3376
Ind. Eng. Chem. Res. 2005, 44, 3376
Reply to the Comments by Kontogeorgis and Coutsikos on “Predictions of Activity Coefficients of Nearly Athermal Binary Mixtures with 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
Sir: We appreciate the comments of Kontogeorgis and Coutsikos on our paper;1 however, we would like to clarify some points. (1) We consider it convenient to show first that, although eq 1 by Kontogeorgis and Coutsikos and eq 4 in our work1 are not apparently equivalent, the former one can be derived from the last one by replacing the definition of the compressibility factors of the pure components and the pressure at infinite dilution from the equation of state. (2) With regard to the term named in the comments by Kontogeorgis and Coutsikos as “res2”, this term comes from the rearrangement of terms that represent only pure component properties (Zi), and we do not consider it residual, as explained below. We find a discrepancy among our definition of “residual” and the one used by Kontogeorgis et al.2 We consider the residual contribution to be the one that accounts for energetic interactions. For instance, even though the fourth term in eq 1
A1 ln B1
[ ] b2 v2 b1 1+ v1 1+
contains the attractive parameter of the diluted component 1, it vanishes at infinite pressure. Assuming that two hypothetical components 1 and 2 had identical “a/b2” parameters, the last term in eq 1 of the comments by Kontogeorgis and Coutsikos will be zero. However, the term written above would still be * To whom correspondence should be addressed. Tel.: 54 291 4861700. Fax: 54 291 4861600. E-mail: ebrignole@ plapiqui.edu.ar.
contributing to the mixture nonideality, unless the pressure of the solution goes to infinity. On the other hand, in ref 2, any term that contains pure-component and mixture energy variables, ai or aij, is considered to be residual. We find this definition too restrictive because, in a fluid represented by an equation of state, both attractive and repulsive terms are needed to represent a condensed state. (3) The mixing rule, eq 2 in the comments by Kontogeorgis and Coutsikos, was formerly presented by Vidal,3 effectively eliminates the residual part of vdWtype equations of state, and predicts an ideal mixture at infinite pressure. Soave4 pointed out that it could be used in nearly athermal mixtures. It is interesting to remark that the combinatorial part of the equation of state vanishes at infinite pressure, the condition at which the free volume vanishes. This fact, assuming such simple behavior at infinite pressure, makes it possible to correctly predict the activity coefficientes, for example, of seven isomers of heptane in n-C32 (see Figure 3 in ref 1), supports the assumption that alkane solutions are effectively ideal at very high pressures. Literature Cited (1) Sacomani, P. A.; Brignole, E. A. Predictions of Activity Coefficients of Nearly Athermal Binary Mixtures Using Cubic Equations of State. Ind. Eng. Chem. Res. 2003, 42, 4143. (2) Kontogeorgis, G. M.; Coutsikos, P.; Harismiadis, V. I.; Fredenslund, A.; Tassios, D. P. A Novel Method for Investigating the Repulsive and Attractive Parts of Cubic Equations of State and the Combining Rules Used with the vdW-1f Theory. Chem. Eng. Sci. 1998, 53, 541. (3) Vidal, J. Mixing Rules and Excess Properties in Cubic Equations of State. Chem. Eng. Sci. 1978, 33, 787. (4) Soave, G. Improvement of the van der Waals equation of state. Chem. Eng. Sci. 1984, 39, 357.
IE0501935
10.1021/ie0501935 CCC: $30.25 © 2005 American Chemical Society Published on Web 03/16/2005
