Equation-of-State Models and Quantum Mechanics Calculations

Ind. Eng. Chem. Res. 2003, 42, 1495-1507

1495

Equation-of-State Models and Quantum Mechanics Calculations Costas Panayiotou† Department of Chemical Engineering, University of Thessaloniki, 54124 Thessaloniki, Greece

An equation-of-state model is proposed, which is based, in part, on the COSMO-RS method for molecular solvation. Free volume is incorporated into the model by consistently combining the surface contact thermodynamics with basic ideas underlying lattice-gas, lattice-fluid, or latticehole models. It is shown that the COSMO-RS method and Guggenheim’s quasi-chemical method produce identical equations for the basic thermodynamic quantities of fluids and their mixtures. In contrast, however, with the lattice-fluid practice, the proposed model attributes a non-zero misfit energy for the interaction of an empty site with a charged surface segment. A method for the evaluation of the chemical potentials of interacting surfaces in terms of the nonrandomness factors is proposed. The model can be applied to the liquid, gas, and supercritical states and to small molecules as well as macromolecules. Some calculations for normal alkanes and polyethylene are presented. Introduction The widely used group-contribution models, such as UNIFAC,1,2 are often very successful in predicting vapor-liquid phase behavior, mutual solubility of partially miscible mixtures, and the partitioning of a solute in two liquid phases at equilibrium. They suffer, however, from significant inaccuracies when compounds with several strong nonalkyl functional groups are considered. In addition, they are not able to distinguish between isomers and to properly account for intramolecular interactions. In a series of recent publications, Klamt and coworkers3-7 have proposed an entirely new approach for the thermodynamics of liquids. Starting from the solvation of molecules in a conductor, they proposed the so-called COSMO-RS (conductor-like screening model for real solvent) model for the determination of the chemical potential of any species in any liquid mixture from ab initio or quantum mechanical calculations. Lin and Sandler8 have derived a thermodynamically consistent variation of the COSMO-RS model, known as COSMO-SAC (segment activity coefficient) model. Very recently, Klamt et al.7 presented a thermodynamically consistent variation of their activity-coefficient model called COSMOSPACE (COSMO surface pair activity coefficient equation), where they also discuss its relation to Quasi-Chemical theory.9 Although the COSMO model is still in its development and the available database is, relatively, nonextensive, it has already attracted very much interest as being a most promising predictive tool for phase equilibria and related properties. Two key limitations of the COSMO-RS or COSMOtherm model are, at present, on one hand its inability to account for high-temperature and -pressure vaporliquid equilibria, for the supercritical state, and for volume changes on mixing, and on the other hand its inability to properly account for the thermodynamics of polymer systems except for some rather limited cases. There are significant difficulties in improving the model for these limitations. As the critical state is approached, the key picture underlying the model, namely, of †

Phone/Fax: +302310-996223. E-mail: [email protected].

molecules surrounded with (or solvated by) other molecules with no noncontacting surface segments ceases to be valid. In the transition from the liquid to the gas state the molecules will continuously adjust their wave functions to minimize the interaction energy with their neighbors but this is not easy to account for. On the other hand, the size of the macromolecules poses a severe problem in the COSMO-type calculations of the screening charge distributions at their surfaces. Regardless of the progress in solving the above quantum mechanical problems, the thermodynamic approach of the COSMO model will certainly continue to attract the strong interest of chemists and chemical engineers involved in process and product design. The present work does not address the quantum mechanical problem. It attempts instead to augment the capacity of the surface-contact thermodynamic approach by incorporating free volume in the formalism and derive an equation-of-state model that is applicable to the subcritical as well as to the supercritical state, able to account for density variations and volume changes, and applicable to polymer systems. In the next paragraph we will lay down the key assumptions and the basic formalism by combining the COSMO approach3-8 with the lattice-fluid approach.9-15 In the Applications section we will use the derived model for the description of some basic thermodynamic quantities of normal alkanes and of the volumetric behavior of polyethylene. Theory Key Assumptions and Free Volume. Let us consider a system containing k types of molecular segments, namely, n1 segments with charge density σ1, n2 segments with charge density σ2..., and nk segments with charge density σk in COSMO’s sense.3-7 These segments belong to t types of molecules. We keep the dielectric continuum and continuum solvation picture. However, we allow for the existence of neutral empty sites or “holes” of no screening charge, whose number depends on the external T and P conditions. In other words, it is assumed that the system contains n0 segments of holes, which for simplicity are assumed to be of equal size (the reference size), are not collapsible, and most

10.1021/ie0207212 CCC: $25.00 © 2003 American Chemical Society Published on Web 03/11/2003

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Ind. Eng. Chem. Res., Vol. 42, No. 7, 2003

importantly, do not modify the molecular cavities or the mean geometrical characteristics of the molecules. Thus, any encounter of a molecular segment with a “hole” segment will lead to an electrostatic misfit energy3-7 plus a segment-specific term incorporating dispersion interactions and any electrostatic energy loss of the contacting portion of the molecule in the transition from the ideal conductor to the vacuum. An encounter of two segments of empty sites will be considered to lead to a zero energy change. This is of course an oversimplistic picture that attempts to reconcile two drastically different situations: the screened molecule in the liquid and the free molecule in a vacuum. As already mentioned, in the transition from the liquid to the gas state the molecules will adjust their electronic charge distribution to minimize the interaction energy with their neighbors, but we will grossly neglect it, although some dependence with density of the misfit energy of a holemolecule encounter could be implemented in the model. Thus, any contact of two interacting surface segments m and n with charge densities σm and σn , respectively, is characterized by a constant total pair interaction energy pair(σm,σn). Chemical Potential of the Surface Segments. A most important ingredient of the COSMO model is the (statistical) thermodynamics of the independent surface segment interactions. Among other features, this approach provides new self-consistency equations for the chemical potential of interacting segments without using any detailed equation for the partition function of the system. Details of two essentially equivalent procedures may be found in two recent references.7,8 We must, first, apply an analogous procedure to our system to derive the key equations for the chemical potential of the surface segments in the presence of empty sites. In the system we will have a total of 1/2(k + 1)(k + 2) types of interacting segment-segment pairs nij. These numbers of pairs must satisfy the following k + 1 conservation equations:

n10 + 2n11 + ... + n1k ) n1 (1)

By combining the last two equations, we obtain

Z(nq - 2) Z(nq)

[

) exp

]

µ(σm) + µ(σn) kT

(4)

The probability p(σm, σn) is obtained as the ratio of the number of states containing at least one (σm,σn) pair over all states of the system. Thus, we may write8

[

]

pair(σm,σn) Z(nq - 2) ) kT Z(nq) -pair(σm,σn) + µ(σm) + µ(σn) (5) 2b exp kT

p(σm,σn) ) 2b exp -

[

]

with b ) 0 when m ) n and b ) 1 when m * n (since fixed positions). To obtain the chemical potential of segments, we consider the probability of finding pairs having at least one segment with charge density σm or, equivalently, the probability p(σm) of finding a segment with charge density σm, namely, k

∑ n)0

1

2b

p(σm,σn) ) nm0/2 + nm1/2 + ... + 2nmm + ... + nmk/2

)

nT,p nm n0 + n1 + ... + nk

µ(σm) )

{∑ [ k

exp

n)0

nk0 + nk1 + ... + 2nkk ) nk It is nij ) nji; thus, the total number of pairs formed in the mixture are

(2)

Consequently, the probability of an encounter of charges σm and σn is

nmn p(σm,σn) ) nT,p

F(nq) - µ(σm) - µ(σn)

-kT ln

........

1 1 nT,p ) (n0 + n1 + ... + nk) ) nq 2 2

F(nq - 2) ) -kT ln Z(nq - 2) )

) p(σm) (6)

By combining eqs 5 and 6 and rearranging, we obtain for the chemical potential of segments

2n00 + n01 + ... + n0k ) n0 n20 + n21 + ... + n2k ) n2

after removal of the m,n pair

]}

-pair(σm,σn) + µ(σn) kT

+ kT ln p(σm) (7)

In our system we have 1/2(k + 1)(k + 2) unknown pairs nij, k + 1 chemical potentials µ(σm), and the number n0 of segments of empty sites or “holes”. If N0 is the number of empty sites in the system, each having ν0 interacting segments, then

n0 ) ν0N0 The total number of unknowns is then

(3)

Let Z(nq) be the partition function of this ensemble of interacting segments and F(nq) the free energy, or

F(nq) ) -kT ln Z(nq) By attributing a chemical potential to the interacting segments, as COSMO does, we may write for our system

(k + 1)(k + 4) 1 (k + 1)(k + 2) + (k + 1) + 1 ) +1 2 2 These unknowns are related by the k + 1 conservation equations (eq 1) and by the 1/2(k + 1)(k + 2) relations by combining eqs 3 and 5, a total of 1/2(k + 1)(k + 4) equations. We need an additional equation to determine all our unknowns. This additional equation will account for the occurrence of empty sites in the system. We will

Ind. Eng. Chem. Res., Vol. 42, No. 7, 2003 1497

treat this occurrence mainly as a combinatorial problem and, thus, we must return to the chemical potential of molecules. Equation of State and Chemical Potential of Molecules. So far, we dealt with the chemical potential of a charged segment at a fixed position. Of interest, however, in phase equilibrium problems is the chemical potential of the molecule, not of its segments. Let in the system be t types of molecules and let νm,i be the number of segments m in each molecule i, the number of molecules i in the system being Ni. We consider then that the chemical potential of molecular species i consists of two parts, one residual due to the contributions of interactions of its fixed charged segments and the other due to the combinatorial contribution of the molecule as a whole entity as it exchanges positions in the volume of the system. For the first part, and on the basis of eq 7, we assume that we may write

considered to be a measure of the total surface area of molecule i, then the total number of surface segments νi on a molecule i is given by

νi )

∑ νm,i[µ(σm) - kT ln p(σm)],

t

Ω)

∏i

( )

Nr! Nq! i δN i N0!N1!N2!...Nt! Nr!

z/2

(9)

where z is the lattice coordination number. As previously,10-14 we will consider each molecule i to be divided into ri volume segments and to have a total of zqi external contacts. Then

Nr ) N0 + r1N1 + r2N2 + ... + rtNt ) N0 + rN

1 ) qi q0′ Reff R0

(13)

t

qN + N0 Nq 1 + N0 ) ) (14) R0 R0 R0 0

qi

t

i ) 0,t (8)

For the combinatorial part we adopt Guggenheim’s counting procedure,9 which leads to the following expression for the pre-exponential part of the partition function or the number of distinguishable configurations of our system12

eff

qi′ q0′

Niνi ) N∑xiνi + n0 ) n + n0 ) ∑ i)0 i)1

nq )

N

m)0

)

Reff being the size of the surface segments treated in COSMO as an adjustable parameter. For consistency, the coordination number z must coincide with 1/R0 ) v0. With this notation, q0 ) 1 as before.10-15 The total number, nq, of surface segments in the system is given by

k

µRi )

qi′

νm,i ) ∑ R m

xi ∑ i)1 R

xi is the mole fraction of component i in the mixture. In the same way, the number of surface segments of type m in the system is given by t

Niνm,i ∑ i)0

nm )

(15)

Following our previous practice,10-14 we define surface area fractions of type m segments in the mixture as

θm )

nm nm nmR0 qN qN ) R ) ) Θm ) nq Nq 0 qN Nq Nq q

/r Θmv˜ + q/r - 1 ) Θm(1 - θ0) (16)

(10) By defining

and

k

Nq ) N0 + q1N1 + q2N2 + ... + qtNt ) N0 + qN (11)

∑ θm

(17)

θm ) Θmθ

(18)

θ)

m)1

N being the total number of molecules in the system or

eq 16 can be written as

N ) N1 + N2 + ... + Nt For consistency, zqi ) qi′ must correspond to the interacting charged segments of each molecule i. Care must be exercised to ensure this consistency. It is convenient to define at this point the reduced volume and reduced density of the system as follows,

v˜ )

V 1 Nr ) ) F˜ rN V*

With these definitions, θm is the surface area fraction of type m in the real mixture with the empty sites, whereas Θm is the corresponding area fraction in the compact system of the same composition but without the empty sites. The total potential energy of our system is

(12)

where V is the total volume of the real system and V* the corresponding volume of a compact system without any empty sites. It should be clear that appropriate interaction parameters from quantum mechanics calculations, such as the COSMO3-8 calculations, are obtained when the COSMO parametrization is done for this reference compact system (reduced volume near onessufficiently low temperature and/or high pressure). Before proceeding to the equation for the potential energy of our system, it is useful to present the relations between the numbers pertaining to surface segments and the numbers pertaining to molecules. If qi′ is

k

E)

k

∑ ∑ i)0 jgi

k

nijpair(σi,σj) ) k

n0ipair(0,i) + ∑ i)0 k

∑ ∑nijpair(σi,σj) ) E0 + E1 i)1 jgi

(19)

The pressure of the system is obtained as follows:

P ) kT

ln Z (∂ ∂V )

T,N

(∂ ln∂VΩ)

) kT

T,N

(∂E ∂V)

T,N

(20)

This is the equation of state of our system and it is the additional equation needed for determining all our

1498


unknowns. By assigning a reference volume v* to each volume (not surface) segment in the system, or by setting V ) (N0 + rN)v*, and by using eq 12, we may write

(

)

∂ ln Z P ) kT ∂V

(

(

)

kT ∂ ln Z rNv* ∂v˜

(

)

T,N

)

kT ∂ ln Ω ) rNv* ∂v˜ T,N

( )

1 ∂E rNv* ∂v˜ T,N

T,N

(21) Detailed solution of all the above-mentioned equations will permit not only the evaluation of the compositions of the phases at equilibrium but also the free volume distribution in each phase. To proceed to the solution of the above system of equations, it is essential to express the involved probabilities in terms of the area fractions of the surface segments or in terms of the reduced volume or density of the system. From the very definition of the probability p(σm), we may write

nm nm ) θm ) Θmθ ) θ, nq n1 + n2 + ... + nk m ) 1, ..., k (22)

p(σm) )

kT

∑ exp

[

]

-pair(0,n) + µ(σn) kT

(28)

µ0 Pv* q z ) + ln(1 - F˜ ) - ln 1 + F˜ - F˜ + kT kT 2 r

[

]

z[µ(σ0) - kT ln θ0] µv0 z pair(σ0,σ0) ) ) (29) kT kT 2 kT

n0 N0 v˜ - 1 ) ) θ0 ) (1 - θ) ) ) nq Nq v˜ + q/ - 1

or

r

1 - F˜ (23) 1 + F˜ q/r - F˜ Combining eqs 7 and 23, we may write

{∑ [ k

µ(σ0) ) -kT ln

exp

]}

-pair(σ0,σn) + µ(σn) kT

n)0

+

kT ln θ0 (24)

)

µ0 ∂(ln Z - PV/kT) )kT ∂N0

(

)-

T,P,N

)

∂(ln Z) ∂N0

+ T,P,N

Pv* kT (25)

or

µR0 µ0 µcomb 0 Pv* ) + + kT kT kT kT

Pv* z q + ln(1 - F˜ ) - ln 1 + F˜ - F˜ + kT 2 r

[

]

z[2µ(σ0) - pair(σ0,σ0) - 2kT ln θ0] ) 0 (29a) 2kT However, since the segments of empty sites have no screening charge, the interaction energy pair(σ0,σ0) is equal to zero. We may then write eq 29 as

We attribute to empty sites the characteristics of molecular entities and, thus, we may also attribute a “chemical potential” to them. This is most easily done in the Gibbs (N,P,T) ensemble as follows:

(

)

kT

It is essential to understand at this point that our system, even when we have one type of molecular components consisting of one type of surface segments, is always a mixture of these molecular segments with segments of empty sites. We may even think of a fictitious mixing process of empty sites with disordered but compact (with no free space between) molecular assemblies. Regardless of the details of this mixing process, we now require that µ0/kT be independent of composition or state of the fluid system. In other words, it has the same value as it would have if the empty sites were in their “pure” state or the absolute vacuum. There is no combinatorial contribution in this “pure” state and the pressure is zero. If we designate it by superscript v, and if we combine eq 7 with the last three equations, we obtain the following expression for the equation of state of our system:

whereas

p(σ0) )

)

ν0[µ(σ0) - kT ln θ0]

-z ln

)

kT ∂ ln Z ) v* ∂N0 T,N

µR0

(26)

µ0 Pv* q z ) + ln(1 - F˜ ) - ln 1 + F˜ - F˜ + kT kT 2 r

[

z[µ(σ0) - kT ln θ0] ) 0 (29b) kT In other words, we may consider as previously10-14 that the number of empty sites, N0, minimizes the free energy of the system and, thus, the equation of state is just the consequence of this minimization condition. The last equation will be further simplified shortly. This is a most important equation that will enable us to solve the above system of equations. To make clear the procedure to be followed, let us consider the case where we have two types of charged molecular segments 1 and 2, respectively. The conservation equationsseq 1sbecome in our case

If we adopt eq 9, the combinatorial contribution is11,12

2n00 + n01 + n02 ) n0

µcomb 0 z q ) ln(1 - F˜ ) - ln 1 + F˜ - F˜ kT 2 r

n01 + 2n11 + n12 ) n1

[

whereas eq 8 gives for the residual part

]

(27)

]

(30)

n02 + n12 + 2n22 ) n2 We have 7 unknowns, the 6 nij numbers of pairs and


n0. Applying eq 7, we may write for the chemical potentials of our surface segments

{

µ(σ0) -pair(σ0,σ0) + µ(σ0) ) ln θ0 - ln exp + kT kT -pair(σ0,σ2) + µ(σ2) -pair(σ0,σ1) + µ(σ1) + exp exp kT kT

}

{

-pair(σ0,σ1) + µ(σ0) µ(σ1) ) ln θ1 - ln exp + kT kT -pair(σ1,σ1) + µ(σ1) -pair(σ1,σ2) + µ(σ2) exp + exp kT kT (31)

}

{

-pair(σ0,σ2) + µ(σ0) µ(σ2) ) ln θ2 - ln exp + kT kT -pair(σ2,σ2) + µ(σ2) -pair(σ1,σ2) + µ(σ1) + exp exp kT kT

}

Equation 29, when coupled with eq 31, can give n0 (or F˜ ), µ(σ0), µ(σ1), and µ(σ2) through an iterative process and starting from an initial value for the reduced density of 1 (if liquid) or 0 (if gas). Having determined n0 and the chemical potentials for the surface segments, we may obtain from eq 3 combined with eq 5 the following equations for the contact pairs

[ [ [

] ] ]

∆Hvap ) (E + PV)vapour - (E + PV)liquid

Prior to use of eq 35, we must calculate the equilibrium pressure for a given temperature or the equilibrium temperature for a given external pressure, as well as the (reduced) densities of the two phases at equilibrium. For phase equilibrium calculations we need expressions for the chemical potentials of the various molecular species of the system. As already indicated, we may write for the chemical potential of each molecular species i:

µcomb µRi µi i Pv* ) + + ri kT kT kT kT

[(

∑

where the segment (volume) fraction of component i is given by

(33)

If some interactions lead to volume change, ∆Vmn, upon formation of the corresponding m,n pair, eq 33 should be corrected to account for that as follows:

V ) rNv* v˜ +

∑nmn∆Vmn m,n

(33a)

k

k

∑ ∑ nmnpair(m,n) m ngm

The heat of vaporization is then obtained as

The compositions of the π phases at equilibrium are then obtained by solving, as usual, the system of equations

i ) 1, 2, ..., t

(39)

z[µ(σ0) - kT ln θ0] 1 Pv* + ln(1 - F˜ ) + 1 - F˜ + )0 kT r kT (LF case15) (40)

(

)

whereas the equation for the chemical potential (eq 37) becomes

µi kT

( )

) ln φi + 1 ri

Pv* kT

+

ri r

+ ln F˜ + ri(v˜ - 1) ln(1 - F˜ ) +

µ(σm) - θm νm,i kT m

∑

(LF case) (41)

In this case we have

θ ) 1 - θ0 ) F˜ (LF case)

Of course, care must be exercised in implementing these volume changes in all thermodynamic quantities affected by them. In a similar manner the total potential energy of our system is given by

E)

(38)

If Flory’s expression16 is used, instead, for Ω in eq 9, the equation of state (eq 29) becomes

By replacing from eq 32 in eq 30, we obtain the remaining numbers of pairs n01, n02, and n12. This procedure can easily be generalized to any number of interacting charged surface segments. Once, however, we have determined the above quantities, we may use them for the evaluation of a number of useful thermodynamic quantities of our system. The total volume of the system is given by (cf. eq 12)

V ) rNv* v˜

riNi xiri ) rN r

π µIi ) µII i ) ... ) µi ,

nq -pair(σ2,σ2) + 2µ(σ2) n22 ) exp 2 kT

)]

φi zqi q ) ln + ln F˜ ln 1 + F˜ - F˜ + kT ri 2 r µ(σm) - kT ln θm Pv* + νm,i (37) ri kT kT m µi

φi )

(32)

(36)

By adopting eq 9 for deriving the combinatorial part and by combining with eq 8 for the residual contribution, eq 36 becomes for open-chain molecules

nq -pair(σ0,σ0) + 2µ(σ0) n00 ) exp 2 kT nq -pair(σ1,σ1) + 2µ(σ1) n11 ) exp 2 kT

(35)

(34)

(42)

Other expressions for Ω, in particular Staverman’s expression,17 may be utilized in an analogous manner. The equations of this section can be turned into more familiar forms if we examine the nonrandom distribution of molecular entities in the system. Nonrandomness Factors. One important feature of the above approach is its inherent capacity to calculate the nonrandom distribution of the various interacting charged surfaces in the system. In this section we will

1500


examine this feature and compare the results with our previous models.10-14 To facilitate the comparison, we will introduce here, as well, the nonrandomness factors Γ. Thus, by using a superscript 0 to indicate the case of perfect random distribution and Γmn to indicate the nonrandomness factor associated with the interaction of surfaces m and n, we may write for the number of m,n contact pairs

nmn ) n0mnΓmn

(43)

θm2Γmm ) exp

1 1 n θ ) a n nθ m a m n 2 2

(44)

θ0Γ00 + θ1Γ01 ) 1

θ0Γ20 + θ1Γ21 + ... + θkΓ2k ) 1

(45)

........

and from eq 50

[ [

θ02Γ00 ) exp

] ]

-pair(σ0,σ0) + 2µ(σ0) ) A0 kT

-pair(σ1,σ1) + 2µ(σ1) θ1 Γ11 ) exp ) A1 kT 2

n10 ) n1θ0Γ10 ) nq exp

θ0Γk0 + θ1Γk1 + ... + θkΓkk ) 1 where the nonrandomness factors obey the symmetry condition

Γmn ) Γnm

(46)

Equivalently, we may introduce local surface fractions10-14 as follows:

θji ) θjΓji

(47)

θjθij ) θiθji

[

]

-pair(σ0,σ1) + µ(σ0) + µ(σ1) kT (54)

θ1θ0Γ10 ) -pair(σ0,σ1) exp kT

[

]x

A0xA1 exp

[

or

Γ10 ) xΓ00xΓ11 exp

[

]

pair(σ0,σ0) + pair(σ1,σ1) - 2pair(σ0,σ1) ) 2kT

(48)

xΓ00xΓ11/xG10 or

θ00 + θ10 + ... + θk0 ) 1

G10Γ102 ) Γ00Γ11

θ01 + θ11 + ... + θk1 ) 1 θ02 + θ12 + ... + θk2 ) 1

(49)

........ θ0k + θ1k + ... + θkk ) 1 With these definitions eq 32 may be rewritten as

] ] ]

-pair(σ0,σ0) + 2µ(σ0) θ0 Γ00 ) exp kT θ12Γ11 ) exp

-pair(σ1,σ1) + 2µ(σ1) kT

θ22Γ22 ) exp

-pair(σ2,σ2) + 2µ(σ2) kT

or, in general,

]

pair(σ0,σ0) + pair(σ1,σ1) 2kT

Replacing from eq 47 in eq 45, we obtain

[ [ [

(53)

Replacing for the chemical potentials from eq 53, the last equation gives

The last two equations, when combined, give

2

(52)

On the other hand, a combination of eqs 3 and 5 for p(σ0,σ1) gives

θ0Γ10 + θ1Γ11 + ... + θkΓ1k ) 1

θij ) θiΓij,

(51)

θ0Γ10 + θ1Γ11 ) 1

where a ) 0 when m * n and a ) 1 when m ) n. Replacing from the last two equations in the conservation equations (1), we obtain

θ0Γ00 + θ1Γ01 + ... + θkΓ0k ) 1

]

-pair(σm,σm) + 2µ(σm) kT

Let us now calculate these nonrandomness factors with the model of the previous paragraph. For this purpose, let us consider first the simplest case of a binary system, namely, a system with surface segments of type 0 and 1, only. In this case we have from eq 45

However, when the surface segment pairing occurs at random, we have

n0mn )

[

(50)

(55)

Equation 55, however, is identical to our previous10-14 corresponding equations for the nonrandom factors and is a well-known result from Guggenheim’s QuasiChemical theory.9 Replacing from eq 52 and solving for Γ10, we obtain, as before,10-14

Γ10 )

2 1 + x1 - 4θ1θ0(1 - G10)

(56)

It is worth observing that for the evaluation of the nonrandomness factors we do not need to specify the interaction energies, pair, separately but rather the interchange energy ∆10 ) 2pair(σ0,σ1) - pair(σ0,σ0) pair(σ1,σ1). When there are three types, 0, 1, and 2, of interacting surface segments, we proceed in a similar manner. However, in this case we have three nonrandomness cross terms to calculate, namely, Γ10, Γ20, and Γ12. The


remaining factors Γ00, Γ11, and Γ22 will be obtained from the conservation equations, which are now

θ0Γ00 + θ1Γ01 + θ2Γ02 ) 1 θ0Γ10 + θ1Γ11 + θ2Γ12 ) 1

(57)

In this case the “quasi-chemical conditions”, that is, the analogues of eq 55, are

) Γ00Γ11

G20Γ202

) Γ00Γ22

G12Γ122

) Γ11Γ22

2µ(σ0) - 2kT ln θ0 - pair(σ0,σ0) ) kT ln Γ00

q z Pv* z + ln(1 - F˜ ) - ln 1 + F˜ - F˜ + ln Γ00 ) 0 (64) kT 2 r 2

[

]

In a similar manner, by using eq 51, we observe that

2µ(σm) - 2kT ln θm ) kT ln Γmm + pair(σm,σm) (65) (58)

Thus, eq 37 for the chemical potential may be written as

[( [

)]

zqi φi q ) ln + ln F˜ ln 1 + F˜ - F˜ kT ri 2 r µi

with

G10 ) exp

2pair(σ1,σ0) - pair(σ1,σ1) - pair(σ0,σ0) kT

G20 ) exp


G12 ) exp


θ1θ0[(1 -

∑

(59)

θ1θ2[(1 -

G12)θ1θ2Γ122

]

The last terms in eqs 64 and 66 should replace the corresponding terms in eqs 40 and 41 in the LF case. Being convinced about the quasi-chemical character of this combined model, we believe a number of comments are in order at this point. We may obtain identical “quasi-chemical” equations to the above equations (like eqs 55 and 58) if we write the full form of the configurational partition function in its maximum term approximation as follows:10-12

Z)

(60)

- (1 - 2θ0)Γ12 + 1]

These are two coupled equations, which must be combined with one of the equation in (58) after substitution from conservation equations, say,

(1 - θ0Γ10 - θ2Γ12)(1 - θ1Γ10 - θ2Γ20) θ1θ0G10Γ102 ) 0 (61) in order to obtain the unknowns Γ10, Γ20, and Γ12. The physically meaningful solutions to this system of equations are10

Γij ) 2(1 - 2λθt) (1 - 2θt) + x(1 - 2θt)2 - 4θiθj(1 - 2λθt)(1 - Gij)

i, j, t ) 0 or 1 or 2 and i * j * t (62)

λ is an intermediate parameter near 1 which must be tried in eq 62 until the solutions satisfy eq 61. Analogous equations are obtained for quaternary and higher systems.

[( ) ] ∏ [( ) ]

(n0mn!) ∏ ∏ m n*m

1 0 nmn ! 2

2

(nmn!) ∏ m n*m

1 nmn ! 2

2

- (1 - 2θ2)Γ10 + 1] )

θ2θ0[(1 - G20)θ2θ0Γ202 - (1 - 2θ1)Γ20 + 1] )

+

pair(σm,σm) Pv* 1 νm,i ln Γmm + (66) ri + kT 2m kT

Equations in (58) are now coupled equations with no simple analytical solution like eq 56. However, as we have shown previously,10-11 the solution may be obtained easily as follows: Substituting eq 57 into eq 58 and rearranging, we obtain

G10)θ1θ0Γ102

(63)

Thus, the equation of state, eq 29a, simplifies to

θ0Γ20 + θ1Γ21 + θ2Γ22 ) 1

G10Γ102

The use of nonrandomness factors simplifies significantly the equation of state. By using the first of eq 50, we observe that

Ω

(

exp -

)

∑ ∑ nmnpair(σm,σn) m ngm kT

(67) The quasi-chemical equations are now obtained by requiring that the numbers of pairs nmn (m*n) minimize the free energy of the system. These minimization conditions lead to the equations

4nmmnnn 2

nmn

) Gmn or GmnΓmn2 ) ΓmmΓnn all m,n pairs

which are precisely the quasi-chemical equations (see eqs 55 and 58). Of course, the fact that the quasichemical equations are identical in the two models does not necessarily imply that the two models are equivalent. However, eq 67 permits the derivation of an analytical equation for the chemical potential of each component i in the mixture. Apart from the combinatorial contribution, which is the same in both models (as long as we adopt the same expression for Ω), the residual contribution obtained from eq 67 is10-11

µRi kT

)

1

k

1

k

∑ νm,i ln Γmm + 2m)0 ∑ νm,i 2m)0

pair(σm,σm) kT

(68)

1502


But this is precisely the last term in eq 66. In other words, we obtain identical expressions for the chemical potential with the two approaches. On the other hand, the equation of state derived in eq 67 is now identical in form to eq 64 (or eq 40 in the LF case) since the term obtained from the terms after Ω in eq 67 is now equal to z/2 ln Γ00. Thus, the previous quasi-chemical approach10-12 and the current approach are entirely equivalent approaches! Consequently, one may calculate all thermodynamic properties without making intermediate calculations of the various chemical potentials for the surface segments, µ(σm). These chemical potentials may be obtained in terms of the nonrandomness factors by using eq 51 or

1 µ(σm) ) kT ln θm + [kT ln Γmm + pair(σm,σm)] (69) 2

∆Hvap ) (E + PV)vapour - (E + PV)liquid ) (n11pair(σ1,σ1) + n10pair(σ1,σ0) + PV)vapor (n11pair(σ1,σ1) + n10pair(σ1,σ0) + PV)liquid ) zq [θ Γ (σ ,σ ) + 2θ0Γ10pair(σ1,σ0)] + PV 2 1 11 pair 1 1 vapor zq ) [θ Γ (σ ,σ ) + 2θ0Γ10pair(σ1,σ0)] + PV 2 1 11 pair 1 1 liquid zq [θ Γ ( (σ ,σ ) - 2pair(σ1,σ0)) + 2 1 11 pair 1 1

{ {

} }

{

2pair(σ1,σ0)] + PV

{zq2 [θ Γ

1 11(pair(σ1,σ1)

}

1 11∆10]

It must be pointed out that the calculated thermodynamic quantities and, in particular, the calculated interaction energies depend on the adopted form for the combinatorial contribution. For completeness and for comparison purposes it is worth reporting the above working equations by adopting Staverman’s17 combinatorial expression for Ω since this appears to be the preferred expression in COSMO calculations.3-8 In this case, eq 64 becomes

z q Pv* + ln(1 - F˜ ) - ln 1 + F˜ - F˜ kT 2 r z F˜ L + ln Γ00 ) 0 (70) 2

[

]

and eq 66 becomes

µi kT

) ln

φi ri

+ ln F˜ -

riF˜ L + ri

[

zqi

]

q ln 1 + F˜ - F˜ 2 r

Pv*

+

kT

µ(σm) - kT ln θm

νm,i ∑ m

kT

(71)

where

L)

∑i

li φi ) ri

∑i

φi

[( ) z

2

1-

qi ri

+

1 ri

]

-1 )

z 2

( ) 1-

q r

+

1 r

- 1 (72)

It is also worth pointing out that for the basic thermodynamic quantities of pure fluids and mixtures we do not need to specify the separate interaction energies pair(σm, σm) and pair(σ0, σm), but rather the interchange energies ∆m0 ) 2pair(σ0,σm) - pair(σ0,σ0) pair(σm,σm). This holds true even in the case of heats of vaporization. To see this, let us consider the simple case of a pure component with one type of charged surfaces with charge density σ1. By combining eqs 34 and 35, we have

vapor

- 2pair(σ1,σ0)) +

2pair(σ1,σ0)] + PV

{zq2 [-θ Γ

}

) liquid

}

+ PV

{zq2 [-θ Γ

vapor

1 11∆10]

}

+ PV

liquid

(73)

As observed, the contribution of the separate term pair(σ1,σ0) cancels out and we are left with the contribution of the interchange energy, ∆10, only. This is an important point and must be taken into account when comparing interaction energies of the COSMO approach with the former quasi-chemical approach12,14 or latticefluid approach.15 One might, of course, treat the COSMO interaction energies as equivalent to pair(σ1,σ1) energies since in the COSMO approach there are no empty sites. Some caution, however, must be exercised as will be clear from the calculations of the next section. Applications In this section we will apply the above formalism to simple fluids. For the quantum mechanics calculations we will confine ourselves to the COSMO-RS approach, although other analogous approaches might be used as well. Thus, all needed information will be obtained by using the TURBOMOLE DFT (density functional theory) quantum chemistry package18 with the BP-TZVP functional/basis set and with default atomic radii (Å) (H, 1.30; C, 2.00) used to define the molecular cavity. Implementing the so-obtained cosmo files into the Cosmotherm-C12 package of Cosmologic GmbH, we may obtain the surface charge distribution and an estimation of the mean intermolecular energy. We will make, first, a most simple application of the above equation-of-state formalism by using experimental PVT data for polymers. We will limit the application to polymers with one type of segments. In this way we will have systems with two segments, the polymer segment (1) and the empty site segment (0). The calculations with the quasi-chemical model are straightforward as shown before.10-14 We solve eq 64 for the reduced density and obtain simultaneously the nonrandomness factors from eqs 56 and 52. The chemical potentials for the surface segments are then obtained from eq 69. Entirely analogous is the procedure when adopting Staverman’s combinatorial term17 and the corresponding equation of state, eq 70. Since this combinatorial term is used by COSMO-RS, all calculations of this section will be done by using the corresponding eqs 70-72. We have applied eq 70 to PVT data for linear polyethylene19 assuming that we have one type of


Figure 1. Experimental19 (symbols) and calculated (lines) specific volumes of linear polyethylene.

Figure 2. Variation of the chemical potential µ(σ0) with temperature at various pressures as calculated by eq 69 for linear polyethylene. Table 1. Characteristic Constants of Pure Fluids *P*-1

fluid methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-dodecane n-hexadecane polyethylene-linear polyethylenebranched

* ) RT* v* ) (J‚mol-1) (cm3 ‚mol-1) 1969 2891 3278 3823 4249 4486 4688 4761 4873 4990 5110 5352 3807 3865

7.516 8.080 9.212 10.96 12.88 13.40 14.13 14.34 14.98 15.31 15.58 16.65 9.422 9.750

v/sp

F*-1

) (cm3‚g-1)

s ) q/r

2.149 1.605 1.448 1.390 1.357 1.303 1.270 1.265 1.269 1.243 1.232 1.216 1.073 1.077

0.900 0.896 0.892 0.887 0.877 0.877 0.877 0.877 0.877 0.877 0.877 0.877 1.330 1.330

charged surface segments. The parameters obtained from correlating the data of Figure 1 with a leastsquares procedure are reported in Table 1. All calculations in this section were done with a lattice coordination number, z ) 10. It is worth pointing out that the calculations shown in Figure 1 cannot be performed by the plain COSMO-RS or COSMOtherm approach. The quantity that can be calculated easily is µ(σ0). In Figure 2 is shown the effect of temperature and pressure on µ(σ0) as calculated by eq 69. As observed, the effect of temperature is relatively small compared to the effect of pressure in this temperature and pressure range.

Figure 3. Effect of temperature on nonrandomness factor Γ00 for linear polyethylene at various pressures as calculated by this model.

Figure 4. Average intermolecular energy per methylene group of linear polyethylene as a function of temperature calculated by the equation-of-state model (EOS) and COSMO-RS.

In Figure 3 is shown the calculated nonrandomness factor Γ00 for the same polymer at various external conditions. It is worth observing that the nonrandomness factor for the distribution of empty sites in the system departs significantly from unity (random distribution), even for this apparently nonpolar system (nonrandom distribution of free volume). From the equation-of-state approach we may obtain the variation of the mean intermolecular interaction energy at each external (T,P) condition from its definition (cf. eqs 34 and 43), which in our case becomes k

E)

k

z

∑ ∑ nmnpair(m,n) ) N2∆10rsθ1Γ11 m ngm

(34a)

In Figure 4 is shown the calculated intermolecular interaction energy per methylene group as a function of temperature at ambient pressure. In the same figure is shown the corresponding energy as calculated by the COSMO approach. For the latter a cosmo file was constructed for a polyethylene molecule of a molecular weight of 1850. As observed, although the trend with temperature is essentially the same in both calculations, there is a significant discrepancy between the two types of calculated energies. Of course, the system treated by the COSMO approach has no empty sites and, thus, the only type of interaction energy is pair(σ1,σ1). By attributing the above discrepancy exclusively to the difference of ∆10 from pair(σ1,σ1), one may have an estimation of the separate components pair(σ1,σ1) and pair(σ0,σ1). From

1504


Figure 5. Experimental20,21 and calculated orthobaric densities of methane as a function of temperature.

Figure 7. Hard core volume of normal alkanes as calculated by the EOS model vs the number of chain carbons.

Figure 6. Experimental20,21 and calculated vapor pressure of methane as a function of temperature.

this procedure we obtain for polyethylene: ∆10 ) 761.5 J/mol; pair(σ1,σ1) ) 421.0 J/mol; pair(σ0,σ1) ) 591.3 J/mol. Once these parameters are known, we may use eq 69 to calculate µ(σ1). It should be stressed once again, however, that we do not need these separate components in any of the calculations of the thermodynamic quantities of our interest. Let us now apply the above formalism to normal alkanes. In this case, besides densities of the liquid and vapor phase we may calculate vapor pressures, heats of vaporization, second virial coefficients, etc. References 20 and 21 may be used as sources for the required experimental data. For simplicity we will also treat alkanes as having two types of charged surface segments as we did for polyethylene. The parameters obtained from correlating the experimental data20,21 with a least-squares procedure are reported in Table 1. In Figure 5 are compared the experimental and calculated orthobaric densities for methane up to the critical point. Again, these calculations cannot be performed by the plain COSMO-RS approach. In Figure 6 are compared the experimental and calculated vapor pressures for the same alkane. From the parameters reported in Table 1 one may obtain the number of segments per molecule as well as the number of external contacts from the defining equations:

r)

Mv*sp v*

)

V* , v*

q ) rs

(74)

Figure 8. Number of segments per molecule (used by the EOS model) vs the number of chain carbons of normal alkanes.

Figure 9. Variation of the mean interaction energy per segment, *, with the volume per segment, v*, for normal alkanes.

In Figure 7 is shown the variation with the number of chain carbons of the hard core volume V* of normal alkanes. A linear fit is obtained with R ) 0.9994. A linear fit, though not as good (R ) 0.9951), is also shown in Figure 8 for the number of molecular segments r as a function of the number of chain carbons of normal alkanes. The plot of * vs v* is also linear as shown in Figure 9.


Figure 10. Hard core volume of normal alkanes as calculated by the EOS model vs the corresponding volume as calculated by COSMO-RS.

Figure 11. Mean intermolecular energy, r*, as calculated by the EOS model vs the corresponding energy as calculated by COSMORS. Table 2. Linear Relations between the Characteristic Constants relation

R

* ) 637.41 + 285.03v* r* ) -3552.76 + 1.0244ECOSMO r* ) -2633.53 + 1.0089ECOSMO, Nc > 1 r* ) 7207.46 + 5084.45Nc, Nc > 1 r ) 4.101 + 0.763Nc r ) 3.4232 + 0.8230Nc, Nc > 4 r ) 2.8648 + 0.0625VCOSMO, Nc > 4 V* ) rv* ) 16.220 + 16.145Nc V* ) 4.9584 + 1.2279VCOSMO

0.9987 0.9994 0.9998 0.9998 0.9951 0.9993 0.9993 0.9999 0.9999

The plots in Figures 10 and 11 are of particular interest. In Figure 10 is shown the variation of the molecular hard core volume V*, obtained by the equation-of-state approach, with the corresponding volume obtained from the cavity volume of the COSMO approach. In Figure 11 is shown the variation of the mean intermolecular energy per molecule, Er ) r*, with the corresponding energy as calculated by the COSMO approach at 25 °C. As observed, in both cases there is a very good linear relation. These linear fits are improved significantly if we disregard the first few alkanes, where our assumption of only one type of charged surface segments (besides the segments of empty sites) is not valid. In Table 2 are summarized these relations. It is worth now testing the predictive capacity of our equation-of-state model. n-Eicosane is well outside the range of Nc’s of alkanes that were treated so far and its

Figure 12. Experimental20 and predicted vapor pressures of n-eicosane as a function of temperature.

Figure 13. Experimental20 and calculated liquid densities of n-eicosane as a function of temperature. Solid line represents predicted densities with the same scaling constants as in Figure 12. Dashed line is obtained with the same parameters except for s, which is set equal to 0.995 instead of 0.877.

scaling constants were not used for obtaining the correlations in Table 2. On the basis of the relations in Table 2, the estimated characteristic constants for n-eicosane are * ) 5477 J/mol, v* ) 17.06 cm3/mol, v/sp ) 1.2004 cm3/g, and s ) 0.877. By using these parameters, we have calculated the vapor pressure of neicosane. Our predictions are compared with the experimental data in Figure 12. As observed, the agreement is very good. By using the very same parameters, we obtain a good description of the heats of vaporization and the solid line in Figure 13 for the liquid densities. Although the densities are well-predicted at low temperatures, their description at high temperatures is rather poor. If one retains the above values for the scaling constants of *, v*, and v/sp and increases s from 0.877 to 0.995, the dashed line in Figure 13 is obtained. It is worth pointing out that the scaling constants in Table 1 for polyethylene were derived by correlating exclusively density data at high temperatures and much higher pressures. In that case, s had to be even higher (equal to 1.330). In Table 1 are also reported the scaling constants for the less compact structure of branched polyethylene. For this polymer the scaling constants of *, v*, and v/sp have higher values compared to linear polyethylene. As expected, these constants have even higher values for the much less dense structures of normal alkanes.

1506


of the charged segments with the empty site, the contribution would be significant since every non-zero charge would contribute positively to the misfit energy via the equation 2

Emisf(σ1,σ0) )

σ1 R′ (σ1 + 0)2 ) R′ 2 2

(75)

R′ being an adjustable parameter. In fact, if we had the σ-profiles for the reference state of density F* (no empty sites), eq 75 would give the component pair(σ1,σ0) of the interchange energy ∆10. Discussion and Conclusions

Figure 14. σ-profiles of polyethylene and ethane as calculated by the COSMO approach.

Before closing this section some additional comments regarding our calculations are in order. For linear polyethylene, the obtained density of the reference state with no empty sites is F* ) 1/v/sp ) 0.932 g/cm3. The Bondi’s volume22 for the -CH2- group is 10.23 cm3/mol, implying a density FBondi ) 1.371 g/cm3. In contrast, the asymptotic value for the cavity volume per methylene group obtained from the COSMO/TURBOMOLE calculations is ca. 13.17 cm3/mol, implying a density FCOSMO ) 0.939 g/cm3. Thus, the estimated “hard-core” densities for polyethylene by the present work and by COSMO differ by about 0.75%, the former being smaller. In other words, the COSMO parametrization was done in a slightly more compact reference state. On the other hand, in the case of ethane the density of the reference state is F* ) 1/v/sp ) 0.623 g/cm3. The cavity volume of the methyl group from COSMO calculations is ca. 17.69 cm3/mol, implying a density FCOSMO ) 0.849 g/cm3, which is ca. 36% higher than F*. In this case, the COSMO parametrization was done in a significantly more compact reference state. Since the cavities differ, the pair interaction energies will also differ accordingly. For all systems treated in this work we have assumed that they are systems with two segments: the segment (1) and the empty site segment (0). A better assumption would be to consider three types of segments: segment (1) corresponding to the -CH2- group, segment (2) corresponding to the -CH3 group, and segment (0). The major contribution to intermolecular interaction energy of alkanes, as calculated by the COSMO approach, is due to the van der Waals interaction energy. This energy for each -CH3 group in ethane is ca. -10080 J/mol, whereas for each -CH2- group in normal alkanes it is ca. -5100 J/mol. However, even this distinction between these two groups is a rather crude one, as is clear from Figure 14, which shows the σ-profiles of polyethylene (mainly -CH2- groups) and ethane (-CH3 groups), as calculated by the TURBOMOLE/COSMOtherm package: In the frame of the COSMO approach we could distinguish as many charged surfaces (groups) as we would like. Although different in shape, in both profiles of Figure 14 the distributed charges are near zero. In fact, the contribution of the misfit energy to the total intermolecular interaction energy for all treated systems is very small, ranging from 1.4% for polyethylene to 2.2% for methane. If, however, we would use these profiles for the calculation of the misfit energy for the contacts

In this work we have shown that the surface contact thermodynamics used in the COSMO-RS model3-5 is, essentially, equivalent to the classical quasi-chemical approach9-14 of group contributions of interacting surfaces. By consistently applying our previous approach,10-14 we were able to incorporate free volume into the COSMO model and derive an equation of state identical in form to the previous one.10-14 In this way the present work extends significantly the pressure and temperature range of applicability of surface-contact thermodynamic models, such as the COSMO-RS3-5 model. We were also able to derive the relation between the nonrandomness factors and the chemical potentials of interacting surfaces. Simple calculations for the PVT behavior of linear polyethylene have shown that even in this rather nonpolar polymer, the free volume is nonrandomly distributed. Besides polyethylene, we have applied our model to normal alkanes, from methane to hexadecane. The scaling constants of normal alkanes obey a number of linear relations with the number of chain carbons or the COSMO volume and mean interaction energy, which are useful for the estimation of these constants for higher alkanes. This has been verified with the prediction of the thermodynamic properties of n-eicosane. In this work we have taken for simplicity the molecular rather than the group-contribution approach in our calculations. Thus, we made the assumption that in polyethylene as well as in normal alkanes we have only one type of interacting charged surfaces of non-zero charge. In other words, we made no distinction between methyl and methylene groups or even smaller groups. One could make this distinction and turn the model into a predictive group-contribution model. The fact that pair(σ0,σm) could be non-zero is quite important. All lattice-fluid type models10-15 are assuming this interaction is zero. The above analysis explains why this assumption does not affect the performance of the lattice-fluid models: In all calculations of the basic thermodynamic quantities, the required parameter is the interchange energy ∆m0 ) 2pair(σ0,σm) pair(σ0,σ0) - pair(σm,σm) and not the separate interaction energies. However, when comparing lattice-fluid interaction energies with energies estimated with quantum mechanics calculations, such as the COSMO approach,3-7 the comparison will be absurd if one does not take into account the difference between interchange energy and interaction energy. The estimated compact or hard core volume by the equation-of-state approach is, in general, different from the volume of the molecular cavity estimated by the COSMO approach. This also implies that the segmentsegment interaction energies of the two approaches


cannot be interchanged. Yet the linear relations of Table 2 indicate that one type of interaction could be used for the estimation of the other. Since in the COSMO approach there are no empty sites in the system, the estimated interaction energy corresponds to pair(σ1,σ1) rather than to ∆10. The other component, pair(σ0,σ1), of the interchange energy could be obtained from eq 75, as indicated in the previous section. The variation of the mean intermolecular interaction energy with temperature in the frame of the COSMO approach is a more or less empirical one. However, Figure 4 in connection with eq 34a indicates that this variation could be associated with the variation in density when an equation-of-state approach is adopted. As mentioned in the Introduction, in the transition from the liquid to the gas state the molecules will continuously adjust their wave functions to minimize the interaction energy with their neighbors. In this work we have neglected this change since our assumption of the existence of empty sites along with the solvation in a perfect conductor is a more crude assumption. One probably better approximation for pair(σ0,σ1) might be pair(σ0,σ1(F)). At present, however, such an attempt to account for this functionality will be purely empirical. Our approach indicates the potential of exploiting quantum mechanics information for the calculation of the thermodynamic properties of fluids over an extended range of temperature and pressure. Of course, a systematic work must be done with various fluids of varying polarity to establish the rules for the appropriate compact state for quantum mechanics calculations and for the translation of these calculations into equation-of-state parameters or scaling constants. Since the model is a group (contact surface) contribution model, it might be used for prediction of thermodynamic properties when, of course, a COSMO-type database for the appropriate compact reference state will be available. Literature Cited (1) Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. GroupContribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures. AIChE J. 1975, 21, 1086. (2) Gmehling, J.; Li, J. D.; Schiller, M. A Modified UNIFAC Model. 2. Present Parameter Matrix and Results for Different Thermodynamic Properties. Ind. Eng. Chem. Res. 1993, 32, 178. (3) Klamt, A. Conductor-Like Screening Model for Real Solvents: A New Approach to the Quantitative Calculation of Solvation Phenomena. J. Phys. Chem. 1995, 99, 2224.

(4) Klamt, A.; Jonas, V.; Burger, Th.; Lohrenz, J. C. W. Refinement and Parameterization of COSMO-RS. J. Phys. Chem. 1998, 102, 5074. (5) 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. (6) Klamt, A. Comments on “A Priori Phase Equilibrium Prediction from a Segment Contribution Solvation Model”. Ind. Eng. Chem. Res. 2002, 41, 2330. (7) Klamt, A.; Krooshof, G. J. P.; Taylor, R. COSMOSPACE: Alternative to Conventional Activity-Coefficient Models. AIChE J. 2002, 48, 2332. (8) Lin, S.-T.; Sandler, S. I. A Priori Phase Equilibrium Prediction from a Segment Contribution Solvation Model. Ind. Eng. Chem. Res. 2002, 41, 899. (9) Guggenheim, E. A. Mixtures; Oxford University Press: Oxford, 1952. (10) Panayiotou, C.; Vera, J. H. The Quasi-Chemical Approach for Non-randomness in Liquid Mixtures. Expressions for Local Surfaces and Local Compositions with an Application to Polymer Solutions. Fluid Phase Equilib. 1980, 5, 55. (11) Panayiotou, C.; Vera, J. H. Local Compositions and Local Surface Area Fractions: A Theoretical Discussion. Can. J. Chem. Eng. 1981, 59, 501. (12) Panayiotou, C.; Vera, J. H. Statistical Thermodynamics of r-mer Fluids and Their Mixtures. Polym. J. 1982, 14, 681. (13) Panayiotou, C. In Handbook of Surface and Colloid Chemistry; Birdi, K. S., Ed.; CRC Press: Boca Raton, FL, 2002. (14) Panayiotou, C. Interfacial Tension and Interfacial Profiles of Fluids and Their Mixtures. Langmuir 2002, 18, 8841. (15) Sanchez, I. C.; Lacombe, R. H. Statistical Thermodynamics of Polymer Solutions. Macromolecules 1978, 11, 1145. (16) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (17) Staverman, A. J. The Entropy of High Polymer Solutions. Recl. Trav. Chim. Pays-Bas 1950, 69, 163. (18) Schaefer, A.; Klamt, A.; Sattel, D.; Lohrenz, C. W.; Eckert, F. COSMO Implementation in TURBOMOLE: Extension of an Efficient Quantum Chemical Code Towards Liquid Systems. Phys. Chem. Chem. Phys. 2000, 2, 2187. (19) Zoller, P.; Walsh, D. PVT Data for Polymers; Technomic Publ. Co.: Lancaster, 1995. (20) Daubert, T. E.; Danner, R. P.; Eds. Data Compilation Tables of Properties of Pure Compounds; AIChE Symposium Series No. 203; American Institute of Chemical Engineers: New York, 1985. (21) Perry, R.; Green, D. Perry’s Chemical Engineers’ Handbook, CD-ROM ed.; McGraw-Hill: New York, 1999. (22) Bondi, A. van der Waals Volumes and Radii. J. Phys. Chem. 1964, 68, 441.

Received for review September 12, 2002 Revised manuscript received January 22, 2003 Accepted January 28, 2003 IE0207212

Equation-of-State Models and Quantum Mechanics Calculations

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