Fluid Phase Equilibria 242 (2006) 10–18

Modification of Simha–Somcynsky equation of state forsmall and large molecules

Ming Wang, Shigeki Takishima ∗, Yoshiyuki Sato, Hirokatsu MasuokaDepartment of Chemical Engineering, Graduate School of Engineering, Hiroshima University,

1-4-1 Kagamiyama, Higashi-Hiroshima 739-8527, Japan

Received 23 November 2005; accepted 8 January 2006Available online 15 February 2006

Abstract

The Simha–Somcynsky equation of state (SS EOS) represents the PVT behavior of polymers quite satisfactorily, but cannot be applied togases at low pressures. This work proposes a modification of the free volume contribution of the SS EOS to allow representation of gaseousstate of low molecular-weight substances by introducing the perturbed hard-chain theory of Beret and Prausnitz into the EOS. In addition tothis modification, two universal constants are introduced to the free volume term for better representation of properties of low molecular-weightsaAp©

K

1

csmpmim

dcsE[[a

0d

ubstances. Characteristic parameters in the modified SS EOS were determined for 44 low molecular-weight substances and 64 polymers. Thebsolute average deviations (AADs) for critical temperature, critical pressure and vapor pressures were 0.86, 2.38 and 2.01%, respectively, whileAD for critical density and saturated liquid density at normal boiling point were somewhat larger, being 20.46 and 5.30%, respectively. The higherformance of the original SS EOS for polymer PVT behavior was maintained in the modified EOS with grand AAD of 0.050% for densities.

2006 Elsevier B.V. All rights reserved.

eywords: Equation of state; Hole theory; Vapor pressure; PVT behavior; Polymer

. Introduction

The knowledge of gas solubilities in molten polymers is ofonsiderable importance for the optimal design and operationsuch as foam plastic production, devolatilization to remove lowolecular-weight substances from polymers and vapor phase

olymerization. Although solubilities have been measured forany gas–polymer systems, equations of state (EOSs) are still

mportant for predicting or correlating the phase behavior ofany systems over a wide range of temperatures and pressures.For polymer melts numerous theoretical EOSs have been

eveloped based on solution theory. Rodgers [1] evaluated theorrelation errors in polymer PVT behavior for five EOSs con-isting of the cell model (CM) EOS of Prigogine [2], the FOVOS of Flory et al. [3], the hole theory of Simha and Somcynsky

4] (SS EOS), the lattice-fluid model of Sanchez and Lacombe5] (SL EOS) and the modified cell model (MCM) EOS of Deend Walsh [6,7]. He found that the SS EOS gave the best corre-

∗ Corresponding author. Tel.: +81 82 424 7713; fax: +81 82 424 7713.

lation of the experimental data and the order of average absolutedeviation in density was SS < MCM < CM < FOV < SL.

Generally, EOSs are desired to be applicable to both small andlarge molecules over a wide range of fluid densities. However,the above equations except for the SL EOS do not satisfy theideal-gas limitation for non-spherical molecules, and hence theycannot be applied to gases at low pressures. Although someEOSs that can be applied to both gases and polymers have beenproposed [8–11], further study is necessary in this region toimprove their applicability.

The present work proposes a modification to the SS EOS,which is one of the best models for the polymer PVT behav-ior among the EOSs based on solution theory. The idea ofBeret and Prausnitz [8] in their perturbed hard-chain the-ory was used to make the SS EOS satisfy the ideal-gaslimit. Moreover, two empirical constants were introduced intothe free volume term for better description of properties oflow molecular-weight substances. Application of the modi-fied Simha–Somcyncky (MSS) EOS to vapor pressures, criticalproperties, and the second virial coefficients of small moleculeswas examined, as well as the PVT behavior of small and large

E-mail address: r736735@hiroshima-u.ac.jp (S. Takishima). molecules.

378-3812/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2006.01.003

M. Wang et al. / Fluid Phase Equilibria 242 (2006) 10–18 11

2. Simha–Somcynsky equation of state

The SS EOS is based on a hole theory, where segments ofmolecules are arranged in well-defined lattice sites. The latticeincludes ‘holes’ that are unoccupied by segments. The change intotal volume of a system is controlled by the cooperative changesin the number of the holes and the volume of a site (cell volume).The configurational partition function, Z, for a pure fluid of Nmolecules was derived by Simha and Somcynsky [4] as follows:

Z = gvcNf exp

(−E0

kT

)(1)

where k is Boltzmann’s constant and T the absolute temperature.The first term in Eq. (1), g, represents the combinatorial factor

arising from the arrangement of segments and holes in the latticesites. The theories of Flory [12] and Huggins [13] were used toderive this term as:

ln g = −N ln y −{

Ns(1 − y)

y

}ln(1 − y) (2)

where y denotes the fraction of occupied sites and s is the numberof segments per molecule.

The second term in Eq. (1), vf, represents the free volumeper each three of the total number 3c of external degrees offreedom attributed to the flexibility of a chain. The free volumemst2w

v

wsie

sa

2

wmp

e

A

Tm(

Consequently, y is obtained by solving the following equation:

s

3c

(s − 1

s− ln(1 − y)

y

)− 2−1/6y(yV )−1/3 − 1/3

1 − 2−1/6y(yV )−1/3

− y

6(yV )2T

(2.409 − 3.039

(yV )2

)= 0 (7)

On the other hand, from the fundamental thermodynamic rela-tion, P = −(∂A/∂V)T,N, the SS EOS is derived as:

PV

T= [1 − 2−1/6y(yV )−1/3]

−1

+ 2y

(yV )2T

[1.011

(yV )2 − 1.2045

](8)

where T , P and V are reduced variables defined by:

T = T

T ∗ , P = P

P∗ , V = V

V ∗ (9)

Three characteristic parameters, T*, P*and V* are defined asfollows:

T ∗ = qzε∗

(ck)(10)

P

V

Wmm

3

oedutpE

W

Z

wBdop

q

ay be considered as the difference between the volume peregment and the hard-core volume of the segment. In the holeheory of Simha and Somcynsky, solid-like free length (ω1/3 −−1/6v∗1/3) and gas-like free length (ω1/3) are linearly averagedith the fraction of occupied site as:

f = [y(ω1/3 − 2−1/6v∗1/3) + (1 − y)ω1/3]3

(3)

here ω = yV/Ns is the cell volume, V the total volume of theystem and v∗ is the characteristic volume of a segment. s = c = 1s used for small spherical molecules, whereas 3c = s + 3 is gen-rally applied for flexible chain molecules.

The last term in Eq. (1), E0, is the interaction energy of theystem and is described by the 12-6 Lennard–Jones potentialmong all segment pairs:

E0 = yNqzε∗[

1.011

(v∗

ω

)4

− 2.409

(v∗

ω

)2]

(4)

here ε* represents the characteristic interaction energy per seg-ent pair, qz = s(z − 2) + 2 the number of nearest neighbor sites

er chain and z the coordination number which is set to 12.From the configurational partition function the Helmholtz

nergy, A, can be directly obtained as:

= −kT ln Z (5)

o obtain the free energy as a function of V, N and T, y is deter-ined by the minimization of the free energy as:

∂A

∂y

)V,T,N

= 0 (6)

∗ = qzε∗

(sv∗)(11)

∗ = Nsv∗ (12)

hen this EOS is used for the PVT relationship of fluids, oneust solve Eqs. (7) and (8) simultaneously by a numericalethod.

. Modification of SS EOS

It is easily found that Eq. (8) becomes PV/NkT = c at the limitf V → ∞. Thus the SS EOS does not satisfy the ideal-gas lawxcept for small spherical molecules (s = c = 1). Furthermore asescribed later, the occupied-site fraction in the SS EOS showsnrealistic behavior at low densities for fluids of c > 1 due tohe expression used for the free volume contribution. Hence, theresent work modifies the free volume contribution of the SSOS.

Vera and Prausnitz [14] expressed the generalized van deraals partition function as follows:

= g

(Vf

Λ3

)N

(qr,v)N exp

(−E0

kT

)(13)

here Vf is the free volume per molecule and Λ is the deroglie’s wavelength that depends only on temperature. qr,venotes the contribution from rotational and vibrational degreesf freedom per molecule, and can be divided into an internalart, qint(T), and an external part, qext(T,V):

r,v = qint(T )qext(T, V ) (14)

12 M. Wang et al. / Fluid Phase Equilibria 242 (2006) 10–18

where qint(T) depends only on temperature, but qext(T,V)depends also on volume. Beret and Prausnitz [8] gave an expres-sion for qext(T,V) as:

qext(T, V ) =(

Vf

V

)c−1

(15)

This equation satisfies four important boundary conditionsincluding the ideal-gas limit and the closest-packed liquid statefor spherical and chain molecules.

Eqs. (13)–(15) can be thought as the basic assumptions alsofor the hole theory. The present work examined several expres-sions for qext(T,V) in the hole theory of Simha and Somcynskyand obtained the best one as:

qext(T, V ) =(vf

ω

)c−1(16)

This expression clearly satisfies the above boundary conditions.Introducing Eq. (16), the partition function Z in this work canbe written as:

Z = g

[(vc

f

Λ3

)ω1−c

]N

exp

(−E0

kT

)(17)

The term of the free volume, vf, in Eq. (3) was derived intu-itively, but it is important in describing the PVT for the smalland large molecules. Therefore, it was modified in this work by

The definition of T*, P* and V* is the same as that in the originalSS EOS. However, the 3c = s + 2 is used for all substances toavoid the discontinuity of c for spherical and chain moleculesin this work. Eqs. (19) and (20) contain additional terms (thelast term in each equation) concerning the external degrees offreedom, c, as compared with the original SS EOS. Because ofthe additional terms the modified EOS satisfies PV/NkT = 1 atthe limit of infinite volume.

The chemical potential for a pure component, µ, which isneeded for phase equilibrium calculations, can be expressed as:

µ

RT= ln

(y

s

)+ sy−1(1 − y) ln(1 − y) − (s − 1) ln

[(z − 1)

e

]

− 3c ln[ya(ω1/3 − b2−1/6v∗1/3) + (1 − ya)ω1/3]

+ 3

2ln

[sh2

2πmkT

]+ (1 − c) ln(ω)

+ c( y

2T

)( 1

yV

)2[

1.011

(1

yV

)2

− 2.409

]

+ c

{1

1 − 2−1/6bya(yV )−1/3

+ 2y2

[1.011

2 − 1.2045

]+ 1 − c

}(21)

(yV ) T (yV ) c

where m is the molecular mass and h is the Planck’s constant.

4. Determination of characteristic parameters

A real fluid can be described by three characteristic parame-ters ε*, v∗, and s or equivalently, T*, P*, and V* in the SS EOS andthe MSS EOS. These parameters in the MSS-I and MSS-II EOSfor 44 low molecular-weight substances were determined usingsaturated vapor pressures (Pvp), critical constants (Tc, Pc, ρc)and saturated liquid density at normal boiling point (ρL

nb,sat). 16data of the saturated vapor pressures were obtained with Wagnerequation [15,16] instead of experimental data in the tempera-ture range from Tc/2 to Tc for substances of Tc/2 <273.15 K,or in the range from 273.15 K to Tc for the other substances.Critical constants were obtained from Poling et al. [16], andsaturated liquid densities at the normal boiling point were calcu-lated from the Rackett equation [17]. The objective function usedwas:

OF =∣∣∣∣∣T

calc − T

expc

Texpc

∣∣∣∣∣2

× 100 +∣∣∣∣∣P

calc − P

expc

Pexpc

∣∣∣∣∣2

+∣∣∣∣∣ρ

calc − ρ

expc

ρexpc

∣∣∣∣∣2

+16∑i=1

∣∣∣∣∣Pcalvp,i − P

expvp,i

Pexpvp,i

∣∣∣∣∣2

+∣∣∣∣∣ρ

L,calnb,sat − ρ

L,expnb,sat

ρL,expnb,sat

∣∣∣∣∣2

(22)

In Eq. (22), critical temperature was more heavily weighed thancritical density and saturated liquid density. Critical pressure andvapor pressures were also heavily weighed using 16 points forvapor pressure. It is because they are more important for phase

introducing two universal constants a and b as:

vf = [ya(ω1/3 − b2−1/6v∗1/3) + (1 − ya)ω1/3]3

(18)

The constant a gives a modification to the linear combination(a = 1) of gas-like and solid-like free lengths with the fractionof occupied sites, y, while the constant b can be a correction tothe hard-sphere diameter owning to the local geometry of close-packed, flexible, polymer molecules and improve the solid-likefree length. The same numerical factor, q, was also postulated inthe MCM EOS [6,7]. The present work examined the two ver-sions of modified SS (MSS) models and compared the results ofthe models. In the MSS-I EOS, a = b = 1 is applied and thereforeEq. (18) is the same as Eq. (3). On the other hand, in the MSS-IIEOS, a = 1.140 and b = 1.257 are treated as universal constants,which are determined from 44 low molecular-weight substancesand 64 polymers as described later.

By the same procedure as used in the derivation of the originalSS EOS, the modified SS EOS for the occupied-site fraction andpressure are obtained as follows:

s

3c

(s − 1

s− ln(1 − y)

y

)− 2−1/6abya(yV )−1/3 − 1/3

1 − b2−1/6ya(yV )−1/3

− y

6(yV )2T

(2.409 − 3.039

(yV )2

)+ 1 − c

3c= 0 (19)

PV

T=[

1 − 2−1/6bya

(yV )1/3

]−1

+ 2y

(yV )2T

[1.011

(yV )2 − 1.2045

]

+1 − c

c(20)

M. Wang et al. / Fluid Phase Equilibria 242 (2006) 10–18 13

equilibrium calculation. The non-linear least square algorithm ofMarquardt [18] was used in the optimization of the characteristicparameters.

On the other hand, the characteristic parameters for 64homopolymers and copolymers were determined from their PVTdata expressed with Tait equation by Rodgers [1], and experi-mental data presented by Zoller and Walsh [19] and Sato et al.

[20]. The objective function used for polymers was:

OF =n∑

i=1

∣∣∣∣∣ρcali − ρ

expi

ρexpi

∣∣∣∣∣2

(23)

The characteristic parameters (ε*, v∗ and s) for the MSS-IEOS were determined for each substance by setting a and b

Table 1Characteristic parameters and deviations of critical constants, saturated vapor pressure, and saturated liquid density at normal boiling point for 44 small moleculesin MSS-I EOS

Substance s e*/k (K) v∗ (cm3/mol) Errora

Tc Pc ρc Pvpb ρnb,sat

L,c

Methane 0.691 184.2 46.55 0.70 2.83 −18.57 1.96 −4.52Ethane 1.159 223.5 38.98 0.75 3.00 −19.15 1.82 −3.92Propane 1.522 235.2 39.03 0.74 3.58 −20.11 1.80 −2.97Butane 1.790 249.8 41.39 0.81 3.44 −20.78 1.74 −3.06Isobutane 1.684 246.9 44.73 0.82 3.10 −19.90 1.81 −3.23Pentane 2.080 257.4 42.71 0.96 4.01 −21.90 1.86 −5.30Isopentane 1.927 261.3 46.16 0.92 3.16 −21.44 1.79 −3.79Neopentane 1.716 262.3 50.81 1.66 8.25 −19.31 1.07 1.60Hexane 2.395 261.3 42.53 1.08 5.54 −21.98 2.46 −8.97Heptane 2.709 263.8 42.65 1.25 6.28 −22.53 2.10 −12.4Octane 3.023 265.5 42.68 1.34 7.21 −22.13 2.28 −14.8Nonane 3.353 266.3 42.11 1.38 8.29 −21.84 2.36 −16.7Decane 3.648 268.0 42.43 1.48 8.83 −21.64 2.45 −19.0Undecane 3.945 269.4 42.31 1.56 8.83 −21.48 2.51 −20.1DCCEP1BTMEP21HHNCopmAD11CCCP11A

odecane 4.266 269.6 42.27yclopentane 1.769 302.3 42.14yclohexane 1.882 318.0 46.06thylene 1.126 209.7 35.86ropylene 1.459 237.0 37.41-Butene 1.740 249.8 39.88enzene 1.865 324.4 39.37oluene 2.216 314.9 39.36

ethanol 3.725 221.7 9.14

thanol 4.558 206.2 9.02ropanol 4.918 207.5 10.18-Propanol 5.189 193.5 9.70-Pentanol 5.032 224.4 14.39

2O 2.485 329.4 7.04

3N 2.050 224.3 10.92

2 0.838 109.7 33.74O2 2.072 167.2 12.30-Xylene 2.423 323.1 41.82-Xylene 2.472 313.4 42.27-Xylene 2.500 312.2 41.39cetone 2.361 264.0 27.34iethyl ether 2.350 241.9 33.87,2-Dichlropropane 2.052 318.9 38.84,2-Dichlropropane 2.769 259.2 28.93S2 1.217 435.7 37.48Cl4 1.792 359.9 40.09hloroform 1.902 337.1 30.69henol 3.370 338.9 17.20,1-Difluoroethane 2.129 230.9 22.76-Chloro-1,1-difluoroethane 2.058 247.4 27.70ADf

a Error = 100 × (cal − exp)/exp.b Error = (

∑|cal − exp|/exp) × (100/n), n: number of data, Wagner parameters werc Calculated from Rackett equation (Daubert et al. [17]).d Liquid density at triple point was used.e Wagner parameters were obtained from Reid et al. [15].f AAD =

∑|Error|/n, n: number of substances.

1.59 10.2 −22.22 2.62 −22.00.84 4.41 −19.59 1.73 −5.571.00 6.30 −19.06 1.92 −12.90.70 3.45 −18.99 1.58 −3.640.71 3.50 −19.14 1.60 −3.090.78 3.50 −19.69 1.55 −3.310.90 5.66 −21.17 1.88 −8.180.94 7.83 −19.86 2.52 −10.9

1.99 6.84 −33.05 5.14 −16.11.36 5.68 −26.91 3.18 −4.780.73 8.08 −21.97 2.18 −2.600.96 8.96 −23.31 1.18 −2.050.54 7.76 −20.30 4.32 −4.211.65 7.19 −31.45 3.56 −18.81.28 6.38 −27.58 2.74 −12.90.60 2.24 −17.22 1.61 −1.851.11 7.51 −17.61 0.94 −0.54d

1.25 6.80 −21.28 2.26 −14.451.33 7.23 −22.35 2.58 −15.781.32 7.17 −22.42 2.48 −15.371.43 8.22 −29.80 3.01 −16.431.08 3.98 −23.53 2.86 −5.371.04 3.78 −19.23 2.23 −8.621.19 7.06 −29.44 2.26 −8.150.92 1.82 −22.14 2.69e −4.120.92 5.12 −20.20 1.95e −8.470.89 2.09 −15.56 1.83e −6.171.13 4.06 −28.82 2.10e −2.331.23 5.61 −25.03 2.09e −10.70.55 6.44 −18.08 1.51e −2.781.07 5.63 21.91 2.22 7.94

e obtained with Poling et al. [16].

14 M. Wang et al. / Fluid Phase Equilibria 242 (2006) 10–18

in Eq. (18) as unity. For the MSS-II EOS, a, b, ε*, v∗ and swere first optimized for each substance. From some additionalinvestigation it was found that the correlation errors for lowmolecular-weight substances were more sensitive to a than b,while correlation errors for polymers were more sensitive to b.Based on these results, the values of a = 1.140 and b = 1.257in the MSS-II EOS were determined as the average values

for the low molecular-weight substances and for the polymers,respectively. Finally, ε*, v∗ and s for the MSS-II EOS wereoptimized again for each substance using the determined valuesof a and b. The value of b = 1.257 is larger than the averagevalue of q = 1.07 in the MCM EOS [6,7], and it may be dueto the difference between the free volume theory and the holetheory.

Table 2Characteristic parameters and deviations of critical constants, saturated vapor pressure, and saturated liquid density at normal boiling point for 44 small moleculesin MSS-II EOS

Substance s e*/k (K) v∗ (cm3/mol) Errora

Tc Pc ρc Pvpb ρnb,sat

L,c

Methane 0.976 175.1 24.67 0.52 0.58 −16.99 1.40 1.45Ethane 1.554 217.8 21.61 0.57 0.32 −17.62 1.44 2.50Propane 2.009 231.2 21.84 0.52 0.65 −18.54 1.27 3.88Butane 2.347 246.6 23.19 0.58 0.39 −19.18 1.50 4.02Isobutene 2.202 244.2 25.17 0.68 0.38 −18.15 1.56 3.87Pentane 2.722 254.6 23.82 0.72 0.88 −20.20 1.51 2.00Isopentane 2.525 258.2 25.82 0.68 0.03 −19.80 1.50 3.34Neopentane 2.371 250.0 27.68 0.51 0.50 −20.46 1.07 5.41Hexane 3.091 260.9 23.78 1.10 3.58 −19.44 1.81 −0.85Heptane 3.522 262.3 23.64 1.04 3.12 −20.54 2.02 −4.90Octane 3.918 264.7 23.61 1.16 3.98 −20.04 2.21 −7.35Nonane 4.325 266.3 23.33 1.24 4.84 −19.79 2.44 −9.25Decane 4.728 267.6 23.22 1.33 5.38 −19.42 2.53 −11.4Undecane 5.110 269.2 23.06 1.43 5.31 −19.20 2.60 −12.4DCCEP1BTM 1.83 3.36 −31.17 5.76 −7.53EP21HHNCopmAD11CCCP11A

odecane 5.555 269.1 22.73yclopentane 2.319 298.6 23.64yclohexane 2.470 314.0 25.60thylene 1,488 206.5 20.03ropylene 1.922 233.4 20.98-Butene 2.283 246.5 22.37enzene 2.450 320.1 21.90oluene 2.859 314.3 22.12ethanol 4.834 221.3 4.99

thanol 5.942 205.6 4.89ropanol 6.398 207.2 5.53-Propanol 6.840 192.0 5.17-Pentanol 6.573 223.9 7.63

2O 3.230 327.5 3.91

3N 2.678 222.2 6.09

2 1.155 105.5 18.33O2 2.590 169.2 7.28-Xylene 3.127 322.4 23.56-Xylene 3.197 312.4 23.66-Xylene 3.229 311.4 23.23cetone 3.114 259.2 15.63iethyl ether 3.053 240.6 18.99,2-Dichloropropane 2.679 315.9 21.73,2-Dimethoxyethane 3.632 255.6 16.63S2 1.569 392.6 24.23Cl4 2.288 327.0 25.67hloroform 2.422 306.8 19.72henol 4.271 311.3 10.83,1-Difluoroethane 2.672 211.8 14.75-Chloro-1,1-difluoroethan 2.608 225.9 17.85ADf

a Error = 100 × (cal − exp)/exp.b Error = (

∑|cal − exp|/exp) × (100/n), n: number of data, Wagner parameters werc Calculated from Rackett equation (Daubert et al. [17]).d Liquid density at triple point was used.e Wagner parameters were obtained from Reid et al. [15].f AAD =

∑|Error|/n, n: number of substances.

1.45 6.71 −19.80 2.86 −14.10.63 1.37 −17.95 1.54 1.410.80 3.66 −16.99 1.66 −5.920.78 2.17 −16.56 0.92 3.810.56 0.90 −17.4 1.20 3.900.56 0.41 −18.13 1.33 3.660.68 2.84 −19.30 1.39 −0.870.93 5.63 −17.44 2.80 −3.05

1.13 0.73 −25.60 3.76 4.400.53 2.43 −20.91 1.55 6.400.56 2.63 −22.50 1.40 6.950.38 3.99 −17.73 3.20 6.631.49 4.40 −29.56 3.59 −11.11.09 3.60 −25.76 2.51 −5.420.42 −0.29 −15.75 1.07 4.441.05 4.47 −15.72 0.44 5.84d

1.12 3.68 −19.43 2.27 −7.501.21 4.31 −20.33 2.47 −8.661.18 4.09 −20.52 2.43 −8.340.69 0.18 −31.02 2.30 −13.30.88 0.58 −22.01 2.57 2.000.83 0.76 −17.50 2.05 1.410.46 −1.66 −31.09 1.73 −5.080.76 1.18 −20.94 2.43e 2.170.72 2.22 −18.45 1.45e 1.370.66 1.05 −13.91 1.58e 1.000.86 0.21 −27.53 1.94e 6.351.09 2.46 −23.47 2.08e 3.660.32 2.93 −16.59 1.35e 4.490.86 2.38 20.46 2.01 5.30

e obtained with Poling et al. [16].

M. Wang et al. / Fluid Phase Equilibria 242 (2006) 10–18 15

5. Results and discussion

5.1. Application to pure low molecular-weight substances

Correlation errors of critical properties, vapor pressures andsaturated liquid density at normal boiling point as well as thecharacteristic parameters in the MSS-I and the MSS-II EOSsare listed in Tables 1 and 2, respectively, for 44 low molecular-weight substances. The average absolute deviations (AAD) forcritical temperature, critical pressure and vapor pressure were1.07, 5.63 and 2.22% for the MSS-I EOS, and 0.86, 2.38 and2.01% for the MSS-II EOS. The AAD for critical density andsaturated liquid density at normal boiling point were somewhatlarger, being 21.91 and 7.94% for the MSS-I EOS and 20.46and 5.30% for MSS-II EOS, respectively. The MSS-II EOS gavesmaller deviations than the MSS-I EOS due to the introductionof the constants a and b in the free volume term.

Fig. 1 shows the correlation results of vapor pressure formethane, carbon dioxide, butane and octane with the MSS-IIEOS. Correlation results were satisfactory for both the EOSs upto the critical point as shown in the figure.

To compare the performance of the original and the modifiedSS EOSs, typical behavior of the cell volume and the occupied-site fraction were plotted as functions of reduced density inFig. 2. As the reduced density approaches zero, the cell vol-ume of the original SS EOS increases suddenly to infinity andtTiEmtvd

Mad

Fb

Fig. 2. Changes of cell volume and the occupied-site fraction with reduceddensity in (a) the original SS EOS and (b) the modified SS EOSs for butane.

large in the liquid region, because the characteristic parameterswere mainly determined from vapor pressures and the criticaltemperature. For comparison, PρT behavior was also predictedwith the SL EOS [5], which is widely used to correlate the

Fig. 3. Comparison of PρT relationships between experiments and predictionsby the MSS-I, the MSS-II, and the SL EOSs for carbon dioxide at temperaturesfrom 280 to 400 K.

he occupied-site fraction has a minimum as shown in Fig. 2(a).his behavior is unrealistic and often leads to failure in numer-

cal vapor–liquid equilibrium calculations with the original SSOS. For this reason, the characteristic parameters for smallolecules could not be determined for the original SS EOS in

his work. In the MSS-I and MSS-II EOSs, however, the cellolume reaches a constant value and the occupied site fractionecreases to zero as shown in Fig. 2(b).

Prediction of PρT relationship for CO2 was carried out by theSS-I and MSS-II EOSs at temperatures from 280 to 400 K and

t pressures up to 20 MPa. The results are shown in Fig. 3. Theeviations between predicted and experimental densities [21] are

ig. 1. Correlation results of vapor pressures for methane, carbon dioxide,utane and octane.

16 M. Wang et al. / Fluid Phase Equilibria 242 (2006) 10–18

Fig. 4. Comparison of the second virial coefficients predicted by the MSS-I andthe MSS-II EOSs with Tsonopoulos equation for carbon dioxide, propane andbutane.

solubility of high-pressure gases in molten polymers [22–24].The characteristic parameters in the SL EOS for CO2 were deter-mined by Wang et al. [25], where vapor pressures and liquiddensities were used in the regression. Consequently, we foundthe largest deviations between the predictions and experimentswere 16% in the MSS-I, 12% in the MSS-II, and 17% in the SLEOS. The average deviations were 5.4, 2.7, and 5.7%, respec-tively. The MSS-II EOS gave better PρT predictions due to theadditional constants a and b.

Fig. 4 shows comparison of the second virial coefficients ofCO2, propane and butane predicted by the MSS-I and the MSS-IIEOS with the Tsonopoulos equation [26]. As seen in Fig. 4, thereis about 10% average deviation between the MSS-I EOS andTsonopoulos equation, while it is within 5% for the MSS-II EOS.From these results, the MSS-I and MSS-II EOSs can be appliedto both gaseous and liquid states of low molecular-weight sub-stances. The MSS-II EOS is better in describing properties oflow molecular-weight substances than the MSS-I EOS.

The relationship between the characteristic parameters of s,v∗ and ε*/k and carbon atoms of normal alkanes can be dis-cussed, as illustrated in Fig. 5. In both MSS-I and MSS-II EOSs,s increased almost linearly with the carbon number, while v∗and ε*/k approached a constant. Thus, introduction of a groupcontribution method may be applied for these characteristicparameters.

5

clcavtcs

Fig. 5. Relationship between characteristic parameters and carbon number ofnormal alkanes for the two MSS EOSs.

behavior of polymers accurately. As an example, Fig. 6 showsthe comparison of correlated and experimental PVT behaviorfor polycarbonate [20]. The three EOSs gave almost the sameaverage absolute derivations (0.078–0.080%) for this polymer.

Thus, the MSS EOSs can be applied to both small and largemolecules without reducing the high performance to correlatepolymer PVT behavior of the original EOS. The extension of theMSS EOSs to mixtures consisting of low and high molecular-weight substances will be discussed in our future work.

Fb

.2. Application to polymers

Average absolute deviations of polymer densities and theharacteristic parameters in the MSS-I and MSS-II EOSs areisted in Table 3 for 64 polymers. For comparison, the same cal-ulations were carried out for the original SS EOS. The grandverage absolute deviation between experimental and correlatedalues for the MSS-II EOS was 0.050% and was slightly betterhan that for the original SS EOS and the MSS-I EOS. However,onsidering the accuracy of experimental data used, it can beaid that these three EOSs have the same ability to correlate PVT

ig. 6. Comparison of experimental and predicted specific volumes of polycar-onate.

M. Wang et al. / Fluid Phase Equilibria 242 (2006) 10–18 17

Table 3Characteristic parameters of polymer liquids determined for the MSS EOSs and deviations of densities of polymer liquids for MSS EOSs and SS EOS

Polymer MSS-I MSS-II SS Reference

P* (MPa) T* (K) V* (cm3/g) AADa P* (MPa) T* (K) V* (cm3/g) AADa AADa

HDPE 538.9 11512 1.208 0.027 807.8 12661 0.984 0.026 0.027 [1]LDPE 616.1 11010 1.192 0.031 923.4 12111 0.972 0.031 0.031 [1]PS 764.2 12277 0.923 0.014 1093.9 13543 0.782 0.013 0.023 [1]PoMS 743.6 12610 0.980 0.012 1111.0 13903 0.798 0.012 0.012 [1]PVAc 953.6 9159 0.814 0.013 1430.2 10083 0.663 0.012 0.013 [1]PMMA 932.5 11658 0.838 0.080 1393.3 12849 0.683 0.078 0.080 [1]BPE 729.0 9840 0.947 0.045 1076.1 10902 0.774 0.045 0.045 [1]PIB 689.6 11151 1.096 0.007 1026.1 12339 0.894 0.008 0.007 [1]PDMS 533.2 7647 0.958 0.030 801.7 8400 0.780 0.022 0.030 [1]PBD 868.1 8895 1.073 0.043 1292.6 9822 0.875 0.042 0.043 [1]PEO 916.3 9921 0.882 0.055 1378.8 10890 0.718 0.052 0.055 [1]PTHF 733.7 10041 1.010 0.032 1100.6 11043 0.822 0.030 0.032 [1]LDPE-A 713.5 10358 1.168 0.044 1072.7 11370 0.951 0.043 0.047 [1]LDPE-B 701.8 10623 1.176 0.040 1054.9 11673 0.958 0.038 0.044 [1]LDPE-C 598.2 10707 1.169 0.041 1070.1 11502 0.954 0.041 0.045 [1]i-PP 555.6 10935 1.195 0.081 836.5 11994 0.972 0.081 0.082 [1]i-PB 598.2 10707 1.169 0.050 899.4 11757 0.952 0.049 0.050 [1]PET 1191.2 11523 0.743 0.055 1797.9 12630 0.605 0.054 0.055 [1]PPO 916.5 10380 0.863 0.075 1382.8 11370 0.702 0.074 0.075 [1]PC 1023.0 11523 0.816 0.042 1538.3 12666 0.664 0.042 0.042 [1]Par 1002.3 12484 0.812 0.033 1507.1 13666 0.661 0.033 0.033 [1]PH 1139.0 11400 0.853 0.035 1709.6 12546 0.695 0.034 0.035 [1]PEEK 1088.7 12240 0.770 0.073 1641.3 13407 0.627 0.074 0.073 [1]PVME 840.9 10170 0.965 0.084 1257.8 11187 0.786 0.078 0.084 [1]PA6 590.7 15841 0.826 0.046 881.0 17464 0.674 0.046 0.046 [1]PA66 733.2 10291 0.816 0.051 1104.3 13285 0.664 0.051 0.051 [1]PMA 940.3 10080 0.842 0.096 1411.2 11100 0.686 0.092 0.096 [1]PEA 797.8 9720 0.877 0.100 1199.1 10656 0.714 0.100 0.100 [1]TMPC 843.8 10860 0.870 0.050 1270.1 11892 0.708 0.050 0.050 [1]HFPC 835.0 10081 0.628 0.096 1116.7 11266 0.515 0.096 0.096 [1]BCPC 899.9 12151 0.702 0.076 1351.6 13363 0.572 0.070 0.076 [1]PECH 848.8 11011 0.735 0.015 1264.4 12145 0.599 0.015 0.015 [1]PCL 780.7 10290 0.912 0.020 1170.6 11319 0.743 0.020 0.020 [1]PVC 763.2 12241 0.727 0.040 1136.1 13528 0.593 0.040 0.040 [1]�-PP 574.6 9510 1.138 0.049 863.4 10467 0.927 0.049 0.049 [1]EP50 615.3 12481 1.236 0.150 922.3 13717 1.006 0.150 0.150 [1]EVA18 707.3 10410 1.136 0.019 1036.0 11427 0.925 0.013 0.019 [1]EVA25 621.6 9150 1.116 0.033 1044.3 11232 0.908 0.033 0.033 [1]EVA28 741.5 10110 1.116 0.040 1114.7 11100 0.908 0.040 0.040 [1]EVA40 746.7 10200 1.184 0.041 1120.0 11190 0.965 0.041 0.041 [1]SAN3 729.1 11680 0.942 0.035 1086.2 12973 0.769 0.035 0.035 [1]SAN6 792.2 11070 0.933 0.031 1187.1 12147 0.760 0.032 0.031 [1]SAN15 772.0 11881 0.928 0.019 1158.3 13069 0.756 0.018 0.019 [1]SAN18 779.0 11971 0.924 0.016 1167.2 13186 0.753 0.016 0.016 [1]SAN70 879.3 13411 0.891 0.038 1311.5 14821 0.726 0.036 0.038 [1]SMMA20 777.4 11395 0.917 0.040 1153.1 12549 0.748 0.038 0.040 [1]SMMA60 817.2 11166 0.868 0.043 1203.9 12520 0.711 0.043 0.043 [1]SAN40 754.1 12520 0.914 0.043 1111.2 14125 0.748 0.042 0.043 [1]PnEA 803.0 9627 0.939 0.058 1207.6 10575 0.764 0.056 0.058 [19]PAC 1477.6 13584 0.707 0.046 2201.0 15009 0.576 0.046 0.046 [19]PnPA 843.9 9756 0.878 0.095 1269.8 10725 0.715 0.093 0.095 [19]PVB 938.4 9873 0.896 0.067 1414.1 10842 0.730 0.065 0.067 [19]PVF 1116.0 10535 0.809 0.044 1670.7 11580 0.658 0.043 0.044 [19]PMMA 1007.1 11391 0.834 0.051 1513.0 12528 0.679 0.050 0.051 [19]PA6 1330.3 11505 0.906 0.061 2003.9 12624 0.738 0.060 0.061 [19]PA9 972.8 11685 0.994 0.085 1466.2 12834 0.810 0.083 0.085 [19]PA12 892.3 11406 1.035 0.073 1345.5 12525 0.843 0.070 0.073 [19]PA6/9 1020.1 12183 0.974 0.046 1535.9 13377 0.792 0.044 0.046 [19]PEO 1103.2 9561 0.870 0.037 1660.6 10497 0.709 0.036 0.037 [19]Pub 835.5 11985 0.985 0.098 1257.2 13173 0.802 0.096 0.098 [19]PES 1290.9 10014 0.755 0.051 1943.6 10989 0.615 0.050 0.051 [19]PCr 950.2 10251 0.912 0.028 1427.3 11268 0.743 0.027 0.028 [19]PP 558.0 10494 1.182 0.150 842.7 11526 0.962 0.150 0.150 [20]PC 956.4 11691 0.817 0.080 1439.2 12831 0.666 0.078 0.080 [20]Grand AAD (%)b 0.052 0.050 0.052

a AAD =∑

|ρcali − ρ

expi |/ρexp

i × (100/n), n: number of data.b Grand AAD =

∑|AAD|/n, n: number of substances.

18 M. Wang et al. / Fluid Phase Equilibria 242 (2006) 10–18

6. Conclusion

The Simha–Somcynsky EOS was modified to allow appli-cation to gaseous state of low molecular-weight substances byintroducing the perturbed hard-chain theory of Beret and Praus-nitz into the free volume term. The modified EOS (MSS-I EOS)could satisfy the ideal-gas limit showing realistic behavior forthe occupied-site fraction and the cell volume, and could beapplied to gases, while the original SS EOS could not except forspherical molecules. In addition to this modification, two univer-sal constants were introduced into the free volume expression forbetter representation of equilibrium properties (MSS-II EOS).The characteristic parameters of 44 low molecular-weight sub-stances and 64 polymers were obtained for the two MSS EOSs.For the low molecular-weight substances, the AADs for criti-cal temperature, critical pressure and vapor pressure were 1.07,5.63 and 2.22% for the MSS-I EOS, and 0.86, 2.38 and 2.01%for the MSS-II EOS, respectively. The AADs for critical den-sity and saturated liquid density at normal boiling point weresomewhat larger, being 21.91 and 7.94% for the MSS-I EOSand 20.46 and 5.30% for MSS-II EOS, respectively. For poly-mers, the MSS EOSs could maintain the high performance inPVT behavior description of the original EOS. Furthermore, PρTbehavior and second viral coefficients for low molecular-weightsubstances were predicted and the MSS-II EOS was found to besuperior to the MSS-I EOS. It is due to the additional universalcdsbm

LaAbcEgGhkmNPqqRsTv

v

VVyzZ

Greek lettersε* parameter for interaction energy of a segment pair (J)Λ de Broglie’s wavelength (m)µ chemical potential (J)ρ density (kg/m3)ω cell volume (m3)

Superscriptscal calculated valueexp experimental value* characteristic parameter∼ reduced property

Subscriptsc critical propertyL liquid statenb,sat saturated state at normal boiling pointvp vapor pressure

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[[

[[[[

[

[

[[

[

[

[

[

[

[

[

onstants a and b in the MSS-II EOS that are more effective inescribing thermodynamic properties for low molecular-weightubstances than that in the MSS-I EOS. The MSS EOSs wille extended to phase equilibrium calculation for gas–polymerixtures in our future publication.

ist of symbolsuniversal constant in Eq. (18)Helmholtz energy (J)universal constant in Eq. (18)parameter for external degree of freedom

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total volume (m3)f free volume of a molecule (m3)

fraction of occupied sitescoordination number (=12)configurational partition function

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