1985
J. Phys. Chem. 1993,97, 1985-1992
Influence of Combining Rules and Molecular Shape on the High Pressure Phase Equilibria of Binary Fluid Mixtures Richard J. Sadus+ Institut f i r Physikalische Chemie und Elektrochemie der UniversitSlt Karlsruhe, Kaiserstrasse 12, 7500 Karlsruhe 1 , Germany Received: August 17, 1992; In Final Form: November 18, 1992
The influence of molecular shape and combining rules for unlike interactions on the critical phase transition of nonpolar type I11 binary mixtures is examined. The critical properties of tetrafluoromethane n-heptane and sulfur hexafluoride either n-octane, n-nonane, or n-undecane are predicted using equations of state based on different assumptions of molecular geometry and compared with experiment. The role of intermolecular separation is studied by applying different combining rules for the contribution of collisions involving unlike molecules to molecular volume. Very good quantitative agreement of theory with experiment is obtained over a wide range of density, temperature, and pressure. It is concluded that modeling the linear n-alkanes as long spherocylinders improves the quality of prediction but the effect is not as dramatic as would be intuitively expected from the nonspherical nature of these molecules. However, correctly determining the parameters characteristic of unlike interactions has a profound influence on the predicted critical properties. An alternative combining rule for the collision volume parameter is proposed which is generally more accurate than either the Lorentz, arithmetic, or geometric combining rules.
+
+
Introduction Equations of state and models of the fluid phase in general are becoming increasingly more sophisticated. Many models' endeavor to explicitly account for such influences as dipole induced interactions or quadrapole moments, in addition to normal dispersion forces. The merits of various perturbation models, equationsof state, and conformal fluid models have recently been extensively reviewed.2 Using an equation of state is the most practical method for phase equilibria calculationsand predictions because it can be directly applied to different liquid, gas, or supercritical phases without any conceptual difficulties. Despite improvements to equations of state to explicitly account for intermolecular interactions, their usefulness ultimately depends on the accurate evaluation of various constants from some experimental data. Successful equations of state require at least two constants which reflect the contribution of molecular interactions and molecular volume. The prediction of mixture phenomena requires a suitable method of extending these parameters which is typically achieved via combining rules and mixture prescriptions. Several theoretically based mixing rules have been based on different assumptions for the radial distribution function of molecules. The van der Waals mixing rules5 are possibly the most widely used and comparison with computer simulation has recently6 confirmed their validity for a far greater range of molecules than would otherwise be expected. However, relatively little is known about thevalidity of combining rules used to obtain the contribution of molecular interactions between unlike molecules to the equation of state parameters of the fluid mixture. This is particularly the case for the combining rule which is used to obtain the contribution of unlike interaction to the molecular volume of the mixture. Most commonly used equations of state are based on the explicit assumption that molecules can be modeled as spheres. This assumption is reasonable for the inert gases and relatively small molecules, or large molecules of spherical symmetry. However, it can be intuitively expected that the phase behavior of fluids containing molecules of nonspherical geometry (e.g., large linear Permanent address: Computer Simulation and Physical Applications Group, Department of ComputerScience, Swinburne University of Technology, P.O. Box 218, Hawthorn, Victoria 3122, Australia. +
0022-3654/93/2097-1985$04.00/0
n-alkanes) would be influenced by molecular shape. Accurate equations of state for molecules of nonspherical geometry have been developed7-10and it is apparent that the prediction of exfunctions1 I and virial coefficientsI2is improved by accounting for molecular shape. However, evidence for the role of molecular shape in the phase behavior of fluids, in general, is inconclusive. For example, the analysis13J4of gas-liquid critical properties using hard sphere and hard convex body equations of state consistently concludes that accounting for molecular geometry does not improve the quality of prediction. The principal aim of this work is to clarify the role of both combiningrulesand molecularshape by predicting type I11critical equilibria of binary mixtures. Type I11 critical behavior15 is manifested as a discontinuity in the gas-liquid critical line. The critical locus commencing from the most volatile component ends at an upper critical end point on the end of a three-phase liquidliquid-vapor line whereas the critical curve emanating from the other component undergoes a continual transition between "gasliquid" and 'liquid-liquid" properties and typically extends to very high pressures. The prediction of type I11critical equilibria represents a stringent test for both the validity of the equation of state and combining rules. It is typically observed for mixtures of molecules of either vastly different sizes,geometries,or polarity. The main critical locus also occurs over a significant range of densities ranging from gas-liquid to liquid-liquid densities. Therefore, the equation of state must be valid over these range of densities in order to satisfactorily predict this phenomenon. Predicting thecritical properties of mixturescan also beansidered as a useful starting point for improving the prediction of fluid phase equilibria in general, because the equation of state parameters used for other equilibria are commonly obtained from the critical properties of the pure fluid. Therefore, any conclusions reached from the analysis of binary mixture critical properties (e.g., validity of combining rules) can be applied to the prediction of noncritical equilibria. Predictions of the critical properties were obtained by using an accurate representation of fluid-fluid interactionsof both hard spheres and nonspherical geometry plus a suitable attractive term. Comparison of the quality of prediction obtained by using these models enables identification of the influence of molecular geometry. The role of intermolecular separation was examined Q 1993 American Chemical Society
Sadus
1986 The Journal of Physical Chemistry, Vol. 97, No. 9, 1993
by applying different combining rules for the contribution of unlike interactions to the molecular volume. It is concluded that the latter quantity influences the phase behavior much more significantly than molecular shape. It appears likely that many large, nonspherical molecules can be legitimately modeled as hard spheres without significantly affecting the quality of prediction. The analysis also suggests a simple improvement to the accuracy of combining rules for molecular volume. k r y
The critical point of a binary fluid mixture is located by determining the temperature (T), volume (v), and composition (x) which satisfy the following conditions:
molecules of arbitrary geometry were used. The van der Waals model (i.e., patt= -alp)was employed for the contribution of attractive interactions. The Pad6 approximation used by Heilig and Franck20q21is an example of a more accurate theoretical model for the contribution of attractive interactions to the pressure of the fluid. Nonetheless, the van der Waals model is a reasonable approximationwhen dispersion forces are the primary contribution to molecular interactions. If the CarnahanStarlingIs representation of hard spheres is used, then the result is the following equation for the fluid
+ +
p = R T ( ~ y y2 - y3)/v(i -y13 - a/V2 (7) whereas using the Bo~blik-Nezbeda~~ representation of molecules of nonspherical geometry yields p = RT(1
+ (3a- 2)y + (3a2- 301 + l)y2 a2y3)/ V( 1 - y)'
The Helmholtz function (A) of the mixture was obtained from conformal solutiontheory.I6 The Helmholtzfunction is identified with the configurational properties of a hypothetical equivalent substance, A*-, which has the configurational properties of the mixture (the asterisk'indicates a configurational property and es denotes equivalent substance). The configurational properties of the equivalent substance are determined from an appropriate reference substance (Po), viz.
A = A*,
+ A,, = f d * & v / h e s , T / f , )- R T l n h, + A,,
properties via conformal parameters fa and h, which are proportional to the strength of intermolecular potential and internuclearseparation,respectively. They can be evaluated from the critical properties of the pure components, i.e.,fi~ = P I I / Poo;h l l = P11/Poo;f22 = 7WPoo; h22 = J%/V~etc. (where the properties of the reference substance are denoted by the 00 subscript and 11 and 22 refer to contributions from components 1 and 2, respectively). It is customaryto use one of the components of the mixture as thereference substance. In this work, component 1 is the reference substance. The configurational properties are obtained from a suitable equation of state. Further details of the calculation and theoretical background are given elsewhere.2 The pressure of a fluid can be explicitly determined from the sum of the contribution of repulsive (Prep)and attractive (Pad contributions, i.e.
= P r e p + Pat1
(6) Good theoretical models are available to represent the repulsion of either hard spheres18 or hard molecules of nonspherical geometry.l9 However, the contribution of attractive forces is inadequately understood. For the purposes of chemical engineering applications, it is often expedient to use an equation of state like the Soavel' equation which consists of an inadequate representation of repulsive interaction but an empirically "improved" attractive term. Repulsive forces can be expected to be the dominant contribution to the fluid at high pressures. Consequently, the CarnahanStarlingls and Boublik-Nezbedal9 equations for the repulsive interactions of hard spheres and P
wherey = b/ Vand b is the volume occupied by 1mol of molecules, a (defined by eq 9 ) is the nonsphericity parameter, and a is the contributionof attractive interactions. Equation 8 will be referred to as the arbitrary shape equation of state to distinguish it from the spherical CarnahanStarling model. The a parameter in eq 8 represents the deviationof the molecule from sphericalgeometry. If r, s, and u represent the mean curvature, surface area, and volume of the molecule, respectively, then the deviation from spherical geometry can be deductdl9 from the following relationship a = rs/3v
(9) It is apparent that when a = 1, eqs 7 and 8 become identical. The equation of state parameters can be obtained by solving the critical conditions of a pure substance, Le.
(4)
where R is the universal gas constant and the A& term is the contribution from combinatorial (cb) mixing obtained from eq 5. The properties of the reference substance are related to mixture
- a/ v2 (8)
(ap/av)T
=
(10)
(a2p/av2)T = 0 (11) It is evident that this procedure can only be used to evaluate two parameters independently of each other, whereas eq 8 contains three adjustable parameters. Svejda and Kohlers have proposed an equation similar to eq 8 for molecules of hard convex bodies (HCB). The geometry of the molecule is examined to deduce the physical dimensions of a hard, invariablecore. This core is covered by an envelope of uniform thickness to obtain the final shape of the hard convex body. The thickness of the convex layer is used to deduce the mean curvature, surface area, and volume of the hard convex body via prescriptionsoriginallyproposed by Kihara.11 Consequently, eq 9 can be used directly to determine the nonsphericity of the hard convex body. Sadus et al." have used this procedure to obtain the thickness of the hard convex layer byevaluating thecriticalconditions(eqs lOand 11). Asdiscussed elsewhere? this negates the usefulness of the three-parameter equation of state because both the volume occupiedby the molecule and the nonsphericity parameter depend on the thickness of the hard convex layer. Therefore, the equation of state behaves like the two-parameter CarnahanStarling equation which may explain why very little discrepancy is observed between results obtained for the critical properties of binary mixtures using the HCB quation and a hard sphere equation, irrespective of the size or shape of the component molecules. A different procedure was used in this work which avoids the above limitation. The equation of state parameters (a and b) can be related to the critical properties of the fluid via the following general relationships
b=yV
(12)
a =B R l T (13) Equation 12 and the definition of y in eqs 7 and 8 are equivalent
The Journal of Physical Chemistry, Vol. 97, NO. 9, 1993 1987
Binary Fluid Mixtures *I
1.70 1.65 1.60 ~
1.55 1S O 1.45 1.40
0.14 0.13
justification whereas the equation of state parameters only Yreflectnintermolecular properties. The prescriptions can be readily applied to the equation of state parameters by recalling that a and b are proportional to f,h, and h,, respectively. The conformal parameters c/o, h,) and remaining equation of state parameters (a, b) for the mixture can be obtained from the van der Waals mixing rules.2.16
b
0.12 0.11
,0.10 0.09 0.08 0.07 0.06 0.05
0.36
C
0.34
2
1
a
3
4
Figure 1. 0 and y coefficients and the critical compressibility factor (2’) obtained by solving the critical conditions of a one-component fluid at different values of a. The line of best fit was obtained by using the polynomial data in Table I.
TABLE I: Polynomhl Coefficients with Respect to u Which C.a Be Used To Calculate the Values of y, 8, and of a Pure Fluid coefficients of a degree
Y
B
0 1 2 3 4 5
0.236 039 4 , 1 6 6 756 0.081 5289 4 , 0 2 3 9585 0.003 7810 2 -0.OOO 2454 36
1.14405 0.314 563 4 . 0 8 9 509 7 0.015 061 4 -0.001 071 08
ZC
0.370 619 4 . 0 1 5 935 9 0.005 112 9 4.OOO 926 689 6.895 344 X
except for the substitution of V = V at the critical point. The values of the dimensionlesscoefficients y and /3 were obtained by solving the critical conditions of a pure fluid at different a values. The /3 and y coefficients have limiting values of 1.3829 and 0.1 3044, respectively, for a hard sphere plus attractive term fluid (i.e., a = 1). The variation of y , @andthe critical compressibility ( Z C ) with respect to the nonsphericity parameter is illustrated in Figure 1. The progressive decline in the magnitude of y as the molecule deviates from a spherical geometry has also been obser~ed~3 for the HCB equation of state. However, in the latter case, the critical compressibility showed very little change from the hard sphere value of 0.359 irrespective of the shape of the molecule, whereas a significant variation is evident in Figure IC. It was found that these parameters could be accurately fitted to either a fourth- or fifth-order polynomial with respect to u.The polynomial coefficients of a are included in Table I. These data form the basis of the line of k t fit illustrated in Figure 1. The equation of state and conformal parameters of the mixture can be obtained from the pure component properties via suitable mixing rules. The nonsphericity parameter of the mixture can be simply obtained by a linear mixing rule proposed by Sadus et a1.13
+
(14) u = xlaII (1 - xI)a22 It is useful to discuss the other mixing rules in terms of the conformal parameters rather than the equation of state parameters. The conformal parameters have a direct theoretical
The van der Waals mixing rules stem from the one-fluid model and assume that the radial distribution functions of the component molecules are all identical. The results compare well with computer simulationdata if the sizedifference between component molecules is not too great. A recent comparison6with computer simulation indicates that the rules are reliable for mixtures exhibiting up to an 8-fold difference in the size of the component molecules. Toevaluate eqs 15 and 16,assumptionsmust be made regarding the nature of molecular interaction between dissimilar molecules. The contribution of unlike interactions to the f conformal parameter is typically evaluated from the following combining rule
Aj
tY;ifj)O.’ (17) where [ is an adjustable parameter which reflects the strength ([ > 1) or weakness (E < 1) of intermolecular interaction. The need for this additional parameter also testifies to the inadequacy of the above combining rule in genuinely reflecting the strength of unlike interactions. Nevertheless, some additional evidence22 for the values genuinely reflecting the strength of unlike interaction has been recently become available. The primary focus of this work is to examine the role of shape and unlike-interaction combining rules on the phase behavior of fluids. The latter contribution is manifested in the combining rule for the h12 parameter which is directly related2 to the conformal parameter for intermolecular separation, g12 (is., h12 = g1z3). The simplestand possibly the most widely used combining rule consists of an arithmetic2J6mean of the conformal parameters of the pure substances (18) hij = (hi( + hjj)/2 The Lorentz rule2J6is a widely used alternative to the arithmetic rule
h, = 0.125(hii’/’
+ hjj1/3)3
Good and Hope23 have also proposed a rule based on a geometric average of the properties of the pure components
h, = (hiih,j)o.’ The justification for using the above combining rules is largely empirical. Their influence on high pressure phase equilibria has not been studied in great detail. The fact that neither of the above options is adequate is testified to by the introduction of an additional coefficient (0to maximize agreement between theory and experiment. However, the analysis of binary mixture data typically reports only the optimum t without indicating the sensitivity of the results to less than optimal conditions. The choice of combining rule for mixtures of similar size is not important because the results obtained from either eqs 17,18, or 19are equivalent. However,a substantial divergence in the value of hij between the various alternatives is apparent for molecules of considerably different sizes (Figure 2). The arithmetic mean produces the largest estimate, whereas the values predicted by the geometric mean are considerably smaller. The predictions of the Lorentz rule are between these extremes, although they
Sadus
1988 The Journal of Physical Chemistry, Vol. 97,No. 9, 1993 5.5
1 ,
::: -2 Y
4.0
withmetie
3.5
3.0 2.5 2.0 1.5
1.o
Figure 2. Variation of h12 with respect to the ratio of h22 relative to h l I predicted by the arithmetic, Lorentz, and geometric combining rules.
are considerably closer to the values predicted by the geometric rule rather than the results obtained for the arithmetic rule.
Results and Discussion The equation of state models used here were specifically developed to identify the influence of molecular shape (a) and intermolecular separation (hij) on the phase behavior of binary fluids. Calculating type I11 equilibria is a good test of the assumptions made about the sphericity and intermolecular separation because the model must be valid for a range of fluid densities in order to yield satisfactory agreement. It is much easier to calculate continuous gas-liquid critical equilibria but it is well-known that many theoretically inadquate models give good agreement at gas-liquid densities. The agreement of theory with experiment for this phenomenon is not substantiallyimproved by using theoretically improved models. The gas-liquid critical transition is also relatively insensitive to the parameters chosen to represent intermolecularseparation. Consequently, the analysis of gas-liquid critical properties alone is likely to be inconclusive. However, a problem arisesin finding experimental data for suitable type I11 mixtures. Both components of the binary mixture must be nonpolar because the theory does not explicitly account for interactions induced by a permanent dipole. At least one of the component molecules must also be demonstrably nonspherical. The most extensive available experimental data2 involvemixtures containing a dipolar component. These systemsare also influenced by dipole-induced interactions in addition to the role of intermolecular separation and molecular shape. The large linear alkanes are good examples of nonspherical molecules. The deviation from nonsphericity of these molecules can also be modeled in a systematic manner. There are many examples2of mixtures containing an alkane and a polar partner (e.g., water, methanol, ammonia, etc.); however, data for type 111 mixtures involving a nonpolar, second component molecule are rarer. Nonetheless, experimental data are available for mixtures of n-alkanes and either tetrafluor~methane,~~ sulfur hexafluoride,25 or carbon dioxide.26 These latter molecules are either small or symmetric, and as such, they can be legitimately modeled as spheres. A comparison between theory and experiment was made for mixtures of heptane + tetrafluoromethane, octane +sulfur hexafluoride, nonane + sulfur hexafluoride, and undecane + sulfur hexafluoride. The n-alkane molecules represent a range of both geometry and molecular size. The hexadecane + carbon dioxide mixture is potentially a very good test of the assumptions made concerning both molecular geometry and combining rules, and experimental data26are available to very high pressures. However, an analysis of this mixture could not be made due to considerable uncertainty in the critical properties of hexadecane. The critical properties of the components studied were obtained from a compilation by Ambrose.27
TABLE Ik Parameters Used To Evaluate the Nonsphericity Factor (a)for b A h n e Molecules P r Z L . L L alkane (K) e (nm2) (nm) (nm) (nm) (L/w) a C7H16 CsHls CsHzo CllH24
540.3 568.83 594.6 638.8
0.348 0.438 0.432 0.426
0.248 0.456 0.511 0.627
0.865 1.043 1.083 1.160
0.998 1.250 1.376 1.628
0.954 1.181 1.278 1.472
2.338 2.894 3.132 3.608
1.392 1.467 1.540 1.692
Determining the Nonsphericity Parameter (a).The nonsphericity factor for the linear alkanes can be deduced from the length (L)and width ( w ) of the molecule by modeling it as a long spherocylinder. This is intuitively a more suitable description of the true shape of the alkane than assuming a spherical geometry. Equation 9 for spherocylinders can be written as28 a = ( ( L / w ) + (L/W)~I/{~W/W) - 11
(21) The width of the molecule was assumed to be equal to the linear H-C-H bond distance (0.408 nm). A linear alkane molecule can be envisaged either in a straight chain (trans) or curled configuration. Toobtain thelength of themolecules, the following average of trans (G) and curled (Lc)configurations suggested by Svejda and Kohlera was used
L = (2/3)L, + (1/3)L,
(22) The length of an all-trans alkane was calculated by using eq 23
+
L,= u(n - 1) sin (8/2) + 2d sin (8/2) 2c (23) where n is the number of C atoms, u is the C-C bond length (0.1 54 nm), d is the C-H bond length (0.109 nm), cis the radius of a hydrogen atom (0.095 nm), and 8 is the C-C-C bond angle (109.5O). The length of the curled configuration was obtained from L, = r + 2d sin (a/2)
+ 2c
(24) Thevariable r is the average distance between the terminal carbon atoms which was evaluated according to the following equation29
3 = (2 + 6e)(n - 1)u2/(2 + 6e)
(25)
where e = 1/{ 1
+ 2 exp(-e/RT)J
(26) and c is the energy difference30 between the gauche and trans conformations (2093.4J mol-'). Thevaluesof aandother related data are summarized in Table 11. Comparison of Experiment with Theory Using the CarnahanStarling (eq 7) and Arbitrary Shape (eq 8) Equations of State. The comparison of theory with experiment for binary mixtures is complicated by the interrelationshipof combining rules, mixture prescriptions, and the equation of state. Therefore, it is often difficult to isolate the relative contributions of these influences on the quality of the predicted phase behavior. The magnitude 0 f f i 2 has the predominant influence in determining the type of critical transition observed. A transition between type I, 11,111, and IV behavior is generally observed by progressively altering the magnitude of the f i 2 parameter. It also has an important influence on the quality of agreement between theory and experiment. A variation of 1% typically alters the observed gasliquid critical temperature by 1 K, whereas the position of liquidliquid equilibria or the pressure-temperature minimum of the main type I11 locus changes by 10 K. The influence of h12on the phase equilibria of fluids is less than adequately documented. Typically, only optimum values are reported without any indication of how sensitive the agreement is between theory and experiment. Christou et al.I4 and Mainwaring et aL3I have reported calculations of type I11 equilibria for binary mixtures containing either tetrafluoromethane or sulfur hexafluoride as
The Journal of Physical Chemistry, Vol. 97,No. 9, 1993 1989
"250
350
300
450
400
5 I
T/K
-
0 '
300
320
360
340
380
400
T/K
Figure 3, Comparison of experiment (0)with theory (+) for the critical locus of tetrafluoromethane n-heptane illustrating the influence of {A (in parentheses) at a constant value of E = 0.82. The CarnahanStarling equation was used for the calculations.
Figure 4. Comparison of experiment (0)with theory for the critical locus of tetrafluoromethane + n-heptane using the arbitrary shape (+, 5 = 0.825, {A = 0.963) and the CarnahanStarling (*, 5 = 0.82, {A = 0.963) equations.
one componentusing the Lorentz and arithmetic combining rules. They concluded that the Lorentz rule was more accurate than the arithmetic combining rule. However, the calculations employed the Guggenheim equation of state32 and the overall agreement between theory and experiment was unsatisfactory. The calculated critical properties typically straddled the experimental data and were far from being quantitatively accurate. The procedure adopted in this work was to obtain an initialfi2 value which could be used to qualitatively predict type 111 equilibria using both the arithmetic and Lorentz combining rules within reasonable proximity (- 10 K)to the experimental data. The geometric rule was not used because it is clear from Figure 2 that in most circumstances, it considerably underestimates the h l z parameter. The initial calculation utilized the CarnahanStarling equation (q7). A dimensionless coefficient (denoted {A and {L for the arithmetic and Lorentz rules, respectively) was introduced into the combining rules in order to improve the qualitative accuracy of the predicted critical properties, Le.
region than is observed experimentally. The influence of the h12 parameter was examined by repeating the calculationswith values intermediate between the values calculated by either the Lorentz or arithmetic rules by introducing the t coefficient into the combining rules. It is apparent from Figure 3 that the overall agreement between theory and experiment can be dramatically improved by choosing a more appropriate h12 value. The prediction of the high-temperature "gas-liquid" portion of the critical locus remains relatively unaffected but the position of the pressuretemperature minimum, and the "liquid-liquid" critical equilibria at higher densities and high pressures, is very sensitive to the h12 value. Figure 4 illustrates the best possible agreement between theory and experiment using the Carnahan-Starling equation and an optimal value of h12. The overall agreement is surprisingly very good inview ofthe nonsphericalnature of n-heptane. Calculations reported elsewhere14using another hard sphere equation of state for this mixture (without optimizing h12),yielded quantitatively poor results. The optimum hl2 value was used in conjunction with the arbitrary shape equation of state (eq 8) to recalculate the critical properties of this mixture to explicitly account for the nonspherical geometry of n-heptane. The results of these calculations are also illustrated in Figure 4. The agreement of theory with experiment is also very good using this procedure. In particular, it is apparent that the prediction of the pressure temperature minimum is substantially improved. The remaining examplesof type I11systems examined all involve sulfur hexafluoride as one component. Mixtures of an n-alkane + sulfur hexafluoride exhibit type 111 equilibria from n-octane onward. The most extensive experimental data is due to Matzik and S ~ h n e i d ewhich r ~ ~ forms the basis of the comparison between theory and experiment presented here. An analysis of these mixtures using a less accurate equation of state is also availableS3l This earlier work reported relatively poor agreement between theory and experiment for these mixtures. Experiment and theory are compared in Figure 5 for the critical locus of sulfur hexafluoride + n-octane using either the Lorentz rule, arithmetic rule, or an optimized value of the hl2 parameter. The calculations using the Lorentz and arithmetic rule intersect each other near the pressure-temperature minimum of the critical
+
The calculationwas subsequentlyrepeated using the optimalvalue of this parameter and the arbitrary shape quation of state (eq 8). Consequently, conclusions could be made regarding the influence of both h12 and molecular shape. Extensive experimental data are available for the critical propertiesof binary mixturesconsistingof a hydrocarbon molecule with tetrafl~oromethane.2~The transition between type I1 and type I11 behavior occurs at the n-butane tetrafluoromethane mixture. The n-heptane + tetrafluoromethane system was chosen as an example of a type 111 mixture involving a demonstrably nonspherical molecule. The properties calculated by using the CarnahanStarling equation of state are compared with experimental values in Figure 3. The critical locus delineatesthe region of two-phase coexistence (left-hand side) from the homogeneous single phase (right-hand side). It is apparent that the Lorentz rule substantially underestimates the extent of two-phase coexistence particularly at high pressure. Conversely, calculations with the arithmetic rule predict a substantially greater two-phase
+
Sadus
1990 The Journal of Physical Chemistry, Vol. 97, No. 9, 1993 150 I
150
I
t
100
m
m
4 2..
B 2
50 -
50
01
0 250
350
300
400
T/K Figure 5. Comparison of experiment (0)with theory (+) for the critical
+
locusofsulfur hexafluoride n-octaneshowing theeffect of using different combining rules and the optimal {A parameter (in parentheses) with 4 = 0.83. The CarnahanStarling equation was used for the calculations.
280
,
300
1
I
I
320
340
360
locus of sulfur hexafluoride + n-nonane illustrating the role of {A (in parentheses) at a constant value of 4 = 0.823. The CarnahanStarling equation was used for the calculations.
are not available, but the calculations clearly support the contention that this is a type I11 system. The calculations with the arbitrary shape equation of state (Figure 6) are also in very good agreement with the experimentaldata. It is clear that either the CarnahanStarling hard sphere equation or the arbitrary shape equation of state yield equally good and accurate predictions of the critical properties of this mixture. The analysis of the n-nonane sulfur hexafluoride system is complicated by the need to estimate the critical volume of nonane (552cm3 mol-’). The agreement between theory andexperiment using the Lorentz rule (Figure 7) is reasonable at relatively low pressures, whereas the trajectory of thecritical locus toward higher temperatures at pressures above 50 MPa is not adequately predicted. There is relatively little difference in the position of the pressure-temperature minimum predicted by the two a p proaohes, but the calculations using the arithmetic rule greatly exaggeratetheextent oftwo-phasecoexistenceatpressures beyond 20 MPa. Neither combining rule can be used to obtain quantitatively accurate agreement of the critical properties close to the pressure-temperature minimum. Nevertheless, it is apparent from Figure 7 that this deficiency can be at least partly remedied by adjusting the h12 parameter. The influence of an optimal h12 value and calculations using the arbitrary shape equation of state are illustrated in Figure 8. The overall agreement between theory and experiment is very good with the CarnahanStarling equation. However, there is some evidence that the representation of the region close to the pressure-temperature minimum of the critical locus can be improved by taking account of the nonspherical nature of nonane. Undecane is a good example of a large (the estimated molar critical volume is 672 cm3)and demonstrably nonspherical (a= 1.692)molecule. Consequently, it is reasonable to expect that both its size and geometry would influence the phase behavior of the binary undecane + sulfur hexafluoride mixture. Calculations using the CarnahanStarling, Lorentz combining rule and other estimates of the h12 parameter are presented in Figure 9. The inadequacy of the Lorentz combining rule is clearly apparent for this mixture. The predicted critical locus is progressivelymoving toward lower temperatures at high pressure whereas, conversely, the availableexperimentaldata indicate that
+
01
280
I
I
I
I
I
I
290
300
310
320
330
340
350
T/K Figure 6. Comparison of experiment (0) with theory for the critical locus of sulfur hexafluoride n-octane using the arbitrary shape (+, 4 = 0.836,{A = 0.95) and the CarnahanStarling (*, 4 = 0.83,{A = 0.95)
+
quations.
curve. This contrasts with the heptane + tetrafluoromethane calculations in which the pressure-temperature minimum predicted by the two combining rules occurred at substantially different temperatures. However, in common with the previous calculations, the Lorentz rule yields results which are in substantially better agreement with the experimental data than is possible with the arithmetic rule. The agreement of theory with experiment can beoptimized by using a more suitable hl2 value. The optimal calculations presented in Figure 6 are probably accurate to within reasonable experimentalerror to pressures of at least 100 MPa. Experimental data for the high temperature “gas-liquid” portion of the curve
2 0
T/K Figure 7. Comparison of experiment (0)with theory (+Tfor the critical
Binary Fluid Mixtures
e $100-
80 -
60 40 -
20 1
01 340
I
I
360
380
I
I
400
420
4 10
T/K
Figure 10. Comparison of experiment (0)with theory for the critical locus of sulfur hexafluoride + n-undecane using the arbitrary shape (+, = 0.816, fp, = 0.928) and the CarnahanStarling (*, E = 0.810, fp, = 0.928) equations.
TABLE IIk Summary of Molecular Interaction Parameters E f hi2 mixture eg 7 eu 8 arith Lorentz expt ca 29
200 180
CFd+ heptane SFb+ octane SFs nonane S F s + undecane
160
+
140
0.825 0.836 0.828 0.816
0.963 0.950 0.945 0.928
1.063 1.015 1.026 1.040
1.967 1.655 1.789 2.038
1.911 1.667 1.792 2.035
Improvements to Unlike-Interaction Combining Rules. The 4, tA,{L, and actual h12 values deduced from the comparison of
120 S
&
q 100 P
80
60 40
20
0 0
0.820 0.830 0.823 0.810
350
400
4
I
T/K Figure 9. Comparison of experiment (0)with theory (+) for the critical locus of sulfur hexafluoride n-undccane illustrating the influence of various combining rule parameters (in parentheses) at a constant value of E = 0.81. The CarnahanStarling equation was used for the calculations.
theory withexperiment arcsummarizedin Table 111. The relative accuracy of the Lorentz and arithmetic combining rules is reflected in the deviation of the coefficientsfrom a value of 1 which indicates themagnitudeof thecorrection required to obtain optimal results. It is apparent that the correction required for the Lorentz rule is considerably smaller than that required for the arithmetic rule. However, calculations using the arithmetic rule correctly predict the tendency of the critical locus to progressivelyextend to higher values of temperature at high pressure. This suggests that the prediction of hl2 values could be improved by taking a weighted averageof the Lorentz and arithmeticcombining rules. The best results were obtained by taking a 2:l geometric average of the Lorentz and arithmetic rules, respectively, i.e.
+
the critical transition occurs at progressivelyhigher temperatures at high pressures. However, more appropriatevalues of h12 yield the correct trajectory. The optimal agreement between theory and experiment illustrated in Figure 10 indicatcs very good quantitative agreement at all pressures expect the region close to the pressuretemperatureminimum. The situation is noticeably improved by employing the arbitrary shape equation of state (eq 8). The agreement of theory and experiment with this equation is also very good along the high pressure part of the critical locus. It is apparent that molecular shape plays a role in determining the phase behavior observed in this mixture.
Values of hlz obtained from this equation can be compared with experimental values in Table 111. It is apparent that in most cases, eq 29 can be used to accurately predict the experimentally obtained h12 parameter. If this conclusion is valid for binary mixtures generally, then it could substantially improvethe a priori prediction of phase equilibria by eliminating the use of the t parameter. The arithmetic combining rule can be considered exact only for the collision volume of two hard spheres. It is plausible that the superiority of the Lorentz rule and eq 29 may be due to their ability to more accurately reflect the "softnessmof intermolecular collision and nonspherical geometry. This may explain why explicitlyaccountingfor the nonsphericity of the alkane molecules via eq 8 does not result in a dramatic improvement in the accuracy
1992 The Journal of Physical Chemistry, Vol. 97, No. 9, 1993
of the predicted phase transitions compared with the results obtained using the Carnahan-Starling model and the Lorentz combining rule. That is, the effect of molecular shape may be significantly manifested in the optimal h12 parameter. This highlights the interdependence of the form of the equation of state, combining rules, and mixture prescriptions. In this context, it would be useful to have some molecular simulation results for type 111equilibria of nonspherical molecules in order to compare with the results for different theoretical models. The simulation of critical properties is a very difficult undertaking, although some results for the helium + hydrogen system have been recently reported.33 It is of interest that one value of h12can be used to give good predictions of phase transitions at both gas-liquid and liquidliquid densities. This indicates that h12, and of course the corresponding equation of state parameter for the contribution of unlike interaction to molecular volume (b12),genuinely characterizes intermolecular interaction. It is common practice to propose that this equation of state parameter is temperaturedependent in order to improve the agreement of theory with experiment. However, on the basis of the results obtained here, this approach cannot be theoretically justified. Conclusion Very good agreement between theory and experiment can be obtained for the type I11 critical equilibria of nonpolar binary mixtures using the CarnahanStarling equation of state and obtaining the optimal hl2 parameter. Taking account of the nonspherical geometry of the component molecules further improves the quality of the predicted phase behavior, but the effect is not as dramatic as would be expected in view of the large deviation from spherical geometry of some of the alkanes. Equation 29, which is more accurate than either the Lorentz, geometric, or arithmetic combining rules, can potentially be used to substantially improve the quality of a priori prediction of phase behavior in general. Acknowledgment. I thank the Alexander von Humboldt Foundation for financial assistanceand I thank Professor E. Ulrich Franck for his gracious hospitality during my visit to Karlsruhe.
Sadus Reference and Notes (1) For a r e n t example see Saager, B.; Fischer, J. Fluid Phase Equilib. 1992, 72, 67. (2) Sadus. R. J. High Pressure Phase Behauiour of Multicomponrnr Fluid Mixtures; Elsevier: Amsterdam, 1992. (3) Mansoori, G.A. ACSSymp. Ser. 1986, No. 300, 314. (4) Ely, J. F. ACS Symp. Ser. 1986, No. 300, 331. (5) Rowlinson,J. S . Liquidsand LiquidMixtures; Buttenvorths: London, 1969; p 250. (6) Harismiadis. V. 1.; Koutras, N. K.; Tassios, D. P.; Panagiotopoulos, A. Z . Fluid Phase Equilib. 1991. 65, 1. (7) Chen, S.S.;Kreglewski,A. Be?. Bunsen-Ges. Phys. Chem. 1977,81, 1048. ( 8 ) Svejda, P.; Kohler, F. Ber. Bunsen-Ges. Phys. Chem. 1983,87,672. (9) Boublik, T. Mol. Phys. 1989, 68, 191. (IO) Walsh, J. M.; Gubbins, K. E. J . Phys. Chem. 1990, 94, 5115. ( I 1) Siddiqi, M. A.; Svejda, P.;Kohler, F. Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 1176. (12) Kihara, T. Adu. Chem. Phys. 1963, 5, 147. (13) Sadus, R. J.; Young, C. L.; Svejda, P. Fluid Phase Equilib. 1988, 39, 89. (14) Christou, G.;Sadus, R.J.; Young, C. L.Ind. Eng. Chem. Res. 1989, 28, 481. (15) van Konynenburg, P. H.; Scott, R.L. Philos. Trans. R . Soc. London 1980, 298A. 495. (16) Hicks, C. P.; Young, C. L. Chem. Reo. 1975, 75, 119. (17) Soave, G. Chem. Eng. Sci. 1972,27, 1197. (18) Carnahan, N. F.; Starling, K . E. J. Chem. Phys. 1969, 51, 635. (19) Boublik, T. Ber. Bunsen-Ges. Phys. Chem. 81,85, 1038. (20) Heilig, M.;Franck, E. U. Be?. Bunsen-Ges. Phys. Chem. 1989,93, 898. (21) Heilig, M.; Franck, E. U. Ber. Bunsen-Ges. Phys. Chem. 1990, 94, 127. (22) Sadus, R. J. J. Phys. Chem. 1989, 93,3787. (23) Good, R. J.; Hope, C. J. J . Chem. Phys. 1970,53, 540. (24) Wirths, M.;Schneider, G. M. Fluid Phase Equilib. 1985,21,257. (25) Matzik, I.; Schneider, G.M.Ber. Bunsen-Ges. Phys. Chem. 1985, 89, 551. (26) Schneider, G.M.; Alwani, Z.; Heim, W.; Horvath, E.; Franck, E. U. Chem. Ing. Tech. 1967, 39, 649. (27) Ambrose, D. Vapour-Liquid Critical Proprries; National Physical Laboratory: Ttddington, U.K., 1980. (28) Wrtler, H.-L.; Heybey, J. Mol. Phys. 1984. 51, 73. (29) Tobolsky, A. V. J. Chem. Phys. 1959,31,387. (30) Pearson, W. B.; Pimentel, G.C. J . Am. Chem. Soc. 1953,75, 532. (31) Mainwaring, D. E.; Sadus, R.J.; Young, C. L.Chem. Eng. Sci. 1988, 43, 459. (32) Guggenheim. E. A. Mol. Phys. 1965.9, 199. (33) Shouten, J. A.; de Kuijpcr, A.; Michels. J. P. J. Phys. Reu. B 1991, 44, 6630.
