The Journal of
Physical Chemistry
~~
~~
0 Copyright, 1992, by [he American Chemical Society
VOLUME 96, NUMBER 13 JUNE 25, 1992
LETTERS Novel Critical Transitions in Ternary Fluid Mixtures Richard J. Sadus Computer Simulation and Physical Applications Group, Department of Computer Science, Swinburne Institute of Technology, PO Box 218, Hawthorn, Victoria 3122, Australia (Received: November 26, 1991; I n Final Form: April 21, 1992)
Recent calculations indicate that a diverse range of critical equilibria are manifested by ternary fluid mixtures. A critical transition is frequently detected between two distinct two-phase regions of equilibria. That is, instead of two phases changing to one homogeneous phase, the critical transition results in the emergence of a different type of equilibria involving another two phases. Different types of transitions are observed depending on whether gas-liquid or lower or upper critical solution equilibria are involved. It is conceivable that some of the reported tricritical equilibria of ternary mixtures or quasi-binary fluids are instead a manifestation of one of these transitions.
Introduction A critical transition occurs when the properties of coexisting phases become identical. The critical properties of either pure substances' or binary mixtures* have been extensively investigated. Indeed, the different types of critical equilibria of binary mixtures forms the basis of a widely accepted classification3 of the phase equilibria of binary fluid mixtures. The different aspects of critical equilibria in binary mixtures invariably involve only two phases. However, a critical transition involving three or more phases can be envisaged in multicomponent mixtures. Extensive calculations have been recently undertaken4 on the nature of the critical equilibria exhibited by ternary mixtures. The investigation of critical phenomena in ternary mixtures frequently identified a transition which does not conform to either normal two-phase criticality or multiphase criticality as implied, for example, by a tricritical point. Instead, a transition is observed between two different dual-phase equilibria. This phenomenon is commonly reported as the experimental manifestation of a tricritical point. It is correctly argued that the observation of the simultaneous disappearance of three phases is difficult to achieve because it occurs at a unique set of physical properties of a ternary mixture. However, it appears plausible that at least in some instances, the 0022-3654/92/2096-5 197$03.00/0
designated tricritical point is in fact a manifestation of a more general class of critical phenomena involving only two phases. It appears likely that a critical transition involving the crassover of two different types of two-phase equilibria is a common aspect of the phase behavior of many ternary and, by extension, multicomponent fluids. The transitions can be categorized in terms of the different types of equilibria involved, Le., between liquid and gas phases or different liquid phases. The type of behavior observed depends on the relative miscibility of the constituent binary mixtures, and the different categories of critical transitions may eventually be useful in developing a general classification for the phase behavior of ternary systems. Critical Transitions in Ternary Mixtures Theory. Previous worksz6 discussed below on critical equilibria in ternary mixtures has identified tricritical phenomena mainly in very carefully selected quasi-binary mixtures. A genuine tricritical point involving the simultaneous transformation of three phases to form a single phase can be observed only in the most experimentally fortuitous circumstances. This can be readily appreciated from the requirement imposed by the phase rule that such a transition is unique to any mixture of three components. 0 1992 American Chemical Society
‘2
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Figure 1. Triangular representation of the composition of a ternary mixture showing the components at each apex and the region of constant x (see text). Cross-sectional profiles of the critical surface of the ternary mixture are obtained by calculating the critical properties at a fixed ratio of component 1 relative to component 2.
represent cross-sectional profiles in the ternary surface and they should not be confused with “critical loci” as manifested by binary fluid mixtures. In addition to normal critical equilibria (Le., involving the transformation of two phases to form a single homogeneous phase) and tricritical equilibria (i.e., the simultaneous transformation of three phases into one phase) a critical transition was often detected whereby two phases became critical with the simultaneous formation of an alternative two-phase region. It is proposed to develop a classification of different types of critical transition based on this division. Of course, a normal transition remains the most common type of critical transition in a ternary mixture. However, it is often necessary to distinguish it from other possibilities such as dual two-phase transitions and multiphase transitions. The terms tricritical, tetracritical, etc., should only be used to denote critical transitions involving the speclfed multiple of phases. These multiphase critical points will occupy the class 1 category corresponding to higher order critical phenomena. Therefore, the classification scheme is likely to retain its generality irrespective of what is subsequently discovered for other multicomponent mixtures. For example, the first opportunity to observe a tetracritical point is in a four-component mixture. Consequently, by including all multiphase transitions in one category, the need for a separate classification scheme for four-component fluids is avoided. Any new phenomena can be incorporated into a new class or subclass within the existing framework. The evidence for the assignment of the critical transition is the same irrespective of the category. Two stable critical points were identified over a narrow range of composition of the critical surface. The difference in the critical volume, temperature, and pressure of these points is progressively diminished as the transition composition is approached whereupon they become identical (Figure 2). It will be recalled that two stable critical points are also sometimes located along the main critical locus, which is characteristic of type I11 phenomena in binary mixtures. However, the distinct critical properties do not approach each other as the composition is varied, and the transition between “gas-liquid” and “liquid-liquid” phenomena (along the p, T minima) occurs over a range of compositions. Three other distinct types of critical phenomena were identified on the basis of the above criteria, and the following classification was adopted: (a) Class 1: Multiphase critical points (Le., tricritical, tetracritical, etc.) involving three or more phases undergoing a simultaneous transition to produce a single homogeneous phase. In the case of a tricritical point, the two menisci signifying the phase boundaries disappear simultaneously. ( b ) Class 2: Lower critical solution phenomena (LCST) and a gas-liquid critical transition meet at relatively low temperatures. A critical transition occurs transforming the two-phase liquidliquid equilibrium into a two-phase gas-liquid equilibrium. That is, the meniscus between the two liquid phases becomes critical
The Journal of Physical Chemistry, Vol. 96, No. 13, 1992 5199
Letters
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400
450
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Figure 2. Critical temperature, pressure and volume of a ternary mixture (at a constant x value, Le., cross-sectional profile of the ternary surface) approaching a class 2 transition denoted by t. At this point all of the physical attributes of the two branches of the critical curve become identical.
simultaneously with the formation of a gas-liquid meniscus. (c) Class 3a: A region of lower critical solution temperature (LCST) and upper critical solution temperature (UCST) phenomena meet. The resulting critical transition is between the different two-phase liquid-liquid equilibria. That is, the lower critical solution meniscus disappears with the simultaneous formation of an upper critical solution meniscus. (6)Class 3b: There is a transition between two different twephase upper critical solution phenomena, Le., one upper critical solution meniscus is replaced by another upper critical solution meniscus. The phenomenological behavior of critical equilibria in the region of these categories of novel critical transitions, are qualitatively illustrated in Figures 3 and 4. Examples of class 2 transitions are found in mixtures containing either water or nitrogen as one component, whereas class 3a and class 3b behavior can be found in mixtures containing carbon dioxide. On first inspection some of the other phase transitions appear to be candidates for tricriticality. This possibility can be excluded on the basis of phenomenological evidence and the phase rule. The ordinary relationship between the number of degrees of freedom f,phases p , and the components c of a fluid is
Figure 3. Critical profile of a ternary mixture at a constant value of x exhibiting a class 2 transition (t) at low temperatures. The region close to the transition point is examined in greater detail in the inserts. The points denoted c l c2 and c3 represent the gas-liquid critical points of a binary mixture of components 1 and 2 and the critical point of component 3, respectively.
+
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(7) However, it is evident that a critical transition has fewer degrees of freedom than noncritical equilibria. For example, the phase rule assigns one degree of freedom for the vapor-liquid equilibrium of a pure substance, Le., the vapor pressure of the fluid can be altered by changing the temperature until the critical point is attained. The critical point is a unique, invariant property of the pure substance. The conditions for a critical point require identical physical properties in the coexisting phases. This imposes p - 1 additional constraints which further reduce the number of degrees of freedom. The result of accounting for these additional constraints alters the phase rule to f - c - p 2 - 0,- 1) = c - 2p 3 (8)
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Figure 4. Critical temperature profile of a ternary mixture at a constant value of x exhibiting a class 3a transition ( t l ) at low temperatures and a class 3b transition (t2) at high temperatures and pressures. The insert examines the region near the class 3b transition in greater detail. Notice that in this instance that the two types of critical transitions occur on the same critical profile.
Consequently, the minimum number of components required to observe critical phenomena involving three phases is three. The tricritical point (of three phases) is invariant in a ternary mixture, whereas a four-component fluid has an additional degree of freedom enabling a line of tricritical points to be observed. The
5200 The Journal of Physical Chemistry, Vol. 96, No. 13, 1992
phase rule can be expected to apply to all systems with the exception of those few examplesS of "special symmetry" such as binary 3He 4He mixtures, paramagnets, or possibly a mixture of dextro and levo enantomers. As discussed below, class 2 and 3 transitions are generally not invariant properties of the fluid and as such they do not correspond to a tricritical transition. Class 1 transitions: A class 1 transition corresponds to a higher order critical point as postulated by van der Waalse6 The first substantial experimental studies of such transitions are found in the Russian EfremovaIz is attributed with the first experimental observation of tricritical transition in a ternary mixture. WidomI3 has provided a historical account of the early work on tricritical phenomena. Knobler and Scotti4have summarized the data for several ternary systems, and the coordinates for a tricritical line in the quaternary carbon dioxide water methanol ethanol mixture. Experimental studies of a small molecule a series of n-alkanes indicate that a tricritical point is obtained for some hypothetical alkane molecule with a noninteger number of carbon atoms. The existence of a tricritical point in unsymmetrical binary fluid mixtures is purely hypothetical because it is specifically precluded by the phase rule. The most extensive studies of tricriticality have involved quasi-binary mixtures. The quasi-binary mixtures are actually three-component fluids (required by the phase rule) in which two of the components have very similar chemical and physical properties. At least one of the binary subsystems which constitute the ternary mixture is a type IV system. The rationale is to contract the region of liquid-liquid-gas equilibrium (Le., bring the lower critical end point (LCEP) and upper critical end point (UCEP) together by adding a suitable third molecule which is typically an isomer of one of the other alkane molecules. A similar technique could also be applied to suitable type I11 systems. Specovius et al.Is and Goh et al.16J7examined the LCEP and UCEP of a series of ethane n-alkane mixtures. The difference between the LCEP and the UCEP becomes progressively reduced between carbon numbers 18-20. Consequently, a tricritical point could be inferred in the quasi-binary ethane + (n-hexadecane + n-eicosane) mixtures1' Peters et a1.I8 have also postulated a hypothetical tricritical point in propane higher n-alkane mixtures at a carbon number between 29 and 30. A three-phase critical point has been r e p ~ r t e dinl ~quasi-binary ~~~ mixtures of methane (n-pentane 2,3-dimethylbutane) and methane (2,2-dimethylbutane + 2,3-dimethylbutane) mixtures. Recently, tricriticality has been observed in polymer mixtures.2' The experimental tricritical points of many of these systems have been compared with calculated values obtained from the quasi-binary theory originally proposed by GriffithsZ2and later extended by An interesting conclusion of this analysis24 is that unlike conventional two-phase critical points, the tricritical points behave classically, Le., the value of the critical exponent is close to the classical value. Indeed, even the simple van der Waals equation can be accurately utilized to study tricritical phenomena. The quasi-binary mixture approach provides a useful method of identifying tricritical points and the theoretical analysis is much simpler than genuine ternary calculations. However, the type and scope of tricritical phenomena that can be studied by this approach is very limited. All of the quasi binary mixtures have by necessity involved a type IV system. In principle, a tricritical point involves the simultaneous transformation of three phases to form a single homogeneous phase of at least three components. This phenomenon should be experimentally manifested by the simultaneous disappearance of two menisci, compared with the disappearance of a single meniscus for a normal critical transition. However, some experimental critical transitions have been designated as "tricritical" points which neither involve three phases nor conform to normal twophase criticality. Lindh et aLZ5have identified three different pathways to a "tricritical" point. They are (i) the simultaneous disappearance of three different phases, (ii) a transition between two different two-phase equilibria, and (iii) the volume of one of two phases contracts to zero. Case i involves three phases which constitute a very rare occurrence, whereas case iii is a calculated
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Letters path which has not been experimentally observed. The second choice has been experimentally observed, and it appears from this work to be a very common feature in ternary mixtures. ScottS has also remarked that case i is very rare and that case ii is much more common. It is misleading to describe the latter transition as a tricritical point because three phases are not involved. Equally, it is clearly not a normal transition. In the context of this work the case ii path behaves like either a class 2 or class 3 transition. Class 2 transitions: The Occurrence of a class 2 transition involves the junction of a gas-liquid critical surface and a surface of lower critical solution equilibria. Figure 3 illustrates a cross-sectional profile of this surface and the equilibria should not be confused with critical lines as manifested in the equilibria of binary mixtures. The concept of a lower critical solution surface is unfamiliar and requires some explanation. Lower critical solution equilibria play a relatively minor role in the phase behavior of binary mixtures. It is commonly manifested in type IV binary mixtures as a lower critical end point (LCEP) on the end of a three-phase line. However, in a ternary mixture, unmixing phenomena occurs over a range of composition, generating a critical surface analogous to the ternary upper critical surface between two component binary mixtures of limited miscibility. The pressure-temperature profile (Figure 3) can give the misleading impression that the transition is a tricritical point. This interpretation is generlly incorrect. A tricritical transition in a ternary mixture is associated with zero degrees of freedom (see eq 9) and must occur at a unique value of composition, temperature, pressure, and volume, whereas the class 2 transition is observed over a range of different compositions, temperatures, and pressures. That is, a line of class 2 transitions can be identified linking different cross sections of the ternary critical surface corresponding to the junction between the lower critical surface and the gas-liquid critical surface. A minimum of four components are required in order to observe a tricritical line. Clearly, the h e of class 2 transitions cannot constitute tricritical equilibria. Critical equilibria involving three components and only two phases have additional degrees of freedom which permits the formation of a critical line as observed for the class 2 transition. However, the calculations do not exclude the possibility of a unique tricritical point occurring somewhere on the class 2 transition curve. In this context it is of interest that experimental investigations of tricritical phenomena typically do not report the simultaneous disappearance of two menisci. Instead, the disappearance of one meniscus occurs simultaneously with the formation of a new meniscus. This observation is consistent with a class 2 transition instead of a genuine tricritical (class 1) transition. However, a tricritical transition can be envisaged at a uniquepoint on the line of class 2 transitions. Some experimental investigations of tricritical equilibria in ternary mixtures probably observe a nontricritical transition along the. class 2 critical line instead of a genuine tricritical point. Class 3 transitions: The class 3 category of transitions involves only equilibria between different liquid phases. Two distinct transitions can be identified involving either lower and upper critical solution equilibria or exclusively upper critical solution equilibria. In both cases, the transition can involve only two phases. The transition occurs as part of the critical surface which divides two-phase coexistence from one homogeneous phase. The transitions were not located at a unique set of physical properties but instead occurred over a range of compositions, temperatures, pressures, and volumes. That is, the transitions were identified for several cross sections of the ternary critical surface. Consequently, a transition involving three phases is also precluded by the phase rule. The different categories of class 3 transitions were observed to occur either in isolation from each other or together on the same profile (see Table I). The occurrence of either a class 2 or class 3a transition is clearly apparent on the pressure-temperature profile as an abrupt intersection between the two different critical surfaces. However, the class 3b transition is very difficult to detect because it occurs over a much narrower range of composition. The critical cross-
The Journal of Physical Chemistry, Vol. 96, No. 13, 1992 5201
Letters
TABLE I: Summary of Critical Phenomena in Ternary Mixtures Observed in This Work" 1
mixture components 2
CHI
n-C6H I 4
C2H6
n-C4H10
n-C6H14 n-C6H14 n-C6H14
n-C5H I2
n-C6Hl4
n-C7H16 n-C8H16
n-C6H14 n-C6H14
n-C9H20
n-C6H14
n-GoH22 CH4 C3H8
n-C6H14
C3H8
C3H8 ClH8
CH4 CHI
C2H6 n-C5H12 C2H6 C2H6
C2H6
n-C5H12 C2H6
C2H6
i-C4HI0 n-C4H10 n-C5H12
C2H6 n-C5H12
N2 N2 C3HB n-C5H12 n-C6H14 n-C6H14
co2
3 CO2
co2 co2 co2 co2 co2 co2 co2 co2 CF4 CF4 CF4 CF4 N2 N2 H20 H20 H2O H2O H2O H20 H20
1&2 IV
binary phase type 1&3
I
1/11
I
1/11 1/11 1/11 1/11 1/11 1/11 1/11 I 1/11 1/11 1/11
I
I I
I
I11 I11 1/11 1/11 1/11 1/11
I
I1 I1 I1 I1 I1 I1 I1 I1 I1 I1 I1 I11 I11 I11 I11 I11 111 I11
critical transition 1(?) & 2 normal normal 3a 3a 3a & 3b 3a & 3b 3a & 3b 3a & 3b normal 3a 3a & 3b 2 2 & 3b 2 2 normal 2 & 3b 2 2 2 2 & 3b
2&3 I1 I1 I1 I1
I1 I1 I1 I1 I1 I1 I1 I1 111 I11 I11 111 I11 I11 I11 111 111 I11
comments tricritical
on the same profile on the same profile on the same profile on the same profile
on the same profile
'The mixtures denoted type 1/11 are formally type I1 mixtures, but any possible liquid-liquid separation occurs at such low temperatures that for practical purposes they can be considered to be in the type I category.
sectional profile appears identical to a type I1 UCST locus of a binary mixture. The distinction only becomes apparent when the variation of a composition along the curve is examined. There is a continuity of composition along the liquid-liquid line of type I1 phenomena with only one stable critical point at each composition. In contrast, there are two critical points at the same composition close to the class 3b transition. The identification of gas-liquid, lower and upper critical solution properties in the above categories is based on the physical properties of the calculated critical points (e.g., the gas-liquid critical volume is usually considerably larger than its critical solution counterpart) and phenomenological evidence (e.g., the pressure along a UCST locus is usually very sensitive to a small change in temperature). Class 3 phenomena were interpreted as involving a transition between two different two-phase equilibria rather than a simultaneous transition involving three phases. The criteria for assigning a class 3 transition does not preclude a multiphase class 1 transition. However, the above interpretation is supported by phenomenological evidence, and it is also a requirement of the restriction imposed by the phase rule. In the case of class 3a and class 3b behavior, the existence of three phases, and therefore a tricritical transition, is incompatible with the critical surface representing the boundary between the two-phase and one-phase region. The fact that this transition is also located at other values of x also precludes three-phase coexistence. It will be recalled that a tricritical point of three phases must be invariant. However, if only two phases are involved, then the transition can be located at other values of x , temperature, and pressure. An unusual feature of ternary phenomena is the possibility of two classes of transitions along the same critical profile. This can occur when an UCST surface bisected by a class 3b transition meets a surface of LCSTs (Figure 4). The variation of critical volume with respect to composition is particularly noteworthy (Figure 5 ) . There is very little variation in the critical volume along most of the profile except for the region close to the transition region where the volume changes rapidly. Summary of Novel Critical Transitions in Ternary Fluids. The data collected in Table I summarize the occurrence of novel critical transitions studied. There is a reasonable variety in the type of mixtures studied, although at least one alkane molecule is common to nearly all of the systems. In every case, one of the components is a relatively small molecule such as carbon dioxide or nitrogen. Consequently, there is a greater variety in the phase behavior types of the constituent binary mixtures than would otherwise be the case if all of the components were of similar size. It would be
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0.69
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Figure 5. Variation of critical volume along a critical profile containing two transitions as exemplified by Figure 4. The insert illustrates the region of the class 3b transition in greater detail. See Figure 4 for legend.
unwise to make any sweeping generalizations about the phase behavior of ternary mixtures from such a limited number of mixtures. Nonetheless, some clear trends are apparent. The ternary mixture must contain at least two pairs of molecules with limited miscibility in order to exhibit class 1-3 critical phenomena. It is notable that these transitions are not observed for mixtures in which two of the constituent binary mixtures are type I systems. This probably also precludes these type of critical transitions in mixtures containing nominally type I1 systems but where any liquid-liquid separation would be expected to occur at temperatures very much less than the critical temperature of the least-volatile component, e.g., alkanes of modest (i.e., C < 10) molecular weight. There is also a greater variability in the type of transition as the constituent molecules become progressively
5202 The Journal of Physical Chemistry, Vol. 96, No. 13, 1992 different. This is particularly apparent in the mixtures containing carbon dioxide. Dual transitions on the same critical profile are only evident in mixtures of type I1 binary systems. A class 2 transition between a gas-liquid line and lower critical solution temperature phenomena apparently requires either a type 111 or IV system. A type IV system also appears to be a necessary requirement for the existence of a multiphase (class 1) transition.
Experimental Evidence How would the above categories of critical phenomena be manifested experimentally? If the sealed tube method29of observation is adopted, then the disappearance of one meniscus accompanied simultaneously by the formation of another meniscus can be expected at the transition point. In the case of class 3a and class 3b phenomena, the two menisci represent two different liquid-liquid phenomena, whereas the liquid-liquid interface is replaced by gas-liquid equilibria in class 2 behavior. This extraordinary crossover between the different two-phase regions has been observed experimentally by Efremova and Shvartsl’ for the carbon dioxide + methanol + water mixture. They observed that below the transition point “only two phases and consequently one meniscus at the interface are visible. On raising the temperature, the meniscus becomes flat, critical opalescence appears, and at the system temperature corresponding to the (critical point) the meniscus vanishes. However, the disappearance of the meniscus does not occur in the centre of the tube as in the ordinary critical phenomena but in the lower third of the tube. Simultaneously a second critical meniscus appears in the upper third of the tube. On further raising of temperature, the upper meniscus becomes noncritical.” This description is completely consistent with the interpretation of the critical phenomena calculated in this work. However, it is in contrast to their earlier measurements9 for the same system in which they came very close to observing a genuine tricritical point. They reported9 “In one of the experiments the gap between the liquid-liquid and liquid-gas critical temperatures was only 0.03 OC; at 44.53 O C critical phenomena were observed between the two liquid phases in the presence of the gas phase, the pressure being 87.3 atm and the volume 62.8 cm3mol-]. On the other hand at 44.56 OC critical phenomena between the liquid and gas phase in the presence of the heavy liquid phase were observed. The pressure was the same as before and the volume changed by only 0.6 cm3 mol-I.” Efremova and Shvarts believed that they were observing a tricritical point on both occasions although they did not believe that the former behavior was typical of tricritical phenomena in general. Later, Lindh et al.25speculated that a small three-phase region had been overlooked. In the context of this work, it is apparent that only the latter observation is a possible tricritical transition, whereas the former observation describes a class 2 transition. It is also possible that other reported measurements of tricritical phenomena are actually manifestations of class 2 behavior. The literature contains no experimental data to either directly support or refute the existence of class 3a and class 3b. This is hardly surprising in view of the limited amount of data available for ternary mixture critical phenomena in general. It is easy in principle to devise an experiment to examine class 2 phenomena. It should be possible to use the sealed tube method,29 although allowance must be made to observe the entire length of the tube and the composition must be carefully determined. Class 3b phenomena, which occur over a very narrow range of composition, will be more difficult to detect. It should be noted that work on double critical points in binary mixtures bears some similarity to the observed novel transitions. In the context of a binary mixture, a double-critical point can be inferred when the main type 111 critical locus just touches a three-phase liquid-liquid-gas line.30 Recently, de Loos et aL3’
Letters measured the composition, temperature, and pressure of the UCST locus of tetrafluoromethane + n-butane. Two critical points were detected at the same composition along the liquid-liquid curve. The two distinct equilibria appear to reach a common value of both temperature and pressure at an identical composition. The critical volume was not measured. The parabolic variation of critical temperature with respect to composition in the vicinity of a higher order transition has also been experimentally observed by Narayanan et al.32for 3-methylpyridine + water + heavy water, near a double-critical point.
Conclusion The calculation of critical equilibria in ternary fluid mixtures indicates that some navel transitions are possible between different types of twephase equilibria. These transitions can form the basis of a classification of phase transitions in fluid mixtures. In view of these calculations, it is plausible that some of the experimentally observed “tricritical” points are instead manifestations of a more general class of transition. References and Notes (1) Ambrose, D. Vapour-Liquid Critical Properties; National Physical Laboratory: Teddington, U.K., 1980. (2) Hicks, C. P.; Young, C. L. Chem. Rev. 1975, 75, 119. (3) van Konynenburg, P. H.; Scott, R. L. Philos. Trans. R. SOC.London 1980, 298A, 495. (4) Sadus, R.J. High Pressure Phase Behauiour of Multicomponent Fluid Mixtures; Elsevier: Amsterdam, 1992. (5) Scott, R. L. Acc. Chem. Res. 1987, 20, 97. (6) Rowlinson, J. S., Ed. J.D. van der Waals: On the Continuity of the Gaseous and Liauid States. North-Holland: Amsterdam. 1988. (7) Radyshedkaya, G. S.; Nikurashina, N. I.; Mertslin, R. V.Zh. Obshch. Khim. 1962, 32, 673 ( J . Gen. Chem. USSR 1962, 32, 673). (8) Krichevskii, I. R.; Efremova, G. D.; Pryanikova, R. 0.;Serebryakova, A. V. Zh. Fiz. Khim. 1%3,37, 1924 (Russ. J. Phys. Chem. 1%3,37, 1046). (9) Efremova, G. D.; Shvarts, A. V. Zh. Fiz. Khim. 1966,40,907 (Russ. J . Phys. Chem. 1966, 40,486). (10) Myasnikova, K. P.; Nikurashina, N. I.; Mertslin, R. V. Zh. Fiz. Khim. 1969, 43, 416 (Russ. J . Phys. Chem. 1969, 43, 223). (1 1) Efremova, G. D.; Shvarts, A. V. Zh. Fir. Khim. 1%9,43,1732 (Russ. J . Phys. Chem. 1969,43, 968). (12) Krichevskii, I. R.; Khazanova, N. E.; Tsekhanskaya, Yu.V.;Linshits,
L. R.; Pryanikova, R. 0.;Sokolova, E. S.; Khodeeva, S. M.; Shvarts, A. V.; Sorina, G. A.; Semenova, A. I. Zh. Fir. Khim. 1972,46,2451 (Russ. J . Phys. Chem. 1972, 46, 1411). (13) Widom, B. In Paulaitis, M. E., Penninger, J. M. L., Gray Jr., R. D., Davidson, P., Eds.; Chemical Engineering at Supercritical Conditions; Ann Arbor Science: Ann Arbor, MI, 1983. (14) Knobler, C. M.; Scott, R. L. In Domb, C., Lebowitz, J. L., Eds.; Phase Transitions and Critical Phenomena; Academic Press: London, 1984; Vol. 9. (15) Specovius, J.; Leiva, M. A,; Scott, R. L.; Knobler, C. M. J . Phys. Chem. 1981,85, 2313. (16) Goh, M. C.; Specovius, J.; Scott, R. L.; Knobler, C. M. J . Chem. Phys. 1987,86, 4120. (17) Goh, M. C.; Scott, R. L.; Knobler, C. M. J . Chem. Phys. 1988,89, 2281.
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