m E

E L S E V I E R Fluid Phase Equlhbrla 102 (1994) 107-120

Solubility in supercritical fluid mixtures with co-solvents: an integral equation approach

Hideki T a n a k a *, Ko ich i ro Nakan i sh i

Department o/Polymer Chemtstry, Graduate School of Engineering, Kvoto ~#m'er~zO', Sakyo-ku, Kvoto 606-01, Japan

Recewed 17 March 1994, accepted m final form 16 June 1994

Abstract

The solubihty of naphthalene in supercrltlcal carbon dioxide flmds has been evaluated by means of the Integral equation method The role of entralner was anvestagated by examining the pressure dependence of solubility curves in fluids with or without entralner The solubility of naphthalene increases with the addition of a small amount of Co-solvent This increase in solubility arises from the attractive interaction between the co-solvent and the solute naphthalene and from a decrease in pressure at the same density due to stronger interactions between carbon dioxide and the co-solvent than that for carbon dioxide dlmer

Keywords Supercrltlcal fluid, Solubility, Entralner, Integral equation

1. Introduction

In the supercritical region, the pressure of any substance changes continuously since there 1s no gas-l iquid phase transition. When a solute with a melting point higher than that of the solvent is dissolved in the supercritlcal fluid, its solubility also changes continuously as a function of the pressure. It is well known th.at the solubility changes abruptly just above the critical temperature as the pressure passes through the critical pressure. Upon the addition of a small amount of some other molecular species or co-solvent, the solubility may increase. This co-solvent is referred to as an "entrainer". This increase in solubility might be accounted for by the conjecture that an attractive interaction between the co-solvent and the solute ~s dominant in the supercritical region.

* Corresponding author

0378-3812/94/$07.00 ,~, 1 9 9 4 - Elsevier Science B V. All rights reserved SSDI 0 3 7 8 - 3 8 1 2 ( 9 4 ) 0 2 5 6 0 - N

108 H Tanaka, K Nakanlshl /Flutd Phase Equthbrla 102 (1994) 107-120

Such an account of the co-solvent effect on the solubility has not, however, been founded on molecular theory In the present paper, the solubility of naphthalene is examined in order to delineate how a small amount of co-solvent increases the solubility. To this end, the chemical potential of the solute at a very dilute concentration should be evaluated.

Supercritical fluids have been used for the extraction of various thermally unstable com- pounds having high melting points. The selection of entralner species is very important for industrial use. If the solubility of various compounds in supercritical fluids is estimated only from intermolecular interactions, it is expected that a relation between the nature of intermolec- ular interactions and the solubility will be easily established. This provides a guideline for the appropriate choice of an entrainer.

One of the best ways to calculate the free energy of solvation in a fluid is undoubtedly computer simulation. In fact, the free energy or chemical potential of solvation has been evaluated in computer simulation by various methods such as the particle insertion method proposed by Widom (1963), Hill's method (1956) and thermodynamic integration (Allen and Tildesley, 1988). In all those methods, computational demand for correct evaluation of the free energy is heavy even in a fairly dilute fluid system composed of only spherical molecules. The free energy might be evaluated erroneously, especially in a mixture unless an appropriate technique is used. When we are interested in the free energy of the solute component in a fluid mixture where the mole fraction of the co-solvent is much lower than that of the solvent (this situation is usual in practical extraction), the statistical error may be too large for obtaining a convergent result by ordinary simulation techniques. Some simulation techniques are well documented in a review article by Shing (1989).

In order to treat mixtures, we take an alternate approach, thereby avoiding the incorrect evaluation of the free energy of solvation due to statistical errors. In the integral equation method, the structural and thermodynamic properties, which are free from statistical errors, are easily calculated for various mtermolecular interaction parameters In addition, the free energy can also be obtained when the hyper netted chain (HNC) approximation (Hansen and McDonald, 1986) is applied, and therefore the solubility is estimated in terms of the intermolecular interactions. The free energy calculation and the prediction of solubility in the supercritical region are made by a somewhat different method, introducing a non-adiabatic coupling between solute and solvent interactions (Munoz and Chimowitz, 1992). Some other publications on supercritical fluids by integral equation study are summarized by Ekart et al. (1989) and Lee et al. (1989).

In the present study, we solve the integral equations with the HNC closure for pure solvent, and mixture with co-solvent. Then, solute-solvent correlation functions are obtained. Those are used in turn for evaluation of the solubility. The entralner effect (Kurnik and Reid, 1982, Gopal et al., 1985; Dobbs et al., 1986, 1987: Joshl and Prausnitz, 1984; Walsh et al., 1987; Akgerman et al., 1989) is examined on the basis of the analysis of these solubility data and the distribution functions.

2. Theory and method

The fluid studied here consists of two or three components: solvent, co-solvent (if any), and solute These are identified as carbon dioxide, octane, and naphthalene. Throughout this study,

H Tanaka, K Nakamsht / Flwd Phase Eqmhbrla 102 (1994) 107-120

Table 1 Intermolecular interaction parameters for carbon d~oxide, octane and naphthalene

109

Molecule ~'~ (~) ~/k b (K)

Carbon dioxide 3 91 225 3 Octane 6 93 421 3 Naphthalene 6 45 554 4

.i Sxze parameter b LJ energy parameter

the intermolecular interactions are assumed to be pariwise additive and the shape of all solvent, co-solvent and solute molcules is approximated to that of a sphere described by the Lennard- Jones (LJ) potential:

, , ,

It is reasonable in the supercritical region (low density and high temperature) that the aspherical nature of the interaction is averaged out owing to rapid rotat ional motions , and so a molecule is well represented by a single-interaction-site model. The energy and size parameters (E and o-) used in the present study are listed in Table 1. These are determined according to the phase d iagram for an LJ fluid (Nicolas et al., 1979) in the usual manner (Eya et al., 1992). In the following sections, several the rmodynamic quantities such as the temperature and density are given in the units reduced by the LJ energy and size parameters of the solvent carbon dioxide.

Structural properties are obtained by solving the Orns te in -Zern ike equat ion in combinat ion with the H N C or Percus -Yevick (PY) closure. The Orns te in -Zern ike equat ion in Fourier space is written in terms of a direct correlation g and a pair correlation function ,17 = ~ - 1 ( g is a

radial distr ibution function (RDF) ) as

t7 . . . . ( 2 ) (1 -pc-')

The H N C closure is written in real space as

h = exp( -flq5 + t) - 1 (3a)

and the PY closure as

h = exp(-fl~b) (t + 1) - 1 (3b)

where p is the number density, t = h - c, and fl is the reciprocal of Bol tzmann's constant times the temperature T. In the case of mixtures, quantit ies such as h, c and p in Eq. (2) are replaced by matrices. The matrix which represents the density has only diagonal elements. The integral equat ion is solved iteratively until the difference in the indirect interaction part t between two successive iterations becomes less than the threshold value, 10 -s, for all distance ranges treated here. The number of grid points is 212 with a discretization size of 0.05 in real space.

Since the solubility of the solute is low, most of its properties at infinite dilution can be substi tuted for the corresponding values of solute in the real system; the number of solute is

110 H Tanaka, K Nakamsht /F lu td Phase Equthbrta 102 (1994) 107 120

unity and the solvent properties on average do not change at all in the presence of this solute species. The solute-solvent correlation functions are written in terms of the solvent-solvent correlation functions as

hu, = Cu, + cu, * pvh,,v (4)

where the subscripts u and v denote the solute and the solvent, respectively, and the asterisk stands for a convolution integral. An iterative scheme is again required to solve Eq. (4) with the HNC or the PY approximation. The extension of Eq. (4) to mixtures with co-solvent is straightforward. The solute-solute correlations for spherical molecules can be obtained without further iteration; the indirect part tuu ~s given simply by Cur * p v h , u . However, little attention will be paid to this solute-solute correlation function in the present study.

It is well known that the PY approximation is superior to the HNC approximation for a system where repulsive interactions dominate. The HNC approximation becomes more accurate with decreasing density. Near the critical temperature of an LJ fluid, we could not solve Eqs. (2) and (3) with the HNC or PY closure by a simple iterative method. It was reported that the PY approximation is more reliable than the HNC near the critical point (Brey and Santos, 1986; McGuigan and Monson, 1990): it is claimed that the HNC approximation does not give a correct critical point. However, in the supercritical region treated here (the LJ reduced temperature is higher than 1.4), the HNC IS also accurate in describing thermodynamic and structural properties since we are not interested in the locus of the critical point. Moreover, the HNC approximation is advantageous since the chemical potential of each component can be easily evaluated; no further iterative scheme is reqmred. Therefore the HNC is adopted m the present study.

Thermodynamic properties can also be calculated from pair correlation functions. For example, the potential energy u and the pressure of the system p are calculated as follows:

u = 2rcp O(r )g ( r ) r 2 dr (5 )

and

2 fo" ?4(r) P -- 1 - ~zp - - g ( r ) r 3 dr (6 ) p k T 3 ~r

In addition, the residual chemical potential of solute/~e can be obtained when the HNC closure is applied (Zichi and Rossky, 1986) according to the equation

!~ ~ = 2 n p k T [h(r) 2 - 2c(r) - c(r)h(r)] r 2 dr (7 )

The solubility of the solute y (as a mole fraction) is calculated in terms of the residual chemical potential of the solute and the saturated vapor pressure of the pure solute species pO as

= p 0

Y p k T exp{ - fl[/~ + v ( p ° - p)] } (8)

where t' is the molecular volume of the pure solute. It should be noted that the pressure p in Eq. (8) is a hydrostatic pressure of the system and is distinguished from the saturated vapor pressure of the solute p0 at a given temperature.

H. Tanaka, K Nakantsht / Fhad Phase Eqmhbrla 102 (1994) 107 120 111

3. Results and discussion

The accuracy of the integral equation method with the HNC closure must be examined prior to its application to the evaluation of solubility. For comparison with an "exact" experimental RDF, we performed a molecular dynamics simulation. The reduced temperature T* was set to 1.35, which is the critical temperature, and the reduced density p* was 0.7, which is fairly high compared with the critical density (p* = 0.35). The RDFs from molecular dynamics simulation and from the integral equation with the HNC closure are depicted m Fig. 1. There is no &fference in the RDFs obtained by these two methods. We therefore expect that the integral equation method is adequately reliable for the calculation of structural and thermodynamic properties in the supercritlcal region.

In the first place, the thermodynamxc and structural properties of pure carbon dioxide were examined. The temperatures of the system were set to 1.4 (315 K) and 1.65 (372 K) in reduced (real) units. The LJ parameters for carbon dioxide and naphthalene were chosen according to the law of corresponding states for an LJ fluid as mentioned above. Unlike interactions were determined according to the usual combining rule. The appropriateness of this choice is examined later. The R D F for CO2-CO2 at a temperature T * = 1.4 and density p* = 0 3 is plotted for both the HNC and the PY approximations in Fig. 2. Again no appreciable difference is found between the two approximations. The energy and pressure from the HNC closure are -2 .1665 and 0.18740, respectively, in reduced units, which should be compared with those from the PY approximation, -2.1421 and 0.187 51. We expect the HNC approximation to describe, at least qualitatively, the structure and thermodynamic properties in the supercritical region except very near the critical point. The (reduced) pressure from the HNC equation is compared with those from the approximate theory, the cubic equation (Peng and Robinson, 1976; Reid et al., 1987), in Fig. 3. Agreement with the cubic equation is good in the lower pressure region but is not so good at a pressure higher than 0.4 (in reduced units).

I I

1

, , I , I i

0 2 3 4 r/sigma

Fig 1 Radial distribution functions for C02 The sohd and broken hnes are from molecular dynamics simulation and the integral equation method with the HNC approximation respectively

112 H Tanaka, K Nakantsht / Fluid Phase Equthbrla 102 (1994) 107 120

1.5

C O

1

0.5

, I i L k

0 2 3 r/sigma

F~g. 2 Radial distribution functions for CO2 at a temperature T * = 1 4 and density p* = 0 3 with the HNC approximation (solid line) and the PY approximation (broken line)

0.4

- I 0 9

0.2

I I I

1 i I i 1 i I i .1 0 , 2 0 . 3 0 . 4 0 . 5

density

Fig 3 Pressure of pure C O 2 a t T = 1.4 calculated by the integral equation with the HNC closure (sohd hne) together with that from a cubic equation of Peng-Robmson type (broken line)

H Tanaka, K. Nakantshl /Flutd Phase Eqmhbria 102 (1994) 107-120 113

I I I

i i I ~ I i

0 2 3 4

r/sigma

Fig 4 Radial distribution functions for CO2 at a temperature T* = 1 4 and densities p* = 0 l, 0 3, 0 5, 0 7 and 0 9 The value of the first minimum of each RDF decreases with increasing density

Other RDFs with the HNC approximation at the same temperature and for various densities (0.1, 0.3, 0.5, 0.7 and 0.9) are shown in Fig. 4. The R D F for a density of 0.1 has a simdar shape to the Boltzmann factor of the pair interaction potential. As expected, near the critical region the value of the R D F beyond the first minimum is higher than unity, reflecting a large density fluctuation as shown in Figs. 2 and 4; there are only two peaks in the RDF. This is typical behavior near the critical point. As the density increases, the shape of the R D F becomes that of the liquid state; the oscillatory character of the R D F gradually dominates and a long-range fluctuation diminishes. It is of interest to examine the magnitude of the density fluctuation to be observed near the critical temperature as shown by Miinster (1969). We obtained the Fourier t ransform of the pair correlation function in the course of the iteration. The values of h (k = 0) for pure CO2 at T* = 1.4, which have a direct connection with the isothermal compressibility, are listed in Table 2. The large fluctuation is indeed seen near the critical density, p* = 0.3.

We chose octane as an entrainer. The mole fraction of the entrainer was 0.035. Because of a convergence problem inherent in the numerical solution of the integral equation, we solved the integral equations for mixtures at a temperature of 372 K (T* = 1.65), which is rather high compared with that for the pure fluid. The results obtained may be extrapolated to the near critical (lower) temperature region. The RDFs at a density p* = 0.4 for CO2 pair in pure carbon dioxide and the mixture are depicted in Fig. 5 together with the R D F for oc tane-CO2. There is only a slight difference in the R D F for CO2 pairs in the pure and mxxture systems. The RDFs for CO2-naphthalene, oc tane -oc tane and oc tane-naph tha lene at the same density are depicted

114 H Tanaka, K Nakamsht / Flutd Phase Equthbna 102 (1994) 107 120

Table 2 Densi ty f luctuat ion term for CO2 fired at T* = 1 4, which 1s represented by h (k = 0), the Four ier t r ans form of the

pair corre la t ion funct ion

Reduced densxty p* h (k = 0) Reduced density p* h (k = 0)

0 1 0 9 9 5 0 3 4 24 19 0 12 11 47 0 36 18 83 0 14 13 44 0 38 12 74 0 16 16.08 0 40 8 50 0 18 19 76 0 4 2 562 0 20 25.26 0 44 3 62 0 22 34 68 0 46 3 62 0 24 66 29 0 48 1 24 0 30 359 62 0 50 0 55 0 32 46 35

1

I I I I

I i k I I 0 2 4

r/sigma

l i

I

I ] ,

Fig 5 R a & a l d is t r ibut ion funct ions for CO 2- C O 2 in pure fluid and for C O 2 C O 2 and CO2 octane m mixture with oc tane at a t empera ture T* = 1 65 and density p* = 0 4 The sohd and broken hnes indicate R D F s for CO2 CO, pairs m the pure fired and mixture respectively The R D F for C O 2 - o c t a n e is ln&cated by a thick b roken hne.

in Fig. 6. These RDFs have only two distinct peaks although the temperature is much higher than the critical value for pure CO2. In particular, the R D F for octane-naphthalene has a long tail. This indicates that there exists a large fluctuation in the co-solvent concentration. The direct correlation functions for CO2-naphthalene and octane-naphthalene are depicted in Fig. 7. The difference in CO, pairs at T* = 1.4 is fairly large over a small range of distance.

The saturated vapor pressure of naphthalene together with the molar volume at a given temperature are required to calculate the solubility. The molar volume is taken from the experimental data at 298.15 K, and its temperature dependence is neglected. The contribution from the saturated vapor pressure is independent of the hydrostatic pressure p of the system.

H Tanaka, K Nakantsht / FIuM Phase Equfftbrta 102 (1994) 107 120 115

1 l i i t f

l! f t

i

/; ,l

i / t

t"l I I

L I I i I i I ,

0 2 4

r/sigma Fig 6 Radaal distribution funcUons for CO2 naphthalene in the pure fluid and for octane octane, CO. naph- thalene and octane naphthalene m mixture with octane at a temperature T* = 1 65 and density p* = 0 4 The sohd. broken, dash-dot and thick broken hnes show RDFs for CO2-naphthalene in the pure fluid, and for COx naph- thalene, octane-octane and octane-naphthalene in the mixture respectively The Lorentz-Berthelot rule is adopted

for all unhke interactions

-6

/ /

/ t

i t

i t

/ t

/

/ i

i ! /

_12 0 , I 2 ~ , I , t 4 r/sigma

1 I , I I

I

,i Fag 7 Direct correlation functions for CO2 naphthalene in the pure fluid and for CO2-naphthalene and octane naphthalene in its maxture with octane at a temperature T* = 1 65 and density p* = 0 4. The solid, broken, and thack broken lanes show the direct correlation functions for CO2-naphthalene in the pure fluad, and for CO2-naphthalene and octane-naphthalene in the mixture respectively The Lorentz-Berthelot rule is adopted for all unhke interactions

This contr ibute s a c o n s t a n t to the so lubi l i ty but it depends o n the temperature , and since we are interested sole ly in the pressure d e p e n d e n c e o f the solubi l i ty , the c o n s t a n t is n o t e s t imated in the present study. Ins tead o f solubi l i ty , an e n h a n c e m e n t factor w h i c h is def ined by

E - y (9) pO/p

116 H Tanaka, K Nakamsht /Flutd Phase Eqmhbrta 102 (1994) 107-120

0 +

0

[ I

© • 2 •

o • © •

2 •

10 20 p/Mpa

Fig 8 Enhancement factor of naphthalene In pure CO~ fluid and its mixture with octane as a function of pressure The Lorentz-Berthelot rule is adopted for all unhke interactions Small black circles represent T* = 1 4 m pure CO~, large black circles T* = 1 65 m pure CO2, and large open circles T* = 1.65 m a mixture with octane

is plotted in Fig. 8 as a function of pressure. The term p0 _ p in Eq. (8) is replaced by - p since p is of the order of 10 i and p0 is roughly 10 - 6 in reduced units even at the higher temperature T * = 1.65. The difference in the solubilities at the lowest and the highest pressures is approxi- mately 102~ in the calculated result. A change of almost two orders of magnitude in the solubility was observed in an experiment near T * = 1.4 (Tsekhanskaya et al., 1964). The dependence of the solubility-on pressure is not perfect but reasonable.

The enhancement factor in a mixture with a co-solvent of octane at temperature T* = 1.65 is also plotted in Fig. 8 against the pressure, together with the factors in pure carbon dioxide. The factor at temperature T* = 1.4 is larger than those at T* = 1.65. However, the saturated vapor pressure p0 at higher temperature T * = 1.65 is about 60-fold higher than that at lower temperature T * = 1.4, and so the magnitude of the solubility is higher at T * = 1.65. The addition of a small amount of octane gives rise to larger enhancement factors over most of the pressure range. This larger enhancement factor (higher solubility) has two origins. One is the lower pressure of the system in the presence of an entrainer, octane, for the same number density as the pure carbon dioxide system. The other is the attractive interactions between the entrainer and the solute. The excess coordination number relative to the uniform distribution defined as

~0 rc "~ 3 Ant = g(r)r-dr/rc - 1 (10)

was calculated, where g(r) is the R D F for either so lu te -carbon dioxide or solute-octane and rc is the upper bound of the coordination shell, the distance to the first minimum of the individual RDF. These quantities are listed in Table 3. It is evident that around a solute molecule there are

H. Tanaka, K Nakamsht / Fluzd Phase Equthbna 102 (1994) 107 120 117

Table 3 The excess coordination number An c (%) (defined by Eq. (10)) relative to the uniform distribution at a temperature T* = 1 65 and density p* = 0 4. The mole fraction of entramer octane is 0 035. The upper bound of integration is

denoted by r~

Molecular pair r c Ant (%)

Pure fluid

C02 C02 1 70 0 6 C02-naph tha l ene 2.25 10 6

M~xture m

C02 C02 1 70 1 6 CO2-naphtha lene 2 25 11 5 CO. octane 2 10 - 2 0 Octane octane 2 50 - 2 0 6 Octane-naphtha lene 2 65 11 9

many more entrainer molecules than carbon dioxide molecules. This stems from the strong Interaction between the entrainer and the solute. However, the entrainer molecules do not tend to form a cluster. The large coordination number of the entrainer molecule comes from the large concentration fluctuation of the entrainer, as is ascertained from the R D F for solute-entrainer.

The interaction between two carbon dioxide molecules is separated into two components: the L J-type interaction and the quadrupole interaction. This electrostatic interaction is taken into account when the intermolecular interaction is approximated to spherical interaction averaged over mutual orientations. The interactions between CO2 and other non-polar molecules are expected to be smaller than those calculated according to the Berthelot rule. Therefore the integral equations were solved using somewhat weaker pair interactions for octane-CO2 and for CO~-naphthalene by the use of the multiplying factors ~ = 0.95 and 0.9 (% = ~(e, ,%) ° 5). The temperature was set to T * = 1.44 (=337 K) for comparison with experimental solubility data (McHugh and Paulaitis, 1980). The solubility curves in pure carbon dioxide are depicted in Fig. 9. As the ~ value decreases, the solubility of naphthalene decreases. In order to compare the calculated solubility with the experimental one, the unknown constant c should be determined. This is chosen so that the solubility calculated with ~ = 1 at 20 MPa (which is found by fitting the solubility data to the third-order polynomial) coincides with that from experiment. The calculated solubilities with different ¢ values seem to differ only by a constant and the differences are almost independent of pressure. The calculated solubility is qualitatively correct and decreases on further increasing the pressure above 40 MPa (not shown in the figure). Small disagreement may arise from the incorrect estimation of the critical point using the calculated phase diagram of the LJ fluid (Johnson et al., 1993), the spherical approximation or the approximation inherent to the integral equation. The factors ~ = 0.95 and 0.9 have no ground, but the smaller interaction parameter is intuitively reasonable. However, further investigation of the appropriate choice of the interaction parameters is required.

118 H Tanaka, K. Nakamsht /Flutd Phase Equlhbrta 102 (1994) 107-120

-1

0 +

~-2

I I I I

o

o O o

c D 0 [3

0 D

0 0 ~

A

O%

d~

% o

©za

©~ Q

o ~

_ 3 J I I I I 10 20 30

p/M Pa

o o o o o

o o o []

o o o o • • [] o [3 o ~ o

o o

40

Fig 9 Solublhty o f naphthalene in pure CO2 fluid as a function of pressure at a temperature T* = 1 44 The unhke energy parameters between CO2 and naphthalene are reduced by the factors ~ = 0.95 and 0 9 to those calculated from the Berthelot rule. Open circles represent ~ = 1, open squares ~ = 0 95, and open triangles ~ = 0 9 The experimental solubd]ty (black circles) is also plotted, e ~s the constant relevant to the saturated vapor pressure of naphthalene.

4. Conclusions

The integral equations with the HNC closure have been solved in order to obtain the structure and thermodynamic properties of carbon dioxide fluid and its mixture with co-solvent octane. The RDF of carbon dioxide calculated from the integral equation was compared with that from molecular dynamics simulation at the critical temperature and the reduced density p* = 0.7. In spite of the fact that the HNC approximation becomes less accurate as the density increases, the agreement of the RDFs was excellent even for a density much higher than the density we are interested in.

The structure and thermodynamic properties of pure carbon dioxide were obtained at a temperature T* -- 1.40 and over most of the density region. The chemical potential of the solute naphthalene was calculated. It is found that the solubility obtained from the integral equation was in reasonable agreement with the experimental solubility. The role of the entrainer was investigated by examining the pressure dependence of the solubility curves of mixtures with an entrainer. The solubility of naphthalene increases with the addition of a small amount of co-solvent. This increase arises from the attractive interaction of the co-solvent with the solute naphthalene and also from a decrease in pressure.

Acknowledgments

The authors thank Ms. Joanne K. Button for a critical reading of the manuscript. They are also grateful to the Computer Center of the Institute for Molecular Science and the Supercom-

H Tanaka, K Nakamsht/Flmd Phase Equdtbrta 102 (1994) 107 120 l l9

puter Laboratory, Institute for Chemical Research, Kyoto University. The present work was supported by a Grant-in-Aid on Supercritical Fluid from the Ministry of Education, Science and Culture, Japan.

List of symbols

c direct correlation function ? Fourier transform of direct correlation function E enhancement factor g radial distribution function h pair correlation function /7 Fourier transform of pair correlation function k Boltzmann's constant k wavenumber for only argument of h Anc relative excess coordination number. p pressure pO saturated pressure of solute rc radius of first coordination shell t indirect part of potential of mean force T temperature T* reduced temperature u potential energy r molecular volume of solute y solubility

Greek letters

E Lennard-Jones energy parameter /~e residual chemical potential

scaling factor of pair potential energy p number density p* reduced number density a Lennard-Jones size parameter q~ lntermolecular interaction

References

Akgerman, A , Roop, R K , Hess, R K and Yeo, S -D, 1989. In" Bruno, T J and Ely, J F (Eds.), Supercntlcal Flmd Technology CRC Press, Boston, MA, p 479.

Allen, M.P. and Tlldesley, D J , 1988. Computer Simulation of Llqmds, Academic Press, London Brey, J J and Santos, A , 1986 Mol Phys, 57 149 Dobbs, J M., Wong, J M and Johnston, R K , 1986. J Chem Eng Data, 31 303 Dobbs, J M Wong, J M , Lahlere, R J. and Johnston, R K., 1987 Ind Eng Chem Res, 26 56

120 H Tanaka, K. Nakantsht / Flutd Phase Equthbrla 102 (1994) 107 120

Ekart, M.P , Brennecke, J F and Eckert, C A , 1989 In: Bruno, T J and Ely, J F (Eds) , Supercritlcal Flmd Technology. CRC Press, Boston, MA, p 163.

Eya, H., Iwai, Y., Fukuda, T and Aral, Y , 1992 Fluid Phase Equilibria, 77 39. Gopal, J S, Holder, G D and Kosal, E , 1985. Ind Eng Chem Process Des Dev, 24 697 Hansen, P and McDonald, I R , 1986 Theory of Simple Liquids Academic Press, London Hill, T L , 1956. Statistical Mechanics McGraw-Hill, New York Johnson, J.K, Zollweg, J A and Gubblns, K.E., 1993. Mol Phys, 78:591 Joshl, D K and Prausnltz, J .M, 1984 AIChE J, 30 522 Kurnik, R T and Reid, R C , 1982 Fluid Phase Equilibria, 8 93 Lee, L L , Debenedettl, P G and Cochran H D , 1989 In: Bruno, T J and Ely, J F (Eds) , Supercritlcal Fluid

Technology CRC Press, Boston, MA, p 193. McGmgan, D B and Monson, P.A., 1990 Fluid Phase Equilibria, 57 227. McHugh, M and PaulaltlS, M E , 1980. J Chem Eng Data, 25. 326 Munoz, F and Chlmowltz, E H , 1992 Fluid Phase EqulhbrIa, 71 237 Munster, A , 1969 Statistical Thermodynamics, Vol 1 Sprmger-Verlag, Academic Press, New York Nicolas, J J , Gubblns, K E , Streett, W.B and Tlldesley, D.J , 1979 Mol Phys, 37 1429 Peng, D - Y and Robinson, D B, 1976 Ind Eng. Chem Fundam, 15 59 Reid, R C., Prausmtz, J M and Sherwood, T K., 1987 The Properties of Gases and Liquids, 4th edn McGraw Hill,

New York Shlng, K.S, 1989 In" Bruno, T J and Ely, J F (Eds) , Supercritical Fluid Technology CRC Press, Boston, MA, p.

227. Tsekhanskaya, Yu V, Iomtev, M B and Mushklna, E V, 1964, Russ J. Phys Chem., 38 1173 Walsh, J M., Ikonomou, G D , and Donohue, M D 1987 Fluid Phase Equilibria, 33 295 Wldom, B, 1963 J Chem Phys, 39 2808 Zlchl, D.A and Rossky, P J , 1986 J Chem Phys, 84. 1712