2029
Znd. Eng. Chem. Res. 1995,34, 2029-2037
A Study on the Models for Infinite Dilution Partial Molar Volumes of Solutes in Supercritical Fluids Hongqin Liut and Eugbnia A. Macedo* Laboratory of Separation and Reaction Engineering, Faculdade de Engenharia, Rua dos Bragas, 4099 Porto Codex, Portugal
Infinite dilution partial molar volumes (PMV) of solutes in supercritical fluids (SCF) are very important thermodynamic properties of SCF mixtures. But until recently very few works have been dedicated to the measurements and the prediction or correlation of these properties. In this paper we present a systematic study on various models for the representation of PMV. First, we tried to predict the PMV using model parameters regressed from solubility data. Four models have been tested, and it is found t h a t all these models can only be used qualitatively, not quantitatively. After that, we used 11 models for the correlations with temperaturedependent parameters. It is found that the model based on Wang and Tavlarides's (WT) dilute solution expression and a three-parameter empirical equation derived in this work give the best results. Some simple correlations also give acceptable results. It is surprising t h a t some theoretical models, e.g., Wheeler's decorated lattice gas model and the models based on the fluctuation theory do not work as well as expected. Moreover, we have used six models for correlations with temperature-independent parameters. We conclude t h a t the model proposed in this work with three parameters based on the WT dilute solution expression gives the best results. The modified Peng-Robinson equation of state (EoS) and a n empirical correlation proposed by O'Connell give acceptable results. The main conclusion of this work is t h a t it is possible to correlate the PMV by using a simple model, although no available model is recommended for quantitative predictions.
Introduction Supercritical fluid technology has received much interest in the last decade because of its unique advantages and potential applications in industry (see, e.g., Brennecke and Eckert, 1989; Johnston and Penninger, 1989). As was pointed out by some authors, the understanding of the phase equilibrium behavior of supercritical fluid (SCF) mixtures is one of the most important problems for the researchers engaged in this field, and therefore, much attention has been given to this subject (Brennecke and Eckert, 1989; Johnston et al., 1989). Up to now, most of the works dealing with the phase behavior of SCF are concerned with the experimental measurements or modeling of the solubilities of a solute in SCF which are, of course, of utmost importance in process design. Among the various properties of SCF mixtures, the infinite dilution partial molar volumes of a solute in SCF (PMV), are also very important in both theoretical and practical aspects. The values of the PMV give the pressure dependence of the solubility or fugacity coefficient 4; of a solute in SCF (see, e.g., Kurnik and Reid, 1981; Prausnitz et al., 1986). Besides this, the PMV is also of fundamental importance in supercritical fluid chromatography (Yonker and Smith, 1988). From the theoretical point of view, the ability to describe the PMV data correctly represents a very stringent test for an equation of state (EoS) or a model describing the solute solubility, since this constraint requires that the differential form must also be correct (Eckert et al., 1983). Moreover, the PMV is related directly with some important molecular properties: (i) the Kirkwood-Buff
E,
* Author t o whom all correspondence should be addressed. ' On leave from the Department of Chemical Engineering, Beijing University of Chemical Technology, 100029, Beijing, China.
integral GL1 (Kirkwood and Buff, 1951; McGuigan and Monson, 1990):
where g;l(r) is the infinite dilution limit of the radial distribution function; (ii) the direct correlation function integral (DCFI) ql,which is a function of Gi1 (O'Connell, 1990; Wooley and O'Connell, 1991)according to
k $ T c l = GYl
(2)
where kr is the isothermal compressibility of the pure solvent (SCF); (iii) the cluster size, which in fact equals the dimensionless total correlation function integral (TCFI)(Debenedetti, 1987;Wooley and O'Connell, 1991); and (iv) the types of molecular interactions: attractive, weakly attractive, and repulsive (Debenedetti and Mohamed, 1989). Unfortunately, up t o now, few works have been dedicated to both measurements and modeling of the PMV in SCF. The earlier works, concerning the measurements of partial molar volumes of mixtures in the whole range of concentrations at supercritical conditions, are those of Khazanova and Sominskaya (19681, Ehrlich and Fariss (19691, Wu and Ehrlich (19731, and Rozen (1976). Since the 198Os, some authors have carried out measurements of infinite dilution PMV of solutes in SCF (van Wasen and Schneider, 1980; Paulaitis et al., 1981; Eckert et al., 1983; Eckert et al., 1986; Foster et al., 1989; Liong et al., 1991; Shim and Johnston, 1991). More recently, and on the other hand, some authors have used Monte Carlo simulation and integral equation methods to calculate the PMV for Lennard-Jones mixtures (Shing and Chung, 1988; McGuigan and Monson, 1990). Since in most practical cases of SCF mixtures
0888-5885/95/2634-2029$09.00/00 1995 American Chemical Society
2030 Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995
the systems treated are dilute solutions, the infinite dilute PMV plays an important role in this field. The models available for the prediction or correlation of the PMV can be divided into five main groups: (1) empirical correlations (Kim and Johnston, 1987; Kumar and Johnston, 1988); (2) EoS methods (e.g., PengRobinson EoS, Eckert et al., 1983); (3) decorated lattice gas models (Wheeler, 1972, 1977; Gilbert and Eckert, 1986); (4)models based on the fluctuation theory of Kirkwood and Buff (Kirkwood and Buff, 1951; Pfund et al., 1988; O’Connell, 1990; Wooley and O’Connell, 1991); and (5) models based on semiempirical dilute solution theories (e.g., Harvey, 1990). The majority of these models have been proposed as complementary results of solubility models, and no systematic studies have been devoted to comparingthese models with each other in their prediction or correlation capabilities. In this paper, we present a systematic study on the models available in the literature or proposed here. Firstly, we give the predicted PMV values calculated for four systems with four representative models. The interaction parameters were regressed from solubility data for the same systems by using the same models. Secondly, correlation results of PMV data for 11 systems using 11models with temperature-dependent parameters are shown and discussed. Finally, six models with temperature-independent parameters were studied in the correlation of PMV data. A comparison between different approaches is given. A brief discussion of the Kirkwood-Buff integrals, the DCFI, and the cluster size, all obtained from the PMV data, is also included. The results obtained in this work can be considered useful in the evaluation of both the PMV and solubility models.
+
+
= Al(T) A2(T)BT12, A3(T)e1RTkT (6)
where Al(T),A2(T),andAB(T) are empirical parameters. 2. Peng-Robinson EoS Method. In recent years, the equation of state method has been widely used for modeling the phase equilibria of dense fluids, especially in SCF mixture systems. Cubic EoS are still mostly being used (see, e.g., Anderko, 1990; Johnston et al., 1989). In a standard, pressure-explicit equation of state, the partial molar volume can be calculated from (7) where VI (= 1/@1)is the molar volume of the solvent. The cubic EoS proposed by Peng and Robinson (1976) is successful in describing the phase equilibrium of SCF mixtures and is widely used in the literature (Anderko, 1990; Johnston et al., 1989). It was adopted in this work. The following expressions can be derived:
2b2alal(Vl- b,) [Vl(V,+ b,) + b1Wl - b,)12 (8) where the mixing rules are as follows:
(10)
PMV Models Used in This Work Correlation Models with Temperature-Dependent Parameters. 1. Empirical Equations. As is well-known, the infinite dilution PMV diverges to either for repulsive and weakly attractive interaction or t o -= for attractive interaction (Debenedetti and Mohamed, 1989) near the solvent critical point. This divergence arises from the long range correlation of the solvent which results in its infinite compressibility. Thus, it is reasonable to relate with k T . A twoparameter empirical equation was proposed by Kim and Johnston (1987):
with kG = 0 when i = j . The temperature-dependence of a(T) is obtained from the literature (Peng and Robinson, 1976). For a binary system, the only adjustable parameter is k12 which is taken as temperaturedependent. 3. The Decorated Lattice Gas Model. The decorated lattice gas model presented by Wheeler (1972, 1977) is successful in representing phase equilibria for SCF mixtures. According to the equation of Wheeler (1972) the PMV is obtained from
where a(T) and b(T) are temperature-dependent parameters. In another work, Kumar and Johnston (1988) proposed two simpler equations for deriving solubility models, which correspond respectively to the so-called log-log and log-linear correlations of the solubility y2 with the solvent density el:
where x is a function of the energy parameter 4, x = exp(-@/RT).To make the correlation more flexible, we propose an expanding expression for Nodqc. By using the conditions at the critical point of the solvent, Ndqc 0.17-0.19 and [a(No~/qc)/a~ll~ 0 (Wheeler, 19721, we get the following equations:
where B(T)and B1(T) are the model parameters. Both eqs 4 and 5 only have one adjustable parameter. In this work we propose a new three-parameter model:
The three parameters of this model are 4, C, and D, which are all taken as temperature-dependent.
+=
-
-
Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995 2031
4. Models Based on the Kirkwood-Buff Fluctuation Theory. The fluctuation theory proposed by Kirkwood and Buff (1951) has recently received much attention for the phase behavior description of fluid mixtures (Cochran et al., 1990). According to this theory, the PMV can be strictly correlated with the fluctuation integral GYl. As mentioned in the Introduction, the fluctuation integral Gil is related with the direct correlation function integral (DCFI) ql.O’Connell (1994) related the DCFI with the reduced Krichevskii parameter AYl, defined as AYl = 1 - ql, and proposed the following equations for AYl:
+
ATl = 1 ael
+ b[exp(cq,) - 11
(15)
where B21(T)and C211(T)are respectively the second and third virial coefficients, and a , b, and c in eq 15 are empirical parameters which are here taken as temperature-dependent. Equations 14 and 15 have been used with success in the correlation of infinite dilution partial molar volumes of gases and salts in water (OConnell, 1994). 5. Models Based on the Dilute Solution Considerations. As is well-known, the EoS method has a shortcoming, i.e., it needs critical property data for all components in the mixtures. However, for some heavy compounds used in the SCF technology, these properties are usually unknown. For this reason, some researchers tried to correlate the solute solubility with the properties of the solvent (SCF)which are generally wellknown. This method is based on the fact that the SCF mixtures are generally dilute solutions. Hence, dilute solution models have been developed. Harvey (1990) proposed a simple model based on the theory of dilute solutions near the critical point of the solvent:
Very recently, Wang and Tavlarides (1994) proposed a dilute solution model for solubility of solutes in compressed gases. They proposed a simple expression for the enhancement factor, from which we can easily derive another two-parameter model for
E:
This model has two adjustable parameters: b(T) and C(T).
In total, 11 models with temperature-dependent parameters have been presented. Among them, five models, eqs 4, 5 , 7-8, 16, and 17, have one parameter; three models, eqs 3, 14, and 18, have two parameters; and three models, eqs 6, 11, and 15, have three parameters. Equations 6, 17, and 18 represent the new models proposed in this work. Correlation Models with Temperature-Independent Parameters, In order to develop a model with temperature-independent parameters, it is fundamental to know the dependence of the parameters with temperature. At present, for an empirical equation, this is difficult since there is not enough information about temperature dependence. In this work, we only test the models which have some theoretical background and a known temperature dependence. We have studied six models with temperature-independent parameters. 1. Model from the Solubility Correlation of Ziger and Eckert (1983). On the basis of the regular solution theory, Ziger and Eckert (1983) proposed a semiempirical equation for the solubility of solids in SCF, which gives an expression for the enhancement factor. The same considerations used for the derivation of eq 18 allow us t o obtain
c=c+-RT P
(16) where the parameter B(T) has been considered as temperature-dependent. This is an extension of the original model, which will allow us to obtain higher accuracy in the correlation of PMV. A comparison with eq 5 shows that the solvent molar volume is added in this model, which usually has a less important weight in the correlations where the absolute values of the PMV are large. Another simple but widely used solubility model for dilute SCF mixtures is the so-called log-log correlation model (Chrastil, 1982; Gurdial et al., 19891, from which it is possible to write =
E
- b(T)RTkT
(17)
This model for has only one parameter b(T),but the solute molar volume data are needed. By comparing eq 17 with eq 3, we can see that eq 17 is a special is taken as a variable. case of eq 3 when Obviously, when
