Ind. Eng. Chem. Res. 1994,33, 1035-1039 1035

Determination of the Domain of the Function F ( P , @ ) = 0 for Wilson, NRTL, LEMF, and UNIQUAC Equations

Maria R. Gennero de Chialvo and Abel C. Chialvo'nt PRELINE, Facultad de Ingenierfa Qulmica, Universidad Nacional del Litoral, Santiago del Estero 2829, (3000) Santa Fe, Argentina

The domain boundary for the simultaneous correlation of the excess molar Gibbs free energy (GE) and the excess molar enthalpy (IIE) of a binary system has been theoretically determined for the Wilson, NRTL, LEMF, and UNIQUAC equations. On this basis, the conditions in which such equations can describe systems with positive and/or negative values of the excess molar entropy (SE) have been evaluated. The results obtained demonstrate that the knowledge of the domain of the function F ( P , G E ) = 0 avoids the inappropriate use of a given correlation equation beyond the corresponding limits.

Introduction

The well-known semitheoretical expressions for the excess molar Gibbs free energy (GE) based on the so-called local composition concept (Wilson, NRTL, LEMF, UNI- QUAC) can be differentiated through the Gibbs-Helm- holtz equation to yield expressions for the excess molar enthalpy (HE). These expressions for both thermodynamic properties GE and HE derived from those models are basically functions of the mole fraction of the solution components (0 < xi < l), the equation constants (Gij > 0), and the temperature. Therefore, for a binary solution at constant temperature, the infinite combinations of these values (xi, GI,, Gzl) produce infinite pairs of values (GE, HE), which can be calculated through each model, defining in each case the domain of the function F(HE,GE) = 0. Consequently, the evaluation of the boundary of such a domain should be important in order to get a complete knowledge of the limits of application of those expressions.

Previous studies have given some indications about such a boundary. In this sense, it has been determined for the Wilson equation that a t T = 298.16 K and XI = xz = 0.5, when the molar volume relation (VI/ V Z ) is equal to 1, the excess molar enthalpy must be less than 165 cal mol-' (Hanks et al., 1978; Vonka et al., 1975) and, for VdV, = 2(0.5), HE < 184 cal mol-' (Hanks et al.). For the same values of temperature and molar fractions and using the NRTL equation with the parameter a equal to 0.3, Hanks et al. have established that, for the region where both GE and HE are positive, simultaneous representation of GE and HE is possible only if GE < 550 cal mol-' and HE < 259 cal mol-'. For the region where GE is positive and HE is negative, they have established that the NRTL equation is limited to data for which HE > -521 cal mol-'. Meanwhile, they have found that the LEMF equation does not have any limiting value. Related to the case of UNIQUAC equation, Skjold-Jorgensen et al. (1980) and Demirel and Gecegormez (1989) have determined that H E must be approximately less than 250 cal mol-'.

The present work deals with the determination of the domain of the function F(HE,GE) = 0 when HE and GE are described through the Wilson, NRTL, LEMF, and UNI- QUAC expressions for binary systems at 298.16 K, considering the interaction parameters to be independent of temperature.

t E-mail: RNFIQUNLs ARCRIDE.EDU.AR.

Theoretical Analysis The function F(HE,GE) = 0 defines implicitly the

relationship GE vs HE. On the other hand, the parametric form can be written

where p represents a specific parameter of a given expression (a in the NRTL equation, etc.). The depen- dence of HE given in eq 1 is evaluated through the Gibbs- Helmholtz relation

(3)

starting from the corresponding equation of GE, where the interaction parameters (gi,-gii) involved in the con- stants Glz and Gzl are considered to be independent of temperature.

Starting from eqs 1 and 2, the determination of the domain boundary of the function F(HE,GE) = 0 was done through the evaluation of the absolute extremes of GE or HE for different values of the relationship GE/HE and a constant value of the parameter p . The calculation of these conditioned extremes was made through a compu- tational multidimensional search technique. During the calculations, the existence of nonabsolute extremes was observed. Consequently, the domain was verified through the evaluation of the function F(HE,GE) = 0 for a grid of GI, and Gzl values. The analysis of the results obtained is given separately for the different equations used.

Wilson Equation. The explicit forms of eqs 1 and 2 developed through the Wilson's model (1964) are

GE = -RT[x, ln(x,+G,,x,) + x , ln(x,+G,,x,)l (5)

where

Vi Gij = - Vj exp[-(gij-gii)/RTl i = 1,2 j = 1,2 (6)

0888-5885/94/2633-1035$04.50/0 0 1994 American Chemical Society

1036 Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994

1

0 2 5 1 /J

-0.25 1

. ._ - 0 6 - 0 3 00 03

HE/Kcal mol-' Figure 1. Domain of the function F ( P , G E ) = 0 for the Wilson equation: (a) Vl/V2 = 1; (b) Vl/Vz = 2(0.5); (dashed lines) Gij 5 3.

The domain of the function F(HE,GE) = 0 corresponding to the Wilson equation is shown in Figure 1, for the following values of the parameter VI/ V2: (a) 1; (b) 2(0.5). The points mentioned in the literature, HE = 165 cal mol-' for case a and HE = 184 cal mol-' for case b, have also been included in the figure. On the other hand, it can be observed that the function F(HE,GE) = 0 shows a discon- tinuity in hTE = 0, with the following GE values: 411 and 215 cal mol-' for case a and 411 and 296 cal mol-' for case b. Besides, a displacement of the limit of the domain toward the region where the excess molar entropy (SEI is positive when VI # VZ should be noticed.

On the other hand, of the over 1446 binary systems correlated by Ohe (1989), only 1.7% of them have Gij > 3. Therefore, when the condition Gij I 3 is imposed, the domain of the points with negative GE and HE values is restricted in both cases to the dashed lines (Figure 1). In this way, the domain of the function F(HE,GE) = 0 corresponding to the Wilson model has been more real- istically delimited.

NRTL Equation. The explicit form of eq 2 corre- sponding to the NRTL theory developed by Renon and Prausnitz (1968) is

where

G, = exp[-a(gij-gij)/RTl i = 1, 2 j = 1,2 (8)

The parametric form of HE is, for convenience, obtained through the expression

H~ = G~ + T S ~ (9)

being the explicit form of the molar excess entropy:

The domain of the function F(HE,GE) = 0 for the present case is shown in Figure 2, with the parameter a equal to 0.3. It should be noticed that the inverse of this parameter (CY-') is an enlargement factor of the domain. The points given in the literature for the extreme values of the excess

0 5

-0.5

G E = O 550

HE=0.259

-10 1 / I I I - I 2 -0 6 00 0 6

Figure 2. Domain of the function F(@,GE) = 0 for the NRTL equation: (dashed line) G,j 5 2.

molar enthalpy, 259 and -521 cal mol-', and the excess molar Gibbs free energy, 550 cal mol-', are also located in the figure. Nevertheless, it must be mentioned that the minimum value of HE in the region where HE < 0 and GE > 0 corresponds to GE = 0 and is equal to -568 cal mol-'. It is important to note that the points which define the boundary in the upper part of the domain correspond to the conditions XI = x 2 and Glz = G21. Besides, it can be observed that in the present case the function F(HE,GE) = 0 is unable to enter in the region where SE > 0, as it was pointed out by Vonka et al. (1975).

On the other hand, from the revision of the extensive compilation made by Ghmeling et al. (1980), it can be concluded that only less than 1% of the binary systems correlated by the NRTL equation have Gij > 2. Conse- quently, the condition Gij I 2 was applied, and the result obtained is shown as a dashed line in Figure 2. This allowed us to define more adequately the usual domain of the function F(HE,GE) = 0 for the NRTL equation.

LEMF Equation. The LEMF (local effective mole fractions) equation arises from a modification of the NRTL theory due to Marina and Tassios (1973). The corre- sponding explicit form of GE is

HE/Kcal mol-'

where

G, = exp[(gij-gii)/RTJ i = 1,2 j = 1,2 (12)

The explicit form of HE can be obtained from eq 9 and the following expression of SE:

Due to the fact that the LEMF equation results from the assignment of the value -1 to the parameter a of the NRTL equation, the domain of the function F(HE,GE) = 0 is located in this case exclusively in the region where S E > 0. Therefore, the shape of the domain is similar to that of the NRTL expression, but rotated 180°, as can be seen in Figure 3. The extreme points equivalent to those of the NRTL case are HE = -77 cal mol-', HE = 156 cal mol-', and GE = -165 cal mol-'. They are included in the figure,

1037 03

0 2

- 5 01 E - 0

Y u' 0 0 W

-0. I

-0 2 I CIE: - 0.165

._

-0 12 0 00 0.12 0 24 0.36 HE / Kcal mol-'

Figure 3. Domain of the function F(HE,GE) = 0 for the LEMF equation: (dashed line) Gij I 2.

in spite of not being reported in the literature. In a form similar to that of the NRTL equation, the points which define the boundary in the lower part of the domain correspond to the conditions XI = x z and GIZ = GZI. In spite of the fact that a systematic study of binary systems correlated by the LEMF equation is not available, it should be inferred from the NRTL results that the domain where Gij I 2 probably represents the application domain of the function F(HE,GE) = 0 for the LEMF expression. There- fore, it is included in Figure 3 as a dashed line. UNIQUAC Equation. Equations 1 and 2 derived on

the basis of the universal quasi chemical (UNIQUAC) theory developed by Abrams and Prausnitz (1975) are

where

rixi r1x1+ rzx2

cpi = i = l , 2

Gij = exp[-(gij-gii)/RTl i = 1,2 j = 1,2 (18)

The parameters qi and ri are characteristic of each compound. In order to determine a relationship between such parameters, the dependence of qi on ri for 65 compounds obtained from the compilation made by Ghmeling et al. has been graphed. As can be observed in Figure 4, the points fit the following linear relationship:

(19)

Therefore, the domain of the function F(HE,GE) = 0 has been established for the following values of the param- eters: (a) r1 = 2 (ql = 1.81751, rz = 10 (qz = 8.1825) and

qi = 0.2265 + 0.7955ri

Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994 IO

/

01 I I 1 0 3 6 9 12

UNIQUAC Parameter r l Figure 4. Relationship between the UNIQUAC parameters qi and ri.

I

0.9

c - 0 E 0 0

-0.9

- 1 0 - I 0 - 1 2 -0 6 -0 0 0 6

HE/Kca l mol- ' Figure 5. Domain of the function F(HE,GE) = 0 for the UNIQUAC equation: (a) rl = 2, r2 = 5; (b) rl = 2, r2 = 10; (dashed lines) Gij I 3.

(b) rl = 2 (q1 = 1.8175), rz = 5 (42 = 4.204). Besides, the value of the parameter z was fixed to 10. The results obtained are shown in Figure 5. The maximum value of HE corresponding to the region where HE > 0 and GE > 0 is 449 cal mol-' for case a and 604 cal mol-' for case b. On the other hand, it can be observed that the function F(HE,GE) = 0 shows a discontinuity in HE = 0, with the following GE values: 1050 and 596 cal mol-' for case a and 1283 and 748 cal mol-' for case b. Besides, it should be noticed that the shape of the domain is similar to that determined for the Wilson equation (figure 1). Further- more, when r1= rz, the UNIQUAC expressions for GE and HE are identical to those of Wilson when V1/Vz = 1, multiplied by a coefficient equal to q1 (=q2).

The revision of the compilation due to Gmehling et al. shows that the percentage of binary systems correlated by the UNIQUAC equation with Gij > 3 is less than 1%. Therefore, the limit of the domain for Gij I 3 has been represented in both cases with a dashed line in Figure 5.

Discussion The domain boundary for the simultaneous correlation

of the excess molar Gibbs free energy (GE) and the excess

1038 Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994

0 2

- - E y" - 0

\ 0 1

I W

0 .o 0 00 0 25 0 50 0 75 1 00

Mole Fraction Benzene Figure 6. Dependence of HE on the mole fraction for the system benzene-n-hexane (parameters taken from Table 3 of Hanks et al., 1978): (--) LEMF equation; (- - -) NRTL equation; (- - -) Wilson equation; (e..) experimental values (Smith and Robinson, 1970).

molar enthalpy (HE) of a binary system has been theo- retically determined for the Wilson, NRTL, LEMF, and UNIQUAC equations. On the basis of the results obtained, it can be concluded that (a) the equations NRTL, Wilson with Vi/ Vj = 1, and UNIQUAC with ri = rj and qi = qj can describe systems with SE < 0 exclusively; (b) the LEMF equation can only describe systems with SE > 0; and (c) the equations Wilson with Vi/ Vj # 1 and UNIQUAC with ri z rj or qi # qj can describe systems with both positive and negative values of SE. Another aspect to be taken into account is the influence of temperature. In all the expressions considered here, the temperature appears always as a scale factor. Therefore, the domains deter- mined at 298.16 K can be corrected for temperature by multiplying by the factor 3.354 X l@3T(K).

The selection of the expressions to be used for the simultaneous description of the excess molar Gibbs free energy and the excess molar enthalpy for a given system should be performed through two main steps. The first one is to determine if the experimental values of HE and GE of the system to be correlated fall inside the domain of the function F(HE,GE) = 0 theoretically established for each one of the equations to be used. In the second step, starting from the equations that fulfill this condition, it must be determined which one correlates more appro- priately the system under study.

The present work mainly deals with the subject related to the first step stated above, which is not usually taken into account in the analysis of the descriptive capability of the correlation equations. Furthermore, it has origi- nated in some cases an inadequate or confusing inter- pretation of such correlation ability. In this sense, the work of Hanks et al. (1978) related to the limits on the simultaneous correlation of GE and HE data can be cited. Table 3 of this work shows the parameters of five binary systems evaluated by the method of Hanks et al. (1971) with the Wilson, NRTL, and LEMF equations. Starting from this information, the dependence of HE on the mole fraction for the system benzene-n-hexane has been calculated for these three equations and compared with the experimental data (Smith and Robinson, 1970). The results obtained are shown in Figure 6. An adequate correlation capability for HE on the part of the NRTL and LEMF equations can be observed. However, if the relationships GE vs P are plotted (Figure 71, it can be

0 3

- 0.2 L 0 E - 0

9 W'

0.1

I I

00 01 0 2

HE/Kcal mol-' Figure 7. GE vs HE plot for the system benzene-n-hexane (param- eters taken from Table 3 of Hanks et al., 1978): (--) LEMF equation; (- - -) NRTL equation; (- - -) Wilson equation; (-e) experimental values (Smith and Robinson, 1970).

observed that the experimental data are only inside the domain established by the LEMF equation. Therefore, further studies in order to analyze the ability of the other equations are completely unnecessary. This fact is re- peated in three of the four remaining systems. On the other hand, it must be mentioned that the method of Hank et al. does not correlate IF: and GE simultaneously.

Another work that should be mentioned is that of Mato et al. (1991) related to the behavior of the LEMF equation for the vapor-liquid equilibrium data treatment of binary systems with negative deviations from ideality. The authors said that the LEMF expression shows an abnor- mally poor performance for these systems and ascribed this behavior to the negative value of the parameter a. They plotted the In yi (xi = x i ) vs In 7; for systems with 77 = y,m (Gij = Gji) and found a minimum value in In yi = 0.2785. Nevertheless, it has been demonstrated for these conditions (Figure 3) that GE has an absolute minimum of -165 cal mol-', which implies that the minimum value of In yi possible of being described by this equation is precisely that mentioned above (In y i = G:,/RT). Con- sequently, the problem lies in the correlation of systems in which GE values are far from those described by the LEMF equation and not in the inconsistency of the negative value of the parameter a with the NRTL theory.

It can be concluded that, for a given correlation equation, the knowledge of the domain of the function F(HE,CE) = 0 avoids the inappropriate use of such an equation beyond the limits established.

Acknowledgment

This work was supported by the Consejo Nacional de Investigaciones Cientlficas y TBcnicas (CONICET, Ar- gentina).

Nomenclature GE = excess molar Gibbs free energy G,, = Wilson, NRTL, LEMF, or UNIQUAC equation pa-

gij - g,, = intermolecular energy interaction parameter IIE = excess molar enthalpy qi = molecular geometric area parameter of pure component

rameter

i

ri = molecular volume parameter of pure component i R = gas constant SE = excess molar entropy T = absolute temperature Vi = molar volume of pure component i xi = mole fraction of component i z = lattice coordination number, here equal to 10

Greek Symbols a = NRTL equation parameter yi = activity coefficient of component i 7; = infinite dilution activity coefficient of component i ai = average segment fraction of component i 19i = average area fraction of component i

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Received for review September 20, 1993 Accepted January 4, 1994'

* Abstract published in Advance ACS Abstracts, March 1, 1994.