2330
Ind. Eng. Chem. Res. 2002, 41, 2330-2331
CORRESPONDENCE Comments on “A Priori Phase Equilibrium Prediction from a Segment Contribution Solvation Model” Andreas Klamt COSMOlogic GmbH&CoKG, Leverkusen, Germany
Sir: Lin and Sandler present in their paper1 a rederivation and a moderate reparametrization of the COSMO-RS model which was originally published2-4 by me and co-workers in 1995, 1998, and 2000. They claim to have removed important thermodynamic inconsistencies of the COSMO-RS model. This is true only in a very limited sense. It is important to note that the central thermodynamic equation proposed in the COSMO-RS paper (eq 6 of that paper) and the one proposed as the COSMO-SAC equation (eq 9 of this paper) are absolutely equivalent if the same interaction energy operator E(σ,σ′) is used. This is shown below. Hence, their criticism regarding thermodynamic inconsistency of this central equation in COSMO-RS is not justified. Only regarding the ad hoc defined combinatorial contribution in COSMO-RS do the authors make a valid point. This term really is not thermodynamically consistent. Nevertheless, this very small inconsistency was removed 2 years ago in the COSMO-RS code, after we received notice of this problem by a customer. The thermodynamic inconsistency of COSMO-RS demonstrated in Appendix II of the paper is solely due to this incorrect combinatorial contribution, because the authors used an old version of the program. Hence, COSMO-RS as used and as distributed by COSMOlogic in the last 2 years is thermodynamically consistent. Considering these facts, it is not acceptable that the model proposed in the present paper is considered as a new model with a new name (COSMO-SAC). It is just a slight reparametrization of COSMO-RS. Details (1) Lin and Sandler argue that the central COSMORS equation has some problems regarding size consistency. To show this, they use eq 17 of their paper, which in some way is analogous to eq 5 of ref 2. However, as was clearly stated there, not eq 5 but eq 6 is the central COSMO-RS equation, which in the more general notation of ref 3 reads
{
}
-E(σ,σ′) + µ′S(σ′) µ′S(σ) ) -kT ln dσ′ p′S(σ′) exp kT
∫
Here p′S(σ) is the normalized σ profile of the ensemble S. The number of segments does not appear in this equation any more, and hence µ′S(σ) does not diverge to negative infinity if the number of segments increases. Hence, the first criticism made by Lin and Sandler is unjustified. Furthermore, this equation gives the result µ′(σ) ) E(σ,σ)/2 if all segments have the same charge
density σ, i.e., if p′S(σ) ) δ(σ-σ′), and it does not yield NE(σ,σ)/2 as Lin and Sandler claim. (2) Lin and Sandler admit that the statistical thermodynamics of COSMO-RS is correct if one assumes that all segments are considered as different. Indeed this is true. It is a physically correct procedure to assume that all segments are distinguishable, even if some of them should have identical properties. This can be shown in two ways: We may assume that there is some hidden property which allows one to distinguish between the objects, let us say some virtual number written on the back of the segments. Such a number would not change the thermodynamics of the system. Hence, if consequently treated in this way, we must get the same physical answers as those in the case when we consider some objects as indistinguishable. In another way, we may consider a series of systems Si, in which the difference of the properties of some objects goes to zero in the limit i f ∞. Then we can treat all of the systems Si correctly with COSMO-RS because the objects are distinguishable. However, because these systems converge to the system of indistinguishable objects, we must have treated that case correctly as well. Finally, it is not even obvious that we have any two identical segments in our ensemble of surface segments considered in COSMO-RS, because there is an infinite number of different ways to cut the surface of a molecule into effective contact segments. Thus, we even do not know whether the statistical thermodynamic picture of distinguishable segments or that of indistinguishable objects is better suited for the problem. Fortunately, both ways lead to the same physical answers. (3) The equivalence of SAC and COSMO-RS can be shown as well by this argumentation: Lin and Sandler claim that the central thermodynamic equation in COSMO-RS is incorrect and that it should be replaced by “their” segment activity coefficient (SAC) equation:
{∑ ( f
µS(σm) ) -kT ln
exp
)}
µS(σn) - E(σm,σn)
n)1
+ kT kT ln pS(σm) (SAC)
However, this is not true. The central COSMO-RS equation is
µ*S(σ) ) -kT ln
{∫
(
dσ′ p′S(σ′) exp
10.1021/ie011031l CCC: $22.00 © 2002 American Chemical Society Published on Web 03/08/2002
)}
µ*S(σ′) - E(σ,σ′) kT (CRS1)
Ind. Eng. Chem. Res., Vol. 41, No. 9, 2002 2331
µ*S(σ) is the chemical potential of a piece of surface of kind σ in the ensemble described by the normalized composition function p′S(σ). Note that, because of our assumption of distinguishability, the chemical potential is just the change of the free energy of the system upon addition of one segment of the unique segment σ. In an ensemble consisting of a finite number f of kinds σm, the quantity µ*S(σm) has to be interpreted as the pseudo chemical potential (according to Ben-Naim5)
µ*S(σm) ) µS(σm) - kT ln p′S(σn)
(CRS2)
because it does not include the entropy ln pS(σm), which results from the permutations of indistinguishable objects. Replacing the integral in CRS1 by a summation over classes, we yield
{
µ*S(σm) ) -kT ln
f
∑ p′S(σn) exp n)1
(
)}
µ*S(σn) - E(σm,σn) kT
(CRS3)
and by use of equation CRS2, this converts easily to
{∑ f
µS(σm) - kT ln p′S(σm) ) -kT ln
(
p′S(σn) exp
)}
n)1
(µS(σn) - kT ln p′S(σn)) - E(σm,σn) kT
(CRS4)
and
{ ( f
µS(σm) ) -kT ln
∑ exp n)1
)}
µS(σn) - E(σm,σn)
+ kT kT ln p′S(σm) (CRS5)
However, CRS5 is just identical with SAC. Hence, SAC is nothing else than a slightly different but equivalent form of the central COSMO-RS equation, and either both are correct or neither of them is. SAC cannot be considered as something new.
(4) The authors are right in the fact that the exact definition of the original COSMO-RS combinatorial contribution, i.e., -λkT ln AS, with AS being the moleaveraged surface area of the components, is thermodynamically inconsistent. This problem was pointed out to us 2 years ago,6 and we removed it by a more sophisticated definition of the average surface area AS. This new definition coincides just with the surface area of the compound for pure compounds, but it is the derivative of a potential and hence obeys Gibbs-Duhem equations. Because the new and correct equation coincides with the old one for pure fluids, all of the calculations done in the original papers remain correct, because only infinite dilution of pure compounds, i.e., only pure liquids, has been considered in these papers quantitatively. Since before Fall 1999, this correction is part of our program COSMOtherm, and hence COSMOtherm is consistent in that regard. (5) The slight reparametrization of COSMO-RS performed by Lin and Sandler (they kept all of our parameters except two) suggests the removal of the smaller averaging radius rav for screening charge densities by the theoretically better justified value of reff. We discussed this topic in ref 2. On the basis of the much broader set of compounds and properties considered in our COSMO-RS parametrization, we found a significant loss of accuracy when going to such large averaging radii. It is likely that the authors will find the same if they expand their work to a broader chemical functionality and to more properties. Literature Cited (1) Lin, S. T.; Sandler, S. A Priori Phase Equilibrium Prediction from a Segment Contribution Solvation Model. Ind. Eng. Chem. Res. 2002, 41, 899. (2) Klamt, A. Conductor-Like Screeing Model for Real Solventss A New Approach to the Quantitative Calculation of Solvation Phenomena. J. Phys. Chem. 1995, 99, 2224. (3) Klamt, A.; Jonas, V.; Bu¨rger, T.; Lohrenz, J. C. W. Refinement and Parametrization of COSMO-RS. J. Phys. Chem. A 1998, 102, 5074. (4) Klamt, A.; Eckert, F. COSMO-RS: A Novel and Efficient Method for the a Priori Prediction of Thermophysical Data of Liquids. Fluid Phase Equilib. 2000, 172, 43. (5) Ben-Naim, A. Solvation Thermodynamics; Plenum Press: New York, 1987. (6) Krooshof, G. Private communication, 1999.
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