Ind. Eng. Chem. Res. 1994,33, 146-150
146
Second Virial Coefficients for Chain Molecules Costas
P.Bokis and M a r c D.Donohue'
Department of Chemical Engineering, Johns Hopkins University, Baltimore, Maryland 21218-2694
Carol K.Hall Department of Chemical Engineering, North Carolina State University, Raleigh, North Carolina 27695- 7905
The importance of having accurate second virial coefficients in phase equilibrium calculations, especially for the calculation of dew points, is discussed. The square-well potential results in a simple but inaccurate equation for the second virial coefficient for small, spherical molecules such as argon. Here, we present a new equation for the second virial coefficient of both spherical molecules and chain molecules which is written in a form similar t o that for the square-well potential. This new equation is accurate in comparison to Monte Carlo simulation data on second virial coefficients for square-well chain molecules and with second virial coefficients obtained from experiments on n-alkanes. Introduction Over the past several years, considerable effort has been focused on the development of theoretically-based equations of state for chainlike molecules. One reason for this is that natural gas and petroleum contain heavy hydrocarbons, which are chain molecules. There are numerous empirical cubic equations of state that can predict the properties of natural-gas mixtures with reasonable accuracy; however, they are unreliable in predicting the dew points for mixtures containing molecules which differ in size. Accurate prediction of dew points for mixtures requires accurate information about the second virial coefficients of the components, because the dew-point pressure and the composition of the vapor phase are strong functions of the second virial coefficient. To illustrate the importance of the second virial coefficients in dew-point calculations, we present a temperature-pressure diagram for the system methane-heptane (Figure 1). In this graph, dew points determined for two sets of calculations are shown. In the first set of calculations, mean-field values of the second virial coefficients (obtained from the van der Waals equation of state) were used. In the second set, more accurate values of the second virial coefficients were used (McGlashan and Potter, 1962; McGlashan and Wormald, 1964). Although these calculations are oversimplified in that the liquid phases were treated as ideal, the large difference in the predicted dew points illustrates the importance of corrections to mean-field behavior in dew-point calculations for highly asymmetric mixtures. The purpose of this paper is to develop a functional form for the second virial coefficient of chain molecules that is both simple and accurate. Comparison is made with simulation data on chain molecules and with experimental data on n-alkanes. Spherical Molecules Molecules have kinetic energy, which is a result of their velocities relative to some fixed point of reference, and potential energy, which is a result of their positions relative to one another (Prausnitz e t al., 1986). The potential energy, I", between two spherically symmetric molecules is a function of their separation distance r.
* Author to whom
correspondence should be addressed.
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Figure 1. Comparisonof predicted dew points using completesecond virial Coefficients (solid lines) and using mean-field values for the secondvirial coefficients(dashed lines) for the binary system methane (1)-heptane (2). The correlation of McGlashan,Potter, and Wormald for the completesecond virial Coefficients for alkanes is used. Henry's law is aesumed for methane.
Several potential energy models have been proposed to take into account the dispersion forcesbetween sphericallysymmetric, nonpolar molecules. One of the simplest models for the potential energy is the square-wellpotential which has the form for r I a for a < r I Ra (1) 0 for r > Ra where r is the distance between molecular centers, e is the depth of the square well, a is the diameter of the spherical molecule, and (R- 1)ais the width of the well. Although the square-well potential is unrealistic (since it has discontinuities), it is mathematically simple and suitable for many practical calculations. The second virial coefficient calculated using the square-well potential is given by
r ( r )=
(" -c
where T is the temperature and bo = (2/3).nN*a3.
0888-5885/94/2633-0146$04.50/00 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 1, 1994 147
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Figu-2. Plotofbow elk forargon,usingeq2 (square-wellpotential).
Hinchfelder et a1, 19S4 Equation 4 Square-Well (Eq. 2)
---_
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Experimentaldata were taken from the Dymond-Smith (1980)data compilation.
It is interesting to investigate how eq 2 compares with experimental data on the second virial coefficientsof small, spherical molecules like argon. Michels et al. (1960)have suggested a simple procedure which involves graphing bo versus e/k at various temperatures. At a given temperature, one assumes an arbitrary value for elk and then calculates bo, so that eq 2 yields the experimental value of B. The calculation is repeated for other values of elk at the same temperature, and a plot of bo versus elk is obtained. Then, one repeats the calculations for other temperatures. This is illustrated in Figure 2, where we have used eq 2 together with experimental second virial coefficient data for argon. If the square-well potential captured the essential features of the real potential, all the curves would intersect at one point whose coordinates would be the "true" values of elk and bo. However, as shown in Figure 2, this is not the case. Therefore, even for a spherically symmetric, nonpolar molecule such as argon, the square-well potential is not adequate. The Lennard-Jones potential is a more realistic representation of intermolecular forces. London (1937) showed that the dispersion energy between two molecules varies proportionally to the inverse separation distance raised to the sixth power. The Lennard-Jones potential has the form
(3) When the Lennard-Jones potential is used in the statistical mechanical equation for the second virial coefficient, the required integration must be performed numerically at each temperature (results are presented by Hirschfelder et al. (1954)). Figure 3 shows a comparison of second virial coefficients calculated using the analytic solution for the square-well potential (eq 2, shown as the dashed line) with numerical results from Hirschfelder et al. (1954) (dark squares). It is clear that the square-well expression does not provide a good representation to the second virial coefficient for the Lennard-Jones potential, and hence it is not surprising that it does not provide a good fit for real molecules. We have examined various ways to improve the performance of eq 2. One possibility is to adjust the values of the well width R and the well depth e; however, this introduces two additional adjustable parameters. One form that seems particularly useful and introduces only one adjustable parameter is
B(7') = bo[ 1 - 3 R 3 - l)(eaf'kT-l)]
(4)
where a is introduced as an adjustable parameter. By
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Figure 3. Lennaid-Jones second virial coefficients. Calculations with the square-well and the modified square well (eq 4) expressions also are shown. 100
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Figure 4. Plot of bo vs elk for argon, using eq 4 (parameter a = 2/3). Experimentaldata were taken from the Dymond-Smith (1980) data compilation.
fitting this parameter to the Lennard-Jones data for the second virial coefficient (Hirschfelder et al., 19541,we find good agreement between eq 4 and the Lennard-Jones second virial coefficient data with a = 0.667for the range of temperatures normally encountered in natural-gas and petroleum processing (0.5 < kT/e < 1.5). This is shown in Figure 3. Though the agreement at high temperatures does not appear to be very accurate, temperatures above 2elk are seldom encountered in actual processes. The good agreement between the Lennard-Jones second virial coefficient data and eq 4 using only one adjustable parameter suggests that eq 4 also may be used to fit experimental data. In Figure 4 we construct a graph of bo vs elk for argon, in order to see how well eq 4 fits experimental second virial coefficient data on argon. We see that, by adjusting the parameter a,all isotherms on the bo vs elk plot can be made to intersect at the same point (in contrast to the case of the square-well potential, shown in Figure 2). For argon, the optimal value of the parameter a! was found to be 0.64,in close agreement with the value for the Lennard-Jones fluid. Chain Molecules In this section we extend the discussion to chain molecules. The idea here is to see whether eq 4 also can
148 Ind. Eng. Chem. Res., Vol. 33, No. 1, 1994 Table 1. Parameters Monte Carlo Data n un *P 2 1.40 1.26 8 2.80 2.91 16 4.05 4.62
u.
and a in Eq 6 from Fitting the Rn
R,b
a
1.29 1.18 1.08
1.40 1.17 1.11
1.89 3.87 4.70
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a Diameter of gyration (Denlinger and Hall, 1990). Based on diameter of gyration.
be used to correlate second virial coefficient data for chainlike molecules. Although at moderate and high densities chain molecules can be treated using a segmentsegment approach, this is not the case at low densities. At low densities interactions occur on a molecule-molecule basis; in this case, the chain molecule can be viewed as a large "equivalent" square-well sphere whose radius is approximately equal to the radius of gyration of the chain molecule. Since the center of a segment on one chain molecule can fall simultaneously into the attractive wells of several segments on a different molecule, the well depth of the large "equivalent" square-well sphere is not simply related to the well depth of a single squqe-well segment. Therefore, there exists an effective value of the &equivalent" square-well sphere's attractive energy. Finally, although the radius of the equivalent sphere is greater than that of the segment, the range of influence (Le., the actual well width) of the equivalent sphere remains the same as that of the segment. Therefore, the width parameter (R in eq2) of the square well surrounding the large equivalent sphere is smaller than the width parameter for a simple segment ( R = 1.5). To test the accuracy of eq 4 for chain molecules, we have attempted to fit the computer simulation data of Wichert and Hall (1993) on second virial coefficients for square-well chains. Wichert and Hall used a Monte Carlo integration technique to calculate the second virial coefficients for square-well dimers, 8-mers, and 16-mers at temperatures that range from near the critical point to infinity. At the infinite-temperature limit, the second virial coefficient has only a repulsive contribution and therefore is positive. At finite temperatures, the attractive contribution becomes significant, resulting in a smaller second virial coefficient (which becomes negative at low temperatures). The expression for the second virial coefficient that we used to fit Wichert and Hall's simulation data is
B,(v =
Fu:[
1 1- -(R3 a - l)(e"f/kT-I)]
where R = 1.5 and is the diameter of the "equivalent sphere" for the chain molecule, and is determined by adjusting the infinite-temperature limit of eq 5 to the infinite-temperature limit of the simulation data on second virial coefficients for chains (Wichert and Hall, 1993).We fit the simulation data for square-well second virial coefficientswith eq 5,using only one adjustable parameter, a. Results are presented in Table 1, which shows values of the parameters o n and a,and also in Figure 5, where we compare eq 5 to the simulation data. One sees that the fits are very good. In order to test the idea of the equivalent sphere, we rewrite eq 5 in the form
&,(n= %3 :[1-
(R: - l)(ea'/kT- I)]
(6)
where R, is the well width of the equivalent sphere, Rn3
- 1= (U3- l ) / a , In the case of 16-mers,the best fit value of R, = 1.08. This suggests that a freely-jointed 16-mer molecule at the second virial coefficient limit is equivalent
S I chain. Dimers (YC) &men (Mc) Id-m.n (UC) Equation 5 SVC for
A
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kT/c Figure 6. Second virial coefficients for square-well dimers, 8-mers, and 16-mers (Wichert and Hall,1993), and correlation using eq 5.
to a spherical molecule with a diameter u n = 4 . 0 5 ~(from Table 1) and a square well of width Rn = 1.08. It is interesting to compare this result with calculations based exclusively on molecular geometry. At the second virial coefficient limit, one would expect that the 16-mer molecule will behave as a spherical molecule with radius equal to the chain radius of gyration, and a well width equal to that of an individual segment. Denlinger and Hall (1990) calculated the mean radius of gyration for isolated n-mers (their results also are presented in Table 1). For a 16-mer, they found the diameter of gyration to 4.62~.The well width of this spherical molecule should therefore beR, = (4.62~+ 0.50)/4.62u = 1.11. Thisnumber (1.11)is reasonably close to the value of 1.08 that was found by fitting the simulation data for 16-mers. Comparison with Experimental Data The functional form for the second virial coefficient developed in this work is compared with experimental data for the second virial coefficients of n-alkanes up to decane. The data are fitted according to eq 5, with R = 1.5. There are three adjustable parameters in the model: the characteristic diameter of the chain molecule, Un,the segmental energy of interaction, tlk, and the parameter a, which describes the temperature dependence of the attractive part of the second virial Coefficient. Results are tabulated in Table 2 and also are presented in Figure 6. An excellent fit is obtained in all cases considered. In Table 2, the errors are presented in terms of both the average absolute percent deviation and the average absolute deviation. The latter is meaningful, since at high temperatures the second virial coefficient is small and the percent errors are verylarge even when the absolute errors are small. Figure 7 shows the variation of the three parameters of the model with the number of carbon atoms in the alkane molecule. One sees that the size parameter un3 does not increase linearly with the carbon number (as does the size parameter u* in PHCT of Donohue and Prausnitz (1978), for example). This is expected since, according to the model developed here, un should behave like the radius of gyration of the molecule, which is not a linear function of the chain length. We also have compared the expression proposed in this paper and the correlation of McGlashan, Potter, and
Ind. Eng. Chem. Res., Vol. 33, No. 1, 1994 149 Table 2. Results from Fitting P u r e Component Second Virial Coefficient Experimental Data T range (K) u,,s (cms/mol) compound clk (K) cl A A D % O 81-4W 16.99 argon 153.1 0.642 0.93 (12) 9&476d 22.75 190.4 0.727 methane 1.79 (19) 268.5 0.860 273-623' 39.77 0.23 (16) ethane 210-85od 51.20 350.1 1.536 1.84 (18) butane 55.50 420.4 1.936 253-91 Id 1.90 (16) hexane 482.5 2.123 280-1136d 65.83 3.30 (17) octane 30&1235d 73.33 494.1 2.684 4.30 (17) decane
A A D b (cm3/mol) 0.349 1-1 0.637 110.81 0.111 12.731 2.65 120.71 9.92 14.191 17.5 161.81 27.73 1335.31
Numbers in parentheses indicate number of data points used in the correlation. * Numbers in square brackets indicate average absolute errors from the McGlashan-Potter correlation. Dymond and Smith (1980). API, project 44,section h. e Douslin and Harrison (1973). 0 2.0
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T (-1 Figure 6. Experimental second virial coefficient data for small and large molecules, and correlation using eq 5.
Wormald (McGlashan and Potter, 1962; McGlashan and Wormald, 1964)that was used to generate the results shown in Figure 1; the results are presented in the last column of Table 2. One sees that the McGlashan, Potter, and Wormald correlation gives much larger errors than our model. The deviation of this correlation from the experimental data is larger in the low-temperature range, which is usually of industrial interest. This is shown in Figure 8, where we plot second virial coefficients for decane calculated with the two equations. The model that we propose in this paper is in excellent agreement with the experimental data, whereas the correlation of McGlashan and co-workers deviates from the data, especially at low temperatures.
Conclusions In this work it is shown that the second virial coefficient has an important role in phase equilibrium calculations, especially calculations of dew points. The square-well potential function provides a simple expression for the second virial coefficient of small spherical molecules. Unfortunately, this expression does not agree with experimental data. We have developed a new expression for the second virial coefficient of chain molecules which is similar to that for the square-well potential. This equation is based on the assumption that chain moleculesin the second virial coefficient limit can be treated as large spherical molecules
2
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which interact with each other via their "surface" segments. Also, because segmental wells overlap, the attractive well depth of the chain molecule is deeper than that of its
160 Ind. Eng. Chem. Res., Vol. 33, No. 1, 1994 individual segments. This behavior is t a k e n i n t o account through the parameter a.
This new equation was tested against Monte Carlo data for square-well dimers, 8-mers, and 16-mers, and the agreement is very good. Very good agreement also is obtained when this model is compared to experimental data on the second virial coefficients for argon and for normal hydrocarbons, up to decane.
Acknowledgment Support of t h i s research b y the Gas Research Institute under Contract No. 5089-260-1888 and the Office of Basic Energy Sciences and Division of Chemical Sciences, U.S. Department of Energy, under Contract No. DE-FGOB87ER13777 and Contract No. DE-FG05-91ER1481 is gratefully acknowledged. We also are thankful to Dr. Arun Thomas for his help with the dew-point calculations.
Nomenclature B(T) = second virial coefficient bo = second virial coefficient for a hard sphere k = Boltzmann’s constant n = number of segments N A = Avogadro’s number r = distance R = range of the attractive well T = temperature u* = size parameter in PHCT Greek Symbols a = parameter in eqs 4 and 5 r(r) = potential energy c = square-well depth B = diameter of segment on = diameter of equivalent sphere
Subscripts g = gyration n = n-mers
Literature Cited Denlinger,M. A.; Hall, C. K. Molecular-DynamicsSimulationResults for the Pressure of Hard-Chain Fluids. Mol. Phys. 1990,71,541. Donohue, M. D.; Prausnitz, J. M. Perturbed Hard-Chain Theory for Fluid Mixtures: Thermodynamic Properties for Mixtures in Natural Gas and Petroleum Technology. AIChE J. 1978,24,849. Douslin, D. R.; Harrison, R. H. Pressure, Volume, Temperature Relations of Ethane. J. Chem. Thermodyn. 1973,4491. Dymond, J. H.; Smith, E. B. The Virial Coefficientsof Pure Gases and Mixtures. A Critical Compilation; Clarendon Press: Oxford, 1980. Hirschfelder, J. 0.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley & Sons, Inc.: New York, 1954. London, F. The GeneralTheory of Molecular Forces. Trans.Faraday SOC.1937, 33, 8. McGhhan,M.L.;Potter,D. J.B. AnApparatusfortheMeasurement of the Second Virial Coefficients of Vapours; the Second Virial Coefficients of Some n-Alkanes and Some Mixtures of n-Alkanes. Proc. R. SOC.(London) 1962, A267,478. McGlashan, M. L.; Wormald, C. J. Second Virial Coefficientsof Some Alk-1-enes, and of a Mixture of Propene+Hept-1-ene. Trans. Faraday SOC. 1964,60,646. Michels, A.; DeGraaff, W.; Ten Seldam, C. A. Virial Coefficients of Hydrogen and Deuterium at Temperatures Between -175 OC and +150 OC. Conclusions from the Second Virial Coefficient with Regards to the Intermolecular Potential. Physica 1960,26,393. Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1986. Wichert, J. M.; Hall, C. K. Computer Simulations of Second Virial Coefficients for Square-Well Chains. 1993, submitted to Macromolecules. Received for review May 19, 1993 Accepted September 29, 1993’ 0 Abstract published in Aduance ACS Abstracts, December 1, 1993.
