3855
J . Phys. Chem. 1992,96, 3855-3860 and more uniform at the expense of enthalpy. Thus, the hydrophobic effect may appear entropic in some instances and enthalpic in others. The numerical results from the present study at the superposition approximation level are judged to be quite satisfactory. We need to note, though, that the empirical potential used for water and the degree has been parametrized with data at 25 “C to which it reproduces the structure of water at other temperatures is not known. This may be one reason for the discrepancies in the calculated entropies at higher temperatures (Figure 6). In any case, more work is needed to ensure that the agreement is not entirely fortuitous. First, the basic assumption, namely, the factorization of the radial distribution function, should be examined independently. Geiger et al.15ahave reported that two different subshells of the first hydration shell are characterized by different orientational distributions. If this is the case, averaging over the whole hydration shell might result in an underestimation of the magnitude of the entropy. In contrast, no subshells in the radial distribution function could be discerned in this work (Figure 2), or in calculations by others.52b Zichi and RosskySZbalso found no differences in orientational preferences within the fmt hydration shell and only a slight broadening of the peaks in the orientational distribution function. Furthermore, the distributions calculated by Geiger et al. appear quite noisy, and therefore the differences they observed may be due to inadequate sampling. The possibility of longer range orientational correlations, e.g. in the second hydration shell, should also be examined. Second, the accuracy of the calculation of the translational entropy should be improved. This contribution is very sensitive to the values of the radial distribution function at relatively long separations, just before it reaches unity. Therefore, one needs (52) (a) Rossky, P. J.; Zichi, D. A. Faraday Symp. Chem. Soc. 1982,17, 69. (b) Zichi, D. A.; Rossky, P. J. J. Chem. Phys. 1985, 83, 797.
very accurate estimates of the RDF a t that range of separations. Finally, the contributions from the terms omitted in the entropy expansion in eq 40 should be examined. Particularly interesting would be the three-particle solutewater-water term, which should contain any “water-structure enhancement” effects. Structure enhancement or increased hydrogen bonding would be manifested in enhanced correlations between water molecules in the first hydration shell.s2 There has been conflicting evidence about this in the simulation literature. Alagona and TanilSchave calculated water-water radial distribution functions in the bulk and in the hydration shell of argon and found no appreciable difference. Differences, however, were observed in the calculations of Geiger et al.lSa Several studies have found a slight enhancement in water-water interaction energy near nonpolar group^.'^^^^^^ From the results of the present work, water-structure enhancement is not required to explain the large entropies of solution, in accord with Stillinger’s view.I4 The major contributions to the entropy come from solute-water correlations. As to the question of utility of the entropy expansion for practical, quantitative calculations, we believe that the conclusion of Baranyai and Evans33needs to be reexamined. In addition to improving the numerical accuracy of the calculations, another possible direction for future work would be to develop and test alternative factorization/reduction schemes for the N-particle correlation function. Acknowledgment. We are grateful to Prof. R. H. Wood for numerous helpful discussions. We also thank Prof. W. L. Jorgensen for making the program BOSS available to us and Dr. Wallace for communicating to us unpublished results. This work was supported by the National Science Foundation (Grant CPE8351228). Partial support was also provided by Union Carbide, Merck, and Exxon. Registry No. Methane, 74-82-8; water, 7732-18-5.
Influence of Quantum Effects on the High-pressure Phase Behavior of Binary Mixtures Containing Hydrogen 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: October 16, 1991; In Final Form: January 3, 1992) The critical properties of binary mixtures containing hydrogen + argon, nitrogen, carbon monoxide, carbon dioxide, methane, and ethane are calculated and compared with experimental data. The properties of these mixtures are likely to be influenced by the quantization of translational motion at high densities. It is shown that the addition of a theoretically based quantum correction term to the equation of state can dramatically improve the quality of the agreement between theory and experiment. Even the simple van der Waals equation can then be used to calculate the critical properties of these mixtures with a reasonable degree of accuracy. The only additional experimental input data are the molecular weights of the constituent components. The experimental E12 parameters obtained from the analysis indicate weak interaction between the different component molecules of the mixture.
Introduction
The phase behavior of fluid mixtures is a direct manifestation of the influence of intermolecular interactions. The diversity of critical phenomenal is perhaps the most interesting aspect of the high-pressure equilibria of binary fluid mixtures. The nature of the critical line can be related, at least qualitatively, to the nature (1) Van Konynenburg, P. H.; Scott, R. L. Philos. Trans. R . SOC.London, Ser. A 1980, 298, 495.
of the components. For example, binary mixtures constituted of components which differ substantially in size often exhibit liquid-liquid criticality,2 and in some cases a discontinuity (i.e., type I11 or type IV behavior) is observed in the gas-liquid critical properties. This discontinuity of gas-liquid critical properties is frequently observed in mixtures containing at least one polar a continuity of component such as water’ or a m m ~ n i awhereas ,~ ( 2 ) Hicks, C. P.; Young, C.
L.Chem. Rev. 1975, 75, 119.
0022-3654/92/2096-3855%03.00/00 1992 American Chemical Society
3856 The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 gas-liquid phenomena between the critical properties of the pure components and no liquid-liquid separation (i.e., type I behavior) is commonly reported2 for mixtures of simple nonpolar molecules of similar size. It is well established that the qualitative features of the various critical equilibria can be calculated by using a simple equation of state.’ The van der Waals equationSwas the first equation to successfully predict the coexistence of liquid and vapor phases, and subsequently improved equations have been developed based on a more accurate representation of the interaction of either hard spheres6*’or hard convex bodies.* Mixing rules are commonly invoked to obtain the equation of state parameters for the mixture from the properties of the constituent pure components. Similarly, combining rules are required in order to determine the contribution of interaction between d i s s i i a r molecules. These combining rules typically require additional adjustable parameters in order to optimize agreement between experiment and theory. Nonetheless, despite these limitations, very good quantitative agreement is often reported9,l0 for the gas-liquid properties of nonpolar or slightly polar mixtures irrespective of the size difference between the component molecules. Good agreement between experiment and theory can often also be obtained” for type I1 liquid-liquid equilibria and the type 111 critical properties of many mixtures. However, poor agreement between theory and experiment is usually reported when one of the constituent molecules is a light gas such as hydrogen, helium, or argon. This apparent anomaly arises because the light gases deviate significantly form the law of corresponding states2 which is approximately obeyed by the heavier gases. This is often attributed to the quantization of translational motion at higher densities. Hooper and Nordholm12 have investigated the influence of quantum effects on the vapor-pressure behavior of pure substances, and DeitersI3 has studied the dew and bubble point loci of the binary hydrogen + methane mixture. These studies indicate the importance of accounting for quantum influences; however, very few data are available on the effect on the critical properties of binary fluids. In this work, the influence of quantum effects on the critical equilibria of several mixtures containing hydrogen, and usually a small nonpolar second component is investigated. Binary mixtures containing hydrogen are possibly the most widely studied class of mixtures in which quantum effects can be expected to have a significant role. The mixtures studied also allow the isolation of the quantum effect from other influences such as dipole moment or the uncertainty in the combining rules and mixture prescriptions due to a disparity between the size of the component molecules. Theory
In terms of the Helmholtz function ( A ) and representing composition by mole fraction ( x ) , the critical point of a binary fluid mixture is represented by the following condition~:’~
(3) Brunner, E. J . Chem. Thermodyn. 1990, 22, 335. (4) Brunner, E. J . Chem. Thermodyn. 1988, 20, 273. (5) Rowlinson, J. S., Ed. J . D. van der Waals: On the Continuity ofthe Gaseous and Liquid States; North-Holland: Amsterdam, 1988. (6) Guggenheim, E. A. Mol. Phys. 1965, 9, 43. (7) Carnahan, N . F.; Starling, K. E. J . Chem. Phys. 1969, 51, 635. (8) Svejda, P.; Kohler, F. Eer. Bunsen-Ges. Phys. Chem. 1983, 87, 672. (9) Sadus, R. J.; Young, C. L.; Svejda, P. Fluid Phase Equilib. 1988,39, 89. (10) Mainwaring, D. E.; Sadus, R. J.; Young, C. L. Chem. Eng. Sci. 1988, 43, 459. (1 1) Christou, G.; Morrow, T.; Sadus, R. J.; Young, C. L. Fluid Phase Equilib. 1985, 25, 263. (12) Hooper, M. A.; Nordholm, S. Ausf. J . Chem. 1980, 33, 2029. (13) Deiters, U . K. Am. Chem. SOC.,Symp. Ser. 1986, 300, 371. (1 4) Sadus, R. J.; Young, C. L. Chem. Eng. Sci. 1987, 42, 17 17.
Sadus
=o
(1)
(3)
The critical point is determined by locating the temperature, volume, and composition which simultaneously satisfy eqs 1 and 2. Details of the algorithm used to solve the simultaneous equation are given elsewhere.l4J5 The located critical point must satisfy the condition Y > 0 in order to be thermodynamically stable. The Helmholtz function can be obtained from conformal solution theory2 and the one-fluid model. The one-fluid model2 is the most widely used averaging procedure, and comparison with computer simulationL6indicates that it is superior to two- and three-fluid models. The configurational Helmholtz function is identified with that of a hypothetical pure substance, the equivalent substance: A = A*,, + Acb = f,A*,(V/h,,T/f,) - RT In h, + Acb (4) The reducing parameters f and h are normally dependent on composition and the contribution of the entropy of mixing can often be determined from eq 5 . A& = RT& In xi (5) There are alternatives”J* to the above equation specifically for molecules of dissimilar size, but they are generally inadequate. The validity of the one-fluid model has been recentlyI9 confirmed by comparison with computer simulation for mixtures of small molecules. The configurational properties of the mixture are determined by employing a suitable equation of state. Mast modem equations of state can be described as “hard body + attractive term” equations, Le., there is distinct separation between the contribution of attractive and repulsive forces. The van der Waals equationS is the simplest realistic equation of state:
P = ~ r e ~ - ~ a t [ = R T / ( V - b ) - a / ~ (6) whereas a more accurate representation of the repulsive interaction of hard spheres is incorporated into the Guggenheim6 equation of state: P = Prep - P a t t = RT/IV(I - b/4U41 - a / p (7) Both equations include two adjustable parameters, a and b, which reflect the contribution of dttractive forces and the volume occupied by the molecule, respectively. To account for the quantization of translational motion at higher densities, the following correctionI2must be applied to the equation of state: (8) P = PrcpQcorr - Patt Hooper and Nordholm12 identified the quantum correction term to be the quotient of two infinite series which approach a value of unity when the classical limit is reached. However, the quantum correction can be approximated byI2 (9) Q w r r = 1/(1 - ~ / V f l ’ ~ ) where V, represents the free volume available to the molecule and (15) Hicks, C. P.; Young, C. L. J . Chem. SOC.,Faraday Trans. 2 1977, 73, 597. (16) Gubbins, K. E. Fluid Phase Equilib. 1983, 13, 35. (17) Flory, P. J. J . Chem. Phys. 1941, 25, 193. (18) Flory, P. J. Discuss. Faraday SOC.1970, 49, 7. (19) Harismiadis, V . I.; Koutras, N. K.; Tassios, D. P.; Panagiotopoulos, A. Z . Fluid Phase Equilib. 1991, 65, 1 .
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3857
Binary Mixtures Containing Hydrogen y = h/{2(2~mkT)O.~j
(10)
The free volume depends on the nature of the equation of state. For the van der Waals equation, it can be approximated by Vf= ( V - b ) / N (11) The method for determining the free volume from other equations of state is described elsewhere." The equation of state parameters can be obtained directly by solving the conditions which govern the critical properties of a pure substance. The quantum correction term in eq 8 has a negligible effect on the calculated value of the covolume parameter. However, the a equation of state parameter for hydrogen (0.02306 Pa m6 mol-2) obtained from eq 8 is substantially different from the classical (0.020 18 Pa m6 mol-2) value. Mixing rules are required in order to obtain the corresponding parameters for the mixture. The van der Waals prescriptions2 were used in this work: a = x2a11 (1 - ~ ) ~2x(l a -~x)a12 ~ (12)
+
+
b =~ ~ 6+ 1 1( 1 - ~ ) ' b 2 2 + 2 ~ ( 1~)bl2
(13) The contribution to the equation of state parameters from interactions between unlike molecules was obtained from the following combining rules2 bl2
= 0.125(611'/~ b221/3)3
(14)
= t 1 2 ~ 1 2 ~ ~ 1 1 ~ 2 2 / ~ ~ l l ~ 2 2 ~ ~ 0 ~ (51 5 ) where tI2is an adjustable parameter which reflects the strength of unlike intermolecular interaction. These combining rules have been used extensivelygJOin calculations of the critical properties of binary mixtures. Equation 14 is substantially more accurate" than other alternatives for the calculation of the critical properties of type I11 mixtures. In some applications, particularly at relatively low temperatures and pressures, it is necessary to apply an additional empirical correction to the covolume combining rule. However, the calculation of the critical properties of a diverse range of mixturesg including type I11 mixturesI0JI consistently indicates that the magnitude of tI2makes the most important contribution. The addition of an empirical parameter to the covolume combining does not significantly improve the agreement if eq 14 is used. The additional parameter is required only if an inferior combining rule (e.g., an arithmetic average of the contribution from the two components) is e m p l ~ y e d .Another ~ alterm from eq 15 and optimize the ternative is to exclude the i12 agreement of theory with experiment by employing eq 14 with an empirical parameter. There are insufficient data in the literature to assess the validity of this approach for predicting critical equilibria. The y term was assumed to vary linearly with composition, i.e. 7 = X Y l l + ( 1 - xh22 (16) Equation 16 is the simplest method for obtaining the quantum correction term for the mixture, and it is consistent with the averaging procedure used in previous13calculations of quantum effects in binary mixtures. A more complicated alternative is to employ a quadratic average similar to that used to obtain the equation of state parameters of the mixtures. This latter approach may be particularly beneficial if the contribution of both yl1and y22are of similar magnitude. The y term of hydrogen is substantially greater than any other second component molecule studied in this work. Consequently, eq 16 is a valid approximation. The molecular weight and the critical properties of the pure substances required to calculate the equation of state parameters were obtained from a compilation by Ambrosee20 012
Results and Discussion Experimental measurements are available for the critical properties of binary mixtures containing hydrogen and (20) Ambrose, D . Yapour-Liquid Critical Properties; National Physical Laboratory: Teddington, 1980.
carbon carbon monoxide,24methane,25and ethane.26 These data form the basis of the comparison of theory with experiment presented in this work. In addition, experimental measurements have been reported for binary mixtures containing hydrogen plus a highly polar component such as methanol,28and water.29 The critical properties of these systems are likely to be substantially influenced by dipole interactions. All the systems studied exhibit type I11 behavior. A critical locus (i.e., the points signifying the critical transition), starting from the critical point of the component with the highest critical temperature, initially approaches the critical point of the other component but then deflects to higher pressures. No experimental critical data could be found at temperatures close to the critical point of hydrogen. The critical properties of the binary mixtures were calculated using the van der Waals equation. The value of the t12parameter was adjusted in order to obtain the optimum agreement between experiment and theory for the pressure-temperature minimum of the critical locus. This value of tI2was subsequently employed in the calculation of the critical properties using the equation of state with the quantum correction. It is also possible to optimize the agreement between theory and experiment for the quantum correction calculations by again using tI2as an adjustable parameter. However, such an approach is clearly not useful in determining the extent of quantum influences. The van der Waals equation was used because it is the simplest possible equation which realistically describes fluid behavior. It can also be used to calculate most aspects of the critical behavior of binary fluid mixtures, and it is the precursor of many "improved" equations of state such as the Redlich-Kwong30 and Peng-Robinson31 equations which have been widely used in chemical engineering applications. However, it is well-known that the contribution of repulsive hard-sphere interaction used in the equation is inadequate at moderate and high densities. Comparison with computer simulation and the virial expansion of a hard-sphere fluid indicate that equations like either the Guggenheim6or Carnahan-Starling7 equations provide a much better representation over the entire range of fluid densities. The main type I11 critical locus extends to a region of moderate density where it is reasonable to infer that the van der Waals is not suitable. Therefore, any change in the discrepancy between theory and experiment by accounting for quantum corrections may be due to compensating for the inherent deficiency of the equation rather than a genuine quantum effect. To test this hypothesis, the calculations were repeated using the Guggenheim equation. The results obtained using the van der Waals and Guggenheim equations for the critical properties of hydrogen argon presented in Figure 1 are typical of calculations of other mixtures containing hydrogen. Overall, the results obtained using the Guggenheim equation are not superior to those obtained from the van der Waals equation. Clearly, the deficiency in the representation of the repulsive interaction of hard spheres is not the main consideration influencing the quality of the agreement between theory and experiment. The role of quantum effects is clearly apparent in the comparison of experiment with theory for the hydrogen + argon mixture illustrated in Figure 2. The calculations using the van der Waals equation are in very poor agreement with experimental results. This is particularly the case at relatively low temperatures and moderate densities whereas the discrepancy is not so serious at higher temperatures. Accounting for the influence of quantum
+
(21) Calado, J. C. G.; Streett, W. B. Fluid Phase Equilib. 1979, 2, 275. (22) Streett, W. B.; Calado, J. C. G. J . Chem. Thermodyn. 1978,10, 1089. (23) Tsang, C. Y.; Streett, W. B. Chem. Eng. Sci. 1981, 36, 993. (24) Verschoyle, T. T. H. Philos. Trans. Roy. SOC.London, Ser. A 1931, 230, 189. (25) Tsang, C. Y.; Streett, W. B. Chem. Eng. Commun. 1980, 6, 365. (26) Heintz, A.; Streett, W. B. Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 298. (27) Brunner, E. J . Chem. Thermodyn. 1988, 20, 1397. (28) Brunner, E. J . Chem. Thermodyn. 1985, 17, 671. (29) Seward, T. M.; Franck, E. U. Ber. Bunsen-Ges. Phys. Chem. 1981, 89, 175. (30) Redlich, 0.; Kwong, J. N. S. Chem. Rev. 1949, 44, 233. (31) Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, IS, 59.
3858 The Journal of Physical Chemistry, Vol. 96, No. 9, 1992
Sadus 70 70
60
50 140
I
40
a E
\
P
30
20
-
o t - 7 - -
0
80
90
100
110 120 T/K
130
140
70
80
90
100
110
120
T/K
Figure 1. Comparison between experiment (Y) and calculationsfor the critical curve of hydrogen + argon using the van der Waals (0)and Guggenheim equations of state (A), without quantum corrections. 60
60
150
Figure 3. Comparison between experiment (Y) and calculations for the critical curve of hydrogen + nitrogen using the van der Waals equation (0)and quantum corrections (A);6 = 0.835.
,
220
7b
50
180
I l6OI
40
0,
2 120
$30
\
m
z
?ooj
20
6o 40
10
20 01
80
I
90
I
100
1
I
110 120 T/K
I
I
130
140
, 150
1
\
-
*
4
0 ,
220
I
240
I
260 T/K
I
280
300
Figure 2. Comparison between experiment (N)and calculations for the critical curve of hydrogen + argon using the van der Waals equation (0) and quantum corrections (A);E = 0.890.
Figure 4. Comparison between experiment (Y) and calculations for the critical curve of hydrogen + carbon dioxide using the van der Waals equation (0) and quantum corrections (A);6 = 0.725.
effects substantially improves the agreement with experiment. Interestingly, there is very little difference in the calculated temperaturmmposition behavior between the two approaches. At most compositions there is also very little distinction between the calculated pressures. However, the quantum correction calculations deviate slightly further away from the experimental values at compositions corresponding to the "liquid-liquid" portion of the equilibria.
The calculations for mixtures of hydrogen and either nitrogen or carbon dioxide are compared with experiment in Figures 3 and 4. If quantum corrections are neglected, then the overall agreement with the experimental pressuretemperature behavior is poor, particularly a t low temperatures. Including quantum effects results in a substantial improvement with experiment. The effect of including the quantum correction term has very little effect on the temperaturecomposition curve, whereas there is a
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3859
Binary Mixtures Containing Hydrogen 40 40
M
150
‘
i
125
i
100
30
m
m
\a
4 \a
a E
2 140
120
75
100
80
0.2
20
50
0.3
0.4
0.5
0.7
0.6
x ( W
25
10
I
I
75
85
80
$
T/K Figure 5. Comparison between experiment (It) and theory for the critical curve of hydrogen + carbon monoxide using the van der Waals equation (0)and quantum corrections (A);f = 0.830.
+
(32) Deiters, U.Chem. Eng. Sci. 1981, 36, 1139.
140 T/K
160
180
2 0
600
550
+
+
120
F i p e 6. Comparison between experiment (It) and calculations for the critical curve of hydrogen + methane using the van der Waals equation (0)and quantum corrections (A);f = 0.855.
small additional discrepancy in the pressure-composition behavior
at compositions corresponding to dense fluid equilibria. Tsang and Streettz3 compared the dew and bubble point loci of the hydrogen carbon dioxide mixture using the Redlich-K~ong,~~ Peng-R~binson,~’and D e i t e r ~equations ~~ of state. They were able to obtain reasonable agreement with experiment by using pseudovalues of the critical properties of hydrogen. It is interesting to contrast the agreement of theory with experiment for hydrogen + carbon monoxide (Figure 5 ) and mixtures containing either carbon dioxide, nitrogen, and argon. There is only a limited amount of data available for this mixture, but it is nonetheless clear that the discrepancy between theory and experiment is worse than was previously observed for the other mixtures. This increased disparity can be attributed to the influence of interactions induced by the permanent dipole of carbon monoxide. Neverthelsss, the agreement between theory and experiment is substantially improved by adding a quantum correction. The hydrogen methane (Figure 6) and hydrogen + ethane (Figure 7) systems are possibly the only examples of binary hydrogen + alkane mixtures for which experimental critical data have been reported. Very good agreement between experiment and theory can be obtained by accounting for quantum corrections in the hydrogen methane system (Figure 6). This conclusion is supported by calculations by Deiters13using his equation of state. However, no data are available for calculations using Deiters’ equation without quantum corrections. No other comparison with theory is available in the literature; however, Tsang and Streettz5 have calculated the phase envelope using the Peng-Robinson, Redlich-Kwong, and Deiters equations with “effective” critical properties to compensate for quantum factors and perturbation theory. The results were also improved by accounting for quantum effects. A similar conclusion was also reached by Chokappa et ai.35 The discrepancy between theory and experiment is considerably greater for the hydrogen + ethane mixture (Figure 7) than was obtained for the hydrogen + methane system. Accounting for
100
500 450 400 350 m a
p
300
250 200 150 100 50
0 1
I
)
I
I
I
I
I
I
120 140 160 180 200 220 240 2 0 T/K
Figure 7. Comparison between experiment (It) and calculations for the critical curve of hydrogen ethane using the van der Waals equation (0)and quantum corrections (A); = 0.700.
+
quantum influences significantly reduces the discrepancy, but it is clear that other factors are also significant. The ethane molecule is substantially larger than hydrogen and therefore, the combining rules used to obtain the contribution to the equation of state parameters are less reliable. In common with the other mixtures, quantum effects have relatively little influence on the temperature-composition behavior, although a slight improvement is observed for the pressurecomposition locus. Heintz and Streett%
3860 The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 Parameter with Respect to the TABLE I: Comparison of the Ratio of the Critical Volumes of the Components in the Binary Hydrogen( 1) Cp9(2) Mixtures component 2 v-771 PII €1 7 argon 1.153 0.890 nitrogen 1.369 0.835 carbon monoxide 1.431 0.830 carbon dioxide 1.446 0.725 methane 1.523 0.855 2.276 0.700 ethane
+
have analyzed the phase envelopes of this mixture with both the Peng-Robinson and Redlich-Kwong equations using “effective” critical constants for the critical properties of hydrogen. They obtained unsatisfactory agreement at high pressures. It is instructive to briefly examine the magnitude of the t12 parameter for the various mixtures. The Hudson and McCoub r e treatment ~ ~ ~ of dispersion forces indicates that t12should have a value of unity for equal size molecules. The parameter has also been widely interpreted as reflecting the strength of unlike interactions. Strong interactions are indicated by tI2> 1, whereas values of tl2< 1 indicate weak interaction. The magnitude of t12relative to the ratio of the critical volumes of the components of the hydrogen mixtures are compared in Table I. Weak interaction between the unlike molecules can be inferred in all cam. It is also apparent that the tI2parameter also reflects the nature of the intermolecular interaction rather than being determined solely by the size difference between the component molecules. For example, there is very little difference in the critical volumes of carbon dioxide and carbon monoxide, but there is a substantial discrepancy between the tlzvalue obtained for the 9teraction with hydrogen. This probably reflects the contribution of the permanent dipole moment of carbon monoxide. This conclusion is consistent with other workg on the sensitivity of the t12to the molecular nature of the constituent components. It should be noted that the modern-day understanding of the critical state indicates that the critical point cannot be treated analyti~ally.~~ Experimental critical exponents do not coincide with the values predicted by classical theory, which assumes the existence of an analytical function for energy. There is no universal critical exponent; instead, the values are very much substance specific. The theory of critical indices is useful in determining various properties such as the compressibility and heat capacities (33) Hudson, G. H.; McCoubrey, J. C. J . Chem. SOC.,Trans. Faraday SOC.1960, 56, 761.
(34) Levelt Sengers, J. M. H. Pure Appl. Chem. 1983, 55, 437. (35) Chokappa, D.; Clancy, P.; Streett, W. B.;Deiters, U. K.; Heintz, A. Chem. Eng. Sci. 1985, 40, 1831.
Sadus near the critical point of a pure substance but not the phase behavior of multicomponent fluids in general. A more practical approach is to modify the equation of state to improve the agreement of theory with experiment near the critical point.
Conclusions It has been demonstrated that quantum effects significantly influence the critical-phase equilibria of binary fluid mixtures containing hydrogen. The quantum effects can be accounted for by the simple addition of a theoretically based quantum correction term to the equation of state. The agreement between theory and experiment is substantially improved by using even the simple van der Waals equation with the quantum correction term. The only additional experimental input parameters required for the quantum calculations are the molecular weights of the component molecules. The tl2parameters obtained from the analysis indicate weak interaction between the unlike molecules. The values of t12are not solely influenced by the size difference between the component molecules: they also reflect the different classes of molecules involved in the interaction.
Notation Helmholtz function attractive equation of state parameter b covolume equation of state parameter conformal parameter f conformal parameter h h Planck‘s constant k Boltzmann’s constant m molecular mass n Avogadro’s number P pressure Q,, quantum correction R universal gas constant T temperature V volume W determinant defined by equation 1 X determinant defined by equation 2 X mole fraction Y determinant defined by equation 3 T 3.14159 quantum correction term defined by equation 10 Y €12 interaction parameter 0 property of the reference substance property of component 1, component 2 1, 2 att attractive property cb combinatorial property es equivalent substance property rep repulsive property Registry No. Hydrogen, 1333-74-0; argon, 7440-37-1; nitrogen, 7727-37-9; carbon monoxide, 630-08-0; carbon dioxide, 124-38-9; methane, 74-82-8; ethane, 74-84-0. A a
