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## Brazilian Journal of Chemical Engineering

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*Print version* ISSN 0104-6632*On-line version* ISSN 1678-4383

### Braz. J. Chem. Eng. vol.16 n.3 São Paulo Sept. 1999

#### https://doi.org/10.1590/S0104-66321999000300001

**Predicting diffusivities in dense fluid mixtures**

**C. DARIVA, L.A.F. COELHO and J. VLADIMIR OLIVEIRA **Programa de Engenharia Química, PEQ/COPPE/UFRJ, 21945-970, Rio de Janeiro - RJ, Brazil,

E-mail: Vladimir@PEQ/COPPE/UFRJ.BR

(Received: September 8, 1998; Accepted: December 8, 1998)

**Abstract - **In this work the Enskog solution of the Boltzmann equation, as corrected by Speedy, together with the Weeks-Chandler-Andersen (WCA) perturbation theory of liquids is employed in correlating and predicting self-diffusivities of dense fluids. Afterwards this theory is used to estimate mutual diffusion coefficients of solutes at infinite dilution in sub and supercritical solvents. We have also investigated the behavior of Fick diffusion coefficients in the proximity of a binary vapor-liquid critical point since this subject is of great interest for extraction purposes. The approach presented here, which makes use of a density and temperature dependent hard-sphere diameter, is shown to be excellent for predicting diffusivities in dense pure fluids and fluid mixtures. The calculations involved highly nonideal mixtures as well as systems with high molecular asymmetry. The predicted diffusivities are in good agreement with the experimental data for the pure and binary systems. The methodology proposed here makes only use of pure component information and density of mixtures. The simple algebraic relations are proposed without any binary adjustable parameters and can be readily used for estimating diffusivities in multicomponent mixtures.

*Keywords:* Statistical mechanics, diffusion coefficient, mixture.

INTRODUCTION

The design of separation processes requires the accurate determination of diffusion coefficients and besides conventional techniques many applications can be envisaged such as tertiary-oil recovery, dense fluid cleaning of solid surfaces and the extraction of natural products from vegetables matrices. As a consequence, in recent years there has been a growing interest in correlating and predicting diffusivities in dense fluids and numerous experimental data concerning this subject have been reported in the literature. However, there is not a rigorous and successful theory for diffusion of dense fluids due to their complex microscopic structures.

In a general sense, the existing methods to correlate or predict diffusivities are based on molecular dynamics results and on the hydrodynamic and statistical mechanical theories of fluids. The hydrodynamic theory, as expressed by the Stokes-Einstein relation, does not correlate the diffusivities well when the solute particles are no longer much larger in size compared to the solvent molecules (Evans et al., 1981). Nonetheless, some empirical correlations based on the Stokes-Einstein equation have been proposed in the literature (Hayduk and Minhas, 1982; Liu and Ruckenstein, 1997; Umesi and Danner, 1981). The free-volume theory has also provided a number of empirical correlations (Guo and Kee, 1991; He et al., 1998 ).

In terms of practical applications and theoretical investigations, the Enskog’s kinetic theory of hard-sphere fluids has been the most fruitful. In this theory, the transport properties are calculated by very simple equations relating the particle mass, temperature, fluid density, particle size and the pair correlation function at the contact point. Based on the smooth or rough-hard-sphere theory, several empirical methods have been developed (Erkey et al., 1990; Sun et al., 1994; Eaton and Akgerman, 1997).

Most of the above mentioned correlations are devoted to pure fluids or infinite dilution diffusion coefficients and hence just a few methods for predicting the concentration dependence of mutual diffusion coefficients have been proposed (He, 1995; Wang et al., 1996). Bandrowski and Kubaczka (1982) pointed out that it is practically impossible to determine exactly from the theory the dependence of mutual diffusion coefficients on concentration. Despite the drawbacks mentioned by Taylor and Krishna (1993), empirical relations such as Vignes or Darken and their modifications (see, for example, Reid et al., 1987 and Pertler et al., 1996), which are based, respectively, on infinite dilution and intradiffusion diffusivities, have been used for this purpose.

In this work our main objective is to show that the Enskog based equations work well for estimating diffusivities of dense pure fluids and fluid mixtures when the hard-sphere diameter is obtained from the Weeks et al. (WCA) (1971) perturbation theory of liquids. Experimental self-diffusion data were used to fit the hard-sphere diameters and these diameters were employed in predicting diffusivities in binary mixtures without any cross parameter. Further, calculations along the gas-liquid boundary were also accomplished in order to investigate the behavior of Fick diffusion coefficients in the proximity of a binary vapor-liquid critical point. We then show that with a very simple equation it is possible to predict diffusivities in pure as well as in multicomponent mixtures without introducing empirical relations for the concentration dependence of mutual diffusion coefficients.

THEORY

Enskog’s solution of the Boltzmann equation for a dense pure hard-sphere fluid is given by (Chapman and Cowling, 1970; Reed and Gubbins, 1973):

(1)

where r * (º r s ^{3}) is the reduced density, g(s ) is the pair correlation function at the contact point for hard-spheres (Carnahan and Starling, 1969) and,

(2)

where, m is the particle mass, T the absolute temperature, r is the number density, k the Boltzmann constant and s the hard-sphere diameter.

The density correction proposed by Speedy (1987) has been used to more accurately account for the density dependence of D, namely,

(3)

where,

(4)

where D_{SHS} is the so-called diffusion coefficient for the smooth hard-sphere (SHS) fluid.

To account for the coupling between translational and rotational motions, Chandler (1975) has proposed the following equation for the self-diffusion coefficient:

(5)

where D_{RHS} is the so-called diffusion coefficient for a rough hard-sphere (RHS) fluid and A is a coupling factor bounded between zero and one (0<A<1), and rigorously independent of density. It is unity for the perfect smooth hard-sphere fluid and should also be independent of temperature if only binary collisions occur. However, the literature data reveal that the coupling parameter is a strong function of temperature, affecting sensibly the diffusion coefficient and, therefore, Eq. (5) may not be a good basis for predictive methods (Rocha et al., 1997). Furthermore, the rough hard-sphere theory is not capable of predicting diffusivities in mixtures using only pure component parameters. For instance, for binary mixtures, another coupling parameter, A_{12 }, which is estimated from the binary data is needed and it is an undesirable feature for applications in engineering design.

Very recently Rocha et al. (1997), based on the idea of Schrodt and Davis (1974), showed that the WCA (Weeks et al., 1971; Andersen et al., 1971) perturbation theory of liquids coupled with the Enskog solution of the Boltzmann equation is a successful theory for estimating diffusivities in liquids and liquid mixtures. Basically, the method presented by Rocha et al. consists in making the coupling parameter, A, unity through the use of an effective hard-sphere diameter, d, that comes from the first-order WCA perturbation theory of liquids; i.e., by solving the following equation:

(6)

where Y^{hs}(r,r,d) is the cavity distribution function for a hard-sphere molecule of diameter d at density r , the hard-sphere potential, , is:

(7a)

(7b)

and,

(8)

(9)

(10)

where f ^{ref}(r) and f ^{P}(r) are the reference (repulsive) and perturbation (attractive) potentials, respectively; r_{m} is the distance where the actual potential, f (r), is a minimum and it is given by the Lennard-Jones (6,12) pair-potential model:

(11)

Instead of solving Eq. (6) at a given temperature and density, we have adopted the recent and accurate proposal of Souza and Ben-Amotz (1993),

(12)

where T* (º T/(e /k)) is the reduced temperature, e is the Lennard-Jones energy parameter and the constants a_{1}=1.5001, a_{2}=-0.03367, a_{3}=0.0003935, a_{4}=-0.09835, a_{5}=0.04937, a_{6}=-0.1415 (Souza and Ben-Amotz, 1993).

By Eq. (11) one can see that the hard-sphere diameter, d, is calculated as a function of temperature and the number density for given values of s and e . In this way, all the s in the diffusion coefficient equations are replaced by the WCA hard-sphere diameter, d.

**RESULTS AND DISCUSSION**

In this work we follow the approach of Rocha et al. (1997) to estimate diffusion coefficients of dense pure fluids and fluid mixtures using the smooth-hard-sphere theory (A is made unity in the rough hard-sphere model) with the effective hard-sphere diameters calculated from the WCA perturbation theory, Eq. (11). Of course, the coupling between rotational angular and translational momenta plays an important role in diffusion but it is shown that the smooth hard-sphere model is sufficient to estimate diffusivities in dense pure fluids and fluid mixtures. As previous mentioned, the coupling parameter is a strong function of temperature, but in our case the variation with temperature is given by the equilibrium perturbation theory and the resulting value of A is, in most cases, nearly unity.

In Table 1 we present the results and conditions of fitting of the self-diffusion coefficients using the density correction of Speedy. When densities for pure fluids were not given along with the diffusion coefficients, they were estimated using the modified Racket equation (Reid et al., 1987). The Lennard-Jones parameters of distance, shown in Table 1, were estimated using the Maximum Likelihood Method (Anderson et al., 1978) weighing experimental self-diffusion data according to its precision. We have found that the energy parameters are insensitive to the experimental self-diffusivity data, therefore, for simplicity, they were fixed in this work equal to those presented in the literature (Wilhelm and Battino, 1971; Reid et al., 1987; Hirschfelder et al., 1954). Thus, the model makes use of only one adjustable parameter, s , which is obtained from pure component information. Note that the components figuring in Table 1 are the most commonly used solvents for supercritical fluid processing purposes and once the size parameters have been obtained safe predictions can be accomplished.

Table 1: Average absolute deviations of the fitted self-diffusion coefficient and Lennard-Jones size and energy parameters.

Substance | Range of T(K) | NP^{+} | AAD%^{*} | s (10^{-10}m)^{++} | e /k (K)^{**} | Data Source^{***} |

Carbon dioxide | 298 | 7 | 3.7 | 3.6283 | 195.2 | a |

Ethane | 294 | 5 | 2.5 | 4.1047 | 215.7 | b |

Propane | 294 | 5 | 4.5 | 4.6398 | 237.1 | b |

n-hexane | 232-333 | 10 | 10.4 | 5.5617 | 517.0 | c |

Cyclohexane | 313.0 | 7 | 2.5 | 5.4050 | 573.0 | d |

Ethylene | 298 | 19 | 9.9 | 3.9671 | 224.0 | e |

Benzene | 298 | 10 | 1.7 | 5.0081 | 531.0 | f |

Toluene | 298 | 13 | 2.6 | 5.2829 | 575.0 | g |

Sulfur hexafluoride | 296 | 6 | 4.8 | 4.7808 | 201.0 | h |

Carbon tetrachloride | 298 | 6 | 5.7 | 5.1761 | 536.0 | i |

Acetone | 237-333 | 4 | 5.6 | 4.6132 | 362.0 | j |

Overall deviation (%) | 5.5 |

* ;

^{+}number of data points

^{++}size parameters estimated from self-diffusion data in this work

^{**}Literature values (Reid et al., 1987; Wilhelm and Battino, 1971; Hirschfelder et al., 1954).

^{***}^{a}Etesse et al. (1992);^{b}Greiner-Schmid et al. (1991);^{c}Douglas and McCall (1958);^{d}Jonas et al. (1980);^{e}Arends et al. (1981);^{f}McColl et al. (1972);^{g}Harris et al. (1993);^{h}DeZwaan and Jonas (1975);^{i}Chandler (1975);^{j}Kruger and Weiss (1970).

As an example, Figure 1 depicts experimental and predicted values in a wide range of density for the self-diffusion coefficient of carbon dioxide, outside the temperature range where the size parameter was fitted. It can be seen from this figure a reasonable agreement between experiment and theory showing that the hard-sphere diameter is capable of accounting very well for changes in temperature and density. Note also that there is some divergence in the experimental data obtained by different authors at similar conditions.

Figure 1:Experimental and predicted self-diffusion coefficient for carbon dioxide at 323.15K (Overall deviation (%)=12.4)

**Binary Systems**

For a binary mixture, the mutual diffusion coefficient is given by (Chapman and Cowling, 1970):

(13)

where r_{m} is the number density of the mixture, d_{12} the cross hard-sphere diameter, and m_{12} (º m_{1}m_{2}/(m_{1}+m_{2})) the reduced mass of solute 1 and solvent 2. The parameter d_{12} is calculated as the arithmetic mean of the pure component hard-sphere diameters, d_{1} and d_{2}, following the approach of Lee and Levesque (1973), at the temperature of the mixture and density of the pure substances:

(14)

Considering the importance of some natural products in the field of food industries, soil decontamination as well as for pharmaceutical applications and trying to test the present approach for a greater number of systems, we have predicted mutual diffusivities of a variety of compounds of prominent interest. For these substances, not appearing in Table 1, the Lennard-Jones energy parameters were estimated from the simple relation:

(15)

and the Lennard-Jones size parameters from:

(16)

and the critical properties were estimated from the Joback method (Reid et al., 1987).

For the density correction of Speedy, we have assumed the van der Waals mixing rule (Grundke and Henderson, 1972) so that the reduced density of the mixture, r^{*}_{m}, is given by:

(17)

In Table 2 and from Figures 2 to 6 the predicted mutual diffusion coefficients at infinite dilution are compared with the experimental data. Though these systems incorporate both great differences in size (asymmetries) and in intermolecular forces (chemical nature), making them difficult to predict, the agreement between experiment and theory can be considered quite good, with an overall average absolute error of 16.9%. Notice that no binary adjustable parameters were employed in the calculations. It should also be noted that good predictions, when compared to those obtained from the most recent methods available in the literature (Eaton et al., 1997; Liu and Ruckenstein, 1997), were achieved for those systems whose solute-size parameters were estimated from Eq. (16) and this may be a useful characteristic of the present model.

Table 2: Average absolute deviations of the predicted values for the mutual diffusion coefficient of species 1 at infinite dilution in component 2 (D_{12}).

Solvent | Solute | Range of T(K) | NP | AAD% | Data Source^{*} |

Carbon dioxide | Acetone | 303-333 | 7 | 5.6 | a |

Benzene | 308-328 313 303-333 313-333 | 17 02 11 15 | 17.7 9.8 16.6 6.2 | b c a d | |

Toluene | 313-333 308-328 | 15 18 | 13.5 26.8 | d e | |

Ethylene | 298-348 | 23 | 16.3 | f | |

Naphthalene | 313 303-333 | 02 11 | 32.2 20.9 | c a | |

Phenanthrene | 303-333 308-328 | 07 14 | 23.0 22.1 | a g | |

1,3,5 trimethylbenzene | 303-333 | 07 | 20.7 | a | |

Ethylbenzene | 313-333 | 15 | 18.6 | d | |

Hexachlorobenzene | 313-333 | 14 | 20.7 | g | |

i-propylbenzene | 313-333 | 15 | 21.3 | d | |

n-propylbenzene | 313-333 | 15 | 21.8 | d | |

benzoic acid | 298-318 | 07 | 17.1 | h | |

behenic acid ethyl ester | 308-318 | 17 | 23.8 | i | |

butyric acid ethyl ester | 308-318 | 16 | 15.6 | i | |

capric acid ethyl ester | 308-318 | 16 | 19.4 | i | |

caprylic acid ethyl ester | 308-318 | 16 | 19.4 | i | |

cis-11-eicosenoic acid methyl ester | 313 | 02 | 19.3 | c | |

docosahexenoic acid ethyl ester | 308-318 | 17 | 21.6 | i |

Table 2 (cont.)

Solvent | Solute | Range of T(K) | NP | AAD% | Data Source |

docosahexenoic acid methyl ester | 308-318 | 17 | 20.9 | i | |

erucic acid methyl ester | 313 | 02 | 21.2 | c | |

linoleic acid methyl ester | 308-328 | 19 | 20.8 | j | |

myristic acid ethyl ester | 308-318 | 16 | 20.7 | i | |

myristoleic acid methyl ester | 313 | 02 | 19.8 | c | |

stearic acid ethyl ester | 308-318 | 17 | 19.2 | i | |

caffeine | 308-328 | 21 | 12.3 | e | |

chrysene | 303-333 | 04 | 24.2 | a | |

d1-limonene | 313 | 02 | 23.4 | c | |

indole | 313 | 02 | 6.5 | c | |

oleic acid | 308 | 02 | 15.7 | h | |

pyrene | 303-333 | 03 | 24.6 | a | |

vitamin A acetate | 313 | 02 | 10.1 | c | |

vitamin E | 313 | 02 | 22.3 | c | |

ethyl acetate | 308-328 | 15 | 26.5 | e | |

phenol | 308-328 | 21 | 13.3 | e | |

Ethylene | carbon dioxide | 298-348 | 27 | 9.2 | f |

Ethane | 1-octene | 296-322 | 06 | 5.8 | k |

1-tetradecene | 293-322 | 09 | 12.6 | k | |

Propane | 1-octene | 296-337 | 08 | 7.7 | k |

1-tetradecene | 292-337 | 08 | 14.8 | k | |

Sulfur hexafluoride | 1,3,5 trimethylbenzene | 328 | 11 | 27.1 | l |

1,4 dimethylbenzene | 328 | 11 | 21.9 | l | |

benzene | 328 | 09 | 28.6 | l | |

toluene | 328 | 11 | 30.6 | l | |

carbon tetrachloride | 328 | 07 | 31.1 | l | |

Cyclohexane | benzene | 298-523 | 08 | 4.8 | m |

toluene | 298-523 | 08 | 3.9 | m | |

naphthalene | 298-523 | 08 | 4.5 | m | |

phenanthrene | 298-523 | 08 | 4.6 | m | |

p-xylene | 298-523 | 08 | 5.7 | m | |

n-hexane | benzene | 333-543 | 15 | 12.5 | n |

toluene | 333-543 | 15 | 11.2 | n | |

naphthalene | 333-543 | 15 | 9.1 | n | |

phenanthrene | 333-543 | 15 | 8.3 | n | |

p-xylene | 333-543 | 15 | 9.6 | n | |

Overall deviation (%) |
| 16.9 |

^{ * aSassiat et al. (1987); bChen (1983); cFunazukuri et al. (1992); dSuarez et al. (1993); eLai and Tan (1995); fTakahashi and Hongo (1982); gAkgerman et al. (1996); hCatchpole and King (1994); iLiong et al. (1992); jFunazukuri et al. (1991); kNoel et al. (1994); lKopner et al. (1987); mSun and Chen (1985a); nSun and Chen (1985b) }

Figure 2:Experimental and predicted values for the mutual diffusion coefficient of benzene (1) at infinite dilution in carbon dioxide (2) – see Table 2.

Figure 3:Comparison of experimental and predicted diffusion coefficients of aromatics at infinite dilution in carbon dioxide – see Table 2.

Figure 4:Comparison of experimental and predicted diffusion coefficients of esters at infinite dilution in carbon dioxide – see Table 2.

Figure 5:Comparison of experimental and predicted values of diffusion coefficients of various solutes at infinite dilution in n-hexane and in cyclohexane – see Table 2.

Figure 6:Comparison of experimental and predicted diffusion coefficients of various solutes at infinite dilution in sulfur hexafluoride – see Table 2.

These results show that very good predictions can be obtained even when the temperature of the system is out of the range where the size parameters were fitted. For instance, diffusivities of solutes in n-hexane were predicted in a wide range of temperature, from sub to the supercritical solvent region. Thus, as found by Rocha et al. (1997), it is not necessary to attribute any reduction in diffusion to inelastic effects as proposed in the rough-hard-sphere theory when the hard-sphere diameter is adequately estimated. Since the WCA theory overestimates the hard-sphere diameter (Kang et al., 1985; Saumon et al., 1989) and based on the Enskog original solution of the Boltzmann equation for hard-sphere fluids, Eqs. (1) and (2), it would imply in an appropriate treatment for interpreting diffusion, as pointed out by Speedy et al. (1989).

As in the case of self-diffusivities of carbon dioxide, one can observe from Figure 2 some divergence in the experimental data leading to some of the scatter in the predictions shown in Figures 3 to 6. It is also worth noticing from Figures 1 to 6 that as the density increases the agreement between experimental and calculated values for both self- and mutual diffusivities is much improved. This fact might be explained in terms of the WCA perturbation theory of liquids which works very well at higher densities where the structure is primarily determined by repulsive forces.

Concerning the infinite dilution diffusivities, some recent developments should be commented such as those presented by Eaton et al. (1997) and Liu and Ruckenstein (1997). Both methods contain two adjustable parameters obtained from experimental mutual diffusion coefficient of solutes at infinite dilution in some supercritical solvents and they cannot be applied to estimating the concentration dependence of mutual diffusivities.

Considering that the present approach is not limited to infinite dilution case and, in fact, has been successfully used to predict mutual diffusion coefficients in liquid mixtures (Rocha et al., 1997), the behavior of Fick diffusivities was investigated in the vicinity of a binary vapor-liquid critical point. Firstly, the vapor-liquid equilibrium data for carbon dioxide-heptane from Kalra et al. (1978) were employed to calculate mutual diffusivities in the liquid phase along the gas-liquid coexistence curve. To do so, Eqs. (12), (13) and (17) were used together with the reported temperature, molar fraction and density (an indirect measurement).

In order to compare the predicted and experimental mutual diffusion values it is important to mention that the phase equilibrium data from Kalra et al. are in the temperature range of 311-477K while the diffusion data from Saad and Gulari (1984) for this system are in the range of 283-323K. We have not attempted to use an equation of state to fit the phase equilibrium data at 311K and perform extrapolations since the model is very sensitive to the mixture density values.

Consequently, we present in Figure 7 the experimental mutual diffusion of Saad and Gulari from 283 to 313K and our predictions at 311K. Though at higher pressures large deviations are observed, the predicted results can be considered encouraging since a qualitative analysis is possible and, to the best of our knowledge, there is no other methodology available in the literature to perform such predictions. It should be stressed again that no binary interaction parameters or empirical mixing rules were employed in the calculations.

Figure 7:Experimental (Saad and Gulari, 1984) and predicted mutual diffusion coefficient for the system carbon dioxide-heptane in the liquid phase along the gas-liquid boundary. Vapor-liquid equilibrium data from Kalra et al. (1978).

According to Saad and Gulari experimental data, two interesting phenomena can be observed: crossover points and the appearance of minima. As the crossover dictates in most cases a compromise between two process variables, namely temperature and pressure, thus affecting the yield and selectivity of the process, an investigation whether the present model is able to predict the occurrence of crossover points was carried out. For this purpose, vapor-liquid equilibrium data for carbon dioxide-acetone from Adrian and Maurer (1997) were used to generate the mutual diffusion coefficients in the liquid phase along the gas-liquid boundary. It can be noticed from Figure 8 a crossover point showing the competition between kinetic (temperature) and structural effects, as given by the density of the mixture (in molecular units). The second effect, the occurrence of minimum in the diffusion curves, is now under investigation.

Figure 8:Predicted mutual diffusion coefficient for the system carbon dioxide-acetone in the liquid phase along the gas-liquid boundary. Vapor-liquid equilibrium data from Adrian and Maurer (1997).

**CONCLUSIONS**

We have predicted diffusion coefficients in dense pure fluids and fluid mixtures using the Enskog kinetic and WCA theories. Mutual diffusion coefficients at infinite dilution and as a function of concentration were also predicted making use of the smooth-hard-sphere theory with a temperature and density dependent hard-sphere diameter, obtained from the WCA theory of liquids. Mutual diffusion coefficients were predicted using only pure component information and density of mixtures. The proposed method with the Speedy density correction correlates the pure component diffusivities well and predicts the binary diffusion coefficients in good agreement with experimental values. The method applied here needs, in fact, only one adjustable parameter that is obtained from self-diffusion data of pure substances. We have achieved good results thanks to the well-established and meaningful effective hard-sphere diameter.

**NOMENCLATURE**

A coupling factor of the rough hard-sphere theory

a_{i} constants of the Souza and Ben-Amotz correlation for the effective hard-sphere diameter

d effective hard-sphere diameter, m

D self-diffusion coefficient for dense hard-sphere fluid, m^{2}/s

D_{0} defined by Eq. (2), m^{2}/s

D_{12} mutual diffusion coefficient of 1 in binary mixture of 1 and 2, m^{2}/s

D_{RHS} self-diffusion coefficient for the rough hard-sphere theory, m^{2}/s

D_{SHS} self-diffusion coefficient for the smooth hard-sphere theory, m^{2}/s

g(s) pair correlation function at the contact point for hard-spheres

k Boltzmann constant, J/(molec.K)

m particle mass, Kg

N number of components

NP number of data points

P correlation of reduced density

r distance center-to-center of two particles

r_{m} distance where the pair potential function is a minimum, m

T temperature, K

T* reduced temperature

v volume, Kgmol/m^{3}

x_{i} mole fraction of component i

Y^{hs} cavity distribution function for hard-sphere fluids

**Greek Letters**

b º (kT)^{-1} , J^{-1}

e energy parameter of the Lennard-Jones model, J

f actual pair potential function, J

hard-sphere potential, J

f^{p} perturbation pair potential function, J

f^{ref} reference pair potential function, J

r number density, number of molecules/m^{3}

r^{*} reduced density

number density of the mixture, number of molecules/m^{3}

reduced density of the mixture, Eq. (17)

s hard-sphere diameter, m

**Subscripts**

RHS rough hard-sphere

SHS smooth hard-sphere

c critical value

Superscripts

SP Speedy

**ACKNOWLEDGEMENT**

This work was supported by PRONEX, grant no. 124/96 (Minitério de Ciência e Tecnologia - Brazil). The authors would like to thank CNPq for the financial support.

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