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Excess molar volumes for binary and ternary mixtures of (N,N-dimethylformamide+ethanol+water) at the temperature 298.15 K
J. Chem. Thermodynamics 1998, 30, 1347]1361 Article No. ct980402
Excess molar volumes for binary and ternary mixtures of ( N, N-dimethylformamide H ethanol H water) at the temperature 298.15 K Tong-Chun Bai, Jia Yao, and Shi-Jun Han a Chemistry Department, Zhejiang Uni¨ ersity, Hangzhou, 310027, China
The results of excess molar volumes for the ternary mixture N, N-dimethylformamide ŽDMF. q ethanol q water4, and for the binary constituents ŽDMF q water., ŽDMF q ethanol., and Žethanol q water. at T s 298.15 K were reported. Several empirical expressions were used to predict and correlate the ternary excess molar volumes. A pseudo-binary mixture approach was used to analyse the system studied. The partial molar volumes of ethanol at infinite dilution, and the apparent molar volumes of ŽDMF q water. mixed solvent at several fixed compositions were evaluated and correlated with the composition of the ŽDMF q water. mixed solvent. q 1998 Academic Press KEYWORDS: excess molar volume; mixture; ethanol; N, N-dimethylformamide; water
1. Introduction The thermodynamic properties of solution containing amides, amino acids, peptides, and their derivatives are of interest. The correlation between solute hydration and solute]solute interaction is complex.Ž1,2. In view of the fact that the interactions occurring between solutes in water and in amidic solvent are different, it would be worthwhile to explore the effect of changing solvent medium on molecular interactions. The aqueous solution of N, N-dimethylformamide ŽDMF. is a model mixed solvent representing the environment of the interior of proteins.Ž2. Alcohols are model molecules for the study of hydrophobic interactions. In ŽDMF q water. mixing solvents, DMF]water association exists and changes with composition.Ž3. In the ŽDMF q alcohol q water. ternary mixture, the properties of alcohol are affected by this DMF]water association. It is of interest to study the effect of changing the medium from water to amidic solvent on the thermodynamic properties of alcohol. Zegers and SomsenŽ4. reported the volumetric properties of ŽDMF q alcohol. mixtures. In order to give a better description of the solvation of a
To whom correspondence should be addressed.
0021]9614r98r111347 q 15 $30.00r0
q 1998 Academic Press
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T.-C. Bai, J. Yao, and S.-J. Han
DMF by water and by alcohol molecules in the ternary mixture ŽDMF q alcohol q water., Zielkiewicz Ž3,5,6. published a series of papers on the excess molar volumes of the ternary mixtures at T s 313.15 K. In the present paper, the results of excess molar volumes for the ternary mixture ŽDMF q ethanol q water., and for the binary constituents ŽDMF q water., ŽDMF q ethanol., and Žethanol q water. at T s 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. In order to obtain some information about the correlation between alcohol]amidic interactions with amide solvation, a pseudo-binary mixture approach was applied to the system studied. In this approach, the alcohol was added to ŽDMF q water. mixed solvents with fixed composition. The excess molar volumes for these pseudo-binary mixtures were evaluated. It is our goal to obtain some information about the changing excess molar volumes with ŽDMF q water. composition. The partial molar volumes of ethanol at infinite dilution, and the apparent molar volumes of ŽDMF q water. mixed solvent at several fixed compositions were evaluated and correlated with the composition of the ŽDMF q water. mixed solvent.
2. Experimental The DMF Žanalytical reagent grade, Shanghai Chemical Co.. was dried over freshly activated Al 2 O 3 for at least 48 h and then fractionally distilled under reduced pressure. The product was stored over freshly activated molecular sieve 0.4 nm. Ethanol Žanalytical regent grade, Shanghai Chemical Co.. was dried over freshly activated CaO for 2 days and fractionally distilled. The product was stored over freshly activated 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. Ternary mixtures were prepared by mixing a measured binary mixture ŽDMF q water at known composition. with a pure liquid Žethanol, as the third component.. All mixtures were prepared by mass. The measured physical properties of the pure substance together with literature Ž5,7 ] 9. values are included in table 1.
TABLE 1. Densities and refractive indices of pure liquids at T s 298.15 K Component
DensityrŽg . cmy3 . this work
DMF
0.94403
Ethanol
0.78498
Refractive indices
literature Ž7.
0.944061 0.94383Ž8. 0.784962 Ž9. 0.78509Ž9. 0.78493Ž9.
this work
literature
1.4282
1.42850 Ž5. 1.4282 Ž7. 1.35941Ž9. 1.35917 Ž5. 1.35937 Ž5.
1.3594
Excess molar volumes of ŽDMF q ethanol q water.
1349
3. Results and discussion EXCESS MOLAR VOLUMES FOR BINARY MIXTURES
Excess molar volumes VmE were calculated from densities r . Results of VmE and r for binary mixtures are given in table 2. A function of the form: VmE r Ž cm3 . moly1 . s x Ž 1 y x .
n
Ý
Bi Ž 2 x y 1 .
iy1
,
Ž 1.
is1
was fitted to the experimental results. Parameters of equation Ž1. were obtained by using Powell’sŽ10. optimization method with equation Ž2. as the objective function: m
Fs
1r2 2
Ý VmE , expt Ž x i . y VmE , calc Ž x i . 4 r Ž m y n .
,
Ž 2.
is1
where m is the number of experimental data points and n the number of parameters used in equation Ž1.. Parameters of equation Ž1. and the standard deviations of the fitting s are given in table 3. Figure 1 shows a comparison of the values of VmE for three binary mixtures xDMF q Ž1 y x . water4 , xethanol q Ž1 y x . water4 , and xDMF q Ž1 y x . ethanol4 at T s 298.15 K with the literature values at T s 313.15 K ŽZielkiewicz Ž5. .. Both binary systems ŽDMF q water. and Žethanol q water. have nearly the same magnitude of VmE values, while for those of ŽDMF q ethanol. are of relatively smaller magnitude. These are indications that the molecular interactions between ethanol]water and DMF]water are stronger than that of DMF]ethanol. A temperature effect on VmE is observed in figure 1. The values of binary VmE for ŽDMF q ethanol. at T s 298.15 K have greater absolute magnitudes than those at T s 313.15 K over the whole composition region. For the systems ŽDMF q water. and Žethanol q water., both have greater absolute magnitude at T s 298.15 K than at T s 313.15 K in the middle concentration region. This indicates that at lower temperature, due to lower molecular thermal agitation, the associations or interactions Žand thus the molecular packing. between the components become stronger. For Žethanol q water. and ŽDMF q water. systems, the temperature effect is hardly observed in the water-rich region; this might be attributed to the hydrophobic interaction and hydrogen bond interaction between the solute and water. The competition of molecular interaction among components and the temperature effect give rise to an asymmetric effect when the ternary mixture is composed of these constituents at T s 298.15 K. EXCESS MOLAR VOLUMES FOR TERNARY MIXTURES
Excess molar volumes were calculated from densities by using the following equation: y1 y1 VmE, 123 s x 1 M1 Ž ry1 y ry1 y ry1 y ry1 1 . q x 2 M2 Ž r 2 . q x 3 M3 Ž r 3 .,
Ž 3.
1350
T.-C. Bai, J. Yao, and S.-J. Han TABLE 2. Densities r and excess molar volumes VmE for binary mixtures at T s 298.15 K
x
rrŽg . cmy3 .
VmE rŽcm3 . moly1 .
x
rrŽg . cmy3 .
VmE rŽcm3 . moly1 .
xDMF q Ž1 y x . water 0.0055 0.0112 0.0142 0.0241 0.0319 0.0382 0.0464 0.0553 0.0620 0.0721 0.0805 0.0907 0.0914 0.1011 0.1125 0.1230 0.1347 0.1478 0.1489 0.1621
0.99667 0.99649 0.99640 0.99620 0.99616 0.99614 0.99620 0.99630 0.99640 0.99657 0.99670 0.99685 0.99686 0.99698 0.99708 0.99713 0.99712 0.99704 0.99703 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
0.0063 0.0123 0.0174 0.0240 0.0305 0.0365 0.0418 0.0469 0.0528 0.0728 0.1159 0.1696 0.2385 0.3224
0.99402 0.99140 0.98927 0.98670 0.98432 0.98220 0.98036 0.97876 0.97690 0.97075 0.95815 0.94223 0.92190 0.89882
y0.0236 y0.0485 y0.0711 y0.1033 y0.1355 y0.1675 y0.1946 y0.2235 y0.2557 y0.3641 y0.5806 y0.7887 y0.9574 y1.0586
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
0.87550 0.84478 0.83780 0.82517 0.81151 0.81136 0.80319 0.80287 0.79644 0.79637 0.79442 0.79273 0.79240 0.78828
y1.0982 y1.0324 y0.9880 y0.8675 y0.6712 y0.6599 y0.5087 y0.5015 y0.3473 y0.3455 y0.2932 y0.2455 y0.2360 y0.1080
0.88822 0.87887 0.86822 0.85777 0.84823 0.83899 0.82820 0.81950 0.81040 0.80160
y0.2498 y0.2557 y0.2589 y0.2567 y0.2490 y0.2390 y0.2189 y0.1980 y0.1688 y0.1298
xethanol q Ž1 y x . water 0.4205 0.5787 0.6199 0.6993 0.7938 0.7974 0.8555 0.8580 0.9078 0.9084 0.9238 0.9372 0.9398 0.9732
xDMF q Ž1 y x . ethanol 0.9366 0.9102 0.8778 0.8498 0.8191 0.7889 0.7583 0.7317 0.7032 0.6381
0.93735 0.93425 0.93035 0.92692 0.92310 0.91928 0.91536 0.91188 0.90810 0.89917
y0.0881 y0.1042 y0.1214 y0.1356 y0.1504 y0.1656 y0.1815 y0.1935 y0.2071 y0.2323
0.5623 0.5007 0.4333 0.3700 0.3145 0.2627 0.2047 0.1597 0.1145 0.0726
where M1 , M2 , and M3 are molar masses of pure components 1, 2, and 3; and r 1 , r 2 , and r 3 are densities of the pure components, and r is the density of the
Excess molar volumes of ŽDMF q ethanol q water.
1351
TABLE 3. Fitting parameters Bi of equation Ž1. and the standard deviations s of the fitting for the excess molar volumes of binary mixtures investigated xDMF q Ž1 y x . ethanol xDMF q Ž1 y x . water xethanol q Ž1 y x . water B1 B2 B3 B4 B5 B6 B7 B8 B9 E s Ž Vm .rŽcm3 . moly1 .
y1.025633 0.138920 y0.432249 0.648744 0.633981 y0.566669 y1.693487 0.00051
y4.383917 1.758919 0.283819 y1.800046 0.560583 1.760020 0.447975 y0.894894 1.039060 0.00042
y4.347337 0.816881 y0.935253 1.588231 y2.013010 y1.357381 3.415028 y1.375997 0.000747
mixture. The values of r and Vm,E 123 for ternary mixtures are given in table 4. Several empirical methods have been suggested by Esteve et al.Ž11. to estimate ternary excess properties from experimental results on the constituents. Some of the predictive methods were originally proposed to predict molar enthalpy, or volume, or Gibbs free energy.Ž12. 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 effect. The ternary excess volume Vm,E 123 is the sum of the
FIGURE 1. The excess molar volumes VmE for three binary mixtures at T s 298.15 K and T s 313.15 K as a function of mole fraction. For xDMF q Ž1 y x . water4: I, T s 298.15 K; curve 1, T s 313.15 K; for xethanol q Ž1 y x . water4: `, T s 298.15 K; curve 2, T s 313.15 K; and for xDMF q Ž1 y x . ethanol4: ^, T s 298.15 K; curve 3, T s 313.15 K.
1352
T.-C. Bai, J. Yao, and S.-J. Han
TABLE 4. Densities r , excess molar volumes for ternary mixture Vm,E 123 , and excess molar volumes for pseudo-binary mixture Vm,E 2q13 at T s 298.15 K, and the values calculated by the correlative equation Ž11. and by PBMA x1
x2
r . Ž g cmy3 .
VmE, 123 Ž cm3 . moly1 .
0.0117 0.0115 0.0114 0.0114 0.0113 0.0112 0.0112 0.0111 0.0108 0.0102 0.0095 0.0087 0.0077 0.0067 0.0050 0.0040 0.0033 0.0023
0.0000 0.0120 0.0192 0.0253 0.0315 0.0373 0.0434 0.0493 0.0760 0.1219 0.1860 0.2558 0.3411 0.4267 0.5673 0.6586 0.7206 0.8034
0.99648 0.99140 0.98880 0.98654 0.98439 0.98255 0.98058 0.97875 0.97084 0.95780 0.93905 0.91828 0.89484 0.87485 0.84750 0.83200 0.82236 0.81100
y0.0375 y0.0888 y0.1260 y0.1559 y0.1884 y0.2205 y0.2537 y0.2864 y0.4314 y0.6599 y0.8896 y1.0208 y1.0815 y1.1013 y1.0481 y0.9333 y0.8248 y0.6782
0.0246 0.0243 0.0240 0.0239 0.0236 0.0235 0.0232 0.0231 0.0227 0.0217 0.0203 0.0185 0.0164 0.0136 0.0101 0.0092 0.0086 0.0042
0.0000 0.0151 0.0250 0.0316 0.0410 0.0461 0.0580 0.0619 0.0775 0.1207 0.1754 0.2477 0.3350 0.4491 0.5919 0.6253 0.6526 0.8283
0.99620 0.99054 0.98714 0.98501 0.98200 0.98040 0.97680 0.97565 0.97105 0.95835 0.94214 0.92079 0.89706 0.87065 0.84354 0.83780 0.83327 0.80760
y0.0850 y0.1588 y0.2093 y0.2446 y0.2942 y0.3214 y0.3846 y0.4058 y0.4848 y0.6808 y0.8673 y1.0096 y1.0841 y1.0951 y1.0100 y0.9681 y0.9285 y0.5966
0.0529 0.0521 0.0515 0.0511 0.0506 0.0502 0.0498 0.0495 0.0486
0.0000 0.0164 0.0270 0.0341 0.0444 0.0528 0.0589 0.0658 0.0819
0.99627 0.99082 0.98744 0.98524 0.98204 0.97957 0.97770 0.97568 0.97095
y0.2017 y0.2845 y0.3387 y0.3740 y0.4249 y0.4677 y0.4962 y0.5294 y0.6031
VmE, 123 Ž cm3 . moly1 . equation Ž11.4
VmE, 2q13 Ž cm3 . moly1 .
VmE, 2q13 Ž cm3 . moly1 . ŽPBMA.
0.0000 y0.0518 y0.0892 y0.1194 y0.1522 y0.1844 y0.2179 y0.2508 y0.3968 y0.6270 y0.8591 y0.9930 y1.0568 y1.0798 y1.0319 y0.9205 y0.8143 y0.6709
0.0000 y0.0534 y0.0886 y0.1198 y0.1526 y0.1839 y0.2174 y0.2501 y0.3977 y0.6273 y0.8579 y0.9933 y1.0585 y1.0777 y1.0330 y0.9205 y0.8137 y0.6711
0.0000 y0.0751 y0.1264 y0.1623 y0.2127 y0.2403 y0.3045 y0.3260 y0.4064 y0.6061 y0.7971 y0.9456 y1.0276 y1.0483 y0.9753 y0.9363 y0.8989 y0.5820
0.0000 y0.0755 y0.1275 y0.1627 y0.2133 y0.2408 y0.3044 y0.3250 y0.4056 y0.6057 y0.7969 y0.9475 y1.0260 y1.0486 y0.9757 y0.9366 y0.8983 y0.5821
0.0000 y0.0861 y0.1425 y0.1792 y0.2322 y0.2767 y0.3064 y0.3409 y0.4179
0.0000 y0.0867 y0.1428 y0.1801 y0.2335 y0.2763 y0.3067 y0.3405 y0.4161
f m s 0.0117 y0.0370 y0.0894 y0.1240 y0.1550 y0.1871 y0.2176 y0.2507 y0.2824 y0.4250 y0.6427 y0.8614 y0.9988 y1.0787 y1.1037 y1.0433 y0.9318 y0.8252 y0.6450 f m s 0.0246 y0.0842 y0.1560 y0.2063 y0.2411 y0.2907 y0.3180 y0.3810 y0.4014 y0.4810 y0.6764 y0.8585 y1.0024 y1.0841 y1.1067 y1.0176 y0.9770 y0.9384 y0.5796 f m s 0.0529 y0.2018 y0.2833 y0.3371 y0.3732 y0.4254 y0.4672 y0.4966 y0.5297 y0.6027
Excess molar volumes of ŽDMF q ethanol q water.
1353
TABLE 4}continued x1
x2
0.0462 0.0428 0.0385 0.0343 0.0288 0.0216 0.0182 0.0131 0.0088
0.1269 0.1925 0.2730 0.3530 0.4567 0.5912 0.6557 0.7519 0.8341
r
Ž g . cmy3 .
VmE, 123 Ž cm3 . moly1 .
VmE, 123 Ž cm3 . moly1 . Žequation Ž11.4
VmE, 2q13 Ž cm3 . moly1 .
VmE, 2q13 Ž cm3 . moly1 . ŽPBMA.
0.95755 0.93804 0.91502 0.89418 0.87070 0.84500 0.83401 0.81899 0.80750
y0.7769 y0.9528 y1.0673 y1.1081 y1.1018 y1.0096 y0.9242 y0.7544 y0.5837
y0.7769 y0.9493 y1.0616 y1.1100 y1.1121 y1.0149 y0.9279 y0.7535 y0.5595
y0.6008 y0.7899 y0.9207 y0.9776 y0.9922 y0.9272 y0.8547 y0.7044 y0.5502
y0.6002 y0.7912 y0.9208 y0.9771 y0.9921 y0.9273 y0.8550 y0.7039 y0.5504
0.0000 y0.1030 y0.1403 y0.1812 y0.2320 y0.2655 y0.3079 y0.3476 y0.4164 y0.5905 y0.7365 y0.8501 y0.9057 y0.9084 y0.8422 y0.7749 y0.6737 y0.4713
0.0000 y0.1033 y0.1411 y0.1825 y0.2318 y0.2658 y0.3065 y0.3475 y0.4159 y0.5906 y0.7369 y0.8504 y0.9052 y0.9084 y0.8428 y0.7744 y0.6738 y0.4713
0.0000 y0.0902 y0.1429 y0.1811 y0.2186 y0.2584 y0.2887 y0.3249 y0.3673 y0.4923 y0.6521 y0.7456 y0.7995 y0.8083 y0.7506 y0.6714 y0.5854 y0.4479
0.0000 y0.0908 y0.1436 y0.1811 y0.2188 y0.2576 y0.2895 y0.3243 y0.3668 y0.4922 y0.6525 y0.7459 y0.7991 y0.8083 y0.7506 y0.6719 y0.5847 y0.4481
f m s 0.0875 0.0875 0.0857 0.0850 0.0843 0.0834 0.0827 0.0819 0.0811 0.0796 0.0752 0.0700 0.0631 0.0553 0.0448 0.0340 0.0283 0.0222 0.0130
0.0000 0.0202 0.0278 0.0363 0.0467 0.0541 0.0632 0.0727 0.0895 0.1402 0.1999 0.2789 0.3676 0.4881 0.6107 0.6764 0.7466 0.8514
0.99682 0.99051 0.98814 0.98554 0.98248 0.98020 0.97755 0.97468 0.96967 0.95460 0.93710 0.91503 0.89250 0.86600 0.84304 0.83200 0.82090 0.80560
y0.3550 y0.4508 y0.4854 y0.5233 y0.5704 y0.6012 y0.6404 y0.6768 y0.7396 y0.8957 y1.0205 y1.1061 y1.1302 y1.0901 y0.9803 y0.8898 y0.7637 y0.5241
0.1321 0.1295 0.1279 0.1267 0.1254 0.1241 0.1229 0.1216 0.1199 0.1141 0.1037 0.0938 0.0827 0.0670 0.0509 0.0399 0.0312 0.0209
0.0000 0.0197 0.0319 0.0410 0.0505 0.0607 0.0695 0.0795 0.0924 0.1363 0.2148 0.2902 0.3740 0.4926 0.6146 0.6977 0.7636 0.8417
0.99712 0.99114 0.98744 0.98468 0.98180 0.97875 0.97604 0.97308 0.96920 0.95613 0.93360 0.91344 0.89290 0.86710 0.84403 0.82990 0.81945 0.80770
y0.5458 y0.6253 y0.6713 y0.7046 y0.7369 y0.7711 y0.7966 y0.8274 y0.8627 y0.9637 y1.0807 y1.1330 y1.1412 y1.0852 y0.9609 y0.8364 y0.7144 y0.5343
y0.3551 y0.4503 y0.4856 y0.5249 y0.5715 y0.6034 y0.6416 y0.6799 y0.7428 y0.8964 y1.0142 y1.0973 y1.1309 y1.1024 y0.9844 y0.8863 0.7563 y0.5070 fm s 0.1321 y0.5455 y0.6254 y0.6729 y0.7069 y0.7409 y0.7760 y0.8045 y0.8354 y0.8724 y0.9757 y1.0878 y1.1367 y1.1479 y1.0981 y0.9678 y0.8373 y0.7099 y0.5254
1354
T.-C. Bai, J. Yao, and S.-J. Han TABLE 4}continued
r
Ž g . cm .
VmE, 123 Ž cm3 . moly1 .
0.99618 0.98917 0.98602 0.98315 0.98004 0.97640 0.97397 0.97050 0.96423 0.94740 0.93565 0.90654 0.88855 0.86340 0.83890 0.83141 0.82160 0.80870
y0.7967 y0.8422 y0.8724 y0.9008 y0.9252 y0.9554 y0.9740 y0.9991 y1.0407 y1.1237 y1.1602 y1.1812 y1.1529 y1.0601 y0.8924 y0.8215 y0.7111 y0.5278
0.0000 0.99012 0.0285 0.98265 0.0423 0.97905 0.0556 0.97551 0.0715 0.97128 0.0835 0.96813 0.0954 0.96505 0.1096 0.96130 0.1303 0.95600 0.2052 0.93730 0.2734 0.92115 0.3649 0.90060 0.4599 0.88040 0.5719 0.85808 0.7073 0.83335 0.7512 0.82583 0.8136 0.81543 0.8591 0.80800 s Ž VmE .rŽcm3 . moly1 .
y1.0657 y1.1064 y1.1237 y1.1346 y1.1470 y1.1552 y1.1630 y1.1689 y1.1780 y1.1933 y1.1896 y1.1556 y1.0797 y0.9461 y0.7410 y0.6656 y0.5449 y0.4439
x1
x2
0.1930 0.1884 0.1863 0.1844 0.1824 0.1799 0.1783 0.1760 0.1717 0.1601 0.1518 0.1298 0.1149 0.0918 0.0662 0.0578 0.0462 0.0302
0.0000 0.0239 0.0347 0.0448 0.0553 0.0679 0.0763 0.0883 0.1103 0.1703 0.2135 0.3273 0.4047 0.5246 0.6569 0.7007 0.7606 0.8433
y3
VmE, 123 Ž cm3 . moly1 . equation Ž11.4
VmE, 2q13 Ž cm3 . moly1 .
VmE, 2q13 Ž cm3 . moly1 . ŽPBMA.
0.0000 y0.0909 y0.1293 y0.1656 y0.1980 y0.2379 y0.2630 y0.2974 y0.3559 y0.4851 y0.5548 y0.6635 y0.6946 y0.6942 y0.6284 y0.5912 y0.5269 y0.4072
0.0000 y0.0907 y0.1293 y0.1640 y0.1986 y0.2382 y0.2634 y0.2978 y0.3560 y0.4851 y0.5542 y0.6637 y0.6949 y0.6938 y0.6285 y0.5913 y0.5268 y0.4072
0.0000 y0.0711 y0.1031 y0.1281 y0.1575 y0.1785 y0.1989 y0.2201 y0.2511 y0.3462 y0.4152 y0.4788 y0.5041 y0.4899 y0.4290 y0.4005 y0.3462 y0.2936
0.0000 y0.0716 y0.1017 y0.1284 y0.1580 y0.1788 y0.1984 y0.2206 y0.2509 y0.3461 y0.4153 y0.4786 y0.5044 y0.4897 y0.4291 y0.4002 y0.3466 y0.2934
f m s 0.1930 y0.7694 y0.8409 y0.8710 y0.8979 y0.9243 y0.9536 y0.9720 y0.9964 y1.0358 y1.1126 y1.1452 y1.1697 y1.1478 y1.0577 y0.8858 y0.8133 y0.6999 y0.5094 f m 0.3195 0.3195 0.3104 0.3060 0.3018 0.2967 0.2928 0.2891 0.2845 0.2779 0.2540 0.2322 0.2029 0.1726 0.1368 0.0935 0.0795 0.0596 0.0450
0.00802 a
y1.0646 y1.1031 y1.1195 y1.1338 y1.1489 y1.1588 y1.1676 y1.1766 y1.1870 y1.2013 y1.1895 y1.1455 y1.0715 y0.9514 y0.7572 y0.6792 y0.5508 y0.4417
0.00074 b
s mole fraction of DMF in the pseudo-binary mixture w f m DMF q Ž1 y f m .water4 q ethanolx. Standard deviations for fitting the excess molar volumes of the ternary mixture by using equation Ž11.. b The mean values of standard deviation for fitting the excess molar volumes by using the pseudobinary mixture approach. fm a
binary excess volumes of its constituents. The simplest expression is of the form: VmE, 123r 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 . , Ž 4 .
Excess molar volumes of ŽDMF q ethanol q water.
1355
where x 1 s 1 y x 2 y x 3 , and the values of VmE,i j Ž x i , x j . are calculated according to equation Ž5.: VmE, i jr Ž cm3 . moly1 . s x i x j
n
Ý Bk Ž x i , x j . ky 1 ,
Ž 5.
k
using values of Bk from table 3 and at the composition Ž x i , x j .. According to the Kohler expression suggested by Esteve et al.,Ž11. the excess molar volume for a ternary mixture is given by: VmE, 123r 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 . q 2
2
2 Ž x 2 q x 3 . VmE, 23 Ž x 20 , x 30 . ,
Ž 6.
in which Vm, i j denotes the excess molar volume for the binary mixture at composition Ž x i0 , x j0 ., such that x i0 s x irŽ x i q x j . s 1 y x j0 with VmE, i j Ž x i0 , x j0 . being calculated by equation Ž5.. Equations Ž4. and Ž6. 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. Contrarily, Tsao and Smith proposed an asymmetrical equation, suggested by Esteve et al.:Ž11. VmE, 123r 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 . ,
Ž 7.
in which x 20 s x 2rŽ1 y x 1 . s 1 y x 30 . Hillert also suggested an asymmetrical equation: Ž11. VmE, 123r 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 3r¨ 23¨ 32 . VmE, 23 Ž ¨ 23¨ 32 . ,
Ž 8.
where ¨ i j s Ž1 q x i y x j .r2. The standard deviations of the fits for excess molar volumes by using predictive equations equations Ž4. and Ž6]8.4 are reported in table 5. Another type of equation is of the form that introduces ternary effect terms. CibulkaŽ13. proposed the following expression: VmE, 123r 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 ??? . , Ž 9.
1356
T.-C. Bai, J. Yao, and S.-J. Han TABLE 5. Comparison of the standard deviations s for fitting the excess molar volumes for the ternary mixture by using several equations
s Ž VmE .rŽcm3 . moly1 .
Equations Equation Ž4. Equation Ž6. Equation Ž7. 1Žwater. 2ŽDMF. 3Žethanol. Equation Ž8. 1Žwater. 2ŽDMF. 3Žethanol. Equation Ž9. Equation Ž10. 1Žwater. 2ŽDMF. 3Žethanol. Equation Ž11. 1Žwater. PBMAa
0.019 0.031 0.030 0.013 0.0092 0.011 0.0080 0.00074
a The mean values of standard deviations for the pseudo-binary mixture at their several fixed compositions f m .
in which Vm,E i j Ž x i , x j . were calculated by using equation Ž5. at composition Ž x i , x j ., and using values of Bi from table 3. The Ci is the fitting parameter of ternary effect. Singh et al.Ž14. proposed an equation of the form: VmE, 123r 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 2
3
C3 x 12 Ž x 2 y x 3 . q C4 x 13 Ž x 2 y x 3 . q ??? 4 .
Ž 10 .
However, Jasinski and Malanowski Ž12. suggested an equation of the form: VmE, 123r 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 Ž 2 x1 y 1. q C3 Ž 2 x1 y 1. 2 q ??? 4 .
Ž 11 .
In equations Ž9]11., ternary effect parameters Ci were obtained by Powell’sŽ10. optimization method. The results are given in table 6. The standard deviations s Ž Vm,E 123 . of the fits by predictive equations equations TABLE 6. Fitting parameters Ci of equations Ž9]11.
C1 C2 C3 C4 C5 C6
Equation Ž9.
Equation Ž10. 1Žwater.2ŽDMF.3Žethanol.
Equation Ž11. 1Žwater.
y4.277903 42.665050 23.189790 y68.955990 y21.790030 y77.449910
1.549553 7.257435 1.740424 y173.2988 257.3791 y443.8122
1.520536 1.399926 4.366468 3.074258 y20.903420 y10.958790
Excess molar volumes of ŽDMF q ethanol q water.
1357
Ž4. and Ž6]8.4 and by correlative equations equations Ž9]11.4 are reported in table 5. In general, correlative equations give lower values of s Ž Vm,E 123 . than those of predictive expressions. For comparison with the experimental results, the values of Vm,E 123 calculated by Jasinski’s equation equation Ž11.4 give the lowest standard deviation among these fits. They are given in table 4. The asymmetrical equations give more weight to the binary constituent 1-2 and 1-3, and therefore component 1 plays the more important role. The rule for selecting the numbering of the components has been given by Pando et al.Ž15. Instead of looking for the most dissimilar component in the ternary mixture, it is necessary to examine the three binary curves involved and look for the two binaries which exhibit the two largest absolute values of excess property in their maxima or minima. The common component of these two mixtures will be designated as component 1. For ŽDMF q ethanol q water. the two larger absolute values of VmE correspond to the minima of ŽDMF q water. and Žethanol q water.. This means that following this rule water must be component 1, which coincides with the results in table 5. The temperature effect on Vm,E 123 for the ternary mixture can be observed by comparing our results at T s 298.15 K Žtable 4. with the literature value at T s 313.15 K. The minimum value of Vm,E 123 at T s 313.15 K is y1.0974 cm3 . moly1 , where the composition is x 1 s 0.2566 Žfor DMF. and x 2 s 0.1484 Žfor ethanol.. The minimum value of VmE123 at T s 298.15 K is y1.1933 cm3 . moly1 , where x 1 s 0.2540 and x 2 s 0.2052. These data indicate that as temperature rises the absolute magnitude of the minima for Vm,E 123 is decreased. PSEUDO-BINARY MIXTURE APPROACH
In measuring densities of the ternary system, mixtures were made of several pseudo-binary mixtures in which component 2 Žpure ethanol. was added to a binary mixture of component 1 ŽDMF. and 3 Žwater. have a fixed composition f m , where f m s x 1rŽ1 y x 2 .. There are three reasons for using a pseudo-binary mixture approach ŽPBMA. to analyse the volume data. First, in the ŽDMF q ethanol q water. mixture, molecular interactions give rise to asymmetric volume effects, which cause the standard deviation of ternary fits Žtable 5. to be larger than those of binary fits Žtable 3.. Among the three binary mixtures, ŽDMF q water. and Žethanol q water. show the largest and nearly similar negative values of VmE . The binary mixture of ŽDMF q ethanol. shows a relatively smaller value of VmE . As a result, we pay more attention to observe the effect of changing composition in ŽDMF q water. mixed solvents on the ternary excess volumes, and to explore improved approaches. Second, aqueous solutions of amides represent a model mixed solvent of the environment of the interior of protein.Ž2. Alcohols are model molecules to study hydrophobic interactions. The interactions occurring between solutes in water and in amidic solvents are different. It is our goal to obtain some information about the effect of changing medium from water to an amidic solvent on the thermodynamic properties of alcohols. Third, in aqueous solution of some amides ŽDMF, N-methylformamide, and N, N-dimethylacetamide., a minimum
1358
T.-C. Bai, J. Yao, and S.-J. Han
value for partial molar volume of amide exists in the water-rich region.Ž16. In this region, some special molecular interaction, such as hydrogen bond interaction and hydrophobic interaction, have an observable influence on the volume effect. For the ternary mixture, the minimum of Vm,E 123 shifts towards the water-rich region. Thus, we obtained more data in the water-rich region. In the ternary mixture, the molar mass M13 of the f m DMF q Ž1 y f m . water4 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 12 .
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 13 .
Note that the mole fraction x 20 of alcohol in the pseudo-binary mixture is equal to x 2 , where M1 , M2 , and M3 are the molar mass of pure components. The excess molar volume Vm,E 2q13 was calculated from measured densities by using equation Ž14.: y1 y ry1 VmE, 2q13 s Ž 1 y x 2 . M13 Ž ry1 y ry1 13 . q x 2 M2 Ž r 2 .,
Ž 14 .
where r 13 is the density of ŽDMF q water. mixed solvent at its fixed composition. The values of Vm,E 2q13 are given in table 4. The values of Vm,E 123 can be obtained from Vm,E 2q13 by using equation Ž15.: VmE, 123 s VmE, 2q13 q Ž 1 y x 2 . VmE, 13 ,
Ž 15 .
where Vm,E 13 is the excess molar volumes of the ŽDMF q water. binary mixture. The values of Vm,E 13 can be found in table 4 where x 2 is zero. The values of Vm,E 2q13 were fitted by using equation Ž1. for just the binary mixtures. The fitting parameters are given in table 7. The standard deviations of the fitting for Vm,E 2q13 for several pseudo-binary mixtures are included in table 7. The mean values of the standard deviations for all pseudo-binary mixtures marked by s ŽPBMA.4 are compared with other correlative equations in table 5. It shows clearly that the pseudo-binary mixture approach gives the minimum value of the standard deviation. For TABLE 7. Fitting parameters Bi of equation Ž1., and the standard deviations s of the fitting for pseudo-binary mixtures at several fixed compositions f m fm :
0.0117
0.0246
B1 B2 B3 B4 B5 B6 E s Ž Vm .rŽcm3 . moly1 .
y4.281615 0.604403 y0.975514 4.451434 y2.114918 y8.300995 0.0011
y4.152739 0.704244 y1.001243 3.311779 y1.216175 y5.589344 0.00091
0.0529
0.0875
0.1321
0.1930
y3.931405 y3.620765 y3.226831 y2.794223 0.660008 0.590436 0.468540 0.289877 y1.133435 y1.372548 y0.994626 y0.906541 2.213060 1.224990 0.299708 0.263107 y0.847718 0.024315 y0.158901 y0.259339 y3.547047 y1.591276 y0.357216 y0.566760 0.00085 0.00067 0.00056 0.00056
0.3195 y2.015957 0.170430 y0.263917 y1.038844 y0.591372 0.775517 0.00053
Excess molar volumes of ŽDMF q ethanol q water.
1359
FIGURE 2. I, Excess partial molar volumes of ethanol at infinite dilution V2`E ; `, excess partial E Ž . molar volumes VDM F of DMF in DMF q water binary mixture; and ^, the fitting parameters B1 for the pseudo-binary mixture approach, as a function of f m .
comparison with the experimental results, the values of Vm,E 2q13 calculated by PBMA from the parameters in table 7 are included in table 4. They are in good agreement with the experimental data. In figure 2, one of the curves shows the dependence of B1 Žin table 7. on f m . The meaning of B1 is the excess volume Vm,E 2q13 when x 2 s 0.5. The curve increases with the increase of f m . This is an indication that the interaction between ethanol and the mixed solvent Žwater q DMF. become weaker as f m rises. PARTIAL MOLAR VOLUMES OF METHANOL IN PSEUDO-BINARY MIXTURE
In our mixture, the values Vm,E 2q13 are smooth functions of x 2 . By using the parameters in table 7, the partial molar volumes V2 of ethanol in pseudo-binary mixtures can be calculated by equation Ž16.: V2 s VmE, 2q13 q V20 q Ž 1 y x 2 . Ž d VmE, 2q13rd x 2 . T , P .
Ž 16 .
When x 2 approaches zero, the partial molar volume of ethanol at infinite dilution 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 ŽDMF q water. mixed solvents with fixed composition f m were evaluated and are listed in table 8. The values of V2`E , the excess partial molar volume of ethanol at infinite dilution, are presented in figure 2 as a function of f m . The values of V2`E first decrease as f m increases, and then reach a minima, where f m s 0.0529. After
1360
T.-C. Bai, J. Yao, and S.-J. Han TABLE 8. The partial molar volumes of ethanol at infinite dilution V2` and the apparent molar volumes of ŽDMF q water. mixed solvent V130 , in pseudo-binary mixtures at several fixed compositions f m V2` rŽcm3 . moly1 .
fm 0.0 0.0117 0.0246 0.0529 0.0875 0.1321 0.1930 0.3195 1.0
V130 rŽcm3 . moly1 .
55.136 54.561 53.891 53.449 53.495 53.897 54.742 55.910 56.392
18.069 18.724 19.447 21.010 22.906 25.365 28.756 35.970 77.428
that minimum V2`E increases with rise in f m . Another curve in figure 2 shows the E Ž . excess partial molar volumes of DMF VDM F in DMF q water binary mixture as a function of f m . It has a minimum at the point where f m is about 0.072. The interesting thing is that this minimum corresponds to the minimum of V2`E in its concentration neighborhood. In this region, ŽDMF q water. mixed solvent is in a E closely packed state.Ž7,16. The curve of VDM F is a reflection of DMF hydration, `E while the values of V2 are reflections for the interactions of the solvated solute E with the mixed solvent. The dependence of V2`E on VDMF at different f m is a reflection of the correlation between solute solvation and solute Žethanol. ]solute E ŽDMF. interactions. The dependence of the B1 curve on the VDMF curve, as well as on f m , is a reflection of the effect of medium change on the properties of ethanol. The apparent molar volumes V130 of ŽDMF q water. mixed solvent at fixed composition f m were evaluated. Their values are listed in table 8. The values of V130 increase with increase of f m . An empirical equation of the form: V2` Ž or V130 . r Ž cm3 . mol -1 . s
Ý
A i f mŽ iy1. ,
Ž 17 .
is1
was used to fit the values of V2` Žor V130 . with f m . The fitting parameters A i in
TABLE 9. Fitting parameters A i of equation Ž17. for partial molar volumes of ethanol at infinite dilution V2` , and for the apparent molar volumes of ŽDMF q water. mixed solvent V130 , the standard deviations s , and the correlative coefficients r of the fitting
A1 A2 A3 A4 A5 srŽcm3 . moly1 . r
for V2`
for V130
55.0237 y47.1663 405.026 y977.25 620.759 0.15 0.9946
18.0735 55.7635 y8.4586 36.8847 y24.8351 0.0063 0.9999
Excess molar volumes of ŽDMF q ethanol q water.
1361
equation Ž17., the standard deviations, and the correlative coefficients of the fitting are given in table 9. The correlations by using equation Ž17. are in good agreement with the experimental results. We thank the Natural Science Foundation of Zhejiang province for their financial support. REFERENCES 1. Lilley, T. H. Pure Appl. Chem. 1994, 66, 429]434. 2. Lilley, T. H. In Biochemical Thermodynamics. Jones, M. N.: editor. Elsevier: Amsterdam. 1998, Ch. 1, p. 48. 3. Zielkiewicz, J. J. Chem. Thermodynamics 1995, 27, 415]422. 4. Zegers, H. C.; Somsen, G. J. Chem. Thermodynamics 1984, 16, 225]235. 5. Zielkiewicz, J. J. Chem. Thermodynamics 1994, 26, 1317]1322. 6. Zielkiewicz, J. J. Chem. Thermodynamics 1995, 27, 225]230. 7. Chu, D. Y.; Zhang, Y.; Hu, I. Y.; Liu, R. L. Acta Physico-chimica Sinica 1990, 6, 203]208. ŽIn Chinese.. 8. Davis, M. I. Thermochimca Acta 1987, 120, 299]314. 9. Benson, G. C.; Kiyohara, O. J. Solution Chem. 1980, 9, 791]804. 10. Powell, M. J. D. Computer J. 1964, 7, 155]162. 11. Esteve, X.; Patil, K. R.; Fernandez, J.; Coronas, A. J. Chem. Thermodynamics 1995, 27, 281]292. 12. Jasinski, B.; Malanowski, S. Chem. Eng. Sci. 1970, 25, 913]920. 13. Cibulka, I. Collection of Czechoslo¨ ak Chem. Commun. 1982, 47, 1414]1419. 14. Singh, P. P.; Nigam, R. K.; Sharma, S. P.; Aggarwal, S. Fluid Phase Equilibria 1984, 18, 333]344. 15. Pando, C.; Renuncio, J. A. R.; Calzon, J. A. G.; Christensen, J. J.; Izatt, R. M. J. Solution Chem. 1987, 16, 503]527. 16. Davis, M. I.; Hernandez, M. E. J. Ch. Eng. Data 1995, 40, 674]678.
(Recei¨ ed 5 January 1998; in final form 5 June 1998)
WE-114
Excess molar volumes for binary and ternary mixtures of ( N, N-dimethylformamide H ethanol H water) at the temperature 298.15 K Tong-Chun Bai, Jia Yao, and Shi-Jun Han a Chemistry Department, Zhejiang Uni¨ ersity, Hangzhou, 310027, China
The results of excess molar volumes for the ternary mixture N, N-dimethylformamide ŽDMF. q ethanol q water4, and for the binary constituents ŽDMF q water., ŽDMF q ethanol., and Žethanol q water. at T s 298.15 K were reported. Several empirical expressions were used to predict and correlate the ternary excess molar volumes. A pseudo-binary mixture approach was used to analyse the system studied. The partial molar volumes of ethanol at infinite dilution, and the apparent molar volumes of ŽDMF q water. mixed solvent at several fixed compositions were evaluated and correlated with the composition of the ŽDMF q water. mixed solvent. q 1998 Academic Press KEYWORDS: excess molar volume; mixture; ethanol; N, N-dimethylformamide; water
1. Introduction The thermodynamic properties of solution containing amides, amino acids, peptides, and their derivatives are of interest. The correlation between solute hydration and solute]solute interaction is complex.Ž1,2. In view of the fact that the interactions occurring between solutes in water and in amidic solvent are different, it would be worthwhile to explore the effect of changing solvent medium on molecular interactions. The aqueous solution of N, N-dimethylformamide ŽDMF. is a model mixed solvent representing the environment of the interior of proteins.Ž2. Alcohols are model molecules for the study of hydrophobic interactions. In ŽDMF q water. mixing solvents, DMF]water association exists and changes with composition.Ž3. In the ŽDMF q alcohol q water. ternary mixture, the properties of alcohol are affected by this DMF]water association. It is of interest to study the effect of changing the medium from water to amidic solvent on the thermodynamic properties of alcohol. Zegers and SomsenŽ4. reported the volumetric properties of ŽDMF q alcohol. mixtures. In order to give a better description of the solvation of a
To whom correspondence should be addressed.
0021]9614r98r111347 q 15 $30.00r0
q 1998 Academic Press
1348
T.-C. Bai, J. Yao, and S.-J. Han
DMF by water and by alcohol molecules in the ternary mixture ŽDMF q alcohol q water., Zielkiewicz Ž3,5,6. published a series of papers on the excess molar volumes of the ternary mixtures at T s 313.15 K. In the present paper, the results of excess molar volumes for the ternary mixture ŽDMF q ethanol q water., and for the binary constituents ŽDMF q water., ŽDMF q ethanol., and Žethanol q water. at T s 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. In order to obtain some information about the correlation between alcohol]amidic interactions with amide solvation, a pseudo-binary mixture approach was applied to the system studied. In this approach, the alcohol was added to ŽDMF q water. mixed solvents with fixed composition. The excess molar volumes for these pseudo-binary mixtures were evaluated. It is our goal to obtain some information about the changing excess molar volumes with ŽDMF q water. composition. The partial molar volumes of ethanol at infinite dilution, and the apparent molar volumes of ŽDMF q water. mixed solvent at several fixed compositions were evaluated and correlated with the composition of the ŽDMF q water. mixed solvent.
2. Experimental The DMF Žanalytical reagent grade, Shanghai Chemical Co.. was dried over freshly activated Al 2 O 3 for at least 48 h and then fractionally distilled under reduced pressure. The product was stored over freshly activated molecular sieve 0.4 nm. Ethanol Žanalytical regent grade, Shanghai Chemical Co.. was dried over freshly activated CaO for 2 days and fractionally distilled. The product was stored over freshly activated 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. Ternary mixtures were prepared by mixing a measured binary mixture ŽDMF q water at known composition. with a pure liquid Žethanol, as the third component.. All mixtures were prepared by mass. The measured physical properties of the pure substance together with literature Ž5,7 ] 9. values are included in table 1.
TABLE 1. Densities and refractive indices of pure liquids at T s 298.15 K Component
DensityrŽg . cmy3 . this work
DMF
0.94403
Ethanol
0.78498
Refractive indices
literature Ž7.
0.944061 0.94383Ž8. 0.784962 Ž9. 0.78509Ž9. 0.78493Ž9.
this work
literature
1.4282
1.42850 Ž5. 1.4282 Ž7. 1.35941Ž9. 1.35917 Ž5. 1.35937 Ž5.
1.3594
Excess molar volumes of ŽDMF q ethanol q water.
1349
3. Results and discussion EXCESS MOLAR VOLUMES FOR BINARY MIXTURES
Excess molar volumes VmE were calculated from densities r . Results of VmE and r for binary mixtures are given in table 2. A function of the form: VmE r Ž cm3 . moly1 . s x Ž 1 y x .
n
Ý
Bi Ž 2 x y 1 .
iy1
,
Ž 1.
is1
was fitted to the experimental results. Parameters of equation Ž1. were obtained by using Powell’sŽ10. optimization method with equation Ž2. as the objective function: m
Fs
1r2 2
Ý VmE , expt Ž x i . y VmE , calc Ž x i . 4 r Ž m y n .
,
Ž 2.
is1
where m is the number of experimental data points and n the number of parameters used in equation Ž1.. Parameters of equation Ž1. and the standard deviations of the fitting s are given in table 3. Figure 1 shows a comparison of the values of VmE for three binary mixtures xDMF q Ž1 y x . water4 , xethanol q Ž1 y x . water4 , and xDMF q Ž1 y x . ethanol4 at T s 298.15 K with the literature values at T s 313.15 K ŽZielkiewicz Ž5. .. Both binary systems ŽDMF q water. and Žethanol q water. have nearly the same magnitude of VmE values, while for those of ŽDMF q ethanol. are of relatively smaller magnitude. These are indications that the molecular interactions between ethanol]water and DMF]water are stronger than that of DMF]ethanol. A temperature effect on VmE is observed in figure 1. The values of binary VmE for ŽDMF q ethanol. at T s 298.15 K have greater absolute magnitudes than those at T s 313.15 K over the whole composition region. For the systems ŽDMF q water. and Žethanol q water., both have greater absolute magnitude at T s 298.15 K than at T s 313.15 K in the middle concentration region. This indicates that at lower temperature, due to lower molecular thermal agitation, the associations or interactions Žand thus the molecular packing. between the components become stronger. For Žethanol q water. and ŽDMF q water. systems, the temperature effect is hardly observed in the water-rich region; this might be attributed to the hydrophobic interaction and hydrogen bond interaction between the solute and water. The competition of molecular interaction among components and the temperature effect give rise to an asymmetric effect when the ternary mixture is composed of these constituents at T s 298.15 K. EXCESS MOLAR VOLUMES FOR TERNARY MIXTURES
Excess molar volumes were calculated from densities by using the following equation: y1 y1 VmE, 123 s x 1 M1 Ž ry1 y ry1 y ry1 y ry1 1 . q x 2 M2 Ž r 2 . q x 3 M3 Ž r 3 .,
Ž 3.
1350
T.-C. Bai, J. Yao, and S.-J. Han TABLE 2. Densities r and excess molar volumes VmE for binary mixtures at T s 298.15 K
x
rrŽg . cmy3 .
VmE rŽcm3 . moly1 .
x
rrŽg . cmy3 .
VmE rŽcm3 . moly1 .
xDMF q Ž1 y x . water 0.0055 0.0112 0.0142 0.0241 0.0319 0.0382 0.0464 0.0553 0.0620 0.0721 0.0805 0.0907 0.0914 0.1011 0.1125 0.1230 0.1347 0.1478 0.1489 0.1621
0.99667 0.99649 0.99640 0.99620 0.99616 0.99614 0.99620 0.99630 0.99640 0.99657 0.99670 0.99685 0.99686 0.99698 0.99708 0.99713 0.99712 0.99704 0.99703 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
0.0063 0.0123 0.0174 0.0240 0.0305 0.0365 0.0418 0.0469 0.0528 0.0728 0.1159 0.1696 0.2385 0.3224
0.99402 0.99140 0.98927 0.98670 0.98432 0.98220 0.98036 0.97876 0.97690 0.97075 0.95815 0.94223 0.92190 0.89882
y0.0236 y0.0485 y0.0711 y0.1033 y0.1355 y0.1675 y0.1946 y0.2235 y0.2557 y0.3641 y0.5806 y0.7887 y0.9574 y1.0586
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
0.87550 0.84478 0.83780 0.82517 0.81151 0.81136 0.80319 0.80287 0.79644 0.79637 0.79442 0.79273 0.79240 0.78828
y1.0982 y1.0324 y0.9880 y0.8675 y0.6712 y0.6599 y0.5087 y0.5015 y0.3473 y0.3455 y0.2932 y0.2455 y0.2360 y0.1080
0.88822 0.87887 0.86822 0.85777 0.84823 0.83899 0.82820 0.81950 0.81040 0.80160
y0.2498 y0.2557 y0.2589 y0.2567 y0.2490 y0.2390 y0.2189 y0.1980 y0.1688 y0.1298
xethanol q Ž1 y x . water 0.4205 0.5787 0.6199 0.6993 0.7938 0.7974 0.8555 0.8580 0.9078 0.9084 0.9238 0.9372 0.9398 0.9732
xDMF q Ž1 y x . ethanol 0.9366 0.9102 0.8778 0.8498 0.8191 0.7889 0.7583 0.7317 0.7032 0.6381
0.93735 0.93425 0.93035 0.92692 0.92310 0.91928 0.91536 0.91188 0.90810 0.89917
y0.0881 y0.1042 y0.1214 y0.1356 y0.1504 y0.1656 y0.1815 y0.1935 y0.2071 y0.2323
0.5623 0.5007 0.4333 0.3700 0.3145 0.2627 0.2047 0.1597 0.1145 0.0726
where M1 , M2 , and M3 are molar masses of pure components 1, 2, and 3; and r 1 , r 2 , and r 3 are densities of the pure components, and r is the density of the
Excess molar volumes of ŽDMF q ethanol q water.
1351
TABLE 3. Fitting parameters Bi of equation Ž1. and the standard deviations s of the fitting for the excess molar volumes of binary mixtures investigated xDMF q Ž1 y x . ethanol xDMF q Ž1 y x . water xethanol q Ž1 y x . water B1 B2 B3 B4 B5 B6 B7 B8 B9 E s Ž Vm .rŽcm3 . moly1 .
y1.025633 0.138920 y0.432249 0.648744 0.633981 y0.566669 y1.693487 0.00051
y4.383917 1.758919 0.283819 y1.800046 0.560583 1.760020 0.447975 y0.894894 1.039060 0.00042
y4.347337 0.816881 y0.935253 1.588231 y2.013010 y1.357381 3.415028 y1.375997 0.000747
mixture. The values of r and Vm,E 123 for ternary mixtures are given in table 4. Several empirical methods have been suggested by Esteve et al.Ž11. to estimate ternary excess properties from experimental results on the constituents. Some of the predictive methods were originally proposed to predict molar enthalpy, or volume, or Gibbs free energy.Ž12. 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 effect. The ternary excess volume Vm,E 123 is the sum of the
FIGURE 1. The excess molar volumes VmE for three binary mixtures at T s 298.15 K and T s 313.15 K as a function of mole fraction. For xDMF q Ž1 y x . water4: I, T s 298.15 K; curve 1, T s 313.15 K; for xethanol q Ž1 y x . water4: `, T s 298.15 K; curve 2, T s 313.15 K; and for xDMF q Ž1 y x . ethanol4: ^, T s 298.15 K; curve 3, T s 313.15 K.
1352
T.-C. Bai, J. Yao, and S.-J. Han
TABLE 4. Densities r , excess molar volumes for ternary mixture Vm,E 123 , and excess molar volumes for pseudo-binary mixture Vm,E 2q13 at T s 298.15 K, and the values calculated by the correlative equation Ž11. and by PBMA x1
x2
r . Ž g cmy3 .
VmE, 123 Ž cm3 . moly1 .
0.0117 0.0115 0.0114 0.0114 0.0113 0.0112 0.0112 0.0111 0.0108 0.0102 0.0095 0.0087 0.0077 0.0067 0.0050 0.0040 0.0033 0.0023
0.0000 0.0120 0.0192 0.0253 0.0315 0.0373 0.0434 0.0493 0.0760 0.1219 0.1860 0.2558 0.3411 0.4267 0.5673 0.6586 0.7206 0.8034
0.99648 0.99140 0.98880 0.98654 0.98439 0.98255 0.98058 0.97875 0.97084 0.95780 0.93905 0.91828 0.89484 0.87485 0.84750 0.83200 0.82236 0.81100
y0.0375 y0.0888 y0.1260 y0.1559 y0.1884 y0.2205 y0.2537 y0.2864 y0.4314 y0.6599 y0.8896 y1.0208 y1.0815 y1.1013 y1.0481 y0.9333 y0.8248 y0.6782
0.0246 0.0243 0.0240 0.0239 0.0236 0.0235 0.0232 0.0231 0.0227 0.0217 0.0203 0.0185 0.0164 0.0136 0.0101 0.0092 0.0086 0.0042
0.0000 0.0151 0.0250 0.0316 0.0410 0.0461 0.0580 0.0619 0.0775 0.1207 0.1754 0.2477 0.3350 0.4491 0.5919 0.6253 0.6526 0.8283
0.99620 0.99054 0.98714 0.98501 0.98200 0.98040 0.97680 0.97565 0.97105 0.95835 0.94214 0.92079 0.89706 0.87065 0.84354 0.83780 0.83327 0.80760
y0.0850 y0.1588 y0.2093 y0.2446 y0.2942 y0.3214 y0.3846 y0.4058 y0.4848 y0.6808 y0.8673 y1.0096 y1.0841 y1.0951 y1.0100 y0.9681 y0.9285 y0.5966
0.0529 0.0521 0.0515 0.0511 0.0506 0.0502 0.0498 0.0495 0.0486
0.0000 0.0164 0.0270 0.0341 0.0444 0.0528 0.0589 0.0658 0.0819
0.99627 0.99082 0.98744 0.98524 0.98204 0.97957 0.97770 0.97568 0.97095
y0.2017 y0.2845 y0.3387 y0.3740 y0.4249 y0.4677 y0.4962 y0.5294 y0.6031
VmE, 123 Ž cm3 . moly1 . equation Ž11.4
VmE, 2q13 Ž cm3 . moly1 .
VmE, 2q13 Ž cm3 . moly1 . ŽPBMA.
0.0000 y0.0518 y0.0892 y0.1194 y0.1522 y0.1844 y0.2179 y0.2508 y0.3968 y0.6270 y0.8591 y0.9930 y1.0568 y1.0798 y1.0319 y0.9205 y0.8143 y0.6709
0.0000 y0.0534 y0.0886 y0.1198 y0.1526 y0.1839 y0.2174 y0.2501 y0.3977 y0.6273 y0.8579 y0.9933 y1.0585 y1.0777 y1.0330 y0.9205 y0.8137 y0.6711
0.0000 y0.0751 y0.1264 y0.1623 y0.2127 y0.2403 y0.3045 y0.3260 y0.4064 y0.6061 y0.7971 y0.9456 y1.0276 y1.0483 y0.9753 y0.9363 y0.8989 y0.5820
0.0000 y0.0755 y0.1275 y0.1627 y0.2133 y0.2408 y0.3044 y0.3250 y0.4056 y0.6057 y0.7969 y0.9475 y1.0260 y1.0486 y0.9757 y0.9366 y0.8983 y0.5821
0.0000 y0.0861 y0.1425 y0.1792 y0.2322 y0.2767 y0.3064 y0.3409 y0.4179
0.0000 y0.0867 y0.1428 y0.1801 y0.2335 y0.2763 y0.3067 y0.3405 y0.4161
f m s 0.0117 y0.0370 y0.0894 y0.1240 y0.1550 y0.1871 y0.2176 y0.2507 y0.2824 y0.4250 y0.6427 y0.8614 y0.9988 y1.0787 y1.1037 y1.0433 y0.9318 y0.8252 y0.6450 f m s 0.0246 y0.0842 y0.1560 y0.2063 y0.2411 y0.2907 y0.3180 y0.3810 y0.4014 y0.4810 y0.6764 y0.8585 y1.0024 y1.0841 y1.1067 y1.0176 y0.9770 y0.9384 y0.5796 f m s 0.0529 y0.2018 y0.2833 y0.3371 y0.3732 y0.4254 y0.4672 y0.4966 y0.5297 y0.6027
Excess molar volumes of ŽDMF q ethanol q water.
1353
TABLE 4}continued x1
x2
0.0462 0.0428 0.0385 0.0343 0.0288 0.0216 0.0182 0.0131 0.0088
0.1269 0.1925 0.2730 0.3530 0.4567 0.5912 0.6557 0.7519 0.8341
r
Ž g . cmy3 .
VmE, 123 Ž cm3 . moly1 .
VmE, 123 Ž cm3 . moly1 . Žequation Ž11.4
VmE, 2q13 Ž cm3 . moly1 .
VmE, 2q13 Ž cm3 . moly1 . ŽPBMA.
0.95755 0.93804 0.91502 0.89418 0.87070 0.84500 0.83401 0.81899 0.80750
y0.7769 y0.9528 y1.0673 y1.1081 y1.1018 y1.0096 y0.9242 y0.7544 y0.5837
y0.7769 y0.9493 y1.0616 y1.1100 y1.1121 y1.0149 y0.9279 y0.7535 y0.5595
y0.6008 y0.7899 y0.9207 y0.9776 y0.9922 y0.9272 y0.8547 y0.7044 y0.5502
y0.6002 y0.7912 y0.9208 y0.9771 y0.9921 y0.9273 y0.8550 y0.7039 y0.5504
0.0000 y0.1030 y0.1403 y0.1812 y0.2320 y0.2655 y0.3079 y0.3476 y0.4164 y0.5905 y0.7365 y0.8501 y0.9057 y0.9084 y0.8422 y0.7749 y0.6737 y0.4713
0.0000 y0.1033 y0.1411 y0.1825 y0.2318 y0.2658 y0.3065 y0.3475 y0.4159 y0.5906 y0.7369 y0.8504 y0.9052 y0.9084 y0.8428 y0.7744 y0.6738 y0.4713
0.0000 y0.0902 y0.1429 y0.1811 y0.2186 y0.2584 y0.2887 y0.3249 y0.3673 y0.4923 y0.6521 y0.7456 y0.7995 y0.8083 y0.7506 y0.6714 y0.5854 y0.4479
0.0000 y0.0908 y0.1436 y0.1811 y0.2188 y0.2576 y0.2895 y0.3243 y0.3668 y0.4922 y0.6525 y0.7459 y0.7991 y0.8083 y0.7506 y0.6719 y0.5847 y0.4481
f m s 0.0875 0.0875 0.0857 0.0850 0.0843 0.0834 0.0827 0.0819 0.0811 0.0796 0.0752 0.0700 0.0631 0.0553 0.0448 0.0340 0.0283 0.0222 0.0130
0.0000 0.0202 0.0278 0.0363 0.0467 0.0541 0.0632 0.0727 0.0895 0.1402 0.1999 0.2789 0.3676 0.4881 0.6107 0.6764 0.7466 0.8514
0.99682 0.99051 0.98814 0.98554 0.98248 0.98020 0.97755 0.97468 0.96967 0.95460 0.93710 0.91503 0.89250 0.86600 0.84304 0.83200 0.82090 0.80560
y0.3550 y0.4508 y0.4854 y0.5233 y0.5704 y0.6012 y0.6404 y0.6768 y0.7396 y0.8957 y1.0205 y1.1061 y1.1302 y1.0901 y0.9803 y0.8898 y0.7637 y0.5241
0.1321 0.1295 0.1279 0.1267 0.1254 0.1241 0.1229 0.1216 0.1199 0.1141 0.1037 0.0938 0.0827 0.0670 0.0509 0.0399 0.0312 0.0209
0.0000 0.0197 0.0319 0.0410 0.0505 0.0607 0.0695 0.0795 0.0924 0.1363 0.2148 0.2902 0.3740 0.4926 0.6146 0.6977 0.7636 0.8417
0.99712 0.99114 0.98744 0.98468 0.98180 0.97875 0.97604 0.97308 0.96920 0.95613 0.93360 0.91344 0.89290 0.86710 0.84403 0.82990 0.81945 0.80770
y0.5458 y0.6253 y0.6713 y0.7046 y0.7369 y0.7711 y0.7966 y0.8274 y0.8627 y0.9637 y1.0807 y1.1330 y1.1412 y1.0852 y0.9609 y0.8364 y0.7144 y0.5343
y0.3551 y0.4503 y0.4856 y0.5249 y0.5715 y0.6034 y0.6416 y0.6799 y0.7428 y0.8964 y1.0142 y1.0973 y1.1309 y1.1024 y0.9844 y0.8863 0.7563 y0.5070 fm s 0.1321 y0.5455 y0.6254 y0.6729 y0.7069 y0.7409 y0.7760 y0.8045 y0.8354 y0.8724 y0.9757 y1.0878 y1.1367 y1.1479 y1.0981 y0.9678 y0.8373 y0.7099 y0.5254
1354
T.-C. Bai, J. Yao, and S.-J. Han TABLE 4}continued
r
Ž g . cm .
VmE, 123 Ž cm3 . moly1 .
0.99618 0.98917 0.98602 0.98315 0.98004 0.97640 0.97397 0.97050 0.96423 0.94740 0.93565 0.90654 0.88855 0.86340 0.83890 0.83141 0.82160 0.80870
y0.7967 y0.8422 y0.8724 y0.9008 y0.9252 y0.9554 y0.9740 y0.9991 y1.0407 y1.1237 y1.1602 y1.1812 y1.1529 y1.0601 y0.8924 y0.8215 y0.7111 y0.5278
0.0000 0.99012 0.0285 0.98265 0.0423 0.97905 0.0556 0.97551 0.0715 0.97128 0.0835 0.96813 0.0954 0.96505 0.1096 0.96130 0.1303 0.95600 0.2052 0.93730 0.2734 0.92115 0.3649 0.90060 0.4599 0.88040 0.5719 0.85808 0.7073 0.83335 0.7512 0.82583 0.8136 0.81543 0.8591 0.80800 s Ž VmE .rŽcm3 . moly1 .
y1.0657 y1.1064 y1.1237 y1.1346 y1.1470 y1.1552 y1.1630 y1.1689 y1.1780 y1.1933 y1.1896 y1.1556 y1.0797 y0.9461 y0.7410 y0.6656 y0.5449 y0.4439
x1
x2
0.1930 0.1884 0.1863 0.1844 0.1824 0.1799 0.1783 0.1760 0.1717 0.1601 0.1518 0.1298 0.1149 0.0918 0.0662 0.0578 0.0462 0.0302
0.0000 0.0239 0.0347 0.0448 0.0553 0.0679 0.0763 0.0883 0.1103 0.1703 0.2135 0.3273 0.4047 0.5246 0.6569 0.7007 0.7606 0.8433
y3
VmE, 123 Ž cm3 . moly1 . equation Ž11.4
VmE, 2q13 Ž cm3 . moly1 .
VmE, 2q13 Ž cm3 . moly1 . ŽPBMA.
0.0000 y0.0909 y0.1293 y0.1656 y0.1980 y0.2379 y0.2630 y0.2974 y0.3559 y0.4851 y0.5548 y0.6635 y0.6946 y0.6942 y0.6284 y0.5912 y0.5269 y0.4072
0.0000 y0.0907 y0.1293 y0.1640 y0.1986 y0.2382 y0.2634 y0.2978 y0.3560 y0.4851 y0.5542 y0.6637 y0.6949 y0.6938 y0.6285 y0.5913 y0.5268 y0.4072
0.0000 y0.0711 y0.1031 y0.1281 y0.1575 y0.1785 y0.1989 y0.2201 y0.2511 y0.3462 y0.4152 y0.4788 y0.5041 y0.4899 y0.4290 y0.4005 y0.3462 y0.2936
0.0000 y0.0716 y0.1017 y0.1284 y0.1580 y0.1788 y0.1984 y0.2206 y0.2509 y0.3461 y0.4153 y0.4786 y0.5044 y0.4897 y0.4291 y0.4002 y0.3466 y0.2934
f m s 0.1930 y0.7694 y0.8409 y0.8710 y0.8979 y0.9243 y0.9536 y0.9720 y0.9964 y1.0358 y1.1126 y1.1452 y1.1697 y1.1478 y1.0577 y0.8858 y0.8133 y0.6999 y0.5094 f m 0.3195 0.3195 0.3104 0.3060 0.3018 0.2967 0.2928 0.2891 0.2845 0.2779 0.2540 0.2322 0.2029 0.1726 0.1368 0.0935 0.0795 0.0596 0.0450
0.00802 a
y1.0646 y1.1031 y1.1195 y1.1338 y1.1489 y1.1588 y1.1676 y1.1766 y1.1870 y1.2013 y1.1895 y1.1455 y1.0715 y0.9514 y0.7572 y0.6792 y0.5508 y0.4417
0.00074 b
s mole fraction of DMF in the pseudo-binary mixture w f m DMF q Ž1 y f m .water4 q ethanolx. Standard deviations for fitting the excess molar volumes of the ternary mixture by using equation Ž11.. b The mean values of standard deviation for fitting the excess molar volumes by using the pseudobinary mixture approach. fm a
binary excess volumes of its constituents. The simplest expression is of the form: VmE, 123r 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 . , Ž 4 .
Excess molar volumes of ŽDMF q ethanol q water.
1355
where x 1 s 1 y x 2 y x 3 , and the values of VmE,i j Ž x i , x j . are calculated according to equation Ž5.: VmE, i jr Ž cm3 . moly1 . s x i x j
n
Ý Bk Ž x i , x j . ky 1 ,
Ž 5.
k
using values of Bk from table 3 and at the composition Ž x i , x j .. According to the Kohler expression suggested by Esteve et al.,Ž11. the excess molar volume for a ternary mixture is given by: VmE, 123r 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 . q 2
2
2 Ž x 2 q x 3 . VmE, 23 Ž x 20 , x 30 . ,
Ž 6.
in which Vm, i j denotes the excess molar volume for the binary mixture at composition Ž x i0 , x j0 ., such that x i0 s x irŽ x i q x j . s 1 y x j0 with VmE, i j Ž x i0 , x j0 . being calculated by equation Ž5.. Equations Ž4. and Ž6. 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. Contrarily, Tsao and Smith proposed an asymmetrical equation, suggested by Esteve et al.:Ž11. VmE, 123r 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 . ,
Ž 7.
in which x 20 s x 2rŽ1 y x 1 . s 1 y x 30 . Hillert also suggested an asymmetrical equation: Ž11. VmE, 123r 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 3r¨ 23¨ 32 . VmE, 23 Ž ¨ 23¨ 32 . ,
Ž 8.
where ¨ i j s Ž1 q x i y x j .r2. The standard deviations of the fits for excess molar volumes by using predictive equations equations Ž4. and Ž6]8.4 are reported in table 5. Another type of equation is of the form that introduces ternary effect terms. CibulkaŽ13. proposed the following expression: VmE, 123r 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 ??? . , Ž 9.
1356
T.-C. Bai, J. Yao, and S.-J. Han TABLE 5. Comparison of the standard deviations s for fitting the excess molar volumes for the ternary mixture by using several equations
s Ž VmE .rŽcm3 . moly1 .
Equations Equation Ž4. Equation Ž6. Equation Ž7. 1Žwater. 2ŽDMF. 3Žethanol. Equation Ž8. 1Žwater. 2ŽDMF. 3Žethanol. Equation Ž9. Equation Ž10. 1Žwater. 2ŽDMF. 3Žethanol. Equation Ž11. 1Žwater. PBMAa
0.019 0.031 0.030 0.013 0.0092 0.011 0.0080 0.00074
a The mean values of standard deviations for the pseudo-binary mixture at their several fixed compositions f m .
in which Vm,E i j Ž x i , x j . were calculated by using equation Ž5. at composition Ž x i , x j ., and using values of Bi from table 3. The Ci is the fitting parameter of ternary effect. Singh et al.Ž14. proposed an equation of the form: VmE, 123r 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 2
3
C3 x 12 Ž x 2 y x 3 . q C4 x 13 Ž x 2 y x 3 . q ??? 4 .
Ž 10 .
However, Jasinski and Malanowski Ž12. suggested an equation of the form: VmE, 123r 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 Ž 2 x1 y 1. q C3 Ž 2 x1 y 1. 2 q ??? 4 .
Ž 11 .
In equations Ž9]11., ternary effect parameters Ci were obtained by Powell’sŽ10. optimization method. The results are given in table 6. The standard deviations s Ž Vm,E 123 . of the fits by predictive equations equations TABLE 6. Fitting parameters Ci of equations Ž9]11.
C1 C2 C3 C4 C5 C6
Equation Ž9.
Equation Ž10. 1Žwater.2ŽDMF.3Žethanol.
Equation Ž11. 1Žwater.
y4.277903 42.665050 23.189790 y68.955990 y21.790030 y77.449910
1.549553 7.257435 1.740424 y173.2988 257.3791 y443.8122
1.520536 1.399926 4.366468 3.074258 y20.903420 y10.958790
Excess molar volumes of ŽDMF q ethanol q water.
1357
Ž4. and Ž6]8.4 and by correlative equations equations Ž9]11.4 are reported in table 5. In general, correlative equations give lower values of s Ž Vm,E 123 . than those of predictive expressions. For comparison with the experimental results, the values of Vm,E 123 calculated by Jasinski’s equation equation Ž11.4 give the lowest standard deviation among these fits. They are given in table 4. The asymmetrical equations give more weight to the binary constituent 1-2 and 1-3, and therefore component 1 plays the more important role. The rule for selecting the numbering of the components has been given by Pando et al.Ž15. Instead of looking for the most dissimilar component in the ternary mixture, it is necessary to examine the three binary curves involved and look for the two binaries which exhibit the two largest absolute values of excess property in their maxima or minima. The common component of these two mixtures will be designated as component 1. For ŽDMF q ethanol q water. the two larger absolute values of VmE correspond to the minima of ŽDMF q water. and Žethanol q water.. This means that following this rule water must be component 1, which coincides with the results in table 5. The temperature effect on Vm,E 123 for the ternary mixture can be observed by comparing our results at T s 298.15 K Žtable 4. with the literature value at T s 313.15 K. The minimum value of Vm,E 123 at T s 313.15 K is y1.0974 cm3 . moly1 , where the composition is x 1 s 0.2566 Žfor DMF. and x 2 s 0.1484 Žfor ethanol.. The minimum value of VmE123 at T s 298.15 K is y1.1933 cm3 . moly1 , where x 1 s 0.2540 and x 2 s 0.2052. These data indicate that as temperature rises the absolute magnitude of the minima for Vm,E 123 is decreased. PSEUDO-BINARY MIXTURE APPROACH
In measuring densities of the ternary system, mixtures were made of several pseudo-binary mixtures in which component 2 Žpure ethanol. was added to a binary mixture of component 1 ŽDMF. and 3 Žwater. have a fixed composition f m , where f m s x 1rŽ1 y x 2 .. There are three reasons for using a pseudo-binary mixture approach ŽPBMA. to analyse the volume data. First, in the ŽDMF q ethanol q water. mixture, molecular interactions give rise to asymmetric volume effects, which cause the standard deviation of ternary fits Žtable 5. to be larger than those of binary fits Žtable 3.. Among the three binary mixtures, ŽDMF q water. and Žethanol q water. show the largest and nearly similar negative values of VmE . The binary mixture of ŽDMF q ethanol. shows a relatively smaller value of VmE . As a result, we pay more attention to observe the effect of changing composition in ŽDMF q water. mixed solvents on the ternary excess volumes, and to explore improved approaches. Second, aqueous solutions of amides represent a model mixed solvent of the environment of the interior of protein.Ž2. Alcohols are model molecules to study hydrophobic interactions. The interactions occurring between solutes in water and in amidic solvents are different. It is our goal to obtain some information about the effect of changing medium from water to an amidic solvent on the thermodynamic properties of alcohols. Third, in aqueous solution of some amides ŽDMF, N-methylformamide, and N, N-dimethylacetamide., a minimum
1358
T.-C. Bai, J. Yao, and S.-J. Han
value for partial molar volume of amide exists in the water-rich region.Ž16. In this region, some special molecular interaction, such as hydrogen bond interaction and hydrophobic interaction, have an observable influence on the volume effect. For the ternary mixture, the minimum of Vm,E 123 shifts towards the water-rich region. Thus, we obtained more data in the water-rich region. In the ternary mixture, the molar mass M13 of the f m DMF q Ž1 y f m . water4 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 12 .
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 13 .
Note that the mole fraction x 20 of alcohol in the pseudo-binary mixture is equal to x 2 , where M1 , M2 , and M3 are the molar mass of pure components. The excess molar volume Vm,E 2q13 was calculated from measured densities by using equation Ž14.: y1 y ry1 VmE, 2q13 s Ž 1 y x 2 . M13 Ž ry1 y ry1 13 . q x 2 M2 Ž r 2 .,
Ž 14 .
where r 13 is the density of ŽDMF q water. mixed solvent at its fixed composition. The values of Vm,E 2q13 are given in table 4. The values of Vm,E 123 can be obtained from Vm,E 2q13 by using equation Ž15.: VmE, 123 s VmE, 2q13 q Ž 1 y x 2 . VmE, 13 ,
Ž 15 .
where Vm,E 13 is the excess molar volumes of the ŽDMF q water. binary mixture. The values of Vm,E 13 can be found in table 4 where x 2 is zero. The values of Vm,E 2q13 were fitted by using equation Ž1. for just the binary mixtures. The fitting parameters are given in table 7. The standard deviations of the fitting for Vm,E 2q13 for several pseudo-binary mixtures are included in table 7. The mean values of the standard deviations for all pseudo-binary mixtures marked by s ŽPBMA.4 are compared with other correlative equations in table 5. It shows clearly that the pseudo-binary mixture approach gives the minimum value of the standard deviation. For TABLE 7. Fitting parameters Bi of equation Ž1., and the standard deviations s of the fitting for pseudo-binary mixtures at several fixed compositions f m fm :
0.0117
0.0246
B1 B2 B3 B4 B5 B6 E s Ž Vm .rŽcm3 . moly1 .
y4.281615 0.604403 y0.975514 4.451434 y2.114918 y8.300995 0.0011
y4.152739 0.704244 y1.001243 3.311779 y1.216175 y5.589344 0.00091
0.0529
0.0875
0.1321
0.1930
y3.931405 y3.620765 y3.226831 y2.794223 0.660008 0.590436 0.468540 0.289877 y1.133435 y1.372548 y0.994626 y0.906541 2.213060 1.224990 0.299708 0.263107 y0.847718 0.024315 y0.158901 y0.259339 y3.547047 y1.591276 y0.357216 y0.566760 0.00085 0.00067 0.00056 0.00056
0.3195 y2.015957 0.170430 y0.263917 y1.038844 y0.591372 0.775517 0.00053
Excess molar volumes of ŽDMF q ethanol q water.
1359
FIGURE 2. I, Excess partial molar volumes of ethanol at infinite dilution V2`E ; `, excess partial E Ž . molar volumes VDM F of DMF in DMF q water binary mixture; and ^, the fitting parameters B1 for the pseudo-binary mixture approach, as a function of f m .
comparison with the experimental results, the values of Vm,E 2q13 calculated by PBMA from the parameters in table 7 are included in table 4. They are in good agreement with the experimental data. In figure 2, one of the curves shows the dependence of B1 Žin table 7. on f m . The meaning of B1 is the excess volume Vm,E 2q13 when x 2 s 0.5. The curve increases with the increase of f m . This is an indication that the interaction between ethanol and the mixed solvent Žwater q DMF. become weaker as f m rises. PARTIAL MOLAR VOLUMES OF METHANOL IN PSEUDO-BINARY MIXTURE
In our mixture, the values Vm,E 2q13 are smooth functions of x 2 . By using the parameters in table 7, the partial molar volumes V2 of ethanol in pseudo-binary mixtures can be calculated by equation Ž16.: V2 s VmE, 2q13 q V20 q Ž 1 y x 2 . Ž d VmE, 2q13rd x 2 . T , P .
Ž 16 .
When x 2 approaches zero, the partial molar volume of ethanol at infinite dilution 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 ŽDMF q water. mixed solvents with fixed composition f m were evaluated and are listed in table 8. The values of V2`E , the excess partial molar volume of ethanol at infinite dilution, are presented in figure 2 as a function of f m . The values of V2`E first decrease as f m increases, and then reach a minima, where f m s 0.0529. After
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T.-C. Bai, J. Yao, and S.-J. Han TABLE 8. The partial molar volumes of ethanol at infinite dilution V2` and the apparent molar volumes of ŽDMF q water. mixed solvent V130 , in pseudo-binary mixtures at several fixed compositions f m V2` rŽcm3 . moly1 .
fm 0.0 0.0117 0.0246 0.0529 0.0875 0.1321 0.1930 0.3195 1.0
V130 rŽcm3 . moly1 .
55.136 54.561 53.891 53.449 53.495 53.897 54.742 55.910 56.392
18.069 18.724 19.447 21.010 22.906 25.365 28.756 35.970 77.428
that minimum V2`E increases with rise in f m . Another curve in figure 2 shows the E Ž . excess partial molar volumes of DMF VDM F in DMF q water binary mixture as a function of f m . It has a minimum at the point where f m is about 0.072. The interesting thing is that this minimum corresponds to the minimum of V2`E in its concentration neighborhood. In this region, ŽDMF q water. mixed solvent is in a E closely packed state.Ž7,16. The curve of VDM F is a reflection of DMF hydration, `E while the values of V2 are reflections for the interactions of the solvated solute E with the mixed solvent. The dependence of V2`E on VDMF at different f m is a reflection of the correlation between solute solvation and solute Žethanol. ]solute E ŽDMF. interactions. The dependence of the B1 curve on the VDMF curve, as well as on f m , is a reflection of the effect of medium change on the properties of ethanol. The apparent molar volumes V130 of ŽDMF q water. mixed solvent at fixed composition f m were evaluated. Their values are listed in table 8. The values of V130 increase with increase of f m . An empirical equation of the form: V2` Ž or V130 . r Ž cm3 . mol -1 . s
Ý
A i f mŽ iy1. ,
Ž 17 .
is1
was used to fit the values of V2` Žor V130 . with f m . The fitting parameters A i in
TABLE 9. Fitting parameters A i of equation Ž17. for partial molar volumes of ethanol at infinite dilution V2` , and for the apparent molar volumes of ŽDMF q water. mixed solvent V130 , the standard deviations s , and the correlative coefficients r of the fitting
A1 A2 A3 A4 A5 srŽcm3 . moly1 . r
for V2`
for V130
55.0237 y47.1663 405.026 y977.25 620.759 0.15 0.9946
18.0735 55.7635 y8.4586 36.8847 y24.8351 0.0063 0.9999
Excess molar volumes of ŽDMF q ethanol q water.
1361
equation Ž17., the standard deviations, and the correlative coefficients of the fitting are given in table 9. The correlations by using equation Ž17. are in good agreement with the experimental results. We thank the Natural Science Foundation of Zhejiang province for their financial support. REFERENCES 1. Lilley, T. H. Pure Appl. Chem. 1994, 66, 429]434. 2. Lilley, T. H. In Biochemical Thermodynamics. Jones, M. N.: editor. Elsevier: Amsterdam. 1998, Ch. 1, p. 48. 3. Zielkiewicz, J. J. Chem. Thermodynamics 1995, 27, 415]422. 4. Zegers, H. C.; Somsen, G. J. Chem. Thermodynamics 1984, 16, 225]235. 5. Zielkiewicz, J. J. Chem. Thermodynamics 1994, 26, 1317]1322. 6. Zielkiewicz, J. J. Chem. Thermodynamics 1995, 27, 225]230. 7. Chu, D. Y.; Zhang, Y.; Hu, I. Y.; Liu, R. L. Acta Physico-chimica Sinica 1990, 6, 203]208. ŽIn Chinese.. 8. Davis, M. I. Thermochimca Acta 1987, 120, 299]314. 9. Benson, G. C.; Kiyohara, O. J. Solution Chem. 1980, 9, 791]804. 10. Powell, M. J. D. Computer J. 1964, 7, 155]162. 11. Esteve, X.; Patil, K. R.; Fernandez, J.; Coronas, A. J. Chem. Thermodynamics 1995, 27, 281]292. 12. Jasinski, B.; Malanowski, S. Chem. Eng. Sci. 1970, 25, 913]920. 13. Cibulka, I. Collection of Czechoslo¨ ak Chem. Commun. 1982, 47, 1414]1419. 14. Singh, P. P.; Nigam, R. K.; Sharma, S. P.; Aggarwal, S. Fluid Phase Equilibria 1984, 18, 333]344. 15. Pando, C.; Renuncio, J. A. R.; Calzon, J. A. G.; Christensen, J. J.; Izatt, R. M. J. Solution Chem. 1987, 16, 503]527. 16. Davis, M. I.; Hernandez, M. E. J. Ch. Eng. Data 1995, 40, 674]678.
(Recei¨ ed 5 January 1998; in final form 5 June 1998)
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