Chemical Physics 356 (2009) 110–120

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Chemical Physics

journal homepage: www.elsevier .com/ locate/chemphys

On the solution of the integral equation of the conductor-like screening modelfor real solvents

Robert Franke a,b,*, Jürgen Friedrich c,1

a Evonik Oxeno GmbH, Paul-Baumann-Straße 1, D-45772 Marl, Germanyb Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, D-44780 Bochum, Germanyc AQura GmbH, Paul-Baumann-Straße 1, D-45764 Marl, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 July 2008Accepted 22 October 2008Available online 26 October 2008

This paper is dedicated to WernerKutzelnigg on the occasion of his 75thbirthday.

Keywords:COSMO-RSCOSMODielectric continuum modelHammerstein equationActivity coefficientChemical potential

0301-0104/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.chemphys.2008.10.025

* Corresponding author. Address: Evonik Oxeno GmD-45772 Marl, Germany.

E-mail address: robert.franke@evonik.com (R. Fran1 Present address: HOB GmbH & Co. KG, D-90556 Ca

This paper presents a new approach to solving the equation which forms the basis for the conductor-likescreening model for real solvents (COSMO-RS). The approach is based on the recognition that the integralequation for the determination of the chemical potential of a surface segment as a function of screeningcharge density r belongs to the class of the non-linear Hammerstein integral equations. The solution ofthis integral equation can be written in closed form in which the numerical values of the coefficients aredetermined by solving a non-linear system of equations. The algorithm for solving the COSMO-RS integralequation is described and illustrated in the case of a model problem. The new approach is then applied forthe calculation of chemical potentials and infinite dilution activity coefficients, and its convergencebehaviour is investigated for the systems 1-butyne/n-butane and n-hexane/water. Calculations of infinitedilution activity coefficients c1 are reported for a representative set of organic compounds in aqueoussolutions and in n-hexane at 298.15 K, and are compared with experimental results and those from cal-culations with a commercial program in order to demonstrate the feasibility of our approach. Finally, asimple method to calculate approximate activity coefficients is proposed and applied to a species subsetof the industrially important C4 fraction.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

In a paper published in 1995, Klamt presented an equationwhich relates the thermodynamics of mixed fluids of arbitrarycomplexity to quantum chemical calculations of single moleculesembedded in a dielectric continuum [1]. Klamt employed thedielectric continuum model COSMO (conductor-like screeningmodel) [2] which, besides other results, yields two quantities:the ideal screening energy, and the screening surface charge densi-ties on a van der Waals-like surface around the molecule. The aver-aging of these screening charge densities over a region with radius0.05 nm and the creation of a histogram representing the distribu-tion of these averaged charge densities results in a function pM(r),which describes the amount of surface with a screening chargedensity in a given interval [r � 1/2 � dr,r + 1/2 � dr]. This function,the so called r-profile [1], offers what is effectively a fingerprint ofa molecule with respect to its solvation behaviour. All the informa-tion about a molecule necessary for COSMO-RS calculations is con-tained in its r-profile.

ll rights reserved.

bH, Paul-Baumann-Straße 1,

ke).dolzburg, Germany.

The r-profiles of water, n-butane, 1-butyne and n-hexane aregiven in Fig. 1; we consider the case of water. As can be seen, ther-profile of water is relatively broad and symmetric, showingtwo maxima: at �1.6 e/nm2 and at +1.8 e/nm2. The latter, at posi-tive screening charge density, corresponds to the lone pairs of thepartially negatively charged oxygen atom, while the former, atnegative screening charge density, corresponds to the two partiallypositively charged hydrogen atoms. It should be noted that in ther-profiles negative partial charges always correspond to positivescreening charge densities and vice versa. The shape of the r-pro-file of water reflects water’s ability to act as a hydrogen bond donoras well as an acceptor. For an illustrative discussion of r-profileswe refer to Ref. [1]. The r-profile of an ensemble S of n differenttypes of molecules Mi with mole fractions xi is given by [1,3]

pSðrÞ ¼Xn

i¼1xi � pMi ðrÞ ð1Þ

From statistical thermodynamics for an ensemble S of pairwiseinteracting contact segments characterised by the r-profile pS(r)Klamt derived an integral equation for the calculation of the chem-ical potential lS of segments with screening charge density r [1]:

lSðrÞ ¼ �RT lnZ b

apsð~rÞ exp

�eðr; ~rÞ þ lSð~rÞRT

� �d~r

" #ð2Þ

Fig. 1. r-Profiles for water, n-butane, 1-butyne and n-hexane.

R. Franke, J. Friedrich / Chemical Physics 356 (2009) 110–120 111

where R is the gas constant and T the temperature. The domain ofintegration is determined by the characteristics of the r-profileand the used units (see Section 2.2 and Section 3.1). In practical cal-culations we always use a value of a = �3 for the lower limit andb = +3 for the upper limit. The term eðr; ~rÞ in Eq. (2) denotes theinteraction energy expression for the segments with screeningcharge density r and ~r, respectively. The first paper on COSMO-RS employs an expression of the form

eðr; ~rÞ ¼ gðrþ ~rÞ2 ð3Þ

This so-called electrostatic misfit energy represents the self-en-ergy of the screening charge density rþ ~r on a contact surface oftwo touching molecules [2]. The empirical constant g is deter-mined with reference to experimental data [1,3]. While optimizingthe COSMO-RS inherent parameters, it was found that the electro-static misfit energy expression offers only a partial description ofthe energetics of hydrogen bonding. To improve the accuracy ofthe model the heuristic expression

cHB �maxð0;racceptor � rHBÞ �minð0;rdonor þ rHBÞ ð4Þ

was added to (3) [3], where rdonor = min (r; ~r) and racceptor ¼maxðr; ~rÞ. The inclusion of this term with its two adjustable param-eters cHB and rHB in the energy expression resulted in the halving ofthe root mean square value of the fit for compounds containing car-bon, hydrogen and oxygen atoms. Despite the greater accuracyachieved by this ad hoc modification, it is somewhat unsatisfactorythat the expression (4) cannot be derived from physical principlesonly.

Once the COSMO-RS integral equation has been solved, the cen-tral property in the thermodynamics of mixing – the chemical po-tential of molecule Mi in solvent S – is determined by

lMiS ¼

Z b

apMi ðrÞ � lSðrÞdrþ lMi

CS ¼ ~lMiS þ lMi

CS ð5Þ

where the combinatorial part of the chemical potential lMiCS de-

scribes contributions resulting from the degeneracy of a molecularconfiguration, and is usually small in comparison to the so calledresidual part ~lMi

S . Since the combinatorial contribution dependson the size and the shape of Mi and S it cannot be determined fromthe r-profile, for the r-profile does not contain any informationconcerning the shape of a molecule. In the parametrisation of theCOSMO-RS model described in Ref. [4], the combinatorial contribu-tion is approximated by the empirical expression

lMiCS ¼ c0 ln½Vi� þ c1 1� Vi

V 0� ln½V 0�

� �� c2 1� Ai

A0� ln½A0�

� �ð6Þ

where Vi is the molecular volume of compound Mi, Ai is its molecu-lar surface area, and the total volume and surface area of all com-pounds in the ensemble are given respectively by

V 0 ¼Xn

i¼1xi � Vi ð7aÞ

and

A0 ¼Xn

i¼1xi � Ai ð7bÞ

The parameters c0, c1 and c2 have to be adjusted to experimentaldata. The molecular surface area and volume are calculated fromthe cavity constructed in the COSMO part of a quantum chemicalcalculation where this molecular-shaped cavity defines the bound-ary to the dielectric continuum. For further details we refer to Ref.[2]. Eq. (5) offers the possibility of calculating the chemical poten-tial of any compound in a fluid of arbitrary composition.

By far the most time-consuming step of the COSMO-RS ap-proach is the underlying quantum chemical calculation, whichhas to be carried out for each compound of the mixture. Theremarkable feature of the COSMO-RS approach is that this calcula-tion is only necessary for individual molecules. The subsequent sta-tistical thermodynamics is thus reduced to calculations for anensemble of decoupled surface segments each carrying a screeningcharge density, which can be performed in an efficient manner byapplying Eq. (2). Klamt and co-workers have provided a carefullyparameterised COSMO-RS model integrated in the program COS-MOtherm [5] which enables the calculation of a great variety ofthermodynamic quantities of mixtures for a wide range oftemperatures.

The method for solving the integral equation numerically as de-scribed in Refs. [1,3] and implemented in COSMOtherm is based ona simple algorithm. In fact, the COSMOtherm program solves a gen-eralisation of Eq. (2) where the one dimensional r-profiles are re-placed by two-dimensional functions in order to account for thecorrelation between screening charge densities on different seg-ments (for details see Ref. [3]).The basic principles of the algorithmare not affected by this enhancement. In a first step, Eq. (2) issolved for each r-value in the interval �3 e/nm2 to +3 e/nm2,beginning with the choice lSð~rÞ ¼ 0 on the right hand side. Thenext step is the integration with respect to ~r which yields the firstvalue for lS(r). Inserting this value for lSð~rÞ and iterating up toself-consistency gives the solution for the r-value under consider-ation. For each discrete value of r, this procedure thus produces acorresponding value for the chemical potential on a surface seg-ment lS(r). Finally, the first term on the right hand side of Eq.(5) is evaluated by numerical integration in order to calculate thechemical potential. The COSMO-RS model as implemented in theCOSMOtherm program is a powerful means for the a priori predic-tion of thermodynamic data of fluid systems. In particular it has gi-ven excellent results both in research and development proceduresand in the process synthesis in industrial speciality chemistry,where reliable experimental data are seldom available in theexplorative phase. To date, COSMO-RS is unrivalled as a generallyapplicable predictive tool in the important field of liquid–liquidequilibria.

However, the disadvantage of solving the integral Eq. (2)numerically as described in the previous paragraph is that it yieldsthe so called r-potential [1] – that is, the chemical potential as afunction of the screening charge density lS(r) – only pointwisefor selected values of r. For some applications, such as the provi-sion of infinite dilution activity coefficients at given temperaturefor solving problems in process design, or the use of r-potentialsin quantitative structure–property relationship (QSPR) studies, ananalytic expression for lS(r) would be valuable. This could thenbe substituted into the equation for the chemical potential (5),making it possible to de-couple not only of the quantum-chemical

112 R. Franke, J. Friedrich / Chemical Physics 356 (2009) 110–120

calculations but also the subsequent procedure for solving the inte-gral Eq. (2) from the calculation of the physico-chemical proper-ties. If the integrand in Eq. (5) could be presented as acontinuous function with a known antiderivative, the calculationof the chemical potential could easily be performed with a spread-sheet program.

Here we present a new approach to solving the COSMO-RS inte-gral equation which produces an analytic expression for lS(r). It isbased on the recognition that the COSMO-RS integral equation be-longs to the class of non-linear Hammerstein equations. The pre-sentation of the material in the paper is given in the followingway. Non-linear Hammerstein equations are briefly introduced inSection 2. An approximate integral equation with degenerate ker-nel is derived which models the COSMO-RS integral equation withnon-degenerate kernel up to arbitrary precision. The solution ofthis equation for the physically important case where the energyexpression is given by the electrostatic misfit energy is shown. Sec-tion 3 presents numerical results for a model problem and for avariety of molecular systems. The conclusions are summarised inSection 4. Mathematical details are documented in Appendix A.

2. Theory

2.1. General solution of the COSMO-RS integral equation

Rewriting Eq. (2) as

yðxÞ ¼ �RT lnZ b

apðzÞXðx; zÞ exp

yðzÞRT

� �dz

" #ð8Þ

and substituting s(x) � exp[�y(x)/(RT)] yields

sðxÞ ¼Z b

apðzÞXðx; zÞ½sðzÞ��1dz: ð9Þ

This is an integral equation of the type

yðxÞ ¼Z b

aKðx; zÞf ðyðzÞÞdz ð10Þ

which belongs to the class of non-linear Hammerstein integralequations [6,7].

Let us first briefly review the solution of equations of type (10)for the case of a degenerate kernel. In this case, the analysis isstraightforward and the principal strategy for solving this type ofequation can be clearly illustrated.

A kernel K(x,z) is degenerate if it obeys:

Kðx; zÞ ¼Xn

l¼0

rlðxÞslðzÞ ð11Þ

for n 2 N0 and n finite. Under the assumption of a degenerate kernelwe can write Eq. (10) as

yðxÞ ¼Z b

a

Xn

l¼0

rlðxÞslðzÞ" #

f ðyðzÞÞdz() yðxÞ

¼Xn

l¼0

rlðxÞZ b

aslðzÞf ðyðzÞÞdz

" #: ð12Þ

The solution of Eq. (12) then has the form

yðxÞ ¼Xn

l¼0

AlrlðxÞ ð13Þ

with coefficients Al given by

Al ¼Z b

aslðzÞf ðyðzÞÞdz ð14Þ

For the determination of Al we substitute (13) into (14) to pro-duce a non-linear system of equations

Al ¼Z b

aslðzÞf

Xn

m¼0

AmrmðzÞ !

dz ð15Þ

which is soluble by finding the fixed point of the mapping

U : Rnþ1 ! Rnþ1;

A0

..

.

An

0BB@1CCA#

R ba s0ðzÞf

Pnm¼0

AmrmðzÞ� �

dz

..

.R ba snðzÞf

Pnm¼0

AmrmðzÞ� �

dz

0BBBBBB@

1CCCCCCA: ð16Þ

Turning to the re-written COSMO-RS equation (8), it is clear thatwith

Xðx; zÞ ¼ expð�Nðx; zÞÞ ð17Þ

where N(x,z) = g(x + z)2 and g is a constant, its kernel cannot be for-mulated according to Eq. (11), so that we cannot directly apply thestrategy of solving the Hammerstein equation reviewed above.However, if the non-degenerate remainder of K(x,z), which we shalldenote as ~Kðx; zÞ, is n + 1 times differentiable with respect to x, it isalways possible to apply Taylor’s Theorem and expand the remain-der around x0 to give

eK ðx; zÞ ¼Xn

k¼0

XðkÞx ðx0; zÞk!

ðx� x0Þk" #

þXðnþ1Þx ðn; zÞðnþ 1Þ! ðx� x0Þnþ1 ð18Þ

with n 2 [x0,x]. For the particular case of ~Kðx; zÞ being differentiablean arbitrary number of times with the condition

8x 2 R limn!1

Xðnþ1Þx ðn; zÞðnþ 1Þ! ðx� x0Þnþ1 ¼ 0 ð19Þ

~Kðx; zÞ can be approximated to arbitrary precision around x0 by theTaylor polynomialXn

k¼0

XðkÞx ðx0; zÞk!

ðx� x0Þk" #

: ð20Þ

With the expansion of the non-degenerate remainder of thekernel according to Eq. (18) we now have instead of Eq. (9) theapproximating integral equation

pðnÞðxÞ ¼Z b

a

Xn

k¼0

XðkÞx ðx0; zÞk!

ðx� x0ÞkpðzÞ" #

½pðnÞðzÞ��1dz ð21Þ

with degenerate kernel, where for p(n)(x): limn!1pðnÞðxÞ ¼ sðxÞ. Thehigher the degree of the Taylor polynomial, the more precisely dothe solutions of Eq. (21) describe the solutions of the COSMO-RSintegral Eq. (9). If condition (19) holds, then Eq. (21) gives the exactsolution. Applying Eqs. (13)–(15) yields for the solution of Eq. (21)

pðnÞðxÞ ¼Xn

k¼0

Akðx� x0Þk ð22Þ

with the coefficients given by

Ak ¼Z b

a

XðkÞx ðx0; zÞpðzÞ

k!Pnv¼0

Avðz� x0Þvdz: ð23Þ

The solution of Eq. (8) can therefore be written:

yðxÞ � yðnÞðxÞ ¼ �RT lnXn

k¼0

Akðx� x0Þk" #

: ð24Þ

The solutions of the non-linear system of equations resultingfrom Eq. (23) depend on the r-profile p(z) and the XðkÞx ðx0; zÞ.

R. Franke, J. Friedrich / Chemical Physics 356 (2009) 110–120 113

2.2. Solution for the energy expression given by the electrostatic misfit

Before we present the solution of Eq. (9) for the particularenergy expression describing the electrostatic misfit, we firstoutline some important characteristics of r-profiles. In physi-cally relevant cases the r-profile p(r) has the followingproperties:

p : R! R; r # pðrÞ ð25aÞ8r pðrÞP 0 ð25bÞ½8r r < a) pðrÞ ¼ 0� ^ ½8r r > b) pðrÞ ¼ 0� ð25cÞ

8pðrÞ 0 <Z 1

�1pðrÞdr <1 ð25dÞ

Eq. (25b) follows from the definition of the r-profile which isthe area of surface having a particular screening charge density r.Eqs. (25c) and (25d) are the consequence of the physical natureof r, implying that its value is finite on segments of finite area ofsurface and that both the surface and the charge of a molecularsystem are finite quantities. In the following we always assumethat Eq. (25) hold. Note, that because Eq. (25d) holds r-profilescan be normalized to 1. For the normalized form Eq. (25b) is like-wise valid. Therefore r-profiles which are normalized to 1 areprobability density functions which describe the probability of find-ing a particular screening charge density of value r on a contactsegment of a molecule.

Let us now investigate the solution of the COSMO-RS integralequation for a particular case of the energy expression, namelythe electrostatic misfit energy [1]. In this case the relevant partof the kernel of the integral equation is given by

Xðx; zÞ � exp � a2RTðxþ zÞ2

h i¼ exp½�bðxþ zÞ2� ð26Þ

where a and b are constants.Eq. (9) now reads:

sðxÞ ¼Z b

apðzÞ exp½�bðxþ zÞ2�½sðzÞ��1dz: ð27Þ

In order to expand the kernel we note the identity

exp½�bðxþ zÞ2� ¼ expð�bx2Þ � expð�bz2Þ � expð�2bxzÞ ð28Þ

and the Maclaurin Series representation

expð�2bxzÞ ¼Xn

k¼0

ð�2bzÞk

k!xk þ Rn ð29Þ

with the property

8ðx; zÞ 2 R2; limn!1

Rn ¼ 0: ð30Þ

This yields the integral equation:

pðnÞðxÞ ¼Z b

a

Xn

k¼0

ð�2bzÞk

k!� xk � expð�bx2Þ � expð�bz2Þ � pðzÞ

" #� ½pðnÞðzÞ��1dz: ð31Þ

From Eqs. (22) and (23), it therefore follows that the solution of Eq.(31) is:

pðnÞðxÞ ¼ expð�bx2Þ �Xn

k¼0

Ak � xk ð32Þ

with coefficients Ak:

Ak ¼ð�2bÞk

k!

Z b

a

zkpðzÞPnm¼0Am � zm dz: ð33Þ

Since Eq. (30) holds, the solutions of Eq. (31) approximate thesolution of the COSMO-RS integral equation (27) up to arbitraryprecision.

For the solution of Eq. (8) with the energy expression (26) wenow have:

yðxÞ � yðnÞðxÞ ¼ RTbx2 � RT lnXn

k¼0

Ak � xk

" #: ð34Þ

Writing Eq. (34) as

yðnÞðxÞ ¼ RTbx2 � RT � n

� ln½jxj� þ ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA0

xnþ A1

xn�1 þA2

xn�2 þ � � �An

���� ����n

s" #( )ð35Þ

it is clear that for not too small arguments x the solution of Eq. (31)must always have the form of a parabola, as hypothesised by Klamt[1].

3. Results and discussion

3.1. Test calculations for a synthetic r-profile

Having presented the general solution of the COSMO-RS inte-gral equation and the discussion of the particular solution forthe electrostatic misfit energy expression, we shall now reporton numerical investigations for a synthetic r-profile to illustratethe performance of our approach and to give a benchmark. Inpractical calculations with equations of type (8) it is necessaryto specify a r-profile and an energy expression. Here we willfocus on the electrostatic misfit energy (26) with the constantb = 1. The molar gas constant R is given as 0.00831451 kJ/(mol K) and the temperature T = 298.15 K. Note we shall omitthe units of x, z, b, R and T in this Section. Hence the functionsp(n)(x) and y(n)(x) are dimensionless. The integration is carriedout for the lower limit a = �3 and the upper limit b = 3. As syn-thetic r-profile we employ the piecewise defined analyticalfunction:

pðzÞ�0 z<�2exp½�ð5zþ2:5Þ2�þ 1

25z2þ1þ½sinð5zþ2:5Þ�2

ð5z�2:5Þ4þ1þqð5zÞ �26 z<2

0 z P 2

8><>:ð36Þ

where

qðzÞ �0 z < 7�ðz� 7Þ � ðz� 9Þ 7 6 z < 90 z P 9

8><>: ð37Þ

The graph of the function (36) is depicted in Fig. 2. Althoughat first glance the shape of our fictitious profile looks like a realworld example we have constructed it primarily in order to con-sider a non-trivial test case. In our day-to-day work with COS-MO-RS we have never found r-profiles with tails far beyondabsolute values of 2 e/nm2 for neutral species. Therefore we haverestricted our test function to this interval, and have designedthe relative maximum at 1.6 to take a value of more than 1.0and the pronounced fine structure in the interval of �0.8 to+0.8 in order to illustrate the performance of our approach inpractical calculations. Typically the r-profiles of most organiccompounds have narrow central peaks near the origin and,depending on the occurrence of polar functional groups, shallowsatellites with maxima of approximately 0.2 nm4/e at screeningcharge densities in the intervals �2 e/nm2 to �1 e/nm2 and1 e/nm2 to 2 e/nm2 (see Fig. 1). We therefore expect our syn-

Fig. 3. Approximate solutions of the COSMO-RS integral equation for the synthetic

Fig. 2. Synthetic r-profile according to Eq. (36).

114 R. Franke, J. Friedrich / Chemical Physics 356 (2009) 110–120

thetic r-profile to be a challenging test case. Suggesting that p(z)in Eq. (36) is given in the units nm4/e and z in the units e/nm2,the integral

R 3�3 pðzÞdz has the value 1.81 nm2. The surface area of

n-octane is 1.97 nm2; according to this criterium our syntheticr-profile could therefore be that of a molecule. As we discussedin Section 2.1, the coefficients Ak of the solution (34) of theapproximating integral equation are determined by solving anon-linear system of equations. This is equivalent to findingthe fixed point of a mapping U, which in the case of Eq. (31)can be given as

U : Rnþ1 ! Rnþ1;

A0

..

.

An

0BB@1CCA#

R ba

pðzÞPn

m¼0

Am �zmdz

..

.

ð�2bÞnn!

R ba

znpðzÞPn

m¼0

Am �zmdz

0BBBBBBB@

1CCCCCCCA: ð38Þ

The straightforward implementation for determining the fixedpoint is based on the following iterative algorithm:

Step 0: Choose � > 0:Choose initial value

Að0Þ0

..

.

Að0Þn

0BB@1CCA 2 Rnþ1:

Set k ¼ 0:

Step 1: Calculate

Aðkþ1Þ0

..

.

Aðkþ1Þn

0BB@1CCA ¼ U

AðkÞ0

..

.

AðkÞn

0BB@1CCA

0BB@1CCA:

If jAðkþ1Þk � AðkÞk j 6 � for all nþ 1 coefficients, then terminate the

algorithm.Otherwise, let k kþ 1 and return to Step 1.Numerical experiments revealed that the iteration function U

does not converge to a fixed point. Since the choice of an itera-tion function is not unique this does not necessarily imply thatthe fixed point iteration method fails to solve the problem. Wefound that an algorithm which makes use of the iterationfunction

W : Rnþ1 ! Rnþ1;

A0

..

.

An

0BB@1CCA#

12

A0

..

.

An

0BB@1CCAþU

A0

..

.

An

0BB@1CCA

26643775 ð39Þ

is very robust and never failed to converge. We always choose thevector (1 1� � �1) as initial value for the fixed point. A formal analysisof Eqs. (38) and (39) is given in Appendix A.

Fig. 3 shows the improvement of the solution y(n)(x) and (inset)the graph of the improvement of p(n)(x) for the synthetic r-profile(36) with increased expansion lengths n + 1. The correspondingcoefficients are documented in Table 1. To quantify the quality ofthe approximation, we define the defect function ~yðnÞðxÞ

~yðnÞðxÞ � �RT lnZ b

apðzÞ exp

yðnÞðzÞRT

� bðxþ zÞ2� �

dz

" #ð40Þ

which is generated by evaluating the right hand side of Eq. (8) forthe approximated solution y(n)(x). To illustrate the deficiencies ofthe approximated solution it is convenient to consider the function

DðnÞðxÞ � ~yðnÞðxÞ � yðnÞðxÞ ð41Þ

which we shall refer to as the difference function. This is displayedin Fig. 4. The solution for n = 28 yields values for the difference func-tion which are smaller than 1 � 10�11 in the interval between �1.7and 0.9. It has a value of 3 � 10�6 for x = �3 and a value of�4.6 � 10�4 at x = 3. A useful way to quantify the quality of anapproximate solution is to determine the distance between the de-fect function ~yðnÞðxÞ and y(n)(x) given by

dN ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ b

aðDðnÞðxÞÞ2dx

sð42Þ

which is zero for the exact solution of the integral equation. Valuesfor dN are documented in Table 1.

The target quantity of the COSMO-RS model is the chemical po-tential ~l, given by the integral

R ba pðxÞyðxÞdx. We immediately see

that the precision of a calculation of ~l depends only on the accu-racy of the solution y(n)(x) in the interval where the r-profile p(z)is non-zero. Since the error of y(n)(x) increases as x tends to theendpoints of the interval [a,b] the accuracy of the calculated chem-ical potential depends on the behaviour of the r-profile at largeabsolute values of x. The main sources of the relatively slow con-vergence of the chemical potential reported in Table 1 are the largevalues of the synthetic r-profile around x = 1.6 and the error ofy(n)(x) for x-values greater than 1.5. An expansion length of 13 isneeded to yield an accuracy of one decimal place for the chemicalpotential. The value calculated for y(32)(x) is accurate to seven dec-imal places. In order to find the limit of the chemical potential wehave performed a series of calculations up to n = 200. Thecorresponding value is documented in Table 1.

r-profile.

Table 1Chemical potential and solution y(n)(x) of the COSMO-RS integral equation employing the synthetic r-profile (36) for different expansion lengths n + 1.

n 2 8 12 22 28 n ?1a

~l 0.76351583 0.80032276 0.92500345 0.93824382 0.93824397 0.93824399dN 10.25403623 4.70405035 3.88887062 0.03382975 0.00012559A0 1.2365838339E+00 1.2766596723E+00 1.3221755428E+00 1.3301420632E+00 1.3301421615E+00A1 �6.6105892459E�01 �8.7960528297E�01 �1.1294221592E+00 �1.1713966955E+00 �1.1713972235E+00A2 1.0033270303E+00 1.1500979925E+00 1.4676122065E+00 1.5237143646E+00 1.5237150876E+00A3 �8.1312520865E�01 �1.1692794941E+00 �1.2326160935E+00 �1.2326169307E+00A4 6.2196778618E�01 8.9930238956E�01 9.4989488283E�01 9.4989557170E�01A5 �3.6510852513E�01 �5.4768271629E�01 �5.8142742809E�01 �5.8142790140E�01A6 1.9490389837E�01 2.9322783036E�01 3.1178600317E�01 3.1178627206E�01A7 �8.7638820679E�02 �1.3402894671E�01 �1.4291471289E�01 �1.4291484600E�01A8 3.5649947571E�02 5.4619160122E�02 5.8328685150E�02 5.8328742743E�02A9 �1.9721427756E�02 �2.1112808917E�02 �2.1112831331E�02A10 6.4871547184E�03 6.9558694606E�03 6.9558773148E�03A11 �1.9335269159E�03 �2.0784802040E�03 �2.0784827330E�03A12 5.3408101305E�04 5.7509956312E�04 5.7510031029E�04A13 �1.4648462424E�04 �1.4648483013E�04A14 3.5004517009E�05 3.5004569744E�05A15 �7.7806272936E�06 �7.7806400298E�06A16 1.6394010488E�06 1.6394039348E�06A17 �3.2369506472E�07 �3.2369568647E�07A18 6.1108952037E�08 6.1109078596E�08A19 �1.0868409598E�08 �1.0868434246E�08A20 1.8620935548E�09 1.8620981149E�09A21 �3.0168513586E�10 �3.0168594710E�10A22 4.7406586670E�11 4.7406724345E�11A23 �7.0618117985E�12A24 1.0268445373E�12A25 �1.4174192155E�13A26 1.9216846145E�14A27 �2.4744988796E�15A28 3.1484672554E�16

a See text.

Fig. 4. Difference functions of the approximate solutions of the COSMO-RS integralequation.

Fig. 5. Difference function of the approximate solution with expansion lengthn = 40 and of the solution with 120 additional coefficients.

R. Franke, J. Friedrich / Chemical Physics 356 (2009) 110–120 115

Finally we shall report in this section on a technique which al-lows the refinement of the solution without solving the non-lin-ear system of equations. If one is interested in an extremelyaccurate y(n)(x), especially for values of x far from the origin, itis necessary to perform calculations with large expansion lengthsn + 1 which may be very time-consuming and have numericalinstabilities which are difficult to control. To illustrate theremaining deficiencies of calculations for y(n)(x) Fig. 5 displaysthe graph of the difference function for a calculation withn = 40. Its value is smaller than |1.0 � 10�11| in the interval[�3,2.3] and increases up to �4.7 � 10�10 at x = 3. The distancebetween the defect function and the solution is 6.7 � 10�5.Fig. 5 also shows the behaviour of y(40)(x) for values of |x| greater

than 3. We see a sharp increase of the defect function for n = 40 atthe endpoints of the interval [�3,3]. Note that the domain of y(x)is R and that the difference function of the exact solution is zeroon R. The solution can be improved by calculating N–n additionalcoefficients �Ak to yield

yðxÞ � Y ðNÞðxÞ ¼ RTbx2 � RT lnXn

k¼0

Ak � xk þXN

k¼nþ1

�Ak � xk

" #ð43Þ

The ith coefficient �Ai is calculated by

�Ai ¼ð�2bÞi

i!

Z b

a

zipðzÞPnm¼0Am � zm dz ð44Þ

Table 2Dependency on the expansion length n + 1 of chemical potentials in kJ/mol and thecorresponding logarithms of infinite dilution activity coefficients for the system 1-butyne/n-butane at 298.15 K.

n ~l1-butyne1-butyne

~ln-butane1-butyne

~ln-butanen-butane

~l1-butynen-butane ln ~c1n-butane

1-butyne ln ~c11-butynen-butane

2 8.5397 3.4619 2.0997 10.1075 0.6324 0.54954 8.2132 3.6497 2.0989 10.0664 0.7476 0.62556 8.1753 3.6698 2.0989 10.0653 0.7624 0.63378 8.1720 3.6714 2.0989 10.0652 0.7637 0.634310 8.1719 3.6715 2.0989 10.0652 0.7638 0.634412 8.1719 3.6715 2.0989 10.0652 0.7638 0.634414 8.1719 3.6715 2.0989 10.0652 0.7638 0.6344

116 R. Franke, J. Friedrich / Chemical Physics 356 (2009) 110–120

where the n + 1 coefficients Am are those determined by solving Eq.(38). The graph of the difference function for the solution (43) inwhich the 41 coefficients are augmented by 120 coefficients(according to Eq. (44)), is displayed in Fig. 5. It can be seen thatthe solution is considerably improved, showing a value of1.86 � 10�8 for the difference function at x = 6 while the corre-sponding value for y(40)(x) is �0.03 (not shown in Fig. 5).

3.2. Calculation of infinite dilution activity coefficients for molecularsystems

In these final sections we report on a selection of pilot calcula-tions for molecular systems. The underlying density functional the-ory (DFT) calculations employing the COSMO solvation model [2]were performed with the TURBOMOLE [8] suite of programs usingthe functional BP86 and, for all atoms, a valence triple zeta basisincluding one set of polarisation functions [9]. The r-profiles weregenerated from the raw data provided by TURBOMOLE using theCOSMOtherm program [5]. Except where otherwise indicated, allreported calculations of thermodynamic data were calculatedusing our own code. In calculations of activity coefficients withCOSMOtherm we used the parameter file BP_TZVP_C12_0102.

Following Ref. [3] we rely our calculations on an integral equa-tion for the chemical potential of a surface segment l(r) with aver-age molecular contact area aeff given as

lðrÞ ¼ � RTaeff

lnZ b

aPð~rÞXðr; ~rÞ exp

aeff � lð~rÞRT

� �d~r

" #: ð45Þ

An augmentation of the energy expression to improve thedescription of hydrogen bonding using an expression like Eq. (3)leads to a kernel which is very tedious to handle. Since the devel-opment of an alternative term in the energy expression is beyondthe scope of this paper we account only for the electrostatic misfitenergy:

Xðr; ~rÞ ¼ exp � aeff � a02RT

ðrþ ~rÞ2�

: ð46Þ

Pð~rÞ in Eq. (45) denotes the normalized r-profile defined as

Pð~rÞ � pð~rÞR ba pð~rÞd~r

: ð47Þ

The discrete r-profile provided by the COSMOtherm program istransformed into an approximate analytic function by using a linearspline interpolation. Eqs. (45) and (46) contain the parameters aeff

and a0 which are determined from fitting experimental data [3].In the calculations reported below, we have used a value of0.071 nm2 for aeff [3] and the completely ad hoc parametera0 = 72.72781616 kJ�nm2

mol�e2 . In all cases the temperature wasT = 298.15 K. With the solution of Eq. (45) we obtain for the residualpart of the chemical potential (5) of a molecule i in the solvent j

~lij ¼

Z b

aPiðrÞ � lðnÞj ðrÞdr

¼ RTaeff

aeff � a02RT

�Z b

aPiðrÞ � r2dr�

Z b

aPiðrÞ � ln

Xn

k

Ajk � rk

" #dr

( ):

ð48Þ

and for the logarithm of the infinite dilution activity coefficientwithout combinatorial contribution ~ci1

j :

ln½~ci1j � �

~lij � ~li

i

RT¼ 1

aeff

Z b

aPiðrÞ � ln

Xn

k

Aik � rk

" #dr

(

�Z b

aPiðrÞ � ln

Xn

k

Ajk � rk

" #dr): ð49Þ

For activity coefficients at finite concentrations we have to employthe corresponding r-profile of the ensemble (1) instead of that ofthe pure component.

3.3. Convergence

The convergence behaviour of calculations for the chemical po-tential depends on the r-profile of solute i and the r-potential ofsolvent j, which is given by the solution of the COSMO-RS integralequation (see Eq. (48)). Table 2 shows the results of a series of cal-culations for the binary system 1-butyne/n-butane.

Before we discuss these, let us consider the r-profiles in Fig. 1.The distribution functions for n-butane and 1-butyne clearly differqualitatively, with n-butane being the less polar compound. This isreflected in the narrow screening charge distribution around zero.The peak at positive charge densities corresponds to the carbonatoms. The peak at negative charge densities can be assigned tothe hydrogen atoms. The r-profile of 1-butyne is much broaderthan that of n-butane. It shows relative maxima around+0.2 e/nm2 and +0.9 e/nm2 corresponding to the sp-carbon atomsand the electron density of the two p-bonds, and a relativemaximum at �1.2 e/nm2 which is caused by the positively charged(acidic) hydrogen atom bonded to the sp-carbon. This chargeseparation – a consequence of the high s-character of the carbonatoms of the triple bond – accounts for the relatively high acidityand polarity of 1-butyne compared to n-butane.

For the chemical potential of n-butane in itself rapid conver-gence rates are observed, leading to a value for n = 4 which is accu-rate to five decimal places (although only four are documented inTable 2). The fast convergence rate results from the narrow r-pro-file of n-butane centered at the origin. The distance between thedefect function and the r-potential y(4)(r) for n-butane is3 � 10�4 in the interval [�0.6 e/nm2,0.5 e/nm2] indicating a veryaccurate solution in the interval where the r-profile is non-zero.

To yield an accuracy of four decimal places for the chemicalpotential of 1-butyne in n-butane an expansion length of nine isnecessary because the r-profile of 1-butyne is non-zero inthe interval [�1.5 e/nm2,1.2 e/nm2] with relative maxima at�1.2 e/nm2 and 0.9 e/nm2. This requires a considerably more accu-rate solution for the r-potential of n-butane.

In the case of the chemical potential of 1-butyne in itself anexpansion length of 11 is needed to reach this precision. For valuesof the logarithm of infinite dilution activity coefficients accurate tofour decimal places an expansion length of 11 is necessary; in thecase of two decimal places n = 6 is required.

These values are representative for relatively unpolar moleculessuch as saturated and unsaturated hydrocarbons. In the class ofpolar neutral molecules, water shows one of the most extendedr-profiles (see Fig. 1). Table 3 documents investigations into theconvergence properties of our approach for the binary systemn-hexane/water. Following Ref. [4] we have not employed ther-profile of water in our calculations exactly as it was constructed

Fig. 7. Approximate solutions of the COSMO-RS integral equation for water.

R. Franke, J. Friedrich / Chemical Physics 356 (2009) 110–120 117

from the raw data of the COSMO-calculation with TURBOMOLE,but have scaled the r-profile by a factor of 0.6. A similar ad hocmodification is used in the COSMOtherm program [4]. For thechemical potentials in n-hexane we found a rapid convergencesimilar to the case of n-butane. To yield an accuracy of four decimalplaces for water we need an expansion length of 11 while n = 8 wassufficient for 1-butyne in n-butane due to the considerably lessextended r-profile in the case of 1-butyne. In the case of waterin n-hexane, the integrand in Eq. (48) PiðrÞlðnÞj ðrÞ shows two sharppeaks at �1.60 e/nm2 and 1.85 e/nm2 respectively (see Fig. 6).Therefore the r-potential of n-hexane must be calculated withhigh precision at least in the interval [�2 e/nm2,2.2 e/nm2] to yieldaccurate results for the chemical potential of water in hexane.

The values for the chemical potential in water show a slow con-vergence for both solutes although their r-profiles are totally dif-ferent (see Fig. 1). An expansion length of 21 is needed to reachan accuracy of four decimal places. Fig. 7 displays the r-potentiall(n)(r) for water up to n = 14. It can be seen that the solutions ofthe integral equation for expansion length smaller than 11 differsignificantly from those for n = 14 in the whole interval [�2.1 e/nm2,2.2 e/nm2] where the r-profile of water is non-zero. The devi-ation is less pronounced near the origin than at high absolute val-ues of r but an inspection of the inset graph of Fig. 7 reveals that itis still substantial. The distance between the defect function andthe r-potential l(14)(r) is 2.74 while it is only 0.002 for n-hexane.

Table 3Dependency on the expansion length n + 1 of chemical potentials in kJ/mol and thecorresponding logarithms of infinite dilution activity coefficients for the systemwater/n-hexane at 298.15 K.

n ~lwaterwater ~ln�hexane

water ~ln-hexanen-hexane

~lwatern-hexane ln ~c1n-hexane

water ln ~c1watern-hexane

2 26.2790 28.1026 2.0091 34.3456 10.5259 3.25404 19.8201 31.6087 2.0085 33.9760 11.9405 5.71046 17.3125 32.0343 2.0085 33.9557 12.1122 6.71388 16.4367 31.7532 2.0085 33.9552 11.9988 7.066810 16.1731 31.5336 2.0085 33.9551 11.9102 7.173112 16.1062 31.4830 2.0085 33.9551 11.8898 7.200114 16.0927 31.4868 2.0085 33.9551 11.8914 7.205616 16.0905 31.4914 2.0085 33.9551 11.8932 7.206418 16.0903 31.4928 2.0085 33.9551 11.8937 7.206520 16.0902 31.4931 2.0085 33.9551 11.8939 7.206522 16.0902 31.4931 2.0085 33.9551 11.8939 7.206524 16.0902 31.4931 2.0085 33.9551 11.8939 7.2065

Fig. 6. The function PwaterðrÞ � lð24Þn-hexaneðrÞ.

Fig. 8. Difference functions of approximate solutions of the COSMO-RS integralequation for water.

Fig. 8 displays the difference function for the r-potential ofwater. It shows steep increases for increasing absolute values ofr with a minimum near the origin. While the slow convergencein the case of water is mainly due to the relatively large errors ofthe r-potential near the shallow maxima of the r-profile of waterat �1.6 e/nm2 and 1.8 e/nm2 the slow convergence for the chemi-cal potential of n-hexane in water has its origin in the deficienciesof the r-potential at small absolute values of r due to the sharppeaked r-profile of n-hexane. Therefore a very accurate r-poten-tial for water near the maximum of the r-profile at �0.1 e/nm2 isnecessary for high precision values of the chemical potential ofn-hexane in water. For values of the logarithm of infinite dilutionactivity coefficients accurate to four decimal places an expansionlength of 21 is necessary; in the case of two decimal placesn = 14 is required.

High precision calculations are not usually necessary in practi-cal applications. In view of the approximations in the COSMO-RStheory and the error bars of experimental activity coefficients atinfinite dilution, it is in most cases sufficiently accurate to employan expansion length of 9. This typically leads to an intrinsic error of1% for values of the logarithm of the infinite dilution activity coef-ficients for polar systems. For non-polar systems the error is signif-icantly smaller. For systems involving only hydrocarbons anexpansion length of seven can be used, producing intrinsic errorsof around 0.2%.

118 R. Franke, J. Friedrich / Chemical Physics 356 (2009) 110–120

3.4. Infinite dilution activity coefficients in water and n-hexane

In this section we compare activity coefficients in infinite dilu-tion for a series of solutes in water and in n-hexane calculated withour own approach with those from calculations with the COS-MOtherm program and with experimental data. In all calculationswith our own code we used an expansion length of nine. The com-binatorial part of the chemical potential has been calculated usingEq. (6) where the parameters c0, c1 and c2 have been taken fromRef. [4]. Thus differences between the calculations with our ap-proach and those with COSMOtherm originate solely from ~l, theresidual part of the chemical potential. All experimental data havebeen taken from the DECHEMA database. The calculated logarith-mic infinite dilution activity coefficients together with the corre-sponding experimental results are listed in Table 4.

It will be observed that in almost every case COSMOtherm yieldsbetter predictions than our own approach. In particular, for activitycoefficients in water COSMOtherm, with its high-quality paramet-risation and its ability to account for the correlation of screeningcharge densities across different segments, yields very good resultswhich are superior to or not very different from the best predic-tions of empirical group contribution models. Our own approachgives poor results for some of the aqueous systems. However, asmentioned above, in contrast to COSMOtherm, we do not hereaccount for contributions to hydrogen bonding beyond the electro-static misfit energy. Therefore our calculations predict excessivelylarge values for the chemical potentials ~li

j and ~lii of protic polar

molecules. In mixtures of protic polar and aprotic polar speciesthe chemical potential ~li

j is systematically overestimated. If thehydrogen bonding-interactions in the solute–solvent mixturediffer significantly from those in the solute, as they do in alco-hol–water mixtures, large errors of calculated infinite dilutionactivity coefficients will be observed; this can be seen for the sys-tem 1-propanol/water in Table 4. Since no cancellation of errorswill occur if we calculate the infinite dilution activity coefficientfor a polar aprotic solute in a polar protic solvent very large devi-ations might occur in these cases. As an example we have docu-mented in Table 4 the logarithmic activity coefficient for acetonein water. For less polar solutes such as nitromethane and diethyl-ether, the strength of hydrogen-bonding is less pronounced; there-fore the deviations of the predicted activity coefficients using ourapproach diminish. Nevertheless they remain large in comparisonto the errors for those systems where no hydrogen-bonding inter-actions take place.

Because we systematically calculate too large ~lii our predictions

for infinite dilution activity coefficients of protic polar species inaprotic unpolar solvents should be poorer than those for aproticpolar and aprotic unpolar molecules in the same solvent. Thiscan be seen in the case of 1-propanol/n-hexane. The analogous er-ror for the system water/n-hexane is not visible due to the ad hoc

Table 4Infinite dilution activity coefficients of organic compounds in water and n-hexane at 298.

Solute i ln ci1water

COSMOtherm Own approach Experim

Water 0.00 0.00 0.001-Propanol 3.32 7.41 2.60Acetone 1.66 6.93 1.95Nitromethane 4.13 5.54 3.45Diethylether 5.89 9.00 4.231-Butyne 6.62 7.97 6.371-Butene 8.09 9.83 8.53n-Butane 9.44 10.23 9.99n-Hexane 12.25 10.91 12.791-Hexene 10.89 9.76 11.93Benzene 7.96 8.46 7.82

scaling of the r-profile of water. For all other solutes in n-hexanewe get reasonable results which are slightly poorer than those cal-culated with COSMOtherm, with the exception only of n-butanewhere both approaches give identical values.

3.5. Infinite dilution activity coefficients for C4 components

In industrial chemistry, research into conversion of inexpensiveolefins for monomer manufacture, oligomerisation and polymeri-sation is an important area. Olefins are virtually absent in fossilfuels and must be manufactured in cleavage or cracking processes.Depending on process design and conditions, naphtha crackingyields different ratios of the industrially most important C2, C3

and C4 olefins. In the case of the C4 fraction obtained from crackingof hydrocarbons, its amount and composition depends on the feed-stock, the type of cracking process and the severity of cracking con-ditions. Since the C4 fraction cannot be separated into itscomponents economically by simple distillation, sophisticatedphysical and chemical separation processes must be developed.An essential part of modern process synthesis relies on computa-tional models. In these models the physical and chemical proper-ties of the species involved are the most important inputparameters. An experimental determination would often be expen-sive and time-consuming; reliable computational methods, espe-cially those which enable the a priori prediction of properties,therefore offer an interesting alternative. We have successfully em-ployed COSMO-RS for the prediction of infinite dilution coefficientsin order to describe liquid–liquid phase equilibria which controlextraction procedures. In this section we shall report on calcula-tions in the field of C4 chemistry to illustrate a scheme for decou-pling the solution of the COSMO-RS integral equation from thecalculation of infinite dilution activity coefficients.

Expanding the logarithmic terms on the right hand side of (49)in a Taylor series

lnXn

k

Ajk;r

k

" #�Xm

j

~Ajjr

j ð50Þ

yields the approximation

ln½~ci1j � � ln½Ci1

j � ¼1

aeff

Xm

j

~AijMi

j �Xm

j

~AjjMj

j

!ð51Þ

where

Mkj �

Z b

apkðrÞ � rjdr ð52Þ

is the so-called jth r-moment of component k, introduced in Ref.[3]. Our approach makes it easy to obtain approximate activitycoefficients Ci

j. Once the COSMO-RS integral Eq. (8) has been solvedfor single molecules or a complex mixture (i.e. for particular

15 K.

ln ci1n-hexane

ent COSMOtherm Own approach Experiment

7.60 6.73 7.414.32 1.84 3.761.76 1.31 1.903.70 3.39 3.680.32 0.240.90 0.600.10 0.09�0.02 �0.02 0.04

0.00 0.00 0.000.09 0.06 0.100.68 0.42 0.75

Table 5Infinite dilution activity coefficients of selected components of a C4 fraction at 298.15 K.

Solvent j Solute i n-Butane 1-Butene 1,3-Butadiene 1-Butyne Butenine

n-Butane ~ci1j , own approach 1.00 1.14 1.48 2.14 2.38

Ci1j , own approach 1.00 1.14 1.48 2.14 2.41

~ci1j , COSMOtherm 1.00 1.14 1.61 2.52 2.98

1-Butene ~ci1j , own approach 1.13 1.00 1.07 1.27 1.25

~Ci1j , own approach 1.13 1.00 1.07 1.26 1.26

~ci1j , COSMOtherm 1.12 1.00 1.11 1.41 1.48

1,3-Butadiene ~ci1j , own approach 1.48 1.07 1.00 1.06 1.04

Ci1j , own approach 1.48 1.07 1.00 1.05 1.04

ci1j , COSMOtherm 1.61 1.11 1.00 1.08 1.09

1-Butyne ci1j , own approach 1.89 1.24 1.05 1.00 0.94

Ci1j , own approach 1.89 1.23 1.05 1.00 0.96

ci1j , COSMOtherm 2.31 1.36 1.06 1.00 0.95

Butenine ci1j , own approach 2.05 1.34 1.03 0.94 1.00

Ci1j , own approach 2.05 1.34 1.03 0.94 1.00

ci1j , COSMOtherm 2.87 1.46 1.07 0.95 1.00

R. Franke, J. Friedrich / Chemical Physics 356 (2009) 110–120 119

r-profiles), yielding n + 1 coefficients Ak, we only need the m + 1corresponding r-moments Mj in order to calculate the approximatechemical potentials and activity coefficients for all possible combi-nations of pure components or their mixtures. These calculationscan easily be carried out with a spreadsheet program since neitherthe solution of an integral equation nor the calculation of integralsis necessary.

Results calculated for infinite dilution activity coefficients at298.15 K for a subset of components of the C4 fraction are listedin Table 5. To the best of our knowledge no experimental valuesfor these infinite dilution activity coefficients can be found in theliterature. We employed an expansion length of 7. For the repre-sentation of the logarithmic terms according to Eq. (50) m = 6 hasbeen chosen. In the calculations with COSMOtherm we switchedoff the calculation of the combinatorial contribution to the chemi-cal potential by using the option combi. With few exceptions, thepredictions from our own approach and COSMOtherm show onlyinsignificant deviations, with COSMOtherm predicting significantlylarger values for the mixtures of n-butane in 1-butyne, butenine in1-butyne, 1-butyne in n-butane, and 1-butyne in butenine. Thelargest difference is found when considering the infinite dilutionactivity coefficient for n-butane in butenine where the value ofCOSMOtherm is 28% larger than the one calculated with our ownapproach. On the other hand the approximate activity coefficientsCi1

j calculated according to our approach show only very small er-rors when compared with those calculated from Eq. (49). Here theworst case percentage error amounts to 2%.

4. Conclusions

It has been shown that the central equation of COSMO-RS be-longs to the class of non-linear Hammerstein integral equations.The kernel of this integral equation is non-degenerate. We havederived an approximate integral equation with degenerate kernelwhich models the COSMO-RS equation with electrostatic misfitenergy up to arbitrary precision. The most significant practicalachievement of our approach is the provision of a simple equa-tion for the chemical potential as a function of the screeningcharge density r, the r-potential. A detailed description andanalysis of an algorithm to determine this function as well asbenchmark calculations for a model problem have been pre-sented. The method has been tested for a representative subsetof molecular systems and applied successfully for the predictionof infinite dilution activity coefficients in cases where onlyaprotic species are involved. The functional form of the hydro-gen-bonding term of the energy expression as published to dateappears to be inadequate for an elegant treatment with ourapproach for solving the COSMO-RS integral equation. However,

we believe that our approach could be extended to calculate ther-potential of protic polar systems through the development of anew energy expression for the description of hydrogen-bondingbeyond electrostatic misfit. Since the existing form of the expres-sion is purely heuristic there is no a priori argument againstalternative expressions. An application in the field of C4 chemis-try is presented to illustrate the possibility of calculating approx-imate infinite dilution coefficients.

Appendix A. Analysis of Eqs. (38) and (39)

Preliminaries

We rewrite Eq. (31) as

sðxÞ ¼Z b

a

X1k¼0

"pðzÞ � ½sðzÞ��1 � zk � expð�bz2Þ � xk

� expð�bx2Þ � ð�2bÞk

k!

#dz ðA1Þ

and assume that:

sðzÞ and pðzÞ are continuous on ½a,b� ðA2aÞ8z 2 R sðzÞ > 0: ðA2bÞ

Therefore the subsequent estimation holds

jpðzÞj � j½sðzÞ��1j � j expð�bz2Þj � j expð�bx2Þj � ð�2b � x � zÞk

k!

����������

6

���pðzmaxÞ��� � ½sðzminÞ��1��� ��� � 1 � 1 � 2k � j � bjk � jxjk � ½maxða; bÞ�k

k!ðA3Þ

The rhs of (A3) can be written as C � ðD�2�b�jxjÞk

k!where C and D are

constants. Furthermore we have the identity

C �X1k¼0

ðD � 2 � b � jxjÞk

k!¼ C � expðD � 2 � b � jxjÞ 8x 2 R:

From Weierstraß’ approximation theorem follows that the series in(A1) is uniformly convergent on [a,b]. From assumptions (A2) wededuce that all terms of (A1) are continuous on [a,b]8x 2 R ^ 8k 2 N0. Hence it furthermore follows that we can inter-change the integration and summation in (A1) to give

sðxÞ ¼ expð�bx2Þ

�X1k¼0

ð�2bÞk

k!�Z b

apðzÞ � ½sðzÞ��1 � zk

expðbz2Þdz

" #� xk ðA4Þ

120 R. Franke, J. Friedrich / Chemical Physics 356 (2009) 110–120

which is equivalent to

sðxÞ ¼ expð�bx2Þ �X1k¼0

ð�2bÞk

k!� ak � xk ðA5Þ

where we denote

ak ¼Z b

azk � pðzÞ � expð�bz2Þ

sðzÞ dz

which is equivalent to

8k 2 N0 ak ¼Z b

azk � pðzÞP1

l¼0ð�2bÞl

l!� al � zl

dz: ðA6Þ

From (A2) we have that the seriesP1

k¼0ð�2bÞk

k!� ak � zk is continu-

ous and positive-valued for all z 2 R.

Remarks on the convergence using mappings U and W

We want to determine the fixed point of

U : RN0 ! RN0 ; hXkik2N0#

Z b

azk � pðzÞP1

l¼0ð�2bÞl

l!Xl � zl

dz

* +k2N0

ðA7Þ

Let hakik2N0be a fixed point of the mapping U and d 2 Rþ it is

U d � hakik2N0

�¼ U hd � akik2N0

�¼ 1

d�Z b

azk � pðzÞP1

l¼0ð�2bÞl

l!al � zl

dz

* +k2N0

¼ 1d� hakik2N0

ðA8Þ

and it follows:

UðUðd � hakik2N0ÞÞ ¼ d � hakik2N0

ðA9Þ

Since (A7) holds for all d 2 Rþ specially for those in an arbitraryvicinity of 1, there exists no vicinity of hakik2N0

where convergenceis guaranteed.

The modified mapping

W : RN0 ! RN0 ; hXkik2N0#

12ðhXkik2N0

þUðhXkik2N0ÞÞ ðA10Þ

obviously has the same fixed point as U. Furthermore it holds

Wðd � hakik2N0Þ ¼ 1

2� ðd � hakik2N0

þUðd � hakik2N0ÞÞ

¼ 12� dþ 1

d

� �� hakik2N0

ðA11Þ

With f ðdÞ � 12� dþ 1

d

� �it is

WðnÞðd � hakik2N0Þ � WðWð. . . W|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

n times

ðd � hakik2N0Þ . . .ÞÞ

¼ f ðnÞðdÞ � hakik2N0: ðA12Þ

We have now to prove that

limn!1

f ðnÞðdÞ ¼ 1 8d 2 Rþ: ðA13Þ

Since (A13) is trivially true for d = 1 and 1d > 18d 2�0;1½ we have

to prove (A13) "d > 1 where it is

f ðdÞ ¼ 12� dþ 1

d

� �<

12� ðdþ 1Þ ðA14Þ

and moreover it holds 1 ¼ffiffiffiffiffiffiffiffid � 1

d

q� 1

2 � dþ 1d

�8d > 0.

If gðdÞ � 12 � ðdþ 1Þ, then it is

1 � f ðf ðn�1ÞðdÞÞ � gðf ðn�1ÞðdÞÞ � gðgðf ðn�2ÞðdÞÞÞ � � � �� gðnÞðdÞ 8n 2 N0 ðA15Þ

where we have 1 < g(d) < d and 1 < gðnÞðdÞ < gðn�1ÞðdÞ 8n 2 N0.Therefore g(n)(d) is a strictly monotonic decreasing and boundedsequence.

We need to show that limn!1gðnÞðdÞ ¼ 1 8d 2 Rþ ^ d > 1. To thisend we assume limn!1gðnÞðdÞ ¼ k > 1. Now we consider

gðkÞ ¼ 12� ðkþ 1Þ < 1

2� ðkþ kÞ ¼ k ðA16Þ

It follows g(k) = k � e1 with k � e1 > 1 and furthermore it holds

gðkþ e2Þ ¼12� ðkþ e2 þ 1Þ ¼ gðkÞ þ 1

2� e2

¼ k� e1 þ12� e2 8e2 > 0 ðA17Þ

If we suppose 2 e1 > e2 > 0 it follows

gðkþ e2Þ ¼ k� e1 þ12� e2 < k� e1 þ

12� ð2 � e1Þ ¼ k ðA18Þ

Since g(n)(d) is for d > 1 a strictly monotonic decreasingsequence, k > 1 can not be a limit of g(n)(d). This completes theproof of (A13). &

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