Modeling and Experimental Evaluation of Thermodynamic Properties

Ind. Eng. Chem. Res. 2007, 46, 7353-7366

7353

Modeling and Experimental Evaluation of Thermodynamic Properties for Binary Mixtures of Dialkylcarbonate and Alkanes Using a Parametric Model Juan Ortega,* Fernando Espiau, Remko Vreekamp, and Jose´ Tojo† Laboratorio Termodina´ mica y Fisicoquı´mica, Parque Cientı´fico-Tecnolo´ gico, UniVersidad de Las Palmas de Gran Canaria, Canary Islands, Spain

This work shows the utility of a parametric model, as a function of concentration and temperature, to correlate isobaric vapor-liquid equilibrium data and other themodynamic properties of solutions. With the model proposed here, a procedure can be designed to correlate properties and estimate parameters by successive steps in the order (x, cEp ) f (x, hE) f (x, gE), applied here to a set of binary systems for which experimental data already exist for these properties. Some of the steps can be omitted if the values for some of these properties are not available. The practical application is carried out on a set of binary systems of (dialkylcarbonate + alkane) for which data have been found for the three quantities. However, new experimental values of hE and gE, obtained isobarically, are also provided for the systems considered necessary to verify the model. The results of this application show a good correlation for these quantities, improving on the results of previous models of NRTL and UNIQUAC. 1. Introduction This work forms part of a project supervised by us, aimed at modeling systems containing a dialkylcarbonate with a second component, such as an alkane, or an alkanol, systems for which complete data are available in the literature,1-6 or esters, for which few studies have been published. In previous works,3,5 data were presented for the thermodynamic properties of binary systems of dimethylcarbonate (DMC) + alkanes. There is a vast amount of experimental information available to date in the literature on mixtures with organic carbonates, because of the industrial interest for these mixtures.7,8 Their high capacity as solvents is noteworthy, as is their application as gasoline additives. The modeling we propose in this work requires experimental values for the greatest number of properties in solution possible, although in an initial objective we propose that only vapor-liquid equilibrium (VLE) data, enthalpies hE, and mixing thermal capacities cEp are required. This is one of the reasons why this set of mixtures was chosen as a working tool, completing where necessary the existing database in order to extensively verify the model with new contributions to the model and procedure, for which the preliminary steps have already appeared in previous works.9 To complete the proposed study with the set of primary dialkylcarbonate + alkane systems chosen and to improve the correlation model, based on the analysis of data published in the literature, this data set was considered to be insufficient to achieve one of the objectives of the model and additional values were required for some properties of these systems. Hence, hE values have been estimated at several temperatures for mixtures of diethylcarbamate (DEC) with alkanes (from hexane to decane) and the VLE for DEC + nonane or + decane, since the properties for the others have already been published in previous papers.2,4 The approach followed here is aimed at adapting a thermodynamic-mathematical model to correlate and estimate the * Corresponding author. E-mail: [email protected]. † Article written in memory of Prof. Tojo, who died before the work we organized together had been completed.

greatest number of thermodynamic properties possible, obtained in this case for isobaric working conditions, using a simple parametric expression, dependent on concentration and temperature. Different possibilities of use are established to optimize application of the model to each case. The utility of the model will be evaluated with the corresponding statistical parameters of goodness-of-fit, comparing the final results with application to real systems of other classical known models, used in recent years to correlate thermodynamic properties, such as NRTL10 and UNIQUAC.11 2. Thermodynamic-Mathematical Approach of the Model In an earlier work,9 a useful mathematical model was proposed for isobaric equilibrium studies of nonelectrolytic mixtures for the Gibbs function gE ) gE(x, T). In its most general form, this model adopted the following polynomial expression,

gE(x, T) RT

) z(1 - z)

x

aizi where z ) ∑ x + k(1 - x) i)0

(1)

and where, starting with a quadratic expression for the coefficients ai as a function of temperature, of the form 3 A Tj-1, the expressions of its derivatives ai ) Ai0T-1 + ∑j)1 ij are

dai dT

)-

Ai0 T2

3

+

(j - 1)AijTj-2 ∑ j)1 d2a i dT

2

)

2Ai0 T

3

3

+

(j - 1)(j - 2)AijTj-3 ∑ j)1

(2)

Equation 1, therefore, represents the simultaneous dependence of Gibbs’ function with the concentration and temperature. The so-called actiVe fraction z depends on the concentration x and

10.1021/ie0707366 CCC: $37.00 © 2007 American Chemical Society Published on Web 09/27/2007

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Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007

Figure 1. Values obtained for (a) kv and (b) kh versus T for binary mixtures formed by DMC (- - -) or DEC (s) with an alkane (hexane (b), heptane (O), octane (1), nonane (4), and decane (9)). Table 1. Physical Properties of Pure Compounds and Comparison with Those from Literature 103R (K -1)

Tob (K) compound

a

purity

exp.

lit.

exp.

399.95a

1.16 1.41

1.39a

1.28

1.29b

398.65

341.89a 341.88c 371.57a 371.60c 398.82a,c

1.18

1.16a

>99%

423.66

423.97a,c

1.12

1.12b

>99%

446.90

447.30a,c

1.05

1.05a

DEC

+99

hexane

>99%

341.93

heptane

>99%

371.18

octane

>99%

nonane decane

F (kg‚m-3)

lit.

nD

T/K

exp.

lit.

exp.

lit.

298.15 318.15 298.15 318.15 298.15 318.15 298.15 318.15 298.15 318.15 298.15 318.15

969.13 946.56 654.59 636.39 679.47 662.13 698.39 682.09 713.85 698.06 726.19 711.14

969.30a 946.43d 654.84a 636.67b 679.46a 662.32b 698.62a 682.09b 713.75a 698.06b 726.35a 711.43b

1.3824

1.3829a

1.3724 1.3614 1.3853 1.3748 1.3951 1.3856 1.4030 1.3939 1.4095 1.4006

1.37226a 1.3615b 1.38511a 1.3750b 1.39505a 1.3855b 1.40311a 1.3939b 1.40967a 1.4008b

Reference 13. b Reference 14. c Reference 15. d Reference 16.

a parameter k, for which the definition is established for each specific case as the work progresses. Thermodynamically,the successive derivatives of Gibbs’ function provide us other mixing quantities cEp (x,

(

) (

∂hE(x, T) T) ) ∂T

)

∂sE(x,T) )T ∂T p,x

) -T

p,x

(

)

∂2gE(x, T) ∂T2

(3)

p,x

and if the number of terms of eq 1 is limited to second order, we obtain

-

sE R

[ ( )] ( ) ∑( ) ( ) ∑ [( ) ( )] [() dai

2

) z(1 - z)

-

hE RT

2

cEp ) -Tz(1 - z)2 R

ai + T ∑ dT i)0 2

) z(1 - z)

i)0

dai dT

zi + T

zi +

dz

dT

dz

dT

Y

Y

(4)

(5)

dai d2ai i 2Y dz +T z - T2 + dT T dT dT2 d2z dz dY Y 2 + (6) dT dT dT

( )

( )( )]

with 2

Y ) (1 - 2z)

aizi + z(1 - z)(a1 + 2a2z) ∑ i)0

(7)

With this general idea of the model, by applying eq 1, the active fraction z can depend on the different working conditions, p and T, and can generally be written as z ) z[x, k(T, p)]. This approach is important because, taking into account the shape of eq 6, if k is not temperature-dependent, then z would not be either, and the second addend of eqs 4-6 would become zero, since (dz/dT) ) 0, producing quite simple final expressions. This is of great interest because it gives rise to a more rigorous and extensive differentiated procedure. Its consideration will be different in each case and will especially depend on the kind of correlation required by each model. The following section analyzes the specific case of the dependence of z on temperature, through k, z ) z[x, k(T)] ) z(x, T), verifying that

x(1 - x) dz dk dk dz )) 2 dT dT dk dT [x + k(1 - x)]

( )( )

( )

(8)

an expression that will be formulated according to the dependence on k(T).

Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007 7355 Table 2. Excess Molar Enthalpies hE for Binary Systems of DEC (1) + an Alkane (2) (Hexane to Decane) at Three Different Temperatures x1

hE (J‚mol-1)

x1

hE (J‚mol-1)

x1

hE (J‚mol-1)

0.7386 0.7992 0.8695 0.9378

904 732 507 263

0.8897 0.9514

492 243

0.8415 0.9013 0.9491

742 497 275

0.8397 0.8973 0.9510

827 566 300

0.8219 0.8854 0.9491

945 657 326

0.6271 0.6864 0.7438 0.8105 0.8787 0.9427

1138 1031 892 712 490 262

0.7229 0.7811 0.8407 0.8968 0.9519

1019 857 668 458 233

0.8989 0.9518

509 264

0.7658 0.8299 0.8979 0.9550

1065 843 547 268

0.8508 0.9051 0.9553

868 605 311

T ) 291.15 K 0.0462 0.1217 0.1873 0.2656 0.3382 0.3981

384 734 957 1147 1235 1258

DEC (1) + hexane (2) 0.4596 1263 0.5007 1240 0.5098 1248 0.5538 1205 0.6128 1140 0.6706 1046

0.0813 0.1639 0.2495 0.3305 0.4033 0.4672

474 804 1065 1230 1313 1338

DEC (1) + heptane (2) 0.5238 1329 0.5740 1296 0.6368 1220 0.6977 1101 0.7607 941 0.8250 738

0.0885 0.1772 0.2614 0.3438 0.4196 0.4842

525 862 1132 1311 1407 1436

0.5409 0.5909 0.6052 0.6598 0.7148 0.7777

0.1009 0.2049 0.3018 0.3897 0.4652 0.5127

611 1001 1291 1457 1530 1529

DEC (1) + nonane (2) 0.5333 1525 0.5592 1497 0.6095 1438 0.6623 1352 0.7191 1225 0.7784 1052

0.0755 0.1822 0.2784 0.3684 0.4428 0.5114

475 922 1264 1475 1579 1618

0.5444 0.5596 0.5930 0.6436 0.6992 0.7597

0.0677 0.1450 0.2303 0.3118 0.3853 0.4461

317 618 883 1090 1215 1250

DEC (1) + hexane (2) 0.4736 1260 0.5061 1246 0.5182 1246 0.5580 1217 0.5701 1207 0.5906 1180

0.0686 0.1450 0.2265 0.2985 0.3652 0.4276

415 732 1002 1174 1275 1322

DEC (1) + heptane (2) 0.4794 1323 0.4840 1326 0.5201 1321 0.5658 1281 0.6152 1241 0.6675 1151

0.0863 0.1789 0.2696 0.3528 0.4297 0.4952

496 854 1135 1308 1395 1419

0.5518 0.6155 0.6589 0.7174 0.7770 0.8372

0.0895 0.1860 0.2710 0.3536 0.4280 0.4933

537 928 1203 1384 1477 1504

DEC (1) + nonane (2) 0.5037 1503 0.5477 1494 0.5510 1486 0.5952 1439 0.6483 1378 0.7057 1256

0.1031 0.2093 0.3055 0.3899 0.4677 0.5268

592 1026 1342 1521 1610 1617

0.5362 0.5825 0.6317 0.6840 0.7381 0.7963

DEC (1) + octane (2) 1426 1388 1346 1257 1136 963

DEC (1) + decane (2) 1593 1597 1551 1468 1348 1169 T ) 298.15 K

DEC (1) + octane (2) 1407 1352 1272 1136 968 760

DEC (1) + decane (2) 1605 1572 1513 1413 1281 1095

7356


Table 2 (Continued) T ) 318.15 K 0.0571 0.1115 0.1794 0.2467 0.3110 0.3727

289 529 783 983 1127 1223

DEC (1) + hexane (2) 0.4262 1251 0.4564 1285 0.4797 1289 0.4924 1288 0.5366 1273 0.5859 1233

0.0682 0.1419 0.2116 0.2841 0.3517 0.4144

344 649 903 1116 1234 1320

0.0705 0.1515 0.2296 0.3043 0.3741 0.4385

0.6416 0.6997 0.7620 0.8238 0.8838 0.9455

1156 1043 885 694 479 232

DEC (1) + heptane (2) 0.4722 1354 0.5220 1354 0.5671 1322 0.5804 1317 0.6286 1262 0.6808 1170

0.7341 0.7912 0.8452 0.8995 0.9513

1048 887 698 482 250

409 753 1035 1229 1350 1413

DEC (1) + octane (2) 0.4947 1426 0.5455 1409 0.5889 1369 0.6011 1359 0.6488 1287 0.6986 1177

0.7502 0.8033 0.8569 0.9088 0.9580

1060 896 701 474 236

0.0802 0.1654 0.2490 0.3273 0.3986 0.4634

465 823 1119 1319 1438 1495

DEC (1) + nonane (2) 0.4991 1502 0.5161 1505 0.5377 1490 0.5789 1470 0.6233 1418 0.6714 1345

0.7205 0.7715 0.8190 0.8700 0.9191 0.9611

1252 1096 913 710 477 240

0.0852 0.1771 0.2629 0.3454 0.4180 0.4825

483 869 1182 1391 1509 1562

DEC (1) + decane (2) 0.5166 1569 0.5356 1573 0.5551 1565 0.5964 1541 0.6399 1497 0.6850 1403

0.7322 0.7798 0.8284 0.8777 0.9241 0.9668

1282 1113 925 711 474 234

Table 3. Coefficients and Standard Deviation, s, Obtained Using Eq 1 to Correlate the Excess Molar Enthalpies hE/RT (YE ) hE/RT (J‚mol-1)) binary mixture

kh

a0

a1

a2

103‚s(hE/RT)/ (J‚mol-1)

DEC (1) + hexane (2) DEC (1) + heptane (2) DEC (1) + octane (2) DEC (1) + nonane (2) DEC (1) + decane (2)

T ) 291.15 K 1.006 3.143 -3.284 1.128 2.845 -1.759 1.251 3.130 -2.058 1.374 3.558 -2.928 1.498 3.773 -3.041

2.130 0.801 0.768 1.294 1.067

8 4 5 5 5


T ) 298.15 K 1.007 1.988 0.312 -0.611 1.128 2.789 -1.780 0.780 1.251 3.155 -1.894 0.720 1.374 3.397 -2.674 1.088 1.497 3.693 -3.108 1.394

4 4 4 4 3


T ) 318.15 K 1.009 1.986 0.058 -0.377 1.129 2.305 -0.512 -0.029 1.250 2.824 -1.859 0.790 1.372 3.107 -2.253 0.985 1.494 3.244 -2.283 0.755

2 2 2 3 4

2.1. Evaluation of the Parameter k. The significance of the parameter k will depend on the property to which it is applied. Hence, for excess volumes VE, this parameter is identified with the quotient of the molar volumes of the pure compounds at the same working conditions and is written as kV, adopting the following form for a binary mixture of two generic compounds i and j,

kV(p, T) ) kij(p, T) )

Vjo(p, T) o

Vi (p, T)

)

Mj Fi(p, T) Mi Fj(p, T)

(9)

Table 4. Experimental Vapor Pressures, poi , for DEC T (K)

poi (kPa)

T (K)

poi (kPa)

T (K)

poi (kPa)

352.51 355.37 358.18 360.75 363.09 365.23 367.35 369.35 371.20 372.98 374.78 376.31 377.98 379.41 380.83 382.29 383.60 384.89

21.33 24.00 26.66 29.33 32.00 34.66 37.33 40.00 42.66 45.33 48.00 50.66 53.33 56.00 56.66 61.33 64.00 66.66

386.19 387.46 388.69 389.76 390.93 392.07 393.13 394.22 395.26 396.48 397.32 398.44 399.32 400.20 401.09 401.94 402.78 403.62

69.33 72.00 74.66 77.33 80.0 82.66 85.33 87.99 90.66 93.32 95.99 98.66 101.32 103.99 106.66 109.32 111.99 114.66

404.46 405.22 406.02 406.78 407.54 408.29 409.03 409.72 410.45 411.16 411.83 412.42 413.18 413.75 414.39 415.03 415.65

117.32 119.99 122.65 125.32 127.99 130.65 133.32 136.00 138.65 141.31 143.99 146.65 149.32 151.98 154.65 157.32 159.98

where Mi, Mj, Fi, and Fj, respectively, correspond to the molecular masses and the corresponding densities for compounds i and j, in specific conditions. It has been demonstrated that the values calculated with eq 9 are quite similar to those obtained with the quotient of volumeparameters ri ) ∑kυ(i) k Rk for each substance i, where Rk are Van der Waals group volume parameters (see ref 12) and where υ(i) k is the number of groups of type k in molecule i. However, it is more recommendable to do the calculation using eq 9, since real values are used for the pure compounds, while the group-contribution method does not take into account either the working conditions or the differences between regioisomers. Also, the simple method proposed can be used in some real

Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007 7357

Figure 2. Equimolar values of hE for binaries DEC (1) + alkane (2) (hexane (b), heptane (2), octane (9), nonane (1), decane (f)) versus temperature. The corresponding open symbols belong to values from ref 17.

cases where it would be more complex to calculate ri values for some substances, as occurs with ionic liquids. In the case of enthalpies hE, the parameter k is renamed kh. Taking into account previous recommendations for volumes, it would be simpler to use the quotient of surface parameters qi, since the energetic effects of mixing are proportional to the contact surfaces of the molecules present. In other words, kh ) qj/qi, where qi can be calculated in an analogous manner to the volume parameters, as the weighted sum of the group area parameters Qk (see ref 12). However, a more accurate approximation of kh would be obtained from the quotient of the real interacting surfaces. Hence, an approximate form of kh can be obtained from the quotient of the theoretical parameters of area qi, weighted with the real and theoretical volumes, obtained by group contribution. This approach would have the following form:

( )( ) ( )

q2 r1 kh ) q1 r2

2/3

Vo2

2/3

Vo1

(10)

The same reasoning can be applied to determine the k parameter when correlating excess thermal capacities cEp , establishing in this case that kc ) cop,2/cop,1. In any case, to correctly apply the parameter k in eqs 4-6, it is necessary to know the variation of k with T, which is represented in eq 8. Therefore, it is first necessary to define the relationship of dependence k(T) using the coefficient of isobaric thermal expansion of each of the pure compounds. The expressions for kV and kh would have the following extended form,

kV(p, T) )

( )( ) Mj Mi

Foi Foj

Figure 3. Vapor pressure lines in reduced coordinates for DMC, from ref 3, and for DEC (this work), and graphic determination of acentric factors.

e(Ri-Rj)(T-T0) ) kV(p, T0) e(Ri-Rj)(T-T0) (11)

Figure 4. Plot of VLE values T-x1-y1 for binary systems (x1DEC + x2alkanes) at 101.32 kPa: (O), hexane; (∆), heptane; (0), octane; (1), nonane; (f), decane.

where now

( )( ) [

q2 r1 kh(p, T) ) q1 r 2

2/3

]

MjFio(p, T0)

MiFjo(p, T0)

2/3

e(2/3)(Ri-Rj)(T-T0)

(13)

where now Foi and Foj are the densities corresponding to a reference temperature T0 and kV(p, T0) is an equivalent expression to eq 9 at this temperature. Likewise, eq 10 can be proposed for kh, including the change in temperature,

resulting in similar relationships, of the type A exp(BT), for parameters kV and kh. From here, this result can be extrapolated to other thermodynamic quantities of solutions, where an analogous form can be proposed for the corresponding parameter k of the Gibbs function, kg, which, for the isobaric case, would be similar to

kh(p, T) ) kh(p, T0) e(2/3)(Ri-Rj)(T-T0)

kg(p, T) ) kg(p, T0) eB(T-T0)

(12)

(14)

7358


Table 5. Experimental and Calculated Values for the Isobaric VLE of Binary Mixtures of DEC (1) + Nonane (2) and + Decane (2) at p ) 101.3 kPa T (K)

x1

422.91 421.55 419.73 417.90 415.87 413.67 411.65 410.06 409.51 408.91 408.25 407.60 406.81 406.07 405.43 404.99 404.33 403.69 403.07 402.53 401.95 401.31 400.90 400.47 400.11 399.80 399.53 399.25 399.03 398.87 398.77 398.74 398.81 398.89 398.95 399.08 399.18

0.0060 0.0180 0.0391 0.0617 0.0902 0.1269 0.1663 0.1958 0.2069 0.2216 0.2398 0.2552 0.2759 0.3003 0.3199 0.3447 0.3708 0.3977 0.4293 0.4592 0.4944 0.5312 0.5613 0.5986 0.6353 0.6721 0.7010 0.7383 0.7787 0.8206 0.8555 0.8867 0.9152 0.9390 0.9575 0.9745 0.9886

445.64 443.23 441.30 440.16 438.52 436.64 435.27 432.72 431.04 428.57 426.68 423.70 422.05 420.54 418.91 417.25 415.70 414.27 412.55 411.15 409.70 408.65 407.28 406.05 404.80 403.75 403.03 402.27 401.60 401.23 400.80 400.52 400.11 400.10 399.75 399.50 399.43

0.0067 0.0205 0.0318 0.0395 0.0503 0.0629 0.0730 0.0923 0.1073 0.1302 0.1509 0.1910 0.2092 0.2291 0.2533 0.2748 0.3012 0.3278 0.3649 0.3967 0.4465 0.4795 0.5189 0.5669 0.6214 0.6748 0.7175 0.7705 0.8269 0.8595 0.8904 0.9144 0.9346 0.9540 0.9702 0.9846 0.9895

γ2

gE/RT

DEC (1) + nonane (2) 0.0219 2.001 0.0628 1.975 0.1271 1.921 0.1857 1.859 0.2510 1.806 0.3244 1.751 0.3869 1.676 0.4286 1.642 0.4423 1.626 0.4611 1.607 0.4798 1.572 0.4987 1.561 0.5196 1.536 0.5417 1.499 0.5594 1.478 0.5739 1.424 0.5932 1.392 0.6111 1.359 0.6321 1.324 0.6495 1.291 0.6685 1.253 0.6919 1.228 0.7084 1.203 0.7279 1.173 0.7467 1.145 0.7648 1.117 0.7825 1.104 0.8046 1.086 0.8278 1.066 0.8526 1.046 0.8680 1.024 0.8880 1.012 0.9072 0.999 0.9289 0.996 0.9486 0.996 0.9674 0.995 0.9847 0.994

0.999 1.003 1.001 1.002 1.002 1.002 1.003 1.012 1.017 1.018 1.025 1.026 1.035 1.043 1.051 1.068 1.082 1.101 1.119 1.143 1.176 1.201 1.229 1.269 1.315 1.371 1.401 1.451 1.522 1.615 1.802 1.952 2.157 2.292 2.375 2.501 2.619

0.004 0.015 0.026 0.040 0.055 0.070 0.088 0.107 0.114 0.119 0.127 0.133 0.143 0.151 0.158 0.165 0.172 0.179 0.185 0.189 0.193 0.195 0.194 0.191 0.186 0.178 0.170 0.158 0.143 0.123 0.106 0.086 0.065 0.046 0.032 0.017 0.005

DEC (1) + decane (2) 0.0374 1.855 0.1085 1.848 0.1602 1.831 0.1923 1.812 0.2319 1.777 0.2770 1.767 0.3090 1.749 0.3658 1.732 0.4017 1.699 0.4537 1.672 0.4877 1.619 0.5448 1.531 0.5702 1.521 0.5957 1.504 0.6231 1.479 0.6467 1.474 0.6717 1.451 0.6916 1.422 0.7189 1.386 0.7391 1.358 0.7615 1.289 0.7784 1.261 0.7999 1.241 0.8210 1.203 0.8422 1.164 0.8613 1.127 0.8777 1.101 0.8948 1.066 0.9144 1.034 0.9259 1.017 0.9378 1.006 0.9492 0.999 0.9599 1.000 0.9707 0.991 0.9802 0.993 0.9892 0.994 0.9926 0.995

0.998 0.994 0.994 0.991 0.993 0.994 0.995 0.997 0.999 1.001 1.012 1.024 1.036 1.044 1.052 1.066 1.075 1.095 1.112 1.134 1.181 1.205 1.228 1.268 1.330 1.407 1.462 1.586 1.749 1.888 2.061 2.175 2.278 2.368 2.499 2.659 2.679

0.002 0.007 0.013 0.015 0.023 0.030 0.036 0.048 0.056 0.067 0.083 0.101 0.116 0.126 0.137 0.153 0.163 0.177 0.187 0.197 0.206 0.208 0.211 0.208 0.202 0.192 0.176 0.155 0.124 0.104 0.085 0.066 0.054 0.031 0.021 0.009 0.005

y1

γ1

where, in this case, kg(p, T0) would be a value obtained at a reference temperature T0. The problem that arises in its application would be to have a value for this parameter at known conditions of temperature and pressure that could be extrapolated to other situations, especially with temperature for the isobaric case, as occurs with VLE, all awaiting a suitable relationship to associate kg, kV, and kh. Although the latter two can be calculated with eqs 9-13, this is not the case with kg. However, there is always the possibility that this could be achieved by a direct adjustment of the Gibbs function. It could also be possible to calculate values of this parameter using the simplest expression possible from eq 1, in an interval corresponding to the peak of gE(p, T, x), which would only have three parameters, a0, a1, and kg. This would allow us to select a small temperature range, at constant pressure, in order to obtain a value of kg(p, T) as close as possible to the data set of the process being treated. In the VLE studies, the best choice would depend on the quality and quantity of data existing for a specific system, and this procedure could be applied to either isobaric or isothermic VLE studies. 3. Experimental Section 3.1. Materials. The DEC and the alkanes used in the experiences are of the highest quality grade supplied by Aldrich. Before use, they were degassed with ultrasound and kept for several hours on molecular sieves, type 0.3 nm. Table 1 shows the values of some of the physical properties, Tob,i, F, and nD, that characterize the purity of some substances, measured directly at two temperatures, and their comparison with data in the literature, where these are available. The F data for the compounds used here and for the DMC obtained from previous publications3,5 were used to obtain an exponential representation as a function of temperature and to obtain the isobaric thermal coefficients of expansion required for eq 11. These values and those found in the literature are also recorded in Table 1. Parts a and b of Figure 1 show, respectively, the variation in parameters kV (eq 9) and kh (eq 10) with temperature. The observation of this figure indicates a slight gradient, positive in DEC mixtures and negative in dimethylcarbonate mixtures with alkanes, becoming more pronounced in the longerchain hydrocarbons. However, these results are largely insignificant in both cases since the absolute value of this slope corresponds to mixtures with decane and is of the order of 5 × 10-4. With this value, it is possible to simplify eqs 4-6, which would be used to correlate the data. 3.2. Apparatus and Procedure. The equipment used for the experimental part of this work and the procedure followed to measure the different properties have been described elsewhere. The VLE enclosure and auxiliary installations were described in a previous work,9 while a calorimeter type Calvet, from Setaram Lyon (France), was used to determine hE and was calibrated electrically at regular time intervals and at the two temperatures applied in this work, with the apparatus constant being independent of T. The densities of the mixtures F were used to determine the VLE concentrations and were measured in an Anton Paar densimeter, model DMA-55, with an uncertainty of (0.02 kg‚m-3. Paired values (x, F) were obtained for each of the synthetic mixtures prepared in the entire concentration range, which present an uncertainty of (3 × 10-4 units. The relationship used is of the form F(x) ) [(F1 - F2)x + F2] - [x(1 - x)(ax2 + bx + c)], where F, F1, and F2 are, respectively, the densities of the mixture and those of pure carbonate and alkanes. The concentrations of the liquid phase x and vapor phase y obtained present an uncertainty of (0.002 units. After


7360


Figure 5. Plots of experimental and calculated excess properties for (DMC + an alkane) binary mixtures. Each horizontal set represents the binaries containing an alkane as a second component, from hexane to decane. In the first column appear the experimental values of gE/RT (b) and γi (2) along the curves obtained by different models: NRTL (- - -), UNIQUAC (s s s), and eq 17 (s). The inset shows the differences between experimental values and those calculated (at T1 ) 298.15 K and T2 ) 318.15 K) by NRTL and eq 18. The second column shows a set of graphics containing experimental hE values at T1 ) 298.15 K (b) and T2 ) 318.15 K (O) along the curves obtained by eq 18, and the TsE curve (s ‚ s ‚ s) obtained by eq 19 at T ) 298.15 K. The third column shows a set of graphics containing experimental cEp at T1 ) 288.15 K (b), T2 ) 298.15 K (O), and T3 ) 308.15 K (1) as well as the curves estimated by eq 20 (s) and those obtained from (∂hE/∂T)p (- - -) at T ) 298.15 K.

reaching V-L equilibria, the densities of the samples from both phases are measured, calculating the respective concentrations from the inverse of the regression curve F ) F(x), obtained as shown previously, for each mixture studied. 4. Presentation of Results 4.1. Excess Properties. Table 2 of this work shows the hE for the binary mixtures DEC (1) + alkane (2) at three temperatures, 291.15, 298.15, and 318.15 K. Each of these sets of experimental data was correlated against concentration x, with a simple polynomial equation similar to eq 1, carrying out the corresponding identification of kh as described. Table 3 shows the values calculated for all the parameters using a least-squares procedure in addition to standard deviations s, for each system. Figure 2 shows the equimolar values of hE for the binary systems presented in this work together with those found in the literature,17 measured at T ) 298.15 K, showing a good agreement between them. Observing the graph, one can see a nonlinear variation in the hE(T) points, which changes according to the alkane concerned. Hence, the slope is positive for the hexane but changes sign for the other alkanes, from negative to positive between the temperatures of 291.15 and 318.15 K. In the literature,18,19 data have been found for the coefficient cEp for some of these mixtures, which have been used in the mathematical-thermodynamic treatment proposed in this work. 4.2. Vapor Pressures. Owing to the influence of vapor pressures poi on the characterization of VLE data, using the same equipment,9,20 new values of (T, poi ) were determined experimentally only for DEC. These are shown in Table 4, since the values of DMC were presented in another article.3 Correlation of the data with Antoine’s equation in the form

log(poi /kPa) ) A - B/[(T/K) - C]

(15)

gave the following values, A ) 6.4296 (a ) 2.8943), B ) 1569.68 (b ) 2.7173), and C ) 44.3 (c ) 0.075). The values in brackets represent the values obtained with eq 15 but using the reduced magnitudes of pressure and temperature. The vapor pressure line is represented in reduced coordinates in Figure 3. The next step was to determine acentric factors for DMC and DEC according to the definition of Pitzer: ω ) -(log o )Tr)0.7 - 1, giving a result for DMC of ω ) 0.382 (0.38515) pi,r

and for DEC of ω ) 0.471 (0.48615), used in the following calculations to characterize VLE. 4.3. VLE Data. Table 5 shows the values of the quantities that characterize each of the liquid and vapor states in isobaric equilibria, T, x, y at p ) 101.32 kPa, for binary mixtures DEC (1) + C9 (2) or +C10 (2). The activity coefficients γi were calculated for the liquid phase considering the nonideality of the vapor phase, by the expression

ln γi ) ln

( ) pyi

poi xi

+

(Bii - Voi )(p - poi ) + RT p (2B - Bii - Bjj)δijyj2 (16) RT ij

determining the second virial coefficients of the pure compounds Bii, and of the mixtures Bij, by the expressions of Tsonopoulos.21 The molar volumes of the pure compounds Voi in the liquid phase at equilibrium temperature were calculated by applying Rackket’s equation modified by Spencer and Danner.22 The vapor pressures poi of eq 15 for each of the substances and at each equilibrium T were calculated with Antoine’s equation using the constants given here for DEC and those obtained in previous works9 for other substances. The values of γi and of adimensional Gibbs function, Q ) (gE/RT), are shown in Table 5. One index of quality for the VLE data used here consisted of prior verification of the point-to-point consistency test h y1 proposed by Fredenslund et al.,23 verifying in all systems δ < 0.01. Figure 4 shows the quantities T vs x, y measured for this work, together with other previously published ones,2 maintaining the regularity of the DEC + alkanes group (C6C10). 5. Procedure to Correlate Thermodynamic Quantities Data were correlated using the corresponding equations proposed in Section 2, which we define more clearly during their application. Hence, starting with a specific expression of eq 1 for Gibbs’ function,

(

)

2 A i0 (x, T) ) z(1 - z) + Ai1 + Ai2T zi RT i)0 T

gE

∑

(17)

we can obtain expressions for enthalpy and other derived


7362


Figure 6. Plots of experimental and calculated excess properties for (DEC + an alkane) binary mixtures. Each horizontal set represents the binaries containing an alkane as the second component, from hexane to decane. In the first column appear the experimental values of gE/RT (b) and γi (2), along the curves obtained by different models: NRTL (- - -), UNIQUAC (s s s), and eq 17 (s). The inset shows the differences between the experimental values and those calculated (at T1 ) 291.15 K, T2 ) 298.15 K, and T3 ) 318.15 K) by NRTL and eq 18. The second column shows a set of graphics containing experimental hE values at T1 ) 291.15 K (2), T2 ) 298.15 K (b), and T3 ) 318.15 K (O), along the curves obtained by eq 18, and TsE curve (s ‚ s ‚ s) obtained by eq 19 at T ) 298.15 K. The third column shows a set of graphics containing experimental cEp at T1 ) 288.15 K (b), T2 ) 298.15 K (O), and T3 ) 308.15 K (1), along the curves estimated by eq 20 (s) and those obtained from (∂hE/∂T)p (- - -) at T ) 298.15 K.

quantities, such as entropy and thermal capacities, eqs 4-7, if we consider, as demonstrated in the previous section for the systems studied here, Figures 1 and 2, that (dz/dT) ) 0, since values for the parameter k do not vary greatly with temperature. According to this, the corresponding equations for the mentioned excess quantities in adimensional form would be represented by hE RT

sE R

() dai

2

) -Tz(1 - z)

∑ dT i)0

2

∑ i)0

) -z(1 - z)

2

zi ) -z(1 - z)

i)0

[ ( )] ai + T

dai dT

∑

(

Ai0 T

)

- Ai2T zi (18)

zi ) 2

-z(1 - z) cEp R

2

) -Tz(1 - z)

∑ i)0

(Ai1 + 2Ai2T)zi ∑ i)0

[ ( ) ( )] dai

2

dT

+T

d2ai dT

2

(19)

zi ) 2

-2Tz(1 - z)

Ai2zi ∑ i)0

(20)

Starting with eq 17, and taking into account that z ) z(x), the activity coefficients can be obtained for each of the compounds, taking into consideration the adimensional form of the Gibbs function, Q, resolving

ln γi ) Q + (1 - xi)

( ) ∂Q ∂xi

) Q + (1 - xi)

p,T

( )( ) ∂Q dz ∂z dxi

(21)

where i ) 1, 2, which would produce in our case the following generic equation, 2

ln γi ) z(1 - z)

∑ i)0

3

aizi + (1 - xi)(

() z

(i + 1)bizi)k ∑ x i)0

substances 1 and 2 could also be determined, respectively, by

ln γ∞1 ) limxf0 ln γ1 ≡ limzf0 ln γ1 )

a0 k

(23)

ln γ∞2 ) limxf1 ln γ2 ≡ limzf1 ln γ2 ) k(a0 + a1 + a2) (24) For this set of relationships, experimental values are only available for three of the properties, hE, cEp , and gE or γi, so these could possibly be solved simultaneously. However, as shown here, a simple correlation procedure was devised by successive steps, easily deduced from the close examination of eqs 17-20. Hence, having paired sets of data (x, cEp ) at one or several temperatures, the Ai2 coefficients can be determined, which, when introduced in eq 18 with values of (x, T, hE), can be used to obtain values for Ai0, and finally with eq 17, the Ai1 can be calculated. It does not seem recommendable to consider a procedure in the opposite direction, since the successive derivations required to carry it out, which produce eq 20 if we start with eq 17, would result in a loss of precision. Another possibility that should not be ruled out is to obtain a simultaneous fit of all the quantities, as mentioned previously, if one has a powerful computer and an equally suitable algorithm that would result in a convergence of solutions. When taking this into consideration and having quality experimental values, the different pathways would lead to the same result. Now, with the experimental values of the systems (x1DMC or x1DEC + x2 alkanes), the articulated correlation procedure was carried out in the three steps indicated. An optimum correlation of the experimental data would require a suitable objective function (OF) to minimize the differences between experimental and theoretical values. Here, this OF is proposed simply as T

OF )

ni

∑ ∑[ME(Ti,x1j) - (ME)ij]2 i)1 j)1

(25)

2

(22)

1

where bi ) ai - ai-1 and 0 e i e 3. The corresponding values for the activity coefficients at infinite dilution for the two

where the subindex i refers to the number of temperatures of the generic excess property ME (in adimensional form) and j refers to the number of experimental data points, at each concentration, in each property studied. The experimental data sets have been recorded in the following way. First, the cEp

Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007 7363 Table 6. Results Obtained in the Correlation of VLE, hE, and cEp for the Binaries of (a Dialkyl Carbonate + an Alkane) Using Different Models eqs 17-20 NRTL

UNIQUAC

A0j

∆g12 ) -12 380.213 ∆g21 ) 13 438.939 R ) -0.05 γ1∞ ) 3.561 γ2∞ ) 4.630 0.014 0.094 171.2

DMC (1) + hexane (2) ∆u12 ) 356.046 1 075.320 ∆u21 ) 951.894 -0.231 -4.177 × 10-3 γ1∞ ) 4.021 γ2∞ ) 4.442 0.008 0.035 852.6

∆g12 ) -20 908.786 ∆g21 ) 21 903.139 R ) -0.02 γ1∞ ) 3.390 γ2∞ ) 4.576

DMC (1) + heptane (2) ∆u12 ) 284.778 1 806.715 ∆u21 ) 959.206 -4.594 2.097 × 10-3 γ1∞ ) 3.297 γ2∞ ) 4.777

0.008 0.086 127.1 6.50

0.007 0.046 953.6 1.47

∆g12 ) -13 300.783 ∆g21 ) 14 588.009 R ) -0.048 γ1∞ ) 2.979 γ2∞ ) 5.392

DMC (1) + octane (2) ∆u12 ) 422.856 1 788.810 ∆u21 ) 837.324 -4.630 3.488 × 10-3 γ1∞ ) 2.920 γ2∞ ) 5.944

0.010 0.201 154.7 6.60

0.012 0.112 906.5 1.79

∆g12 ) 20 866.131 ∆g21 ) -20 463.340 R ) -0.03 γ1∞ ) 3.007 γ2∞ ) 3.409

DMC (1) + nonane (2) ∆u12 ) -1 001.917 2 188.536 ∆u21 ) 2 624.496 -6.355 4.288 × 10-3 ∞ γ1 ) 3.201 γ2∞ ) 3.289

0.035 0.283 239.0 8.23

0.038 0.250 817.4 1.69

∆g12 ) 34 547.649 ∆g21 ) -33 027.051 R ) -0.01 γ1∞ ) 3.390 γ2∞ ) 4.605

DMC (1) + decane (2) ∆u12 ) -782.001 2799.168 ∆u21 ) 2 551.607 -7.798 7.702 × 10-3 γ1∞ ) 3.426 γ2∞ ) 5.061

s(gE/RT) s(γi) s(hE) s(cEp )

0.033 0.138 238.2 8.92

0.028 0.120 819.5 2.99

s(gE/RT) s(γi) s(hE)

∆g12 ) 4 903.217 ∆g21 ) -4 818.476 R ) -0.27 γ1∞ ) 2.866 γ2∞ ) 1.640 0.008 0.061 94.9

DEC (1) + hexane (2) ∆u12 ) -1 263.206 1 035.840 ∆u21 ) 2 383.333 -3.573 3.352 × 10-3 γ1∞ ) 3.239 γ2∞ ) 1.676 0.016 0.044 263.1

∆g12 ) -14 317.084 ∆g21 ) 15 065.990 R ) -0.03 γ1∞ ) 2.201 γ2∞ ) 2.259

DEC (1) + heptane (2) ∆u12 ) 169.233 1 008.233 ∆u21 ) 440.429 -2.779 2.220 × 10-3 γ1∞ ) 2.287 γ2∞ ) 2.181

0.017 0.097 90.1 2.66

0.015 0.090 587.1 0.43






A1j

A2j

-1 504.791 3.864 8.739 × 10-4 γ1∞ ) 4.231 γ2∞ ) 4.336

44.459 3.802 -1.032 × 10-2 kg ) 1.030 kh ) 1.384 0.008 0.029 28.8

-3 649.852 16.836 -1.857 × 10-2 γ1∞ ) 3.357 γ2∞ ) 4.336

3 001.766 -14.572 1.912 × 10-2 kg ) 0.868 kh ) 1.552 kc ) 1.362 0.008 0.036 42.2 0.20

-3 768.695 19.785 -2.520 × 10-2 γ1∞ ) 2.940 γ2∞ ) 5.741

3 362.230 -19.060 2.510 × 10-2 kg ) 1.547 kh ) 1.720 kc ) 1.553 0.010 0.028 47.9 0.17

-5 438.496 26.291 -3.391 × 10-2 γ1∞ ) 3.912 γ2∞ ) 3.310

4 589.269 -22.764 3.405 × 10-2 kg ) 0.460 kh ) 1.889 kc ) 1.724 0.035 0.165 61.4 0.43

-8 524.707 41.283 -5.738 × 10-2 γ1∞ ) 3.094 γ2∞ ) 5.656

7 486.588 -40.203 5.781 × 10-2 kg ) 1.600 kh ) 2.059 kc ) 1.909 0.031 0.156 42.4 0.94

-764.340 4.127 -4.977 × 10-3 γ1∞ ) 3.181 γ2∞ ) 1.520

-244.217 2.456 -4.049 × 10-3 kg ) 0.513 kh ) 1.007 0.012 0.037 36.1

-1 152.810 5.602 -8.120 × 10-3 γ1∞ ) 2.851 γ2∞ ) 2.377

790.199 -3.655 6.916 × 10-3 kg ) 0.728 kh ) 1.126 kc ) 1.022 0.010 0.056 17.8 0.22

7364


Table 6 (Continued) eqs 17-20 NRTL

UNIQUAC

A0j


∆g12 ) -14 346.401 ∆g21 ) 15 385.527 R ) -0.03 γ1∞ ) 2.169 γ2∞ ) 2.469 0.008 0.055 81.6

DEC (1) + octane (2) ∆u12 ) 418.973 986.247 ∆u21 ) 216.909 -2.291 8.516 × 10-4 ∞ γ1 ) 2.152 γ2∞ ) 2.540 0.008 0.054 554.0


∆g12 ) -12 930.945 ∆g21 ) 13 690.089 R ) -0.04 γ1∞ ) 1.977 γ2∞ ) 2.474 0.008 0.049 81.3

DEC (1) + nonane (2) ∆u12 ) 483.932 1 222.106 ∆u21 ) 117.284 -2.817 2.200 × 10-3 ∞ γ1 ) 1.918 γ2∞ ) 2.617 0.007 0.036 616.8


∆g12 ) -10 957.149 ∆g21 ) 11 613.000 R ) -0.06 γ1∞ ) 1.869 γ2∞ ) 2.590 0.007 0.041 79.8

DEC (1) + decane (2) ∆u12 ) 303.504 1 112.898 ∆u21 ) 289.790 -2.051 2.492 × 10-4 γ1∞ ) 1.828 γ2∞ ) 2.699 0.009 0.027 707.8

belong to the set {(Ti, x1j, (cEp /R)ij,), j ) 1, ..., n}, where (cEp /R)ij refers to values of heat capacity of the adimensional excess obtained at the concentration of the first component x1j and at the different temperatures i. Second, we have the enthalpy data expressed as {(Ti, x1j, (hE/RT)ij), j ) 1, ..., n}, where the adimensional term of the enthalpy refers to the number of points j at each temperature i, 291.15, 298.15, and 318.15 K, in this work. Finally, the isobaric VLE data represented by the set {(Ti, x1j, ln γ1j, ln γ2j), j ) 1, ..., n}, where ln γ1j and ln γ2j are the natural logarithms of the activity coefficients for the concentration of the first component, x1j, at the temperature Ti of each equilibrium state. In all cases, a genetic algorithm was used24 with a minimization procedure of the OF established in eq 25. The treatment can be, therefore, divided into the three following steps: (1) The first process, or correlation step, is carried out on heat capacity data from the literature16,18,19 by using eq 20, with which values can be obtained for the coefficients Ai2, for a constant value of kc, obtained as indicated in Section 2.1. (2) Next, in a second correlation process, the values of Ai2 are introduced in eq 18, and with a constant value for each binary system of the parameter kh, calculated by eq 12 and presented in Table 3, the corresponding coefficients, Ai0, are calculated. (3) Finally, in the last step, Ai2 and Ai0 are introduced in eq 17 and can be used to determine the Ai1 and the value of kg by optimization in the regression process. The final results of the correlation applied to each system are presented in Table 6, where we can observe, in the parameters of goodness-of-fit and the graphs in Figures 5 and 6, good correlations of the thermodynamic properties. In spite of the sigmoidal form of the representation of cEp ) f1(x), the fit of data to eq 20 is acceptable; see parts f, i, l, and o of Figure 5 or part f of Figure 6. There is also a good representation of hE ) f2(x) at the two temperatures and the estimation of

A1j

A2j

-735.376 2.332 -1.706 × 10-3 γ1∞ ) 2.466 γ2∞ ) 2.482

231.680 0.625 1.034 × 10-4 kg ) 0.578 kh ) 1.251 0.009 0.041 10.3

-1 112.559 3.241 -3.452 × 10-3 γ1∞ ) 1.969 γ2∞ ) 2.811

248.083 0.291 -1.032 × 10-3 kg ) 1.481 kh ) 1.372 0.007 0.028 12.6

-375.425 -1.534 5.254 × 10-3 γ1∞ ) 1.907 γ2∞ ) 2.593

-426.010 5.032 -8.218 × 10-3 kg ) 0.889 kh ) 1.497 0.009 0.014 17.0

enthalpies, which are positive with a peak in concentrations nearer to pure dialkyl carbonate, possibly due to dipole-dipole interactions of the dialkylcarbonate molecules in those regions. Finally, the representations of the figures labeled a, d, g, j, and m show a good estimation of the functions gE/RT and γi. In the inserts, the differences between estimated and experimental values, or direct fit, are shown, revealing small differences that are included in the lower error zone of 10%. This stepwise procedure also permits us to estimate the cEp when there are no experimental values, estimations which, at least quantitatively, seem to be not different from experimental values, in spite of the fact that they do not, in any case, present the double inflection shown in real behavior. Table 6 also shows the corresponding coefficients extrapolated to an infinite dilution, obtained with eqs 23 and 24, compared with those obtained using other equations. The high values of the γi∞ would suggest the interactional effects mentioned previously. 5.1. Correlation Results with NRTL and UNIQUAC. In order to validate the model and procedure proposed in the previous section, local composition models NRTL10 and UNIQUAC11 were chosen, which can be used to simultaneously correlate the VLE, hE, and cEp data, for sets of binary systems dialkylcarbonate + alkanes selected in this work. By assigning the NRTL model or the UNIQUAC model to the adimensional Gibbs function, with the successive derivatives with respect to temperature according to the thermodynamic relationships shown in eq 3, expressions for hE and for cEp can be obtained. These do not present a simple, useful way to carry out a stepwise correlation for each of the properties, as was done with the parametric model proposed here. This explains why a simultaneous correlation is done for these properties, establishing only one objective function that optimizes the minimum deviations for each of these. In other words,

Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007 7365 2

OF )

Nomenclature

n

∑ ∑[(ln γi(Tj, x1j) - ln γij)/ln γij]2 + i)1 j)1 2

ni

∑ ∑ i)1 j)1

[(

hE

(Tj, x1j) -

RT

mj 3

∑ ∑ i)1 j)1

[(

( ) )( ) ] hE

hE

+

RT ij cEp cEp / (Tj, x1i) R R ij R

cEp

RT

/

Latin Letters

2

ij

( ) )( ) ]

2

(26)

ij

Obviously, for mixtures for which there are no values of cEp or hE, the corresponding addend must be eliminated. This is the case of the mixtures of DMC + hexane and DEC + alkane(C6, C8-C10) for which values of (x, cEp ) have not been found, simultaneously correlating VLE and hE data but allowing the model to give approximate estimations of the thermal coefficients. Table 6 shows the results obtained by applying the NRTL and UNIQUAC models to the set of mixtures chosen for this work. In general, we can consider both applications to be deficient, although with some noteworthy characteristics. Graphically, the results can also be analyzed in the matrix corresponding to Figures 5 and 6. Hence, the NRTL model offers better estimations when only the quantities of excess enthalpies are correlated and values of the Gibbs function are used and does not effectively predict heat capacities. When there is a combined correlation of the three quantities, there is considerable loss of accuracy. On the other hand, the UNIQUAC model does not offer good representations of the thermodynamic quantities considered together, offering in this case a poorer fit than NRTL. In both cases, the parameters of goodness-of-fit are compiled in Table 6 and confirm these observations. Because of the poor correlations, the hE curves for the systems studied have not been depicted in Figures 5 and 6.

A, B, C ) Antoine constants ai ) coefficients of eq 1 Aij ) coefficients of eqs 17-19 Bij ) second virial coefficients cEp ) excess thermal capacity gE/RT ) adimensional excess Gibbs function hE ) excess enthalpy k ) parameter of eq 1 Mi ) molecular weight for compound i nD ) refractive index OF ) objective function p ) total pressure poi ) vapor pressure for compound i qi ) surface parameter for compound i ri ) volume parameter for compound i R ) gas constant s ) standard deviation for the property M, s(M) )

[

N

]

(1/N)1/2 ∑ (Mi,exp - Mi,cal)2 1/2 i)1 T ) temperature L Vi ) saturated liquid molar volume for compound i xi ) mole fraction in the liquid phase for compound i yi ) mole fraction in the vapour phase for compound i zi ) active fraction for compound i Greek Letters Ri ) expansivity coefficient for component i F ) density γi ) activity coefficients of compound i υ(i) k ) number of functional groups of type k in the molecule of component i Literature Cited

6. Conclusions In this work, several applications have been developed for a parametric model and a correlation procedure, which improves the estimation of thermodynamic properties of binary solutions, such as VLE, hE, and cEp data at isobaric conditions, also permitting estimation of entropies of the systems studied. The procedure and the model permit a stepwise correlation, even when experimental values are not available for some of the properties. In this work, cEp values were not available for some systems, but values of this coefficient could be estimated by correlating the other quantities. A set of mixtures was chosen comprising (dialkylcarbonate + alkanes), for which the literature presents experimental values for the three thermodynamic properties required to verify the model. Nonetheless, the database in the literature was increased by providing experimental measurements made in this study for the group of mixtures containing dialkylcarbonate. Application of the model and the procedure produced some excellent representations of the quantities, significantly improving on correlations made with known local composition models of NRTL and UNIQUAC. For all cases, a genetic algorithm and a suitable objective function were used. Acknowledgment The authors gratefully acknowledge the financial support received from the Ministerio de Educacioń y Ciencia for the Project CTQ2006-12027.

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ReceiVed for reView May 24, 2007 ReVised manuscript receiVed July 18, 2007 Accepted August 1, 2007 IE0707366

Modeling and Experimental Evaluation of Thermodynamic Properties

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