Ind. Eng. Chem. Res. 1988,27, 664-671
664
Kim, C. H.; Vimalchand, P.; Donohue, M. D.; Sandler, S. I. AZChE J. 1986,32, 1726. Knapp, H.; Doring, R.; Oellrich, L.; Plocker, U.; Prausnitz, J. M. VapopLiquid Equilibria for Mixtures of Low Boiling Substances; DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, 1982;Vol. VI. Lee, K. H.; Lombardo, M.; Sandler, S. I. Fluid Phase Equilib. 1985, 21, 177. Maloney, D. P.;Prausnitz, J. M. J. Appl. Polymer Sei. 1974,18, 2703.
Panagiotopoulos, A. Z.; Reid, R. C. ACS Symp. Ser. 1986,300,571. Rigas, T.J.;Mason, D. F.; Thodos, G . J . Chem. Eng. Data 1969,4, 201. Vairogs, J.; Klekers, A. J.; Edmister, W. C. AZChE J. 1971,17, 308. Yarborough, L. J . Chem. Eng. Data 1972,17,129. Received f o r review June 17,1987 Revised manuscript received November 3, 1987 Accepted November 20, 1987
Thermodynamic Excess Properties in Binary Fluid Mixtures Keshawa P. Shukla, Ariel A. Chialvo, and James M. Haile* Department of Chemical Engineering, Clemson University, Clemson, South Carolina 29634
This paper illustrates how a plot of the excess Gibbs free energy versus excess enthalpy can be used to organize our knowledge of the nonideal solution behavior in binary liquid mixtures. Such a plot is particularly suited to presenting temperature effects, and we therefore use the plot to show (a) how temperature changes influence the strength and kind of nonideal solution behavior and (b) how temperature changes promote various kinds of liquid-liquid immiscibility. We also show that such a plot can be used to quantify how changes in intermolecular forces affect nonideal solution behavior. T o accomplish this, statistical mechanical perturbation theory calculations of the excess free energy and excess enthalpy were performed on a number of Lennard-Jones mixtures. The calculations were performed at three state conditions and over a range of size ratios 1 I g g g / g u I 2 and over a range ~ 4. The results demonstrate how energetic and entropic effects can of energy ratios 0.2 I E B B / E I either compete or cooperate to produce particular kinds of nonidealities. 1. Introduction
The myriad varieties of thermodynamic and phase equilibrium behavior displayed by fluid mixtures present a significant challenge to the organizational concepts of classical and statistical thermodynamics. Attempts to organize the observed equilibrium behavior of fluid mixtures have been couched in both molecular and thermodynamic terms. Molecular schemes classify mixtures according to the kinds of intermolecular forces acting among the constituent molecules (see, e.g., Rowlinson and Swinton (1982)). Thermodynamic schemes focus on a progression in the behavior of particular macroscopic properties. For example, a popular thermodynamic classification scheme is that of Scott and van Konynenburg (1970, 1980) who divide binary mixtures into six primary classes, according to the behavior of mixture critical lines in pressure-temperature space. Another system of thermodynamic classification can be based upon the thermodynamic excess properties. For a generic, extensive thermodynamic property M , the excess property ME is defined by M E = M - Mis (1) where M is the mixture property at a chosen temperature, pressure, and composition and Misis the ideal solution value of M at the same state condition. The appeal of (1) is that M" contains ideal gas contributions plus entropic contributions due to the distinguishability of molecular species, so that the excess property ME contains only contributions originating from differences in intermolecular interactions. As far as we are aware, the first systematic classification of binary mixtures based on excess properties was made by Malesinski (1965). Subsequently, the approach has evolved along the lines of fiding the most informative pair of excess properties to plot on a set of two-dimensional, orthogonal axes (Kauer et al., 1966). This evolution culminated in the work of Gaube and his colleagues (Gaube and Koenen, 1979; Kohler and Gaube, 1980; Koenen and 0888-5885/88/2627-0664$01.50/0
Gaube, 1982) who collected from the literature excess property data for about 200 equimolar binary mixtures and presented the data on a plot of g E versus hE. Here g E is the excess Gibbs free energy and hE is the excess enthalpy. The gE-hE diagram embodies, in a particularly informative way, the 12 classes of binary mixtures identified by Malesinski. In this paper we attempt to provide a comprehensive summary of the kinds of mixture behavior that can be deduced from such a diagram. In the next section we describe the g E-hEdiagram, including the mixture characteristics represented by its principal features. We then show the kinds of trajectories that a particular mixture can follow on the diagram in response to a change in temperature. The analysis of temperature effects leads to consideration of liquid-liquid immiscibility and the g E-hEdiagram proves particularly suited to describing the behavior of binary mixtures that exhibit either closed solubility loops or miscibility gaps. In the last section we present gehE diagrams for mixtures of Lennard-Jones (U) atoms, for which we have calculated the excess properties via statistical mechanical perturbation theory. The diegrams resulting from those calculations provide new information on how changes in the LJ size and energy parameters affect excess properties in simple mixtures. 2. The g E - E~ Diagram Classical thermodynamics gives the following fundamental relations among the excess properties for closed systems: gE = hE - TsE (2) sE = - ( a g E / a n p ,
(3)
hE = - T ~( a ( g E / T ) / a n p X (4) cpE= ( a h E / a q p X= T ( a S E / a n p , = -T ( a 2 g E / a T 2 ) , (5) In these equations Tis the absolute temperature, s is the 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 665 hE
hE
I0
/
ID0
Figure 1. Basic g diagram with ita six principal regions. The three regions IB, 11, and IIIA lying above the diagonal have sE> 0 and hence dgE/aT < 0. Conversely, regions IA, IIIB, and IV below the diagonal have sEC 0 and hence agE/aT > 0.
entropy, and C p is the isobaric heat capacity. The subscripts P and x mean the derivatives are to be taken at constant pressure and constant composition. Figure 1shows the basic gE-hEdiagram. Mixtures lying on the righbhalf plane have g E values that exceed the ideal solution value; e.g., in vapor-liquid equilibrium situations, if such mixtures have azeotropes, they will be minimum boiling point azeotropes. Conversely, mixtures in the left-half plane that have azeotropes have only maximum boiling point azeotropes. Mixtures lying at the origin and having zero excess volume (uE = 0) are ideal solutions. Mixtures lying on the abscissa (hE= 0) are the athermal solutions of Guggenheim (1952), while mixtures lying on the ordinate (gE = 0) but not at the origin Malesinski termed pseudoideal solutions. From (2) we see that the diagonal (gE = hE)has sE = 0. A mixture lying on the diagonal in the first quadrant (gE= hE> 0 ) and also having uE = 0 is a regular solution of Hildebrand (Hildebrand et al., 1970). From (2) we find that points lying in the half-plane above the diagonal, Le., points in quadrant I1 and in octants IB and IIIA, have sE > 0. Conversely, points in the half-plane below the diagonal, i.e., points in quadrant IV and in octants IA and I W , have sE< 0. From (3), knowing these signs for sEalso gives us the signs for the temperature derivative of g E . The three lines g E = 0, hE = 0, and sE = 0 divide the diagram into six regions, which we label as octants IA and IB, quadrant 11, octants IIIA and IIIB, and quadrant IV. Unfortunately, the diagram gives no information on the temperature derivative of hE;i.e. CpE can be of either sign anywhere on the plot. Thus, we may conceive two such diagrams, one for CpE > 0 and another for CpE < 0. The 12 regions on these two diagrams correspond to the 12 kinds of binary mixtures identified by Malesinski (1965). The diagram immediately conveys the dominant source of nonidealities in g E , i.e., whether nonideal values of g E primarily originate from entropic or energetic effects. Thus,in quandrants 11and IV,the sign of g E is determined by sEand the nonideality of g E is primarily entropic. In octants IB and IIIB, the converse occurs. In octants IA and IIIA, both energetic and entropic effects contribute to nonideal values of g E . The great majority of real binary mixtures fall in the first quandrant of the diagram (Kohler and Gaube, 1980; Koenen and Gaube, 1982). 3. Temperature Effects
From (4), hEprovides the response of g E to an isobaric
I0
IIIB CFrO Figure 2. Three classes of trajectories that a binary mixture could follow in response to increasing temperature at fixed pressure and composition. These trajectories assume the mixture has CpE > 0 at all temperatures. Arrows indicate the direction of increasing T.
change in temperature, while CpE, from (5), gives the curvature of the variation of g E with temperature. Thus, the gE-hEdiagram is particularly suited to describing how changes in temperature affect nonideal solution behavior. We consider here the effects of isobarically increasing the temperature on a binary mixture of constant composition. The response is shown as a trajectory on the gE-hEdiagram. We divide the response into two types, depending upon whether CpE is positive or negative. CpEPositive. For a mixture with positive excess heat capacity, we identify three classes of temperature trajectories, which are shown in Figure 2. If the mixture falls in octant IA, then both ag E/dTand dhE/dTare positive and increasing the temperature moves the mixture away fom the origin-it becomes more nonideal. In addition, if agE/dT> dhE/dT,then the mixture remains in octant IA as T increases; this situation is labeled trajectory 1 in Figure 2. However, if dgE/dT< ahE/dT,then the mixture moves toward the diagonal sE = 0 and it may cross into octant IB if Tis increased sufficiently (trajectory 2). At the point of crossing the diagonal, g E is a maximum with respect to temperature and the slope of the trajectory on the gE-hE diagram must be infinite; Le., because of (3)
(ahE/8gEip,= (ahE/anpx(aT/agE)px =m
(6)
Since we are assuming here that CpE > 0, by (5) the extremum in g E versus T must be a maximum at sE = 0. In octant IB, agE/aT < 0, so that increasing T moves the mixture toward g E = 0, and if T increases sufficiently, the mixture may cross g E = 0 and enter quadrant 11. With CpE > 0, a mixture in quadrant I1 cannot leave that quadrant in response to an increase in temperature; it only moves further from the origin, exhibiting stronger negative deviations from ideality. A mixture in octant IIIA has dgE/aT< 0, so increasing T moves the mixture toward hE = 0 and at sufficiently high T , it can leave IIIA and enter quadrant 11. We label this behavior as trajectory 3 in Figure 2. In octant IIIB of the diagram dgE/aT> 0, we are still assuming ahE/dT> 0, and therefore mixtures lying in this region have two possible trajectories. (a) If ahE/aT > agE/aT,then the trajectory moves toward the diagonal sE = 0. If it crosses the diagonal, then by the argument of (6) g E passes through a maximum and the trajectory is one
666
Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 hE
IB
Figure 3. Same as Figure 2, but for CpE < 0 at all temperatures. Arrows again indicate the direction of increasing temperature. Note the trajectories on this diagram are merely those of Figure 2 rotated 180° about the origin.
of class 3. (b) Conversely, if dhE/aT < dgE/aT,then the mixture moves toward g E = 0 as T increases. A t sufficiently high T,the trajectory may enter quadrant IV. Such a trajectory may be of either class 1 or class 2, depending on its behavior in octant IA. A variant of trajectory 3 occurs in octant IIIB if the magnitudes of dhE/dT and agE/aT are appropriately balanced; i.e., the trajectory may pass through the origin and enter the upper half-plane as T increases. Assuming CpE remains greater than zero as the trajectory passes through the origin, then g E is a maximum at the origin and the trajectory emerges into quadrant 11. Trajectories of class 3 are the only ones having CpE > 0 that can pass through the origin. Such systems, while in octant IIIB, become more ideal with increasing temperature. In quadrant IV, mixtures have dgE/dT> 0, so increasing T moves the mixture toward hE = 0. At sufficiently high T , the mixture may pass into the first quadrant, where it can be identified as either class 1 or 2, depending on its behavior there. CpENegative. For mixtures having negative excess heat capacities, three other temperature trajectories occur, as shown in Figure 3. The analyses of these trajectories correspond to those given above for CpE > 0, so we do not repeat the exercise here. Trajectory N in Figure 3 corresponds to trajectory ( N - 3) in Figure 2. Analogous to the argument of (6), when either trajectory 5 or 6 crosses sE = 0, g E passes through a minimum with temperature and the trajectory has an infinite slope at the crossing. Only trajectories of class 6 can possibly pass through the origin as T incrases, when CpE < 0. Table I collects those mixtures whose behavior, as indicated by Figures 2 and 3, consistently becomes more nonideal or more ideal with increasing temperature. We measure the strength of the nonideality by the distance All2 that a mixture lies from the origin on the g E-hEdiagram; thus, A= + (hE)2
If the measure A always changes in the same way with increasing temperature, that is, if a region of the diagram forces (aA/anpz = 2 g E ( a g E / a n p x+ 2hE (ahE/anpx= 2[-gEsE + hECpE]
Table I. Classes of Binary Mixtures Whose Nonideality Changes Uniformly When the Temperature Is Increased at Fixed Pressure and Composition region of g E-hE diagram (Figure type of behavior 1) when T increased sign of CpE CpE > 0 octant IA always more nonideal always more nonideal quadrant I1 always more ideal octant IIIB CpE < 0 octant IB always more ideal octant IIIA always more nonideal always more nonideal quadrant IV
to always have the same sign, then the region appears in Table I. CpE= 0. Real mixtures are not necessarily confined to either Figure 2 or Figure 3; Le., the sign of CpE can change as the temperature is increased. When this occurs, the mixture moves from a trajectory on Figure 2 to one on Figure 3, or vice versa. A t CpE = 0, from (5), hE and sE will be at extrema, and, from (3) and (5), g Ewill be at a point of inflection with respect to temperature. If CpE = 0 (i.e., CpE changes sign) along the diagonal sE = 0, then hE is still at an extremum and g E is still at a point of inflection with respect to temperature. In this latter case, the temperature trajectory does not cross the diagonal. In any event, the behavior of real mixtures may be more complex than that appearing in either Figure 2 or 3 since temperature trajectories can be some combination of those from both figures. 4. Liquid-Liquid Immiscibility According to classical stability analysis (Prigogine and Defay, 1954), at constant temperature, pressure, and composition, a binary liquid mixture of components A and B will spontaneously split into two liquid phases when (a2Ag/aXA2),
c, where c is a positive constant. For example, applying (10) to quadratic mixtures (those that obey gEIRT = bxAxB,where b is a temperature-dependent parameter) yields gE/RT> ' I zfor a liquid-liquid phase split to occur. Mathematically, (10) can be satisfied for negative values of gE/RT,but we are not aware of any real binaries that do so. Negative values of g E are typically caused by attractive forces among molecules of different species being
Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 667 h E / RT
hE/ RT
I
I
O/
I
4’‘
/
” I I
I
I no LLEl
‘4.
LLE gElRT = c
9EIRT.c
Figure 4. Basic gE/RT-hE/RTdiagram for identifyingliquid-liquid immiscibility in a system at fixed pressure and composition. In the upper half-plane a @ E / R T ) / a T< 0, while in the lower half-plane a(gE/RT)/aT> 0. Applying the assumptions described in the text, a binary mixture exhibits liquid-liquid equilibrium if g E / R T> c, where c is a positive constant.
stronger than those among molecules of the same species. Such interactions tend to deter phase separation. Conversely, positive values of g E are typically caused by stronger attractions among molecules of the same species; if such interactions are strong enough, a phase split occurs. To interpret liquid-liquid immiscibility on a plot of excess properties, (10) suggests that the plot should be in the variables gE/RTvs hE/RT,rather than g E vs hE. The temperature effects on this diagram differ somewhat from those on the gE-hEdiagram of Figure 1. Of importance for the present discussion is that, from (4), d(gE/RT)/dT is negative in the upper half-plane and positive in the lower half-plane. The principal features of the diagram are shown in Figure 4. Assuming gE/RT > c for liquid-liquid immiscibility, then a phase split can only occur in quadrants I and IV of Figure 4. Let us rewrite (2) in the form
g E / R T = hE/RT - s E / R
(11)
and substitute (11) into (10). The criterion for immiscibility then becomes
(
d2(hE/RT ) dX2A
(
aZ(sE/R)
)Tp-
dXA2
)Tp
1
(12)
0, isobarically increasing the temperature moves the mixture toward the immiscibility boundary and when gE/RT = c, the phase split occurs. A further increase in T drives the mixture to larger values of g E / R T ,the mixture becomes more nonideal, and the immiscibility gap in composition widens. A schematic of such a temperature trajectory is shown on Figure 5.
Figure 5. Schematic representations of liquid-liquid immiscibility in binary mixtures, induced by changes in temperature at fixed pressure and fixed overall mixture composition. UCSTs can only occur in quadrant I and LCSTs only in quadrant IV. Four possible trajectories are shown, with arrows indicating the direction of increasing temperature. Dashed lines are for d(hE/RT)/dT> 0, while long and short dashes are for d(hE/RT)/dT< 0. If the diagram is for the mixture having the critical composition, then the squares would be UCSTs and the triangles LCSTs.
If the composition of this mixture happens to be the consolute composition, then when gE/RT= c, the mixture will exhibit a critical point, and since the mixture is two phase for temperatures higher than the critical one, the critical point is a lower critical solution temperature. At the temperature of the phase split, (12) must be satisfied and hence #(sE/R)/dxA2 must be > 0; Le., sEC 0. However, note that a LCST can occur regardless of the signs of either d(hE/RT)/dTor CpE. Upper Critical Solution Temperatures (UCST). Now consider the possibility that d2(hE/RT)/axA2 is negative; Le., hE > 0. To be a candidate for liquid-liquid immiscibility, the mixture now lies in quadrant I. In this case, condition (12) can be satisfied by an excess entropy of either sign. If the mixture is initially single phase, then it lies to the left of g E / R T= c in quadrant I of Figure 5. If the temperature is now increased isobarically, the mixture moves toward g E = 0, away from the two-phase region, because d(gE/RT)/aTC 0. Thus, the temperature must be decreased to approach gE/RT= c. If the mixture happens to have the consolute composition, then at gE/RT = c the mixture has an upper critical solution temperature. As for a LCST, a UCST can occur regardless of the signs of either d(hE/RT)/aTor CpE. In summary, prouided g E ( X A ) , hE(XA), and S ~ ( X A )have no points of inflection, the conditions for binary mixtures to exhibit consolute points are UCST (quadrant I, Figure 4): g E > 0,
hE > 0 ,
sE > 0 or C 0
(13)
LCST (quadrant IV, Figure 4): g E > 0,
hE C 0 , SE c 0 (14) No restrictions apply to the signs of uE or CpE. Conditions (13) and (14) have long been available in the literature (Coop and Everett, 1953; Prigogine and Defay, 1954; Rowlinson and Swinton, 1982). The point here is that the gEIRT-hE/RTdiagram provides a useful mechanism for understanding and retaining these conditions. Closed Solubility Loops. Some binary mixtures exhibit both a UCST and a LCST; i.e., for T > TucsTand
668 Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 hE/ R T
I
hE/ R T
T
I
I
gE/RT=c
7
I XA
gE/ RT = C
x i
Figure 6. Behavior of excess properties associated with binary mixtures that form closed solubility loops. A typical T - X A diagram is shown on the right for a particular mixture a t fixed pressure. The overall mixture composition is xA0. The corresponding g E/RThE/RTdiagram shows schematically the mixture’s trajectory as the temperature is increased (at fixed pressure and overall mixture composition) from a value initially less than T L C S T If the chosen mixture composition were that of one of the consolute points, then the trajectory would pass through that consolute point.
Figure 7. Behavior of excess properties associated with binary mixtures that form miscibility gaps. A typical 7’-XA diagram is shown on the right for a particular mixture at fixed pressure. The overall mixture composition is xA0. The corresponding g E/RThE/RTdiagram shows schematically the mixture’s trajectory as the temperature is increased (at fixed pressure and overall mixture composition) from a value initially less than TucsT. If the chosen mixture composition were that of one of the consolute points, then the trajectory would pass through that consolute point.
T < TXST,the components are miscible in all proportions. If TUCST > TLcST,then the mixture is said to have a closed solubility loop. The g Ef RT-hEf RT diagram is useful for understanding the relations among the excess properties that promote formation of closed solubilityloops. Consider such a mixture, initially one phase at T TLCST. On Figure 6 we trace the trajectory followed by this mixture as the temperature is increased isobarically. Assume for this mixture that a liquid-liquid phase split occurs when g E f RT > c. Since we intend to encounter a phase split on raising the temperature, the mixture must initially fall in quadrant IV of Figure 6. Now, as T is increased, the mixture moves toward the two-phase region until at some temperature gE/RT > c and the mixture splits into two liquid phases. If the mixture has the consolute composition, then this temperature will be TLCSp Otherwise the mixture encounters the lower branch of the solubility loop, as in Figure 6. If T is increased further, gE/RTcontinues to increase, and f dT > 0, the system can cross hE = 0 provided d(hE/RT) and enter quadrant I. As hE = 0 is crossed, gE/RT is a maximum and thereafter further increases of T cause gEf RT to decrease back toward g E f RT = c. A t some T , if the condition gERT = c is again met, the mixture reverts to a single phase. For a closed solubility to occur, the mixture must have d(hE/RT) f dT > 0 , at least while in quadrant IV and for some way into quadrant I. Now (d(hE/RT)/dT)px = -hE/RT2 CpE/RT > 0 (15)
perature is raised isobarically, g E / R Tdecreases until at some temperature g E / R T = c and the mixture becomes one phase. If d(hE/RT)/dT< 0, then further increases in Twill drive the mixture toward hE = 0, at which point gE/RT is a minimum, and the mixture enters quadrant IV. Further increases in T cause g E/RTto increase back toward g E f RT = c. When this point is reached, the mixture again splits into two liquid phases. Analogous to the situation for closed loops, d(hE/RT)/aTmust be negative, at least in quadrant I and for some way into quadrant IV, for the miscibility gap to form. From (15), CpE can be of either sign in quadrant I, but near hE = 0, CpE must be negative. Whether or not a closed solubility loop or a miscibility gap forms depends crucially on (a) whether the mixture lies in quadrant I or IV at low temperatures (i.e., T TUCST and TLCST),(b) whether the mixture has gE/RT< c or > c at low temperatures, and (c) whether the sign of d(hE/ RT)/dT is positive or negative at low temperatures. Thus, if at T < TucsTand TLcsT the mixture lies in quadrant IV, has gE/RT < c , and has d(hE/RT)/dT> 0, then the mixture could exhibit a closed solubility loop. Conversely, if at T < TucsT and TLCsT the mixture lies in quadrant I, has g E / R T> c, and has d(hE/RT)/dT< 0, then a miscibility gap could possibly form. A few binary mixtures exhibit both miscibility gaps and closed solubility loops. An interpretation of this behavior in terms of excess properties can be obtained by combining the diagrams from Figures 6 and 7 .
+
suggests that CpEcould be of either sign in quadrant IV. But in the region close to and on either side of hE = 0, C p E must be positive to satisfy (15) and hence cause a closed solubility loop. Once the mixture is in quadrant I and well away from hE= 0, then d(hEf RT)/dT could become negative and the UCST still occur. But, in any case, it is more likely that a closed solubility loop will form if CpE is positive for all temperatures TucsT > T > TLcsT. Miscibility Gaps. Other binaries have both UCSTs and LCSTs, but with TucsT< TLCST. Such mixtures are said to exhibit miscibility gaps. The behavior of these mixtures on a gE/RT-hEf RT diagram is to some extent the reverse of that described above for closed solubility loops. If we start with such a mixture at temperature T < TUCST, then the mixture is already in the two-phase region of quadrant I, as shown on Figure 7 . As the tem-
5. Lennard-Jones Mixtures
In this section we describe how changes in energy and size parameters affect the excess properties in binary mixtures of Lennard-Jones (LJ) atoms. The LJ pair potential used was u,p(r) = 4 e , p [ ( ‘ ~ , ~ / r ) ’-~ ( ‘ ~ , p / r ) ~ l
Here the energy parameter cap is the depth of the a@-pair potential at its minimum and the size parameter uapis the distance at which the @pair potential crosses zero. We used the usual Lorentz-Berthelot combining rules for the unlike interactions, CAB
=
(CAACBB)”’
‘JAB
=
+ aBB1
l/,[aAA
Ind. Eng. Chem. Res., Vol. 27, No. 4,1988 669 1000
Figure 8. Behavior of excess properties in equimolar LJ mixtures at 240 K and 1047 bar, aa calculated from WCA perturbation theory (Shukla, 1987). Lines are at constant size ratios X = U B ~ / U U , with energy ratios = tBB/ tM varying along each line between the values indicated at the end points of the lines. The dotted line is the locus of points having equal energy ratios, [ = 1.
The excess properties for equimolar LJ mixtures were computed by using a form of Weeks-Chandler-Andersen perturbation theory (Weeks et al., 1971a,b) (WCA-PT) based on hard-sphere mixture reference fluids. This version of WCA-PT differs from previous versions in two important ways: (a) The reference hard-sphere diameters (du, dBB, and da) are chosen so that the blip integrals AB, satisfy EA& = 0, and simultaneously the diameters obey In this way, the fist-order correction to the reference-fluid free energy is identically zero. (b) A highly accurate equation of state, due to Boublik (1986), was used to compute the properties of the hard-sphere reference mixtures. A more detailed description of the theory in provided by Shukla (1987). By comparison with numerous computer simulation results, this version of WCA-PT has been shown to predict accurate values for the total and excess thermodynamic properties of LJ mixtures (Shukla, 1987; Shukla and Haile, 1987) over a wide range of state conditions and potential parameters. The theory was used to compute the excess Gibbs free energies and excess enthalpies for equimolar W mixtures at three state points. At each state, mixtures were studied for size ratios X = aggf au in the range 1 IX I2 and for energy ratios E = cBBfEu in the range 0.2 IE I4. Throughout the calculations, the values of the component A parameters were fixed at c u f k = 119.8 K and u u = 3.405 A. These values establish a system of units by which dimensional values of the excess properties can be obtained. The resulting g Ef hE diagrams are presented in Figures 8-11. Referring to these figures, we identify the following characteristic behavior for binary atomic mixtures: (a) = 1, X # 1. The nonideality in these mixtures is due solely to differences in atomic sizes, gE and hE are both negative, and the mixtures lie in quadrant 111. At low pressure, size differences give rise to energetic effects that dominate the nonideal solution behavior, and the mixtures lie in octant IIIB. However, increasing the pressure enhances entropic contributions and can push the mixtures into odant IIIA,where both energy and entropy contributq to the nonideality. This behavior is illustrated in Figure
I
T = 360K
P
-
1047 bar
E q u i m o l a r LJ Mixtures
,
Figure 9. Same as Figure 8, but for 360 K and 1047 bar. Key as in Figure 8.
800 600
-fo!q!o
/-Ao
1
a;-0
-4h
I
-2bo
do
2o;
600
do
i,1,0
Excess G (J/moll
Figure 10. Same as Figure 8, but for 360 K and 628 bar. hE (J/mol)
- 800
-400
gE
/ /
Figure 11. Effect of pressure on the locus of points having equal energy ratios at 360 K for equimolar U mixtures. The closed circles on each isobar are for mixtures having size ratio X = 2.
11 which shows the locus 5 = 1computed at 360 K and at three pressures: 209, 628, and 1047 bar. (b) [ > 1, X = 1. In Figure 8, for atoms of equal size (A = I), mixtures having 5: > 1 are nearly regular solutions. This behavior is consistent with regular solution theory
670 Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988
in that the theory assumes molecules of equal size. But the regular solution behavior in Figure 8 is, in fact, coincidentally a result of the chosen state condition. Thus, when the temperature and/or pressure are changed (see Figures 9 and lo), mixtures having equal sizes are no longer regular solutions. (c) 5 < 1, X 5 1.25. These mixtures are characterized by small size ratios, with the smaller atom having the deeper minimum in its potential. Such mixtures tend to lie in octant IB, where energy effects dominate entropic effects. Since g E, hE,and sEare all positive, the mixtures display positive deviations from ideality. (d) 5 > 1 , l < X I 1.25. These are again mixtures with small size ratios, but now the larger atom has the deeper potential minimum. The mixtures fall in octant IA, both energetic and entropic effeds contribute to the nonideality, and g E and hE are positive, while sE is negative. This behavior is typical of real mixtures. (e)5 < 1, X > 1.25. For large size ratios, with the deeper potential minimum on the smaller atom, the mixtures tend to lie in octant IB, quadrant 11, and octant IIIA, depending on the energy ratio 6. For small energy ratios, the mixtures occupy octant IB; for moderate energy ratios, the mixtures occupy quadrant 11; and for energy ratios near unity, size effects dominate and the mixtures lie in octant IIIA. (f) 5 > 1, X > 1.25. For mixtures having large size ratios and the deeper potential minimum on the larger atom, the tendency toward negative deviations is strong and such mixtures tend to lie in octant IIIB. For energy ratios only somewhat larger than unity, the energy effects actually seem to enhance the size differences, and the mixtures exhibit even iarger negative deviations. Large energy ratios must be imposed to overcome that tendency (note the trajectory in Figure 8 for X = 1.5, which curves back toward g E = 0 when 6 > 2.5). These same six kinds of qualitative behavior are observed if the temperature is changed isobarically (cf. Figure 9 with Figure 8) or the pressure is changed isothermally (cf. Figure 10 with Figure 9). We cannot make definitive statements about the sign of CpE at the conditions of either Figure 8 or 9, but the average behavior over the 120 K temperature range spanned by the two figures is as follows: (a) X = 1, Octant IB. CpE > 0, and the temperature trajectory is of class 2, as in Figure 2. (b) 5 > 1, X 5 1.25, Octant IA. CpE > 0, and the temperature trajectory is of class 1. (c)6 > 1, X > 1.25, Octants IIIA and IIIB. CpE 0, and the temperature trajectory is of class 5. (d) 5 < 1, X > 1, Octant IB and Quadrant 11. CpE 0, and the temperature trajectory is of class 5 . A major conclusion of Figures 8-10 is that all regions of the g E-hE diagram are accessible by manipulating the relative magnitudes of size and energy parameters in simple Lennard-Jones mixtures. Note however that few sets of LJ parameters place mixtures in quadrant IV, and no fluid LJ mixtures were found far removed from the origin in quadrant IV. When trying to move mixtures with large 6 and X values from quadrant I11 to quadrant IV, we typically encountered a phase transition before quadrant IV was reached. We caution that the behavior of LJ mixtures shown in Figures 8-11 is indicative of the behavior of real mixtures of simple molecules and cannot necessarily be used to interpret the observations of Gaube and co-workers (Gaube and Koenen, 1979; Kohler and Gaube, 1980; Koenen and Gaube, 1982) in their sampling of 200 mixtures of inorganic and aqueous/organic mixtures. Many of those mixtures involve strong polar molecules, hydrogen bonding, chain
molecules, or other kinds of structural and orientational-dependent forces that, of course, are excluded from the Lennard-Jones model of intermolecular forces. Thus, for example, although Gaube et al. find very few mixtures in quadrant IV of the diagram, we are not justified in saying that the sparsity of LJ mixtures in that region “explains” their absence in nature. 6. Conclusions
In this paper we have tried to demonstrate how plots of gEversus hEcan be used to organize our understanding of relations among thermodynamic excess properties and particular equilibrium behavior of binary mixtures. The g E-hEdiagram is particularly suited for describing temperature effects, because hEis related to the temperature derivative of g E. Thus, we have pointed out how changes in temperature can be interpreted on the diagram in terms of the type and strength of the nonidealities. Moreover, we have illustrated how the diagram helps interpret various kinds of temperature effects that promote liquid-liquid immiscibility. The diagram can only portray pressure effects in a parametric fashion, as for example in Figure 11, because the pressure derivative of g E , the excess volume, is not simply related to hE. We have also shown how the diagram can be used to correlate changes in intermolecular forces with excess properties. The exercise here was done using simple atomic mixtures, but the gE-hE diagram reveals in a quantitative and striking way what range of size and energy parameters are required to produce particular kinds of nonideal solution behavior. Such plots should prove especially informative when developed for mixtures of molecular fluids, with their multitude of possible variations in asymmetric intermolecular forces. In this paper we have not used the gE-hE diagram to make new discoveries or new predictions about mixture behavior. We have merely tried to illustrate the utility of the diagram using behavior well-established in the literature. Our own particular interest is in studying mixtures by molecular-scale computer simulation. We find the g E-hE diagram valuable in choosing the kinds and strengths of intermolecular forces used in the simulations, in interpreting the excess property results obtained from the simulations, and in testing theories that predict excess properties. Experimentalists who measure excess properties might find the diagram useful in similar ways. Acknowledgment
We are pleased to thank Professors M. M. Abbott, J. Fischer, and J. P. O’Connell for helpful discussions. This work was supported in part by a National Science Foundation Presidential Young Investigator Award (1984-1989) to J.M.H. Literature Cited Boublik, T. Molec. Phys. 1986, 59, 371. Coop, J. L.; Everett, D. H. Discuss. Faraday Soc. 1953, 15, 174. Gaube, J.; Koenen, E. Chem.-1ng.-Tech. 1979,51, 496. Guggenheim, E. A. Mixtures; Clarendon: Oxford, 1952. Hildebrand, J. H.; Prausnitz, J. M.; Scott, R. L. Regular and Related Solutions; Van Nostrand: New York, 1970. Kauer, V. E.; Bittrich, H. J.; Krug, K. Wiss. 2.Tech. Hochsch. “Carl Schorlemmer” Leuna-Merseburg 1966,8, 139. Koenen, H. E.; Gaube, 3. Ber. Bunsenges. Phys. Chem. 1982,86,31. Kohler, F.; Gaube, J. Pol. J. Chem. 1980, 54, 1987. Malesinski, W. Azeotropy and Other Theoretical Problems i n Vapor-Liquid Equilibrium; Wiley Interscience: London, 1965; Chapter 3. Prigogine, I.; Defay, R. Chemical Thermodynamics; Everett, D. H., Transl.; Longmans, Green, & Co.: London, 1954.
Ind. Eng. Chem. Res. 1988,27, 671-679 Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.; Butterworths: London, 1982;Chapter 1. Scott, R. L.; van Konynenburg, P. H. Discuss. Faraday SOC.1970, 49,81. Shukla, K. P. Molec. Phys. 1987, 62,1143. Shukla, K. P.;Haile, J. M. Molec. Phys. 1987, 62,617.
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van Konynenburg, P. H.; Scott, R. L. Phil. Trans. 1980, A298,495. Weeks, J. D.;Chandler, D.; Andersen, H. C. J . Chem. Phys. 1971a, 54,5237;Phys. Rev. 1971b, A4, 1597.
Received for review July 17, 1987 Accepted December 7,1987
Generalized Multiparameter Correlation for Nonpolar and Polar Fluid Transport Properties Ting-Horng Chung,+Mohammad Ajlan,t Lloyd L. Lee,* and Kenneth E. Starling School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, Oklahoma 73019
Correlations are presented for the viscosity and thermal conductivity of nonpolar, polar, and associating fluids over the wide ranges of PVT states. Empirically correlated density-dependent functions were developed to extend the kinetic gas theory to include dense fluids. Extensive comparisons with experimental data of pure fluids are made. The average absolute deviation is 4% for viscosity predictions and 6 % for thermal conductivity predictions. The conformal solution model mixing rules have shown to yield predictions of viscosity and thermal conductivity for nonassociating mixtures of sufficient accuracy for most industrial uses. The viscosity and thermal conductivity predictive procedure is simple and straightforward. It requires only critical constants and acentric factors for nonpolar fluids. For polar and associating fluids, the dipole moment and an empirically determined association parameter, in addition, are required. Transport properties are important quantities required in engineering design for production, transportation, and processing. For example, viscosity is an important parameter for the determination of pipeline size and the power required to pump fluids through it. Viscosity also enters into heat-exchanger and separation equipment sizing and is a critical parameter for the recovery efficiency of reservoir oils. However, our understanding of the transport properties is far behind that of equilibrium properties. The difficulties we face in the study of transport properties are twofold: one is the inherent difficulties involved in accurate measurements, and the other is the complexity involved in theoretical treatments. Applications of theoretical developments in either the distribution function approach-a generalized Boltzmann equation method (Chapman and Cowling, 1952)-or the time-correlation function approach (Zwanzig, 1965; Steele and Hanley, 1969) have only been possible for simple cases such as dilute gases of simple molecules. For dense gas properties, one of the very few theoretical results is the Enskog dense gas theory for the hard-sphere-potential model (Chapman and Cowling, 1952). Following this theory, many investigators have proposed correlations for the viscosity ratio ( v / q o )as a function of reduced density and reduced temperature (Reid et al., 1977). Most of the available methods for the prediction of transport properties are empirical correlations and are limited to specified state regions and fluids. A good review of these methods was given by Reid et al. (1977). They concluded that none of the methods available are particularly reliable, especially for polar fluids. Methods based on the principle of corresponding states (Helfand and Rice, 1960; Gubbins, 1973) have been widely used for the determination of the transport properties of simple dense fluids. For simple liquids (Ar, Kr, Xe, and CHI), the 'Current address: National Institute for Petroleum and Energy Research (NIPER), Bartlesville, OK 74003. Current address: Department of Chemical Engineering, University of Petroleum and Minerals, Dhahran, Saudi Arabia.
principle has been shown to be accurately obeyed (Tham and Gubbins, 1969). A conformal solution theory has also been developed (Mo and Gubbins, 1976) for mixtures of such simple fluids. For more complicated molecules, the liquid viscosity has been shown not to obey the simple corresponding states principle (Tham and Gubbins, 1970). The discrepancy is more obvious for thermal conductivity since polyatomic fluids do not obey the principle even at dilute gas conditions (Chung et al., 1984b). This is because the corresponding states arguments do not correctly take into account the effect of internal degrees of freedom on the thermal conductivity (Hanley, 1977). Recently, the applicability of corresponding states for thermodynamic properties has been broadened considerably by the introduction of state-dependent shape factors (Leland et al., 1968; Rowlinson and Watson, 1969) to include substances of more complicated molecules (Haile et al., 1976; Ely and Hanely, 1981, 1983). Unfortunately, this method is currently applicable to nonpolar fluids only, and its use requires complicated procedures. The methods presented in this work for the prediction of transport properties are relatively simple and can be applied to both polar and associating fluids. The viscosity and thermal conductivity of dense fluids are empirically correlated as functions of density and temperature. For mixtures, the predictive method is similar in essence to the conformal-solutionmodel. Based on this model, mixing rules are developed for all parameters; the viscosity and thermal conductivity correlations are the same as those for pure fluids. The correlation of low-pressuregas viscosity and thermal conductivity based on the kinetic gas theory has been published earlier (Chung et al., 1984b), where accurate models were presented for dilute gas viscosity and thermal conductivity of nonpolar, polar, and associating fluids. The low-pressuregas viscosity and thermal conductivity models are now extended to fluids at high densities by introducing empirically correlated, density-dependent functions. As fluid density approaches zero, these correlations will reduce to the low-pressure gas expressions. These correlations use
0888-5885/88/2627-0671~01.50/0 0 1988 American Chemical Society
