962
J. Chem. Eng. Data 2001, 46, 962-966
Property Changes of Mixing for the 1-Butanol + Methanol + 2-Methoxy-2-methylbutane System at 298.15 K and Atmospheric Pressure Alberto Arce,* Eva Rodil, and Ana Soto Department of Chemical Engineering, University of Santiago de Compostela, E-15706 Santiago, Spain
Densities, refractive indices, and speeds of sound at 298.15 K and atmospheric pressure for the 1-butanol + methanol + 2-methoxy-2-methylbutane ternary system have been measured. Excess molar volumes, molar refraction, and isentropic compressibility changes of mixing were determined from experimental measurements. The results were fitted to the Redlich-Kister equation to obtain the coefficients. These property changes were also predicted using several empirical equations from the values of their constituent binary subsystems, and the best results were obtained with the Kohler equation.
Introduction Reformulated gasolines generally require high proportions of oxygenated compounds. Most of those oxygenates are alcohols or ethers containing one to six carbons. 2-Methoxy-2-methylbutane (TAME) has a high blending octane number and appears to be a good candidate for use in gasolines. As part of an ongoing study of the thermophysical properties of mixtures containing TAME, we present some data for the ternary system 1-butanol + methanol + TAME. Molar volume, molar refraction, and isentropic compressibility changes of mixing were calculated from experimental measurements of densities, refractive indices, and speeds of sound at 298.15 K and atmospheric pressure for the 1-butanol + methanol + 2-methoxy-2-methylbutane (TAME) ternary system and for the 1-butanol + TAME binary system. No comparable data in the surveyed literature for the ternary and this binary system have been found. In previous works we have published these properties for TAME + methanol and 1-butanol + methanol systems.1,2 We will also evaluate several empirical equations that allow the prediction of the properties of the ternary system from the corresponding properties of the constituent binary subsystems. If this approach works, an enormous experimental effort could be saved.
Table 1. Densities G, Refractive Indices nD, and Speeds of Sound u of the Pure Components at 298.15 K and Atmospheric Pressure component
F/g‚cm-3 exptl
lit.
u/m‚s-1
nD exptl
lit.
exptl
lit.
1-butanol 0.8060 0.805 75(3) 1.3975 1.397 41(3) 1241 1240(3) methanol 0.7866 0.786 37(3) 1.3264 1.326 52(3) 1102 1102(4) TAME 0.7658 0.765 77(5) 1.3858 1.385 80(5) 1115 not found
water. Refractive indices were measured to within (0.0001 in an Atago RX-1000 refractometer. A Hetotherm thermostat was used to maintain the temperature at (298.15 ( 0.02) K. Results The measured densities (F), speeds of sound (u), and refractive indices (nD) of the 1-butanol + methanol + TAME ternary system and the 1-butanol + TAME binary system at 298.15 K and atmospheric pressure are listed in Table 2, together with the quantities calculated from them: excess molar volumes (VE), molar refraction (∆R), and isentropic compressibility changes of mixing (∆κs). Excess molar volumes were calculated using the expression
VE ) V -
∑x V i
i
(1)
i
Experimental Section Materials. Methanol was supplied by Merck with nominal purity >99.8 mass %. 1-Butanol was supplied by Aldrich with nominal purity >99.9 mass %, and TAME was supplied by Fluka with nominal purity >98.9 mass %. They were degassed using ultrasound. The water contents of the methanol, 1-butanol, and TAME (determined with a Metrohm 737 KF coulometer) were 0.03, 0.1, and 0.02 mass %, respectively. Table 1 lists the densities, speeds of sound, and refractive indices measured for the pure components, together with published values for these properties.3-5 Apparatus and Procedure. The mixtures were prepared by mass, using a Mettler AE240 balance, having a repeatibility of (0.0001 g. The densities and speeds of sound of the mixtures were measured to within (0.0001 g‚cm-3 and (1 m‚s-1, respectively, in an Anton Paar DSA48 densimeter and sound analyzer calibrated with air and
where V is the molar volume of the mixture and Vi and xi are the molar volume and mole fraction, respectively, of component i. The molar refractions (R) were calculated using the Lorentz-Lorenz equation
n2D - 1 V R) 2 nD + 2
(2)
with nD and V being the refractive indices and the molar volume, respectively, and the molar refraction changes of mixing (∆R) were obtained from
∆R ) R -
∑x R i
i
i
where Ri is the molar refraction of pure component i.
10.1021/je010033+ CCC: $20.00 © 2001 American Chemical Society Published on Web 06/19/2001
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Journal of Chemical and Engineering Data, Vol. 46, No. 4, 2001 963 Table 2. Densities G, Speeds of Sound u, Isentropic Compressibilities Ks, Refractive Indices nD , Excess Molar Volumes VE, and Changes of Mixing ∆Ks and ∆R for Mixtures of 1-Butanol (1) + Methanol (2) + TAME (3) at 298.15 K and Atmospheric Pressure x1
x2
F/g‚cm-3
u/m‚s-1
κs/T Pa-1
nD
VE/cm3‚mol-1
∆κs/T Pa-1
∆R/cm3‚mol-1
0.9267 0.8825 0.8392 0.7855 0.7396 0.6731 0.6452 0.5751 0.5297 0.5243 0.4405 0.3827 0.3652 0.2994 0.2687 0.1904 0.1615 0.1135 0.0579 0.1034 0.0929 0.0832 0.0734 0.0641 0.0555 0.0442 0.0354 0.0263 0.0167 0.2000 0.1788 0.1603 0.1465 0.1233 0.1036 0.0900 0.0771 0.0559 0.0380 0.3131 0.2795 0.2496 0.2233 0.1926 0.1606 0.1359 0.1098 0.0855 0.0511 0.3907 0.3429 0.3073 0.2699 0.2367 0.2084 0.1720 0.1323 0.0951 0.0667 0.4812 0.4310 0.3768 0.3335 0.2937 0.2516 0.2136 0.1562 0.1254 0.0663 0.5975 0.5373 0.4638 0.4384 0.3593 0.2840 0.2411 0.1808 0.1339 0.0804
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.8966 0.8052 0.7212 0.6366 0.5559 0.4812 0.3837 0.3066 0.2284 0.1451 0.8000 0.7153 0.6412 0.5861 0.4932 0.4144 0.3601 0.3082 0.2235 0.1519 0.6869 0.6132 0.5475 0.4898 0.4225 0.3524 0.2982 0.2408 0.1877 0.1122 0.6093 0.5348 0.4792 0.4208 0.3692 0.3250 0.2682 0.2063 0.1483 0.1041 0.5188 0.4647 0.4062 0.3596 0.3167 0.2713 0.2303 0.1684 0.1352 0.0714 0.4025 0.3620 0.3124 0.2953 0.2421 0.1913 0.1624 0.1218 0.0902 0.0542
0.7691 0.7710 0.7727 0.7748 0.7766 0.7792 0.7803 0.7830 0.7848 0.7869 0.7883 0.7907 0.7914 0.7941 0.7953 0.7985 0.7996 0.8015 0.8037 0.7899 0.7867 0.7840 0.7815 0.7792 0.7773 0.7750 0.7732 0.7714 0.7695 0.7926 0.7891 0.7862 0.7841 0.7808 0.7782 0.7766 0.7751 0.7726 0.7706 0.7954 0.7917 0.7883 0.7855 0.7825 0.7795 0.7774 0.7752 0.7732 0.7704 0.7971 0.7928 0.7895 0.7862 0.7834 0.7811 0.7783 0.7754 0.7727 0.7707 0.7988 0.7949 0.7908 0.7877 0.7848 0.7819 0.7794 0.7758 0.7739 0.7703 0.8010 0.7971 0.7923 0.7907 0.7858 0.7815 0.7791 0.7758 0.7733 0.7704
1126 1132 1137 1144 1149 1157 1160 1168 1173 1180 1184 1192 1194 1203 1207 1218 1221 1227 1234 1125 1130 1132 1132 1131 1130 1129 1128 1125 1122 1144 1145 1145 1143 1140 1137 1135 1133 1129 1125 1164 1162 1158 1154 1149 1144 1140 1136 1132 1126 1176 1171 1166 1160 1154 1150 1144 1138 1132 1128 1189 1183 1175 1168 1161 1155 1149 1140 1136 1127 1204 1196 1185 1181 1168 1157 1151 1142 1136 1128
1026 1013 1001 987 975 959 953 936 926 913 904 890 886 870 863 845 838 829 817 1001 996 996 999 1003 1007 1012 1017 1024 1032 965 966 971 976 986 994 1000 1005 1015 1025 928 936 945 956 968 980 990 999 1009 1024 908 919 931 945 958 968 981 995 1009 1021 886 899 916 931 945 959 972 991 1001 1022 862 878 899 907 933 956 969 988 1002 1020
1.3870 1.3877 1.3882 1.3889 1.3894 1.3902 1.3905 1.3913 1.3918 1.3924 1.3928 1.3934 1.3936 1.3943 1.3947 1.3955 1.3958 1.3963 1.3969 1.3403 1.3529 1.3611 1.3673 1.3719 1.3752 1.3787 1.3809 1.3827 1.3842 1.3510 1.3606 1.3667 1.3703 1.3750 1.3781 1.3798 1.3812 1.3831 1.3843 1.3614 1.3682 1.3725 1.3755 1.3783 1.3806 1.3820 1.3833 1.3842 1.3852 1.3676 1.3730 1.3760 1.3785 1.3802 1.3815 1.3829 1.3841 1.3849 1.3853 1.3738 1.3772 1.3799 1.3816 1.3827 1.3837 1.3845 1.3853 1.3856 1.3860 1.3807 1.3826 1.3841 1.3845 1.3855 1.3861 1.3863 1.3864 1.3864 1.3863
-0.207 -0.297 -0.365 -0.428 -0.468 -0.508 -0.520 -0.537 -0.541 -0.537 -0.529 -0.508 -0.499 -0.454 -0.427 -0.337 -0.297 -0.221 -0.119 0.037 -0.163 -0.298 -0.396 -0.461 -0.497 -0.510 -0.489 -0.435 -0.332 0.062 -0.162 -0.302 -0.380 -0.469 -0.509 -0.518 -0.511 -0.463 -0.377 0.074 -0.166 -0.314 -0.405 -0.477 -0.517 -0.525 -0.510 -0.468 -0.352 0.074 -0.198 -0.338 -0.438 -0.495 -0.522 -0.529 -0.499 -0.430 -0.345 0.069 -0.158 -0.340 -0.438 -0.494 -0.525 -0.529 -0.498 -0.460 -0.323 0.055 -0.166 -0.349 -0.395 -0.492 -0.528 -0.523 -0.482 -0.417 -0.298
-11 -17 -21 -25 -27 -29 -30 -31 -32 -32 -32 -31 -31 -29 -28 -23 -20 -15 -8 4 -14 -23 -26 -28 -28 -27 -25 -21 -15 5 -15 -24 -28 -30 -30 -29 -28 -23 -18 5 -15 -26 -29 -31 -31 -30 -27 -24 -17 4 -19 -28 -31 -32 -32 -31 -27 -22 -17 2 -16 -28 -32 -33 -33 -31 -28 -25 -16 1 -16 -28 -30 -33 -32 -31 -27 -23 -15
-0.006 -0.009 -0.013 -0.017 -0.021 -0.025 -0.027 -0.031 -0.033 -0.035 -0.036 -0.036 -0.036 -0.034 -0.033 -0.027 -0.024 -0.019 -0.010 0.001 0.001 0.001 0.002 0.002 0.002 0.002 0.002 0.001 0.001 0.002 0.001 0.001 0.001 0.000 -0.000 -0.001 -0.001 -0.001 -0.001 0.004 -0.001 -0.004 -0.006 -0.008 -0.008 -0.009 -0.008 -0.007 -0.005 0.004 -0.013 -0.019 -0.023 -0.023 -0.022 -0.019 -0.014 -0.009 -0.006 0.002 -0.003 -0.008 -0.011 -0.012 -0.013 -0.013 -0.012 -0.011 -0.007 -0.001 -0.008 -0.015 -0.017 -0.021 -0.022 -0.021 -0.018 -0.015 -0.010
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Journal of Chemical and Engineering Data, Vol. 46, No. 4, 2001
Table 2. (Continued) x1
x2
F/g‚cm-3
u/m‚s-1
κs/T Pa-1
nD
VE/cm3‚mol-1
∆κs/T Pa-1
∆R/cm3‚mol-1
0.6763 0.5892 0.5328 0.4730 0.3984 0.3364 0.2976 0.2242 0.1615 0.1090 0.8014 0.7049 0.6303 0.5576 0.4779 0.4272 0.3473 0.3072 0.1926 0.1124 0.8960 0.7943 0.7099 0.6243 0.5508 0.4659 0.3812 0.2774 0.2168 0.1182
0.3237 0.2820 0.2550 0.2264 0.1907 0.1610 0.1425 0.1073 0.0773 0.0522 0.1986 0.1747 0.1562 0.1382 0.1184 0.1059 0.0861 0.0761 0.0477 0.0278 0.1040 0.0922 0.0824 0.0725 0.0639 0.0541 0.0442 0.0322 0.0252 0.0137
0.8021 0.7971 0.7939 0.7906 0.7865 0.7833 0.7813 0.7776 0.7744 0.7717 0.8038 0.7991 0.7954 0.7918 0.7879 0.7856 0.7819 0.7801 0.7749 0.7713 0.8049 0.8004 0.7968 0.7930 0.7897 0.7859 0.7823 0.7779 0.7755 0.7714
1212 1201 1192 1183 1172 1163 1158 1148 1139 1132 1225 1212 1201 1191 1179 1172 1161 1156 1142 1131 1233 1220 1208 1196 1186 1175 1164 1151 1144 1132
848 870 886 904 926 944 955 976 995 1011 830 852 871 891 913 927 948 959 990 1013 818 840 859 881 900 922 944 970 985 1013
1.3846 1.3860 1.3865 1.3869 1.3872 1.3874 1.3874 1.3873 1.3871 1.3868 1.3901 1.3904 1.3903 1.3901 1.3898 1.3895 1.3891 1.3888 1.3880 1.3872 1.3938 1.3934 1.3930 1.3924 1.3918 1.3910 1.3902 1.3892 1.3886 1.3875
0.043 -0.221 -0.345 -0.443 -0.524 -0.558 -0.563 -0.535 -0.465 -0.365 0.023 -0.228 -0.362 -0.454 -0.518 -0.542 -0.553 -0.544 -0.457 -0.326 0.010 -0.215 -0.366 -0.471 -0.523 -0.546 -0.540 -0.501 -0.458 -0.334
0 -20 -27 -32 -34 -34 -33 -30 -25 -19 -0 -18 -26 -31 -32 -33 -32 -30 -25 -17 -0 -16 -25 -30 -32 -32 -31 -28 -25 -17
-0.003 -0.013 -0.018 -0.021 -0.024 -0.024 -0.024 -0.021 -0.017 -0.013 -0.006 -0.014 -0.018 -0.020 -0.022 -0.022 -0.021 -0.020 -0.015 -0.001 -0.006 -0.013 -0.017 -0.020 -0.022 -0.022 -0.021 -0.018 -0.015 -0.009
The speeds of sound through the mixtures (u) and the corresponding densities (F) were used to calculate isentropic compressibilities (κs) using the equation
κs ) u-2F-1
(4)
and the isentropic compressibility changes of mixing (∆κs) were obtained using the expression
∆κs ) κs -
∑φ κ
(5)
i si
i
where κs and κsi are the isentropic compressibilities of the mixture and component i, respectively, and φi is the volume fraction of component i in the mixture as given by
∑x V
φi ) xiVi/
j
(6)
j
j
where j refers to all components of the mixture. For the ternary system, Figure 1 shows the plotted density and refractive index isolines versus the mole fraction of the mixture and the plotted sound velocity isolines versus volume fraction. Figure 2 shows excess molar volume isolines versus mole fraction and isentropic compressibility changes of mixing isolines versus volume fraction. Correlation. The VE, ∆R, and ∆κs data were correlated with the composition data by means of the Redlich-Kister polynomial,6 which for binary mixtures is
∆M ) xixj
∑A (x - x ) K
i
j
k
(7)
K
where ∆M is VE or ∆R and x is the mole fraction, or ∆M is ∆κs and x is the volume fraction, AK is the polynomial coefficient, and K is the number of the polynomial coef-
ficient. For ternary systems the corresponding equation is
∆M123 ) ∆M12 + ∆M32 + ∆M13 + x1x2x3(A + B(x1 - x2) + C(x3 - x2) + D(x1 - x3) + E(x1 - x2)2 + F(x3 - x2)2 + G(x1 - x3)2 + ... (8) where ∆M123 is VE, ∆κs, or ∆R; xi is the mole fraction or volume fraction of component i, according to the parameter being correlated as previously indicated; and ∆Mij is the value of the Redlich-Kister polynomial for the same property fitted to the data for the binary system (i, j). Equations 7 and 8 were fitted to the appropriate parameter-composition data for the binary and ternary systems by least-squares regression, applying Fisher’s F-test to establish the number of coefficients. These coefficients and their mean standard deviations are listed in Table 3 for the binary systems (data for TAME + methanol and 1-butanol + methanol systems were taken from previously papers1,2) and in Table 4 for the ternary system. Prediction. Although it would be desirable to be able to estimate the thermodynamic properties of multicomponent systems from the properties of their pure components, in practice such estimates are often inaccurate due to the effects of mixing. An attractive alternative that limits experimental work to binary mixtures is to evaluate the property changes of mixing of the multicomponent system from the properties of its constituent binary subystems. To assess the viability of this approach for the ternary system studied here, their VE, ∆R, and ∆κs values were predicted from the properties of their constituent binary subsystems by means of empirical equations available in the literature. The equations used were from refs 7-14. These equations are common in the bibliography, have been used in previous papers,15 and are not therefore repeated unnecessarily. For each ternary system, Table 5 lists the standard deviations in the values of VE, ∆R, and ∆κs predicted by each equation.
Journal of Chemical and Engineering Data, Vol. 46, No. 4, 2001 965
Figure 1. Density (g‚cm-3)-mole fraction (a), refractive index-mole fraction (b), and speed of sound (m‚s-1)-volume fraction (c) isolines for 1-butanol + methanol + TAME mixtures at 298.15 K and atmospheric pressure.
Figure 2. Excess molar volumes (cm-3‚mol-1)-mole fraction (a) and isentropic compressibility (T Pa-1) changes of mixing (b) isolines for 1-butanol + methanol + TAME mixtures at 298.15 K and atmospheric pressure.
Conclusions A thermodynamic contribution to the study of 1-butanol + methanol + TAME mixtures has been carried out with the determination of molar volume, molar refraction, and isentropic compressibility changes of mixing from experi-
mental measurements of densities, refractive indices, and speeds of sound at 298.15 K and atmospheric pressure. Excess molar volumes were negative, reaching a minimum around -0.56 cm3‚mol-1; only the binary system 1-butanol + methanol shows positive but very low values
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Journal of Chemical and Engineering Data, Vol. 46, No. 4, 2001
Table 3. Polynomial Coefficients (AK) and Standard Deviations (σ) Obtained for the Fits of Eq 7 to the VE-, ∆Ks-, and ∆R-Composition Data for the Binary Systems (for ∆Ks, System Compositions Were in Volume Fraction, Oi) property VE/cm3‚mol-1 ∆κs/T Pa-1 ∆R/cm3‚mol-1
A0
A1
A2
A3
A4
1-Butanol + Methanol 0.2673 -0.2032 18.9 -19.7 -7.2 0.0079 0.0445 -0.0582
σ 0.001 0.1 0.001
TAME + Methanol VE/cm3‚mol-1 -1.8559 -0.1787 -0.3252 -0.2319 0.001 ∆κs/T Pa-1 -84.6 -52.6 -62.4 -128.4 0.2 ∆R/cm3‚mol-1 -0.0857 0.001 VE/cm3‚mol-1 ∆κs/T Pa-1 ∆R/cm3‚mol-1
1-Butanol + TAME -2.1579 0.1624 -0.6287 0.4640 -126.1 19.0 -52.2 67.0 -0.1366 -0.0620
0.001 0.1 0.001
Table 4. Polynomial Coefficients and Standard Deviations (σ) Obtained for the Fits of Eq 8 to the VE-, ∆Ks-, and ∆R-Composition Data for the Ternary System 1-Butanol (1) + Methanol (2) + TAME (3) (for ∆Ks, System Compositions Were in Volume Fraction, Oi) property
A
B
C
D
σ
VE/cm3‚mol-1 ∆κs/T Pa-1 ∆R/cm3‚mol-1
-1.2775 -60.64 0.3407
0.6392 -12.04 -0.2812
0.5977 23.84 0.0284
0.0415 -35.88 -0.3096
0.008 0.54 0.005
Table 5. Standard Deviations in the Excess Molar Volumes and Changes of Mixing in Molar Refraction and Isentropic Compressibility Predicted for the Ternary Mixture at 298.15 K and Atmospheric Pressure
Radojkovic Rastogi Kohler Jacob and Fitzner Tsao and Smitha Tsao and Smithb Tsao and Smithc Colinet Toopa Toopb Toopc Scatcharda Scatchardb Scatchardc
VE123
E ∆Ks123
∆RE123
cm3‚mol-1
T Pa-1
cm3‚mol-1
0.027 0.165 0.011 0.030 0.063 0.071 0.087 0.231 0.036 0.037 0.080 0.164 0.197 0.111
1 10 2 2 5 6 5 14 3 4 5 10 10 7
0.010 0.006 0.010 0.009 0.013 0.014 0.011 0.013 0.011 0.011 0.011 0.017 0.021 0.011
a 1-Butanol is the asymmetric component. b Methanol is the asymmetric component. c TAME is the asymmetric component.
of this property. Isentropic compressibility changes of mixing are also negative, reaching a minimum of approximately -34 T Pa-1 except for the previously cited binary system. Values of molar refraction changes of mixing are so small that it is difficult to establish any conclusion. In all cases the data were satisfactorily correlated with the composition data by the Redlich-Kister polynomial.
The equation of Kohler, followed by those of Radojkovic and Jacob and Fitzner, is the most adequate for the prediction of the properties considered in this paper and usable in the absence of ternary data. The equations of Colinet, Rastogi, and Scatchard produce high deviations and are not recommended. Limitations are found using the equations of Toop and Tsao and Smith. The conclusions obtained for the prediction of the property changes of mixing (and therefore for the direct physical properties also) from the constituent binary systems are repetitive for mixtures with ethers and alcohols.2,15,16 Literature Cited (1) Arce, A.; Mendoza, J.; Ageitos, J.; Soto, A. Densities, Refractive Indices, Speeds of Sound, and Isentropic Compressibilities of Water + Methanol + 2-Methoxy-2-methylbutane at 298.15 K. J. Chem. Eng. Data 1996, 41, 724-727. (2) Arce, A.; Rodil, E.; Soto, A. Physical Properties of the Ternary System 1-Butanol + Methanol + 2-Methoxy-2-methylpropane at 298.15 K: Measurement and Prediction. J. Chem. Eng. Data 1999, 44, 1028-1033. (3) Riddick, J. A.; Bunger, W. B.; Sakano, T. Organic Solvents, 4th. ed.; John Wiley: New York, 1986. (4) Aminabhavi, T. M.; Aralaguppi, M. Y.; Harogoppad, Sh. B.; Balundgi, R. H. Densities, viscosities, refractive indices, and speeds of sound for methyl acetoacetate + aliphatic alcohols (C1C8). J. Chem. Eng. Data 1993, 38, 31-39. (5) Linek, J. Excess volumes and refractive indices in benzene-tertamyl methyl ether and cyclohexane-tert-amyl methyl ether systems at 298.15 K. Collect. Czech. Chem. Commun. 1987, 52, 2839-43. (6) Redlich, O. J.; Kister, A. T. Algebraic representation of thermodynamic properties and the classification of solutions. Ind. Eng. Chem. 1948, 40 (2), 345-348. (7) Radojkovic, N.; Tasic, A.; Grozdanic, D.; Djordjevic, B.; Malic, M. Mixing cell indirect measurements of excess volume. J. Chem. Thermodyn. 1977, 9, 349-352. (8) Rastogi, R. P.; Nath, J.; Das, S. S. Thermodyamic properties of ternary mixtures. 2. The excess volumes of mixing of ternary mixtures of cyclohexane, aromatics, and halomethanes. J. Chem. Eng. Data 1977, 22, 249-252. (9) Kohler, F. Estimation of the thermodynamic data for ternary system from the corresponding binary systems. Monatsh. Chem. 1960, 91, 738-740. (10) Jacob, K. T.; Fitzner, K. The estimation of the thermodynamic properties of ternary alloys from binary data using the shortest distance composition path. Thermochim. Acta 1977, 18, 197-206. (11) Tsao, C. C.; Smith, J. M. Heat of mixing of liquids. Applied thermodynamics. Chem. Eng. Prog., Symp. Ser. 1953, 49, 107117. (12) Colinet, C. Ph.D. Thesis, University of Grenoble, France, 1967. (13) Toop, G. W. Predicting ternary activities using binary data. Trans. TMS-AIME 1965, 223, 850-855. (14) Scatchard, G.; Ticknor, L. B.; Goates, J. R.; McCartney, E. R. Heats of mixing in some nonelectrolyte solutions. J. Am Chem. Soc. 1952, 74, 3721-4. (15) Arce, A.; Rodil, E.; Soto, A. Molar Volumes, Molar Refractions and Isentropic Compressibilities of (Ethanol + Methanol + 2-Methoxy-2-methylpropane) and (Ethanol + Methanol + 2-Methoxy-2-methylbutane) at 298.15 K. J. Chem. Eng. Data 1997, 42, 721-726. (16) Arce, A.; Martı´nez-Ageitos, J.; Rodil, E.; Soto, A. Thermodynamic Behaviour of Ethanol + Methanol + 2-Ethoxy-2-Methylpropane System. Physical Properties and Phase Equilibria. Fluid Phase Equilib. 1999, 165, 121-139. Received for review February 12, 2001. Accepted April 30, 2001. This work was partly supported by the Ministerio de Ciencia y Tecnologı´a (Spain) under Project PPQ2000-0969.
JE010033+
