, < 4 [ 3 (+ ~ 1) - ~ r 1
Weakly attractive behavior occurs when 0 3pra'/2(3 - p r y
(15)
4[3 + - 1)1 Equations 13 and 14 can be used to map the boundaries between the three regimes in p,, T, space. If, for example, we consider a solute-solvent pair with cy = 10.73 and y = 4.36 we obtain Figure 1. These values of a and y correspond to a van der Waals representation of the naphthalene-C02 system. The differences between the three types of behavior are significant only in the near-critical region due to the divergence of the solute's partial molar properties and of the quantity r, the sign of these diver(28) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G.Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1986.
The Journal of Physical Chemistry, Vol. 95, No. 1 , 1991 389
Behavior of Dilute Supercritical Mixtures
0, /u,
3.0
Tr
~
= 1.0
CONTOURS AT .IC Ih'ERVALS
3
1.41
/ / // /
Figure 3. Three-dimensionalrepresentation of the effect of the energy ratio on the extent of the attractive region. van der Waals model. T,
\'
ATTRACTIVE
and pr are solvent reduced quantities (1 = solute; 2 = solvent). Arrows indicate unity. E, / E 2
Pr Figure 2. Effect of the energy ratio on the boundaries between attractive, weakly attractive, and repulsive regimes for a solutesolvent pair of equal size. van der Waals model. Labels on curves are energy ratios (Z = t l / c 2 ) . Axes are solvent reduced quantities.
gencies diffcring in the attractive, weakly attractive, and repulsive cases. I t can be seen from Figure I that, according to the van der Waals model, the naphthalene-C02 system exhibits attractive behavior in the near-critical region, as expected. The dotted lines in Figure 1 correspond to the boundaries between attractive, weakly attractive, and repulsive behavior as calculated from experimental measurements of VIm for the naphthalene-CO, system for several densities at T , = 1.014 and T , = I .046.4 These points were obtained from the identity CI2= = 1 - VI"/(kTKT), by finding the experimental densities at which the conditions CI2== 1 .O and CI2-= 0.0 are satisfied. In performing these calculations, both experimental4 ( VImand p ) and literature29 (KT)values have been used. At a given temperature, the actual boundaries occur at a higher density than that predicted by the van der Waals model. Nevertheless, it is significant that even for a highly nonspherical and asymmetric system such as naphthalene-C02 we can still obtain a reasonable estimate for the boundary locations using the van der Waals model with no binary interaction coefficient. The van der Waals parameters a and b can be related to an idealized intermolecular potential u(r) with hard-core repulsion and dispersive (Le., r4) attraction and thus acquire molecular ~ignifican.cce3~
Assuming a uniform distribution of molecules, and approximating the configurational integral by the position-independent Boltzmann factor of a mean interaction energy per molecule, integrated over the effective volume available for molecular motion, one obtains b = 2ra3/3, a = With this molecular interpretation for the a and b parameters, we have y = (a1/a2)3and a = ( ~ ~ / u ~ ) ~ ( tUsing ~ / t ~these ) . relationships in eqs 13-15, we can study the effect of the size and energy ratios, D = al/a? Z cl/c2, on the boundaries between the three types of behavior. In deriving eqs 13-1 5 we used the geometric mean for u I 2and ~ _ _ _ _ _ _ _ ~
_____
~
Angus, S.;Armstrong, B.; de Reuck, K. M. IUPAC International Thermodynamic Tables of the Fluid State-Carbon Dioxide; Pergamon: (29)
New York. 1976. (30) Hill, T. L.An Introduction to Statistlcal Thermodynamics; Dover: New York, 1986.
T,.
=1.0
CONTOURS AT . 0 5 INTERVALS
Figure 4. Three-dimensional representation of the effect of size ratio on the extent of the attractive region. van der Waals model. T, and p r are solvent reduced quantities ( I = solute; 2 = solvent). Arrows indicate
unity.
therefore we may write e I 2 = (t1t2)1/2and thus Z may also be expressed as the square of the ratio of the solute-solvent to the solvent-solvent interaction energy ([t12/t2] 2). In Figure 2 we show the effect of changing the energy ratio, Z, for a solutesolvent pair of equal size ( D = 1.O). As the energy ratio is increased so that the solute-solute interaction energy, el, is greater than the solvent-solvent interaction energy, €2. the attractive region expands (Le., attractive behavior is found over a wider range of pr and T,), while the repulsive region becomes smaller. If we focus on the attractive region, trends can be more easily seen in Figure 3. The base plane corresponds to T, = 1. The surface (excluding the base plane) is the upper limit of temperature for which attractive behavior is possible. For a size ratio of I , there is no attractive region if Z < 1. As Z increases, attractive behavior is possible over a larger range of pr and T,. This increase in the attractive region due to the increase in Z is unbounded, as can be seen from eq 13, where this quantity appears only in the numerator (a = D3Z). If the solute is smaller than the solvent, the attractive region shifts to the right (higher density) and toward higher energy ratios ( 2 > 1; the minimum energy ratio required for attractive behavior is now greater than 1). Conversely, if the solute is larger than the solvent, the attractive region shifts to the left (lower density) and toward lower energy ratios (I: < 1; the minimum energy ratio required for attractive behavior is now less than 1). Thus we see that the range of pr and T, where attractive behavior is exhibited increases as the energy ratio Z increases, while the smallest I: for which near-critical attractive behavior is possible is a function of the size ratio D. We now consider the effect of D in more detail. Figure 4 illustrates the effect of the size ratio, D, for a mixture with Z = 1. The attractive region is bounded in that if the ratio of solute to solvent diameter is less than 1 or greater than 2.2, attractive
390 The Journal of Physical Chemistry, Vol. 95, No. 1. 1991
Petsche and Debenedetti
behavior is not possible. When a solute has a smaller well depth than the solvent, the size of the attractive region decreases. In fact, if Z < 0.89 an attractive region does not exist. When a solute has a larger well depth than the solvent, the size of the attractive region increases but it is still bounded. The fact that attractive behavior, for a given 2. is possible only for a limited range of D can be seen from eq 13 where D3/2is in the numerator while D3 is in the denominator (a = D3Z, y = D3). In physical terms, this suggests that the relevant parameter in determining a system's attractive or repulsive character should be a specific energy. In fact, according to eq 13, the appropriate quantity is
which is simply the square root of the ratio of component energies per molecular volume. Equivalently, since Z = (t12/c2)2, the relevant parameter may also be expressed as the ratio of solutesolvent to solventsolvent well depths, referred in each case to the square root of the relevant molecular volume (Le., u ~for~c12,/ u23/2 for t2). Thus far we discussed the boundaries between attractive, weakly attractive, and repulsive behavior with temperature and density as dependent variables. We explored, in other words, the effects of molecular asymmetry on the temperature and density limits within which attractive behavior is possible. We now consider how these boundaries depend on Z and D for given T,, pr values. By expressing eq 13 in terms of molecular parameters (Z and D ) we obtain
4'0
I AlTRACTIVE
~
D
r I t t
.O .9
t
.7
NL
We use the symbol Z, to denote the ratio of specific energies defined above (Z, = 2 / D 3 ) and thus
W
For a given T, and p , the right-hand side of eq 18 gives the lower bound on Z, above which attractive behavior is possible. Similarly, from eq 14 we can calculate an upper bound on Z below which we find repulsive behavior Z
I . Thus a system whose component critical property ratios fall within the attractive region fulfills a set of necessary criteria for supercritical solubility. The criteria are necessary but not sufficient since, for example, the solubility of a solid in a supercritical fluid is of course dependent upon its vapor pressure. Measured infinite dilution solute partial molar volumes for all of the representative points shown in Figure 6 were negative in the solvent's near-critical r e g i ~ n ~(Le., . ~ ' attractive behavior) except for the ethane-C02 system, for which a positive solute partial molar volume was measured at T , = 1.0005.32 Note that the representative point for this mixture is very close to the boundary between the attractive and weakly attractive regimes. Deviations from the Berthelot combining rule may be considered, for example, by introducing an interaction parameter k I 2 [al2 = (1 k12)(ala2)1/2]. In eqs 13, 14, and 15, a is replaced by ( 1 - k12)2a. It follows that in all figures and equations we replace with (1 - k12)2Band in Figure 6 we multiply the ordinate by (1 - k 1 2 ) 2 . Returning to the ethane-C02 system, a very small deviation from Berthelot behavior ( k I 2= 0.05)is sufficient to move the system into the weakly attractive region of Figure 6 .
The Lattice-Gas Model The van der Waals picture applies, strictly speaking, to spherically symmetric molecules. In order to investigate the effects of molecular size and energetics upon the attractive, weakly attractive, or repulsive character of nonspherical systems, we use the lattice gas model developed by Panayiotou and Vera.24 This model is very useful for mixtures whose components differ greatly in size and has been successfully used to model the solubility of small nonvolatile organic solutes (such as benzoic acid and (31) Wu, P. C.; Ehrlich, P. AIChE J . 1973, 19, 533. (32) Khazanova, N. E.; Sominskaya, E. E.Russ. J . Phys. Chem. 1971,45, 1485.
acridine) in supercritical C02,33as well as of macromolecules in supercritical fluids, (i.e., polystyrene in ethane or C0234and poly(ethy1ene glycol) in C023s). The model considers a lattice with a coordination number z , and a unit cell size uh. There are N I molecules of type 1, and each type 1 molecule occupies rl lattice sites. There are N2 molecules of type 2, each occupying r2 lattice sites. There are NH empty lattice sites or holes. Thus the total number of lattice sites, N,, is given by N , = NH + r l N 1+ r2N2 (22) To account for connectivity of molecular segments, we define the quantity qi such that the product zqi represents the total number of external contacts per molecule of type i. If ri = 1 the number of external contacts is simply the coordination number z; each additional segment adds 2-2 external contacts. For an open molecule, (Le., a simple linear chain or a branched molecule with no closed rings) it is easy to show that zqi = ri(z - 2 ) + 2 (23) The total number of external contacts for the lattice, zNq, is then given by zNq = zNH + ZqINI + z ~ ~ N Z (24) The energy associated with a given lattice configuration is determined by assuming nearest-neighbor interactions and pairwise additivity. The interaction energy between segments of molecules of type i and j is ti., where the interaction energy of any segment with a hole, or with other segments of the same molecule, is zero. The configurational energy, E, is given by (25) -E = Nlltll + N12t12 + N22t22 where Nij represents the number of nearest-neighbor intermolecular i-j contacts. The canonical partition function Q([Nl,V,T) may be written as Q([Nl,V,T) = [ N l , V , E ) e - E ( [ ' ' ' l ~ ~ / k T(26)
xQ( E
where [ N ] represents Nh,N l , N , and the summation is over all energy levels E consistent with [Nl and V. Q([N-J,V,E)is the number of configurations with energy E . Approximating the partition function by its maximum term, we rite^^-^^
The number of ways of orienting a molecule of type i after one of its segments has been placed on a lattice site is given by hi,and ui is the summetry number of the molecule, or number of equivalent ways of placing the molecule on a given set of lattice sites ( u = 2 for linear molecule^).^^ Thus the (tji/ui) terms take into account the flexibility and symmetry of the molecules and do not contribute to the equation of state since they are assumed not to depend on the system's volume. The number of configurations with a total interaction energy E is determined by gg,,, where g, is the random combinatorial term and g,, is the nonrandom term which corrects for the fact that holes are assumed to mix randomly, but molecular segments mix nonrandomly. Following Guggenheim, g, may be written as36
The g,, term is given by Panayiotou and Vera24as
(33) Kumar, S.K.; Suter, U. W.; Reid, R. C. Ind. Eng. Chem. Res. 1987, 26, 2532. (34) Kumar, S.K. Ph.D. Thesis, Massachusetts Institute of Technology, 1986. (35) Daneshvar, M.; Kim, S.;Gulari, E. J . Phys. Chem. 1990, 94, 2124. (36) Guggenheim, E. A. Mixtures; Clarendon Press: Oxford, U.K., 1952.
392
The Journal of Physical Chemistry, Vol. 95, No. I , 1991
Petsche and Debenedetti
We define a molecular surface fraction 8, for each component as ZqiNi ei = CZqjNj
r>= 2
A= 1 Tr
CONTOURS A T C 2 lh-ERVALS
then, Noijis the number of i-j contacts on a "hole-free" basis for the case where all molecular segments mix randomly, Noij = (zqiNi/2)Bj.Nonrandom mixing is introduced by the factors rij ,." v = n",.r., JJ JJ (30) 2
The nonrandom factors are discussed in detail el~ewhere.~'They do not contribute to CI2*and will not be discussed further. We obtain the equation of state in the usual way 0
from which we derive the following expression for CI2- (see Appendix): C12m= 1 -
I+
Ar2w wz(Ar2- 1) - w 1 - r2w + 2r2w - 2w - z z(2 - 2Ar2 - 2r2w + 2Ar2w + Ar2z) ( 2 - 2r2 + r2z)(2w- 2r2w + z )
'[
~
I+
2
Figure 7. Three-dimensional representation of the effect of energy ratio on the extent of the attractive region. Solute and solvent of equal chain length. Lattice gas model. T, and p , are solvent reduced quantities ( 1 = solute; 2 = solvent). Arrows indicate unity. T,
CONTOURS A T 0.2 INTERVALS
where
x=
+ r2(z - 2 ) ] ( 2 w - 2r2w + z )
w[2
and A is the ratio of solute to solvent chain length ( r l / r 2 ) .T is the ratio of solutesolvent to solventsolvent energy ( c ~ ~ / c ~ Note ~). that the relevant energy ratio is not e I , / c z 2 as in the van der Waals case. We have made no assumption regarding the combining rule for t 1 2 . The reduced density is contained in the parameter w p r ( p g H )and the reduced temperature is included in the parameter T* T , [ 2 k T , / ( z ~ ~where ~ ) ] , pc and T, are the solvent's critical density and temperature, and reduced quantities refer to the solvent's critical point. The quantities p g H and 2 k T c / ( ~ ~are 22) functions of r2 and z and are obtained numerically by using the criticality conditions for the pure solvent (see Appendix). Thus, CI2-is a function of pr,Tr,A,T,r2,z.Given a lattice coordination number, knowledge of the solvent chain length, the ratio of solute to solvent chain length, and the ratio of interaction energies enables the classification of the mixture into the three types of behavior for any solvent reduced density and temperature. Note that, unlike the van der Waals model, the lattice-gas model requires explicit information about the solvent ( r 2 )in addition to solute-solvent size and energy ratios. This is due to the fact that the lattice-gas model interaction energy is a specific energy (energy per contact) whereas the interaction energy for the van der Waals model pertains to the entire molecule. Thus for the lattice-gas model we need information about the solvent size in order to determine the solvent-solvent interaction energy per molecule. The lattice coordination number, z , may vary from 6 (simple cubic lattice) to 12 (face-centered cubic lattice). No higher value of z is geometrically p~ssible.'~The model is insensitive to the value of z within this range33and for the following discussion we consider z = 10 which is the value commonly used.24 The attractive, weakly attractive, or repulsive behavior of a mixture therefore depends explicitly upon molecular chain length (A, r2) and energetics (T). We now explore the effect of these parameters on the attractive or repulsive character of dilute supercritical mixtures. The effect of interaction energy on the size (in p,,T, space) of the attractive region, given a solute-solvent pair of equal chain length, is illustrated in Figure 7. As was found to be the case (37) Panayiotou, C.; Vera, J. H. Fluid Phase Eguilib. 1980, 5, 55
Figure 8. Three-dimensional representation of the effect of chain length ratio (A) on the extent of the attractive region. Lattice-gas model. T, and pr are solvent reduced quantities ( 1 = solute; 2 = solvent). Arrows indicate unity.
for the van der Waals model, the energy ratio I ' has an unbounded effect on the size of the attractive region. Increasing the solute-solvent interaction energy, c12, relative to the solventsolvent interaction energy, e22, raises the upper temperature limit within which attractive behavior is still possible. If the solute chain length is smaller than the solvent chain length ( A < 1) the attractive region shifts to the right (higher density) and the minimum energy ratio required to cause attractive behavior is greater than l . This is exactly the same behavior exhibited by the van der Waals model. It is much more difficult for the larger solvent to aggregate around the solute, and therefore a larger energy ratio is required to overcome this steric hindrance. For a solute that is larger than the solvent the attractive region shifts to the left (lower density) and the minimum T required to obtain attractive behavior is now less than 1. The lattice model, therefore, predicts an energy effect entirely analogous to the one resulting from the van der Waals treatment. Thus, the maximum temperature allowed for attractive behavior increases with the strength of the solute-solvent interactions, and the minimum strength of solute-solvent interactions required for attractive behavior decreases as the solute becomes larger. In Figure 8, we show the effect of chain length ratio, A, given that the energy ratio is 1 .O. The temperature limit for attractive behavior approaches an asymptotic value as the solute chain length is increased. This is evident from the form of the equation for CI2-: rearranging eq 31 to solve for T, we find that both the numerator and denominator are first order in A. If c 1 2 < (Le.,
Behavior of Dilute Supercritical Mixtures 1.2
The Journal of Physical Chemistry, Vof.95, No. 1, 1991 393 for repulsive behavior. Thus it is the structure of the solvent that determines how weak the unlike interactions must be for repulsive behavior. The effect of solvent chain length is considered by comparing the solid lines (r2 = 1) to the dashed lines (r2= 10). We see that increasing the solvent chain length increases the maximum allowable T for repulsive behavior (Le., the mixture is repulsive for higher unlike interaction energies). Also, as r2 increases, the minimum T required for attractive behavior increases. Intuitively, we can imagine that as the solvent chain length increases it becomes more difficult to pack around the solute. In order to overcome the steric effects of long solvent chains we need a higher solute-solvent interaction energy. Note that the curves for r, = 1 end at A = 1. In the lattice-gas model, a unit cell holds only one small molecule or chain segment and therefore it is meaningless for the solute to occupy less than one lattice site.33 Note that using a fixed cell size implies the unphysical constraint whereby all monomers have equal volume. To overcome this limitation, we treat the size ratio as a continuous variable. This is akin to treating the lattice as a q~asi-lattice?~ Le., as a convenient means for counting configurations.
;
..-.-.
ATTRACTIVE ...._ ..-..._.. ......... ........ o
WEAKLY ATTRACTIVE
REPULSIVE
WEAKLY AllRACTIVE
REPULSIVE
0
1
2
3
4
5
6
7
8
9
10
A Figure 9. Boundaries between the attractive, weakly attractive, and repulsive regimes along the solvent’s critical isochore, in terms of chain length and energy ratios, A and T. Solid line: solvent chain length = I , dashed line: solvent chain length = IO. Lattice-gas model. 1) the size of the attractive region decreases while if c12 > the size of the attractive region increases. The shape of the region remains similar to that shown in Figure 8 and it follows that, as the solute becomes longer relative to the solvent, attractive behavior persists indefinitely. This appears to contradict the van der Waals predictions. This apparent contradiction, however, arises from the different definition of the interaction energy in each model. For the lattice-gas model the energy is per contact while in the van der Waals model the energy is per molecule. Thus, increasing the solute size relative to the solvent’s while keeping the energy ratio constant leads to a decrease in the strength of solutesolvent interactions per molecular volume in a van der Waals mixture. In the lattice case, the energy per contact is fixed by definition. As with the van der Waals model, we can establish criteria for attractive and repulsive near-critical behavior in terms of molecular parameters. In Figure 9 we show the boundaries between the three regimes in terms of A and T for T, = 1.05, at the solvent’s critical density. The solid lines correspond to a solvent chain length of I . The dashed lines are the boundaries for a solvent chain length of IO. First, consider the dashed lines. The upper curve represents the minimum T necessary for attractive behavior. We can understand the effect of A on the minimum T by noting that attractive behavior is characterized by a higher local density around the solute (Le., closer packing). When the solute is shorter than the solvent ( A < 1) the minimum T is very large. It is difficult to pack the large solvent around the small solute; thus we need a large unlike interaction energy to overcome this steric effect. As the chain length ratio is increased, it becomes easier for the solvent to pack around the solute and thus the minimum T decreases. We see from Figure 9 that the minimum T for attractive behavior approaches an asymptotic limit. This effect results from the treatment of molecules as chains. As the solute chain length becomes much greater than the solvent chain length, the packing around the solute is only determined by “monomer”-solvent interactions. In effect the solvent molecule does not “see” the entire solute molecule. For very long solute molecules, the packing of the solvent at one end of the solute will be independent of the packing at the other end. The boundary between weakly attractive and repulsive behavior represents the maximum T,below which repulsive behavior occurs. Note that this limit is nearly a horizontal line, indicating that it is only r2 and not A that determines the maximum allowable T
T
c22
Comparison of Model Predictions A comparison of the predictions of the van der Waals and lattice gas models reveals some interesting similarities and differences. For both models the energy effect is quite similar. In the case of van der Waals model, as cl/c2 is increased the upper temperature limit for attractive behavior increases. For the lattice-gas model, the same effect is achieved by increasing c12/c22. Both predictions are consistent with the fact that strong solute-solvent interactions result in attractive behavior-a clustering of the solvent around the solute. Turning now to the predictions regarding the effect of molecular size, it would at first appear that the two models predict quite different behavior. The lattice gas model predicts an asymptotic temperature limit as the size ratio increases. Thus, the mixture cannot lose attractive behavior as the solute size increases. According to the van der Waals model, as the solute-solvent size ratio increases, the upper temperature limit for attractive behavior initially increases but then decreases. As Figure 5a illustrates, it is possible for the mixture to lose its attractive character as the solute size increases. However, in the van der Waals model the interaction energy is per molecule while in the lattice model the energy is per contact. Thus in order to compare the two models we must use a specific energy for the van der Waals model. As Figure 5b shows, for a constant specific energy ratio (with respect to molecular volume) the mixture remains attractive as the solute size increases, in complete agreement with the lattice model predictions. The direct comparison of sizeshape effects of the van der Waals and lattice-gas models is impossible due to differences in the treatment of molecular shape for each model. The van der Waals model assumes spherically symmetric molecules; therefore, an increase in size is three-dimensional. The lattice model is applicable to open-chain molecules. The addition of a molecular segment results in a one-dimensional increase in size. In the lattice model, the cell size is fixed and therefore there is no way of representing a system of spherical molecules of different diameters. Thus there is no way of directly comparing the van der Waals size prediction with the lattice-gas chain length prediction. The bridge between the two models may be provided by an off-lattice model which takes into account segment interaction energy, segment size, and chain length. We expect such a model to match the van der Waals predictions when the solute and solvent are single-segment molecules while in the case of solute and solvent molecules made up of segments of similar size we expect agreement with the lattice-gas predictions. An appropriate model to consider is the perturbed hard chain theory (PHCT) developed by Beret and Pra~snitz,~* extended to mixtures by Donohue and P r a u s n i t ~ , ~ ~ (38) Beret, S.; Prausnitz, J. M. AIChE J . 1975, 21, 1123. (39) Donohue, M. D.; Prausnitz, J. M. AIChE J . 1978, 24, 849.
394
Petsche and Debenedetti
The Journal of Physical Chemistry. Vol. 95, No. 1. 1991 CU/ea
TABLE I: Relevant Ratios for Attractive/Repulsive Behavior
mixture model
relevant ratios
van der Waals
D(u1/u2);x(cI/c2)
lattice
A(ri/r2); r ( e i 2 / c 2 2 ) ;
SPHCT
'2;
=1
CONTOURS A T 0 1 INTERVALS
Tr
2
D ( u ~ / u ~S) (; S ~ / SE(tiIt2); ~); C(CI/C~); c2 3
and simplified (SPHCT) by Kim et The PHCT considers a molecule to behave like a chain of spherical segments interacting with a square-well potential. Rotational and vibrational degrees of freedom, which are important for large molecules, are incorporated into the model by using the Prigogine c parameter40 which represents equivalent translational degrees of freedom (c = 1 for small spherical molecules, c > 1 for more complex molecules). The repulsive term in the configurational partition function is obtained from the hard-sphere Carnahan-Starling equation:' while the attractive term is a perturbation expansion for square-well molecules based upon the molecular dynamics results of Alder.42 The SPHCT uses a simple theoretical expression for the attractive term based upon a combination of lattice statistics of chain molecules and the radial distribution function of square-well molecules. The important parameters that characterize a molecule in the SPHCT are the chain length s, the segment diameter u, the segmental interaction energy t, and the density-dependent number of degrees of freedom 3c. Note that we use ts instead of the quantity t*q used by Kim et aLz5 For a pure fluid these molecular quantities combine to give three adjustable parameters: T* = ts/ck, u* = 2-'f2su3,and c, where k is Boltzmann's constant. For a mixture, the compressibility factor is
0
Figure 10. Three-dimensional representation of the effect of size ratio (D)on the extent of the attractive region. SPHCT. T, and pr are solvent reduced quantities ( 1 = solute; 2 = solvent), C = I , S = I . Arrows indicate unity. 1.41
1.2
1 .o
.8 W
.6
-
.2 0
I
The repulsive (hard-chain) contribution to the compressibility factor, 1 + (c)zreP,is given by the Carnahan-Starling equation, where p
p
=
473 - ~
(TP)~
(33)
(1 - T b ) 3
with 2, = (u*)/u and T = 0.7405. ZMis the maximum coordination number and depends upon the width of the attractive part of the square-well potential and for this analysis is held constant at 36 which is the value frequently used.25,43 The bracketed quantities are mixture parameters obtained from the mixing rules proposed by Kim et aL2$
[ ( )
(cu*Y) = Cxixjciu*ji exp 2qkT tiJsi -I]
(34)
(c) = cxici
(35)
ij
i
( u * ) = C2-'/2xiu;Si i
+
where tu = (citj)1/2(l - kij),u * = ~ 2-1/2u,3s. 11 J' and uij = (ai uj)/2. From these equations we can derive an expression for C,2m(see Appendix) which is a function of the solvent's reduced density and temperature, the solvent's c parameter (c2), and the ratios of interaction energies ( E = t l / t 2 ) , molecular complexity ( C = cI/c2).segment diameters (D = u , / u Z ) ,and chain lengths (S = sI/s2).In Table I we summarize the ratios that define a system's attractive or repulsive character according to the three models investigated here. (40) Prinonine. I . The Molecular Theory . of- Solutions; North Holland: Amsterdam, i951. (41) Carnahan, N . F.; Starling, K. E. AIChE J . 1972, 18, 1184. (42) Alder, B. J.: Young, - D. A.; Mark, M. A. J . Chem. Phys. 1972, 56,
301'3. (43) Peters, C. J.; de Swaan Arons, J.; Levelt-Sengers, J. M. H.; Gallagher, J. S. AIChE J . 1988, 34, 834.
ATTRACTIVE
0
WEAKLY ATTRACTIVE
REPULSIVE
1
2
3
4
5
6
7
8
9
10
S Figure 11. Boundaries between the attractive, weakly attractive, and repulsive regimes along the solvent's critical isochore, in terms of chain length and energy ratios, S and E . SPHCT. C = 1, D = 1.
By allowing for solute-solvent differences in not only chain length but also segment diameter the SPHCT combines features from both van der Waals and lattice theories and as such can be used to compare the predictions of these theories. For example, Figure 10 illustrates the effect of varying the ratio of segment diameters D,maintaining all other ratios constant. Notice the similarity between the SPHCT segment diameter effect (Figure 10) and the van der Waals diameter effect (Figure 4). The attractive region is bounded and thus, as was the case with the van der Waals model, a mixture may lose its attractive character as the solute segment diameter increases. It is interesting to note that this prediction is not limited to spherical molecules since for the SPHCT CIzmis a function of the ratio of chain lengths and not of the lengths themselves. The SPHCT may be compared to the lattice model if the solute and solvent are composed of segments of equal diameter. Figure 11 shows the necessary energy ratios for attractive and repulsive behavior as a function of chain length ratio S for pr = 1, T, = 1. These curves are similar to the necessary criteria in Figure 9 for the lattice model. Thus the SPHCT indicates that the van der Waals and lattice model predictions are complementary in that one considers the segment diameter effect exclusively while the other considers only the chain length effect. For the SPHCT, the behavior of a mixture (Le., attractive, weakly attractive, or repulsive) is affected by several parameters. Neither van der Waals model nor the lattice model considers molecular complexity (as quantified by the c parameter). A preliminary investigation of the effect of this parameter indicates
The Journal of Physical Chemistry, Vol. 95, No. 1 , 1991 395
Behavior of Dilute Supercritical Mixtures that the region for attractive behavior (in p , T r coordinates) increases slightly if the solute is structurally more complex than the solvent (c, > c2). A detailed investigation of the interplay of energetics, size, shape, and molecular complexity is in progress. Thermodynamic Implications
We have used the van der Waals model to classify near-critical mixtures on the basis of differences in size and interaction energies between solute and solvent. It is interesting to compare this classification with that of Scott and van Konynenburg" who have used the van der Waals equation to predict several types of phase diagrams for binary mixtures. They assumed quadratic mixing rules for the mixture parameters. a, =
X12Ull
+ 2x,x2a,2 + x22a22
un
(37) (38)
In addition, they assumed that b12= ( b , , + b2,)/2 which reduces eq 38 to the one used in our analysis bm = Xlb,, + x2b22
(39)
b22 + bl,
b2z2
hI2
-.8
-.6
-.4
-.2
0
.2
.4
-6
c
l = - b22 - b , , ( a2 2- - a , , ) / (
I
" -1.0
Scott and van Konynenburg defined three parameters:
f =
.1
-a22+ -
all)
b2z2
bl12
(41)
Figure 12. Boundaries between the attractive, weakly attractive, and repulsive regimes in terms of Scott and van Konynenburg's [ and { parameters along the solvent's critical isochore. Solid line, TI = I .O; dashed line, T, = 1.05. van der Waals model.
The parameters C; and {can be expressed in terms of the a and y parameters defined earlier to derive the van der Waals criteria for attractive, weakly attractive, or repulsive behavior,
Their work dealt almost exclusively with = 0, (Le., solute and solvent are of equal size), for which they were able to identify five major types of phase behavior based upon the locus of critical poi n ts.44-45 Type I. One continuous gas-liquid critical line between the critical points of the two components (C,to C2) and complete miscibility of the liquid phases at all temperatures. Type I I . Two critical lines: C, to C2 and a locus of critical solution points starting at the upper critical end point (UCEP) and rising to high pressures. Type 111. A discontinuous gas-liquid critical line: one branch starting at C, and ending at the U C E P the other branch starting at C2 and rising to high pressures. Type IV. Three critical lines: C, to UCEP; LCEP (lower critical end point) to C2; starting at UCEP and rising to high pressures. Type V. Two critical lines: C, to UCEP; LCEP to C2. An additional type of phase behavior (type VI) is experimentally observed for mixtures of complex molecules where strong intermolecular interactions result in two distinct critical curves; one connects C,and C2,the other connects UCEP and LCEP.46 Such behavior cannot be predicted by the van der Waals equation.45 In the above classification, 1 denotes the more volatile component. If, in addition to = 0, the geometric mean is valid for u12(Le., u I 2= ( u , , u ~ ~ only ) ~ / types ~ ) IT and 111 result. Thus, in order to find the other types of phase behavior for a solute-solvent pair of equal size, the Berthelot combining rule for u12must be relaxed. Scott and van Konynenburg also investigated systems where the solute and solvent have different sizes (5 # 0) and where the Berthelot rule is used for q2.For these systems, we can eliminate A from the set of parameters by using the identity (1 - A ) ' ~
+ f2
= 1
(43)
~~
(44) Scott, R. L.; van Konynenburg, P. H. Discuss. Faraday SOC.1970, 49, 87 (45) van Konynenburg, P. H.; Scott, R. L. Phil. Tram, R . SOC.1980,298, 495. (46) Rowlinson, J . S.; Swinton, F. L. Liquids and Liquid Mixtures; But-
terworths: London, 1982.
(44) (45)
Using this transformation of parameters, we can express the necessary criteria for attractive, weakly attractive, or repulsive behavior in terms of the Scott and van Konynenburg parameters C; and f. In Figure 12 we illustrate the boundaries between the three regimes along the critical isochore. The solid and dashed lines correspond to reduced temperatures of 1.0 and 1.05, respectively. Note that as we move away from the critical point (higher Tr)attractive behavior is found for a smaller set of 5 and {. The effect of increasing the reduced density is similar in that the attractive region decreases. Therefore, the boundary between attractive and weakly attractive behavior calculated at the solvent's critical point can be considered an upper limit for attractive behavior. If the C; and {of a system map onto the weakly attractive or repulsive region, that system can never exhibit attractive behavior under supercritical conditions. If the system maps onto the attractive region it will exhibit attractive behavior over a limited supercritical region depending upon its proximity to the boundary between the weakly attractive and attractive regimes. We can compare this mapping of the three near-critical regimes in (f, 5 ) space with the mapping of the possible types of phase behavior. Scott and van Konynenburg showed that systems for which 5 # 0 and uI2 = can exhibit three out of the five major types of phase behavior (11, 111, IV). In Figure 13 we superimpose Figure 12 ( T , = 1, pr = 1) onto the Scott and van Konynenburg phase behavior mapping. It is interesting to note that the boundary between attractive and weakly attractive behavior coincides exactly with the boundary separating types I1 and I11 from types IIA and IIIA (where A indicates that the mixture forms an azeotrope). Here we see that, for systems for ) ' / ~ , behavior is possible which 4 # 0 and uI2= ( a , , ~ ~ ~attractive only for mixtures that are of type I1 or I11 which do not form an azeotrope (recall that the attractive-weakly attractive boundary is always contained between the limiting curve [evaluated at T, = 1 , pr = 11 and the coordinate axes).
396 The Journal of Physical Chemistry, Vol. 95, No. 1, 1991
Petsche and Debenedetti
vi = u + (1 - xi)( 9 ) ax,
P,T
where xi is the solute mole fraction and u is the molar volume of the mixture. In the infinite dilution limit we have the thermodynamic identity lim
''I
ATTR~CTIVE '. II
(*) P,T
-:E
-:6
-:4
(47)
(%) 3x1 v,T
Recall that the infinite dilution partial molar volume may also be written as
\
.1
-?'.O
= uKT XI+ lim
-:2
>
