J. Phys. Chem. 1996, 100, 17365-17372
17365
An Equation of State for Real Fluids Based on the Lennard-Jones Potential Tongfan Sun and Amyn S. Teja* Fluid Properties Research Institute, School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0100 ReceiVed: July 10, 1996X
A new set of constants for the modified Benedict-Webb-Rubin (MBWR) equation based on the LennardJones (LJ) potential were obtained in the temperature range T* ) 0.45 to T* ) 6.0 by supplementing computer simulation data with data obtained by recalculating the first five virial coefficients of the LJ potential. These virial coefficients are presented in a detailed table that is suitable for the calculation of the internal energy and pressure of the LJ fluid in the vapor state. The MBWR equation was also extended to the alkanes by including a temperature-dependent energy parameter in the LJ potential. The resulting equation was found to be satisfactory for correlating experimental data for the liquid density and vapor pressure and in predicting the heat capacity of n-alkanes upto n-hexadecane. Only three parameters are required to characterize each n-alkane, and furthermore, the parameters show regular behavior with the number of carbon atoms in the n-alkane.
Introduction The calculation of the thermodynamic properties of fluids and fluid mixtures is important in a number of applications, including the design of separators and other process equipment. Such calculations also provide information on intermolecular interactions and their effect on the nonideal behavior of the system. The most common approach for calculation of thermodynamic properties is by use of cubic equations of state and/ or the corresponding state principle.1,2 Both these methods are simple but may require arbitrary constants to characterize each pure fluid. An alternative approach is by use of an equation of state based on an intermolecular potential. This type of equation may, for example, require only size and energy parameters for each fluid and, as a result, allow extrapolation of the equation to fluids from a knowledge of their structure. The Lennard-Jones (LJ) fluid has been widely studied using both theoretical and computer simulation methods. The simulation results have been used to obtain a number of analytical equations of state for the LJ fluid,3-6 the most recent of which is the modified Benedict-Webb-Rubin (MBWR) equation with 33 parameters by Johnson et al.3 The large numbers of parameters allows sufficient flexibility to fit simulation data accurately over a wide range of conditions. The equation of state of Johnson et al. may be written as follows:
P* ) F*T* + F*2 (c1T* + c2T*1/2 + c3 + c4T*-1 + c5T*-2) + F*3(c6T* + c7 + c8T*-1 + c9T*-21) + F*4(c10T* + c11 + c12 T*-1) + F*5(c13) + F*6(c14T*-1 + c15T*-2) + F*7 (c16T*-1) + F*8(c17T*-1 + c18T*-2) + F*9(c19T*-2) + {F*3(c20T*-2 + c21T*-3) + F*5(c22T*-2 + c23T*-4) + F*7(c24T*-2 + c25T*-3) + F*9(c26T*-2 + c27T*-4) + F*11(c28T*-2 + c29T*-3) + F*13(c30T*-2 + c31T*-3 + c32T*-4)} exp(- γF*2) (1) where P* is the reduced pressure ()pσ3/), F* is the reduced density ()Nσ3/V), T* is the reduced temperature ()kT/), p is X
Abstract published in AdVance ACS Abstracts, October 1, 1996.
S0022-3654(96)02047-3 CCC: $12.00
the pressure, N is Avogadro’s number, V is the molar volume, T is the temperature, is the depth of the LJ potential well, σ is the separation distance at zero energy, and k is Boltzmann’s constant. In Johnson’s equation γ ) 3, so that there are 32 universal constants, c1-c32, which were obtained by fitting computer simulation data in the temperature range 0.7 < T* < 6 (corresponding to 0.55 < T/Tc < 4.5 where Tc is the LJ critical temperature). However, when applying this equation to real fluids, we have found that the equation diverges at reduced temperatures T/Tc < 0.65. This greatly limits its application, because the triple point temperatures of real fluids are generally much lower than T/Tc ) 0.65. (For example, T/Tc ) 0.38 for ethane.) Therefore, in order to work with real fluids, the equation must be extended to lower temperatures. The purpose of the present work was therefore to extend the MBWR equation of Johnson et al. to lower temperatures suitable for vapor-liquid equilibrium calculations involving real fluids. A second objective was to extend the equation to a series of real fluids such as the n-alkanes. Extension of Simulation Data for the LJ Fluid The literature on the LJ fluid lists approximately 1500 data points for pressure and 1000 data points for internal energy over a range of temperatures and densities.3 However, not all simulation data are consistent and suitable for the present work. Earlier simulations data, in particular, were obtained for small systems and for short times with a small Lennard-Jones cutoff distance. Therefore, in the present work, we chose the recent simulation data of Johnson et al. which cover the reduced temperature range from 0.7 to 6. We also combined Miyano’s Monte Carlo simulations6 in the low-temperature range from 0.45 to 0.65. At these temperatures, the LJ liquid is in a subcooled state, and the precision of the simulations is poor, as stated by the author. However, Miyano’s data are the only simulation data available in this temperature range. Special attention was paid to simulation data below the critical temperature and at pressures in the range p* ) 0.001-1 (corresponding to 0.3-300 bar for methane). This represents the most important region for practical applications in chemical engineering. However, simulation results for the vapor phase in this region are not very precise, and results for the liquid state are © 1996 American Chemical Society
17366 J. Phys. Chem., Vol. 100, No. 43, 1996
Sun and Teja
TABLE 1: Reduced Virial Coefficients for the LJ Potential T*
B2*
B3*
B4 *
B5*
T*
B2*
B3*
B4*
B5*
0.390 0.400 0.410 0.420 0.430 0.440 0.450 0.460 0.470 0.480 0.490 0.500 0.510 0.520 0.530 0.540 0.550 0.560 0.570 0.580 0.590 0.600 0.610 0.620 0.630 0.640 0.650 0.660 0.670 0.680 0.690 0.700 0.710 0.720 0.730 0.740 0.750 0.760 0.770 0.780 0.790 0.800 0.810 0.820 0.830 0.840 0.850 0.860 0.870 0.880 0.890 0.910 0.920 0.930 0.940 0.950 0.960 0.970 0.980 0.990 1.000 1.020 1.040 1.060 1.080 1.100 1.120 1.140 1.160 1.180 1.200 1.220 1.240 1.260 1.280 1.300
-14.5936 -13.7988 -13.0769 -12.4188 -11.8166 -11.2639 -10.7550 -10.2851 -9.8500 -9.4462 -9.0705 -8.7202 -8.3928 -8.0863 -7.7987 7.5285 -7.2741 -7.0342 -6.8076 -6.5934 -6.3904 -6.1980 -6.0152 -5.8415 -5.6761 -5.5185 -5.3682 -5.2246 -5.0875 -4.9562 -4.8305 -4.7100 -4.5945 -4.4836 -4.3770 -4.2745 -4.1759 -4.0810 -3.9896 -3.9014 -3.8163 -3.7342 -3.6549 -3.5783 -3.5042 -3.4325 -3.3631 -3.2959 -3.2308 -3.1677 -3.1065 -2.9895 -2.9335 -2.8791 -2.8263 -2.7749 -2.7249 -2.6763 -2.6290 -2.5830 -2.5381 -2.4518 -2.3697 -2.2917 -2.2173 -2.1464 -2.0787 -2.0139 -1.9520 -1.8928 -1.8360 -1.7815 -1.7292 -1.6789 -1.6306 -1.5841
-240.776 -199.123 -165.801 -138.923 -117.076 -99.1897 -84.4489 -72.2246 -62.0282 -53.4768 -46.2685 -40.1633 -34.9690 -30.5310 -26.7242 -23.4466 -20.6145 -18.1592 -16.0239 -14.1614 -12.5322 -11.1033 -9.8469 -8.7396 -7.7614 -6.8954 -6.1273 -5.4446 -4.8367 -4.2945 -3.8101 -3.3767 -2.9883 -2.6399 -2.3268 -2.0451 -1.7915 -1.5628 -1.3565 -1.1701 -1.0015 -0.8491 -0.7110 -0.5859 -0.4725 -0.3696 -0.2763 -0.1915 -0.1145 -0.0446 0.0189 0.1291 0.1768 0.2201 0.2594 0.2952 0.3276 0.3570 0.3837 0.4078 0.4297 0.4673 0.4978 0.5225 0.5422 0.5577 0.5696 0.5787 0.5853 0.5897 0.5924 0.5937 0.5937 0.5926 0.5907 0.5882
-20740.6 -15202.1 -11288.5 -8483.45 -6446.23 -4948.49 -3834.81 -2997.93 -2362.84 -1876.46 -1500.74 -1208.18 -978.642 -797.281 -653.027 -537.570 -444.616 -369.361 -308.115 -258.024 -216.864 -182.891 -154.733 -131.303 -111.732 -95.3259 -81.5272 -69.8837 -60.0289 -51.6638 -44.5437 -38.4676 -33.2694 -28.8120 -24.9813 -21.6823 -18.8355 -16.3743 -14.2427 -12.3935 -10.7868 -9.3887 -8.1705 -7.1077 -6.1793 -5.3675 -4.6570 -4.0345 -3.4886 -3.0097 -2.5892 -1.8953 -1.6100 -1.3592 -1.1386 -0.9447 -0.7743 -0.6245 -0.4929 -0.3774 -0.2761 -0.1098 0.0172 0.1134 0.1853 0.2384 0.2766 0.3031 0.3207 0.3311 0.3362 0.3370 0.3347 0.3300 0.3236 0.3158
-2874900 -1834810 -1195110 -793134 -535490 -367317 -255675 -180395 -128890 -93172.6 -68089.3 -50265.7 -37460.6 -28165.7 -21353.2 -16314.6 -12556.1 -9729.84 -7588.39 -5954.18 -4698.62 -3727.78 -2972.55 -2381.68 -1916.87 -1549.35 -1257.33 -1024.23 -837.325 -686.844 -565.202 -466.501 -386.125 -320.449 -266.608 -222.335 -185.820 -155.622 -130.580 -109.763 -92.4158 -77.9277 -65.8017 -55.6321 -47.0872 -39.8944 -33.8297 -28.7081 -24.3766 -20.7084 -17.5981 -12.7132 -10.8042 -9.1791 -7.7948 -6.6150 -5.6088 -4.7506 -4.0183 -3.3934 -2.8603 -2.0182 -1.4064 -0.9636 -0.6447 -0.4168 -0.2557 -0.1435 -0.0671 -0.0167 0.0149 0.0330 0.0417 0.0438 0.0415 0.0364
1.320 1.340 1.360 1.380 1.400 1.420 1.440 1.460 1.480 1.500 1.520 1.540 1.560 1.580 1.600 1.620 1.640 1.660 1.680 1.700 1.720 1.740 1.760 1.780 1.800 1.820 1.840 1.860 1.880 1.900 1.920 1.940 1.960 1.980 2.000 2.100 2.200 2.300 2.400 2.500 2.600 2.700 2.800 2.900 3.000 3.100 3.200 3.300 3.400 3.500 3.600 3.800 3.900 4.000 4.100 4.200 4.300 4.400 4.500 4.600 4.700 4.800 4.900 5.000 5.100 5.200 5.300 5.400 5.500 5.600 5.700 5.800 5.900 6.000 6.100
-1.5394 -1.4962 -1.4547 -1.4146 -1.3758 -1.3385 -1.3023 -1.2674 -1.2336 -1.2009 -1.1692 -1.1385 -1.1088 -1.0799 -1.0519 -1.0247 -0.9984 -0.9727 -0.9478 -0.9236 -0.9001 -0.8772 -0.8549 -0.8332 -0.8120 -0.7914 -0.7714 -0.7518 -0.7327 -0.7142 -0.6960 -0.6783 -0.6610 -0.6441 -0.6276 -0.5506 -0.4817 -0.4197 -0.3636 -0.3126 -0.2661 -0.2236 -0.1845 -0.1485 -0.1152 -0.0844 -0.0558 -0.0291 -0.0043 0.0190 0.0407 0.0803 0.0984 0.1154 0.1315 0.1467 0.1611 0.1747 0.1876 0.1999 0.2116 0.2227 0.2333 0.2433 0.2530 0.2622 0.2709 0.2794 0.2874 0.2951 0.3025 0.3096 0.3164 0.3229 0.3292
0.5850 0.5813 0.5773 0.5729 0.5683 0.5635 0.5586 0.5536 0.5485 0.5434 0.5382 0.5331 0.5280 0.5230 0.5180 0.5131 0.5082 0.5035 0.4988 0.4942 0.4897 0.4853 0.4810 0.4768 0.4727 0.4687 0.4648 0.4610 0.4573 0.4537 0.4502 0.4468 0.4435 0.4402 0.4371 0.4225 0.4099 0.3990 0.3894 0.3810 0.3737 0.3673 0.3617 0.3567 0.3523 0.3484 0.3449 0.3417 0.3389 0.3363 0.3340 0.3300 0.3282 0.3266 0.3251 0.3237 0.3224 0.3211 0.3200 0.3189 0.3179 0.3169 0.3160 0.3151 0.3142 0.3134 0.3126 0.3118 0.3111 0.3104 0.3096 0.3090 0.3083 0.3076 0.3070
0.3073 0.2981 0.2887 0.2791 0.2696 0.2601 0.2509 0.2420 0.2334 0.2251 0.2172 0.2097 0.2026 0.1958 0.1895 0.1835 0.1779 0.1726 0.1676 0.1630 0.1587 0.1547 0.1509 0.1475 0.1442 0.1412 0.1384 0.1359 0.1335 0.1313 0.1293 0.1275 0.1258 0.1242 0.1228 0.1175 0.1145 0.1130 0.1127 0.1131 0.1141 0.1153 0.1168 0.1183 0.1198 0.1213 0.1228 0.1242 0.1255 0.1267 0.1277 0.1296 0.1304 0.1311 0.1318 0.1323 0.1327 0.1331 0.1334 0.1337 0.1339 0.1340 0.1341 0.1341 0.1341 0.1341 0.1340 0.1339 0.1337 0.1335 0.1333 0.1331 0.1328 0.1326 0.1323
0.0295 0.0218 0.0137 0.0058 -0.0018 -0.0088 -0.0151 -0.0207 -0.0256 -0.0298 -0.0332 -0.0360 -0.0382 -0.0397 -0.0408 -0.0414 -0.0415 -0.0413 -0.0407 -0.0398 -0.0386 -0.0373 -0.0357 -0.0340 -0.0321 -0.0301 -0.0280 -0.0258 -0.0235 -0.0212 -0.0189 -0.0166 -0.0142 -0.0119 -0.0095 0.0019 0.0124 0.0217 0.0298 0.0368 0.0427 0.0477 0.0518 0.0552 0.0580 0.0603 0.0622 0.0636 0.0647 0.0656 0.0662 0.0668 0.0669 0.0669 0.0667 0.0665 0.0662 0.0659 0.0655 0.0650 0.0645 0.0640 0.0635 0.0630 0.0624 0.0618 0.0612 0.0606 0.0600 0.0594 0.0589 0.0582 0.0577 0.0571 0.0565
Equation of State for Real Fluids
J. Phys. Chem., Vol. 100, No. 43, 1996 17367
TABLE 2: Comparison of Calculated Reduced Enthalpies with Simulation Results T*
P*
F*cal
F*exp
AAD1, %
H*cal
H*exp
AAD2, %
0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30
0.00131 0.002 64 0.004 70 0.007 69 0.011 68 0.017 41 0.025 05 0.033 84 0.045 10 0.059 74 0.077 18 0.973 0 0.120 40
0.001 91 0.003 64 0.006 17 0.009 71 0.014 28 0.020 84 0.029 69 0.039 83 0.053 46 0.073 01 0.099 56 0.137 03 0.197 82
0.001 93 0.003 63 0.006 17 0.009 70 0.014 26 0.020 81 0.029 64 0.039 74 0.053 30 0.072 67 0.098 70 0.133 90 0.195 00
-1.18 0.17 0.07 0.14 0.13 0.15 0.17 0.23 0.30 0.47 0.86 2.28 1.43
-0.036 48 -0.065 80 -0.106 39 -0.160 15 -0.226 05 -0.317 71 -0.436 70 -0.565 40 -0.732 89 -0.964 90 -1.264 87 -1.660 70 -2.247 47
-0.0367 -0.0651 -0.1052 -0.1580 -0.2228 -0.3128 -0.4295 -0.5562 -0.7210 -0.9500 -1.2430 -1.6310 -2.2060
-0.61 1.06 1.12 1.34 1.44 1.55 1.65 1.63 1.62 1.54 1.73 1.79 1.85
a H*) U* + P*/F* - T*. H* cal ) H* calculated from virial expansion, this work. H*exp ) H* from simulation. AAD1 ) ∑(F*cal - F*exp)/F*cal × 100. AAD2 ) ∑(H*cal - H*exp)/H*cal × 100.
not always available, particularly at T* < 1. In order to supplement these data, therefore, we added new virial series expansion results for the vapor state and employed a perturbation approach to calculate the pressure and internal energy for the liquid state. Nicolas et al.4 used the virial coefficients of an LJ fluid to supplement computer simulation data at low densities. They evaluated the reduced pressure and internal energy according to the relations
p* ) F*T*{1 + ∑(2π/3)n-1Bn*F*n-1}
(2)
U* ) -T*2∑(2π/3)n-1(dBn*/dT*)F*n-1/(n - 1)
(3)
where Bn* ) Bn/b0n-1 with b0 ) 2/3πσ3. The first five virial coefficients of the LJ fluid were used in their calculations, and the virial coefficients were obtained from a table reported by Barker et al.7,8 Since the temperature interval in Barker’s table is very large, interpolations of the temperature required in the determination of dBn*/dT* are not precise. Therefore, the evaluation of U* by Nicolas et al. was not accurate. Moreover, in the region T* < 0.6, which is important for vapor pressure calculations involving real fluids, virial coefficients were not available in Barker’s table. We therefore reevaluated the first five virial coefficients of a LJ fluid using the Mayer f function defined as
fij ) f(rij) ) exp[-u(rij)/kT] - 1
(4)
which leads to the reduced third virial coefficient B3* as follows:
B3* ) -1/3∫∫f01f12f02 dr1 dr2 ) -1/3(C3)
(5)
The integration is carried out over all positions of molecules 1 and 2 with molecule 0 fixed at the center of the coordinate system. Vector distances r1 and r2 start from molecule 0 and end at 1 and 2, respectively. In the (C3) term, the C stands for three molecules and the subscript 3 means there are three pairs of intermolecular interactions. Similarly, the fourth and fifth virial coefficients are given as
B4* ) (-1/8)[3(D4) + 6(D5) + (D6)]
(6)
B5* ) (-1/30)[12(E5) + 60(E6a) + 10(E6b) + 60(E7a) + 30(E7b) + 10(E7c) + 15(E8a) + 30(E8b) + 10(E9) + (E10)] (7) where the notation for the cluster integrals (in brackets) is adopted from Rowlinson9 and Barker et al.8 The integrals
can be classified into two groups. The first group is of the convolution type and is calculated accordingly. The second group includes integrals (D6), (E8a), (E9), and (E10) that are difficult to calculate directly due to the high order of multi-integration required. However, by expanding every Mayer f function of the integrand into Legendre polynomials and using the addition theorem and the orthogonal relation of spherical harmonics, multiintegrations can be simplified into 3-dimensional integrations. These integrations can then be easily treated with the Gauss-Legendre numerical integration method.. All cluster integrals were carefully evaluated and their values compared with the results reported by Barker at six selected temperatures (T* from 0.75 to 20.0). Excellent agreement was obtained to four significant digits for all cluster integrals except E10, where differences of 5-15% were found. The E10 integral is difficult to calculate because it requires the evaluation of 17 483 4-dimensional terms when the first 8 orders of Legendre polynomials are expanded for the integrand. A simplified treatment that differs from the simplified method used by Barker was employed in the present work and resulted in values for this integral that differed by 5-15% from those reported by Barker et al. On the other hand, results for B2*, B3*, and B4* agreed with those of Barker et al. to four significant digits and those for B5* agreed to three digits. Complete results are given in Table 1, which covers the reduced temperature range 0.4-6 with a minimum interval DT* ) 0.01. Nicolas et al.4 used a convergence criterion to examine the density range where the first five virial coefficients give satisfactory results. Convergence of the virial series was tested by comparing values of Z2, Z3, Z4, and Z5, where Z ) P*/F* T* and Zn is the estimate of Z obtained by truncating eq 3 at the Bn* term. Convergence was regarded as satisfactory if |Z4 Z5| e 0.005. In addition to this criterion, a direct comparison of the virial series with computer simulation results was also made. Recently, Lotfi et al.10 have used an isothermal-isobaric ensemble simulation in conjunction with Widom’s particle insertion method and calculated vapor pressures of the LJ fluid in the temperature range T* ) 0.70-1.30. Since Lotfi’s simulation data provide a vapor pressure relation at high density for the vapor phase, they may be used to compare the reevaluated virial series at these densities. Such comparisons are shown in Table 2, which also shows excellent agreement for all data points except at the highest temperature. Uncertainties at high temperatures are due to uncertainties in the simulations10 at these temperatures. These calculations demonstrate that the virial expansion is valid for the whole range of vapor density. Above the critical temperature (T* g 1.3),
17368 J. Phys. Chem., Vol. 100, No. 43, 1996
Sun and Teja
TABLE 3: New Data Obtained from the Virial Series Expansion at Low Pressures and the Ross Perturbation Method at High Pressures T*
F*
U*
P*
T*
F*
U*
P*
0.450 0.450 0.450 0.450 0.450 0.450 0.450 0.450 0.450
0.001 00 0.00300 0.005 57 0.925 00 0.950 00 0.975 00 1.025 00 1.050 00 1.075 00
-0.021 42 -0.070 03 -0.153 40 -6.974 73 -7.100 87 -7.145 02 -7.205 47 -7.237 04 -7.195 68
0.000 44 0.001 25 0.002 14 -0.456 89 0.229 53 1.289 89 4.083 92 5.742 41 7.714 85
0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500
0.001 00 0.004 00 0.008 45 0.800 00 0.850 00 0.900 00 0.950 00 1.000 00 1.050 00
-0.018 07 -0.077 68 -0.193 49 -6.029 94 -6.376 31 -6.710 64 -6.996 45 -7.098 07 -7.156 45
0.000 49 0.001 85 0.003 47 -1.735 89 -1.265 29 -0.480 10 0.743 58 3.056 88 6.176 92
1.250 1.250 1.250 1.250 1.250 1.250 1.250 1.250 1.250 1.250 1.250 1.250 1.250 1.250
0.005 00 0.010 00 0.050 00 0.100 00 0.125 00 0.400 00 0.500 00 0.700 00 0.750 00 0.800 00 0.850 00 0.900 00 0.950 00 1.000 00
-0.041 11 -0.082 20 -0.410 53 -0.819 42 -1.023 03 -2.802 52 -3.442 35 -4.726 17 -5.034 83 -5.322 88 -5.582 76 -5.778 54 -5.894 88 -5.981 60
0.006 14 0.012 06 0.051 78 0.084 04 0.093 87 0.060 94 0.082 04 0.829 09 1.381 93 2.180 66 3.300 12 4.875 11 6.998 87 9.619 52
0.550 0.550 0.550 0.550
0.001 00 0.005 00 0.010 00 0.012 02
-0.015 84 -0.083 95 -0.186 71 -0.237 09
0.000 54 0.002 53 0.004 58 0.005 23
1.300 1.300 1.300 1.300 1.300 1.300
0.005 00 0.010 00 0.050 00 0.100 00 0.130 00 1.000 00
-0.040 35 -0.080 68 -0.402 07 -0.799 66 -1.035 19 -5.913 09
0.006 39 0.012 57 0.054 66 0.090 61 0.104 59 10.02 07
0.600 0.600 0.600 0.600 0.600 0.600 0.600
0.001 00 0.005 00 0.010 00 0.016 27 0.850 00 0.950 00 1.050 00
-0.014 25 -0.073 90 -0.156 97 -0.283 43 -6.255 27 -6.811 33 -6.991 20
0.000 59 0.002 80 0.005 18 0.007 45 -0.546 19 1.720 04 7.104 64
1.350 1.350 1.350 1.350 1.350 1.350
0.005 00 0.010 00 0.050 00 0.100 00 0.135 00 1.000 00
-0.039 65 -0.079 25 -0.394 31 -0.782 29 -1.048 64 -5.845 45
0.006 65 0.013 09 0.057 52 0.097 09 0.115 90 10.41 72
0.650 0.650 0.650 0.650 0.650 0.650 0.650 0.650 0.650 0.650
0.005 00 0.010 00 0.015 00 0.021 21 0.800 00 0.850 00 0.900 00 0.950 00 1.000 00 1.050 00
-0.066 93 -0.138 92 -0.218 41 -0.332 51 -5.865 94 -6.196 66 -6.487 96 -6.726 75 -6.874 78 -6.907 77
0.003 06 0.005 75 0.008 01 0.010 17 -0.825 05 -0.210 95 0.754 38 2.185 78 4.420 56 7.582 28
1.400 1.400 1.400 1.400 1.400 1.400 1.400 1.400
0.005 00 0.010 00 0.050 00 0.100 00 0.135 00 0.750 00 0.850 00 0.950 00
-0.039 02 -0.077 99 -0.387 59 -0.767 70 -1.027 78 -4.930 57 -5.444 69 -5.715 74
0.006 90 0.013 60 0.060 37 0.103 49 0.125 21 1.938 22 4.073 63 8.049 78
0.700 0.700 0.700 0.700 0.700 0.700 0.700
0.005 00 0.010 00 0.020 00 0.026 83 0.850 00 0.950 00 1.050 00
-0.061 78 -0.126 58 -0.270 02 -0.383 99 -6.139 78 -6.646 05 -6.824 39
0.003 33 0.006 30 0.011 10 0.013 39 0.116 99 2.638 39 8.062 99
0.750 0.750 0.750 0.750 0.750 0.750 0.750 0.750
0.005 00 0.010 00 0.020 00 0.033 14 0.850 00 0.950 00 1.000 00 1.050 00
-0.057 80 -0.117 51 -0.244 98 -0.437 95 -6.083 39 -6.568 39 -6.718 85 -6.741 34
0.003 59 0.006 84 0.012 30 0.017 17 0.436 50 3.078 88 5.332 02 8.544 33
1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500
0.005 00 0.010 00 0.050 00 0.100 00 0.150 00 0.200 00 0.300 00 0.400 00 0.500 00 0.600 00 0.700 00 0.750 00 0.800 00 0.850 00 0.900 00 0.950 00 1.000 00
-0.037 90 -0.075 74 -0.375 83 -0.743 06 -1.099 15 -1.425 77 -2.100 96 -2.704 93 -3.315 68 -3.959 95 -4.573 82 -4.859 64 -5.123 82 -5.355 74 -5.519 78 -5.600 49 -5.647 78
0.007 41 0.014 63 0.066 03 0.116 15 0.153 69 0.180 94 0.231 48 0.279 03 0.414 35 0.769 70 1.608 89 2.308 69 3.267 86 4.571 20 6.360 35 8.726 52 11.57 96
0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800
0.005 00 0.010 00 0.020 00 0.030 00 0.040 13 0.750 00 0.850 00 1.000 00 1.050 00
-0.054 63 -0.110 51 -0.227 21 -0.353 11 -0.494 12 -5.380 86 -6.029 87 -6.641 19 -6.658 76
0.003 84 0.007 37 0.013 46 0.018 20 0.021 53 -0.448 16 0.748 95 5.782 81 9.024 36
1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600 1.600
0.005 00 0.010 00 0.050 00 0.100 00 0.160 00 0.650 00 0.750 00 0.850 00 0.950 00
-0.036 93 -0.073 80 -0.365 91 -0.723 00 -1.137 18 -4.222 16 -4.794 79 -5.267 57 -5.517 63
0.007 91 0.015 65 0.071 66 0.128 65 0.182 35 1.341 91 2.652 86 5.061 35 9.369 90
0.850 0.850 0.850
0.005 00 0.010 00 0.020 00
-0.052 04 -0.104 92 -0.213 82
0.004 10 0.007 90 0.014 59
1.800 1.800 1.800 1.800 1.800 1.800 1.800 1.800
0.010 00 0.050 00 0.100 00 0.150 00 0.180 00 0.750 00 0.850 00 0.950 00
-0.070 65 -0.350 19 -0.692 16 -1.025 11 -1.220 28 -4.667 10 -5.096 51 -5.270 63
0.017 70 0.082 83 0.153 35 0.214 84 0.248 88 3.334 99 6.008 50 10.66 23
Equation of State for Real Fluids
J. Phys. Chem., Vol. 100, No. 43, 1996 17369
TABLE 3 (Continued) T*
F*
U*
P*
T*
F*
U*
P*
0.850 0.850 0.850 0.850 0.850 0.850 0.850
0.030 00 0.047 81 0.750 00 0.850 00 0.950 00 1.000 00 1.050 00
-0.328 29 -0.552 62 -5.335 76 -5.977 89 -6.420 68 -6.564 20 -6.576 86
0.020 04 0.026 51 -0.230 37 1.054 49 3.928 13 6.229 13 9.501 84
1.800
1.050 00
-5.177 18
17.77 09
0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900
0.005 00 0.010 00 0.020 00 0.030 00 0.045 00 0.056 17 0.950 00 1.000 00 1.050 00
-0.049 89 -0.100 34 -0.203 29 -0.309 71 -0.478 27 -0.613 20 -6.349 91 -6.488 03 -6.495 82
0.004 36 0.008 43 0.015 70 0.021 82 0.028 77 0.032 16 4.338 39 6.671 11 9.975 97
2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000
0.010 00 0.050 00 0.100 00 0.150 00 0.200 00 0.250 00 0.350 00 0.550 00 0.950 00
-0.068 17 -0.338 06 -0.668 87 -0.992 32 -1.308 63 -1.639 06 -2.201 79 -3.448 79 -5.062 09
0.019 74 0.093 92 0.177 77 0.254 90 0.329 01 0.404 45 0.583 74 1.351 79 11.88 60
0.950 0.950 0.950 0.950 0.950 0.950 0.950 0.950 0.950 0.950
0.005 00 0.010 00 0.020 00 0.040 00 0.050 00 0.065 22 0.750 00 0.850 00 1.000 00 1.050 00
-0.048 06 -0.096 51 -0.194 73 -0.398 33 -0.505 00 -0.675 89 -5.251 93 -5.875 92 -6.412 68 -6.415 54
0.004 61 0.008 95 0.016 80 0.029 21 0.033 76 0.038 52 0.192 31 1.646 76 7.107 89 10.44 59
2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500
0.010 00 0.050 00 0.100 00 0.150 00 0.200 00 0.250 00 0.450 00 0.550 00 0.650 00 0.750 00
-0.063 67 -0.316 25 -0.627 41 -0.934 02 -1.236 89 -1.537 14 -2.701 84 -3.286 92 -3.808 47 -4.245 50
0.024 84 0.121 45 0.238 09 0.353 72 0.472 68 0.599 87 1.388 84 2.138 95 3.434 20 5.576 61
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.005 00 0.010 00 0.020 00 0.040 00 0.060 00 0.074 95 0.850 00 1.000 00 1.050 00
-0.046 51 -0.093 27 -0.187 66 -0.380 90 -0.582 52 -0.740 86 -5.825 18 -6.338 34 -6.336 03
0.004 87 0.009 47 0.017 89 0.031 60 0.041 19 0.045 67 1.934 44 7.539 31 10.91 22
3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000
0.010 00 0.050 00 0.100 00 0.150 00 0.200 00 0.250 00 0.350 00 0.450 00 0.550 00 0.650 00
-0.060 32 -0.300 02 -0.596 34 -0.889 53 -1.180 31 -1.469 57 -2.025 10 -2.582 84 -3.119 14 -3.593 18
0.029 93 0.148 79 0.297 76 0.451 28 0.614 48 0.793 36 1.242 11 1.899 25 2.881 90 4.493 66
1.050 1.050 1.050 1.050 1.050 1.050 1.050 1.050 1.050 1.050 1.050
0.005 00 0.010 00 0.020 00 0.040 00 0.060 00 0.085 37 0.550 00 0.750 00 0.850 00 1.000 00 1.050 00
-0.045 14 -0.090 44 -0.181 59 -0.366 60 -0.556 49 -0.807 43 -3.827 68 -5.179 68 -5.774 97 -6.265 01 -6.257 48
0.005 12 0.009 99 0.018 97 0.033 95 0.045 05 0.053 68 -0.205 92 0.599 49 2.216 57 7.965 57 11.37 39
4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000
0.010 00 0.050 00 0.100 00 0.150 00 0.200 00 0.250 00 0.350 00 0.450 00 0.550 00 0.650 00 0.750 00 0.850 00
-0.055 40 -0.275 96 -0.549 39 -0.820 53 -1.089 60 -1.356 81 -1.871 67 -2.362 08 -2.821 54 -3.195 88 -3.442 74 -3.493 78
0.040 10 0.203 17 0.415 93 0.643 93 0.893 88 1.173 82 1.875 11 2.877 12 4.318 54 6.514 78 9.850 75 14.85 66
1.100 1.100 1.100 1.100 1.100 1.100 1.100 1.100 1.100 1.100 1.100 1.100
0.005 00 0.020 00 0.040 00 0.060 00 0.080 00 0.096 47 0.400 00 0.500 00 0.750 00 0.850 00 1.000 00 1.050 00
-0.043 96 -0.176 48 -0.354 95 -0.536 14 -0.721 01 -0.876 82 -2.874 82 -3.475 68 -5.143 89 -5.725 78 -6.192 65 -6.179 88
0.005 38 0.020 04 0.036 27 0.048 80 0.057 80 0.062 65 -0.077 45 -0.129 34 0.800 78 2.493 44 8.386 87 11.83 07
5.000 5.000 5.000 5.000 5.000 5.000 5.000 5.000 5.000 5.000 5.000
0.010 00 0.050 00 0.100 00 0.150 00 0.200 00 0.250 00 0.350 00 0.450 00 0.550 00 0.650 00 0.750 00
-0.051 55 -0.256 80 -0.511 15 -0.762 91 -1.011 84 -1.257 58 -1.727 00 -2.155 00 -2.543 29 -2.809 14 -2.954 01
0.050 26 0.257 27 0.533 07 0.834 24 1.169 01 1.547 22 2.487 74 3.808 24 5.711 91 8.485 28 12.47 99
1.150 1.150 1.150 1.150 1.150 1.150 1.150
0.005 00 0.020 00 0.040 00 0.060 00 0.080 00 0.108 26 1.000 00
-0.042 89 -0.171 89 -0.344 77 -0.518 94 -0.694 81 -0.947 20 -6.121 37
0.005 63 0.021 11 0.038 56 0.052 48 0.063 07 0.072 68 8.802 58
6.000 6.000 6.000 6.000 6.000 6.000 6.000 6.000 6.000 6.000
0.010 00 0.050 00 0.100 00 0.150 00 0.200 00 0.250 00 0.350 00 0.450 00 0.550 00 0.650 00
-0.048 20 -0.239 95 -0.477 00 -0.710 70 -0.940 42 -1.165 39 -1.587 77 -1.956 86 -2.270 29 -2.460 02
0.060 41 0.311 20 0.649 47 1.022 82 1.440 88 1.915 09 3.083 59 4.701 89 7.026 76 10.28 37
1.200 1.200 1.200 1.200 1.200 1.200
0.005 00 0.010 00 0.050 00 0.100 00 0.120 00 1.000 00
-0.041 97 -0.083 95 -0.420 50 -0.843 81 -1.014 41 -6.050 99
0.005 89 0.011 54 0.048 88 0.077 35 0.083 72 9.213 63
17370 J. Phys. Chem., Vol. 100, No. 43, 1996
Sun and Teja
TABLE 4: Universal Constants c1-c32 in the MBWR Equation
TABLE 5: Comparison of U* and P* Calculated from Equations of State with Computer Simulation Data (All Values in percent)
i
cia
i
ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0.32082971D+00 0.56182644D+01 -0.11956597D+02 0.21320615D+01 -0.14048879D+01 0.18845463D+01 -0.12652950D+01 0.19193601D+01 -0.76863559D+03 -0.13269096D+00 0.15270486D+01 -0.12047540D+01 0.17699304D+02 -0.10261966D+03 0.30276386D+03 0.12886754D+03
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
-0.43416263D+02 -0.15375059D+03 0.68941891D+02 0.76972530D+03 0.10548956D+01 0.22888083D+04 -0.93416742D+00 0.25881616D+04 -0.13842593D+02 0.36560986D+04 0.14155545D+02 0.36833152D+02 -0.12720281D+03 0.19502676D+04 0.70927218D+02 -0.39858862D+01
a
The letter “D” stands for double precision. D+02, for example, indicates that the preceding number should be multiplied by 102.
the virial expansion is applicable at much higher densities (F* ) 0.15 at T* ) 1.5, F* ) 0.20 at T* ) 2.0, and F* ) 0.25 at T* g 2.5). In addition to the virial coefficient data for the gas phase, Ross’s variational perturbation method11 was used to calculate the pressure and internal energy for the liquid state. Although the Ross method is accurate at densities between F* ) 0.7 and F* ) 0.9, only small corrections are necessary for the perturbation results to match the simulations at other densities. The bicubic spline interpolation algorithm was therefore used to apply these corrections. Altogether, 560 experimental data points covering 29 isotherms were included in the data fitting described below, with about half the data points being obtained in this work and the remaining from the work of Johnson et al. The new data of this work are presented in Table 3 Data Fitting As mentioned previously, the MBWR equation of state with 33 constants was used to fit the LJ fluid data. The nonlinear constant γ was set equal to 3 (Nicolas et al., Adachi et al., and Johnson et al.), and the remaining 32 constants were determined by a least-squares fit using an algorithm given by Hust and McCarty.12 The following objective function was minimized:
Min
4∑(U*exp - U*cal)2 + ∑(1 - P*cal/P*exp)2 +
∑(B2*exp - B2*cal)2
(8)
A larger weight was used for U* because of the higher precision of the data for this property obtained by simulation. Data on P* (relative deviation) were included because this favors lowpressure data. The following constraints were imposed at the critical point: T* ) 1.313, F* ) 0.31, P* ) 0.1299, ∂P*/∂F* ) 0, and ∂2P*/∂F*2 ) 0. The critical point values were adopted from Johnson et al. The results for the 32 universal constants for the MBWR are given in Table 4. Results and Comparisons Four equations of state based on the LJ potential were selected for comparison with computer simulation results using the following criteria: (1) correlation of the internal energy and pressure; (2) prediction of vapor pressure; and (3) prediction of vapor-liquid equilibria of mixtures. The four equations are
this work
Johnson
Nicolas
Adachi et al.
U* U* P* P*
AAD MAD AAD MAD
2.01 51 5.43 207
T* ) 0.75 - 6.0 1.49 2.19 51 57 7.43 17.43 666 3534
5.21 80 34.27 3307
U* U* P* P*
AAD MAD AAD MAD
2.17 51 5.93 207
T* ) 0.45 - 6.0 8.95 2.68 518 57 31.84 16.99 4496 3534
6.85 80 32.39 3307
TABLE 6: Comparisons of Vapor Pressures Calculated Using Equations of State with Computer Simulation Data (P*cal - P*exp)/P*exp × 100% T*
P*exp
this work
Johnson
Nicolas
Adachi
0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 AAD, %
0.00131 0.00264 0.00470 0.00769 0.01168 0.01741 0.02505 0.03384 0.04511 0.05974 0.07718 0.0973 0.1204
4.69 -0.19 -1.24 -0.84 1.33 0.69 -0.32 1.73 2.35 1.27 0.72 1.01
5.40 0.73 -0.11 0.36 2.49 1.72 0.57 2.51 3.02 1.83 1.17 1.33
-0.18 -4.63 -5.59 -5.30 -3.42 -4.27 -5.49 -3.83 -3.54 -4.85 -5.68 -5.73
-14.81 -12.40 -8.40 -4.22 0.84 2.43 3.09 6.54 8.25 7.96 8.15
1.37
1.77
4.38
7.01
the LJ equations of Johnson et al., of Nicolas et al., of Adachi et al., and of this work. The internal energy and pressure were compared over two temperature ranges: T* ) 0.7-6 (where all the four equations are applicable) and T* ) 0.45 -6 (where three of the four equations must be extrapolated). Results are given in Table 5 with the average absolute percentage deviation (AAD) and maximum absolute percentage deviation (MAD) between the experimental and calculated values. Table 5 shows that the new equation and the equation of Johnson provide the best fit for both U* and P*, whereas the equation of Nicolas et al. exhibits large errors at some pressures. The equation of state of Adachi et al. exhibits the largest errors in U*, probably because Adachi et al. fit their equation only to pVT data. For the temperature range T* ) 0.45-6, Table 5 shows that the new equation gives the best results for these two properties, whereas the equations of Johnson et al. and Adachi et al. give very large errors. In the case of vapor pressures, the molecular dynamics results of Lotfi et al.10 were chosen for comparison because they provide the vapor-liquid coexistence properties of the pure LJ fluid in the temperature range T* ) 0.70 to T* ) 1.30. The comparisons for the four equations are given in detail at each temperature in Table 6. The equation of Nicolas et al. shows systematic deviations between calculated and experimental values, whereas that of Adachi et al. exhibits large random errors. The new equation gives the best prediction of the vapor pressure curve, with an average error of 1.4%, which is within the uncertainty of the computer simulations. The MBWR equation was extended to mixtures using the van der Waals one-fluid theory for the LJ parameters. Thus
σ3 ) ∑ ∑xixjσ3ij
(9)
σ3 ) ∑ ∑xixjijσ3ij
(10)
Equation of State for Real Fluids
J. Phys. Chem., Vol. 100, No. 43, 1996 17371
Figure 1. Calculations of vapor-liquid equilibria for LJ mixtures with σ22 /σ11 ) 1 and 22 /11 ) 0.5 at T* ) kT/11 ) 1.0. The solid line represents calculated results of this work, whereas the dashed lines represent calculations using the equation of Johnson et al. (- - -) and Adachi et al. (- - --). Simulation data are shown by filled squares.
Figure 2. LJ diameters of the n-alkanes as a function of carbon number.
TABLE 7: LJ Parameters and Correlations of Vapor Pressures and Liquid Densities of n-Alkanes in the Temperature Range T/Tc ) 0.45 - 0.95 for C1 to C10 and T/Tc ) 0.50-0.80 for C11 to C16 n-Alkanes ΑΑD (%) (K)0.5
Pvap
VL
0.47 0.16 0.28 0.24 0.53 0.62 0.89 1.26 1.54 2.13 1.27 1.09 0.87 1.29 0.94 1.36
0.40 1.09 1.08 1.03 1.44 1.60 2.03 1.80 2.26 2.53 1.59 1.74 2.23 1.14 2.30 2.37
substance
σ (Å)
0/k (K)
1/k
methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-undecane n-dodecane n-tridecane n-tetradecane n-pentadecane n-hexadecane
3.7282 4.2338 4.7002 5.0906 5.4409 5.7549 6.0471 6.3130 6.5617 6.7982 7.0345 7.2490 7.4732 7.6006 7.8522 8.0340
152.72 290.79 386.81 476.97 567.69 653.50 741.56 828.20 912.04 993.87 1083.7 1156.7 1231.7 1308.1 1380.2 1459.6
-0.315 -2.953 -5.010 -6.887 -9.057 -11.123 -13.423 -15.709 -17.918 -20.071 -22.739 -24.574 -26.571 -28.523 -30.523 -32.786
where xi is the mole fraction of component i in the mixture. For prediction of vapor-liquid equilibria, simulation results obtained by the Gibbs ensemble method13 were chosen for comparison with the phase equilibrium calculations from the four equations of state. The cross interaction terms in eqs 9 and 10 were obtained from the Lorenz-Berthelot rules without any adjustable parameters. Several calculations have been done with different pairs of LJ parameters, and typical results are given in Figure 1. The figure shows that the new equation gives good results, whereas the equation of Adachi et al. gives large errors. Extension to Real Fluids (n-Alkanes) As is well-known, the LJ potential is accurate only for simple fluids such as argon and neon and is a poor representation for fluids such as cyclohexane and n-decane. However, the LJ potential may be extended to such fluids if effective parameters, which incorporate nonspherical effects as well as the effect of many-body interactions, are employed. (See, for example, the work of Rowlinson,14 who used a spherical LJ potential to treat nonspherical molecules such as nitrogen, propylene, and carbon dioxide.) Following the work of Rowlinson, the LJ energy parameter was made temperature dependent for real fluids, with the LJ diameter σ left unchanged. Thus
) 0 + 1T0.5
(11)
The n-alkanes, CH4 to C16H34, were chosen for study because
Figure 3. LJ parameter 0 of the n-alkanes as a function of carbon number.
Figure 4. LJ parameter 1 of the n-alkanes as a function of carbon number.
they form the simplest class of molecular fluids and because their properties change regularly with chain length. Moreover, thermodynamic properties of the n-alkanes have already been studied using methods such as cubic equations of state. Vapor pressures and liquid densities of the n-alkanes were obtained from the compilation of Daubert and Danner15 and were used to obtain the LJ parameters σ, 0, and 1 for each n-alkane. The parameters are given in Table 7 together with the AAD in the vapor pressure Pvap and AAD in liquid volume VL. The fit for the smaller n-alkanes were found to be better than those for the larger molecules. The fact that the energy parameter becomes more temperature dependent (1 become larger) as carbon number increases represents a significant nonspherical effect for large n-alkane molecules. It should also be mentioned that some scatter is observed for C14H30, probably due to uncertainty in the experimental data. The parameters are plotted as functions of the carbon number in Figures 2-4 and show regular behavior with carbon number. To test the predictive capability of the equation of state, the heat capacity at constant pressure Cp was calculated. Such calculations involve second-order derivatives of the equation of state and represent a severe test of its ability to predict thermodynamic properties. Experimental Cp data for the n-alkanes were obtained from the compilation of Daubert and
17372 J. Phys. Chem., Vol. 100, No. 43, 1996
Sun and Teja
TABLE 8: Comparison of Cp of n-Alkanes Predicted Using the MBWR and P-T Equations of State in the Temperature Range T/Tc ) 0.45-0.95 (Cp,cal - Cp,exp)/Cpexp × 100% C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
MBWR EOS 3.8 1.7 1.6 1.9 1.9 3.2 2.4 2.9 5.5 9.0 P-T EOS 13.1 9.0 7.0 6.2 4.0 3.4 4.3 3.3 5.5 6.0
Danner,15 and calculated values were obtained from the relation
Cp ) Cp0 - T(∂p/∂T)V2/(∂p/∂V)T - T(∂2A/∂T2)V - R
characteristic parameters for each alkane and is able to predict vapor pressures, liquid densities, and heat capacities of the alkanes. Furthermore, the three parameters show regular behavior with the number of carbon atoms in the alkane and thus show promise for extrapolation to alkanes for which no data are available. Acknowledgment. This research was supported by members of the Fluid Properties Research Industrial Associates Program, based at Georgia Tech. References and Notes
(12)
Comparisons in the temperature range T/Tc from 0.45 to 0.95 are presented in Table 8. Also tabulated are comparisons using the Patel-Teja cubic equation of state2 (P-T EOS). For C1 to C8 n-alkanes, the MBWR EOS predicts better results than the P-T EOS, with an average deviation of 2.47% vs 6.34% for the P-T EOS. Summary New constants of the MBWR equation of state are presented, based on computer simulation data for the LJ fluid, supplemented with data obtained from the virial expansion and the Ross perturbation method. Comparison of the new EOS with existing equations shows that the new EOS is able to correlate computer simulation data over a wide temperature range from T* ) 0.45 to T* ) 6. An extension of the new EOS to real fluids has been tested and found to be satisfactory for n-alkanes up to C16H34. The equation requires a knowledge of only three
(1) Walas, S. M. Phase Equilibria in Chemical Engineering; Butterworth: Stoneham, 1985. (2) Patel, N. C.; Teja, A. S. Chem. Eng. Sci. 1982, 37, 463. (3) Johnson, J. K.; Zollweg, J. A.; Gubbins, K. E. Mol. Phys. 1993, 78, 591. (4) Nicolas, J. J.; Gubbins, K. E.; Streett, W. R.; Tildesley, D. J. Mol. Phys. 1979, 37, 1429. (5) Adachi, Y.; Fijihara, I.; Takamiya, M.; Nakanishi, K. Fluid Phase Equilib. 1988, 39, 1. (6) Miyano, Y. Fluid Phase Equilib. 1993, 85, 71. (7) Barker, J. A.; Monaghan, J. J. J. Chem. Phys. 1962, 36, 2564. (8) Barker, J. A.; Leonard, P. J.; Pompe, A. J. Chem. Phys. 1966, 44, 4206. (9) Rowlinson, J. S. Proc. Roy. Soc. London 1964, A279, 147. (10) Lotfi, A.; Vrabec, J.; Fischer, J. Mol. Phys. 1992, 76, 1319. (11) Ross, M. J. Chem. Phys. 1979, 71, 1567. (12) Hust, J. G.; McCarty, R. D. Cryogenics 1967, 2, 200. (13) Harismiadis, V. I.; Koutras, N. K.; Tassios, D. P.; Panagiotopoulos, A. Z. Fluid Phase Equilib. 1991, 65, 1. (14) Rowlinson, J. S. Trans. Faraday Soc. 1954, 50, 647; 1955, 51, 1317. (15) Daubert, T. E.; Danner, R. P. Data Compilation Tables of Properties of Pure Compounds; Design Institute for Physical Properties Data, American Institute of Chemical Engineering: New York, 1985.
JP9620476
