1962
J . Phys. Chem. 1994,98, 1962-1967
Linear Isotherms for Dense Fluids: Extension to Mixtures Cholamabbas Parsafar*Jand E. A. Mason' Department of Chemistry, Brown University, Providence, Rhode Island 0291 2 Received: August 24, 1993; In Final Form: November 16, 1993'
A new linear regularity recently reported for pure dense fluids, that (Z - l)02 is linear with respect to p2, is found from experiment to be valid for mixtures as well. A simple model that mimics the regularity is used to predict the composition and temperature dependences of the two parameters of the linear isotherms. The results are used to predict the densities of some binary mixtures a t different compositions and temperatures; agreement with experiment is better than 1%. Also, the density of a ternary system is calculated and agrees with experiment within 1.5%. The predicted composition dependences of the parameters of the linear isotherms are found to be accurate even for systems in which a stable complex forms, if the complex is considered as a separate species. Experimental Tests for Binary Mixtures
Introduction A general regularity was recently reported for dense fluids, both compressed liquids (T< Tc)and dense supercritical fluids ( T > Tc).l The regularity is that (Z- l ) v 2 is linear in p2, where Z pvlRT is the compression factor and p = l / v is the molar density. The regularity was tested for 13 fluids, including nonpolar, polar, hydrogen-bonded, and quantum fluids, and found to be valid for all of them. Experimentally, the regularity holds for liquid isotherms from thevaporization line to the freezing line and for supercritical isotherms for densities greater than the Boyle density and for temperatures less than twice the Boyle temperature. The Boyle temperature, TB, and Boyle density, p~ = 1lug, are defined in terms of the second virial coefficient, B2: Bz(TB) = 0 and U B = T$\( TB),where B'z = dB2ldT. An upper density limit of the regularity for supercritical isotherms was not apparent from the experimental data; it is at least twice the Boyle density, if indeed there is any limit before freezing. A simple molecular model was shown to mimic the regularity and, more importantly, to predict the temperature dependences of the intercept and slope of (Z-l)v2 vs p2. The prediction was that the intercept varies linearly with 1 / T and that the slope is proportional to 11T. The available experimental data were found to follow the prediction quite accurately. The purpose of this paper is to examine whether the regularity extends to mixtures. The reason for expecting it to hold for mixtures is the success of one-fluid approximations for mixtures, in which mixtures follow the same equation of state as single substances, but with parameters that depend on the mixture composition.2 In particular, we address the following points: (1) experimental tests of the regularity for binary fluid mixtures; (2) if the linear regularity holds, determine the composition dependences of the intercept and slope of the line; (3) prediction of mixture densities from the composition and temperature dependences of the intercepts and slopes; (4) extension to multicomponent mixtures. We find that the same simple model that was used to predict the temperature dependences of the intercept and slope for single substances can also be used to predict their composition dependences for mixtures, with good agreement with experiment. An apparent exception occurs for the mixture C2H5OH CHCl', in which a stable complex appears to form, but even this system conforms to prediction if the complex is included as a separate species.
+
Permanent address: Department of Chemistry, Isfahan University of Technology, Isfahan, Iran. 0 Abstract published in Aduance ACS Abstracts, February 1, 1994.
We first test the ability of the linear relation
(Z-l)02=A+Bp2 (1) to represent the volumetric behavior of dense fluid mixtures for 10 pairs of substances a t various temperatures and compositions. As an illustration, we show in Figure 1 the results at 298.15 K for the strongly-hydrogen-bonded system CHJOH H20, a t mole fractions 0,0.25,0.50,0.75, and 1. It can be seen that the measurements3 are well represented by straight lines over the entire pressure range 0.1-200 MPa and that the representation is just as good for the mixtures as for the pure components. As expected, the intercept ( A ) and the slope ( B ) both depend on the composition of the mixture. Results for this system at other temperatures,' and for nine other systems,C1l are summarized in Table 1, including the intercept and slope of the fitted straight line at each temperature and composition and the pressure range of the measurements. Because both (Z-l ) v 2 and p z are subject to experimental error, we measure the goodness of fit by the coefficient of determination R2, which is the square of the correlation coefficient.12 In the present cases, R2 should be within 0.005 of unity for the fit to be considered good. In all cases, the densities calculated from the straight lines agree with the measurements to better than 1%. Except for C2H6 C02, all the data shown are for the subcritical region (i.e., for liquids). The linearity holds for the mixtures as well as it does for the pure components in all cases.
+
+
Composition Dependences of Intercept and Slope In order to predict the composition dependences of the parameters A and B of eq 1, we use the same simple model that was used for pure substances to mimic the linear regularity and to predict the temperature dependences of A and B. It is to be emphasized that this procedure is not meant to be a fundamental theory of the equation of state, but only uses a model that mimics the linearity in order to infer how A and Bdepend on temperature and composition. Details are given in ref 1; here we give just the key points needed to infer composition and temperature dependences. This model approximates the internal pressure of a fluid by modeling the average configurational potential energy and then taking its derivative with respect to volume. For a repulsive attractive (n,m)potential, theaverage potential energy for a single substance is approximately
where Uis the total potential energy among Nmolecules counting
0022-365419412098- 1962%04.50/0 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 7, 1994 1963
Linear Isotherms for Dense Fluids 3
effectively pairwise and random. Notice also that the predicted temperature dependence of B proportional to 11T is preserved. The intercept parameter A for a pure substance has two contributions, an internal pressure contribution from U and a nonideal thermal pressure contribution, which leads to a temperature dependence of the form
2
a
ba A sa
A A“- A’/RT (5) The contribution A’arises from the attractive part of U and, by the same argument given above for B, also has a quadratic dependence on composition for a binary mixture,
1
The contribution A” arises from the nonideal thermal pressure and is given explicitly by
d
*
n
+ ~ X , X ~ A+’ ~x;Ai2 ,
A‘= x12A;,
4
~ [ )d- I]
o
Y
A”= 1 1 *
8
p -1
-2 0.6
0.7
0.8
0.9
1.0
1.1
1.2
p2, kg2 L-2 Figurel. (Z-1)u2vsp2forxCHjOH+(1 -x)H20mixturesat298.15 K, for x = 0 (o),0.25 (m), 0.5 (0),0.75 ( O ) , and 1 (A).
nearest-neighbor interactions only, z ( p ) is the average number of nearest neighbors, t is the average distance between nearest neighbors, and C, and C,,, are constants. The obvious generalization for a binary mixture of N1 molecules of species 1 and N2 molecules of species 2 is
a similar expression for the attraction (3) where zi(p) is the coordination number for species i, xi is its mole fraction, the (C,Ji/)sare repulsive interaction constants for the ij pair, and til is the average nearest-neighbor separation for the ij pair. We assume that zi(p) is independent of the composition but varies linearly with density as for pure liquids and that til = kilt, where k,j is a constant and t is the average distance between nearest neighbors. This is a random distribution assumption that will not be valid for strongly associated mixtures, which are discussed later. The somewhat artificial nature of the model is indicated by the fact that the linear relation of eq 1 requires n = 9 and m = 3 according to the model, whereas the linearity experimentally holds much more generally. However, the essential point is the explicit quadraticdependence of U on composition, since the slope parameter B for a single substance derives entirely from the repulsive part of U.’ We may therefore immediately conclude that B for a binary mixture has the following composition dependence:
+
B = xl2Bl1 2x1x2Bl2+ xB :22
(4)
where BII and B22 are the slope parameters of the pure components and B12is the slope parameter for a hypothetical pure fluid having 12 interactions. Notice that this composition dependence is the same as that for the second virial coefficient and, for the same fundamental reason, that the interactions are represented as
P R ~ T P
(7)
For single substances this turned out to be approximately constant in the density and temperature range of interest, but for a mixture it is not clear how it varies with composition. We can get some idea of the likely composition dependence of A” by considering a van der Waals fluid and then checking any result with experimental data. For a van der Waals fluid, A”is equal to b / p ( 1 - bp), and the density range of interest occurs around the minimum of this function, which is at bp = 112. In this region A”varies only weakly with density and is independent of temperature. The value of A”at the minimum is 462, which leads to the tentative conclusion that the composition dependence of A”is given by the square of the composition dependence of the van der Waals b. Since b must be quadratic in composition to lead to the correct second virial coefficient for a mixture, we may conclude that A” is a quartic function of composition. This result can also be obtained from the more accurate IhmSong-Mason equation of state,13 subject to a few minor approximations. The resulting composition dependence of A for a binary mixture is therefore a complicated quartic function requiring six parameters for its specification (three each for A”and A?. To simplify matters, we therefore try the plausible assumption that the ratio AIB can be represented by only a quadratic expression in the mole fraction,
where (A/B)I and (A/B)22 are the parameter ratios for the pure componentsand (A/B)12 is the parameter ratio for the hypothetical pure fluid having 12 interactions. The hope is that a quartic divided by a quadratic can be approximated by a quadratic. We can test the foregoing results for B and AIB, eqs 4 and 8, with experimental data on real binary fluid mixtures. As an example, we show in Figure 2 the values of B and of A I B as a function of composition for C6H6 C6HQ at 303.15 K. The curves are quadratic fits to the experimental datas and are not forced to go through the end points. They can be seen to represent the data quite well, within about 1%. Similar tests were carried out for other binary mixtures, for which the results are summarized in Table 2. Several features can be seen in Table 2. The slope parameter Bis generally fitted well by a quadratic (about 1%average absolute deviation), with two notable exceptions: the strongly-hydrogenbonded system CH30H H20, especially at low temperatures, and the system C2H50H+ CHCI3, which is believed to form a stable complex containing at least three alcohol molecules for each chloroform molecule.11J4J5 Complex formation is considered explicitly in a following section. However, the ratio A I B is fitted well by a quadratic for all the systems (better than l%),showing that any peculiarities in B aremirrored in A and essentially cancel out of the ratio.
+
+
1964 The Journal of Physical Chemistry, Vol. 98, No. 7, 1994
Parsafar and Mason
TABLE 1: Intercept ( A ) , Slope (B),Coefficient of Determination (@), and Pressure Range of the Data (Ap) for Binary Mixtures mixture T, K 9 -A, L2k g 2 B, L4k g 4 R2 Ap, M Pa airb
C2H6 + COf CS2 + CH2C12d
90 120 310.93 298.15
C~H~+C-C~HI~'
298.15 348.15
C6H6 + CsHsCY
303.15
CsH6 + CsHsClg
298.15 398.15
CaHsN02 + C6Hsh
303.15
CH3OH + H20'
323.14
C2HsOH + CHCld
298.15
0.1777 0 0.5 1856 1 0 0.4997 1 0 0.4997 1 0 0.6 1 0 0.5 1 0 0.5 1 0 0.4 1 0 0.5 1.o 0 0.0987 0.2002 0.2956 0.401 1 0.4983 0.5986 0.6959 0.8024 0.9005 1
11.346 6.392 3.541 8.733 8.738 9.689 37.461 33.190 28.340 28.942 24.347 21.837 25.576 26.425 27.697 26.147 28.196 28.676 14.797 15.029 15.906 27.247 28.723 30.046 9.257 12.579 13.195 8.921 9.264 9.751 10.275 1 1.026 11.903 12.967 14.355 16.311 18.721 22.257
14.645 10.463 4.847 4.708 5.023 5.736 59.775 47.774 35.367 51.266 38.728 30.213 20.564 26.756 34.943 20.862 27.380 35.830 14.030 17.538 24.220 34.346 26.958 20.581 8.446 15.116 19.639 3.848 4.28 1 4.867 5.555 6.574 7.854 9.625 12.150 16.321 22.476 33.305
1.000 1.000 1.ooo 1.ooo 1.000 1.000 1.ooo 0.999 1.000 0.999 0.999 0.999 1.ooo 0.999 1.000 0.999 0.999 1.000 0.999 0.999 0.999 1.ooo 1.ooo 1.000 1.000 1.000 1.ooo 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999
0.76-14 2-14 13-68 0.6-102 0.3-100 0.1-101 0.1-35 0.1-105 0.1-70 0.1-105 0.1-105 0.1-105 0.1-200 0.1-200 0.1-90 0.2-97 0.2-208 0.2-68 0.5-98 0.6-98 1.4-98 0.1-90 0.1-200 0.1-100 0.1-208 0.1-208 0.1-208 0.1-250 0.1-250 0.1-250 0.1-250 0.1-250 0.1-250 0.1-250 0.1-250 0.1-250 0.1-250 0.1-250
a Mole fractionof first component. Reference4. Reference 5. Reference6. Reference 7. /Reference& E Reference9. h Reference 10. i Reference 3. Reference 1 1.
Prediction of Mixture Densities Since the main purpose in knowing the composition and temperature dependences of the parameters A and B is to predict mixture densities from limited experimental data, it is important to check the accuracy of such predictions. For a given isotherm, measurements are needed on the pure components and one binary mixture in order to determine the values of All, A12, 4 2 , Bll, B I Z ,and B22. (In principle, measurements on any three mixtures would suffice.) From these results the densities of mixtures of any composition at the same temperature can be predicted. The accuracy of such predictions for four binary mixtures, using the parameter values from Table 1, is summarized in Table 3. The accuracy is better than 1%. A more severe test is to predict densities at temperatures different from those of the measurements, which requires knowledge of the temperature dependences of A and B. The simple model predicted that for pure components A is linear in 1/ T and B is proportional to 1/ T, and this result was verified by experimental data.' For mixtures the prediction is the same, a t least within the one-fluid approximation, which suggests that the contribution of the nonideal thermal pressure given by eq 7 can be taken as a constant. The results of this prediction are illustrated in Figures 3 and 4 for C6H6 + c-CsHl2 mixtures. Both A and B are seen to vary linearly with 1 / T a t all compositions, but the intercept for the slope parameter B is not zero for the mixtures as it is for the pure components. In other words, B for a mixture is linear in 1/ T, but not proportional to 1/ T. This behavior means that the nonideal thermal pressure term, namely, (l/p2) [( 1/pR). ( a p / d T ) , - 11 of eq I, is not a constant but depends also on
density, a dependence that contributes a nonzero intercept to B vs 1/ T. This result suggests a small breakdown in the one-fluid approximation. In any case, whether B is proportional to 1 / T or only linear in 1/T, measurements are needed at least at two temperatures in order to determine the temperature dependences of the six parameters, three Aiis and three BOIS. We illustrate the results of such a prediction for C6H6 + C6HsCl mixtures, using the results given in Table 1 for 298.15 and 398.15 K and assuming that both A and B are linear in 1/T. From these results we predict the densities of this mixture at two other temperatures and compositions, for comparison with experiment. The results are summarized in Table 4,where it is seen that the accuracy is a little poorer than for density predictionsat thesame temperature (hardly a surprise) but is still better than 1%. We conclude that mixture densities can be predicted from limited measurements to a level of accuracy of about 1%.
Multicomponent Mixtures We expect that the linearity of (Z- 1)02vs p2 should hold for multicomponent mixtures as well as for binary mixtures. Moreover, using a similar argument as for binary mixtures, we expect B and A / B to be quadratic functions of composition and also B and A to be linear in 1/ T. Hence for a multicomponent mixture we write
The Journal of Physical Chemistry, Vol. 98, No. 7, 1994 1965
Linear Isotherms for Dense Fluids 40
TABLE 2 Accuracy of Representing the Parameters B and A / B by Quadratic Functions of Composition av abs 5% dev mixture T, K B AIB C6H6 + C-CgH12 298.15 1.6 0.04 323.15 348.15 293.15 303.15 313.15 283.15 298.15 298.15
C6H6 + C6H5N02 CHoOH + H2O I
20
1
-
0.0
0.2
0.4
0.6
0.8
1 .o
X Figure 2. Quadratic composition dependence of the parameters B (0) and A / B (W) for XC&Cl+ (1 - x)C6H6 mixtures at 303.15 K. The values of A / B have been multiplied by a factor of 20 to bring them on the scale.
CzHsOH + CHCl3
1.2 1.1 1.2 1.3 1.o 3.2 1.9 15.7
0.02 0.10 0.45 0.38 0.42 0.50 0.48 0.20
TABLE 3 Accuracy of hedicting Binary Mixture Densities from Measurements at the Same Temperature mixture T, K XO 100(lApllp)avb AP, M Pa CS2 + CH2C12 298.15 0.30014 0.02(0.03) 0.699.9 C6H6 + C&Cl
303.15
CsHsN02 C6H6
303.15
CHoOH + H20
323.15
0.40137 0.59792 0.70178 0.2 0.6 0.8
0.2 0.6 0.8 0.25 0.75
0.02(0.07) 0.02(0.03) 0.02(0.05) 0.18(0.32) 0.23(0.38) 0.1 l(0.18) 0.18(0.24) 0.75(0.80) 0.07(0.15) 0.53(0.59) 0.14(0.24)
0.7-100 0.1-100.2 0.9-100.8 0.1-180 0.1-180 0.1-180 0.1-140 0.1-200 0.1-1 70 0.1-207.8 0.1-221.5
Mole fraction of first component. Maximum deviations in paren-
theses. Experimental densities from the references cited in Table 1.
(9)
Thus, the densities of multicomponent mixtures should be predictable from measurements on the pure components and all the binary pairs of the mixture. To test the foregoing expectation, we use the experimental data on pure n-octane,'6 1-octene,17and 2,2,4-trimethylpentane,l8 plus the data on their equimolar binary mixtures,lg to calculate the density of the ternary system at the same temperature. The results fortheequimolar ternarysystemat 298.15 Karecompared with the experimental datal9 in Table 5. The deviations are less than 1.5%. Although further work on multicomponent mixtures would be required to draw firm conclusions, it appears from this comparison that such mixtures fall within the scope of the linear regularity and the predicted behavior of its parameters.
Complex Formation The slope parameter B for the system C~HSOH + CHCL shows marked deviations from a quadratic dependence on composition (Table 2). It has been suggested, on the basis of the unusual behavior of the vapor-liquid equilibrium, that this mixture forms a complex containing at least three ethanol molecules for each chloroform molecule.14J5 Such complex formation would be expected to cause deviations from the expected quadratic composition dependence, inasmuch as the system really contains a third component rather than only the initial two. In this section we examine whether complex formation can account quantitatively for the unusual composition dependence of B for this system. Suppose that each molecule of component X reacts with a
7n
0.0028
0 0030
6.0032
0.0034
1/T, K-1 Figure 3. Temperature dependence of the intercept parameter A for XC6H6 + (1 -X)C-C6H12 miXtUrS, for X 0 (n),0.2504 (m), 0.4997 (e), 0.7497 (0),and 1 (A).
molecules of component Y to form a stable complex C
X+aY+C (1 1) with a 1 1. We make the simplifying extreme assumption that C is so stable that the reaction goes to completion, so that the initial solution of X + Y is really a binary solution of C plus
Parsafar and Mason
1966 The Journal of Physical Chemistry, Vol. 98, No. 7, 1994
TABLE 4 Accuracy of Predicting Binary Mixture Densities from Measurements at Other Temperatures: xC& + (1 x)C&CI’ x = 0.25
T,K
p , M Pa
323.15
0.2 15.0 43.7 69.1 96.6 124.2 152.0 179.4 206.0 0.2 27.0 57.0 84.0 111.7 138.7 166.6 192.0
348.15
a
palc,
x = 0.75
kg L-I Wdev p , M Pa
1.0275 1.0396 1.0605 1.0769 1.0931 1.1079 1.1216 1.1342 1.1457 1.0020 1.0262 1.0493 1.0676 1.0845 1.0995 1.1138 1.1259
0.47 0.34 0.22 0.21 0.23 0.28 0.34 0.41 0.48 0.68 0.39 0.32 0.34 0.40 0.47 0.57 0.66
0.2 14.3 41.8 69.9 97.7 124.6 152.4 179.0 204.2 0.2 27.8 55.7 83.5 110.8 138.8 166.4 187.2
palcr
kg L-’ %dev
0.9152 0.9268 0.9469 0.9649 0.9807 0.9947 1.0079 1.0196 1.0300 0.8900 0.9152 0.9366 0.9551 0.9713 0.9864 1.0000 1.0095
0.62 0.48 0.34 0.32 0.35 0.40 0.46 0.53 0.60 0.89 0.56 0.50 0.50 0.56 0.65 0.73 0.82
Experimental data from ref 9.
TABLE 5: Accuracv of hedictine Ternarv Mixture Densities: Equimoiai @Octane lSOctene ”+ 2,2,4-Trimethylpentane at 298.15 K.
+
P , kg
1/T, K-1 Figure 4. Same as Figure 3, for the slope parameter B.
excess X or Y. If we mix Nx molecules (moles) of X with Ny molecules (moles) of Y, we can distinguish two cases. If Ny > aNx, then all the X molecules react to form NXmolecules of C, leaving (Ny - aNx) molecules of Y. The true mole fractions in the mixture are xc = l / k and
x ; = (k- l ) / k
(12)
calcd
measd
% dev
0.1 26.2 51.4 102.9 149.0 198.8 249.4 303.7 349.6 399.1 449.4 499.6
0.7028 0.7210 0.7360 0.7617 0.7808 0.7988 0.8149 0.8304 0.8423 0.8543 0.8655 0.8760
0.6925 0.7146 0.7309 0.7574 0.7761 0.793 1 0.8082 0.8224 0.8335 0.8440 0.8542 0.8637
1.5 0.9 0.7 0.6 0.6 0.7 0.8 1.o 1.1 1.2 1.3 1.4
Experimental data from ref 19.
where
k = 1-
+ (Ny/N,)
(13)
and the slope parameter B for the mixture is
or B = (Byy - 2Byc + Bcc)(1/k2)
+ 2(Byc - Byy)(l/k) +
By, (15) Thus, B is a quadratic function of l / k , not of the initial mole fraction (before reaction to form C) of X or Y. If Ny < aNx, all the Y molecules react to form Ny/a molecules of C, leaving (Nx -NY/a)molecules of X. The true mole fractions in the mixture are (16)
x C = k ’ and x ’ , = l - k ’ where k’= Ny/aNx and the slope parameter is given by B = (Bcc- 2Bcx
+ Bxx)k” + 2(Bcx-Bx,)k’+
(17) B,
(18)
Thus, B is a quadratic function of k’, not of the initial mole fraction. We have analyzed the values of B given in Table 1 for C2H5OH CHC13 mixtures according to the foregoing scheme,
+
L-I
P,M Pa
assuming initially that a = 3. The results are shown in Figure 5 as a function of the initial mole fraction of C ~ H S O H .The fitted curve has two branches, one a quadratic in l / k and the other a quadratic in k’,that intersect at 0.75mole fraction C2H2OH to give a value of Bcc = 14.1L4kg4. The resulting fit gives a considerable overall improvement over that obtained without the assumption of complex formation. However, the fit for x > 0.75C2HsOH is decidedly peculiar and much worse than for x < 0.75C2H5OH. This peculiarity can be remedied by choosing a = 4, leading to the nearly perfect fit shown by the broken curve. This is consistent with the vapor-liquid equilibrium results.14.15 It can be objected that a better fit is inevitable because five parameters are available with the complex (Bxx,BYY,BCC,BXC, BYC)instead of only three without the complex (Bm,Byy, Bxy). To investigate this point, we have examined the range between 0 and 0.75 mole fraction C2H5OH, where only three parameters are available in either case. Here the fit is excellent in terms of a quadratic in k’ = x / 3 ( 1 - x ) , where x is the mole fraction of C Z H ~ O Hbut , very poor in terms of a quadratic in x , suggesting that the complex formation is real. It is worth mentioning that the other anomalous system, CH3O H + H20, is also believed to form complexes, at least in the gas phase,ZO but these complexes are sufficiently weakly bound that the foregoing analysis is probably inapplicable.
Discussion We have shown that (2- l)u2 varies linearly with pz for all types of dense fluid mixtures, even the strongly-hydrogen-bonded
The Journal of Physical Chemistry, Vol. 98, No. 7, 1994 1967
Linear Isotherms for Dense Fluids
substances and mixtures. This permits, for example, supercritical properties to be calculated from subcritical measurements. The temperature dependence of the parameters of the TME are not really known, although the slope parameter for a Lennard-Jones (12,6) fluid has recently been found empirically to be linear in 11T.2' ( 3 ) Composition dependence of parameters. The parameters of the present regularity for mixtures can be described by simple quadratic functions of composition, so that mixture properties can be predicted on the basis of only a few measurements. The composition dependences of the T M E parameters have been measured for a number of mixtures and usually vary smoothly with compo~ition,2~ but there is no theoretical or model basis for describing the composition dependences. As a final comment, we note that it would be of great practical value if the mixture interaction parameters B, and (A/B)l,could be predicted from the corresponding parameters for pure components i and j . The simple model suggests, but does not strongly predict, that a geometric-mean combination rule might be suitable, but development of useful combination rules remains for future work.
n
0.0
I
0.2
.
I
.
I
0.6
0.4
.
,
0.8
Acknowledgment. G.A.P.thanks theauthorities of the Isfahan University of Technology for financial support during his sabbatical leave.
.
1.0
References and Notes X Figure 5. Composition dependence of the slope parameter B when a stable complex forms, according to eqs 15-18: the mixture xC2HsOH + (1 - x)CHCls. Solid curve calculated for an assumed 3:l complex (x = 0.75); broken curve calculated for an assumed 4:l complex (x = 0.80).
(1) Parsafar, G. A,; Mason, E. A. J. Phys. Chem. 1993, 97, 9048. (2) Rowlinson, J. S.;Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.;Butterworth: London, 1982; pp 287-8. (3) Kubota, H.; Tsuda, S.;Murata, M.; Yamamoto, T.; Tanaka, Y.; Makita, T. Rev. Phys. Chem. Jpn. 1979, 49, 59. (4) Younglove, B. A.; Frederick, N. V. Int. J. Thermophys. 1992, 13, 1033.
system CH30H + HzO (Figure 1) and the complex-forming system C ~ H J O H+ CHC13 (Table 1). The composition and temperature dependences of the intercept and slope parameters have been obtained from a simple model and tested against experimental data. Both parameters are linear in 1/ T . The slope parameter B is a quadratic function of mole fractions, as is the parameter ratio AIB. It is interesting to compare the present linear isotherms with other equations of state for compressed dense fluids. One of the most successful of these, which we may call the Tait-Murnaghan equation (TME), takes the isothermal differential bulk modulus (reciprocal compressibility) to be linear in the pressure. This linearity and the present linearity are mutually compatible, inasmuch as both have been shown to be obtainable (to a suitable degree of accuracy) from general equations of state based on statistical mechanic~.lJ'-~~ However, the new linearity of ( Z l)u2vs p* has some important advantage over the TME regularity, as follows: (1) Greater range of ualidity. The TME holds only in the subcritical region ( T < Tc),and the higher-temperature isotherms sometimes show a small curvature, for which a term in p 2 may be introduced. In contrast, the present linearity holds far into the supercritical region, up to about ~ T (roughly B ST,).' (2) Temperature dependence of parameters. The intercept and slope of the present regularity are linear in 11T, for both pure
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