Ind. Eng. Chem. Res. 1999, 38, 4951-4958
4951
GENERAL RESEARCH A Molecular-Based Equation of State for Simple and Chainlike Fluids Y. C. Chiew,*,†,‡ D. Chang,† J. Lai,† and G. H. Wu‡ Department of Chemical Engineering, National University of Singapore, Kent Ridge Crescent, Singapore 119260, Singapore, and Department of Chemical and Biochemical Engineering, Rutgers University, Piscataway, New Jersey 08855-0909
A perturbed Lennard-Jones chain (PLJC) equation of state is developed for simple chainlike fluids and polymers. The PLJC equation is formulated on the basis of a well-defined molecular modelsthe Lennard-Jones chains. Analytical expressions for the Helmholtz energy and compressibility factor were derived on the basis of a first-order variational perturbation theory using the hard-sphere chain as reference. The PLJC equation of state is characterized by three segment-based parameters, namely, the number of segments per molecule m, segment size σ, and segment energy /k. Unlike other perturbation theory-based models, the diameter of the hard-sphere chain segments depends not only on temperature kT/ but also on chain length m. Model parameters for nonassociating molecular fluids were obtained by regressing against vapor pressure and saturated liquid density data. The PLJC model was also applied to high molecular weight polymers; model parameters were obtained by fitting measured specific volumes. 1. Introduction Equations of state models are practical tools for estimating thermodynamic properties and phase behavior of fluids. They have been widely used in process simulation and design. In recent years, a great number of theoretical studies aimed at developing molecularbased equations of state have been reported in the literature. Specifically, there has been increased interest in developing molecular-based equation of state models for chainlike molecules. These analytical equations of state possess physically significant parameters which capture the effects of molecular size, shape, and energy.1,2 Most of these analytical models are derived on the basis of perturbation theories formulated in statistical thermodynamics of dense fluids. Perturbation theories for chain molecules fall into two categories: (1) those whose reference system constitutes nonassociated, nonbonded segments and (2) those whose reference system constitutes bonded hard-sphere chains. Wertheims’s first-order thermodynamic perturbation theory (TPT1) is an example of category (1) where the reference system comprises nonbonded monomers whose potential of interaction is the same as that for the segment in a chain molecule. Banaszak, Chiew, and Radosz,3,4 employed the first-order thermodynamic perturbation theory to determine the pressure of adhesive hard chains, square-well chains, and Lennard-Jones chains. By contrast, the reference system used in theories belonging to category (2) comprises preconnected hard* To whom correspondence should be addressed. E-mail:
[email protected]. † National University of Singapore. ‡ Rutgers University.
sphere chains. The reference system is perturbed solely by the pair potential energy. An example of this approach is the variational perturbation theory of O’Lenick and Chiew.5 In this perturbation approach, the freely jointed hard-sphere chain fluid is used as the reference in the same way the hard-sphere fluid is used as the reference in most equations of state for simple spherical fluids. The hard-sphere chain model has been studied by a number of investigators.6-12 Each molecule in this model fluid consists of a series of linearly jointed hardsphere segments that are free to rotate around each other. Despite its simplicity, it captures connectivity effects and excluded-volume interactions in chain molecules. In this paper we present the perturbed Lennard-Jones chain (PLJC) equation of state applicable to real simple and chain molecules. The theoretical foundation for the proposed PLJC equation of state is the variational perturbation theory for Lennard-Jones chain fluid developed by O’Lenick and Chiew.5 Recently, Von Solms, O’Lenick, and Chiew13 extended this theory to LennardJones chain mixtures. The goal of this work is to derive a theoretically based practical equation of state applicable to simple molecules and chain fluids over a wide range of densities and temperatures. In the PLJC model, simple and chain fluids are characterized by three parameters: the number of segments in a molecule m, segment size σ, and nonbonded segment energy . These segment-based parameters are obtained from available data such as vapor pressure and saturated vapor and liquid densities. The PLJC equation of state presented in this paper is similar to the statistical associating fluid theory (SAFT),1 the perturbed hard-sphere chain (PHSC)2 and the generalized flory dimer (GFD)14 equations of state
10.1021/ie990208x CCC: $18.00 © 1999 American Chemical Society Published on Web 11/05/1999
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Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999
in that all four models use three parameters: chain length, segment size, and segment energy to characterize real fluids, and the hard-sphere chain fluid to account for repulsion between chain molecules. Note that unlike the SAFT, PHSC, and PLJC equations which were derived via perturbation theory, the GFD was obtained through a mean field treatment. The PLJC equation differs from the SAFT, PHSC, and GFD equations in the following ways. First, in the SAFT equation, the hard-sphere chain equation was obtained from the first-order thermodynamic perturbation theory for associating fluids (TPT1), while the hard-sphere chain term used in the PHSC equation2 was derived from a modification of the Percus-Yevick (PY) solution obtained by Chiew.9 In the GFD, the hard-sphere chain term was obtained from a generalized dimer theory.11 In the PLJC equation, the hard-sphere term comes from the PY result of Chiew9 directly. Second, the perturbation contribution that accounts for nonbonded attraction between molecules is treated differently in these models. The SAFT equation1 uses the power series initially fitted by Alder et al.15 to molecular dynamics data for squarewell spheres, and subsequently modified by Chen and Kreglewski,16 to model attraction between chain segments. The PHSC uses a van der Waals-type perturbation term coupled with the Song-Mason method17 to model attraction. The attractive contribution in the GFD was based on a square-well chain model and derived in the context of the generalized dimer mean field theory. In the PLJC model the attractive perturbation term is determined through a first-order variational perturbation scheme. Its calculation takes into account the chain connectivity effect in a self-consistent manner through the interchain segment-segment correlation function determined from the Percus-Yevick theory.9 Third, in both the SAFT and PHSC equations, the effective hardsegment diameter of the reference hard-sphere chain fluid depends on temperature only. In contrast, the hard-segment diameter in the PLJC equation depends on both temperature and chain length. From a theoretical viewpoint, this chain length dependence is not unexpected since it captures chain end effects and interaction between molecules. This paper is organized as follows. In section 2 we present the first-order variational theory which forms the theoretical background of the PLJC equation. Section 3 gives the analytical expressions for the compressibility factor and Helmholtz energy of the PLJC model. Results of the application of the PLJC equation to real molecules and polymers are reported and discussed in section 4. Conclusions of this study are summarized in section 5. 2. Theoretical Background The perturbed Lennard-Jones chain (PLJC) equation of state model presented here is based on the first-order variational perturbation theory for Lennard-Jones chains (LJC) developed by O’Lenick and Chiew.5 A LennardJones chain molecule consists of a series of m freely jointed segments that interact via the Lennard-Jones potential φ(r) ) 4{(σ/r)12 - (σ/r)6}. Each segment is therefore characterized by the size parameter σ and energy parameter . In this theory the Helmholtz free energy ALJC/NckT of Lennard-Jones chains are calculated by separating the Lennard-Jones potential into a hard-sphere reference potential and an attractive perturbation part:5
φ(r) ) φHSC(r) + w(r)
(1)
On the basis of the separation of the Lennard-Jones potential in eq 1 above, the Helmholtz free energy of Lennard-Jones chains ALJC/NckT is shown to follow the inequality5
ALJC AHSC APERT e + NckT NckT NckT
(2)
Here, AHSC/NckT refers to the Helmholtz free energy of the hard-sphere chain reference fluid, while the perturbation term APERT/NckT is given by
( )[ (
APERT 2πmF σ ) NckT T* a
3
IA +
)]
1 - 1 IB a6
(3)
Here, and σ represent the characteristic energy and size of the chain segments; F ) N/V is the total number of chain segments per unit volume; N denotes the total number of chain segments in the system and is related to Nc, the total number of chain molecules, by the simple relation N ) mNc; T* ) kT/ is the dimensionless temperature; the dimensionless parameter a ) d/σ where d is the diameter of the reference hard-sphere chain segment. The diameter d is state-dependent and can be expressed as a function of F, T, and chain length m. In eq 3 above, the quantities IA and IB can be evaluated from the equations below:5
IA )
(
)
∫0∞ ζ412 - ζ46
IB )
3 12 gHSC dζ inter(ζ; F*a ,m)ζ
3 2 ∫0∞ζ412 gHSC inter(ζ; F*a ,m)ζ dζ
(4a) (4b)
where gHSC inter is the interchain segment-segment correlation function of the hard-sphere chains. The correlation function gHSC inter has been solved for in the Percus-Yevick approximation by Chiew9 and was used in the evaluation of integrals IA and IB. The optimal values of the hard-segment diameter are obtained by minimizing the right-hand side of eq 2 with respect to the parameter a ) d/σ. It was found that this optimal value of aopt depends on temperature, density, and chain length m.5,13 The variational theory described above constitutes the foundation of the perturbed LennardJones chain equation of state model presented in the following section. 3. Analytical Perturbed Lennard-Jones Chains (PLJC) Equation of State In the perturbed Lennard-Jones chain model, the Helmholtz energy A/NckT of the chain fluid is given by
AREF APERT A ) + NckT NckT NckT
(5)
where the hard-sphere chain reference term is
AREF m m+3 3 ) + - (m + 1) + NckT (1 - η)2 2(1 - η) 2 (m - 1) ln(1 - η) (6) Here, Nc represents the total number of chain molecules
Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 4953
( ) P FckT
REF
1 + η + η2 - η3 1 + η/2 - (m - 1) (1 - η)3 (1 - η)2
)m
(9b)
and
( ) P FckT
PERT
)
(
)
12mη 12mη 1 J (η,m) + - 1 JB(η,m) 6 A T*a T*a6 a6 (9c)
where
( )
∂IA ) [(-1.0755 - 0.22169m) + ∂η m η(2.077 - 4.5236m) - η2(1.6623 - 4.41m)]/ [0.4571 + m] (9d)
JA(η,m) ≡ IA + η
Figure 1. Variation of dimensionless hard-segment diameter a ) d/σ with dimensionless temperature T* ) kT/ for different chain lengths m.
per unit volume in the system; k is Boltzmann’s constant; T is absolute temperature; m denotes the number of segments in a chain molecule; η is the hardsegment packing fraction defined as η ) πFd3/6 ) πF*a3/6 with F* ) Fσ3. The perturbation term APERT/ NckT is given by eq 3 above. The perturbation integrals IA and IA (given in eq 3), originally evaluated by O’Lenick and Chiew,5 are fitted to analytical expressions for use here:
IA ) [(-0.22169m - 1.0755) - (2.2618m -
( )
∂IB ) [(0.42130 + 0.03171m) + ∂η m η(0.1974 + 1.3253m) + 5.3598mη2]/m (9e)
JB(η,m) ≡ IB + η
The PLJC equation of state is applied to nonassociating molecular fluids and polymers. As given by eq 9, the model is not directly applicable to associating molecular fluids such as alcohols, acids, etc., for which specific interactions (e.g., hydrogen bonding) play an important role in their thermodynamic properties and structure. For these molecular fluids, modification of the theory to accommodate the effect of the interactions is required. Molecules where these interactions exist but are not significant (e.g., aromatics and the chloro alkanes) can be accommodated by the model.
1.0385)η + (1.47m - 0.55411)η2]/[0.4571 + m] (7a)
(
IB ) 0.03171 +
0.4213 0.0987 + 0.66264 + η+ m m 1.7866η2 (7b)
) (
)
With values of IA and IB, optimal values of dimensionless hard-segment diameter a ) d/σ are obtained numerically. We note that the density dependence of a ) d/σ is small over the density range of interest. To avoid cumbersome computations and facilitate the efficient application of the equation of state model to phase equilibria and thermodynamic property calculations of real fluids, we found that the dimensionless hardsegment diameter a ) d/σ can be accurately represented by the density independent equation:
(
21/6 -1 a
)
3
)
(0.005397m + 0.006354) kT + (9.44 + m) (0.01222m + 0.005102) (8) (9.947 + m)
Shown in Figure 1 is the variation of a ) d/σ with temperature kT/ and chain length m. The effective hard-segment diameter decreases with increasing temperature and chain length. To summarize, the PLJC equation of state model is given by
( )
P P ) FckT FckT where
REF
+
( ) P FckT
PERT
(9a)
4. Results and Discussion 4.1. Segment-Based Parameters for Nonassociating Molecular Fluids. We proceed to consider how to use the equation of state to correlate the properties of pure liquids. The model consists of three parameters: segment size σ, segment energy /k, and the number of segments m; it accounts for chain connectivity, repulsion, and attraction between chain segments. We apply the PLJC equation of state model to nonassociating normal fluids including normal alkanes, branched alkanes, normal alkenes, cyclic alkanes, aromatics, and chlorinated hydrocarbons. Temperatureindependent segment-based parameters σ, /k, and m) are regressed from vapor pressures and saturated liquid densities as a function of temperature. Values of these three parameters were obtained by minimizing the objective function S which measures the sum of squares of the relative deviation between calculated and experimental properties: NP
S)
∑
k)1
(
)
sat Pcalc. (Tk) - Psat expt.(Tk)
Psat expt.(Tk)
2
+ Nd
∑
k)1
(
)
sat Fcalc. (Tk) - Fsat expt.(Tk)
Fsat expt.(Tk)
2
where Np and Nd, respectively, represent the number of vapor pressure and saturated liquid density data points used. Equal weight was given to vapor pressure data and saturated liquid density data in the regression. The results are given in Table 1. The root-mean-square
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Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999
Table 1. Segment-Based Parameters for Nonassociating Fluids % root-mean-square dev. % average absolute dev. molar mass (g/mol) T range (K) chlorine carbon monoxide carbon dioxide sulfur dioxide hydrogen sulfide carbon disulfide
70.906 28.010 44.010 64.063 34.080 76.131
180-400 72-121 218-298 283-413 284-369 278-533
methane ethane propane butane pentane hexane heptane octane nonane decane dodecane tetradecane hexadecane eicosane octacosane hexatriacontane
16.043 30.07 44.097 58.124 72.146 86.172 100.198 114.224 128.259 142.276 170.34 198.39 226.43 282.56 394.77 506.99
92-180 160-300 190-360 220-420 233-450 243-493 273-523 303-543 303-503 313-573 313-523 313-533 333-593 393-573 449-704 497-768
cyclopentane methylcyclopentane ethylcyclopentane propylcyclopentane butylcyclopentane pentylcyclopentane cyclohexane methylcyclohexane ethylcyclohexane propylcyclohexane butylcyclohexane pentylcyclohexane
70.135 84.162 98.189 112.216 126.244 140.272 84.162 98.189 112.216 126.243 140.270 154.297
253-483 263-483 273-513 293-423 314-458 333-483 283-513 273-533 273-453 313-453 333-484 353-503
2-methylpropane 2-methylbutane neopentane
58.124 72.146 72.146
193-393 193-433 282-410
ethylene propylene butene hexene
28.054 42.081 56.108 84.156
133-263 140-340 203-383 213-403
chloromethane chloroethane 1-chloropropane 1-chlorobutane 1-chloropentane 1-chlorohexane dichloromethane trichloromethane tetrachloromethane
50.488 64.515 78.542 92.569 106.596 120.623 84.933 119.378 153.823
213-333 211-440 238-341 262-375 283-407 306-435 229-333 244-343 273-523
benzene toluene ethylbenzene propylbenzene butylbenzene chlorobenzene m-xylene tetralin biphenyl
78.114 92.141 106.168 120.195 134.212 112.559 106.168 132.205 154.213
300-540 293-533 293-573 323-573 293-523 273-543 308-573 293-673 433-653
naphthalene 1-methylnaphthalene 1-ethylnaphthalene 1-propylnaphthalene 1-butylnaphthalene phenanthrene anthracene pyrene triphenylene
128.174 142.201 156.228 170.255 184.282 178.234 178.234 202.255 228.293
373-693 383-551 393-563 403-545 43-537 373-633 493-673 513-673 553-693
m
σ (Å)
/k (K)
1.394 3.608 286.515 1.331 3.316 92.456 3.7676 2.3036 133.5771 3.814 2.485 190.150 1.421 3.342 250.969 1.714 3.728 339.285 Normal Alkanes 1.000 3.776 154.639 1.682 3.578 190.035 2.200 3.613 203.317 2.504 3.742 222.380 2.908 3.766 232.871 3.370 3.781 238.320 3.994 3.724 238.640 4.442 3.728 243.811 4.563 3.834 255.538 5.343 3.741 251.340 6.068 3.787 260.614 6.807 3.825 267.252 7.575 3.849 272.346 9.525 3.802 276.560 14.184 3.745 274.490 16.903 3.783 283.874 Cyclic Alkanes 2.323 3.811 279.215 2.651 3.893 275.225 2.893 3.987 285.580 3.517 3.912 277.098 3.534 4.081 292.800 4.128 4.003 287.069 2.536 3.926 290.471 2.707 4.094 292.535 2.868 4.206 306.735 3.298 4.173 302.732 3.836 4.090 296.672 4.373 4.063 292.691 Branched Alkanes 2.221 3.890 226.061 2.451 4.007 246.101 2.878 3.778 213.117 Alkenes 1.502 3.583 186.232 1.697 3.847 229.522 2.257 3.770 231.438 2.796 3.974 258.196 Chloro Alkanes 1.464 3.664 289.432 2.207 3.534 257.661 2.083 3.933 294.876 2.307 4.057 304.264 2.560 4.130 309.505 3.420 3.902 285.767 2.311 3.411 280.574 2.336 3.686 293.040 2.388 3.888 298.687 Aromatics 2.712 3.615 285.430 2.943 3.756 293.731 3.194 3.853 298.367 3.638 3.857 294.142 3.805 3.956 302.862 2.527 3.942 337.282 3.237 3.837 298.782 3.424 3.944 339.146 4.118 3.818 344.250 Polynuclear Aromatics 3.187 3.943 362.381 3.177 3.998 384.044 3.561 4.061 369.645 3.656 4.140 371.957 3.795 4.311 365.949 3.181 4.366 450.877 3.302 4.300 444.236 4.953 3.895 399.057 3.344 4.770 508.569
Psat
Fliq
Psat
vliq
3.90 1.98 1.74 2.07 5.17 5.50
2.10 1.43 4.29 4.62 5.74 4.14
3.61 1.78 1.48 1.83 4.42 4.71
1.65 1.18 3.47 3.78 4.87 2.40
4.84 3.62 4.13 5.40 4.24 4.78 4.37 3.57 1.51 3.88 1.15 0.56 1.96 2.46 1.95 1.51
2.69 3.66 3.50 5.74 2.49 3.56 3.53 2.60 0.74 1.70 1.07 1.12 1.07 0.69 3.25 3.46
3.49 3.06 3.60 4.70 3.63 4.01 3.77 3.03 1.30 3.30 0.96 0.45 1.57 2.09 1.54 1.35
2.37 2.27 2.61 3.53 1.82 2.48 2.48 1.77 0.61 1.29 0.91 0.93 0.98 0.61 2.32 2.31
3.59 2.73 2.88 2.90 0.75 0.62 2.25 3.28 1.07 0.54 1.59 0.62
3.86 2.64 2.32 1.15 0.68 0.42 2.59 1.50 1.17 0.72 1.65 1.29
3.01 2.40 2.47 2.61 0.62 0.55 1.87 2.76 0.93 0.49 1.45 0.55
3.30 2.21 1.95 1.11 0.56 0.33 2.19 1.29 1.02 0.66 1.43 1.18
4.677 3.707 1.832
2.193 2.067 1.640
3.994 2.976 1.471
1.689 1.717 1.114
3.51 4.84 3.08 1.41
1.82 2.07 1.38 1.41
3.12 4.04 2.48 1.20
1.59 1.78 1.21 1.22
2.13 5.18 1.76 2.77 0.26 0.48 0.93 0.63 2.78
0.72 1.98 0.82 0.86 0.84 0.66 1.05 1.01 1.63
1.81 4.35 1.51 2.39 0.22 0.44 0.83 0.52 2.33
0.53 1.67 0.70 0.70 0.72 0.60 0.93 0.89 1.27
3.64 4.47 2.79 2.97 1.17 2.73 4.11 3.90 3.65
2.53 1.18 1.65 1.37 1.21 1.45 1.74 2.34 0.90
2.87 3.83 2.41 2.48 1.00 2.25 3.42 3.41 3.08
1.70 1.00 1.30 1.22 0.98 1.17 1.28 2.01 0.78
3.01 1.27 1.72 3.46 4.93 0.27 0. 90 5.27 4.63
1.79 0.64 2.88 5.96 0.47 1.36 0.82 0.63 1.31
2.65 1.13 1.43 2.46 4.36 0.22 0.78 4.50 3.95
1.37 0.47 2.59 5.95 0.38 1.21 0.75 0.51 1.13
Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 4955 Table 1 (Continued) % root-mean-square dev. molar mass (g/mol)
T range (K)
trimethylamine triethylamine
59.111 101.192
193-277 323-368
dimethyl ether methyl ethyl ether methyl propyl ether diethyl ether phenyl ether
46.069 60.096 74.123 74.123 170.212
179-265 233-299 267-335 273-453 523-633
58.080 72.107 86.134 86.134
273-492 289-376 274-399 275-399
acetone methyl ethyl ketone methyl propyl ketone diethyl ketone
m
σ (Å)
/k (K)
Tertiary Amines 1.726 3.972 282.624 3.625 3.525 249.344 Ethers 2.261 3.561 217.799 4.986 2.844 169.845 2.954 3.754 235.203 3.355 3.459 219.070 5.087 3.587 316.968 Ketones 2.880 3.262 262.229 3.276 3.628 254.049 3.280 3.835 268.308 4.047 3.547 243.101
Figure 2. Vapor pressures plotted as a function of temperature for several normal alkanes. Symbols are experimental data while solid curves represent calculations from the PLJC equation of state.
% average absolute dev.
Psat
Fliq
Psat
vliq
0.35 1.48
1.83 0.93
0.32 1.24
1.49 0.75
2.48 4.67 3.71 3.40 1.81
3.00 3.08 3.83 3.89 0.34
1.82 3.70 3.02 2.68 1.57
2.64 2.46 3.38 2.53 0.27
4.28 4.35 2.53 4.41
5.80 1.04 0.93 0.96
3.73 3.37 2.19 3.73
5.37 0.97 0.65 0.89
Figure 3. Vapor-liquid-phase coexistence for octane. Symbols represent experimental data while solid curves are calculations from the PLJC equation of state.
deviations for vapor pressure and saturated liquid density are given by
[ ( [ (
percent rms deviation P
sat
) 100 ×
1
Np
∑
Npk)1
Psat expt.(Tk)
percent rms deviation Fliq ) 100 ×
1
Nd
∑
Ndk)1
)] )]
sat Pcalc. (Tk) - Psat expt.(Tk)
sat Fcalc. (Tk) - Fsat expt.(Tk)
Fsat expt.(Tk)
2 1/2
2 1/2
To facilitate comparison with other models, the average absolute deviation (AAD%) in vapor pressure and saturated liquid density are also calculated using the regressed parameters. The quality of fit (as measured by rms% deviations) ranges from 1 to 6% and is as good as usually expected for a three-parameter equation of state. It compares favorably with the SAFT and PHSC equations of states. Figure 2 shows measured vapor pressures of several n-alkanes (C6 - C10) as a function of temperature. The curves represent vapor pressure calculated using the best fit parameters from the PLJC model. Excellent agreement between experimental and calculated values is observed. Figure 3 shows the vapor-liquid-phase coexistence curve for octane. Again, good agreement
Figure 4. Critical temperatures of alkanes plotted as a function of carbon number. Open squares, predictions from the PLJC equation; filled squares, experimentally measured values.
between calculated and experimental saturated densities is observed, except in the vicinity of the critical point. The deficiency near the critical point can be attributed to the fact, like other engineering equations of state, the PLJC model is an analytical theory and fails to capture the nonclassical critical behavior of real fluids. Figure 4 shows the critical temperatures of alkanes predicted by the PLJC equation and those obtained experimentally. The PLJC tends to overpredict the critical temperatures and the discrepancy between
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Figure 5. Root-mean-square percent deviations in vapor pressure and saturated liquid density of PLJC and PHSC equations of state.
Figure 7. Variation of segment number m with molar mass (g/ mol) for normal alkanes.
Figure 8. Variation of mσ3 with molar mass (g/mol) for normal alkanes. Figure 6. Absolute added deviations (AAD%) deviations in vapor pressure and saturated liquid density of PLJC and SAFT equations of state.
the predicted and experimental values increases with carbon number. Figure 5 show rms% deviations in vapor pressure and saturated liquid density for n-alkanes for PLJC and perturbed hard-sphere chain (PHSC) equations of state. Although the average rms% deviations for the PLJC are smaller than those obtained for the PHSC equation, a direct comparison of the relative accuracy of these two equations cannot be made because different experimental data were used in the regression. However, it is noteworthy that the rms% deviations of the PLJC model decrease as the carbon number increases, suggesting that the performance of the PLJC model improves with chain length, in sharp contrast to the PHSC equation. This may be attributed to the fact that, unlike the PHSC equation, the diameter d of hard-chain segments in the PLJC depends not only on temperature but also on chain length m. A similar comparison is made with the statistical associating fluid theory (SAFT). Figure 6 shows the AAD% deviations in vapor pressure and saturated liquid volume. SAFT gives smaller vapor pressure AAD%s at low carbon number; however, for carbon number greater than 12, the reverse trend holds with PLJC giving smaller AAD%s. In the case of saturated liquid volume, the PLJC gives better fit to
the data compared with SAFT except for low carbon numbers less than 5, as reflected in the smaller AAD%. Again, the behavior of SAFT is similar to that of PHSC in that the deviations between experimental data and model predictions for SAFT increase at large carbon numbers in contrast to those of PLJC. We now focus on the behavior of the model parameters. For normal alkanes, Figures 7-9 show relationships between these three molecular parameters and molar mass Mw (in g/mol). In all cases, simple linear relationships are observed (for m > 1):
m ) 0.032Mw + 0.620
(10a)
mσ3 ) 1.749Mw + 29.682
(10b)
m/k ) 9.393Mw - 1.050
(10c)
These simple linear relationships can be used to estimate PLJC model parameters for n-alkanes of different molar mass. We note that the value of m is systematically smaller than the number of carbon atoms in the alkane molecule. This is consistent with the molecular feature of the Lennard-Jones chain model (that constitutes the theoretical foundation of the PLJC model) in which “covalently bonded” segments are nonoverlapping and are free to rotate around each other. The carbon
Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 4957 Table 2. PLJC Model Parameters for Several Polymers polymer
no. of data points
Mw
m
σ
/k (K)
rms % dev specific vol.
polystyrene high-density polyethylene low-density polyethylene poly(ethylene glycol) poly(vinyl acetate) poly(dimethyl siloxane) poly(tetrahydrofuran) poly(methyl methacrylate)
138 67 42 68 63 63 47 106
104 K 52 K 25 K 7.5 K 84 K 47 K 40 K 100 K
2908.193 2565.03 690.34 310.813 3517.633 1222.787 1221.1412 3228.7176
3.8045 3.2817 4.0843 3.2062 3.1149 3.9087 3.7568 3.455
605.902 396.036 504.364 411.298 403.816 386.0655 498.8315 558.7225
0.207 0.180 0.151 0.117 0.0687 0.140 0.101 0.159
Figure 9. Variation of m/k with molar mass (g/mol) for normal alkanes.
segments in an alkane, say, a hexane molecule, is not freely rotating, and hence the number of segments m in a hexane molecule predicted by the PLJC equation of state must necessarily be less than 6 (in fact, m ) 3.407 for hexane), indicating that the segments are overlapping segments. In addition, the segment energy /k increases with molar mass or carbon number but becomes independent of molar mass and approaches an asymptotic value of (m/k)/m = 297 in the limit of Mw f ∞. The parameter mσ3 increases with molar mass linearly and the parameter σ3 becomes independent of molar mass for large values of Mw. Overall, the variations of m, mσ3, and m/k with molecular weight are well-behaved and follow predictable trends. The molecular parameters m, σ, and /k for other nonassociating fluids such as cyclic alkanes, branched alkanes, alkenes, chloro alkanes, aromatics, and polynuclear aromatics can be found in Table 1. Polymers Molecular parameters for polymer liquids can be determined from experimental pressure-volume-temperature P-v-T data. P-v-T experimental data used in our nonlinear least-squares regression are taken from Zoller and Walsh.18 The three PLJC molecular parameters are determined by minimizing the objective function defined as the sum of squares of relative deviation between calculated and experimental polymer specific volumes at a specified temperature and pressure: N
OF )
∑
k)1
(
)
liq vliq calc(Tk,Pk) - vexpt(Tk,Pk)
vliq expt(Tk,Pk)
2
Figure 10. Specific volume (in cm3/g) of polystyrene is plotted as a function of temperature and pressure. Symbols represent experimentally measured data while the solid curve is calculated from the PLJC equation.
Very good agreement between the measured and calculated specific volumes is obtained for a wide range of polymers. The rms% deviation for these polymer systems ranges from 0.07% to 0.16%. A typical result is shown in Figure 10 where the specific volume of polystyrene is plotted as a function of temperature and pressure. The curves represent best fit to the data. Values of the three PLJC parameters m, σ, and /k for polystyrene and other polymers are displayed in Table 2. Conclusions The perturbed Lennard-Jones chain (PLJC) equation of state presented in this work is based on the firstorder variational perturbation theory for Lennard-Jones chains.5 The equation of state model uses the hardsphere chain as reference and utilizes the interchain segment-segment correlation function of hard-sphere chains to calculate the attractive contribution to the Helmholtz energy. Unlike other theoretically based equation of states for chainlike fluids, the diameter of the hard-sphere segments of the reference system depends on temperature as well as chain length. Three temperature-independent parameters are segment number m, segment size σ, and segment energy /k. These molecular parameters were determined for some nonassociating fluids from experimental thermodynamic properties of saturated liquid. For different classes of normal fluids such as n-alkanes, n-alkenes, cyclic alkanes, and aromatics, the three parameters were found to be well-behaved and follow simple functions of molar mass or molecular weight.
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Acknowledgment Partial support from the Petroleum Research Fund administered by the American Chemical Society is gratefully acknowledged. We thank the academic research fund (RP3981605) National University of Singapore for partial support of this research. Literature Cited (1) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284. (2) Song, Y.; Lambert, S. M.; Prausnitz, J. M. A Perturbed Hard-Sphere Chain Equation of State for Normal Fluids and Polymers. Ind. Eng. Chem. Res. 1994, 33, 1047. (3) Banaszak, M.; Chiew, Y. C.; Radosz, M. Thermodynamic Perturbation Theory: Sticky Chains and Square-Well Chains. Phys. Rev. E 1993, 48, 3760. (4) Banaszak, M.; Chiew, Y. C.; O’Lenick, R.; Radosz, M. Thermodynamic Perturbation Theory: Lennard-Jones Chains. J. Chem. Phys. 1994, 100, 3803. (5) O’Lenick, R.; Chiew, Y. C. Variational Theory for LennardJones Chains. Mol. Phys. 1995, 85, 257. (6) Bokis, C. P.; Cui, Y. P.; Donohue, M. D. Thermodynamic Properties of Hard-Chain Molecules. J. Chem. Phys. 1993, 98, 5023. (7) Chang, J.; Sandler, S. I. The Correlation Functions of HardSphere Chain Fluids: Comparison of the Wertheim Integral Equation Theory with the Monte Carlo Simulation. J. Chem. Phys. 1995, 102, 437. (8) Chapman, W. G.; Jackson, G.; Radosz, M.; Gubbins, K. E. Phase Equilibria of Associating Fluids: Chain Molecules with Multiple Bonding Sites. Mol. Phys. 1988, 65, 1057.
(9) Chiew, Y. C. Percus-Yevick Integral Equation Theory for Athermal Hard-Sphere Chains. Part I: Equations of State. Mol. Phys. 1990, 70, 129. (10) Dickman, R.; Hall, C. K. Equation of State for Chain Molecules: Continuous-Space Analogue of Flory Theory. J. Chem. Phys. 1986, 85, 4108. (11) Honnel, K. G.; Hall, C. K. A New Equation of State for Athermal Chains. J. Chem. Phys. 1989, 90, 1841. (12) Wertheim, M. S. Thermodynamic Perturbation Theory of Polymerization. J. Chem. Phys. 1987, 87, 7323. (13) Von Solms, N.; O’Lenick, R.; Chiew, Y. C. Lennard-Jones Chain Mixtures: Variational Theory and Monte Carlo Simulation Results. Mol. Phys. 1999, 96, 15. (14) Bokis, C. P.; Donohue, M. D.; Hall, C. K. Application of a Modified Generalized Flory Dimer Theory to Normal Alkanes. Ind. Eng. Chem. Res. 1994, 33, 1290. (15) Alder, B. J.; Young, D. A.; Mark, M. A. Studies in Molecular Dynamics. X. Corrections to the Augmented van der Waals Theory for the Square Well Fluid. J. Chem. Phys. 1972, 56, 3013. (16) Chen, S. S.; Kreglewski, A. Applications of the Augmented van der Waals Theory of Fluids. I. Pure Fluids. Ber. Bunsen-Ges. Phys. Chem. 1977, 81, 1048. (17) Song, Y.; Mason, E. A. Statistical-Mechanical Theory of a New Analytical Equation of State. J. Chem. Phys. 1989, 91, 7840. (18) Zoller, P.; Walsh, D. J. Standard Pressure-VolumeTemperature Data for Polymers; Technomic Publ. Co.: Lancaster, PA, 1995.
Received for review March 22, 1999 Revised manuscript received August 20, 1999 Accepted August 24, 1999 IE990208X
