Ind. Eng. Chem. Res. 2003, 42, 4143-4145
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CORRELATIONS Predictions of Activity Coefficients of Nearly Athermal Binary Mixtures Using Cubic Equations of State Pablo A. Sacomani and Esteban A. Brignole* PLAPIQUI, Universidad Nacional del SUR-CONICET, Km 7 Camino La Carrindanga, CC 717 Bahı´a Blanca, 8000, Argentina
It is shown that cubic equations of state of the van der Waals family predict the ideal behavior of athermal solutions at infinite pressure. An expression for the infinite-dilution activity coefficient in binary mixtures is derived from the Soave-Redlich-Kwong (SRK) equation of state. The nonresidual contribution of this expression is used to predict the activity coefficients of athermal mixtures. In this work, we compare experimental infinite-dilution activity coefficients with predictions from different models. The comparison shows better agreement of the present model for light linear and branched alkanes diluted in normal alkanes of 20-36 carbon atoms. 1. Introduction The nonideality of mixtures having low-energy interactions is mainly determined by the differences in size and shape of the components. The basic approach used to estimate activity coefficients in athermal mixtures is the Flory-Huggins model.1,2 This model is the simplest approximation to the athermal Gibbs function of mixing, and it gives the activity coefficient of component 1 infinitely diluted in component 2 as
ln γ∞1 ) ln(1/r) + 1 - 1/r
(1)
where r is the ratio of the molar volumes r ) v2/v1. Hildebrand3 proposed a model for the entropy of mixing of asymmetric molecules based on free volume differences. For a binary mixture, this model gives
[
] [
V - N1b1 - N2b2 ∆S M + ) N1 ln R N1(v1 - b1) V - N1b1 - N2b2 N2 ln (2) N2(v2 - b2)
]
where V is the total volume of the mixture and N1 and N2 are the numbers of moles, b1 and b2 the van der Waals volumes, and v1 and v2 the molar volumes of the pure components. Applying eq 1 to athermal mixtures and neglecting excess volume effects, the following expression is derived for the mole-fraction-based activity coefficient of component 1 infinitely diluted in component 2
(
)
v1 - b1 v1 - b1 +1ln(γ∞1 ) ) ln v2 - b2 v2 - b2
This equation was proposed by Elbro et al.4 to predict activity coefficients in polymer solutions. 2. Activity Coefficients from Cubic Equations of State The following expression for the activity coefficient of component 1 infinitely diluted in component 2 can be derived from the Soave-Redlich-Kwong (SRK) equation of state (EoS), after assuming a binary interaction parameter, k12, equal to zero
(
)
b1 Z 2 - b2 (Z2 - 1) - (Z1 - 1) - ln b2 Z1 - b1 b2 1+ 2 b2 v2 b1 xa2 xa1 A1 ln + ln 1 + b1 B1 RT b2 b1 v2 1+ v1
ln γ∞1 )
( )[(
) ( )]
(4)
The residual term (the last term within the brackets in eq 4) goes to zero in athermal mixtures. In addition, when the limiting conditions of infinite pressure are applied to this equation, the following results are obtained: (a) The third and fourth terms in eq 3 are zero because the molar volume is equal to the covolume at infinite pressure (b parameter in cubic EoS). (b) At very high pressures, the SRK EoS can be simplified to
P)
a RT RT ≈ v - b v(v + b) v - b
(5)
Therefore
(3)
* To whom correspondence should be addressed. Tel.: 54 291 4861700. Fax: 54 291 4861600. E-mail: brigno@ criba.edu.ar.
Z)1+
Pb )1+B RT
(6)
Using this result, at infinite pressure, the first and second terms in eq 4 cancel each other, and a zero value is obtained for the right-hand side of eq 4. As a
10.1021/ie030028u CCC: $25.00 © 2003 American Chemical Society Published on Web 07/18/2003
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Ind. Eng. Chem. Res., Vol. 42, No. 17, 2003
consequence, an ideal value of the infinite-dilution activity coefficient for binary mixtures is obtained at infinite pressure. In general, when the approach of Huron and Vidal5 for computing the limiting value of the excess Gibbs function at infinite pressure is applied to the van der Waals family of equations of state, one finds that ideal behavior is predicted for athermal solutions at infinite pressure. This is a consequence of the following assumptions: (a) the condition of zero excess enthalpy for athermal solutions and (b) the requirement that the excess volume be equal to zero at infinite pressure, to ensure a finite limit in the value of the excess Gibbs function. Consequently, in cubic equations of state, the “combinatorial” or “free volume” contribution to the solution nonideality vanishes at infinite pressure. In other words, if the free volumes of the mixture components fall to zero at infinite pressure, it is not possible to account for free volume effects at this limit. As a result, in cubic equation-of-state models, the nonideality at infinite pressure can be attributed only to the difference in interaction energy (or chemical nature) of the mixture components. Soave et al.6 applied the SRK EoS to calculate, from experimental data at atmospheric pressure, activity coefficients in alkane-alkane binary mixtures at infinite pressure. They found that the activity coefficients at infinite pressure were very close to 1. These results are in agreement with the previous discussion about the ideal behavior of athermal mixtures at infinite pressure. In eq 4, we can distinguish between (a) the “residualcontribution”, which does not vanish at infinite pressure, and (b) the “nonresidual contribution”, which does go to zero at infinite pressure. The last term in eq 4 represents the residual contribution, because the logarithmic factor is nonzero at infinite pressure. All other terms vanish at infinite pressure, so they are responsible for the combinatorial free volume (nonresidual) contributions. Therefore, the expression ∞ ln γ1,SRK-NR )
b1 (Z - 1) - (Z1 - 1) b2 2
ln
(
)
Z2 - B2 Z1 - B1
Table 1. SRK Equation-of-State Pure-Component Parameters alkane
Tc (K)
Pc (bar)
Zc
ω0.5
ω0.7
n-C20 n-C22 n-C24 n-C28 n-C30 n-C32 n-C34 n-C36
768.59 788.15 806.67 836.24 848.02 859.74 870.47 880.94
11.27 10.21 9.35 7.89 7.36 6.87 6.43 6.08
0.210 0.2033 0.1973 0.1848 0.1795 0.1743 0.1694 0.1652
2.6511 2.8653 3.0548 3.4574 3.6272 3.7977 3.9642 4.1068
0.8860 0.9577 1.0211 1.1558 1.2126 1.2697 1.3254 1.3731
Table 2. Average and Maximuma Absolute Percent Errors in Calculated Activity Coefficients for All Solutes in n-Alkane Solvents absolute error in calculated activity coefficients (%)
n-C20 n-C22 n-C24 n-C28 n-C30 n-C32 n-C34 n-C36 mean maximum a
SRK-NR (eq 4)
SRK, k12 ) 0 (eq 3)
FloryHuggins
free volume (eq 2)
4.88 (7.43) 6.59 (9.36) 3.31 (9.27) 3.25 (8.80) 2.86 (7.52) 3.27 (11.21) 3.14 (11.37) 3.20 (11.87) 3.81 (11.87)
13.27 (22.29) 14.65 (27.73) 24.69 (38.63) 35.59 (46.94) 40.97 (49.97) 46.53 (58.81) 53.27 (63.75) 61.01 (72.17) 36.25 (72.17)
17.12 (22.65) 19.34 (22.43) 16.84 (20.45) 19.22 (22.77) 20.06 (23.38) 21.08 (24.24) 21.62 (24.97) 22.34 (25.86) 19.70 (25.86)
9.61 (13.71) 10.95 (15.04) 7.65 (12.06) 8.87 (13.37) 9.18 (13.84) 10.09 (14.87) 10.23 (14.92) 11.01 (15.79) 9.70 (15.79)
Value between parentheses.
( )
b2 A1 v2 ln (7) b1 B1 1+ v1 1+
is called the SRK nonresidual (SRK-NR) contribution. 3. Results and Discussion In this work, we compare experimental infinitedilution activity coefficients for athermal solutions with predictions obtained with the following models: (a) Flory-Huggins (eq 1), (b) Elbro et al.4 (eq 3), (c) SRK with k12 ) 0 (eq 4), and (d) SRK-NR contribution (eq 7). The experimental data of Parcher7 et al. on infinitedilution activity coefficients for normal and branched light alkanes in normal alkanes having 20-36 carbon atoms at 80 °C were used to test the different models. Critical properties of solutes were taken from Reid et al.8 The critical properties and the acentric factors (ω0.5 and ω0.7) of the solvents were obtained from the characterization procedure of Twu et al.9,10 and are given in Table 1.
Figure 1. Infinite-dilution activity coefficients of n-butane in n-alkanes: 0, experimental; /, SRK nonresidual; ×, SRK zero interaction coefficient; +, Flory-Huggins; ], Elbro (Hildebrand free volume).
In this procedure, the acentric factor is obtained at Tr ) 0.5 to improve the vapor pressure predictions for heavy alkanes using a generalized vapor pressure correlation. For all compounds, the EoS parameters were calculated from critical properties and from vapor pressure information. From boiling points and critical properties, accurate equation-of-state parameters can be obtained for a wide range of temperatures.11 The solvent molar volumes were calculated from tabulated density data at the temperature of interest7 and from the corresponding molecular weights. For solutes, tabulated density data extrapolated to the temperature of interest by means of the modified Racket
Ind. Eng. Chem. Res., Vol. 42, No. 17, 2003 4145
Figure 2. Infinite-dilution activity coefficients of n-octane in n-alkanes: 0, experimental; /, SRK nonresidual; ×, SRK zero interaction coefficient; +, Flory-Huggins; ], Elbro (Hildebrand free volume).
accurate than those of the Elbro et al.4 model. The use of eq 4, the SRK model with interaction parameters equal to zero, produces the greatest deviations. A comparison between the experimental data and predictions based on the SRK-NR expression (eq 6) and the complete activity coefficient expression (eq 4) indicates that the EoS residual contribution worsens the quality of activity coefficient predictions in nearly athermal mixtures. The residual term generally gives activity coefficient values that are too high. It has been stated that cubic EoS are not able to describe mixtures of compounds that have large differences in molecular size. This limitation of cubic EoS might be due to the prediction of residual contributions that are too large, even for nearly athermal mixtures. These results explain the need for rather large binary interaction parameters to compensate for the EoS residual term deficiency. Figures 1 and 2 illustrate the quality of the SRK-NR predictions for n-butane and n-octane respectively, infinitely diluted in different normal alkanes. Figure 3 shows the performance of the different models in the prediction of infinite-dilution activity coefficients of several heptane isomers in n-dotriacontane. The values obtained from eq 4 were excluded because the errors were too high. Even though the effect of branching is somewhat subtle, the nonresidual model (SRK-NR) follows the experimental trend better than the other models. Acknowledgment The authors acknowledge the support of CONICET(Argentina) and the Universidad Nacional del Sur for this work. Literature Cited
Figure 3. Infinite-dilution activity coefficients of branched heptanes in n-C32 alkane: s, experimental; /, SRK nonresidual; +, Flory-Huggins; ], Elbro (Hildebrand free volume). Solutes: 1, 3,3-dimethylpentane; 2, 2,3-dimethylpentane; 3, 2,2,3-trimethylbutane; 4, 3-methylhexane; 5, n-heptane; 6, 2-methylhexane; 7, 2,2-dimethylpentane.
equation8 were used. van der Waals volumes (vVdW) were obtained from molecular weights (Mw) using the equation
vVdW (cm3/mol) ) 5.4097 + 0.729 31 × Mw
(8)
This equation reproduces exactly the van der Waals volumes of n-alkanes as given by Bondi;12 the effect of branching on these volumes is negligible. For example, for 2,2,3-trimethylbutane, the equation overpredicts the van der Waals volume by 0.038%. For simplicity, infinite-dilution activity coefficients and liquid-phase molar volumes were calculated with the SRK EoS at zero pressure, because of the negligible effect of pressure on the liquid-phase activity coefficients at low pressures.6 Table 2 reports the average and maximum absolute percent deviations between the experimental and calculated infinite-dilution activity coefficients for all of the models studied. It can be seen that the SRK-NR equation gives predictions that are similar to and even more
(1) Flory, P. J. Thermodynamics of High Polymer Solutions. J. Chem. Phys. 1941, 9, 660. (2) Huggins, M. L. Solutions of Long Chain Compounds. J. Chem. Phys. 1941, 9, 440. (3) Hildebrand, J. H. The Entropy of Solution of Molecules of Different Size. J. Chem. Phys. 1947, 15, 225. (4) Elbro, H. S.; Fredenslund, Aa.; Rasmussen, P. A New Simple Equation for the Prediction of Solvent Activities in Polymer Solutions. Macromolecules 1990, 23, 4707. (5) Huron, M. J.; Vidal, J. New Mixing Rules in Simply Equations of State for Representing Vapor-Liquid Equilibria of Strongly non Ideal Mixtures. Fluid Phase Equilib. 1979, 3, 325 (6) Soave, G.; Bertucco, A.; Vecchiatto, L. Equation-of-State Group Contributions from Infinite-Dilution Activity Coefficients. Ind. Eng. Chem. Res. 1994, 33, 975. (7) Parcher, J. F.; Weiner, P. H.; Hussey, L.; Westlake, T. N. Specific Retention Volumes and Limiting Activity Coefficients of C4-C8 Alkane Solutes in C22-C36 n-Alkane Solvents. J. Chem. Eng. Data 1975, 20, 145. (8) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases & Liquids, 4th ed.; McGraw-Hill: New York, 1987. (9) Twu, C. H. An Internally Consistent Correlation for Predicting the Critical Properties and Molecular Weights of Petroleum and Coal-tar Liquids. Fluid Phase Equilib. 1984, 16, 137. (10) Twu, C. H.; Coon, J. E.; Cunningham, J. R. A Generalized Vapor Pressure Equation for Heavy Hydrocarbons. Fluid Phase Equilib. 1994, 96, 19. (11) Sacomani, P. A.; Gros, H. P.; Brignole, E. A. Computation of the Attractive Energy Parameter of Normal Fluids in Cubic Equations of State. Fluid Phase Equilib. 1996, 12, 27. (12) Bondi, A. Physical Properties of Molecular Crystals, Liquids and Gases; John Wiley & Sons: New York, 1968.
Received for review January 13, 2003 Revised manuscript received June 9, 2003 Accepted June 13, 2003 IE030028U
