Ind. Eng. Chem. Res. 2005, 44, 6973-6980
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Extension of a New Equation of State to the Liquid Mixtures Elaheh K. Goharshadi* and Majid Moosavi Department of Chemistry, Ferdowsi University of Mashhad, Mashhad 91779, Iran
A new equation of state recently reported for pure liquids has been extended to predict the volumetric and thermodynamic properties of 10 binary mixtures and 1 ternary liquid mixture at different temperatures, pressures, and compositions. A wide comparison with literature experimental data shows the ability of this equation of state to reproduce and predict different volumetric and thermodynamic properties of liquid mixtures. For the binary and ternary mixtures, the composition dependences of the parameters of the equation of state are assumed as quadratic functions of mole fraction. Using these mixing rules, the agreement between the calculated and experimental densities is better than 0.8% for binary mixtures and 3.1% for ternary mixtures. Introduction Accurate thermodynamic properties play a vital role in nearly every area of chemical engineering, as well as in our fundamental understanding of the fluid phase behavior of pure fluids and mixtures. A good understanding of the PVT (pressure-volume-temperature) behavior of pure compounds and mixtures is of great importance in many fields of research as well as in industrial practice. The densities of fluids as a function of temperature, pressure, and composition are particularly important for the design of industrial plants, pipelines, and pumps. These data are needed for solving material and energy balances required for the design and optimization of chemical processes.1 The most convenient form for representation of PVT behavior for process design and optimization calculations has long been recognized as that of analytic equations of state. The equation of state (EoS) has a central importance in the treatment of the thermodynamic properties of fluids, particularly mixtures. A general equation of state for liquids has been recently derived by Goharshadi et al.2 (GoharshadiMorsali-Abbaspour, GMA EoS) which has been found to be valid for polar, nonpolar, and hydrogen-bonded fluids.2,3 The equation of state is based on the average potential energy and is given as
(2Z - 1)Vm3 ) A(T) + B(T)F
(1)
where Z, Vm, and F are compressibility factor, molar volume, and density, respectively. The intercept and slope of and this equation depends on the temperature via the equations
A(T) ) Ao -
2A1 2A2 ln T + RT R
2B1 2B2 ln T + B(T) ) Bo RT R
(2) (3)
where Ao-A2 and Bo-B2 are constants. To use the equation of state for a liquid, the A and B parameters must be known. To find these parameters, we may plot * Corresponding author. Tel.: +98511-8432022. Fax: +985118438032. E-mail:
[email protected].
Figure 1. (2Z - 1)Vm3 versus F for the 1,3,5-trimethyl benzenebenzene mixture at 298.15 K and different mole fractions.5
(2Z - 1)Vm3 versus F for different isotherms. The slope and intercept of the straight lines can be fitted to eqs 2 and 3, from which Ao-A2 and Bo-B2 can be found, respectively. According to one-fluid approximation, this EoS can be extended to mixtures, but the parameters of the EoS (A and B) are dependent on composition as well as temperature.4 In the present work, we try to reach the following aims: (1) examine the validity of eqs 1-3 for mixtures; (2) calculate the volumetric properties of different binary mixtures and one ternary mixture at various temperatures, pressures, and compositions; and (3) determine the composition dependence of the parameters of the equation of state. Results and Discussion Experimental Test of GMA EoS. We have used the experimental PVT data of dense fluid mixtures at various temperatures and compositions to examine the linearity of (2Z - 1)Vm3 versus F (eq 1). Figure 1 shows the results at 298.15 K for the 1,3,5-trimethyl benzenebenzene mixture at different mole fractions, and Figure 2 shows the results for a mixture of 1-chloropentaneoctane (xoctane ) 0.50421) at different temperatures. As the figures show, the linearity holds very well for all mixtures and the slope and the intercept both depend
10.1021/ie050158y CCC: $30.25 © 2005 American Chemical Society Published on Web 07/07/2005
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Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005
Table 1. Intercept (A), Slope (B), Square of Correlation Coefficient (R2) of Eq 1, and Pressure Range of the Data for Binary Mixtures x
T (K)
B(T) -A(T) (10-9 L3/mol3) (10-12 L4/mol4)
R2
∆P (MPa)
1,3,5-trimethylbenzene (1 - x)-benzene (x) 5 0.00000 298.15 0.393927 0.0546153 0.9999 0.1-39.62 308.15 0.373131 0.0521888 0.9998 318.15 0.354155 0.0499826 1.0000 328.15 0.33563 0.0478003 0.9999 0.10849 298.15 0.330522 0.0440728 0.9999 0.1-39.62 308.15 0.316695 0.0426339 0.9998 318.15 0.299321 0.0406572 0.9999 328.15 0.283059 0.0388015 0.9999 0.19391 298.15 0.288013 0.0372078 0.9999 0.1-39.62 308.15 0.27695 0.0361244 1.0000 318.15 0.262178 0.0345085 0.9999 328.15 0.247986 0.0329444 0.9999 0.29799 298.15 0.245764 0.030496 0.9999 0.1-39.62 308.15 0.232842 0.0291651 0.9999 318.15 0.220374 0.0278614 0.9999 328.15 0.206793 0.0263861 0.9999 0.43460 298.15 0.193162 0.0226633 0.9999 0.1-39.62 308.15 0.182457 0.0216099 0.9998 318.15 0.173088 0.0207001 0.9999 328.15 0.163061 0.0196854 0.9999 0.49717 298.15 0.171323 0.0195722 0.9999 0.1-39.62 308.15 0.162525 0.0187455 0.9999 318.15 0.153846 0.0179158 0.9999 328.15 0.144647 0.0170032 1.0000 0.60681 298.15 0.139109 0.015117 0.9998 0.1-39.62 308.15 0.130453 0.0143147 0.9999 318.15 0.123834 0.013728 0.9999 328.15 0.117398 0.013142 0.9999 0.70885 298.15 0.112321 0.0116238 0.9999 0.1-39.62 308.15 0.106579 0.0111402 0.9999 318.15 0.100724 0.010636 1.0000 328.15 0.0949319 0.0101241 1.0000 0.79512 298.15 0.0922518 0.00913644 0.9999 0.1-39.62 308.15 0.0881112 0.00881795 1.0000 318.15 0.083481 0.008442 1.0000 328.15 0.078872 0.008058 1.0000 0.89947 298.15 0.072265 0.006763 1.0000 0.1-39.62 308.15 0.06939 0.006566 1.0000 318.15 0.065553 0.00627 1.0000 328.15 0.062258 0.00602 1.0000 1.00000 298.15 0.057708 0.005096 0.9999 0.1-39.62 308.15 0.05456 0.004872 1.0000 318.15 0.051694 0.004668 1.0000 328.15 0.048772 0.004454 1.0000 1-chloropentane (1 - x)-octane (x)6 0.00000 298.15 0.17354 0.020879 0.9999 2.13-38.6 308.15 0.164298 0.019976 0.9999 318.15 0.155195 0.01907 0.9999 328.15 0.147048 0.018263 0.9999 0.10255 298.15 0.194891 0.024294 0.9999 2.13-38.6 308.15 0.184508 0.023244 0.9999 318.15 0.174112 0.022168 0.9999 328.15 0.16434 0.021147 0.9999 0.19057 298.15 0.215217 0.027627 0.9998 2.13-38.6 308.15 0.20373 0.026433 0.9999 318.15 0.192039 0.025182 0.9999 328.15 0.181088 0.024001 0.9999 0.31109 298.15 0.245536 0.03277 0.9999 2.13-38.6 308.15 0.231663 0.031247 0.9999 318.15 0.218694 0.029813 0.9999 328.15 0.205926 0.028375 0.9999 0.39496 298.15 0.268312 0.036757 0.9999 2.13-38.6 308.15 0.253876 0.035148 0.9999 318.15 0.238798 0.033415 0.9999 328.15 0.224795 0.031795 0.9999 0.50421 298.15 0.300642 0.04256 0.9999 2.13-38.6 308.15 0.283994 0.040635 0.9999
x
T (K)
0.50421 318.15 328.15 0.57786 298.15 308.15 318.15 328.15 0.60675 298.15 308.15 318.15 328.15 0.67832 298.15 308.15 318.15 328.15 0.78698 298.15 308.15 318.15 328.15 0.88835 298.15 308.15 318.15 328.15 1.00000 298.15 318.15 328.15 0.000
0.227
0.410
0.410
0.601
0.799
303.15 323.15 353.15 393.15 303.15 323.15 353.15 393.15 303.15 323.15 353.15 393.15 303.15 323.15 353.15 393.15 303.15 323.15 353.15 393.15 303.15 323.15 353.15 393.15
0.25197 288.15 298.14 313.14 323.14 338.13 0.50139 288.15 298.14 313.14 323.14 338.13 1.00000 278.15 288.15 298.14 313.14 323.14
-A(T) B(T) (10-9 L3/mol3) (10-12 L4/mol4)
R2
∆P (MPa)
0.267096 0.038629 0.9999 0.251874 0.0368191 0.9999 0.324139 0.0468884 0.9999 2.13-38.6 0.305393 0.0446425 0.9999 0.288235 0.0425956 0.9999 0.270696 0.0404296 0.9999 0.33376 0.0486806 0.9999 2.13-38.6 0.314635 0.046378 0.9999 0.297383 0.0443117 1.0000 0.278606 0.0419537 0.9999 0.358443 0.053353 0.9999 2.13-38.6 0.338909 0.0509848 0.9999 0.318234 0.0483874 0.9999 0.298844 0.0459217 0.9999 0.399031 0.0611894 0.9999 2.13-38.6 0.375968 0.0582639 0.9999 0.353606 0.0553842 0.9999 0.332267 0.0525968 0.9999 0.439007 0.069145 0.9999 2.13-38.6 0.413716 0.0658658 0.9999 0.3893 0.0626419 0.9999 0.365744 0.0594741 0.9999 0.4865 0.0788486 0.9999 2.13-38.6 0.430955 0.0713493 0.9999 0.404339 0.0676544 0.9999 n-decane (1 - x)-methane (x)7 1.15048 0.225126 0.9989 0.1-76.19 1.01490 0.20256 0.9995 0.864919 0.178001 0.9990 0.686016 0.147602 0.9997 0.478407 0.0776164 0.9989 9.828-76.132 0.426724 0.0708920 0.9994 0.352203 0.0606961 0.9990 0.287403 0.0518761 0.9995 0.185088 0.0252123 0.9994 20.02-75.61 0.159138 0.0223044 0.9984 0.352203 0.0606961 0.9990 0.287403 0.0518761 0.9995 0.185088 0.025212 0.9994 20.02-75.61 0.159138 0.022304 0.9984 0.137771 0.020054 0.9983 0.108607 0.016745 0.9986 0.049778 0.005527 0.9997 24.84-75.15 0.042397 0.004878 0.9985 0.033924 0.004137 0.9973 0.029442 0.003809 0.9984 0.006862 0.000594 0.9933 40-75.25 0.005128 0.000481 0.9993 0.003767 0.000399 0.9968 0.002841 0.000353 0.9851 water (1 - x)-tetrahydrofuran (x)8 0.001947 6.30E-05 1.0000 100-275 0.001851 6.04E-05 1.0000 0.001723 5.70E-05 1.0000 0.001645 5.48E-05 1.0000 0.001531 5.17E-05 1.0000 0.007308 0.00035 0.9998 0.1-300 0.006912 0.000334 0.9998 0.006385 0.000313 0.9998 0.006055 0.0003 0.9997 0.005607 0.000283 0.9997 0.045676 0.003601 0.9995 0.1-300 0.042959 0.003425 0.9995 0.040467 0.003261 0.9993 0.036933 0.003027 0.9994 0.035293 0.002927 0.9996
Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005 6975 Table 1 (Continued) x
T (K)
-A(T) B(T) (10-9 L3/mol3) (10-12 L4/mol4)
R2
water (1 - x)-trifluoroethanol (x)9 0.006815 0.000358 0.9993 0.006654 0.000354 0.9985 0.005878 0.000319 0.9991 0.005326 0.000296 0.9991 0.0048296 0.00027559 0.9991 0.0043856 0.00025704 0.999 0.003978 0.00023999 0.999 heptane (1 - x)-pentane (x)10 298.15 0.079307 0.00902625 0.9999 323.15 0.0672389 0.00792723 0.9997 343.15 0.0576302 0.00706529 0.9996 298.15 0.250978 0.0355654 0.9999 323.15 0.210922 0.0307952 0.9995 348.15 0.184697 0.0278396 0.9996 298.15 0.186889 0.0250237 1.0000 323.15 0.15785 0.0217934 0.9997 348.15 0.137249 0.0195925 0.9996 298.15 0.159208 0.02064 0.9999 323.15 0.133638 0.0178811 0.9997 348.15 0.116016 0.0160641 0.9996 298.15 0.0980553 0.0114722 0.9999 323.15 0.0827448 0.0100071 0.9997 348.15 0.0713006 0.00894014 0.9996 298.15 0.0953814 0.0112379 0.9999 323.15 0.0808451 0.00985462 0.9996 348.15 0.0694332 0.00878707 0.9995 298.15 0.0951863 0.0112013 0.9999 323.15 0.0804876 0.00980148 0.9996 348.15 0.0690958 0.00873671 0.9996 298.15 0.158728 0.0205821 0.9999 323.15 0.134676 0.0180236 0.9996 348.15 0.117108 0.0162211 0.9996 heptane (1 - x)-hexane (x)10 298.15 0.288115 0.0420096 0.9999 323.15 0.245296 0.0368237 0.9997 348.15 0.214443 0.0332145 0.9996 298.15 0.269749 0.0387783 0.9999 323.15 0.227659 0.0337104 0.9997 348.15 0.199188 0.0304409 0.9996 298.15 0.233024 0.032559 0.9999 323.15 0.197214 0.028399 0.9997 348.15 0.171805 0.025546 0.9996 298.15 0.200296 0.027192 0.9999 323.15 0.169891 0.023773 0.9997 348.15 0.147814 0.021368 0.9996 298.15 0.185954 0.024859 0.9999 323.15 0.157267 0.021678 0.9997 348.15 0.136899 0.019509 0.9996 298.15 0.175499 0.023205 0.9998 323.15 0.14831 0.020229 0.9996 348.15 0.129213 0.01823 0.9996 298.15 0.171752 0.022618 0.9999 323.15 0.14546 0.019757 0.9996 348.15 0.126276 0.017742 0.9996 298.15 0.158728 0.020582 0.9999 323.15 0.134676 0.018024 0.9996 348.15 0.117108 0.016221 0.9996 hexane (1 -x)-pentane10 (x) 298.15 0.158728 0.020582 0.9999 323.15 0.134676 0.018024 0.9996 348.15 0.117108 0.016221 0.9996 298.15 0.147299 0.018814 0.9999 323.15 0.124417 0.016409 0.9997 348.15 0.10745 0.014669 0.9993 298.15 0.147578 0.018846 0.9999 323.15 0.124462 0.016415 0.9997 348.15 0.107871 0.014725 0.9996
∆P (MPa)
x 0.248
0.65330 310 320 340 360 380 400 420
0.1-200
1.000
0.1-40
0.374
0.1-40
0.499
0.1-40
0.622
0.1-40
0.75
0.1-40
0.874
0.1-40
1.000
0.126
0.371
0.502
0.751
0.874
0.877
1.000
0.000
0.125
0.375
0.625
0.750
0.840
0.874
1.000
0.000
0.122
0.124
0.250
0.372
T (K) 298.15 323.15 348.15 298.15 323.15 348.15 298.15 323.15 348.15 298.15 323.15 348.15 298.15 323.15 348.15 298.15 323.15 348.15 298.15 323.15 348.15 298.15 323.15 348.15 298.15 323.15 348.15
0.1-40
0.1-40
0.1-40
0.1-40
0.1-40
0.1-40
0.1-40
0.1-40
0.1-40
0.1-40
0.1-40
0.1-40 0.1-40
0.2001 273.21 293.21 313.16 333.15 353.14 373.14 393.13 413.12 428.12 448.12 473.11 0.3508 273.21 293.21 313.15 333.15 353.14 373.13 393.12 413.12 428.12 0.5001 273.21 293.22 313.15 333.15 353.14 373.14 393.13 413.12 428.12 0.8000 273.21 293.21 313.16 333.15 353.14 373.14 393.14 413.13 428.13 448.13 473.13
-A(T) B(T) (10-9 L3/mol3) (10-12 L4/mol4)
R2
0.135929 0.017086 0.9999 0.11492 0.014922 0.9996 0.100983 0.013567 0.9998 0.135337 0.017005 0.9999 0.113904 0.014783 0.9996 0.098666 0.013261 0.9996 0.124974 0.015469 0.9999 0.105606 0.0135094 0.9997 0.0913268 0.012111 0.9996 0.124829 0.0154371 0.9999 0.105229 0.0134488 0.9996 0.0909185 0.0120455 0.9996 0.114414 0.0139279 0.9998 0.09664 0.0121602 0.9997 0.0832955 0.0108693 0.9996 0.104377 0.0125024 0.9999 0.0887694 0.0109957 0.9996 0.0762523 0.00980078 0.9996 0.0951709 0.0112079 0.9999 0.0806685 0.00983094 0.9997 0.0681239 0.00861937 0.9995 0.0868362 0.0100533 0.9999 0.0736047 0.0088229 0.9997 0.0632572 0.00787731 0.9995 0.079307 0.00902625 0.9999 0.0672389 0.00792723 0.9996 0.0576302 0.00706529 0.9996 1-butanol-diisopropyl ether (x)11 0.0987429 0.00964883 0.9994 0.0876846 0.00872636 0.9995 0.0782778 0.00793906 0.9995 0.0700406 0.00724566 0.9995 0.0626624 0.00662092 0.9994 0.0563751 0.00609435 0.9994 0.0502455 0.00556873 0.9994 0.0447759 0.0050992 0.9993 0.0410108 0.00477421 0.9993 0.0365723 0.00439413 0.9992 0.0312723 0.00392472 0.9990 0.11867 0.0123998 0.9994 0.105064 0.0111946 0.9994 0.093345 0.0101498 0.9995 0.0830046 0.00921906 0.9995 0.0735817 0.00835994 0.9996 0.0658275 0.00766412 0.9994 0.0585115 0.00699547 0.9994 0.0518144 0.00637671 0.9994 0.0477097 0.00600975 0.9994 0.144285 0.0160543 0.9991 0.124687 0.0141723 0.9994 0.11088 0.0128805 0.9996 0.0976452 0.0116041 0.9995 0.0864988 0.0105315 0.9995 0.0778018 0.00972182 0.9996 0.0679998 0.00874016 0.9993 0.0602188 0.00798165 0.9993 0.0549018 0.00745671 0.9995 0.196129 0.0246351 0.9991 0.171289 0.0220291 0.9992 0.154277 0.0203323 0.9993 0.132556 0.0179222 0.9993 0.116751 0.0162187 0.9993 0.102888 0.0147115 0.9992 0.091247 0.0134565 0.9995 0.079514 0.0121232 0.9990 0.072172 0.0113015 0.9988 0.0629667 0.0102463 0.9985 0.0527228 0.00905125 0.9990
∆P (MPa) 0.1-40
0.1-40
0.1-40
0.1-40
0.1-40
0.1-40
0.1-40
0.1-40
0.1-40
0.355-34.979
0.329-34.975
0.354-34.980
0.355-34.979
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Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005
Table 1 (Continued) x
T (K)
0.000 129.32 134.32 142.68 147.08 0.277 134.32 0.485 129.32 134.32
-A(T) B(T) (10-9 L3/mol3) (10-12 L4/mol4) Kr (1 - x)-Ar (x)12 0.0015465 0.000053595 0.0014497 0.000050922 0.0012987 0.000046744 0.0012251 0.000044687 0.0010631 0.000036732 0.0008812 0.000029694 0.0008193 0.000028126
R2
∆P (MPa)
T (K)
x
0.485 142.68 1.0000 1.04-114.67 147.08 0.9996 0.698 134.32 0.9998 0.787 134.32 0.9999 1.000 129.32 0.9989 4.62-66.97 134.32 0.9998 3.3-150.65 142.68 1.0000 147.08
on the temperature as well as the composition of the mixture. The results are summarized in Table 1, including the intercept and slope of the fitted straight line (eq 1) at each temperature and composition, the square of the correlation coefficient (R2), and the pressure range of the experimental data.
-A(T) B(T) (10-9 L3/mol3) (10-12 L4/mol4) 0.0007235 0.0006785 0.0006347 0.0005602 0.0004537 0.000414 0.0003529 0.0003241
0.000025653 0.000024504 0.000021702 0.000019138 0.000015159 0.000014194 0.000012728 0.000012065
R2
∆P (MPa)
0.9998 0.9999 1.0000 5.52-134.32 0.9998 3.66-76.6 1.0000 7.77-142.23 0.9999 0.9997 0.9999
composition with relatively high precision. Figures 3 and 4 show the three-dimensional plots of percent deviations of calculated densities for 1,3,5-trimethyl benzene-benzene and chloropentane-octane mixtures, respectively. Derived Properties from GMA EoS. The expansion coefficient, R ) 1/V(∂V/∂T)P; the isothermal compressibility, κT ) -1/V(∂V/∂P)T; and the internal pressure, Pi ) (∂U/∂V)T, of some mixtures have been calculated at different temperatures, pressures, and compositions. The functions used for calculating these properties using GMA EoS are given as eqs 5-7, respectively. R) (2B1 + 2B2T)F5 + (2A1 + 2A2T)F4 + 2P 5
2
5F (RT Bo - 2B1T + 2T2B2 ln T) + 4F4(AoRT2 - 2A1T + 2A2T2 ln T) + RT2F
(5) κT ) 2 FRT + 4F4(RTAo - 2A1 + 2TA2 ln T) + 5F5(BoRT - 2B1 + 2B2T ln T)
(6) Figure 2. Isotherms of (2Z - 1)Vm3 versus F for a mixture of 1-chloropentane-octane (xoctane ) 0.50421).6
Considering the values of R2 of the different mixtures, it seems that this linearity is a universal feature of all liquid mixtures. Table 2 shows the values of the constants of eqs 2 and 3 for tested liquid mixtures and the square of the correlation coefficients of eqs 2 and 3. The ranges of pressure and temperature for this table are the same as those for Table 1. As the values of R2 in this table show, the temperature dependence of the parameters of the equations of state, namely A and B, for all mixtures is the same as that of the pure case. A more sensible test for the equation of state is to calculate the density at different temperatures, pressures, and compositions and to compare this with the corresponding experimental data. The density of 10 binary mixtures in the wide ranges of temperature, pressure, and composition has been calculated using GMA EoS. The function used for calculating density using GMA EoS is given as
B(T,x)F5 + A(T,x)F4 +
FRT -P)0 2
(4)
The ability of this EoS to predict density at different temperatures, pressures, and compositions for all mixtures may be evaluated by absolute average deviation (AAD). The number of calculated densities (NP) and the AAD values between the calculated and experimental densities have been reported in Table 2. The values of AAD suggest that GMA EoS can predict the density of liquid mixtures at any temperature, pressure, and
Pi ) (B1 + B2T)F5 + (A1 + A2T)F4
(7)
Table 3 tabulates the values of AAD between the calculated and literature experimental data of these thermodynamic properties. As the AAD values of this table show, GMA EoS can predict the thermodynamic functions well. Composition Dependence of the Parameters of GMA EoS. To obtain reliable predictions of mixture thermodynamic properties from equations of state, suitable mixing rules are required. In this work, the composition dependence of the parameters can be assumed as the quadratic functions of mole fraction:
Amix )
∑i ∑j xixjAij
(8)
Bmix )
∑i ∑j xixjBij
(9)
The values of Aij and Bij when i ) j can be obtained from experimental PVT data of pure fluids. In this work, the Aij and Bij for i * j have been calculated using one binary mixture. We have examined the quadratic composition dependence of the parameters A and B (eqs 8 and 9) using experimental data for different binary fluid mixtures. The primary criterion for the performance of these mixing rules is to plot the parameters A and B versus composition, fit the plot by quadratic functions of mole fraction, and examine the values of the correlation coefficients. As an example, we show the values of A and B as a function of composition for 1-chloropentane-octane at different temperatures in Figures 5 and
Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005 6977 Table 2. Values of the Constants of GMA EoS, the Correlation Coefficients of Eqs 2 and 3, the Number of Calculated Densities, and the AAD in Density for Mixtures Ao (L3 mol-3)
A1 (L4 atm mol-4)
0.00000 0.10849 0.19391 0.29799 0.43460 0.49717 0.60681 0.70885 0.79512 0.89947 1.00000
-0.38659 -14.7878 -16.0512 -7.39326 -1.75138 -4.9224 4.47521 -2.54442 -3.40258 -2.83691 -0.45267
6.58749 -22.077 -25.4052 -9.2061 0.416095 -6.02771 11.2082 -2.6579 -4.77747 -4.10955 0.255025
1,3,5-trimethylbenzene (1 - x)-benzene (x) 0.003825 1.0000 0.122021 -0.63985 0.091111 0.9992 2.14079 3.38419 0.098555 0.9991 2.20997 3.64359 0.046051 0.9998 1.02973 1.43806 0.011467 0.9996 0.267673 0.132205 0.030665 0.9999 0.632219 0.875668 -0.02663 0.9994 -0.45458 -1.11404 0.015949 1.0000 0.300425 0.379586 0.021025 0.9999 0.36684 0.557866 0.017489 0.9981 0.287065 0.447404 0.002994 0.9999 0.05337 0.019595
0.00000 0.10255 0.19057 0.31109 0.39496 0.50421 0.57786 0.60675 0.67832 0.78698 0.88835 1.00000
-0.04169 -2.17798 -2.92728 -1.69937 -4.58163 -2.02839 -3.16673 -6.15574 -7.75243 -3.39006 -4.21863 -6.00876
3.25437 -0.31107 -1.28952 1.70409 -3.26484 2.24824 0.612057 -4.87701 -7.29784 1.86151 1.09249 -1.21625
1-chloropentane (1 - x)-octane (x) 0.000966 0.9999 0.048084 -0.24516 0.014097 1.0000 0.34013 0.244082 0.01877 0.9999 0.463263 0.417165 0.011473 1.0000 0.319909 0.050426 0.029138 0.9999 0.730003 0.7548 0.013765 0.9999 0.423678 0.072791 0.020831 0.9999 0.603818 0.332769 0.039055 0.9997 1.07555 1.19858 0.048948 0.9999 1.35972 1.64038 0.022634 1.0000 0.72304 0.286973 0.027861 1.0000 0.956584 0.594639 0.03905 1.0000 1.29423 1.04611
0.000 0.227 0.410 0.601 0.799
3.34895 4.99325 1.97253 2.56311 0.581635
30.1451 20.0429 7.8299 6.32095 1.39302
A2 (L4 atm mol-4 K-1)
Bo (L4 mol-4)
x
R2
B1 (L5 atm mol-5)
n-decane (1 - x)-methane (x) 0.9994 -0.19575 -4.34494 0.9990 -0.52693 -2.42995 0.9953 -0.1796 -0.79504 0.9989 -0.25234 -0.6108 0.9983 -0.04346 -0.10158
-0.01489 -0.02773 -0.01096 -0.01511 -0.00342
B2 (L5 atm mol-5 K-1)
R2
NP
AAD
-0.00086 -0.01311 -0.0135 -0.00635 -0.00169 -0.0039 0.002726 -0.00186 -0.00225 -0.00176 -0.00034
0.9999 0.9988 0.9986 0.9998 0.9994 0.9999 0.9990 1.0000 0.9998 0.9968 0.9998
92 92 92 92 92 92 92 92 92 92 92
0.008 0.025 0.014 0.010 0.008 0.007 0.011 0.008 0.012 0.028 0.008
-0.00034 -0.00213 -0.00289 -0.00204 -0.00455 -0.0027 -0.00381 -0.00669 -0.00844 -0.0046 -0.00604 -0.00814
0.9999 0.9999 0.9999 1.0000 0.9998 0.9998 0.9998 0.9994 0.9998 1.0000 1.0000 1.0000
20 20 20 20 20 20 20 20 20 20 20 15
0.008 0.009 0.009 0.012 0.012 0.011 0.016 0.013 0.014 0.014 0.013 0.010
0.00051 0.00294 0.001011 0.001499 0.000258
0.9992 0.9985 0.9935 0.9984 0.9978
40 32 28 24 20
0.049 0.061 0.070 0.072 0.037
0.25197 0.50139 1.00000
-0.00215 0.020543 0.661348
0.027438 0.16688 2.01951
water (1 - x)-tetrahydrofuran (x) 1.83E-05 0.9999 9.60E-05 -0.00069 -9.95E-05 1.0000 -0.00074 -0.00659 -0.00386 0.9993 -0.04769 -0.14122
-6.62E-07 3.84E-06 0.000284
0.9999 1.0000 0.9989
25 45 45
0.004 0.057 0.087
0.65330
0.004095
0.141324
water (1 - x)-trifluoroethanol (x) 9.15E-07 0.9973 -0.00014 -0.00603
2.04E-07
0.9957
170
0.020
-2.72471 -3.34035 -1.80546 -1.68559 -0.66147 -0.48676 -0.52817 -0.35521
0.006138 0.008334 0.004172 0.004026 0.001385 0.000864 0.000989 0.000585
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
15 15 15 15 15 15 15 15
0.020 0.020 0.023 0.024 0.030 0.030 0.034 0.033
0.000 0.126 0.371 0.502 0.751 0.874 0.877 1.000
7.3135 10.0557 5.35965 5.35243 2.1204 1.3488 1.54397 0.956688
20.0877 24.8293 14.3624 13.8326 6.23863 4.67218 5.0623 3.57094
-0.04291 -0.0596 -0.03149 -0.03155 -0.0123 -0.00765 -0.00882 -0.00536
heptane (1 - x)-pentane (x) 1.0000 -1.0331 1.0000 -1.39479 1.0000 -0.70195 1.0000 -0.67625 1.0000 -0.23495 1.0000 -0.1485 1.0000 -0.16928 1.0000 -0.1012
0.000 0.125 0.375 0.625 0.750 0.840 0.874 1.000
7.3135 9.45233 6.60136 4.99288 5.27309 5.24981 4.2057 3.82346
20.0877 23.9967 17.7009 13.8773 14.15 13.886 11.7862 10.7398
-0.04291 -0.05588 -0.0388 -0.02923 -0.03098 -0.03089 -0.02458 -0.02235
heptane (1 - x)-hexane (x) 1.0000 -1.0331 1.0000 -1.32399 1.0000 -0.87952 1.0000 -0.65135 1.0000 -0.68991 1.0000 -0.68079 1.0000 -0.54135 1.0000 -0.48332
-2.72471 -3.25245 -2.27945 -1.73926 -1.77583 -1.72801 -1.44812 -1.29417
0.006138 0.007899 0.005226 0.003862 0.004102 0.004052 0.003209 0.002867
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
15 15 15 15 15 15 15 15
0.020 0.022 0.023 0.022 0.026 0.026 0.026 0.024
0.000 0.122 0.248 0.250 0.372 0.374 0.499 0.622 0.750 0.874 1.000
3.82346 3.33389 5.049 3.9187 2.92118 3.12714 2.38962 1.12614 -0.05916 1.27624 0.956688
10.7398 9.6368 12.6055 10.5254 8.34501 8.76492 7.1076 4.41169 2.00698 4.34449 3.57094
-0.02235 -0.0194 -0.02992 -0.023 -0.01702 -0.01826 -0.01385 -0.00626 0.000922 -0.00726 -0.00536
hexane (1 - x)-pentane (x) 1.0000 -0.48332 1.0000 -0.39774 1.0000 -0.61357 1.0000 -0.47422 1.0000 -0.34898 1.0000 -0.37312 1.0000 -0.27906 1.0000 -0.12257 1.0000 0.032626 1.0000 -0.13685 1.0000 -0.1012
-1.29417 -1.1085 -1.48086 -1.22532 -0.95061 -0.99999 -0.78992 -0.45665 -0.14363 -0.44206 -0.35521
0.002867 0.002347 0.00367 0.002816 0.002065 0.002209 0.001645 0.000704 -0.00024 0.000798 0.000585
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
15 15 15 15 15 15 15 15 15 15 15
0.024 0.028 0.047 0.025 0.023 0.029 0.036 0.028 0.056 0.038 0.033
0.032371 0.192126 0.588082 0.495409
1.72469 2.43583 3.72567 4.55645
0.000167 -0.00068 -0.00292 -0.00209
-5.26E-05 1.20E-05 0.000207 -2.16E-05
1.0000 0.9999 0.9994 0.9993
85 68 64 83
0.059 0.039 0.057 0.089
0.007424 0.008495 0.005835
4.87E-05 3.13E-06 -9.46E-07
-1.67E-06 -2.87E-07 4.95E-07
1.0000 0.9999 1.0000
65 57 68
0.027 0.044 0.063
0.2001 0.3508 0.5001 0.8000 0.000 0.485 1.000
-0.00591 0.000349 0.000758
1-butanol-diisopropyl ether (x) 1.0000 0.004998 -0.13254 1.0000 -0.00733 -0.20281 0.9995 -0.04209 -0.33303 0.9995 -0.0087 -0.40702 Kr (1 - x)-Ar (x) 1.0000 0.000221 1.0000 2.80E-05 1.0000 -8.21E-05
-0.00016 -0.00019 -0.0002
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Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005
Table 3. Absolute Average Deviationsa (AAD) of Calculated and Experimental Expansion Coefficient (r), Isothermal Compressibility (KT), and Internal Pressure (Pi) of Some Binary Mixtures C7H14 (1 - x)-C6H12 (x)
C6H3(CH3)3-C6H6
water (1 - x)-tetrahydrofuran (x)
water (1 - x)-trifluoroethanol (x)
x
AAD(R)
x
AAD(κT)
x
AAD(R)
AAD(κT)
AAD(Pi)
x
AAD(R)
AAD(κT)
AAD(Pi)
0.000 0.125 0.375 0.625 0.750 0.840 0.874 1.000
1.75 2.45 1.88 1.46 1.84 1.83 1.43 1.45
0.00000 0.10849 0.19391 0.29799 0.43460 0.49717 0.60681 0.70885 0.79512 0.89947 1.00000
1.31 1.42 0.98 1.03 0.92 1.21 1.44 1.71 1.28 0.96 0.63
0.25197 0.50139 1.00000
0.30 0.65 1.52
0.49 2.07 3.05
0.859 3.749 4.820
0.65330
1.21
4.11
5.139
a
N AAD ) 1/N ∑i)1 100|F
exp
- Fcal/F
exp|.
Table 4. Absolute Average Deviations between Experimental and Calculated Densities Based on Mixing Rules x
AAD
NP
1,3,5-trimethylbenzene-benzene (x) 0.10849 0.831 92 0.4346 0.701 92 0.60681 1.808 92 1-chloropentane-octane (x) 0.10255 0.704 15 0.31109 0.582 15 0.60675 0.212 15 0.88835 0.227 15 Kr-Ar (x) 0.277 0.224 10 0.698 0.367 11 0.787 0.418 12
x
AAD
NP
hexane-pentane (x) 0.122 0.048 15 0.374 0.061 15 0.622 0.094 15 0.874 0.133 15 heptane-hexane (x) 0.125 0.060 15 0.375 0.076 15 0.750 0.039 15 0.874 0.055 15
Table 5. AAD Values between Experimental Density Data and Those Predicted Using Our Work and COSTALD Method13,14 x Figure 3. Three-dimensional deviations between our calculated and experimental liquid densities of the 1,3,5-trimethyl benzenebenzene mixture at 298.15 K for different pressures and mole fractions.5
AAD(GMA)a
AAD(COSTALD
NP
0.122 0.374 0.622 0.874
hexane-pentane (x) 0.048 (0.093) 0.707 (1.222) 0.061 (0.119) 0.61 (1.121) 0.094 (0.210) 0.538 (1.067) 0.133 (0.294) 0.445 (0.922)
15 15 15 15
0.125 0.375 0.75 0.874
heptane-hexane (x) 0.060 (0.119) 0.364 (0.815) 0.076 (0.143) 0.441 (0.948) 0.039 (0.122) 0.580 (1.126) 0.055 (0.148) 0.64 (1.195)
15 15 15 15
a The numbers in parentheses show the maximum absolute percent deviation.
Figure 4. Three-dimensional deviations between our calculated and experimental liquid densities of the 1-chloropentane-octane mixture at 2.13 MPa for different temperatures and mole fractions.6
6, respectively. For all the systems, A and B are fitted well by quadratic functions (Rmin2 ) 0.9996 and Rmax2
) 1.0000 for A, and Rmin2 ) 0.9991 and Rmax2 ) 1.0000 for B). The second criterion is to calculate density using these mixing rules. The accuracy of the prediction of density can be described by AAD. Table 4 shows the values of AAD of calculated densities using the mixing rules (eqs 8 and 9). The ranges of pressure and temperature for this table are the same as those for Table 1. As the values of AAD show, the mixing rules proposed in this paper yield good predictions of density. To assess the performance of GMA EoS in the prediction of the density of liquid mixtures, the AAD values (between the calculated densities of some mixtures and the corresponding experimental data) from this equation of state and from the COSTALD method13,14 have been compared in Table 5. Again, the lower AADs obtained by GMA EoS support the ability of this EoS to predict the density of liquid mixtures. Ternary Mixtures. The number of studies on thermodynamic properties of ternary mixtures has increased
Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005 6979 Table 6. Parameters and Square of Correlation Coefficient (R2) of Eq 1 for Ternary Mixtures of Pentane (x1), Hexane (x2), and Heptane10 x1
x2
-A(T) B(T) (10-9 m3 mol-3) (10-12 m4 mol-4)
T (K)
0.245 0.393 298.15 323.15 348.15 0.374 0.267 298.15 323.15 348.15 0.374 0.377 298.15 323.15 348.15 0.250 0.254 298.15 323.15 348.15 0.136 0.125 298.15 323.15 348.15 0.747 0.125 298.15 323.15 348.15 0.125 0.254 298.15 323.15 348.15 0.123 0.374 298.15 323.15 348.15 0.126 0.499 298.15 323.15 348.15 0.125 0.622 298.15 323.15 348.15 0.125 0.751 298.15 323.15 348.15
0.17151 0.1444 0.12387 0.15713 0.13298 0.11453 0.14693 0.12308 0.10464 0.18587 0.15633 0.13601 0.22918 0.19471 0.16934 0.10537 0.08853 0.07613 0.21571 0.18278 0.15871 0.20047 0.17042 0.14841 0.1861 0.15689 0.13659 0.17218 0.14553 0.12655 0.15824 0.13426 0.11661
0.022564 0.019602 0.017391 0.020347 0.017768 0.015817 0.018744 0.016211 0.014277 0.024828 0.021543 0.019384 0.031961 0.02797 0.025114 0.012623 0.01097 0.009784 0.030259 0.026472 0.023799 0.027227 0.023856 0.021472 0.024867 0.021624 0.019467 0.022678 0.019776 0.017789 0.020519 0.017964 0.016146
R2 0.9999 0.9996 0.9982 0.9999 0.9996 0.9995 0.9999 0.9996 0.9974 0.9999 0.9996 0.9996 0.9999 0.9996 0.9996 0.9999 0.9996 0.9994 0.9999 0.9996 0.9996 0.9999 0.9997 0.9996 0.9999 0.9996 0.9997 0.9999 0.9996 0.9995 0.9999 0.9996 0.9996
x1
x2
T (K)
0.246 0.127 298.15 323.15 348.15 0.252 0.501 298.15 323.15 348.15 0.249 0.624 298.15 323.15 348.15 0.376 0.123 298.15 323.15 348.15 0.375 0.500 298.15 323.15 348.15 0.500 0.127 298.15 323.15 348.15 0.498 0.251 298.15 323.15 348.15 0.497 0.378 298.15 323.15 348.15 0.625 0.127 298.15 323.15 348.15 0.621 0.257 298.15 323.15 348.15
-A(T) B(T) (10-9 m3 mol-3) (10-12 m4 mol-4) 0.20068 0.16999 0.14815 0.15788 0.13396 0.11645 0.14774 0.12399 0.10746 0.17109 0.14533 0.12654 0.13583 0.11372 0.09842 0.14661 0.12356 0.10711 0.13578 0.11395 0.09856 0.1253 0.1054 0.09093 0.12394 0.10491 0.09037 0.11456 0.09639 0.08305
0.027255 0.0238 0.021438 0.020471 0.017925 0.01612 0.018871 0.016349 0.014672 0.022543 0.019749 0.017789 0.01707 0.014761 0.013228 0.018725 0.016289 0.014619 0.017069 0.014798 0.013256 0.015503 0.013476 0.012049 0.015332 0.013406 0.011971 0.01394 0.012129 0.010838
R2 0.9999 0.9997 0.9996 0.9999 0.9997 0.9995 0.9999 0.9996 0.9995 0.9999 0.9996 0.9996 0.9999 0.9996 0.9995 0.9999 0.9996 0.9996 0.9998 0.9997 0.9996 0.9999 0.9997 0.9995 0.9999 0.9997 0.9996 0.9999 0.9996 0.9995
Table 7. AAD of Calculated and Experimental Densities Using the Mixing Rules (eqs 10 and 11) and on the Basis of GMA EoS Directly (eq 1) for the Ternary Mixtures of Pentane (x1), Hexane (x2), and Heptane x1
x2
AADGMA EoS
AADmixing
x1
x2
AADGMA EoS
AADmixing
0.245 0.374 0.374 0.250 0.136 0.747 0.125 0.123 0.126 0.125 0.125
0.393 0.267 0.377 0.254 0.125 0.125 0.254 0.374 0.499 0.622 0.751
0.042 0.028 0.048 0.023 0.026 0.032 0.026 0.023 0.024 0.034 0.026
2.019 2.653 2.380 2.227 1.431 2.411 0.454 1.507 1.495 1.285 0.923
0.246 0.252 0.249 0.376 0.375 0.500 0.498 0.497 0.625 0.621
0.127 0.501 0.624 0.123 0.500 0.127 0.251 0.378 0.127 0.257
0.028 0.028 0.025 0.025 0.035 0.026 0.026 0.03 0.024 0.039
2.182 1.774 1.362 2.815 1.754 3.144 2.732 2.028 3.055 2.216
in recent years because of industrial applications and the theoretical interest in studying the nature of molecular interaction and packing phenomena in ternary mixtures.15 We expect GMA EoS to hold for ternary mixtures as well as for binary mixtures and the parameters of EoS to be quadratic functions of composition, as in the following equations: 3
Amix )
∑ ∑xixjAij i)1 j)1 3
Bmix )
3
(10)
3
∑ ∑xixjBij i)1 j)1
(11)
The temperature dependences of these parameters are also the same as those of pure and binary mixtures, namely, eqs 2 and 3. We have used the experimental PVT data of the ternary mixtures of pentane, hexane, and heptane10 at various temperatures and pressures
to examine eq 1 for a ternary mixture. The results of this examination are given in Table 6, including the intercept and slope of the fitted straight line (eq 1) at each temperature and composition and the square of the correlation coefficient (R2). As was the case for binary systems, the ternary liquid mixtures obey GMA EoS very well. To calculate the density of a ternary system, we need the experimental data on pure components (pentane, hexane, and heptane) plus the data on their binary mixtures at one composition at the same temperature. Table 7 shows the average absolute percent deviations of density calculated using the mixing rules and calculated on the basis of GMA EoS directly (eq 1) in comparison with corresponding experimental densities for the ternary mixtures of pentane, hexane, and heptane. Again, the data of this table support the ability of GMA EoS to predict the volumetric properties of liquid mixtures.
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Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005
3. It can predict the volumetric and thermodynamic properties of the liquid mixtures studied in any temperature, pressure, and composition using Table 2. Literature Cited
Figure 5. Quadratic composition dependence of the parameter A for chloropentane (1 - x)-octane (x) at different temperatures.
Figure 6. Quadratic composition dependence of the parameter B for chloropentane (1 - x)-octane (x) at different temperatures.
Concluding Remarks We have extended GMA EoS to the liquid mixture case. The parameters of the equation of state are both temperature and composition dependent. The excellent mutual agreement between the calculated and experimental different properties such as density, expansion coefficient, isothermal compressibility, and internal pressure supports the reliability of GMA EoS to reproduce and predict the experimental volumetric and thermodynamic properties of liquid mixtures. It seems to us that GMA EoS contains three points of particular interest: 1. The form of GMA EoS is generalized and very simple. Evaluation of the coefficients is very easy, so that it is practical and convenient. 2. It can be used to calculate volumetric properties and differential properties such as expansion coefficient, isothermal compressibility, and internal pressure with relatively high precision.
(1) Lo´pez, E. R.; Lugo, L.; Comun˜as, M. J. P.; Garcı´a, J.; Ferna´ndez, J. Liquid Density Measurements of Diethylene Glycol Monoalkyl Ethers as a Function of Temperature and Pressure. J. Chem. Eng. Data 2004, 49, 376-379. (2) Goharshadi, E. K.; Morsali, A.; Abbaspour, M. New Regularities and an Equation of State. Fluid Phase Equilib. 2005, 230, 170-175. (3) Goharshadi, E. K.; Moosavi, F. Prediction of Thermodynamic Properties of Some Hydrofluoroether Refrigerants Using GMA Equation of State. Presented at the 18th IUPAC Conference on Chemical Thermodynamics (ICCT), Beijing, China, 2004. (4) Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.; Butterworth: London, 1982. (5) Mora´vkova´, L.; Wagner, Z.; Linek, J. (P, V, T, x) Measurements of the System Benzene + 1,3,5-Trimethylbenzene at Temperatures from 298.15 to 328.15 K and at Pressures up to 40 MPa. Fluid Phase Equilib. 2003, 209, 81-94. (6) Moravkova, L.; Linek, J. Excess Molar Volumes of (Octane + 1-Chloropentane) at Temperatures between 298.15 and 328.15 K and at Pressures up to 40 MPa. J. Chem. Thermodyn. 2003, 35, 1119-1127. (7) Audonnet, F.; Pa´dua, A. A. H. Viscosity and Density of Mixtures of Methane and n-Decane from 298 to 393 K and up to 75 MPa. Fluid Phase Equilib. 2004, 216, 235-244. (8) Back, P. J.; Woolf, L. A. (P, V, T, x) Measurements for Tetrahydrofuran and {x C4H8O + (1-x) H2O}. J. Chem. Thermodyn. 1998, 30, 353-364. (9) Kabata, Y.; Sonobe, R.; Sugiura, I.; Uematsu, M. Properties of {x CF3CH2OH + (1-x) H2O} at temperatures from 310 to 420 K. III. Densities at Pressures up to 200MPa for x ) 0.6533. J. Chem. Thermodyn. 1993, 25, 1005-1009. (10) Pecar, D.; Dolecek, V. Isothermal Compressibilities and Isobaric Expansibilities of Pentane, Hexane, Heptane and their Binary and Ternary Mixtures from Density Measurements. Fluid Phase Equilib. 2003, 211, 109-127. (11) Ihmels, E. C.; Gmehling, J. Liquid Densities and Excess Volumes of Diisopropyl Ether (DIPE) + 1-Butanol Mixtures from 273 to 473 K and Pressures up to 35 MPa. J. Chem. Eng. Data 2002, 47, 1314-1319. (12) Barreiros, S. F.; Calado, J. C. G.; Clancy, P.; Nunes da Ponte, M.; Streett, W. B. Thermodynamic Properties of Liquid Mixtures of Argon + Krypton. J. Phys. Chem. 1982, 86, 17221729. (13) Hankinson, R. W.; Thomson, G. H. A New Correlation for Saturated Densities of Liquids and Their Mixtures. AIChE J. 1979, 25, 653-663. (14) Thomson, G. H.; Brobst, K. R.; Hankinson, R. W. An Improved Correlation for Densities of Compressed Liquids and Liquid Mixtures. AIChE J. 1982, 28, 671-676. (15) Savaroglu, G.; Aral, E. Densities, Speed of Sound and Isentropic Compressibilities of the Ternary Mixture 2-Propanol + Acetone + Cyclohexane and the Constituent Binary Mixtures at 298.15 and 313.15 K. Fluid Phase Equilib. 2004, 215, 253262.
Received for review February 9, 2005 Revised manuscript received May 31, 2005 Accepted June 7, 2005 IE050158Y
