The solid curve is a plot of the FENE dumbbell results with the time constant X = 0.1405 sec obtained from eq 29 with 111 taken t o be the experimental value of M,, with the assumption that the solution is monodisperse ( M w / M n = l), and with b taken to be equal to 0.1, the best fit value for the high frequency limit of [?‘I. Ac knowledgmenf
The author wishes to acknowledge the financial support received from the National Science Foundation in the form of a graduate fellowship. The author is indebted to Professor R. B. Bird, Professor W. E. Stewart, and the members of the Rheology Research Center at the University of Wisconsin for many helpful discussions. literature Cited
Bird, R. B., Johnson, M. W., Stevenson, J. F., Proc. Int. Congr. Rheol., 5th 4, 159 (1970). Bird, R. B., Warner, H. R., Trans. SOC.Rheol. 15,741 (1971). Bird, R. B., Warner, H. R. Jr., Evans, D. C., Advan. PoEUm. Sci. 8;i (197’1). Cottrell, F. R., Merrill, E. W., Smith, K. A,, J . Polym. Sci., Part A-b 8,289 (1970). Ferry, J. D., “Viscoelastic Properties of Polymers,” 2nd ed, Wiley, New York, N. Y., 1970.
Giesekus, H., Rheol. Acta 5, 29 (1966). Harpst, J. A., Khasna, A. I., Zimm, B. H., Biopolymers 6, 585, 59.5 - - - (1968). Hermans, J. J., Physica 10, 777 (1943). Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids, Second Printing with Corrections,” p 905, Wiley, New York, N. Y., 1964. Janeschitz-Kriegl, H., Kolloid-2. 2. Polym. 203, 119 (1965). Kramers, H. A., Physica 11, 1 (1944). Lodge, A. S., “Elastic Liquids,” Academic Press, Sew York, N. Y., 1964. Massa, D. J., Ph.D. Thesis, University of Wisconsin, 1970. Mikhlin, S. G., “Variational Methods in Mathematical Physics,” (English translation), Chapter IX, Pergamon Press, Oxford, \ - - - - I
1964
Morse, P. M., Feshbach, H., “Methods of Theoretical Physics,” p 780, McGraw-Hill, New York, N. Y., 1953. Noda, I., Yamada, Y., Nagasawa, M., J . Phys. Chem. 72, 2890 (1968). \ - - - - I
Oseen, C. W., Ark. Mat. Astron. Fys. 6 (29), 1 (1910). Peterlin, A., J . Chem. Phys. 39, 224 (1963). Schrag, J. L., private communication, May 1971. Stevenson, J. F., Ph.D. Thesis, University of Wisconsin, 1970. Stewart, W. E., Scffensen,J. P., Trans. SOC.Rheol. in press, 1972. Treloar, L. R. G., The Physics of Rubber Elasticity,]’ University Press, London, 1950. Warner, H. R., Jr., Ph.D. Thesis, University of Wisconsin, 1971. Zimm, B. H., J. Chem. Phys. 24,269 (1956). RECEIVED for review August 20, 1971 ACCEPTEDApril 12, 1972
Correlation and Prediction of Excess Properties for Hydrocarbon Liquid Mixtures Wallace 1 . 4 . Yuan,l David A. Palmer,2 and Buford D. Smith* Thermodynamics Research Laboratory, Washington University, St. Louis, J4o. 63130
A method capable of correlating simultaneously the vapor-liquid equilibrium, the enthalpy of mixing, and the volume change of mixing data for hydrocarbons is presented. The correlation is based on a modified conformal solution theory and requires only four correlation constants for each binary to represent all three types of data as functions of composition and temperature. The correlation is tested on four binary systems: benzene; carbon tetrachloride cyclohexane; benzene cyclohexane; and carbon tetrachloride benzene 4- normal heptane. Carbon tetrachloride is included because of the scarcity of hydrocarbon binaries with a sufficient amount of good GE and HE data in the literature. Most of the VE data used were measured specifically for this work. The correlation constants obtained from GE and VE data at two temperatures accurately predict the HE and activity coefficient values as a function of composition over a wide temperature range.
+
T h e suitability of the conformal solution formalism (Brown, 1957a,b; Longuet-Higgins, 1951; Pitzer, 1939) as a basis for a general correlation of the excess properties of nonelectrolyte liquid mixtures has been investigated. The objective has been the development of a correlation structure which would simultaneously correlate the vapor-liquid equilibrium (VLE or GE) data, the heat of mixing (HE) data, and the volume change of mixing (VE) data for mixtures of industrial interest. 1 Present address, Chemical Engineering Department, Tunghai University, Republic of China. Present address, Amoco Chemicals Corporation, Naperville, Ill.
+-
+
There are several reasons for the interest in all three excess properties despite the fact that users of such a correlation will be primarily interested in VLE data. Inclusion of the HE and VE data provides advantages in the basic correlation work itself; those advantages will be described below a t the pertinent points in the development. Other advantages are gained in the use of the finished correlation. For example, if the correlation is successful in accurately relating all three properties, it will be possible to predict V L E data from the more easily measured HE and VE data. Convenient experimental techniques are now available for the latter two propInd. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
387
perties for low pressure systems, and Stookey and Smith (1971) have reported initial success in such predictions. This possibility is of great practical importance because any working correlation accurate enough for close separations will usually require some input mixture data for the binaries of interest. The conformal solution formalism presented by Brown (1957a,b) had to be extended empirically and its application modified in several important respects before the correlation objective could be achieved. Those changes are presented in this paper. Some are also described in two recently published papers by Calvin and Smith (1971) and Houng and Smith (1971) which cover work done more than 2 years ago. Five important improvements over the procedures described in those papers have been : (1) the elimination of the need for an intermolecular potential equation ; (2) the elimination of the Taylor series expansions; (3) the calculation of the pure component conformal parameters from the vapor pressure and density data (instead of the enthalpy and density); (4) the inclusion of certain temperature derivatives previously neglected; and ( 5 ) the recognition of the temperature dependence of the unlike-pair parameters. These changes have produced a correlation structure which appears to be adequate for the excess property data for nonpolar systems ranging from the simple fluids covered in the two previous papers to the Ce and C?hydrocarbons covered in this paper.
previous conclusion. Therefore, it will be necessary to correlate the temperature dependence of the empirical combination rules if the correlation is to have any temperature extrapolation capabilities. 4. The general validity of any correlation model can be established only by predicting all three excess properties over the entire composition range and as functions of temperature. Because of the powerful effect of the unlike-pair parameters on the predicted results, it is usually possible to make any correlation model fit only one or two properties over the binary composition range, or all three properties a t only the midpoint, if the unlike-pair parameters are treated simply as regression parameters. Hence the need for the next conclusion. 5. The unlike-pair parameter values must be essentially independent of composition if one pair of values is to correlate the excess properties Rithin experimental error over the binary composition range. T h i s iequirement i s a valuable criterion for the adequacy of a n y correlation model. Its satisfaction requires that experimental data (at the same temperature and composition) for two excess properties (GE and VE, or HE and VE) be used simultaneously to solve for unlike-pair parameter values a t each composition where data points are available. 6. It is essential that the GE (VLE),HE, and VE data be carefully evaluated for accuracy before being used to test any correlation or theoretical model. The HE and VE are “slope” properties; Le., they are related t o GE by temperature and pressure differentiations. For HE, for example,
General Considerations
From the experience gained in this correlation project, several general conclusions have been drawn concerning theoretical and correlation studies of liquid mixtures. Great care has been taken to minimize the effect of experimental data errors and numerical “static” on these conclusions. Some of the conclusions may reflect inadequacies in the conformal solution formalism, but it is believed that most of the numbered comments below will apply to any correlation project. 1. Whenever the corresponding states principle is invoked in mixture work, the reducing parameters must be recognized as functions of temperature and pressure if all three excess properties are to be correlated accurately as functions of composition and temperature. Inspection of any two purecomponent pVT diagrams shows that they cannot be made to correspond (or conform) a t all points with constant reducing parameters such as T, and p,. “Local correspondence” or “local conformality” must be forced on the two components by using the necessary parameter values a t each ( T , p ) point. Otherwise, the errors in the representation of the pure component surfaces will be as large or larger than the mixture quantities to be correlated. This necessity has been recognized also by Leach (1967) and by Leland, et al. (1969). 2. The predicted results are extremely sensitive to the unlike-pair parameter values. It was necessary to know the unlike-pair energy parameter within 0.15% and the unlikepair size parameter within 0.0270 in order to correlate within experimental error all three excess properties over the entire composition range and as a function of temperature. XO existing theoretical combining rules can even approach these required accuracies for any large number of systems, and it is not likely that theoretically based rules of sufficient scope will be developed in the near future. Consequently, this crucial step in any correlation of practical value must be handled empirically. 3. It is probably naive to hope that the combination rules be independent of temperature to the extent noted in the 388
Ind. Eng. Chem. Fundam., Vol. 11, No. 3, 1972
Small errors in the GE values used to obtain the correlation constants can cause large errors in the numerical values of HE predicted with those constants. This is illustrated in Figure 1 where a 3% change in the GE value a t 2OoC caused the HE prediction to change about 20%. As shown in Figure 2, the GE datum points a t 20°C were not sufficient to define the curve exactly; the point shown is the actual midpoint experimental value, whereas the solid GE curve was drawn through the midpoint value calculated from a polynomial fit of the GE data over the entire composition range. Obviously, one cannot use inaccurate GE data to determine the correlation constants if HE is t o be predicted accurately. I n fact, this magnification of error in predicting HE from GE is a major justification for reversing the procedure, Le., to GE from HE. The above considerations determined to a large extent the structure of the correlation presented in this paper. Thermodynamic Consistency of the Correlation
Inclusion of the excess enthalpy and volume data in the correlation ensures the thermodynamic consistency of the correlation structure. The correlation is based on an equation for the excess Gibbs function provided by the conformal solution model. The working equations for the activity coefficient, the excess enthalpy, and the excess volume are obtained by the various differentiations indicated by the general thermodynamic relationship (Van Ness, 1964) d(GE/RT)
=
( V E / R T )dp - (HE/RT*)dT
+
i
(In y i dzi) (1)
Fitting experimental activity coefficient data with the derived activity coefficient equation checks the accuracy of the composition dependency of the GE correlation. In a similar fashion, fitting VE and HE data checks the pressure and temperature
25
20
-a =P 15 3
uI 0
tm c
W
10
5
0.2
0
0.4 n
40
20
0
1,
“c
60
Figure 1 . Two midpoint HE predictions based on the experimental GE and VE data at 20 and 7OoC for carbon tetrachloride benzene. The solid points represent the data sets used to determine the correlation constants
+
0.8
0.6 ~
h
n
\It
Figure 2. Reproduction of the correlated GE and VE data and prediction of the HE data for carbon tetrachloride benzene at 20°C
+
dependencies. Fitting the VE and HE data available a t temperatures and pressures close to that of the limited GE data thus provides a substitute for GE data over wide temperature and pressure ranges. The extrapolation capabilities of the correlation are thereby enhanced. Pure Component Relationships
Corresponding States Equations. The conformal solution model provides the following algebraic statement of the two-parameter corresponding states principle.
The equation relates the configurational Gibbs function of the pure substance “a” a t the mixture T and p to the same property for a reference substance “0” a t its corresponding temperature and pressure, T/faa and phaa/faa. In the original conformal solution theory, the pure component conformal parameters, faa and ha,, were defined as constants in terms of the minimum point coordinates of the intermolecular potential energy curves for components “a” and “o.” I n the application described in this paper, faa and ha, and faa/haa are simply the scale-reducing parameters for temperature, volume, and pressure. Their functions are analogous to those of T, and p, in the familiar generalized corresponding states charts. However, they must be allowed to vary with temperature and pressure if the pure component property surfaces are to correspond exactly (local correspondence) a t the mixture conditions. Differentiation of eq 2 with respect to temperature a t constant p gives the configurational enthalpy relationship between “a” and “0.”
All of the four derivative terms were neglected in the work on the simple fluids but the first two had to be retained for the systems covered below. The last two terms were negligible and the equation used in this work was
Differentiation of eq 2 with respect to pressure a t constant T gives the volume relationship between ‘h”and “0.” V’,(T,p)
=
haaV’,
(-T -)
phaa
faa’
+
U‘O
(- -) ; ;+
faa
T phaa bhaa
PV’O
(- -) faa’ faa
T phaa
faa’
dp
faa
b In ha,
- R T ~- (4a) 3P
All of the pressure derivative terms in eq 4a were numerically negligible for the systems under consideration and the working equation was
The primes on the volumes can be dropped because the configurational arid total volumes are the same. Calculation of faa and ha, Values. The two earlier papers calculated the pure component conformal parameter values Ind. Eng. Chem. Fundam., Vol. 11, No. 3, 1972
389
the vapor pressure and liquid density data, the two pure component properties which are most likely to be known accurately. Yuan’s method required the heat of vaporization to calculate H’(T,p) from
Table 1. Typical Pure-Component Conformal Parameters for Cyclohexane with Benzene as the Reference t, OC fw. has af,m b In hs&T
0.0 20.0 40.0 60.0 80.0 100.0 120.0,
0.982008 0.982326 0.982530 0.982604 0.982528 0.982287 0.981867
1,211829 1.210125 1,208785 1,207808 1.207186 1,206907 1.206962
0.0000174 - 0.0000777 0.0000128 -0.0000629 0.0000071 -0.0000489 0.0000021 -0.0000331 - 0.0000088 -0.0000172 -0.0000167 -0.0000041 - 0.0000274 0.0000090
for component ‘la’’ by the simultaneous solution of eq 3a and 4a but with all of the temperature and pressure derivative terms deleted. Yuan (1971) developed a solution method based on eq 3b and 4b. Meanwhile, Palmer (1971) was investigating the use of eq 2 and 4b in the following direct iteration.
and
=
V ( l - a T ) ( p - p’) - A H w
+
This equation assumes the second virial coefficient, B , is sufficient to describe the gas pVT behavior. Typical fa, and ha, values, plus the important derivatives, are shown in Table I. The derivatives were obtained by applying Newton’s formula to fa, and h,, values calculated a t 1.O”C intervals. The computer programs calculated the parameter and derivative values a t the mixture conditions as needed from vapor pressure, density, and second virial coefficient data stored in the computer in equation form. Available data sets for the pure components were carefully evaluated for accuracy. The references for the pure component data used in the test cases described later are listed in Table 11. The Pitzer-Curl correlation was used to predict the second virial coefficients for cyclohexane and carbon tetrachloride because of uncertainties in the available experimental values. It will be noted in Table I that fa, goes through a maximum and ha, through a minimum. This is common in close boiling cyclohexane. The vapor pressure systems such as benzene and density data must be fitted very carefully in such cases if accurate values of the derivatives are to be obtained. Test of Correlation Structure for Pure Components. The validity of the conformal solution equations, the accuracy of the pure component data, and the precision of the numerical calculation procedures can all be checked by using the parameters calculated from the G’, and V , surfaces to predict the H’, surface. H’, values were predicted with eq 3b a t 25, 50, and 75°C for carbon tetrachloride, cyclohexane, and nheptane. Benzene was the reference substance. Six of the nine predicted values were within 1% of the values calculated from eq 8 using experimental AHc values from the sources listed in Table 11. The largest difference was 1.6’%, which is less than the probable experimental error in the 4HCdata.
+
where
=
H’(T,p)
+
G’G“,P) = G(T,p) - G*(T,P) G*’(T,p) V ( p - p’) Bp’ RT In (p’/RT) RT In N
(7b)
G”(T,p) = G’(T,p) - RT In N
(7c)
+
+
+
(74
and
The use of G” instead of G’ in eq 5 reduced numerical difficulties by eliminating the RT In N term. The G” and V values in eq 5 and 6 could be evaluated as functions of T only because the V ( p - p’) term in eq 7b was always completely negligible. Equation 7b is the working equation for G’(T,p) when the gas pVT behavior can be adequately represented by the virial equation of state truncated after the second coefficient. The equation is more complicated when equations like the Redlich-Kwong or Benedict-Webb-Rubin must be used. Palmer’s method of calculation proved to be superior because it involves no temperature derivatives and it utilizes
Mixture Relationships
Definition of the Pseudosubstance. The pure-component forms above are extended to mixtures by the definition of a
Table II. literature References for Pure Component Data for the Four Test Systems Component
Benzene Cyclohexane
n-Heptane
Vapor pressure
liquid density
Rossini (1953) Young (19.10) Bender, et at. (1952) Willingham, et al. (1945) Timmermans (1950) Stull (1947) Reamer and Sage (1957) Glaser and Ruland (1957) Rao, et al. (1957) Young (1910) Young (1910) Rossini (1953)
Carbon tetrachloride Timmermans (1950) Young (1910) Jordan (1954) Washburn (1926) Stull (1947) Killian and Bittrich (1965)
390 Ind. Eng. Chem. Fundam., Vol. 11, No. 3, 1972
Heat of vaporization
Washburn (1926) Timmermans (1950) Kozicki and Sage (1961) Rossini (1953)
Second virial coefficient
Francis, et al. (1952)
Waddington, et al. (1947) McGlashan and Potter Rossini (1953) (1962) Chen (1965) Weast (1966) Washburn (1926)
pure pseudosubstance “p,” which differs from the mixture “x” only by the ideal entropy change of mixing, i.e., G’x(T,P,Xa,%. . .)
=
G‘p(T,p)
+ RT
In
Xa
(9)
a
where G’, and G‘, refer to the mixture and the pseudosubstance, respectively. The conformal parameters of the pure pseudosubstance are denoted by fxx and axand referred to as the mixture parameters. The fxx and h,, parameters characterize the hypothetical pure pseudosubstance whose thermodynamic properties are identical to those of its related mixture (at fixed T, p, xa values) except for the ideal entropy of mixing. The necessary fxx and hxx values a t the given T, p, xa values are obtained by a suitable weighting of the various pure-component (faa, haa) and unlike-pair (jab, hab) conformal parameters. fxx =
hxx =
C C wy(a,b)fab a b a
b
wh(a,b)hab
WP - CZall.l’, a
R T In y1 = G’,(T,p,xJ
[”.(E, 2(zlh11
- G’l(T,p)
2 t z h ( f i 1 - fxx) F)/hXX
+
- PVO
+ -
X Z ~ Z ( ~ I Zf x x ) ]
X
(E,F)/fXX] +
+ x h z - hxx) [ pVo (-T
fxx’
--) phxx fxx
- RT/hxx]
(15)
(10)
The equation for component 2 is obtained by substituting 2 for 1 and vice versa, noting that f1z = fzl and h12 = hzl. The pV0 terms are completely negligible for the systems considered here.
(11)
Correlation of Mixture Parameters
The w,(a,b) and wh(a,b) represent pair weighting functions for the energy and size parameters, respectively. The pseudosubstance concept was suggested by Wheeler (1964) and Wheeler and Smith (1967) as an extension of Brown’s (1957a,b) equivalent substance, which was based on a random-mixture model and used the product Xazb for both weighting functions. It is an artifice which shifts the nonrandom mixture correlation problem in conformal solution theory from the nontherm6dynamic terms in Brown’s mixture expansion to the weighting functions used to define the pseudosubstance. The appropriate weighting functions can be more readily deduced from mixture properties. Corresponding-States Equations. The pseudosubstance, although hypothetical, is a pure substance by definition. Its potential energy curve (which is a weighted average of all the pair-interaction curves) can be conformal to the reference substance’s curve in the same way that the “a” and “b” curves are. It follows then that eq 2 through 4 can be written also for the pseudosubstance “p” with its parameters fxx and laxsubstituted forfaa and baa. Excess Properties. An excess property for the mixture is given by X E =
and eq 13. Since fxx and G’,(T/fxx, plax/fxx) are functions of composition, the resulting equation depends upon the weighting functions, wyand w h , used in eq 10 and 11. The binary form when Leland’s (Leland, et al., 1962) equations (eq 19 and 20) are used for$, and kxis as follows.
(12)
where 134 represents a molar property. Substitution of the corresponding-states expressions for the M’, and M’,gives the equations used to calculate the excess properties. For example
The use of the corresponding-states relationships to calculate the excess properties eliminates the Taylor series expansions used previously. This greatly increases the number of components which can be grouped with a given reference substance and does not require as much thermodynamic data for the reference substance. Activity Coefficients. The expression for the activity coefficients is obtained from (14)
The correlation problem for mixtures centers on the averaging equations used to relate the mixture parameters, fxx and k,,to the like- and unlike-pair parameters, faa, fab, fbb, etc. Several theoretical equations based on various models for the liquid state have been proposed for this task (Brown, 1957a,b; Leland, et al., 1962, 1969; Fredenslund and Sather, 1969; and others) but only three forms have been investigated thoroughly in this study. These include the random forms, the equations first suggested by Leland, et al. (1962), and the equations of Brown (1957a,b). The random equations worked for three of the test binaries but failed on the benzene n-heptane system. The Leland equations worked well for all four of these nonpolar systems. The Brown equations work only if the Lennard-Jones exponents are treated as adjustable parameters; values around 3,6 are usually best. Empirical Weighting Functions. I t is anticipated that none of the presently available theoretical equations will be adequate when the correlation is extended to polar systems. This tentative conclusion is based on the experience of Palmer (1971) with his data on the acetonitrile benzene nheptane system. If this conclusion is proved correct, then it will be necessary to rely on the following empirical form suggested by Wheeler (1964) and Wheeler and Smith (1967) for the weighting functions for each a-b pair in eq 10 and 11.
+
+
a
+
b
where the sa and s b give numerical representation to the size and shape effects of the “a” and “b” molecules, and the I(a,b) functions weight the various molecular pairs according to the strength of their specific interactions. When all of the S’s and Z’s are 1.0, the weighting function reduces to the random formw(a,b) = zaxb. The empirical approach methodically determines the necessary weighting functions for large numbers of binary systems in the hope that large groups of similar components can be represented by the same form. This approach has been successful so far. Houng (1969) and Houng and Smith (1971) found the random forms
+ 2z1xzf12 + x2%2 ZIZhll+ 2~122h12+ zz2hz2
fxx = Zl%l
(17)
hxx
(18)
=
to be sufficient for the simple liquids (argon to methane). A large-scale test has not been possible for the light hydroInd. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
391
carbons (C, to C4) because of the lack of excess property data. New data are being taken for some C4 binaries which, hopefully, will permit a meaningful test for those components sometime in the future. Yuan (1971) found the random form to be adequate for all of the binaries discussed in this paper except the benzene n-heptane system. The averaging equations
+
fxx
=
(212fllhll
hx, =
+ 22122fl2h12 + + 22122h12 +
222f22h22)/hxx
212hll
222h22
(19) (20)
suggested by Leland, et al. (1962), work well for that system as well as the others under consideration here and have been used for all of the results presented below. Leland’s equations, although obtained in a different manner, represent a nonrandom form of eq 16 for the energy parameter fxx. They can be obtained from eq 10, 11, and 16 by setting all the S’s and Z’s in wh(a,b) and all the S’s in w,(a,b) to 1.0, and then assuming the I(a,b)’s in w,(a,b) to be h11, h12, and h22. The h’s are calculated from the liquid molar volumes and their use is equivalent to the use of molar volumes to weight the various pair interactions. The molar volumes provide sufficient weightings for the molecular pairs under consideration here but Palmer (1971) has found them t o be insufficient when a polar molecule is involved. Combination Rules. Since none of the theoretically based combination rules can predict the fab and hab values within the necessary 0.15 and 0.02Cr,, it was necessary t o use the empirical rules and
The kab and Cab are the binary correlation constants and, a t the present time, must be obtained from mixture data. They are small, ranging from about -0.0067 to +0.03 for the test systems used below. They were made to be essentially independent of composition by the proper choice of the averaging equations; Le., the averaging equations were accepted as adequate only if they reduced the correlated experimental binary data to fab and h&b values which were essentially constant from 21 = 0.0 to 21 = 1.0. Temperature Dependence. Unfortunately, the kab and Cab values vary with temperature. This variation is very small but if the required 0.15 and O.OZ’% accuracies are to be maintained over the temperature range of interest, this temperature dependency must be correlated. No theoretical forms have ever been proposed for this task, so the following Arrhenius-type equations were assumed.
(1
- Cab)
=
c,t,t?(DabiT)
(24)
Determination of the constants A , B, C,and D for any binary requires either GE and VE or HE and VE data a t two temperatures. However, once they are known, these four constants permit the prediction of all three excess properties (and the activity coefficients) as a function of composition over a temperature range extending well below and above the original correlated data. Test Systems
A complete test of the correlation structure requires both GE and VE data a t two different temperatures plus HE data 392
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
a t one temperature. Even though most of the required VE data were measured (Yuan, 1971), the GE and H E data available in the literature for the molecular types of interest permitted a complete test on only three binaries and a partial test on a fourth one. The literature was searched for binary systems in the C5-Cs hydrocarbon range with isothermal GE ( V L E ) data a t two or more temperatures and HE data a t one or more temperatures. Only three hydrocarbon binaries were found that met these criteria and it was decided to include carbon tetrachloride to increase the number to the following five: carbon tetrachloride benzene; carbon tetrachloride cyclohexane; benzene cyclohexane; benzene n-heptane; and cyclohexane + isooctane. Thirty-four sets of isothermal vapor-liquid equilibrium data were retrieved from the literature for these five binaries. Because of the implications of Figure l and its related discussion, these data sets were subjected to fairly rigorous screening. Eleven of the 34 sets were eliminated immediately for excessive scatter; the points scattered so badly that either the In (yl/yz) or GE us. 21 curve could not be located with any certainty. The Gibbs-Duhem consistency test eliminated four more sets and identified five others as marginal. A final check with the Gibbs-Helmholtz equation for mutual consistency between the GE and HE data eliminated the five marginal sets plus three others. This reduced the V L E data suitable for testing the correlation to the 11 sets listed in Table 111. The available HE and VE data were checked for scatter and for agreement between the various sources. The HE data, as mentioned above, were checked for mutual consistency with the GE data. All of the H E data used came from the literature; the sources are listed in Table 111. Almost all of the VE data used were measured by Yuan (1971), who took data a t each of the temperatures a t which good GE values were available up to 8OoC.This provided eight sets of good isothermal GE and VE data with which to test the correlation. The data evaluation methods used, plus a comprehensive compilation of binary GE, HE, and VE data for the C5-Ca hydrocarbon systems, are given in a report to the Office of Standard Reference Data by Smith, et al. (1971).
+ +
+
+
Data Reduction
Mixture Parameters. The mixture parameters,
fxx
and
k,, were calculated from the binary mixture data asfunctions of comvosition by solving the GE and V E equations simultaneously a t each mixture composition where GE was known from experimental data. VE values a t the same compositions were obtained by interpolation. The VE data must be used in this calculation if accurate hxx values are to be obtained because the GE and HE values are relatively insensitive to the size parameter. Either GE or H E can be used for the other needed property. GE data were used in this work because they can be more accurately checked for accuracy. The fxx, h,, values for the benzene n-heptane system a t two temperatures are shown in Figure 3. The end-point values are the faa, ha, values; the end-point values for benzene are 1.0 because it was the reference substance. The lack of dependence of h,, on temperature shown in Figure 2 is not typical of all systems. It reflects the fact that VE for benzene + n-heptane is almost constant over the temperature range covered. Test of Basic Correlation Structure at the Mixture Level. Once the GE and VE data have been reduced to fxx and hxx values, the validity of the corresponding-states equations for
+
Table 111. Excess Property Data Used to Test the Correlation System
OC
V I E Data Reference
Schulze (1914) Scatchard, et al. (1940) Scatchard, et al. (1939a,b) Brown and Ewald (1950)
HE Data
VE Data
OC
19.9 40 60 69.8 39.9 60 69.9
Reference
Yuan Yuan Yuan Yuan Yuan Yuan Yuan
(1971) (1971) (1971) (1971) (1971) (1971) (1971)
OC
Reference
15 25 35 40 10
Vilcu and Stanciu (1966) Bennett and Benson (1965) Bennett and Benson (1965) Goates, et al. (1959) Adcock and ,McGlashan (1954) Adcock and McGlashan (1954) Adcock and McGlashan (1954) Adcock and McGlashan (1954) Goates, et al. (1959) Grosse-Wortman, et al. (1966) Lundberg (1964) Noordtzig (1956) Mrazek and Van Ness (1961) Lundberg (1964) Nicholson (1961) Vilcu and Stanciu (1966) Vilcu and Stanciu (1966) Brown, et al. (1955) Lundberg (1964) Lu and Jones (1966) Vilcu and Stanciu (1966) Lundberg (1964)
25 40 55
Boublik (1963) Scatchard, et al. (1939a,b) Boublik (1963) Scatchard, et al. (1939) Kortum and Freier (1954)
80
25 30 40
Powell and Swinton (1968) Yuan (1971) Powell and Swinton (1968) Yuan (1971) 60 69.8 Yuan (1971)
Yuan (1971) Werner and Schuberth (1966) 20 F u and Lu (1966) Yuan (1971) 60 Brown (1952) 79.9 Yuan (1971)
mixtures can be checked by using these “experimental” mixture parameter values to predict HE a t the same temperature. No adjustable parameters (such as k12 and cI2,or the A , B , C, and D constants) are involved in this prediction. The test was passed successfully. KO separate proof of this is given here because the HE curves predicted directly from the f,,,h,, values were almost indistinguishable from those shown later for the final correlation. This test of the basic correlation structure is analogous to the test performed previously a t the pure component level when the H’, surface was predicted with pure-component parameters obtained from the G’, and V , surfaces. Success in these predictions is not guaranteed by the fact that the enthalpy and volume equations were obtained from the Gibbs function equations by thermodynamically consistent derivations. If such were the case, it would always be possible to fit activity coefficient data accurately with fitting equations derived from any GE equation in a thermodynamically consistent manner. Unlike-Pair Parameters. The fxx, h,, values for each binary a t each temperature were further reduced to j a b and hab values at each composition for which the fgx, h,, values were known. Three forms-the random equations (eq 17 and 18), Leland’s equations (eq 19 and 20), and Brown’s (1957a,b) equations-were tested. A particuIar set of averaging equations was judged to be adequate if they provided isothermal f,b and hab values across the composition range which were constant through a t least two decimal digits and preferably three. The fab and hab values cannot vary more than this if the correlated GE and VE data are to be reproduced within experimental error across the entire composition range with single fab and h,b values. Binary Constants. The “essentially constant” fab and h,b values obtained with the Leland equations for each binary at each temperature were averaged arithmetically to obtain
h
15 20 25 30 35 50 90 15 20 20 25 25 35 50
‘dH6
1.02
- nc7H16
Reference is Benzene
1.60
1.50
1.40 hxx
‘XX
1.30
1.20
1.10
0 . d 0
I
0. 2
1.00
0.4
0.6
0.8
J
“dH6 Figure 3. Mixture conformal parameter values at two n-heptane system obtained temperatures for the benzene from GE and VE data with benzene as the reference substance
+
the final fab, hab sets. Only the values between x1 = 0.15 and 0.85 were used in these averages in order to eliminate the larger scatter in the experimental data near the end points. Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
393
I
might more clearly reflect molecular characteristics than do the fab and h a b values. Table IV lists the kab, Cab sets for all of the available good sets of GE and V Edata. Note that the binary constants could be determined a t more than one temperature for only three of the four binaries. Temperature Correlations. The slight variation in the kab, Cab values with temperature must be correlated if the required 0.15 and 0.0201, accuracies in the predictedfab and h a b values is to be achieved. The three sets listed in Table IV for benzene cyclohexane indicated that plots of In (1 - kab) and In (1 - Cab) us. 1/T might be straight lines; hence the Arrhenius forms shown as eq 23 and 24. These equations performed well in that (as will be shown below) the VLE data a t 10 and 119.3OC are predicted accurately with constants determined from data in the 40 to 7OoC range. Also, the available H E data are predicted quite accurately; this prediction is a more sensitive test of the temperature correlations because a differentiation is involved. The final step in the experimental data reduction was the determination of the A,B,C,D set for the three systems where kab, Cab values were available a t two or more temperatures. The final correlation constants are shown in Table V. The accuracy with which these constants reproduce the correlated data and predict other data not involved in the correlation is shown below.
I
Reference is Benzene GEat 40"C, Scatchard, et al. (1939) 0 VE at 40°C, Powell, et al. (1968) A HE at 35"C, Mrazek, et at. (19611 0
+
"gH6 Figure 4. Reproduction of the correlated GE and V E data and prediction of the HE data for benzene cyclohexane a t 40°C
+
Table IV. Binary Correlation Constants wAa,b) =
XaXb
k sb
cab
0.004876 0.005152 0.005095 0.019925 0.019270 0.018614 0.029785 0.025747
0.000358 0.000057 -0.000446 - 0.002658 - 0.002521 -0.002447 - 0.006681 - 0.005776
The j a b , h& constants were converted to kab, Cab values via eq 21 and 22. There were two reasons for this. First, the presence of the temperature-dependent pure component parameters in the combination rules makes k a b and Cab less temperature dependent than j a b and h a b . Second, there is some slight chance that the deviation from the geometric and arithmetic means
Predictions at Correlation Temperatures
Figures 2, 4,and 5 show how well the constants listed in Table V reproduce the correlated GE and VE data a t one of the correlation temperatures for each system. Similar results were obtained a t the other correlation temperatures shown in Table V. Figures 2, 4, and 5 also show how well the HE data a t the same temperature can be predicted. These are true predictions because the HE data were not involved in the determination of the correlation parameters (Table V) in any way. The HE curves shown are almost identical to those obtained directly from thef,,, Lxcurves; this verifies the accuracy of the correlation of the "experimental" fxx, h,, values. The importance of the accurate H E predictions cannot be overemphasized. They verify that the temperature dependence built into the GE equation is a t least partially correct. This is necessary if VLE data are to be predicted a t temperatures other than the correlation temperature. Figure 5 shows the results obtained with two different reference substances. Both gave the same GE correlation. The differences in the VE and H E curves probably reflect discrepanices in the pure component properties; the benzene properties in general are more accurately known than the cyclohexane properties. It appears that any component in the corresponding-states class can be used as the reference substance if its properties are known accurately.
Table V. Temperature Correlation Constants Binary
CCl&&& C~H~-C~HI~ C~HG-?L-C~HV, CsHe-n-C7His
Reference substance
CBHG CsH6 CBHG C6Hi2
Binary temperature correlation constants
A
0.993225 0.996571 0.994226 0.995139
394 Ind. Eng. Chem. Fundam., Vol. 11, No. 3, 1972
B
0.560136
- 5.306103 -7.166731 -7.472261
c 1.001710 1.000234 1.001367 1.000486
D
- 0.605742 0.760234 1.551946 1.886031
Correlation temperatures, OC
20,70 40, 60,70 20,80 20,80
250 0.8
m
1.6
P 3 ui d x
P
150
---
1.4 I Y
Reference is Benze Reference is
I
8
I
0
2loo
c
48 3 50
5-
C, Werner, et al. (1966)
0
0
0.2
0.4
0.6
I. 2
Reference is Benzene
6
w-
2
I
I
I
20
40
M)
t, "C
I
I
I
80
100
120
Figure 6. Prediction of midpoint GE,VE, and HE values as functions of temperature for benzene f cyclohexane. The solid points denote data used to determine the correlation constants
0. a
'gH6
250
- nC7H16
"gH6 Figure 5. Reproduction of the correlated GE and VE data and prediction of the HE data for benzene n-heptane at 20°C with two diffeient reference substances
+
200 Referencei s Benzene
Predictions at Other Tempercttures
Figures 1, 6, and 7 show how the predicted midpoint values a t various temperatures agree with the available good data for all three excess properties. The benzene cyclohexane system (Figure 6) provides the best check on the extrapolation capabilities of the correlation because good GE data are available 30" below and 50" above the correlation temperature range. The first flaw in the predictions becomes apparent in the slope of the predicted HE curves in Figures 1, 6, and 7 . The slope appears to be off somewhat, signifying that the simple temperature correlations may not represent the second temperature derivative of GE accurately. The discrepancies are small in Figures 6 and 7 and might be attributed to possible inaccuracies in a few midpoint HE values if the same trend were not apparent in all three binaries. The discrepancy is the greatest in Figure 1 but, as shown in Figure 2 and discussed earlier, the uncertainties in the GE data a t 2OoC for carbon tetrachloride benzene make the HE predictions suspect. Definite conclusions concerning this possible inadequacy in the temperature correlations must await the experimental data necessary for wider testing. Despite this possible discrepancy, Figure 8 shows good predictions of the activity coefficients over a wide temperature range for benzene cyclohexane. Good predictions were also obtained for the other two binaries but data were not available over such a wide range. The accurate predictions of the activity coefficients a t the correlation temperatures (open points in Figure 8) verify that the composition dependency of the GE correlation is accurate, while the predictions a t temperatures outside the correlation range check the temperature dependency. The correlation constants were also used to predict y and p values a t all of the available experimental 2 and t values for
+
+
+
Reference i s Cyclohexane
$1
--.
VE
1.6
Y
0)
0.4-9 .E m' 0. U
0.2; Y
>
01
I
M
I
I
t,
"c
I
80
0
Figure 7. Prediction of midpoint GE, VE, and HE values as a function of temperature far benzene n-heptane. The solid points denote data used to determine the correlation constants
+
the four test systems. Since the accuracy of the activity coefficient predictions predetermines the accuracy of the y and p predictions, it would be redundant to illustrate the latter. Multicomponent Predictions
Unfortunately, no ternary data are available for the binaries to which the GE and HE data requirements restricted this study. However, all the equations are multicomponent in form and handle multicomponent systems as readily as binary ones if the ksb, Cab constants are available for each binary pair in the mixture. Ternary predictions have been illustrated by Calvin and Smith (1971) and Houng and Smith (1971) for the simple liquids. Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
395
the Washington University Computing Facilities are gratefully acknowledged. 1 . 6 1 Y o o coat 4O.O0C, Scatchard, et al. (1939) oat 70.O’C, Scatchard, et al. (1939) at 60.OoC, Boublik (19631 at 10. O°C, Boublik (19631 1.5 at 119.3”C, Kortum, etal. (1954
A , B , C, D
fined by eq 23 and 24
molar Gibbs function distance conformal parameter, h’/a molar enthalpy molar heat of vaporization size conformal parameter, 93 pair weighting factor for specific interactions between molecular species a and b = binary correlation constant defined by eq 21 = natural logarithm = any molar property = Avogadro’s number = number of moles in the mixture = total system pressure, atm = component vapor pressure, atm = gas law constant = size-shape weighting factor for a given molecular species = absolute temperature, O K = temperature, “ C = molar internal energy = molar volume = pair weighting functions for energy parameters. Defined by eq 10 = pair weighting functions for size parameters. Defined by eq 11 = liquid mole fraction = vapor mole fraction = = = = =
1.2
1.1
0
0. 2
0.4
0.6
22
=
1.3
1.0
= binary temperature correlation constants de= second virial coefficient = binary correlation constant defined by eq = energy conformal parameter
Reference is Benzene
1.4
*
Nomenclature
0.8
1
XC6H6 Figure 8. Predicted activity coefficient curves for benzene cyclohexane at 10, 40, 60,70,and 1 19.3”C. The open points denote the temperatures at which the GE and VE data were correlated
+
Future Work
The correlation framework appears to work well for the Ce-C7 hydrocarbons. These results plus those obtained earlier on the simple liquids indicate that a correlation covering all nonpolar molecules may be possible. An extensive literature data retrieval and evaluation program is underway as the first step in the development of that correlation. The C6-Cs hydrocarbon segment of the retrieval and evaluation program has been covered in the report by Smith, et al. (1971); that report includes 266, 197, and 255 sets of binary VLE, HE, and VE data involving 91 systems and 31 pure components. Extensive HE and VE measurements are being made to supplement the literature data with the intention of basing the correlation constants on HE and VE data and using the available VLE data to test the accuracy of the predictions. Similar data retrieval and evaluation projects are presently underway for the C3 and C4 hydrocarbons and for bhe alcohol water systems. The empirical temperature correlations, eq 23 and 24, appear to extrapolate the VLE data quite well but may not accurately represent the derivatives
+
GREEKLETTERS = liquid thermal expansion coefficient = activity coefficient = partial derivative operator
(Y
;
SUBSCRIPTS = denotes a property of molecular species ‘‘a” or a or b “bll aa or bb = denotes a pair of like molecules = denotes a pair of unlike molecules ab = denotes the reference substance 0 = denotes a property of the hypothetical pseudoP substance = denotes a property of the mixture X = denotes a pair of hypothetical pseudosubstance xx molecules = used instead of “a” and “b” when the equa1 and 2 tions are restricted to a binary SUPERSCRIPTS = denotes an excess property I = denotes a configurational property. Also used in symbol for vapor pressure, p’ // = denotes the modified configurational Gibbs function. See eq 7c * = denotes an ideal gas property
E
literature Cited
It may be advisable to modify the temperature correlations
Adcock, D. S., hicGlashan, M.L., Proc. Roy. SOC.,Ser. A 226,266
when sufficient experimental data become available to adequately define the way the k& and cab constants vary over wider temperature ranges.
Bender, P., Furukawa, G. T., Hyndman, J. R., Znd. Eng. Chem.
Acknowledgment
This research was supported partially by industrial funds in the Thermodynamics Research Laboratory and by Sational Science Foundation grant GK-1971. The services rendered by 396 Ind. Eng. Chern. Fundam., Vol. 1 1, No. 3, 1972
(1954).
44, 387 (1952).
Bennett, J. E., Benson, G. C., Can. J . Chem. 43, 1912 (1965). Boublik, T., Collect. Czech. Chem. Commun. 28, 1771 (1963). Brown, I., Aust. J . Sci. Res., Ser. A 5, 530 (1952). Brown, I., Ewald, A. H., Aust. J . Sci. Res., Ser. A 3, 306 (1950). Brown, C. P., Mathieson, A. R., Thynne, J. C. J., J. Chem. SOC. London 4141 (1985).
Brown, W. B., Phil. Trans. Roy. SOC.London, Ser. A 250, 175 (1957a).
Brown, W. B., Proc. Roy. SOC.,Ser. A 240, 561 (1957b).
Calvin, W. J., Smith, B. D., A.I.Ch.E. J. 17, 191 (1971). Chen, N. H., J. Chem. Eng. Data 10, 207 (1965). Francis, P. G. McGlashan, M. L., Hamann, S. D., McManamey, W. J., J. Chem. Phys. 20, 1341 (1952). Fredenslund, A,, Sather, G. A., IND.ENG.CHEM.,FUNDAM. 8,
Rossini, F. D., Ed., “Selected Values of Properties of Hydrocarbons and Related Compounds,” American Petroleum Research Project 44, Carnegie Press, Pittsburgh, Pa., 1953. Scatchard, G., Wood, S. E., Mochel, J. M., J . Amer. Chem. SOC.
Fu, S. J., Lu, B. C. Y., J. Appl. Chem. 16, 324 (1966). Glaser, F., Ruland, H., Chern.-Zng.-Tech. 29, 772 (1957). Goates, J. R., Sullivan, R. J., Ott, J. F., J . Phys. Chem. 63, 589
Scatchard, G., Wood, S. E., Mochel, J. M., J . Amer. Chem. SOC.
718 (1969).
(1959).
Grosse-Wortman, H., Jost, W., Wagner, H. G., 2. Phys. Chem., Frankfurt Am. Main 49(1-2), 74 (1966). Houng, J. J., D.Sc. Dissertation, Washington University, St. Louis, -Missouri, 1969. Houng, J. J., Smith, B. D., A.I.Ch.E. J . 17, 1102 (1971). Jordan, T. E., “Vapor Pressures of Organic Compounds,’’ Interscience Publishers, New York, N. Y., 1954. Killian, H., Bittrich, H. J., 2. Phys. Chem., L e i p i g 230(5-6), 383 (1965).
Kortum, G., Freier, H. J., Chem.-Ing.-Tech. 26, 670 (1954). Kozicki, W., Sage, B. H., Chem. Eng. Sci. 15, 270 (1961). Leach, J. W., Ph.D. Dissertation, Rice University, Houston, Texas, 1967. Leland, T. W., Chappelear, P. S., Gamson, B. W., A.1.Ch.E. J . 8, 482 (1962).
Leland, T. W., Rowlinson, J. S., Sather, G. A., Watson, I. D., Trans. Faradau SOC.65. 2034 (1969). Longuet-HigginG H. C., h o c . Roy. Soh., Ser. A 205, 247 (1951). Lu, B. C. Y., Jones, €1. K. D., Can. J . Chem. Eng. 44,251 (1966). Lundberg, G. W., J. Chem. Eng. Data 9, 193 (1964). McGIashan, M. L., Potter, D. J. B., Proc. Roy. SOC.,Ser. A 267, 478 (1962).
Mrazek, R. V., Van Ness, H. C., A.I.Ch.E. J. 7, 190 (1961). Nicholson, D. E., J. Chem. Eng. Data 6, 5 (1961). Noordtzig, R. M. A., Helv. Chim. Acta 39,637 (1956). Palmer, D. A., D.Sc. Dissertation, Washington University, St. Louis. Missouri. 1971. Pitzer, K. S., J. Chem. Phys. 7,583 (1939). Powell, R. J., Swinton, F. L., J . Chem. Eng. Dota 13, 260 (1968). Rao, V. N. K., Swami, D. R., Rao, ?VI.N., A.I.Ch.E. J. 3, 191
61, 3206 (1939a).
Scatchard, G., Wood, S. E., Mochel, J. AI., J. Phys. Chem. 43, 119 (193913).
62, 712 (1940).
Schulze, A., 2. Phys. Chem., Leipzig 86, 309 (1914). Smith, B. D., Holt, D. L., Stookey, D. J., Yuan, I-C., “Evaluation of Thermodynamic Excess Property Data for Nonelectrolyte Mixtures-Subcritical, Miscible, Binary, Mixtures of Hydrocarbons with Five to Eight Carbon Atoms,” Thermodynamics Research Laboratory, Washington University, St. Louis. Missouri. 1971. Stookey, D’. J., Smith, B. D., paper presented a t the Houston Meeting of the A.1.Ch.E. in March 1971. Stull, D. R., Ind. Eng. Chem. 39, 517 (1947). Timmermans, J., Ed., “Physico-Chemical Constants of Pure Organic Compounds,” Vol. 1, Elsevier Publishing Co., New York, N.Y., 1950. Van Ness, H. C., “Classical Thermodynamics of Non-Electrolyte Solutions,” Pergamon Press, New York, N. Y., 1964, p 79. Vilcu, R., Stanciu, F., Rev. Roum. Chim. 1 1 , 175 (1966). Waddington, G., et al., J . Amer. Chem. SOC.69, 27 (1947). Washburn, E., Ed., “International Critical Tables,’’ McGrawHill, New York, N. Y., 1926. Weast, R. C., Ed., “Handbook of Chemistry and Physics,” 47th ed., The Chemical Rubber Co., 1966. Werner, V. G., Schuberth, H., J . Prakt. Chem. 31, 225 (1966). Wheeler, J. D., Ph.D. Dissertation, Purdue University, Lafayette, Indiana, 1964. Wheeler, J. D., Smith, B. D., A.Z.Ch.E. J . 13, 303 (1967). Willingham, C. B., Taylor, W. J., Pignocco, J. &I.,Rossini, F. D,, J . Res. S a t . Bur. Stand. 35, 219 (1945). Young, S., Sci. Proc. Roy. Dublin SOC.,S.S . 12, 374 (1910). Yuan, I-C., D.Sc. Dissertation, Washington University, St. Louis, Missouri, 1971.
,.
RECEIVED for review October 1, 1971 ACCEPTED June 2, 1972
(1067) \A””.
Reamer, H. H., Sage, B. H., J . Chem. Eng. Data2, 9 (1957).
Correlation of a Partially Miscible Ternary System with Conformal Solution Theory David A. Palmer,l Wallace I.-C. Yuan,* and Buford D. Smith* Thermodynamics Research Laboratory, Washington University, St. Louis, Mo. 63230
+
+
Extensive GE,HE, and VE data for the partially miscible acetonitrile benzene n-heptane system have been correlated successfully with a formalism based on conformal solution theory. The GE correlation was superior to that obtained with the three-parameter NRTL equation. The general weighting function approach suggested in 1964 b y Wheeler was used. The radius of gyration was used to represent size-shape effects. Pair weighting parameters were determined for those molecular pairs with identifiable special interactions. A general correlation based on this approach may be feasible for mixtures containing polar molecules.
A
previous paper (Yuan, et a ~1972) , described a correlation structure for hydrocarbon (nonpolar) systems based on the conformal solution formalism. Two binary constants, obtained from GE and VE data at one temperature, were required to correlate the binary GE and VE data a t one temperature. 1
111.
Present address, Amoco Chemicals Corporation, Naperville,
Present address, Chemical Engineering Department, Tunghai University, Republic of China.
Four binary constants, obtained from either GE and VE data or H E and VE data a t two temperatures, were required to predict GE, HE, and VE data as a function of both composition and temperature. The correlation utilized the Leland, et ai. (1962), equations to relate the mixture conformal parameters to the like- and unlike-pair parameters. The present paper extends the correlation structure to the partially miscible acetonitrile (1) benzene (2) n-heptane (3) system which includes a polar molecule. (The numbers
+
+
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
397
