Second through Fifth Virial Coefficients for Model Methane-Ethane ...
Second through Fifth Virial Coefficients for Model Methane-Ethane Mixtures
Hye Min Kim, Andrew J. Schultz, and David A. Kofke Department of Chemical and Biological Engineering University at Buffalo, The State University of New York Buffalo, NY 14260-4200
Corresponding author: David A. Kofke Phone: 716-645-1173 Fax: 716-645-3822 [email protected]
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Abstract We report the virial coefficients for model methane-ethane mixtures up to the fifth order from 190K to 320K. United-atom models with modified Buckingham exp-6 sites are used for both methane and ethane. Comparison of computed second- and third-order coefficients to experimental values from the literature reveals some shortcomings in the molecular models, showing differences of order 5-10% for B2, and 20-30% for B3. We use the virial equation of state (VEOS) to obtain spinodal curves and critical points as a function of mole fraction. Despite the inaccuracies in the models’ virial coefficients, we find that the VEOS provides accurate critical points for the methane-ethane mixture, in comparison to experiment.
1. Introduction Virial equation of state (VEOS) describes the thermodynamic behavior of gases at low density[1]:
p = 1 + B2 ρ + B3 ρ 2 + B4 ρ 3 + B5 ρ 4 + ρkT
(1)
where p is the pressure, ρ is the number density, k is the Boltzmann constant, T is the absolute temperature and Bn is the nth virial coefficient. The virial series must be truncated to be used practically, and we use VEOSn to represent the virial series truncated after Bnρn-1. Mixture virial coefficients are expressed rigorously in terms of virial coefficients of the pure components (Bα0, B0α), cross virial coefficients (Bαβ), and the mole fraction of each component (xα) [2]. For a binary mixture:
B2 = x12 B20 + 2x1x2 B11 + x22 B02 2
B3 = x13 B30 + 3x12 x2 B21 + 3x1x22 B12 + x23 B03 B4 = x14 B40 + 4x13 x2 B31 + 6x12 x22 B22 + 4x1x23 B13 + x24 B04 B5 = x15 B50 + 5x14 x2 B41 + 10x13 x22 B32 + 10x12 x23 B23 + 5x1x24 B14 + x25 B05
(1)
In general, coefficient Bαβ is given as a configurational integral involving α molecules of species 1 and β molecules of species 2. [1, 3] In this work, we calculated the virial coefficients of methane-ethane mixture up to the fifth order. These virial coefficients are used to predict the location of the critical point.
2. Model and method 2.1. Model The modified Buckingham exp-6 potential is used to describe the n-alkane molecules [4-7]. This model was employed by Lenart and Panagiotopoulos [5] for the methane-ethane system as part of a larger study on tracing critical lines of binary mixtures by molecular simulation. The exp-6 potential is parameterized to the vapor-liquid coexistence properties for a range of the alkane chain lengths and has higher accuracy than the usual Lennard-Jones intermolecular potential for the alkane molecules.
⎧ εα 6 r r [ exp(α [1− ]) − ( m )6 ] ⎪ rm r U disp (r) = ⎨ α − 6 α ⎪∞ ⎩
r > rmax
(5)
r ≤ rmax
where ε is the minimum potential energy, α is the repulsive steepness of the potential and rm is the separation distance between the alkane molecules when the exp-6 potential is the lowest 3
value. To prevent the exp-6 potential being negative at very small radial distance, we set rmax as the distance where the potential is a maximum and set the potential to be positive infinity when the separation distance is less than rmax. Methane and ethane molecules are represented using the united-atom approximation (i.e. no explicit hydrogen atoms are included), so the only interaction sites are located on the carbon atoms. The bond length separating the two ethane sites is fixed at 1.839 Å. For the mixtures, we use Lorentz-Berthelot combining rules to develop the cross-species interactions from those for the pure components. In particular:
1 2
σ ij = (σ ii + σ jj )
α ij = α iiα jj
ε ij = ε iiε jj
(6)
The quantity σ represents a size parameter, specifically where the potential energy becomes zero, and can be numerically determined. The model is defined by specifying σ values, which are then used to calculate rm values. The potential parameters are listed in Table 1. These are the same as used by Lenart and Panagiotopoulos in their Monte Carlo study of this system [5]. Table 1. Potential parameters Species
ε/k (K)
σ (Å)
α
rm (Å)
rmax (Å)
methane[7] 160.3
3.741
15
4.184
0.704
ethane[4]
129.6
3.679
16
4.094
0.575
mixture
144.13
3.71
15.49
4.139
0.64
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Nuclear quantum effects can make a significant contribution to the virial coefficients, particularly for methane, which has a fairly small moment of inertia. At temperatures considered in the present study, quantum contributions to B2 for methane are on the order of 2% [8]. However, inasmuch as the models examined here are determined by fitting to experiment, and are not derived from ab initio calculations, any such quantum contributions should be considered as included implicitly in the parameterization of the model. Accordingly, in this work we do not attempt to include such contributions separately.
2.2. Critical properties In this study, we focus on locating the critical point of methane-ethane mixture, and then calculate the critical properties of methane(1)-ethane(2) mixtures from the virial coefficients. As described in [3], we begin by considering the spinodal curve defined by the criterion 2 ⎛ ⎛ ∂µ ⎞ 1 ⎛ ∂ p ⎞ ⎞ 1− x1 ⎛ ∂ p ⎞ 1 − ⎜ ⎜⎜ ⎟ ⎟⎟ − ρ 2 ⎜ ∂x ⎟ = 0 ⎜⎝ ⎝ ∂x1 ⎟⎠ ρ ∂x ⎝ ⎠ T ,ρ ⎠ ⎝ 1 ⎠ T ,ρ 1 T ,ρ 1
⎛ ∂p⎞ ⎜⎝ ∂ ρ ⎟⎠ T ,x
(7)
where µα is the chemical potential of species α and xα is the mole fraction of species α. We trace points along the spinodal curve until we reach a point that also satisfies the critical criterion [3], ( ρµ xx − pxx ) + κ ( px2 − 3(1− x1 ) pxx px ) + κ 2 (1− x1 )(3ρ px2 pxρ − 2 px3 ) −κ 3 (1− x1 ) ρ 2 px3 pρρ = 0
(8)
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where the subscripts are partial derivatives with respect to x ( ≡ x1) and/or ρ, and κ is the compressibility
1 ⎛ ∂ρ ⎞ ⎜ ⎟. ρ ⎜⎝ ∂p ⎟⎠
In some cases, the critical criteria are not satisfied at some temperatures. In this situation, we can estimate the critical point using a near-miss criterion [3, 9].
2.3. Computational Methods For the calculation of cluster integrals shown in VEOS, we applied the Mayer sampling Monte Carlo (MSMC) simulation method [10]. Via Mayer sampling, the average values of the ratio of the cluster integrals of the system of interest (or target system) to that of an already known cluster integral (reference system) is calculated. In the present work, we employed the overlap sampling implementation of MSMC [11]. We use hard-spheres of diameter 0.7(σmethane+ σethane) as a reference system.
3. Results and discussion We calculated the virial coefficients of methane-ethane mixture from B2 to B5. A total of 18 Bαβ coefficients are evaluated to describe the virials B2 to B5. The virial coefficients are presented in Tables 2-5, and data are plotted against temperature in Figure 1, which for comparison also gives available values derived from experimental data [12-18]. In the tables and figures, Bαβ represents the virial coefficient of the mixture containing α molecules of methane and β molecules of ethane. For the calculation of every virial coefficient, we performed 10 MSMC simulations with 109 Monte Carlo steps each. We obtained good precision for all temperatures. In general, the virial coefficients are very negative at low temperatures, pass 6
through a positive maximum at intermediate temperatures before decaying toward zero at high temperatures. For many methane-rich coefficients, this behavior would occur at temperatures below 190K, while for the second virial coefficient, the high-temperature behavior would occur at temperatures above 320K. The mixture coefficients tend to adopt behavior that is qualitatively between that of the pure components. The computed B2 values for pure components and their mixtures agree well with the experiment, methane slightly more so than ethane. B3 values for pure methane are slightly lower than the experimental data, and more so at lower temperature, but the agreement is reasonably good. However, B3 for pure ethane is much lower than the experimental data over the entire temperature range. B4 data derived from experiment have been published for methane [18], and not surprisingly they differ from the values calculated for the model, which are about a factor of 7 larger; high-order data are difficult to extract from PVT measurements, so this comparison should not be used to form conclusions about the model. The comparison is at least consistent in suggesting a relatively weak dependence on temperature for B40 in this narrow range where the data overlap. Table 2. Second virial coefficients calculated from MSMC for methane-ethane mixtures. Here and in subsequent tables, numbers in parentheses indicate the 68% confidence limits in the last digit(s) of the reported values.
T (K) 190 200 210 220 230 240 250 260 270
B20 -107.011(8) -96.90(2) -88.03(3) -80.17(4) -73.09(4) -66.76(5) -61.04(6) -55.82(4) -51.03(4)
B2 (cm3/mol) B11 -208.05(2) -188.88(2) -172.20(7) -157.47(4) -144.41(7) -132.89(7) -122.40(8) -112.92(6) -104.48(4)
B02 -415.30(5) -374.99(3) -340.42(16) -310.63(8) -284.6(1) -261.79(8) -241.54(7) -223.5(1) -207.12(8) 7
280 290 300 310 320
-46.66(4) -42.70(2) -38.98(6) -35.54(4) -32.432(6)
-96.66(4) -89.53(3) -82.99(5) -76.97(6) -71.421(5)
-192.54(9) -179.32(4) -167.23(6) -156.23(2) -146.063(16)
Table 3. Third virial coefficients calculated from MSMC for methane-ethane mixtures.
T (K) 190 200 210 220 230 240 250 260 270 280 290 300 310 320
B3 (cm3/mol)2 B30 2575.9(15) 2527(3) 2459.3(11) 2381.9(14) 2303.5(8) 2226.6(13) 2152.5(6) 2083.9(4) 2021.3(6) 1964.3(6) 1910.9(7) 1863.7(5) 1820.1(3) 1780.7(7)
B21 3627(5) 3779(7) 3809(2) 3761(3) 3672(3) 3572(4) 3451(3) 3331.0(6) 3213(2) 3106(2) 3003(2) 2905.7(17) 2819(2) 2736.4(10)
B12 2125(7) 3715(5) 4611(7) 5085(7) 5300(5) 5345(3) 5307(4) 5201(4) 5066.8(15) 4916(4) 4758(3) 4600(2) 4449(4) 4301(2)
B03 -18223(27) -8547(26) -2509(13) 1294(7) 3703(10) 5209(9) 6105(11) 6611(9) 6868(7) 6958(7) 6944(4) 6855(2) 6720(8) 6554(2)
Table 4. Fourth virial coefficients calculated from MSMC for methane-ethane mixtures.
T (K) 190 200 210 220 230 240 250 260 270 280 290
B40 9.51(3)×104 8.59(4)×104 7.50(4)×104 6.639(11)×104 5.813(15)×104 5.15(3)×104 4.62(2)×104 4.18(2)×104 3.882(16)×104 3.635(16)×104 3.430(13)×104
B31 1.63(3)×105 1.661(12)×105 1.557(10)×105 1.371(11)×105 1.195(3)×105 1.043(3)×105 8.99(2)×104 7.92(3)×104 6.96(3)×104 6.181(19)×104 5.620(15)×104
B4 (cm3/mol)3 B22 2.3(14)×104 1.93(2)×105 2.51(2)×105 2.546(19)×105 2.390(9)×105 2.114(16)×105 1.875(6)×105 1.635(9)×105 1.407(4)×105 1.220(4)×105 1.077(3)×105
B13 -2.349(16)×106 -9.16(7)×105 -2.03(7)×105 1.29(7)×105 2.74(2)×105 3.22(4)×105 3.22(2)×105 3.02(3)×105 2.707(16)×105 2.43(10)×105 2.110(11)×105
B04 -2.062(12)×107 -1.077(7)×107 -5.599(12)×106 -2.86(2)×106 -1344(6)×106 -5.31(4)×105 -9.1(8)×104 1.41(4)×105 2.51(4)×105 2.86(3)×105 2.931(10)×105
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300 310 320
3.307(9)×104 3.218(10)×104 3.167(8)×104
5.163(16)×104 4.799(11)×104 4.545(11)×104
9.48(3)×104 8.46(4)×104 7.658(16)×104
1.874(9)×105 1.634(5)×105 1.452(5)×105
2.81(2)×105 2.613(17)×105 2.407(7)×105
Table 5. Fifth virial coefficients calculated from MSMC for methane-ethane mixtures.
T (K) 190 200 210 220 230 240 250 260 270 280 290 300 310 320
B50
6.5(10)×105 2.4(7)×105 -3.6(9)×105 -7.3(7)×105 -9.3(5)×105 -9.4(4)×105 -9.2(3)×105 -8.3(4)×105 -7.0(4)×105 -4.7(1)×105 -3.4(2)×105 -1.7(2)×105 -1.9(12)×104 1.3(2)×105
B41
2.7(4)×106 2.8(5)×106 2.16(15)×106 7(2)×105 -3.0(7)×105 -1.17(9)×106 -1.59(6)×106 -1.61(9)×106 -1.58(6)×106 -1.51(4)×106 -1.33(12)×106 -1.10(4)×106 -8.0(2)×105 -5.6(2)×105
B5 (cm3/mol)4 B32 B23
-9.3(17)×106 7.3(9)×106 1.01(8)×107 7.1(3)×106 4.1(11)×106 1.4(4)×106 -4(2)×105 -1.12(12)×106 -1.90(14)×106 -2.58(11)×106 -2.64(10)×106 -2.36(8)×106 -2.37(5)×106 -1.95(5)×106
-3.73(7)×108 -7.0(4)×107 -5.4(19)×106 1.36(12)×107 1.41(13)×107 1.19(7)×107 8.2(4)×106 2.8(5)×106 1.3(4)×106 -1.0(2)×106 -2.2(2)×106 -3.0(2)×106 -3.68(9)×106 -3.63(13)×106
B14
-2.595(15)×109 -1.055(12)×109 -4.09(9)×108 -1.39(4)×108 -2.9(7)×107 4(2)×106 1.63(11)×107 1.30(17)×107 1.25(9)×107 8.4(8)×106 1.9(6)×106 -4(4)×105 -2.1(4)×106 -3.7(4)×106
B05
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-2.108(10)×10 -9.31(5)×109 -4.28(5)×109 -1.984(16)×109 -9.4(3)×108 -4.31(11)×108 -1.99(5)×108 -8.1(7)×107 -3.0(2)×107 -1.47(16)×107 -4.4(13)×106 -5.0(12)×106 -2.6(8)×106 -3.8(6)×106
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Figure 1. Virial coefficients for methane-ethane mixture for n = 2–5 computed from MSMC; a, Hoover et al. [12]; b, Hou et al. [13]; c: McElroy et al. [14]; d: Dantzler et al. [15]; e: Funke et al. [16]; f: Kleinrahm et al. [17]; g: Douslin et al. [18]. Bαβ represents the virial coefficient of the mixture containing α molecules of methane and β molecules of ethane. For the values calculated in this work, error bars (68% confidence) are shown where they are discernable on the scale of the figure.
Deviations of the pressures for pure methane and ethane from experimental data [19] at 300K are presented in Figs. 2 and 3, respectively. For methane, the deviations are less than 1%, except
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VEOS2 which deviates by 5% at high density. The discrepancy is not due to the use of the VEOS per se, as the comparison of VEOS4 and VEOS5 suggests that the series has converged for the state conditions shown in the figure. Rather the deviations come primarily from the slightly too-large value of B2, which is evidenced by the non-zero slope of the VEOS deviation lines at zero density, and reflects on the accuracy of the model. This is also visible in the direct comparison of B2 in Fig. 1, where without the enlarged scale given by Fig. 2, the discrepancy does not seem as significant. B3 for pure methane is smaller than the experimental data, and this tends to compensate for the slight error in B2. Overall, the deviation in pressure still is small out to the highest densities shown (about half the critical density). For ethane in Fig. 3, the pressures from experimental data deviate by 5% at high density. Like pure methane, a too-large value of B2 for ethane contributes most to the deviation for all the VEOSn, and this error is offset partially by a too-small value of B3.
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Figure 2. Deviations of the pressure for methane from experimental data [19] at 300K. The VEOS4 and VEOS5 lines are not distinguishable on the scale of this figure. VEOS lines for Hou use the virial coefficients from Ref. [13].
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Figure 3. Deviations of the pressure for ethane from experimental data [19] at 300K. VEOS lines for Hou use the virial coefficients from Ref. [13]. VEOS3 through VEOS5 are used to compute density- and pressure-composition spinodal lines, plotted in Fig. 4 and Fig. 5, respectively. Every spinodal line ends at the critical point. Again, the consistency between VEOS4 and VEOS5 suggests convergence. Good agreement is seen in comparison with the methane-ethane pressure-composition critical line as given by experiment [20]. Experimental data for the density are not available. It is likely that good agreement would not be seen with VEOS, as previous work consistently finds that the critical density is underestimated by VEOS [21-24].
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Figure 4. Density-composition projection of methane spinodal lines and the critical locus for a methane-ethane mixture: critical locus computed from VEOS3 (black long dash dot line); VEOS4 (red dotted line); VEOS5 (blue dash dot line); spinodal lines computed from VEOS3 (dash lines) and VEOS5 (solid lines) are shown.
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Figure 5. Pressure-composition projection of methane spinodal lines and the critical locus for a methane-ethane mixture: critical locus computed from VEOS3 (black long dash dot line); VEOS4 (red dotted line); VEOS5 (blue dash dot line); spinodal lines computed from VEOS3 (dash lines) and VEOS5 (solid lines) are shown. Experimental data [20] are indicated with open circles. In Fig. 6, the critical temperature as a function of methane composition is presented. Van Konynenburg and Scott [25] introduced six types of mixture critical behavior. The methaneethane mixture is classified as type I, meaning the mixture exhibits complete mixing between the two components over the whole composition range. The locus of critical points from VEOS4 shows significant improvement over VEOS3, while VEOS5 shows little deviation from VEOS4,
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suggesting that the series has converged. Again, both VEOS4 and VEOS5 have very good agreement with the experimental data [20].
Figure 6. Temperature-composition projection of methane critical locus for a methaneethane mixture: critical locus computed from VEOS3 (black dash line); VEOS4 (red dotted line); and VEOS5 (blue solid line); literature data for the model used here [5] (green diamond); and experimental data [20] (black cross). Thinner lines and open symbols indicate critical point is obtained using near-miss critical condition.
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4. Conclusions The comparison of the virial coefficients for the model to experimental values indicates some imperfection in the model, which is not surprising given its simple form. The model was parameterized to describe vapor-liquid coexistence data, and thus one should expect it to perform reasonably well in describing higher-density states. The compensating errors in B2 and B3 show in part how this is done. The good performance of the VEOS in describing the critical line is a consequence of the ability of the VEOS to describe the model properties at the relevant conditions, combined with the ability of the empirically-determined model to characterize this behavior even though its description of two- and three-body interactions is imperfect. In future studies it would be of interest to examine more complex alkane mixtures such as more asymmetric systems such as ethane-decane, or ternary or higher-component alkane mixtures. Calculations for such systems are made more difficult by the need to incorporate contributions related to molecular flexibility [24].
Acknowledgements This work is supported by the National Science Foundation under grant CBET-0854340. Calculations were performed using resources from the University at Buffalo Center for Computational Research. We acknowledge improvements to this manuscript resulting from very helpful comments contributed by an anonymous reviewer.
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Kleinrahm, R., W. Duschek, W. Wagner, and M. Jaeschke, Measurement and Correlation of the (Pressure, Density, Temperature) Relation of Methane in the Temperature-Range from 273.15-K to 323.15-K at Pressures up to 8 Mpa. Journal of Chemical Thermodynamics, 1988. 20(5): p. 621-631. Douslin, D.R., R.H. Harrison, R.T. Moore, and J.P. MuCullough, P-V-T Relations for Methane. Journal of Chemical & Engineering Data, 1964. 9(3): p. 358-363. Lemmon, E.W., M.O. McLinden, and D.G. Friend, Thermophysical Properties of Fluid Systems., in NIST Chemistry WebBook, NIST Standard Reference Database Number 69 (http://webbook.nist.gov/chemistry/fluid), P.J. Linstrom and W.G. Mallard, Editors. Bloomer, O.T., D.C. Gami, and J.D. Parent, Physical-chemical properties of methaneethane mixtures. Inst. Gas Technol., Research Bull, 1953. No. 22: p. 39 pp. Schultz, A.J. and D.A. Kofke, Sixth, seventh and eighth virial coefficients of the LennardJones model. Mol. Phys., 2009. 107(21): p. 2309-2318. Schultz, A.J. and D.A. Kofke, Virial coefficients of model alkanes. The Journal of Chemical Physics, 2010. 133(10): p. 104101. Shaul, K.R.S., A.J. Schultz, and D.A. Kofke, The effect of truncation and shift on virial coefficients of Lennard–Jones potentials. Collection of Czechoslovak Chemical Communications, 2010. 75(4): p. 447-462. Shaul, K.R.S., A.J. Schultz, and D.A. Kofke, Mayer-sampling Monte Carlo calculations of uniquely flexible contributions to virial coefficients. Journal of Chemical Physics, 2011. 135(12): p. 124101. Van Konynenburg, P.H. and R.L. Scott, Critical lines and phase equilibriums in binary Van der Waals mixtures. Philos. Trans. R. Soc. London, Ser. A, 1980. 298: p. 495-540.
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Figure Captions Figure 1. Virial coefficients for methane-ethane mixture for n = 2–5 computed from MSMC; a, Hoover et al. [12]; b, Hou et al. [13]; c: McElroy et al. [14]; d: Dantzler et al. [15]; e: Funke et al. [16]; f: Kleinrahm et al. [17]; g: Douslin et al. [18]. Bαβ represents the virial coefficient of the mixture containing α molecules of methane and β molecules of ethane. For the values calculated in this work, error bars (68% confidence) are shown where they are discernable on the scale of the figure. Figure 2. Deviations of the pressure for methane from experimental data [19] at 300K. The VEOS4 and VEOS5 lines are not distinguishable on the scale of this figure. VEOS lines for Hou use the virial coefficients from Ref. [13]. Figure 3. Deviations of the pressure for ethane from experimental data [19] at 300K. VEOS lines for Hou use the virial coefficients from Ref. [13]. Figure 4. Density-composition projection of methane spinodal lines and the critical locus for a methane-ethane mixture: critical locus computed from VEOS3 (black long dash dot line); VEOS4 (red dotted line); VEOS5 (blue dash dot line); spinodal lines computed from VEOS3 (dash lines) and VEOS5 (solid lines) are shown. Figure 5. Pressure-composition projection of methane spinodal lines and the critical locus for a methane-ethane mixture: critical locus computed from VEOS3 (black long dash dot line); VEOS4 (red dotted line); VEOS5 (blue dash dot line); spinodal lines computed from VEOS3 (dash lines) and VEOS5 (solid lines) are shown. Experimental data [20] are indicated with open circles.
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Figure 1. Temperature-composition projection of methane critical locus for a methaneethane mixture: critical locus computed from VEOS3 (black dash line); VEOS4 (red dotted line); and VEOS5 (blue solid line); literature data for the model used here [5] (green diamond); and experimental data [20] (black cross). Thinner lines and open symbols indicate critical point is obtained using near-miss critical condition.
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Hye Min Kim, Andrew J. Schultz, and David A. Kofke Department of Chemical and Biological Engineering University at Buffalo, The State University of New York Buffalo, NY 14260-4200
Corresponding author: David A. Kofke Phone: 716-645-1173 Fax: 716-645-3822 [email protected]
1
Abstract We report the virial coefficients for model methane-ethane mixtures up to the fifth order from 190K to 320K. United-atom models with modified Buckingham exp-6 sites are used for both methane and ethane. Comparison of computed second- and third-order coefficients to experimental values from the literature reveals some shortcomings in the molecular models, showing differences of order 5-10% for B2, and 20-30% for B3. We use the virial equation of state (VEOS) to obtain spinodal curves and critical points as a function of mole fraction. Despite the inaccuracies in the models’ virial coefficients, we find that the VEOS provides accurate critical points for the methane-ethane mixture, in comparison to experiment.
1. Introduction Virial equation of state (VEOS) describes the thermodynamic behavior of gases at low density[1]:
p = 1 + B2 ρ + B3 ρ 2 + B4 ρ 3 + B5 ρ 4 + ρkT
(1)
where p is the pressure, ρ is the number density, k is the Boltzmann constant, T is the absolute temperature and Bn is the nth virial coefficient. The virial series must be truncated to be used practically, and we use VEOSn to represent the virial series truncated after Bnρn-1. Mixture virial coefficients are expressed rigorously in terms of virial coefficients of the pure components (Bα0, B0α), cross virial coefficients (Bαβ), and the mole fraction of each component (xα) [2]. For a binary mixture:
B2 = x12 B20 + 2x1x2 B11 + x22 B02 2
B3 = x13 B30 + 3x12 x2 B21 + 3x1x22 B12 + x23 B03 B4 = x14 B40 + 4x13 x2 B31 + 6x12 x22 B22 + 4x1x23 B13 + x24 B04 B5 = x15 B50 + 5x14 x2 B41 + 10x13 x22 B32 + 10x12 x23 B23 + 5x1x24 B14 + x25 B05
(1)
In general, coefficient Bαβ is given as a configurational integral involving α molecules of species 1 and β molecules of species 2. [1, 3] In this work, we calculated the virial coefficients of methane-ethane mixture up to the fifth order. These virial coefficients are used to predict the location of the critical point.
2. Model and method 2.1. Model The modified Buckingham exp-6 potential is used to describe the n-alkane molecules [4-7]. This model was employed by Lenart and Panagiotopoulos [5] for the methane-ethane system as part of a larger study on tracing critical lines of binary mixtures by molecular simulation. The exp-6 potential is parameterized to the vapor-liquid coexistence properties for a range of the alkane chain lengths and has higher accuracy than the usual Lennard-Jones intermolecular potential for the alkane molecules.
⎧ εα 6 r r [ exp(α [1− ]) − ( m )6 ] ⎪ rm r U disp (r) = ⎨ α − 6 α ⎪∞ ⎩
r > rmax
(5)
r ≤ rmax
where ε is the minimum potential energy, α is the repulsive steepness of the potential and rm is the separation distance between the alkane molecules when the exp-6 potential is the lowest 3
value. To prevent the exp-6 potential being negative at very small radial distance, we set rmax as the distance where the potential is a maximum and set the potential to be positive infinity when the separation distance is less than rmax. Methane and ethane molecules are represented using the united-atom approximation (i.e. no explicit hydrogen atoms are included), so the only interaction sites are located on the carbon atoms. The bond length separating the two ethane sites is fixed at 1.839 Å. For the mixtures, we use Lorentz-Berthelot combining rules to develop the cross-species interactions from those for the pure components. In particular:
1 2
σ ij = (σ ii + σ jj )
α ij = α iiα jj
ε ij = ε iiε jj
(6)
The quantity σ represents a size parameter, specifically where the potential energy becomes zero, and can be numerically determined. The model is defined by specifying σ values, which are then used to calculate rm values. The potential parameters are listed in Table 1. These are the same as used by Lenart and Panagiotopoulos in their Monte Carlo study of this system [5]. Table 1. Potential parameters Species
ε/k (K)
σ (Å)
α
rm (Å)
rmax (Å)
methane[7] 160.3
3.741
15
4.184
0.704
ethane[4]
129.6
3.679
16
4.094
0.575
mixture
144.13
3.71
15.49
4.139
0.64
4
Nuclear quantum effects can make a significant contribution to the virial coefficients, particularly for methane, which has a fairly small moment of inertia. At temperatures considered in the present study, quantum contributions to B2 for methane are on the order of 2% [8]. However, inasmuch as the models examined here are determined by fitting to experiment, and are not derived from ab initio calculations, any such quantum contributions should be considered as included implicitly in the parameterization of the model. Accordingly, in this work we do not attempt to include such contributions separately.
2.2. Critical properties In this study, we focus on locating the critical point of methane-ethane mixture, and then calculate the critical properties of methane(1)-ethane(2) mixtures from the virial coefficients. As described in [3], we begin by considering the spinodal curve defined by the criterion 2 ⎛ ⎛ ∂µ ⎞ 1 ⎛ ∂ p ⎞ ⎞ 1− x1 ⎛ ∂ p ⎞ 1 − ⎜ ⎜⎜ ⎟ ⎟⎟ − ρ 2 ⎜ ∂x ⎟ = 0 ⎜⎝ ⎝ ∂x1 ⎟⎠ ρ ∂x ⎝ ⎠ T ,ρ ⎠ ⎝ 1 ⎠ T ,ρ 1 T ,ρ 1
⎛ ∂p⎞ ⎜⎝ ∂ ρ ⎟⎠ T ,x
(7)
where µα is the chemical potential of species α and xα is the mole fraction of species α. We trace points along the spinodal curve until we reach a point that also satisfies the critical criterion [3], ( ρµ xx − pxx ) + κ ( px2 − 3(1− x1 ) pxx px ) + κ 2 (1− x1 )(3ρ px2 pxρ − 2 px3 ) −κ 3 (1− x1 ) ρ 2 px3 pρρ = 0
(8)
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where the subscripts are partial derivatives with respect to x ( ≡ x1) and/or ρ, and κ is the compressibility
1 ⎛ ∂ρ ⎞ ⎜ ⎟. ρ ⎜⎝ ∂p ⎟⎠
In some cases, the critical criteria are not satisfied at some temperatures. In this situation, we can estimate the critical point using a near-miss criterion [3, 9].
2.3. Computational Methods For the calculation of cluster integrals shown in VEOS, we applied the Mayer sampling Monte Carlo (MSMC) simulation method [10]. Via Mayer sampling, the average values of the ratio of the cluster integrals of the system of interest (or target system) to that of an already known cluster integral (reference system) is calculated. In the present work, we employed the overlap sampling implementation of MSMC [11]. We use hard-spheres of diameter 0.7(σmethane+ σethane) as a reference system.
3. Results and discussion We calculated the virial coefficients of methane-ethane mixture from B2 to B5. A total of 18 Bαβ coefficients are evaluated to describe the virials B2 to B5. The virial coefficients are presented in Tables 2-5, and data are plotted against temperature in Figure 1, which for comparison also gives available values derived from experimental data [12-18]. In the tables and figures, Bαβ represents the virial coefficient of the mixture containing α molecules of methane and β molecules of ethane. For the calculation of every virial coefficient, we performed 10 MSMC simulations with 109 Monte Carlo steps each. We obtained good precision for all temperatures. In general, the virial coefficients are very negative at low temperatures, pass 6
through a positive maximum at intermediate temperatures before decaying toward zero at high temperatures. For many methane-rich coefficients, this behavior would occur at temperatures below 190K, while for the second virial coefficient, the high-temperature behavior would occur at temperatures above 320K. The mixture coefficients tend to adopt behavior that is qualitatively between that of the pure components. The computed B2 values for pure components and their mixtures agree well with the experiment, methane slightly more so than ethane. B3 values for pure methane are slightly lower than the experimental data, and more so at lower temperature, but the agreement is reasonably good. However, B3 for pure ethane is much lower than the experimental data over the entire temperature range. B4 data derived from experiment have been published for methane [18], and not surprisingly they differ from the values calculated for the model, which are about a factor of 7 larger; high-order data are difficult to extract from PVT measurements, so this comparison should not be used to form conclusions about the model. The comparison is at least consistent in suggesting a relatively weak dependence on temperature for B40 in this narrow range where the data overlap. Table 2. Second virial coefficients calculated from MSMC for methane-ethane mixtures. Here and in subsequent tables, numbers in parentheses indicate the 68% confidence limits in the last digit(s) of the reported values.
T (K) 190 200 210 220 230 240 250 260 270
B20 -107.011(8) -96.90(2) -88.03(3) -80.17(4) -73.09(4) -66.76(5) -61.04(6) -55.82(4) -51.03(4)
B2 (cm3/mol) B11 -208.05(2) -188.88(2) -172.20(7) -157.47(4) -144.41(7) -132.89(7) -122.40(8) -112.92(6) -104.48(4)
B02 -415.30(5) -374.99(3) -340.42(16) -310.63(8) -284.6(1) -261.79(8) -241.54(7) -223.5(1) -207.12(8) 7
280 290 300 310 320
-46.66(4) -42.70(2) -38.98(6) -35.54(4) -32.432(6)
-96.66(4) -89.53(3) -82.99(5) -76.97(6) -71.421(5)
-192.54(9) -179.32(4) -167.23(6) -156.23(2) -146.063(16)
Table 3. Third virial coefficients calculated from MSMC for methane-ethane mixtures.
T (K) 190 200 210 220 230 240 250 260 270 280 290 300 310 320
B3 (cm3/mol)2 B30 2575.9(15) 2527(3) 2459.3(11) 2381.9(14) 2303.5(8) 2226.6(13) 2152.5(6) 2083.9(4) 2021.3(6) 1964.3(6) 1910.9(7) 1863.7(5) 1820.1(3) 1780.7(7)
B21 3627(5) 3779(7) 3809(2) 3761(3) 3672(3) 3572(4) 3451(3) 3331.0(6) 3213(2) 3106(2) 3003(2) 2905.7(17) 2819(2) 2736.4(10)
B12 2125(7) 3715(5) 4611(7) 5085(7) 5300(5) 5345(3) 5307(4) 5201(4) 5066.8(15) 4916(4) 4758(3) 4600(2) 4449(4) 4301(2)
B03 -18223(27) -8547(26) -2509(13) 1294(7) 3703(10) 5209(9) 6105(11) 6611(9) 6868(7) 6958(7) 6944(4) 6855(2) 6720(8) 6554(2)
Table 4. Fourth virial coefficients calculated from MSMC for methane-ethane mixtures.
T (K) 190 200 210 220 230 240 250 260 270 280 290
B40 9.51(3)×104 8.59(4)×104 7.50(4)×104 6.639(11)×104 5.813(15)×104 5.15(3)×104 4.62(2)×104 4.18(2)×104 3.882(16)×104 3.635(16)×104 3.430(13)×104
B31 1.63(3)×105 1.661(12)×105 1.557(10)×105 1.371(11)×105 1.195(3)×105 1.043(3)×105 8.99(2)×104 7.92(3)×104 6.96(3)×104 6.181(19)×104 5.620(15)×104
B4 (cm3/mol)3 B22 2.3(14)×104 1.93(2)×105 2.51(2)×105 2.546(19)×105 2.390(9)×105 2.114(16)×105 1.875(6)×105 1.635(9)×105 1.407(4)×105 1.220(4)×105 1.077(3)×105
B13 -2.349(16)×106 -9.16(7)×105 -2.03(7)×105 1.29(7)×105 2.74(2)×105 3.22(4)×105 3.22(2)×105 3.02(3)×105 2.707(16)×105 2.43(10)×105 2.110(11)×105
B04 -2.062(12)×107 -1.077(7)×107 -5.599(12)×106 -2.86(2)×106 -1344(6)×106 -5.31(4)×105 -9.1(8)×104 1.41(4)×105 2.51(4)×105 2.86(3)×105 2.931(10)×105
8
300 310 320
3.307(9)×104 3.218(10)×104 3.167(8)×104
5.163(16)×104 4.799(11)×104 4.545(11)×104
9.48(3)×104 8.46(4)×104 7.658(16)×104
1.874(9)×105 1.634(5)×105 1.452(5)×105
2.81(2)×105 2.613(17)×105 2.407(7)×105
Table 5. Fifth virial coefficients calculated from MSMC for methane-ethane mixtures.
T (K) 190 200 210 220 230 240 250 260 270 280 290 300 310 320
B50
6.5(10)×105 2.4(7)×105 -3.6(9)×105 -7.3(7)×105 -9.3(5)×105 -9.4(4)×105 -9.2(3)×105 -8.3(4)×105 -7.0(4)×105 -4.7(1)×105 -3.4(2)×105 -1.7(2)×105 -1.9(12)×104 1.3(2)×105
B41
2.7(4)×106 2.8(5)×106 2.16(15)×106 7(2)×105 -3.0(7)×105 -1.17(9)×106 -1.59(6)×106 -1.61(9)×106 -1.58(6)×106 -1.51(4)×106 -1.33(12)×106 -1.10(4)×106 -8.0(2)×105 -5.6(2)×105
B5 (cm3/mol)4 B32 B23
-9.3(17)×106 7.3(9)×106 1.01(8)×107 7.1(3)×106 4.1(11)×106 1.4(4)×106 -4(2)×105 -1.12(12)×106 -1.90(14)×106 -2.58(11)×106 -2.64(10)×106 -2.36(8)×106 -2.37(5)×106 -1.95(5)×106
-3.73(7)×108 -7.0(4)×107 -5.4(19)×106 1.36(12)×107 1.41(13)×107 1.19(7)×107 8.2(4)×106 2.8(5)×106 1.3(4)×106 -1.0(2)×106 -2.2(2)×106 -3.0(2)×106 -3.68(9)×106 -3.63(13)×106
B14
-2.595(15)×109 -1.055(12)×109 -4.09(9)×108 -1.39(4)×108 -2.9(7)×107 4(2)×106 1.63(11)×107 1.30(17)×107 1.25(9)×107 8.4(8)×106 1.9(6)×106 -4(4)×105 -2.1(4)×106 -3.7(4)×106
B05
10
-2.108(10)×10 -9.31(5)×109 -4.28(5)×109 -1.984(16)×109 -9.4(3)×108 -4.31(11)×108 -1.99(5)×108 -8.1(7)×107 -3.0(2)×107 -1.47(16)×107 -4.4(13)×106 -5.0(12)×106 -2.6(8)×106 -3.8(6)×106
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Figure 1. Virial coefficients for methane-ethane mixture for n = 2–5 computed from MSMC; a, Hoover et al. [12]; b, Hou et al. [13]; c: McElroy et al. [14]; d: Dantzler et al. [15]; e: Funke et al. [16]; f: Kleinrahm et al. [17]; g: Douslin et al. [18]. Bαβ represents the virial coefficient of the mixture containing α molecules of methane and β molecules of ethane. For the values calculated in this work, error bars (68% confidence) are shown where they are discernable on the scale of the figure.
Deviations of the pressures for pure methane and ethane from experimental data [19] at 300K are presented in Figs. 2 and 3, respectively. For methane, the deviations are less than 1%, except
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VEOS2 which deviates by 5% at high density. The discrepancy is not due to the use of the VEOS per se, as the comparison of VEOS4 and VEOS5 suggests that the series has converged for the state conditions shown in the figure. Rather the deviations come primarily from the slightly too-large value of B2, which is evidenced by the non-zero slope of the VEOS deviation lines at zero density, and reflects on the accuracy of the model. This is also visible in the direct comparison of B2 in Fig. 1, where without the enlarged scale given by Fig. 2, the discrepancy does not seem as significant. B3 for pure methane is smaller than the experimental data, and this tends to compensate for the slight error in B2. Overall, the deviation in pressure still is small out to the highest densities shown (about half the critical density). For ethane in Fig. 3, the pressures from experimental data deviate by 5% at high density. Like pure methane, a too-large value of B2 for ethane contributes most to the deviation for all the VEOSn, and this error is offset partially by a too-small value of B3.
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Figure 2. Deviations of the pressure for methane from experimental data [19] at 300K. The VEOS4 and VEOS5 lines are not distinguishable on the scale of this figure. VEOS lines for Hou use the virial coefficients from Ref. [13].
12
Figure 3. Deviations of the pressure for ethane from experimental data [19] at 300K. VEOS lines for Hou use the virial coefficients from Ref. [13]. VEOS3 through VEOS5 are used to compute density- and pressure-composition spinodal lines, plotted in Fig. 4 and Fig. 5, respectively. Every spinodal line ends at the critical point. Again, the consistency between VEOS4 and VEOS5 suggests convergence. Good agreement is seen in comparison with the methane-ethane pressure-composition critical line as given by experiment [20]. Experimental data for the density are not available. It is likely that good agreement would not be seen with VEOS, as previous work consistently finds that the critical density is underestimated by VEOS [21-24].
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Figure 4. Density-composition projection of methane spinodal lines and the critical locus for a methane-ethane mixture: critical locus computed from VEOS3 (black long dash dot line); VEOS4 (red dotted line); VEOS5 (blue dash dot line); spinodal lines computed from VEOS3 (dash lines) and VEOS5 (solid lines) are shown.
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Figure 5. Pressure-composition projection of methane spinodal lines and the critical locus for a methane-ethane mixture: critical locus computed from VEOS3 (black long dash dot line); VEOS4 (red dotted line); VEOS5 (blue dash dot line); spinodal lines computed from VEOS3 (dash lines) and VEOS5 (solid lines) are shown. Experimental data [20] are indicated with open circles. In Fig. 6, the critical temperature as a function of methane composition is presented. Van Konynenburg and Scott [25] introduced six types of mixture critical behavior. The methaneethane mixture is classified as type I, meaning the mixture exhibits complete mixing between the two components over the whole composition range. The locus of critical points from VEOS4 shows significant improvement over VEOS3, while VEOS5 shows little deviation from VEOS4,
15
suggesting that the series has converged. Again, both VEOS4 and VEOS5 have very good agreement with the experimental data [20].
Figure 6. Temperature-composition projection of methane critical locus for a methaneethane mixture: critical locus computed from VEOS3 (black dash line); VEOS4 (red dotted line); and VEOS5 (blue solid line); literature data for the model used here [5] (green diamond); and experimental data [20] (black cross). Thinner lines and open symbols indicate critical point is obtained using near-miss critical condition.
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4. Conclusions The comparison of the virial coefficients for the model to experimental values indicates some imperfection in the model, which is not surprising given its simple form. The model was parameterized to describe vapor-liquid coexistence data, and thus one should expect it to perform reasonably well in describing higher-density states. The compensating errors in B2 and B3 show in part how this is done. The good performance of the VEOS in describing the critical line is a consequence of the ability of the VEOS to describe the model properties at the relevant conditions, combined with the ability of the empirically-determined model to characterize this behavior even though its description of two- and three-body interactions is imperfect. In future studies it would be of interest to examine more complex alkane mixtures such as more asymmetric systems such as ethane-decane, or ternary or higher-component alkane mixtures. Calculations for such systems are made more difficult by the need to incorporate contributions related to molecular flexibility [24].
Acknowledgements This work is supported by the National Science Foundation under grant CBET-0854340. Calculations were performed using resources from the University at Buffalo Center for Computational Research. We acknowledge improvements to this manuscript resulting from very helpful comments contributed by an anonymous reviewer.
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Kleinrahm, R., W. Duschek, W. Wagner, and M. Jaeschke, Measurement and Correlation of the (Pressure, Density, Temperature) Relation of Methane in the Temperature-Range from 273.15-K to 323.15-K at Pressures up to 8 Mpa. Journal of Chemical Thermodynamics, 1988. 20(5): p. 621-631. Douslin, D.R., R.H. Harrison, R.T. Moore, and J.P. MuCullough, P-V-T Relations for Methane. Journal of Chemical & Engineering Data, 1964. 9(3): p. 358-363. Lemmon, E.W., M.O. McLinden, and D.G. Friend, Thermophysical Properties of Fluid Systems., in NIST Chemistry WebBook, NIST Standard Reference Database Number 69 (http://webbook.nist.gov/chemistry/fluid), P.J. Linstrom and W.G. Mallard, Editors. Bloomer, O.T., D.C. Gami, and J.D. Parent, Physical-chemical properties of methaneethane mixtures. Inst. Gas Technol., Research Bull, 1953. No. 22: p. 39 pp. Schultz, A.J. and D.A. Kofke, Sixth, seventh and eighth virial coefficients of the LennardJones model. Mol. Phys., 2009. 107(21): p. 2309-2318. Schultz, A.J. and D.A. Kofke, Virial coefficients of model alkanes. The Journal of Chemical Physics, 2010. 133(10): p. 104101. Shaul, K.R.S., A.J. Schultz, and D.A. Kofke, The effect of truncation and shift on virial coefficients of Lennard–Jones potentials. Collection of Czechoslovak Chemical Communications, 2010. 75(4): p. 447-462. Shaul, K.R.S., A.J. Schultz, and D.A. Kofke, Mayer-sampling Monte Carlo calculations of uniquely flexible contributions to virial coefficients. Journal of Chemical Physics, 2011. 135(12): p. 124101. Van Konynenburg, P.H. and R.L. Scott, Critical lines and phase equilibriums in binary Van der Waals mixtures. Philos. Trans. R. Soc. London, Ser. A, 1980. 298: p. 495-540.
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Figure Captions Figure 1. Virial coefficients for methane-ethane mixture for n = 2–5 computed from MSMC; a, Hoover et al. [12]; b, Hou et al. [13]; c: McElroy et al. [14]; d: Dantzler et al. [15]; e: Funke et al. [16]; f: Kleinrahm et al. [17]; g: Douslin et al. [18]. Bαβ represents the virial coefficient of the mixture containing α molecules of methane and β molecules of ethane. For the values calculated in this work, error bars (68% confidence) are shown where they are discernable on the scale of the figure. Figure 2. Deviations of the pressure for methane from experimental data [19] at 300K. The VEOS4 and VEOS5 lines are not distinguishable on the scale of this figure. VEOS lines for Hou use the virial coefficients from Ref. [13]. Figure 3. Deviations of the pressure for ethane from experimental data [19] at 300K. VEOS lines for Hou use the virial coefficients from Ref. [13]. Figure 4. Density-composition projection of methane spinodal lines and the critical locus for a methane-ethane mixture: critical locus computed from VEOS3 (black long dash dot line); VEOS4 (red dotted line); VEOS5 (blue dash dot line); spinodal lines computed from VEOS3 (dash lines) and VEOS5 (solid lines) are shown. Figure 5. Pressure-composition projection of methane spinodal lines and the critical locus for a methane-ethane mixture: critical locus computed from VEOS3 (black long dash dot line); VEOS4 (red dotted line); VEOS5 (blue dash dot line); spinodal lines computed from VEOS3 (dash lines) and VEOS5 (solid lines) are shown. Experimental data [20] are indicated with open circles.
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Figure 1. Temperature-composition projection of methane critical locus for a methaneethane mixture: critical locus computed from VEOS3 (black dash line); VEOS4 (red dotted line); and VEOS5 (blue solid line); literature data for the model used here [5] (green diamond); and experimental data [20] (black cross). Thinner lines and open symbols indicate critical point is obtained using near-miss critical condition.
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