3374
Ind. Eng. Chem. Res. 2005, 44, 3374-3375
CORRESPONDENCE Comments on “Predictions of Activity Coefficients of Nearly Athermal Binary Mixtures Using Cubic Equations of State” Georgios M. Kontogeorgis*,† and Philippos Coutsikos‡ IVC-SEP Engineering Research Centre, Department of Chemical Engineering, Technical University of Denmark, Building 229, DK-2800 Lyngby, Denmark, and Diadikassia Business Consultants SA, 180 Kifissias Avenue, 15231 Halandri, Athens, Greece
Sir: Sacomani and Brignole1 have shown, in a recent very interesting paper, that when using what they call the “nonresidual” part of the Soave-Redlich-Kwong (SRK) equation of state (EoS), excellent agreement is obtained between the experimental and calculated activity coefficients for alkane systems. Their expression for the infinite-dilution activity coefficient of SRK is
( )
ln γ∞1 ) ln
v1 - b1 v1 - b1 +1+ res2 + v2 - b2 v2 - b2
(
a1 b1 xa2 xa1 f(b1,v1) + ln b1RT f(b2,v2) RT b2 b1
)
We believe indeed that this mixing rule (eq 2) eliminates the energetic (“residual”) contribution of SRK, of other van der Waals (vdW) type cubic EoSs, and of the van Laar activity coefficient model. The van Laar equation can be written as
gE,vanLaar
)
RT
res2 )
(
[(
2
ln f(b2,v2) (1)
gE,EoS,∞P RT
)
a2 v1b2 - v2b1 b2RT v 2 + b v 2 2 2
Sacomani and Brignole performed activity coefficient calculations for athermal alkane solutions using only the first terms of eq 1 (upper line) because they considered the last term of eq 1 (bottom line) to be the “residual” energetic contribution to the activity coefficient. For nearly athermal solutions such as a mixture of alkanes, this “energetic” contribution should be of minor importance. It can be shown that eq 1 without the last “residual” term can be derived for the SRK EoS using a specific mixing rule for the equation’s parameters:
a b
)
ai
∑i xib
(
ai
∑ xi RT i b
-
b
i
)
a
(3)
while the vdW, SRK, and Peng-Robinson (PR) EoSs at infinite pressure are
)c
where
vi + bi f(bi,vi) ) vi
1
(2)
i
The derivation is a bit tedious by straightforward and is not reproduced here. The well-known arithmetic mean rule is used for the covolume parameter b. * To whom correspondence should be addressed. Tel.: 45252859. Fax: 45882258. E-mail:
[email protected]. † Technical University of Denmark. ‡ Diadikassia Business Consultants SA.
1
ai
∑xi RT i b
-
i
)]
a b
(4)
The constant c depends on the exact functional form of the EoS. For the vdW equation, c ) 1 (the same result as that in the van Laar theory), for PR, c ) 0.623, and for SRK, c ) ln 2. Both the van Laar model and the various cubic EoSs at infinite pressure contain no “combinatorial/free volume contributions”; thus, the use of eq 2 eliminates the residual part of these models. Thus, the use of SRK or PR together with the mixing rule of eq 2 permits testing of these EoSs for asymmetric athermal systems where only combinatorial/free volume effects dominate and no significant energetic interactions are present. Alternatively, the typical choice is the vdW one-fluid mixing rules (vdW1f):
a)
∑i ∑j xixjaij
b)
∑i xibi
(5)
with the geometric mean rule used for the cross energy parameter:
aij ) xaiaj(1 - kij)
(6)
Experimental and predicted (kij ) 0) activity coefficients at infinite dilution of n-hexane in n-alkanes and heavy alkanes in n-hexane are shown in Tables 1 and 2. The PR is used here, but similar results are obtained with SRK and for other asymmetric alkane solutions including gas alkanes. Very good agreement is obtained
10.1021/ie0500063 CCC: $30.25 © 2005 American Chemical Society Published on Web 03/16/2005
Ind. Eng. Chem. Res., Vol. 44, No. 9, 2005 3375 Table 1. Experimental and Predicted Activity Coefficients at Infinite Dilution of n-Hexane (n-C6) in n-Hexane/n-Alkane Systems Using the PR EoS, with the vdW1f Mixing Rules (Eq 5) and the a/b Mixing Rule (Eq 2) systema
T (K)
expt value
PR-vdW1f (eq 5)
PR-a/b rule (eq 2)
n-C6/n-C12 n-C6/n-C16 n-C6/n-C18 n-C6/n-C20 n-C6/n-C22 n-C6/n-C24 n-C6/n-C28 n-C6/n-C32 n-C6/n-C36
293.15 293.15 308.15 353.15 373.15 373.15 373.15 373.15 373.15
0.961 0.905 0.877 0.872 0.859 0.801 0.736 0.689 0.639
1.011 1.030 1.030 1.031 1.039 1.059 1.091 1.133 1.182
0.941 0.883 0.855 0.832 0.777 0.738 0.662 0.593 0.528
a
n-Cx indicates a normal alkane with x carbon atoms.
Table 2. Experimental and Predicted Alkane Activity Coefficients at Infinite Dilution for n-Alkane/n-Hexane Systems Using the PR EoS, with the vdW1f Mixing Rules (Eq 5) and the a/b Mixing Rule (Eq 2) systema
T (K)
expt value
PR-vdW1f (eq 5)
PR-a/b rule (eq 2)
n-C12/n-C6 n-C16/n-C6 n-C20/n-C6 n-C22/n-C6 n-C32/n-C6
293.15 293.15 269.10 281.40 288.30
0.961 0.892 0.850 0.756 0.471
1.025 1.091 1.402 1.895 22.43
0.887 0.717 0.490 0.329 0.031
a
n-Cx indicates a normal alkane with x carbon atoms.
between the experimental and predicted activity coefficients only when the mixing rule of eq 2 is used. The use of the classical mixing rules, eq 5, yields even qualitatively wrong results (positive deviations from Raoult’s law). The activity coefficients of heavy alkanes are much more difficult to be accurately represented, but even in this extreme case, PR with eq 2 yields predictions in qualitative agreement with the experiment.
Our results are in agreement with the conclusions of Sacomani and Brignole.1 The major conclusion is that cubic EoSs like SRK and PR can represent asymmetric athermal systems very accurately when a suitable mixing rule (eq 2) is used. This rule has the property that only combinatorial/free volume effects are accounted for, which is a reasonable assumption for alkane solutions. The use of eq 2 is equivalent to using the socalled “nonresidual part” of the SRK of Sacomani and Brignole. It has often been reported that a cubic EoS cannot adequately represent asymmetric systems. These problems seem to be more due to the vdW fluid mixing rules (eq 5), especially the geometric-mean rule for the crossenergy parameter (eq 6) rather than the functional form of the EoS and especially of the vdW-based repulsive term, despite what is often stated. Previous studies2 indicate that the classical vdW one-fluid mixing rules (eqs 5 and 6) yield very large values of the “residual” contributions even for athermal mixtures, in accordance to what Sacomani and Brignole1 state toward the end of their paper: “the limitation of cubic EoS might be due to the prediction of residual contributions that are too large even for nearly athermal mixtures”. 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, Ph.; Harismiadis, V. I.; Fredenslund, Aa.; 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 (3), 541.
IE0500063
