12644
J. Phys. Chem. 1996, 100, 12644-12648
Density Calculation of Compressed Liquid Mixtures Using LIR along with Mixing and Combining Rules Gholamabbas Parsafar* and Nassrin Sohraby Department of Chemistry, Isfahan UniVersity of Technology, Isfahan, Iran 84156 ReceiVed: January 24, 1996; In Final Form: April 28, 1996X
In this work the Lennard Jones (12-6) potential and the Lorentz-Bertholet combining rules have been used to calculate the linear isotherm regularity (LIR) parameters for a hypothetical pure fluid in which all the molecular interactions are unlike one another. It can be shown that whenever the sizes and potential well depths are the same for the two components, these parameters are obtained from the “mean geometric approximation” or simply “MGA”. The values of these parameters have been used to calculate the density of binary and ternary mixtures, for mixtures of inert gases, nonpolar, polar, and strongly hydrogen-bonded systems, at different temperatures, pressures, and mole fractions. The calculated results show a maximum deviation less than 1.6%, except for the strongly hydrogen-bonded system of CH3OH/H2O for which the maximum deviation is 2.4%. For the cases in which there is a remarkable difference between molecular sizes and/or the potential well depths of the components, or when a more accurate result is needed, the parameters have been calculated by using an equation more accurate than the MGA. However, the results reveal that the MGA is an excellent approximation for all systems including atomic-atomic, polar-nonpolar, hydrocarbons, and strongly hydrogen-bonded mixtures. Even in the cases where the value of the molecular sizes and/or the potential well depths are quite different, the deviation given by MGA is small. By use of the temperature dependence of LIR parameters, the density has also been calculated at temperatures other than that of the pure components. In this case the calculated values are in excellent agreement with the experimental values.
Introduction
The EOS is
Despite the fact that the description of dense fluids at the molecular level is very complicated, they obey some relatively simple rules (regularities). Among these we may refer to (1) the Tait equation that was presented by P. G. Tait about 100 years ago,1 (2) the near linearity of the vapor pressure diagram from the triple point to the critical point,2 (3) the linearity of bulk modulus vs pressure for each isotherm,3 (4) the near linear plot of the average density of a saturated liquid and its vapor vs temperature,4 (5) the Zeno line or the linear plot of T vs density at unit compression factor,5 (6) the existence of a common bulk modulus point in the plot of the bulk modulus vs molar volume that is known as Huang and O’Connell’s law,6 (7) the near linear plot of pressure vs temperature for each isochore in a wide density range from ideal gas to the dense liquid,7 (8) the common compression factor point in which the isotherms of (Z - 1)V2 against F2 intersect at a single point,8 (9) the common bulk modulus point,8 (10) the linear isotherm regularity (LIR),9 and (11) the linearity of the inverse isobaric expansivity vs pressure for each isochore of dense fluid.10 Recently, a new equation of state (EOS) was introduced for dense fluids, which has been successfully applied for nonpolar, polar, quantum, and strongly hydrogen-bonded fluids. The EOS has been also applied for dense fluid mixtures.11 This regularity states that the plot of (Z - 1)V2 vs F2 is linear for each isotherm where Z ) pV/RT is the compressibility factor and F ) 1/V is the molar density. This rule can be used for liquids (T < Tc) and for supercritical dense fluids (T > Tc) and is valid for densities greater than the Boyle density and for temperatures less than twice the Boyle temperature.9 X
Abstract published in AdVance ACS Abstracts, June 15, 1996.
S0022-3654(96)00239-0 CCC: $12.00
(Z - 1)V2 ) A + BF2
(1)
in which A and B are temperature dependent parameters. The temperature dependencies of A and B are given as
A ) A′′ - A′/RT
(2a)
B ) B′/RT (2b) where A′ and B′ are related to the intermolecular attraction and repulsion forces, respectively, while A′′ is related to the nonideal thermal pressure.9 The one-fluid approximation shows that this EOS can be used for mixtures. However, it must be borne in mind that in this case Amix and Bmix are both composition and temperature dependent parameters. The temperature dependencies of these two parameters for a mixture is11 Amix ) A′′ - A′/RT
(3a)
Bmix ) B′′ + B′/RT
(3b)
Equation 3b shows a small breakdown in the one-fluid approximation that is unlike that of the pure fluid; the intercept of B against 1/T, (i.e., B′′) is not zero for a mixture. The composition dependencies of the parameters are as follows:11
Bmix ) ∑ Bijxixj
(4a)
(A/B)mix ) ∑(A/B)ijxixj
(4b)
ij
ij
The values of (A/B)ij and Bij when i ) j can be obtained from experimental p-V-T data of pure fluids. In this work the Bij and (A/B)ij, for i * j, have been related to the pure components © 1996 American Chemical Society
Compressed Liquid Mixtures
J. Phys. Chem., Vol. 100, No. 30, 1996 12645
EOS parameters and their Lennard-Jones (LJ) potential parameters. The parameters A′ and B′ are proportional to σ6 and σ12, respectively, and A′′ is related to the nonideal thermal pressure, where and σ are the LJ(12,6) potential parameters. The purpose of this work is to calculate the density of dense fluid mixtures by using the experimental data of their pure components within a reasonable precision. Mixing Rules. Recently, much attention has been paid to the process of generalizing a pure component equation of state to mixtures. This generalization may be applied by using mixing and combining rules. The mixing rule may be defined as a set of rules needed to generalize the EOS of pure fluids to their mixtures. The one-fluid mixing rule of van der Waals is the most commonly used mixing rule, by which the van der Waals EOS can be used for fluid mixtures as well, but the parameters of the EOS are composition dependent. This article uses this mixing rule along with the LIR EOS for mixtures. Combining Rules. The rules that relate the interaction parameters of a mixture to those of its components are called combining rules, some of which are as follows. 1. The Lorentz semiempirical combining rule12 is
σij ) (σii + σjj)/2
(5a)
TABLE 1: Values of the H Parameter for Different Binary Mixtures at T ) 298.15 K mixture Ne-Ara Ar-Kra Kr-Xea N2-O2a CH4-Ara CO-CO2a CF4-SF6a C2H6-C3H8a N2-COa H2O-CH3OHb a
H
[(σ11 + σ22)/2σ]-6H
[(σ11 + σ22)/2σ]12
1.0772 1.0297 1.0542 1.0047 1.0091 1.0913 1.0351 1.0591 1.0015 1.0104
1.0409 1.0273 1.0409 1.0039 0.9993 1.0663 1.0157 0.9664 1.0013 0.9600
1.071 1.005 1.026 1.002 1.020 1.047 1.039 1.206 1.000 1.099
Data from ref 25. b Data from ref 26.
A′12/(A′11A′22)0.5 ) [(σ11 + σ22)/2σ]6
The parameter A′′ is proportional to b2 for a van der Waals fluid,11 where b is the van der Waals covolume. Thus, A′′ is proportional to σ6. Therefore,
A′′12/(A′′11A′′22)0.5 ) [(σ11 + σ22)/2σ]6 A12/(A11A22)0.5 ) [(σ11 + σ22)/2σ]6H
(5b)
where η is σjj/σii. This rule is accurate for hard spheres mixtures. 2. The mean geometric rule for potential well depths12 is
ij ) (iijj)1/2
(6a)
(11b)
Using eqs 11a and 11b, we find that
or
σij ) (1 + η)σii/2
(11a)
(11c)
where
H ) (1 - 12/kT)/[(1 - 11/kT)(1 - 22/kT)]0.5 By use of eq 9 along with eqs 11a-c the following relation can be deduced:
(A/B)12/[(A/B)11(A/B)22]0.5 ) [(σ11 + σ22)/2σ]-6H (12)
or
ij ) iiR1/2
(6b)
where R is defined as jj/ii. A more accurate form of this rule is13
(7) 12 ) (1 - k12)(1122)1/2 where k12 shows the deviation from the hard sphere model. The value of this parameter is close to zero for similar components. However, its value may be as large as 0.1 for quite different components. Calculation of LIR Parameters for Dense Fluid Mixtures Owing to the fact that the LJ(12,6) potential describes the effective pair potential of a dense fluid fairly well,9 this potential is used in this work. The parameter B in eq 1 is related to the repulsion term of the potential. Therefore, it is proportional to σ12. By application of the Lorentz combining rule to a binary mixture, the following relations are obtained:
B12/(B11B22)0.5 ) {[(σ11 + σ22)/2]12/(σ11σ22)6}[12/(1122)0.5] (8)
If the molecular sizes and the potential well depths of the mixture components are approximately the same, eqs 9 and 12 are simplified as
B12 ) (B11B22)0.5
(13a)
(A/B)12 ) [(A/B)11(A/B)22]0.5
(13b)
These results will be refered to as the “mean geometric approximation” or “MGA” from now on. However, the results given in Table 1 show that the approximation is reasonable for many different binary mixtures (the molecular parameters are taken from ref 25). This work calculates the density of some binary mixtures and a ternary mixture using this approximation and the LIR. The results will be presented in the following section. Comparison with Experiment The relation between parameters Bmix and (A/B)mix in LIR and the composition of a binary mixture can be shown to be
Bmix ) B11x12 + 2x1x2B12 + B22x22
(14a)
By use of the Bertholet combining rule, eq 8 reduces to
B12/(B11B22)0.5 ) [(σ11 + σ22)/2σ]12
(9)
σ ) (σ11σ22)0.5
(10)
where
As it was mentioned before, the parameter A of the LIR is related to A′ and A′′ (see eq 13), where A′ is related to the attraction term of the effective pair potential. If the LJ(12,6) potential along with the Lorentz and Bertholet combining rules are used simultaneously, the following relation can be written:
(A/B)mix ) (A/B)11x12 + 2x1x2(A/B)12 + (A/B)22x22 (14b) Equations 14a and 14b are accurate if the molecules are randomly distributed in the mixture. By use of the values obtained by the MGA for the Bmix and (A/B)mix (see eqs 13a and 13b), the values of Amix and Bmix can be calculated from the pure component LIR parameters at the same temperature by inserting the MGA values for B12 and (A/B)12 into eqs 13a and 13b, respectively. The calculated density for an Ar/Kr mixture at 134.32 K has been compared with the experimental data in Table 2 (ref 16). In this table the deviation in density
12646 J. Phys. Chem., Vol. 100, No. 30, 1996
Parsafar and Sohraby
TABLE 2: Comparison between the Calculated and Experimental Values of an Argon-Krypton Mixture Density at 134.32 K (All Densities are in mol cm-3) Fexpa
p, MPa
Fcal
4.62 8.74 15.66 24.03 28.36 36.29 44.70 50.97 59.31 66.97
0.028 03 0.028 37 0.028 91 0.029 42 0.029 66 0.030 08 0.030 48 0.030 76 0.031 11 0.031 40
x ) 0.277 0.028 00 0.028 31 0.028 78 0.029 28 0.029 52 0.029 93 0.030 32 0.030 60 0.030 94 0.031 23
3.30 7.68 10.41 24.21 30.57 37.25 46.90 52.65 59.95 65.37
0.027 82 0.028 32 0.028 60 0.029 75 0.030 18 0.030 60 0.031 12 0.031 41 0.031 76 0.031 99
x ) 0.485 0.027 88 0.028 31 0.028 55 0.029 60 0.030 01 0.030 40 0.030 91 0.031 20 0.031 53 0.031 77
a
[|Fcal - Fexp|/Fcal]100 0.07 0.19 0.44 0.47 0.51 0.46 0.51 0.53 0.54 0.54 0.22 0.02 0.18 0.50 0.57 0.65 0.68 0.70 0.71 0.71
TABLE 4: Comparison for an Equimolar Ternary Mixture of Octane, Isooctane, and 1-Octene at 298.15 K p, MPa
Fexpa
Fcal
[|Fcal - Fexp|/Fcal]100
0.1 26.2 51.4 102.9 149.0 198.8 249.4 303.7 349.6 399.1 449.6 499.6
0.6926 0.7146 0.7309 0.7574 0.7761 0.7931 0.8082 0.8224 0.8335 0.8440 0.8542 0.8637
0.7046 0.7212 0.7350 0.7589 0.7768 0.7938 0.8092 0.8240 0.8355 0.8470 0.8579 0.8681
1.71 0.92 0.56 0.19 0.10 0.09 0.12 0.20 0.24 0.36 0.43 0.51
a
Reference 16.
is observed to be less than 0.72%. The same calculations have been done for different binary mixtures, specifically for the inert
Reference 24.
gases, polar, nonpolar, and strongly hydrogen-bonded molecules. The results are tabulated in Table 3. On the whole the maximum deviation observed in the density is less than 1.6%, with the exception of the strongly hydrogen-bonded system (i.e., CH3OH/H2O) where the maximum deviation is 2.5 percent and C6H14/C16H34 with 3.4%. The same calculations have been done for a ternary mixture, i.e., n-C8H18, iso-C8H18, and 1-C8H16 having an equimolar composition at 298.15 K (ref 24). The results shown in Table 4, which are in excellent agreement with the experimental data, indicate that the maximum deviation in the calculated density is 1.71%. Such calculations show that
TABLE 3: Comparison for Various Binary Mixtures at Given Temperatures mixture Ar + Kra
CS2 + CH2Cl2b
C6H6 + C6H5Clc
CH3CN + C6H6d
a
xi
T, K
0.277 0.698 0.787 0.485 0.485 0.30014 0.40137 0.51858 0.59792 0.70178 0.25 0.50 0.75 0.25 0.50 0.75 0.22439 0.30296 0.39311 0.48345 0.61058 0.71127 0.81607 0.22439 0.30296 0.39311 0.48345 0.61058 0.71127 0.81607 0.22439 0.30296 0.39311 0.48345 0.61058 0.71127 0.81607 0.22439 0.30296
134.32 142.68 147.08 298.15
298.15 323.15 298.15
308.15
318.15
328.15
100[|Fexp - Fcal|/Fcal]avj
∆p, MPa
0.36(0.48) 0.35(0.54) 0.26(0.51) 0.44(0.71) 0.50(0.83) 0.67(0.93) 0.80(1.04) 0.83(1.12) 0.78(1.06) 0.73(0.96) 0.48(0.59) 0.72(0.84) 0.57(0.67) 0.85(1.01) 1.17(1.34) 0.97(1.16) 0.90(0.95) 1.12(1.19) 1.25(1.35) 1.36(1.44) 1.34(1.43) 1.22(1.31) 0.91(0.95) 0.95(0.97) 1.10(1.14) 1.26(1.32) 1.36(1.41) 1.39(1.44) 1.21(1.31) 0.90(0.95) 0.93(0.98) 1.10(1.17) 1.28(1.36) 1.36(1.44) 1.46(1.49) 1.28(1.36) 0.92(0.98) 0.93(0.98) 1.21(1.27)
4.62-66.97 5.52-68.07 3.66-76.60 4.49-150.65 6.11-120.45 0.83-99.94 0.71-100.02 0.31-100.53 0.09-100.22 0.85-100.9 0.2-206.5 0.2-207.6 0.2-180.1 0.2-200.0 0.2-206.9 0.2-204.2 1.297-39.3 0.003-38.605 0.349-37.821 0.371-39.168 0.891-39.5 0.253-37.097 0.544-39.469 0.327-20.8 0.368-20.738 0.349-22.860 0.497-22.383 0.117-18.152 0.280-39.311 0.319-39.48 0.521-39.302 0.375-39.29 0.391-39.17 0.276-39.47 0.321-39.393 0.442-39.556 0.388-39.534 0.497-39.206 0.332-39.5
mixture
C6H5NO2 + C6H6e
C6H6 + C6H5Clf
C16H34 + C6H14g
CH3OH + H2Oh
xi 0.39311 0.48345 0.61058 0.71127 0.81607 0.2 0.4 0.8 0.2 0.4 0.8 0.2 0.4 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
T, K
293.15 303.15 313.15 303.15
298.15
323.15
348.15
283.15 298.15 323.15
100[|Fexp - Fcal|/Fcal]avj
∆p, MPa
1.32(1.42) 1.44(1.54) 1.44(1.54) 1.29(1.41) 0.94(1.00) 1.00(1.02) 1.38(1.41) 0.73(0.75) 1.26(1.32) 1.44(1.51) 0.70(0.75) 1.22(1.31) 1.34(1.48) 0.51(0.81) 1.10(1.19) 1.50(1.54) 1.07(1.17) 0.64(0.73) 2.19(2.46) 2.60(2.86) 2.28(2.48) 1.25(1.38) 2.20(3.07) 2.63(3.05) 2.13(2.59) 1.20(1.49) 2.16(2.98) 2.44(3.34) 1.84(2.67) 1.09(1.59) 1.71(2.14) 1.87(2.19) 1.21(1.34) 1.86(2.22) 1.99(2.24) 1.49(1.63) 1.58(2.20) 2.25(2.36) 1.80(1.90)
0.123-39.444 0.441-39.58 0.345-39.6 0.313-39.400 0.410-39.514 0.1-30.0 0.1-30.0 0.1-30.0 0.1-140.0 0.1-200.0 0.1-170.0 0.1-30.0 0.1-30.0 0.1-30.0 0.1-180 0.1-200 0.1-200 0.1-200 0.1-143.4 0.1-89.1 0.1-61.6 0.1-46.0 0.1-301.3 0.1-202.0 0.1-185.4 0.1-158.3 0.1-490.8 0.1-352.1 0.1-325.0 0.1-304.3 0.1-206.5 0.1-206.3 0.1-205.5 0.1-207.3 0.1-207 0.1-207 0.1-207 0.1-207 0.1-221
Reference 16. b Reference 17. c Reference 18. d Reference 19. e Reference 20. f Reference 21. g Reference 22. h Reference 23. i First components mole fraction. j Maximum deviation in parentheses.
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J. Phys. Chem., Vol. 100, No. 30, 1996 12647
TABLE 5: Comparison for Temperature(s) Other Than That of the Pure Components mixture C6H5NO2 + C6H6
b
C6H6 + C6H5Cla
CH3OH + H2Oc
xd
T, K
100[|Fexp - Fcal|/Fcal]ave
∆p, MPa
0.2 0.4 0.6 0.8 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
303.15
0.95(1.18) 0.89(1.34) 1.54(1.87) 0.01(0.56) 1.03(1.15) 1.29(1.42) 1.06(1.22) 1.53(1.82) 1.64(1.96) 1.54(1.73) 2.70(3.96) 1.61(1.96) 0.06(1.17)
0.1-140.0 0.1-200.0 0.1-200.0 0.1-170.0 0.2-206.0 0.2-206.9 0.2-204.2 0.2-192.0 0.2-206.0 0.2-187.2 0.1-207.3 0.1-207.3 0.1-207.7
323.15 348.15 298.15
TABLE 7: Comparison between the Molar Volume Given by Baonza, LIR, and Experiment, in cm3 mol-1, for Different Mixtures of Krypton-Methane at 142.68 K, Where x Is Mole Fraction of Methanea p, MPa
a
Reference 18. b Reference 20. c Reference 23. d First components mole fraction. e Maximum deviation in parentheses.
TABLE 6: Calculated Density Given by MGA (eqs 13a and 13b) and Lorentz-Bertholet Combining Rules (eqs 9 and 12) for Methanol/Water Mixture at x ) 0.25 and T ) 283.15 K p, bara
[100(Fcal - Fexp/Fcal)]MGAb
[100(Fcal - Fexp/Fcal)]L,Bc
1.0 33.1 168.9 514.3 687.7 859.4 1029.7 1193.8 1373.8 1551.0 1720.6 1891.9
1.07 1.10 1.21 1.50 1.59 1.67 1.75 1.83 1.87 1.90 1.94 1.97
0.51 0.54 0.64 0.90 0.98 1.05 1.12 1.19 1.23 1.26 1.29 1.31
a
b
c
Reference 23. Mean geometric approximation. Lorentz-Bertholet combining rules.
a
Vexp
Vcal (LIR)
Vcal (Baonza)
2.52 7.41 24.91 38.97 45.03 58.41 72.19 85.98
39.08 38.47 36.84 35.91 35.59 34.91 34.32 33.81
x ) 0.355 38.90 38.35 36.82 35.89 35.55 34.88 34.28 33.76
2.60 8.11 21.74 36.35 44.14 57.72 72.88 85.29
39.98 39.23 37.83 36.73 36.26 35.54 34.74 34.24
x ) 0.525 39.78 39.10 37.78 36.71 36.23 35.50 34.82 34.32
39.98 39.23 37.81 36.68 36.19 35.44 34.74 34.24
11.79 15.01 28.77 36.35 43.52 50.13 77.70 98.67
39.34 39.11 37.85 37.29 36.83 36.44 35.12 33.93
x ) 0.661 39.36 39.01 37.80 37.25 36.78 36.40 35.07 34.32
39.44 39.07 37.77 37.19 36.70 36.30 34.94 33.76
39.08 38.47 36.81 35.83 35.47 34.78 34.16 33.63
The experimental data are taken from ref 27.
by Baonza et al.27 and with the experimental values in Table 7. The results given in both methods are comparable. Results and Discussion
the MGA is a reasonable approximation. Since the temperature dependencies of Amix and Bmix parameters are known (see eqs 3a and 3b), then the density of mixtures can be calculated even at temperatures different from that of the pure components. We have used the experimental data for C6H6 and C6H5NO2 for 293.15 and 313.15 K isotherms to calculate the density of their mixture at 303.15 K (ref 20). The calculated results are compared with the experimental results in Table 5. Such calculations have also been performed for C6H6/C6H5Cl mixtures using the experimental data of the pure component at 298.15 and 398.15 K to predict the density at temperatures of 323.15 and 348.15 K (see Table 5 and ref 18). Such calculations for the CH3OH/H2O mixture with different mole fractions have been done by using the 283.15 and 373.15 K data for the 298.15 K isotherm, for which the results are given in Table 5 (ref 23). For the last mixture the maximum deviation is about 2.7%. As mentioned before, in the cases in which the molecular sizes and/or the potential well depths of the components are remarkably different from each other, or if a more accurate result is needed, the parameters B12 and (A/B)12 may be calculated without using MGA, i.e., using the Lorentz-Bertholet combining rules (eqs 9 and 12). In Table 6 the calculated values of the density using the Lorentz-Bertholet combining rules and the MGA for a methanol-water mixture have been compared for different pressures at 283.15 K. It can be seen that the use of the Lorentz-Bertholet combining rules reduces the density deviation. Finally, we have calculated the molar volume for different mixtures of Kr-CH4 at 142.68 K using the mean geometric approximation. The results are compared with those calculated
An EOS that is generally used for dense fluids (liquids) is the Tait-Marnaghan equation of state (TME). Even though this EOS has been widely applied for subcritical fluids, it is valid for supercritical dense fluids even in a wider pressure range.10 The LIR EOS that has been recently introduced for dense fluids (F > FB, where FB is the Boyle density) has some advantages over the TME, some of which have been already introduced11 and others that are as follows. (i) The LIR enables us to predict many empirically known regularities and has been used to predict some new one’s.10 (ii) By application of the LIR to mixtures as shown in this article, the density and the EOS parameters for the mixture equation of state are calculated simply from pure component data. This calculation can be done at the same temperature (Tables 2-4) and also at temperatures different from that of the pure components (Table 5). The results given in Table 5 show that the average deviation in density is less than 3%. In the cases where more accurate results are required, the molecular parameters, i.e., and σ of the LJ(12,6) potential, should be known so that the eqs 11c and 12 can be used. In this case the deviation is reduced (see Table 6). From the calculated results the following conclusions may be obtained. n-Hexane-n-Hexadecane. Although there is a fairly large difference between the vaporization enthalpies (which is related to the well depths14) and between the molecular sizes of the components, the maximum deviation in the calculated density using MGA is 3.4%. [The vaporization enthalpies of n-hexane and n-hexadecane are 29.353 and 51.469 kJ mol-1, respectively.15 Therefore, the difference between the values of vaporization enthalpies is about 43%.] From the investigation
12648 J. Phys. Chem., Vol. 100, No. 30, 1996 of this mixture, we may conclude that the MGA is useable for most mixtures. By use of this approximation, the mixture density can be calculated over large pressure and temperature ranges. The deviation observed is only a few percent. Methanol-Water. In this mixture the calculated density using the MGA has a maximum deviation of 2.4%. This deviation is due to the following factors. First, if the interaction energy between different pairs has a significant differences, it is expected that the molecules are not randomly distributed in the systems, especially at low temperatures, and hence the quadratic dependencies of (A/B)mix and Bmix on mole fraction are not held, and use of eqs 4a and 4b leads to a remarkable deviation. In this system the hydrogen bonding between two water molecules is stronger than that in methanol-methanol and in methanol-water. This differences in the strength of hydrogen bonding is due to the differences in the vaporization entropies. The values of vaporization entropies for methanol and water at the normal boiling point are 106 and 109 J mol-1 K-1, respectively.15 From the greater value of the vaporization entropy of water, we may conclude that water molecules in the liquid state are more ordered than methanol. Such a highly structural ordering is due to a stronger hydrogen bonding in water. It is expected that the aggregation of water molecules with each other reduces when its mole fraction becomes less. Therefore, we may expect that the reduction of water concentration causes a more random distribution of molecules. The calculated density then has a better agreement with the experimental data. Such an expectation is in accordance with the results given in Table 5. Other factors that may cause the deviation in the calculated density are the differences between the well depths of two kinds of molecules and their sizes, which are 6% and 22%, respectively.26 However, from the calculated results given in Table 5 for different binary mixtures of methanol-water, we may conclude that the deviation due to such factors is not significant.
Parsafar and Sohraby References and Notes (1) Dymond, J. H.; Malhotra, R. Int. J. Thermophys. 1988, 9, 941. (2) Scott, R. L. J. Chem. Educ. 1953, 30, 542. (3) Song, Y.; Caswell, B. Int. J. Thermophys. 1991, 12, 855. (4) Rowlinson, J. S.; Swinton, F. L. Liquid and Liquid Mixtures, 3rd ed.; Butterworth: London, 1982; pp 72-73. (5) Xu, J.; Herschbach, D. R. J. Phys. Chem. 1992, 96, 2307. (6) Boushehri, A.; Tao, F.-M.; Mason, E. A. J. Phys. Chem. 1993, 97, 2711. (7) Battite, J. A.; Stockmayer, W. H. Rep. Prog. Phys. 1940, 7, 195. (8) Najafi, B.; Parsafar, G. A.; Alavi, S. J. Phys. Chem. 1995, 99, 9248. (9) Parsafar, G. A.; Mason, E. A. J. Phys. Chem. 1993, 97, 9048. (10) Alavi, S.; Parsafar, G. A.; Najafi, B. Int. J. Thermophys. 1995, 16, 1421. (11) Parsafar, G. A.; Mason, E. A. J. Phys. Chem. 1994, 98, 1962. (12) Wakayama, Shin-ichi; Koshi, M.; Matsui, H. Bull. Chem. Soc. Jpn. 1991, 64, 3329. (13) Geoffry, C. M. Intermolecular forces; Oxford University Press: Oxford, 1987; p 519. (14) Boushehri, A.; Mason, E. A. Int. J. Thermophys. 1993, 14, 685. (15) West, R. C. Handbook of Chemistry and Physics, 55th ed.; CRC Press: Boca Raton, FL, 1974. (16) Barrelo, S. F.; Calado, J. C. G.; Clancy, P. J. Phys. Chem. 1982, 86, 1722. (17) Takagi, T.; Teranishi, H. J. Chem. Thermodyn. 1982, 14, 1167. (18) Takagi, T.; Teranishi, H. J. Chem. Thermodyn. 1982, 14, 577. (19) Colin, A. C.; Cancho, S.; Rubio, R. G.; Copostizo, A. J. Phys. Chem. 1993, 97, 10796. (20) Garcia Baonza, V.; Caceres Alonso, M.; Arsuaga ferreras, J.; NunezDelgado, J. J. Chem. Thermodyn. 1991, 23, 231. (21) Kubta, H.; Tsuda, S.; Murata, M. Phys. Chem. Jpn. 1979, 2, 59. (22) Kashiwagi, H.; Fukunaga, T.; Tanaka, Y.; Kubota, H.; Makita, T. J. Chem. Thermodyn. 1983, 15, 567. (23) Dymond, J. H.; Young, K. J. J. Chem. Thermodyn. 1979, 11, 887. (24) Dymond, J. H.; Malhotra, R. J. Chem. Thermodyn. 1988, 20, 60. (25) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids, 2nd printing; Wiley: New York, 1964; pp 1110-1112. (26) Leewen, M. E. V. Fluid Phase Equilib. 1994, 99, 1. (27) Baonza, V. G.; Orbis, F.; Cace´res, M.; Nunez, J. J. Phys. Chem. 1995, 99, 5166.
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