Excess molar volumes for ternary mixture (N,N-dimethylformamide+1-propanol+water) at the temperature 298.15 K

Fluid Phase Equilibria 152 Ž1998. 283–298

Excess molar volumes for ternary mixture ŽN, N-dimethylformamideq 1-propanolq water . at the temperature 298.15 K Tong-Chun Bai, Jia Yao, Shi-Jun Han

)

Chemistry Department, Zhejiang UniÕersity, Hangzhou, 310027, China Received 26 November 1997; accepted 5 August 1998

Abstract The results of excess molar volumes for ternary mixture N, N-dimethylformamide ŽDMF. q 1-propanolq water and for binary constituents, DMF q water, DMFq 1-propanol and 1-propanolq water at 298.15 K are reported. Several empirical expressions were used to predict and correlate the ternary excess molar volumes from experimental results on the constituent binaries. A pseudo-binary mixture approach ŽPBMA. was used to analyze the system studied. The partial molar volumes of 1-propanol at infinite dilution in w f m DMF q Ž1 y f m .waterx mixed solvents at their several fixed composition fm were evaluated and correlated with the composition f m . q 1998 Elsevier Science B.V. All rights reserved. Keywords: Density; Excess molar volume; Mixture; 1-propanol; N, N-dimethylformamide; Water

1. Introduction Aqueous solution of N, N-dimethylformamide ŽDMF. is a model mixed solvent to represent an environment of protein’s interiors w1x. In this binary mixture, DMF–water association exists and changes with the composition. Alcohols are model molecules to study the hydrophobic interactions w2x. In the DMF q alcoholq water ternary mixture, the properties of alcohol is affected by DMF–water association. It is our interest to study the effect of medium changing from water to amidic solvent on the thermodynamic properties of alcohol, and on the hydrophobic interactions. In present paper, the results of densities, excess molar volumes for ternary mixture DMF q 1propanolq water and for binary constituents, DMFq water, DMF q 1-propanol and 1-propanolq water at 298.15 K are reported. Several empirical expression were used to predict and correlate the )

Corresponding author. E-mail: [email protected]

0378-3812r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 8 . 0 0 4 0 2 - 6

T.-C. Bai et al.r Fluid Phase Equilibria 152 (1998) 283–298

284

ternary excess molar volumes from experimental results on the constituent binaries. In order to obtain some information about the correlation between alcohol–amidic interactions with amide solvation, a pseudo-binary mixture approach ŽPBMA. was applied to the system studied. The values of excess molar volume for the pseudo-binary mixture w f m DMF q Ž 1 y f m .waterx q 1-propanol were evaluated and fitted by empirical equation to a good accuracy. The partial molar volumes at infinite dilution V2` of 1-propanol in pseudo-binary mixture were evaluated and correlated with the composition f m of the DMF q water mixed solvent. A correspondence between V2` and partial molar volume of DMF in DMF q water binary mixture was observed.

2. Experimental section DMF Žanalytical reagent grade, Shanghai. was dried over freshly ignited Al 2 O 3 for at least 48 h, and then fractionally distilled under reduced pressure. The product was stored over freshly ignited molecular sieve 0.4 nm. 1-Propanol Žanalytical regent grade, Shanghai. was dried over anhydrous K 2 CO 3 , refluxed with pieces of sodium, fractionally distilled. The product was stored over freshly ignited molecular sieve 0.3 nm. Water was de-ionized and distilled twice before use. Densities of the pure liquids and mixtures were measured with an Anton Paar DMA 602 densimeter, thermostated by a circulating-water bath with a precision of 0.01 K. Dry air at atmosphere and pure water were used to calibrate the densimeter. All mixtures were prepared by mass. Ternary mixtures were prepared by mixing a measured binary mixture ŽDMF q water at known composition. with a pure liquid Ž 1-propanol, as the third component. . Ternary system were composed of a series of these pseudo-binary mixtures. The compositions of the mixtures were determined by weighing, their mole fraction being generally reliable to 1 = 10y4. The sensitivity of the densimeter corresponds to a precision of 1 = 10y6 g cmy3. The reproducibility of the density estimates was found to be of the order 2 = 10y5 g cmy3. The measured physical properties of the pure material together with literature w3–7x values are given in Table 1. Literature values of density for pure 1-propanol were different by their estimating method. Benson and Kiyohara’s w3x value was estimated by extrapolating to zero content of water. Table 1 Densities and refractive indices of pure liquids at 298.15 K Component

Density Žg cmy3 . This work

DMF

0.94403

1-Propanol

0.79965

a

Chu et al., 1990 w5x. Davis, 1987 w6x. c Zielkiewicz, 1995 w7x. d Wilhoit and Zwolinski, 1973 w4x. e Benson and Kiyohara, 1980 w3x. b

Refractive indices Literature a

0.944061 0.94383 b 0.79976 d 0.799353 e

This work

Literature

1.4282

1.42850 c 1.4282 a 1.38324 c 1.38348 c

1.3833

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285

Wilhoit and Zwolinski’s w4x value was the one from experimental. Our value was lower than that of Wilhoit and Zwolinski’s. That might mean that the water content was much lower. The values of density for pure DMF is different by literature also. In spite of our precautions, the results indicate that our value is better agreement with the value of Chu et al. w5x.

3. Results and discussion 3.1. Excess molar Õolumes for binary mixtures Excess molar volumes VmE were calculated from densities r . Results of r and VmE for binary mixtures of the investigated constituents are given in Table 2. A function of the form n

VmE.i j

3

y1

Ž cm mol . s x i x j

Ý Bk Ž x i y x j .

ky1

Ž1.

ks1

was used to fit the experimental results. Parameters of Eq. Ž1. were obtained by the optimization method of Powell w8x with Eq. Ž 2. as the objective function. m

Fs

½Ý

is1

1r2 2

VmE.exp Ž x i . y VmE.cal Ž x i . r Ž m y n .

5

Ž2.

where, m is the number of experimental data points, and n the number of parameters used in Eq. Ž 1. . The values of n were selected by the minimum of the objective function F in Eq. Ž2.. The values of the parameter Bk in Eq. Ž 1. and the standard deviations of the fitting are given in Table 3. Fig. 1 show the comparison of the VmE for three binary mixtures, x DMF q Ž1 y x .water, x1-propanolq Ž1 y x .water and x DMF q Ž1 y x . 1-propanol, at 298.15 K with the literature values at 313.15 K Ž from the work of Zielkiewicz w7x. . For DMFq water mixture, Fig. 1 present the largest negative value, which indicate a stronger molecular interaction between the two components. For 1-propanolq water mixture, it presents a mediate magnitude, and for DMFq 1-propanol mixture it shows a smaller magnitude. These are reflections of that the molecular interaction between 1-propanol and water is stronger than that of DMF–1-propanol, but weaker than that of DMF–water. In water-rich region, when the mole fraction of water is higher than 0.9, both binary constituents, DMF q water and 1-propanolq water have nearly the same magnitude, while those of DMF q 1propanol are still of relatively smaller magnitude. These might be attributed to the hydrophobic interaction occurred in the water-rich region. A temperature effect on VmE could be observed from Fig. 1. The values of VmE at 298.15 K have all magnitude greater than those at 313.15 K. This is a reflection that at a lower temperature due to lower molecular thermal agitation, the associations or interactions Ž and then the molecular packing. between the components become stronger. But in water-rich region, mole fraction of water higher than 0.9, temperature effect is hardly observable for both DMF q water and 1-propanolq water mixtures. The temperature effect varies with the water content. The competition of molecular interaction among components and the changing of temperature effect with solution composition give rise to some asymmetric effect when the ternary mixture is composed of these constituents.

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286

Table 2 Densities Ž r . and excess molar volumes Ž VmE . for binary mixtures at 298.15 K VmE Žcm3 moly1 .

x

r Žg cmy3 .

VmE Žcm3 moly1 .

xDMF q (l y x)water 0.0055 0.99667 0.0112 0.99649 0.0142 0.99640 0.0241 0.99620 0.0319 0.99616 0.0382 0.99614 0.0464 0.99620 0.0553 0.99630 0.0620 0.99640 0.0721 0.99657 0.0805 0.99670 0.0907 0.99685 0.0914 0.99686 0.1011 0.99698 0.1125 0.99708 0.1230 0.99713 0.1347 0.99712 0.1478 0.99704 0.1489 0.99703 0.1621 0.99682

y0.0158 y0.0357 y0.0463 y0.0828 y0.1137 y0.1390 y0.1736 y0.2119 y0.2414 y0.2864 y0.3237 y0.3689 y0.3720 y0.4147 y0.4640 y0.5085 y0.5565 y0.6084 y0.6127 y0.6613

0.1783 0.1941 0.2042 0.2078 0.2361 0.2623 0.2907 0.3610 0.3651 0.4429 0.4623 0.5202 0.6582 0.6893 0.7392 0.7943 0.8240 0.8885 0.9235 0.9606

0.99655 0.99612 0.99584 0.99570 0.99463 0.99340 0.99185 0.98727 0.98698 0.98104 0.97951 0.97485 0.96435 0.96218 0.95880 0.95522 0.95334 0.94940 0.94739 0.94551

y0.7202 y0.7724 y0.8052 y0.8156 y0.8965 y0.9605 y1.0181 y1.1099 y1.1132 y1.1295 y1.1227 y1.0760 y0.8645 y0.8034 y0.6937 y0.5577 y0.4773 y0.2880 y0.1812 y0.0776

xDMF q (l y x)1-propanol 0.9544 0.93830 0.9286 0.93478 0.9107 0.93231 0.8847 0.92870 0.8535 0.92432 0.8347 0.92170 0.8035 0.91731 0.7848 0.91470 0.7521 0.91011 0.6838 0.90047

y0.0551 y0.0642 y0.0685 y0.0733 y0.0758 y0.0792 y0.0820 y0.0857 y0.0908 y0.0994

0.6299 0.5878 0.5039 0.4338 0.3729 0.3219 0.2466 0.1982 0.1538 0.0910

0.89277 0.88670 0.87452 0.86431 0.85545 0.84800 0.83694 0.82976 0.82310 0.81360

y0.1008 y0.0986 y0.0908 y0.0853 y0.0844 y0.0831 y0.0796 y0.0732 y0.0622 y0.0417

x1-propanolq (l y x)water 0.0049 0.99429 0.0057 0.99391 0.0082 0.99264 0.0096 0.99190 0.0107 0.99140 0.0122 0.99068 0.0139 0.98980 0.0186 0.98780 0.0234 0.98570 0.0276 0.98382 0.0320 0.98187 0.0358 0.98025

y0.0223 y0.0271 y0.0403 y0.0470 y0.0538 y0.0622 y0.0703 y0.1003 y0.1289 y0.1521 y0.1760 y0.1972

0.1293 0.1952 0.2692 0.3669 0.5132 0.5688 0.6339 0.7010 0.7545 0.7570 0.7932 0.7991

0.94169 0.91700 0.89450 0.87230 0.84787 0.84030 0.83250 0.82542 0.82027 0.82004 0.81674 0.81621

y0.5409 y0.6080 y0.6315 y0.6606 y0.6459 y0.6146 y0.5724 y0.5214 y0.4694 y0.4668 y0.4224 y0.4142

x

r Žg cmy3 .

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Table 2 Žcontinued. x

r Žg cmy3 .

x1-propanolq (l y x)water 0.0407 0.97818 0.0567 0.97148 0.0921 0.95680

VmE Žcm3 moly1 .

x

r Žg cmy3 .

VmE Žcm3 moly1 .

y0.2240 y0.3052 y0.4469

0.8463 0.8953 0.9239

0.81205 0.80801 0.80580

y0.3385 y0.2536 y0.2036

3.2. Excess molar Õolumes for ternary mixtures The experimental values of densities for the ternary system were obtained by adding the third component Ž1-propanol. to a constant ratio of the other two Ž DMFq water.. Excess molar volumes were calculated by using the following equation: y1 y1 y ry1 y ry1 VmE.123 s x 1 M1 Ž ry1 y ry1 1 . q x 2 M2 Ž r 2 . q x 3 M3 Ž r 3 .

Ž3.

where M1, M2 and M3 are molar masses of components 1, 2 and 3, respectively. The r 1, r 2 and r 3 E are densities of pure component and r the density of mixture. The values of r and Vm.123 for ternary mixtures are given in Table 4. Several empirical method have been suggested by Esteve et al. w9x, to estimate ternary excess properties from experimental results on the constituents. Some of the predictive method were originally proposed to predict molar enthalpy, volume or Gibbs free energy. Nevertheless, they should be applicable to any other excess property. The first type of expressions are of the form that there are no parameters to include the ternary E is predicted from binary data. The simplest expression is of effect. The ternary excess volume Vm.123 the form w10x VmE.123 Ž cm3 moly1 . s VmE.12 Ž x 1 , x 2 . q VmE.13 Ž x 1 , x 3 . q VmE.23 Ž x 2 , x 3 .

Ž4.

E Ž . Ž . where x 1 s 1 y x 2 y x 3 and the values of Vm.i j x i , x j being calculated according to Eq. 1 using Ž . values of Bk from Table 3 and at the composition x i , x j .

Table 3 Fitting parameters Ž Bk . of Eq. Ž1. and the standard deviations Ž s . of the fitting for three binary mixtures investigated B1 B2 B3 B4 B5 B6 B7 B8 B9 s Ž VmE . Žcm3 moly1 .

x DMFqŽ1y x .1-propanol

x DMFqŽ1y x .water

x1-propanolqŽ1y x .water

y0.361119 y0.192992 y0.595511 0.870314 1.236784 y1.329132 y1.561227

y4.384739 1.762886 0.327434 y1.845555 0.264230 1.883635 1.075158 y0.987477 0.632431 0.00043

y2.609885 0.873844 y0.322025 y3.041861 y4.562641 13.81551 2.641219 y11.47942

0.00063

0.0019

288

T.-C. Bai et al.r Fluid Phase Equilibria 152 (1998) 283–298

Fig. 1. Excess molar volumes as a function of mole fraction for binary mixtures, x DMFqŽ1y x .water, I, at 298.15 K and curve 1 at 313.15 K; x Ž1-propanol.qŽ1y x .water, `, at 298.15 K and curve 2 at 313.15 K and x ŽDMF.qŽ1y x .1-propanol, ^, at 298.15 K and curve 3 at 313.15 K. The values of VmE at 313.15 K were taken from literature Žfrom the work of Zielkiewicz w7x..

According to the expression of Kohler w11x, the excess molar volume for a ternary mixture is given by 2

2

VmE.123 Ž cm3 moly1 . s Ž x 1 q x 2 . VmE.12 Ž x 10 , x 20 . q Ž x 1 q x 3 . VmE.13 Ž x 10 , x 30 . 2

q Ž x 2 q x 3 . VmE.23 Ž x 20 , x 30 .

Ž5.

E Ž 0 0. in which, Vm.i j denotes the excess molar volume for the binary mixture at composition x i , x j , such E Ž 0 0. Ž . that x i0 s x irŽ x i q x j . s 1 y x j0. With Vm.i j x i , x j being calculated by Eq. 1 . Eqs. Ž4. and Ž5. are symmetrical in the sense that all the three binary mixtures are treated identically. Their numerical predictions do not depend on the arbitrary designation of component numbering. On the contrary, Tsao and Smith w12x proposed an asymmetrical equation

VmE.123 Ž cm3 moly1 . s x 2r Ž 1 y x 1 . VmE.12 Ž x 1 ,1 y x 1 . q x 3r Ž 1 y x 1 . VmE.13 Ž x 1 ,1 y x 1 . q Ž 1 y x 1 . VmE.23 Ž x 20 , x 30 .

Ž6.

where x 20 s x 2rŽ1 y x 1 . s 1 y x 30. Hillert w13x also proposed an asymmetrical equation, VmE.123 Ž cm3 moly1 . s x 2r Ž 1 y x 1 . VmE.12 Ž x 1 ,1 y x 1 . q x 3r Ž 1 y x 1 . VmE.13 Ž x 1 ,1 y x 1 . q Ž x 2 x 3rn 23 n 32 . VmE.23 Ž n 23 , n 32 .

Ž7.

E . of the predictions by Eqs. where Õi j s Ž1 q x i y x j .r2. In Table 5, the standard deviations s Ž Vm.123 Ž4. – Ž7. are shown.

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289

Table 4 E E ., for pseudo-binary mixture Ž Vm.2q13 . at 298.15 K, and the Densities Ž r ., excess molar volumes for ternary mixture Ž Vm.123 Ž Ž .. values calculated by correlative equation Eq. 10 and by PBMA x1

x2

r Žg cmy3 .

E Vm.123 Žcm3 moly1 .

E Vm.123 Žcm3 moly1 . Eq. Ž10.

E Vm.2q13 Žcm3 moly1 .

E Vm.2q13 Žcm3 moly1 . PBMA

f m s 0.0117 0.0117 0.0116 0.0116 0.0116 0.0116 0.0114 0.0113 0.0112 0.0111 0.0104 0.0101 0.0092 0.0084 0.0072 0.0057 0.0044 0.0040 0.0029 0.0028 0.0019 0.0016 0.0010

0.0000 0.0036 0.0050 0.0075 0.0098 0.0197 0.0287 0.0384 0.0510 0.1064 0.1325 0.2083 0.2821 0.3800 0.5099 0.6197 0.6593 0.7525 0.7614 0.8333 0.8667 0.9147

0.99648 0.99465 0.99398 0.99283 0.99218 0.98787 0.98405 0.97977 0.97432 0.95102 0.94073 0.91403 0.89275 0.87070 0.84895 0.83464 0.83009 0.82047 0.81965 0.81334 0.81068 0.80693

y0.0375 y0.0566 y0.0638 y0.0781 y0.0995 y0.1593 y0.2106 y0.2595 y0.3184 y0.5070 y0.5637 y0.6518 y0.6756 y0.6768 y0.6487 y0.5858 y0.5509 y0.4505 y0.4412 y0.3574 y0.3193 y0.2484

y0.0372 y0.0570 y0.0649 y0.0792 y0.0924 y0.1490 y0.2001 y0.2534 y0.3185 y0.5236 y0.5762 y0.6351 y0.6538 y0.6802 y0.6552 y0.5840 y0.5559 y0.4725 y0.4622 y0.3589 y0.3022 y0.2184

0.0000 y0.0187 y0.0266 y0.0414 y0.0626 y0.1221 y0.1745 y0.2235 y0.2830 y0.4729 y0.5308 y0.6218 y0.6491 y0.6534 y0.6306 y0.5717 y0.5383 y0.4409 y0.4324 y0.3513 y0.3142 y0.2450

0.0000 y0.0237 y0.0328 y0.0487 y0.0631 y0.1223 y0.1723 y0.2223 y0.2814 y0.4728 y0.5311 y0.6227 y0.6491 y0.6522 y0.6318 y0.5716 y0.5377 y0.4415 y0.4317 y0.3524 y0.3134 y0.2451

f m s 0.0246 0.0246 0.0245 0.0245 0.0244 0.0244 0.0243 0.0242 0.0240 0.0239 0.0237 0.0236 0.0234 0.0231 0.0222 0.0209 0.0197 0.0180 0.0156 0.0125 0.0110

0.0000 0.0055 0.0069 0.0093 0.0110 0.0123 0.0187 0.0245 0.0315 0.0370 0.0424 0.0500 0.0633 0.0974 0.1506 0.1989 0.2716 0.3683 0.4926 0.5525

0.99620 0.99373 0.99315 0.99212 0.99184 0.99129 0.98865 0.98625 0.98330 0.98105 0.97875 0.97546 0.96975 0.95534 0.93434 0.91743 0.89599 0.87380 0.85238 0.84385

y0.0850 y0.1175 y0.1265 y0.1413 y0.1596 y0.1681 y0.2057 y0.2406 y0.2769 y0.3069 y0.3325 y0.3657 y0.4169 y0.5189 y0.6105 y0.6496 y0.6727 y0.6815 y0.6682 y0.6421

y0.0844 y0.1168 y0.1252 y0.1391 y0.1493 y0.1567 y0.1943 y0.2273 y0.2666 y0.2960 y0.3241 y0.3615 y0.4212 y0.5370 y0.6287 y0.6566 y0.6734 y0.6958 y0.6725 y0.6353

0.0000 y0.0327 y0.0417 y0.0564 y0.0756 y0.0836 y0.1229 y0.1570 y0.1951 y0.2250 y0.2511 y0.2844 y0.3376 y0.4422 y0.5379 y0.5814 y0.6105 y0.6280 y0.6253 y0.6037

0.0000 y0.0385 y0.0479 y0.0639 y0.0750 y0.0833 y0.1229 y0.1567 y0.1949 y0.2230 y0.2491 y0.2835 y0.3371 y0.4424 y0.5387 y0.5818 y0.6102 y0.6276 y0.6258 y0.6032

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290

Table 4 Žcontinued. x1

x2

r Žg cmy3 .

E Vm.123 Žcm3 moly1 .

E Vm.123 Žcm3 moly1 . Eq. Ž10.

E Vm.2q13 Žcm3 moly1 .

E Vm.2q13 Žcm3 moly1 . PBMA

f m s 0.0246 0.0093 0.0059 0.0038 0.0030

0.6235 0.7616 0.8458 0.8765

0.83473 0.81987 0.81254 0.81012

y0.5847 y0.4350 y0.3438 y0.3103

y0.5845 y0.4615 y0.3362 y0.2834

y0.5529 y0.4145 y0.3306 y0.2997

y0.5534 y0.4140 y0.3313 y0.2993

f m s 0.0529 0.0529 0.0522 0.0516 0.0512 0.0508 0.0502 0.0495 0.0475 0.0448 0.0444 0.0409 0.0380 0.0376 0.0366 0.0317 0.0315 0.0247 0.0245 0.0218 0.0184 0.0129

0.0000 0.0137 0.0264 0.0337 0.0407 0.0524 0.0654 0.1031 0.1546 0.1611 0.2272 0.2824 0.2901 0.3091 0.4007 0.4053 0.5326 0.5367 0.5882 0.6520 0.7569

0.99627 0.99115 0.98614 0.98314 0.98040 0.97556 0.97005 0.95445 0.93460 0.93228 0.91073 0.89565 0.89375 0.88911 0.86980 0.86895 0.84796 0.84736 0.84024 0.83245 0.82120

y0.2017 y0.2902 y0.3612 y0.3967 y0.4309 y0.4779 y0.5208 y0.6122 y0.6766 y0.6822 y0.7135 y0.7261 y0.7279 y0.7272 y0.7185 y0.7186 y0.6571 y0.6540 y0.6117 y0.5618 y0.4530

y0.2018 y0.2827 y0.3531 y0.3905 y0.4242 y0.4759 y0.5257 y0.6286 y0.6925 y0.6963 y0.7148 y0.7228 y0.7241 y0.7270 y0.7288 y0.7280 y0.6613 y0.6582 y0.6182 y0.5680 y0.4659

0.0000 y0.0919 y0.1651 y0.2015 y0.2377 y0.2874 y0.3326 y0.4310 y0.5056 y0.5126 y0.5576 y0.5810 y0.5846 y0.5878 y0.5979 y0.5984 y0.5628 y0.5604 y0.5288 y0.4918 y0.4039

0.0000 y0.0932 y0.1668 y0.2042 y0.2370 y0.2857 y0.3320 y0.4297 y0.5058 y0.5125 y0.5591 y0.5812 y0.5835 y0.5885 y0.5977 y0.5975 y0.5622 y0.5602 y0.5319 y0.4895 y0.4043

f m s 0.0875 0.0875 0.0862 0.0855 0.0849 0.0842 0.0837 0.0829 0.0823 0.0813 0.0773 0.0728 0.0669 0.0598 0.0503 0.0380 0.0330 0.0278 0.0207

0.0000 0.0148 0.0226 0.0292 0.0376 0.0432 0.0521 0.0588 0.0707 0.1164 0.1677 0.2347 0.3165 0.4253 0.5655 0.6222 0.6821 0.7637

0.99682 0.99115 0.98815 0.98565 0.98237 0.98015 0.97656 0.97375 0.96891 0.95080 0.93214 0.91106 0.88990 0.86750 0.84520 0.83765 0.83040 0.82150

y0.3550 y0.4359 y0.4750 y0.5067 y0.5429 y0.5646 y0.5949 y0.6136 y0.6451 y0.7243 y0.7642 y0.7786 y0.7742 y0.7411 y0.6594 y0.6110 y0.5518 y0.4552

y0.3550 y0.4346 y0.4730 y0.5034 y0.5395 y0.5617 y0.5940 y0.6159 y0.6504 y0.7339 y0.7671 y0.7751 y0.7772 y0.7511 y0.6458 y0.5976 y0.5454 y0.4547

0.0000 y0.0858 y0.1276 y0.1621 y0.2013 y0.2251 y0.2589 y0.2793 y0.3150 y0.4110 y0.4685 y0.5068 y0.5312 y0.5375 y0.5050 y0.4769 y0.4389 y0.3714

0.0000 y0.0898 y0.1311 y0.1632 y0.2005 y0.2233 y0.2564 y0.2790 y0.3148 y0.4110 y0.4692 y0.5074 y0.5303 y0.5380 y0.5046 y0.4769 y0.4392 y0.3713

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291

Table 4 Žcontinued. x1

x2

r Žg cmy3 .

E Vm.123 Žcm3 moly1 .

E Vm.123 Žcm3 moly1 . Eq. Ž10.

E Vm.2q13 Žcm3 moly1 .

E Vm.2q13 Žcm3 moly1 . PBMA

f m s 0.1321 0.1321 0.1300 0.1290 0.1280 0.1270 0.1259 0.1249 0.1239 0.1220 0.1161 0.1087 0.1000 0.0885 0.0765 0.0570 0.0485 0.0402 0.0266

0.0000 0.0160 0.0236 0.0313 0.0386 0.0470 0.0548 0.0625 0.0765 0.1211 0.1772 0.2428 0.3303 0.4207 0.5687 0.6326 0.6954 0.7989

0.99712 0.99110 0.98814 0.98514 0.98230 0.97900 0.97595 0.97292 0.96743 0.95059 0.93120 0.91148 0.88950 0.87083 0.84647 0.83770 0.82993 0.81870

y0.5458 y0.6175 y0.6456 y0.6733 y0.6965 y0.7192 y0.7404 y0.7562 y0.7829 y0.8337 y0.8533 y0.8478 y0.8174 y0.7696 y0.6522 y0.5865 y0.5163 y0.3953

y0.5455 y0.6155 y0.6449 y0.6722 y0.6956 y0.7201 y0.7403 y0.7581 y0.7848 y0.8340 y0.8492 y0.8449 y0.8284 y0.7831 y0.6521 y0.5936 y0.5330 y0.3970

0.0000 y0.0807 y0.1132 y0.1441 y0.1715 y0.1997 y0.2239 y0.2450 y0.2782 y0.3539 y0.4039 y0.4342 y0.4522 y0.4534 y0.4170 y0.3860 y0.3503 y0.2854

0.0000 y0.0804 y0.1138 y0.1447 y0.1715 y0.1996 y0.2233 y0.2445 y0.2784 y0.3540 y0.4047 y0.4336 y0.4516 y0.4543 y0.4165 y0.3858 y0.3507 y0.2853

f m s 0.1930 0.1930 0.1895 0.1877 0.1860 0.1844 0.1830 0.1813 0.1796 0.1767 0.1661 0.1538 0.1406 0.1271 0.1050 0.0822 0.0664 0.0521 0.0405

0.0000 0.0180 0.0274 0.0366 0.0445 0.0519 0.0606 0.0693 0.0844 0.1394 0.2031 0.2713 0.3417 0.4558 0.5743 0.6559 0.7299 0.7904

0.99618 0.98951 0.98600 0.98256 0.97960 0.97680 0.97358 0.97034 0.96481 0.94562 0.92563 0.90695 0.89023 0.86755 0.84830 0.83694 0.82780 0.82090

y0.7697 y0.8249 y0.8484 y0.8694 y0.8842 y0.8960 y0.9095 y0.9205 y0.9346 y0.9559 y0.9442 y0.9119 y0.8664 y0.7797 y0.6692 y0.5786 y0.4935 y0.4122

y0.7695 y0.8226 y0.8458 y0.8657 y0.8804 y0.8926 y0.9049 y0.9152 y0.9291 y0.9473 y0.9365 y0.9117 y0.8736 y0.7752 y0.6534 y0.5727 y0.4878 y0.3994

0.0000 y0.0692 y0.1004 y0.1280 y0.1493 y0.1664 y0.1865 y0.2034 y0.2297 y0.2940 y0.3308 y0.3506 y0.3599 y0.3607 y0.3412 y0.3136 y0.2855 y0.2507

0.0000 y0.0694 y0.1003 y0.1276 y0.1487 y0.1669 y0.1863 y0.2037 y0.2301 y0.2939 y0.3307 y0.3503 y0.3604 y0.3608 y0.3403 y0.3151 y0.2844 y0.2510

f m s 0.3195 0.3195 0.3124 0.3094 0.3057 0.3019 0.2984

0.0000 0.0222 0.0318 0.0434 0.0553 0.0661

0.99012 0.98272 0.97954 0.97575 0.97185 0.96830

y1.0657 y1.0885 y1.0942 y1.1013 y1.1048 y1.1033

y1.0643 y1.0851 y1.0912 y1.0964 y1.0995 y1.1006

0.0000 y0.0460 y0.0627 y0.0819 y0.0976 y0.1082

0.0000 y0.0468 y0.0634 y0.0809 y0.0964 y0.1086

T.-C. Bai et al.r Fluid Phase Equilibria 152 (1998) 283–298

292

Table 4 Žcontinued. x1

x2

r Žg cmy3 .

E Vm.123 Žcm3 moly1 .

E Vm.123 Žcm3 moly1 . Eq. Ž10.

E Vm.2q13 Žcm3 moly1 .

E Vm.2q13 Žcm3 moly1 . PBMA

f m s 0.3195 0.2960 0.2923 0.2867 0.2661 0.2427 0.2173 0.1861 0.1525 0.1099 0.0952 0.0680 0.0567 s Ž VmE . Žcm3 moly1 .

0.0738 0.0854 0.1029 0.1671 0.2405 0.3201 0.4175 0.5228 0.6561 0.7022 0.7872 0.8225

0.96582 0.96212 0.95665 0.93767 0.91820 0.89950 0.87940 0.86059 0.84030 0.83405 0.82330 0.81906

y1.1031 y1.1006 y1.0963 y1.0601 y1.0036 y0.9306 y0.8279 y0.7075 y0.5459 y0.4894 y0.3756 y0.3205

y1.1006 y1.0992 y1.0948 y1.0619 y1.0059 y0.9304 y0.8224 y0.7033 y0.5581 y0.5002 y0.3725 y0.3142 0.0090 a

y0.1160 y0.1264 y0.1401 y0.1726 y0.1937 y0.2066 y0.2074 y0.1986 y0.1797 y0.1721 y0.1484 y0.1314

y0.1163 y0.1268 y0.1401 y0.1726 y0.1937 y0.2061 y0.2081 y0.1980 y0.1800 y0.1722 y0.1481 y0.1316 0.0015 b

f m : The mole fraction of DMF in DMFqwater mixed solvent. Standard deviation for fitting excess molar volumes by Eq. Ž10.. b E The mean value of standard deviations of all the pseudo-binary mixtures for fitting the excess molar volumes Vm.2q13 .

a

Another type of equations is of the form that introduces ternary effect terms. Equations are correlation over entire ternary concentration area. Cibulka w14x proposed the following expression, VmE.123 Ž cm3 moly1 . s VmE.12 Ž x 1 , x 2 . q VmE.13 Ž x 1 , x 3 . q VmE.23 Ž x 2 , x 3 . q x 1 x 2 Ž 1 y x 1 y x 2 . = Ž C1 q C2 x 1 q C3 x 2 q C4 x 12 q C5 x 22 q C6 x 1 x 2 q . . . .

Ž8.

E Ž x i , x j . were calculated by Eq. Ž 1. at composition Ž x i , x j . and using values of Bk from where Vm.i j Table 3. Ci is the fitting parameter of ternary effect.

Table 5 Comparison of the standard deviations Ž s . for fitting the excess molar volumes of the ternary mixture by several equations Equation

s Ž VmE . Žcm3 moly1 .

Eq. Ž4. Eq. Ž5. Eq. Ž6. 1Žwater. 2ŽDMF. 3Ž1-propanol. Eq. Ž7. 1ŽDMF. 2Ž1-propanol. 3Žwater. Eq. Ž8. Eq. Ž9. 1ŽDMF. 2Ž1-propanol. 3Žwater. Eq. Ž10. 1Žwater. PBMAa

0.015 0.021 0.027 0.023 0.0099 0.011 0.0090 0.0015

a

The mean values of standard deviations of several pseudo-binary mixtures.

T.-C. Bai et al.r Fluid Phase Equilibria 152 (1998) 283–298

293

Singh et al. w15x proposed an equation of the form VmE.123 Ž cm3 moly1 . s VmE.12 Ž x 1 , x 2 . q VmE.13 Ž x 1 , x 3 . q VmE.23 Ž x 2 , x 3 . q x 1 x 2 Ž 1 y x 1 y x 2 . = C1 q C2 x 1 Ž x 2 y x 3 . q C3 x 12 Ž x 2 y x 3 .

ž

3

qC4 x 13 Ž x 2 y x 3 . q . . .

2

Ž9.

/

However, Jasinski and Malanowski w16x suggested an equation of the form VmE.123 Ž cm3 moly1 . s VmE.12 Ž x 1 , x 2 . q VmE.13 Ž x 1 , x 3 . q VmE.23 Ž x 2 , x 3 . q x 1 x 2 2

= Ž 1 y x 1 y x 2 . C1 q C 2 Ž 2 x 1 y 1 . q C 3 Ž 2 x 1 y 1 . q . . .

ž

/

Ž 10.

In Eqs. Ž 8. – Ž10., ternary effect parameters Ci are obtained by the optimization method by Powell w8x, their values are given in Table 6. E ., of fits by predictive equations, Eqs. Ž4. – Ž 7. , and by correlative The standard deviations, s Ž Vm.123 Ž . Ž . equations, Eqs. 8 – 10 are reported in Table 5. Among predictive equations, Eq. Ž4. gives the lower value. For correlative equations, Eq. Ž10. gives the lower one. In general, the standard deviations of the fits are higher than those of binary fits ŽTable 3. . The difference between the results by these two types of equations are not significant. But, the accuracy of the fits can be examined by comparing E values of the standard deviation with the minimum Ž or maximum. values of Vm.123 , as suggested by E E Pando et al. w17x. The typical ratio of s Ž Vm.123 .rVm.123.min for Eq. Ž4. is 1.4%, and for Eq. Ž10. is 0.81%. By comparing these results with that of Zielkiewicz’s w7x at 313.15 K, it is found that our fits E .s can be considered as acceptable. The best results at 313.15 K for predictive equation is s Ž Vm.123 E Ž . 0.013, correspondingly the ratio is 1.5%, and for correlative method s Vm.123 is 0.0068, correspondingly the ratio is 0.80%. Apparently, the results of correlative equation is better than the one of E . for correlative equation at predictive equations in the case of 313.15 K. Despite the fact that s Ž Vm.123 298.15 K is poor than at 313.15 K, the ratios are very close each other. For comparison with the E experimental results, the values of Vm.123 calculated by Jasinski and Malanowski’s equation, Eq. Ž 10. , are given in Table 4. The asymmetrical equations give more weight to the binary constituents 1–2 and 1–3, and therefore component 1 plays the more important role. The rule for selecting the numbering of the component has been given by Pando et al. w17x. Instead of looking for the most dissimilar component in the ternary mixture, it is necessary to examine the three binary curves involved and looking for the Table 6 Fitting parameters Ž Ci . of Eqs. Ž8. – Ž10.

Ci C2 C3 C4 C5 C6

Eq. Ž8.

Eq. Ž9. 1ŽDMF. 2Ž1-propanol. 3Žwater.

Eq. Ž10. 1Žwater.

y2.882855 y10.41207 12.65332 43.62748 y13.33908 y3.783421

y0.906166 4.316511 y1.448286

y0.816560 y2.066322 y0.746483 16.56666 y6.516324 y35.23529

294

T.-C. Bai et al.r Fluid Phase Equilibria 152 (1998) 283–298

two binary which exhibit the two larger absolute value of excess property in their maxima or minima. The common component of these two mixtures will be designated as component 1. For DMFq 1propanolq water ternary mixture, the two larger absolute value of VmE correspond to the minima of DMF q water and 1-propanolq water. This means that following this rule water must be component 1, which coincides with the results of Eqs. Ž 6. and Ž 10. in Table 5. But the minimum deviation prefer DMF as component 1 to water in Eqs. Ž 7. and Ž 9. . The reason for this might be the molecular interaction intensified at the lower temperature, so that it is hard to get an perfect ternary fits. The temperature effect on VmE for ternary mixture can be observed by compare our results at 298.15 K ŽTable 4. with the literature value at 313.15 K Ž from the work of Zielkiewicz w7x. . The minima value of VmE at 313.15 K is y0.8550 cm3 moly1, where the composition is x 1 s 0.1678 Ž for DMF. and x 2 s 0.2551 Žfor 1-propanol. . In the neighborhood of this composition, Ž x 1 s 0.1538 and x 2 s 0.2031. , the value at 298.15 K is y0.9442 cm3 moly1. The minima value of VmE at 298.15 K is y1.1048 cm3 moly1, where x 1 s 0.3019 and x 2 s 0.0553. These data indicate that when temperature rising, the absolute magnitude of the minima of VmE is decreased and its composition Ž x 1, x 2 . is shifted. 3.3. Pseudo-binary mixture approach In measuring densities of the ternary systems, the mixtures were made of several pseudo-binary mixtures in which component 2 Ž 1-propanol. , was added to binary mixture of component 1 Ž DMF. and 3 Žwater. having a fixed composition f m , where f m s x 1rŽ1 y x 2 .. There are three reasons for us to use the PBMA to analyze the volume data. Firstly, in DMFq 1-propanolq water mixture, molecular interaction give rise to asymmetric volume effect, which cause the standard deviation of ternary fits ŽTable 5. larger than those of binaries Ž Table 3. . Among their three binary mixtures, DMF q water exhibit the largest negative VmE. So, we paid more attention to observe the effect of composition changing in DMFq water mixed solvents on the ternary excess volumes, and to explore improved approaches along the pseudo-binary line. Secondly, aqueous solution of DMF is a model mixed solvent to represent an environment of protein’s interiors. Alcohols are model molecules of study the hydrophobic interactions. The interactions occurring between solutes in water and in amidic solvent are different. It is our goal to study the effect of medium from water to amidic solvent on the change of thermodynamic properties of alcohols. Thirdly, in aqueous solution of some amides Ž DMF, N-methylformamide and N, N-dimethylacetamide. , a minima value of partial molar volume of amides exist in the water-rich area. ŽDavis and Hernandez w18x. In this region, some special molecular interactions, such as hydrogen bond interactions and hydrophobic interactions, have observable E shift toward the water-rich influence on the volume effect. For ternary mixture, the minima of Vm.123 region. So, we measured more data in this region and look for some information. In the ternary mixture, the molar mass M13 of DMF q water mixed solvent at a fixed composition f m was the mean molar mass of its constituents, M13 s f m M1 q Ž 1 y f m . M3 .

Ž 11 .

According to mass conservation, x 1 M1 q x 2 M2 q Ž 1 y x 1 y x 2 . M3 s Ž 1 y x 20 . M13 q x 20 M2 .

Ž 12.

T.-C. Bai et al.r Fluid Phase Equilibria 152 (1998) 283–298

295

Note that the mole fraction x 20 of alcohol in the pseudo-binary mixture is equal to x 2 , where M1, M2 E and M3 are molar mass of component, respectively. The excess molar volume Vm.2q13 was calculated from measured densities by Eq. Ž13., y1 VmE.2q13 s Ž 1 y x 2 . M13 Ž ry1 y ry1 y ry1 Ž 13. 13 . q x 2 M 2 Ž r 2 . where r 13 is the density of DMFq water mixed solvent at its fixed composition. The values of r and E E E Vm.2q13 are given in Table 4. The values of Vm.123 can be obtained from Vm.2q13 by Eq. Ž14.. E E E Vm .123 s Vm .2q13 q Ž 1 y x 2 . Vm .13 Ž 14. E where Vm.13 is the excess molar volumes of the binary mixture, which can be found in Table 4 where E x 2 is zero. The values of Vm.2q13 were fitted by Eq. Ž1. just as the binary mixtures. The fitting E E . for fitting the values of Vm.2q13 parameters are given in Table 7. The standard deviations s Ž Vm.2q13 for several pseudo-binary mixtures are included in Table 7. The mean values of the standard deviations for all the pseudo-binary mixtures Ž marked by PBMA. is compared with the results of other correlative equations in Table 5. It shows clearly that PBMA gives the minimum value of the E calculated by PBMA from the parameters in Table 7 are standard deviation. The values of Vm.2q13 included in Table 4. They are in good agreement with the experimental data. In Fig. 2, one of the curves shows the dependence of B1 Žin Table 7. on f m . The meaning of B1 is E the excess volume Vm.2q13 when x 2 s 0.5. The curve increase with the increase of f m . This is an indication that the interaction between 1-propanol and water q DMF mixed solvent become weaker as f m increases. 3.4. Partial molar Õolumes for 1-propanol at infinite dilution in pseudo-binary mixture E are smooth functions of x 2 . By using the parameters in Table 7, In our mixture, the values Vm.2q13 the partial molar volumes V2 of 1-propanol in pseudo-binary mixtures can be calculated by Eq. Ž15.. V2 s VmE.2q13 q V20 q Ž 1 y x 2 . Ž E VmE.2q13rE x 2 . T .P . Ž 15 . ` When x 2 approaches to zero, the partial molar volume of 1-propanol at infinite dilution Ž V2 . is obtained. The properties of solute at infinite dilution reflects, at least to a good approximation, how the solute interacts with the solvent. The values of V2` in DMF q water mixed solvents with fixed composition f m were evaluated and listed in Table 8. Fig. 2 present the values of excess partial molar volumes at infinite dilution, V2`E , as a function of f m . The values of V2`E decease firstly with f m

Table 7 Parameters Ž Bk . of Eq. Ž1. and the standard deviations Ž s . for fitting the pseudo-binary mixtures w f m DMFqŽ1y f m .waterx q1-propanol4 at their several fixed composition f m fm

0.0117

0.0246

0.0529

0.0875

0.1321

0.1930

0.3195

B1 B2 B3 B4 B5 B6 E . s Ž Vm.2q13 Žcm3 moly1 .

y2.539570 0.584322 y0.858232 2.375453 y2.282022 y1.963954 0.0028

y2.495718 0.528926 y0.148361 2.059746 y3.431980 y1.486473 0.0027

y2.304893 0.763052 y0.758011 y0.099864 y1.911044 1.743811 0.0015

y2.106212 0.502988 y0.793564 0.379894 y1.410482 1.416956 0.0018

y1.762990 0.542470 y0.511725 0.204468 y1.894520 0.577919 0.00054

y1.422181 0.304823 y0.669693 0.115702 y1.012120 0.729832 0.00067

y0.803329 0.236710 y0.568856 y0.511220 y0.084699 1.234704 0.00065

296

T.-C. Bai et al.r Fluid Phase Equilibria 152 (1998) 283–298

E . of Fig. 2. Excess partial molar volumes of 1-propanol at infinite dilution Ž V2`E ., I; excess partial molar volumes Ž VDMF DMF in DMFqwater binary mixture, `; and fitting parameters B1 for pseudo-binary mixture, ^ as a function of f m .

increase, and then reach to a minima, where f m s 0.0529. After that minima, V2`E increase with f m E increase. Another curve in Fig. 2 shows the excess partial molar volumes of DMF, V DMF , in DMF q water binary mixture as a function of f m . It has a minima point at the region where f m is about 0.072. The interesting thing is that this minima corresponding to the minima of V2`E in its concentration neighborhood. In this range, DMFq water mixed solvent is in a closely packing state E w5,18x. The curve of VDMF is a reflection of DMF hydration, while the values of V2`E are reflections for the interactions of the solvated solute Ž1-propanol. with the DMFq water mixed solvent. The E at different f m is a reflection of the correlation between solute solvation dependence of V2`E on VDMF E and solute ŽDMF. –solute Ž1-propanol. interactions. The dependence of B1 curve on the V DMF curve as while as on f m is a reflection of the effect of medium on 1-propanol’s property at different concentration region.

Table 8 The partial molar volumes for 1-propanol at infinite dilution Ž V2`. in w f m DMFqŽ1y f m .waterxq1-propanol4 pseudo-binary mixture at their several fixed composition f m fm

V2` Žcm3 moly1 .

0.0 0.0117 0.0246 0.0529 0.0875 0.1321 0.1930 0.3195 1.0

70.131 68.477 67.974 67.771 68.542 69.658 70.898 72.735 73.220

T.-C. Bai et al.r Fluid Phase Equilibria 152 (1998) 283–298

297

Table 9 Parameters Ž A i . of Eq. Ž16. for fitting partial molar volumes of 1-propanol at infinite dilution Ž V2`., the standard deviations Ž s . and the correlative coefficients Ž r . of the fitting A1 A2 A3 A4 A5 s Žcm3 moly1 . r

70.2085 y28.4883 102.854 y87.3043 15.9475 0.24 0.9964

An empirical equation of the form V2` Ž cm3 moly1 . s

Ý A i fmŽiy1.r2

Ž 16.

is1

was used to fit the values of V2` with f m . The fitting parameters A i in Eq. Ž16., the standard deviations and the correlation coefficients of the fitting are given in Table 9. The calculated values by Eq. Ž16. are in better agreement with the experimental results. 4. Conclusion Excess molar volumes for ternary mixture DMFq 1-propanolq water and for binary constituents, DMF q water, DMF q 1-propanol and 1-propanolq water at 298.15 K are obtained from density measurement. Several empirical expressions were used to predict and correlate the ternary excess molar volumes from experimental results on the constituent binaries. By comparison of the results of ternary and binary fits, an asymmetric effect of binary constituent on ternary volume was noticed. The complexity of molecular interactions causes the standard deviations of ternary fits higher than those of binaries. By comparison our data at 298.15 K with that of Zielkiewicz at 313.15 K, temperature effect was observed. In lower temperature, molecular interaction is intensified, and the asymmetric volume effect on ternary mixture become more observable. In order to explore some information about molecular interaction, and some relationship between volume properties and ternary composition, a PBMA was proposed to analyze the system studied. The partial molar volumes of 1-propanol at infinite dilution in DMFq water mixed solvents at their several fixed composition f m were evaluated and correlated with the composition f m . The correspondence of V2`.E , PBMA parameter B1 and excess partial molar volume of DMF in DMFq water binary E mixture, VDMF , on the composition f m were observed. These are reflections about the effect of medium on 1-propanol’s property at different concentration region. 5. List of symbols A B C F fm

Adjusting parameters in Eq. Ž 16. Adjusting parameters for binary system and for pseudo-binary system Adjusting parameters for ternary system Objective function Mole fraction of DMF in DMF q water mixed solvent

T.-C. Bai et al.r Fluid Phase Equilibria 152 (1998) 283–298

298

Mi m n Vi Vi` Vi`E VmE xi x i0

Molar mass of component i Number of experimental data point Number of parameters used in fitting equation Partial molar volume of component i Partial molar volume of component i at infinite dilution Excess partial molar volume of component i at infinite dilution Excess molar volume Mole fraction Mole fraction scale used in pseudo-binary mixture and fitting expressions

Subscripts 1,2,3 2 q 13

Component 1 Ž DMF. , 2 Ž1-propanol. , 3 Ž water. Pseudo-binary mixture composed of Ž1 y x 2 .w f mŽDMF. q Ž1 y f m .waterx qx 2 Ž1-propanol.

Greek letters s Standard deviation r Density n Mole fraction scale used in Eq. Ž7., n i j s Ž1 q x i y x j . r2 Acknowledgements We thank the Natural Science Foundation of Zhejiang Province for their financial support.

References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x

T.H. Lilley, in: M.N. Jones ŽEd.., Biochemical Thermodynamics, Elsevier, Amsterdam, 1988, pp. 1–52. G. Perron, J.E. Desnoyers, J. Chem. Thermodyn. 13 Ž1981. 1105–1121. G.C. Benson, O. Kiyohara, J. Solution Chem. 9 Ž1980. 791–804. R.C. Wilhoit, B.J. Zwolinski, J. Phys. Chem. Ref. Data 2 Ž1973. 1, Suppl. D.Y. Chu, Y. Zhang, I.Y. Hu, R.L. Liu, Acta Physico-Chimica Sinica 6 Ž1990. 203–208, in Chinese. M.I. Davis, Thermochimica Acta 120 Ž1987. 299–314. J. Zielkiewicz, J. Chem. Thermodyn. 27 Ž1995. 225–230. M.J.D. Powell, Comput. J. 7 Ž1964. 155–162. X. Esteve, K.R. Patil, J. Fernandez, A. Coronas, J. Chem. Thermodyn. 27 Ž1995. 281–292. O. Redlich, A.T. Kister, Ind. Eng. Chem. 40 Ž1948. 345. F. Kohler, Monatsch. Chem. 91 Ž1960. 738. C.C. Tsao, J.M. Smith, Applied thermodynamics, Chem. Eng. Prog. Symp. Ser. 49 Ž1953. 107. M. Hillert, Calphad 4 Ž1980. 1–12. I. Cibulka, Collection of Czechoslovak Chem. Commun. 47 Ž1982. 1414–1419. P.P. Singh, R.K. Nigam, S.P. Sharma, S. Aggarwal, Fluid Phase Equilibria 18 Ž1984. 333–344. B. Jasinski, S. Malanowski, Chem. Eng. Sci. 25 Ž1970. 913–920. C. Pando, J.A.R. Renuncio, J.A.G. Calzon, J.J. Christensen, R.M. Izatt, J. Solution Chem. 16 Ž1987. 503–527. M.I. Davis, M.E. Hernandez, J. Chem. Eng. Data 40 Ž1995. 674–678.