2284
I n d . Eng. Cheni. Res. 1990, 29, 2284-2294
Equation of State for Small, Large, Polydisperse, and Associating Molecules Stanley H. Huang and Maciej Radosz* Exxon Research and Engineering Compan), Annandale. IVeu. Jersey 08801
Statistical Associating Fluid Theory (SAFT) has been extended to many real, molecular, and macromolecular fluids, such as chain, aromatic, and chlorinated hydrocarbons, ethers, alkanols (aliphatic alcohols), carboxylic acids, esters, ketones, amines, and polymers, having molar mass u p t o 100000. T h e key result of this work is a n accurate and physically sound equation of state for predicting density, vapor pressure, and other fluid properties. Practical calculations require three nonspecific parameters: segment number, segment volume, and segmentsegment interaction energy (segment energy for short). For chain molecules, the segment volume and segment energy are found to be nearly constant upon increasing the molar mass, while the segment number is found t o be a linear function of molar mass. As a result, this equation of state represents a useful, predictive correlation for many compounds, such as polymers, where no extensive experimental data are available and where parameters have to be estimated based on molar mass and chemical structure only. Specific interactions, such as hydrogen bonding, are characterized by two association parameters, the association energy and volume, characteristic of each site-site pair. These parameters, having well-defined physical meaning, control the bond strength and hence the degree of association. Introduction
(1975) in the Perturbed Hard Chain Theory (and in more recent PHCT versions) and by Chen and Kreglewski (1977) in their equation of state, also known as BACK and extended to mixtures by Simnick et al. (1979). Our goal is to develop a practical but physically sound equation of state, applicable to small, large, chain, and associating molecules over the whole density range. While we are concerned with the quality of fit for various fluid properties, such as vapor pressures and liquid densities, our primary concern is the ease and reliability of extrapolating to large, often poorly defined and polydisperse pseudocomponents of real polymer and oil solutions. From this point of view, the key to success lies in well-behaved and hence easy to estimate equation of state parameters. We will define model molecules, bulk fluid properties, and equation of state parameters to be derived from fitting real fluids. We will then define our equation of state in terms of the Helmholtz energy and compressibility factor. After presenting many correlation results and equation of state parameters for vapor pressures and liquid densities of over 100 real fluids, we will show an example of how a reliable equation of state can be used to estimate critical constants of large molecules, which cannot be determined experimentally due to thermal decomposition.
Molecularly based equations of state not only provide a useful thermodynamic basis for deriving chemical POtentials or fugacities that are needed for phase equilibrium simulations but also allow for separating and quantifying the effects of molecular structure and interactions on bulk properties and phase behavior. Examples of such effects are the molecular size and shape (e.g., chain length), association (e.g., hydrogen bonding) energy, and mean field (e.g., dispersion and induction) energy. Ideally, a single equation of state should incorporate all these effects. A concept of such an equation of state has recently been proposed by Chapman et al. (1989, 1990) based on perturbation theory. The essence of their approach, referred t o as the Statistical Associating Fluid Theory (SAFT): is to use a reference fluid that incorporates both the chain length (molecular size and shape) and molecular association, in place of the much simpler hard sphere reference fluid used in most existing engineering equations of state. Chapman et al. (1990) developed Helmholtz energy expressions accounting for the chain and association effects based 011 Wertheim's (1984, 1986a,b) cluster expansion theory. Wertheim derived his theory by expanding the Helmholtz energy in a series of integrals of molecular distribution functions and the association potential. On the basis of physical arguments, Wertheim showed that many integrals in this series must be zero and, hence, a simplified expression for the Helmholtz energy can be obtained. This expression is a result of resummed terms in the expansion series (cluster expansion). The key result of Wertheim's theory is a relationship between the residual Helmholtz energy due to association and the monomer density. This monomer density, in turn, is related to a function 1 characterizing the "association strength". A reference equation of state used in this work is based on the SAFT concept that captures the hard sphere, chain, and association effects. Effects due to other kinds of intermolecular forces (dispersion, induction, etc.), usually weaker, can be included through a mean field perturbation term. Our mean field term is similar to that proposed by Alder et al. 11972) and is creatively used in many recent equations of state, most notably by Beret and Prausnitz 0888-5885190; 2629-2283302.~50/0
Real Fluids: Nonassociating and Self-Associating We will treat the effective molecular size (through hard sphere and chain terms) and molecular association as two major effects on the bulk properties of real fluids. This choice can be justified by using a simplified example for the vapor pressure. It is easy to show that the vapor can be approximately correlated with respect pressure (psst) to temperature ( T )by using a Clausius-Clapeyron type of equation given below: In p S a t = A(') - A(Z)/T where A ( * )is a constant proportional to the enthalpy of vaporization. A ( 2 )obtained , from fitting to experimental vapor pressure data, is log-log plotted against the molar mass in Figure l a for nonassociating fluids and in Figure 1 b for associating fluids. The data points shown in these figures are defined in Table I, along with symbols and data points used throughout this paper. C
1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 11, 1990 2285 o a v
n-alkanes n-alkylcyclopentanes n-alkylcyclohexanes / O
2 .
lo'
.-E> f
.
o n-alkanes 0 alkllnoii alkanoia acids t primary amines athen
/
O/O
ketones
5 -
"':,a
Ob)
;/O
- In A@) =
5.0237
+
0.72702 In MM
Table I. Nonassociating and Self-Associatina Real Fluids fluids data points nonassociating n-alkanes 0 polypropylene 0 polyethylene 0 pol yisobutylene 0 n-alkylcyclopentanes A n-alkylcyclohexanes V benzene derivatives 0 polynuclear aromatics 0 ethers A ketones r tertiary amines 4 esters alkenes chlorinated hydrocarbons self-associating acids alkanols (aliphatic alcohols) 0 primary amines 4 secondary amines 4
components only, associating means self-associating and nonassociating means non-self-associating. We should note that some non-self-associating molecules can be cross-associating in mixtures with other molecules. The reduced fluid density 7 (segment packing fraction), the same for segments and molecules, is defined as
where p is the molar density, m is the segment number (number of segments in each molecule, our first pure component parameter), and d is the effective segment diameter that is temperature dependent. This definition is equivalent to We will note that A(2)for nonassociating fluids in Figure l a is a strong, essentially linear function of the molar mass and that all the points representing hydrocarbons of many different structures (chainlike, branched, and ringlike) tend to cluster around the average line. This means that the effect of structure, however significant and accounted for, is much smaller than the effect of molecular size. This is true even though we can observe a slight deviation from the average line for polynuclear aromatics, which can be explained by a small degree of n-n-induced aggregation. By contrast, we will note that A@)for associating fluids in Figure l b shows strong deviations from A(2)for n-alkanes. Expectedly, the magnitude of these deviations depends on the degree of association. For example, carboxylic acids and aikanols (aliphatic alcohols), which are known to be strongly associated, exhibit greater deviations from the alkane behavior than tertiary amines and ethers do. Therefore, we will only treat acids, alkanols, primary amines, and secondary amines as significantly self-associating compounds.
Model Fluid Properties and Parameters Our model pure fluid is a mixture of equi-sized spherical segments interacting according to a square-well potential. In addition, we superimpose two kinds of bonds between these segments, covalent-like bonds to form chains and association bonds to interact specifically. As a result, our model molecules can approximate a broad range of molecules, from nonassociating near-spherical (for example, methane, neopentane) and nonspherical (chain alkanes, polymers) to associating near-spherical (methanol) and nonspherical (alkanols). Since this paper deals with pure
7 =
TpmuO
(2)
where T = 0.740 48 and uo is the segment molar volume in a close-packed arrangement, Le., the volume occupied by NA, closely packed segments, in milliliters per mole of segments. Hence, on the basis of eqs 1 and 2, we can express uo as
(3) Since uo is implicitly temperature dependent, because d is temperature dependent, it is useful to introduce a corresponding, temperature-independent segment molar volume at T = 0, which will be denoted uw and referred to as the segment uolume, our second pure component parameter: uoo
nNAv =u3
61
(4)
where u is a temperature-independent segment diameter. Since a characteristic volume, rather than diameter, is traditionally selected as a pure component parameter, for example, b in cubic equations of state, V* in the Perturbed Hard Chain Theory of Beret and Prausnitz (1975), and uW in the Chen-Kreglewski (1977) equation of state, uoowill also be used in this work as one of the pure component parameters. However, while all the volume parameters were defined by others per mole of molecules, our uoo is defined per mole of segments. Temperature dependence of the segment diameter d in eq 3 is based on the Barker-Henderson approach (1967).
2286 Ind. Eng. Chem. Res., Vol. 29, No. 11, 1990
case in our example. Similarly, the fraction of molecules that are present as dimers is given by 2(XA)2(1 - XA).The general result for the fraction of molecules present as m-mers is
Specific equations for d and uo given below, d = o[ 1 - C exp[
s] ]
X(m-mer) = m(XA)2(1- X A ) " - '
(9)
and the average chain length is given by mave= l / X A
are based on the work of Chen and Kreglewski (1977), who obtained eq 6 by solving the Barker-Henderson integral equation for d d = J d [ l - exp(-u(r)/kT)] dr 0
(10)
Equation 9 expresses the "most probable distribution" of Flory (1953), which is in agreement with experimental results for polymer polydispersity.
(7)
using a square-well potential; uo/k in eqs 5 and 6 is the well depth, a temperature-independent energy parameter, characteristic of nonspecific segment-segment interactions, which will be referred to as the segment energy, in kelvin, our third pure component parameter. Similar to Chen and Kreglewski (1977),we set the integration constant C = 0.12 and use their temperature dependence of u
where e l k is a constant that was related to Pitzer's acentric factor and the critical temperature (Chen and Kreglewski, 1977; Kreglewski, 1984) for various molecules. Since our energy parameter is for segments rather than for molecules, we set e l k = 10 for all the molecules. The only exceptions are a few small molecules where the e / k values close to those derived by Kreglewski have been used ( e l k = 0 for argon; 1 for methane, ammonia, and water; 3 for nitrogen; 4.2 for CO; 18 for chlorine; 38 for CS,; 40 for CO,; and 88 for SO,); larger values of e l k for some of the molecules treated here as nonassociating may be caused by their weak self-association. In addition to the three pure component parameters characterizing nonassociating molecules, uoo, m, and uo,we . assouse two association parameters, eAA and K ~ These ciation parameters have been proposed for a square-well model of specific interactions between two sites A (Chapman et al., 1990). The parameter eAA characterizes the association energy (well depth), and the parameter K~ characterizes the association volume (corresponds to the well width rAA). In general, the number of association sites on a single molecule is unlimited, but it has to be specified for each specific molecule. However, we do not specify the location of association sites. We label these sites with superscripts A, B, C, etc., to keep track of the specific site-site interactions. Each association site is assumed to have a different interaction with the various sites on another molecule. Cluster structure limitations, steric hindrance approximations, and size distributions are explained in detail elsewhere (Chapman et al., 1990). In brief, the fraction of clusters of a given size can be estimated by using general statistical arguments (Flory, 1953). As an example, we shall consider a pure component system composed of molecules having two sites A and B, where only AB bonding and only chain clusters are allowed. The fractions of each chain length present (dimers, trimers, etc.) are functions of X Aand X B (the fractions of molecules NOT bonded at sites A and B, respectively) that, in turn, are calculated from the equation of state described in section 3. Assuming that the activity of a site is independent of bonding at the other sites on the same molecule, the fraction of molecules that are present as monomers is X AXB, or (XA)2 if X A = X B, which is the
Equation of State The theory results underlying our equation of state are given in this section in terms of the residual Helmholtz energy aresper mole, defined as ares(T,V,N) = a(T,V,N) - aided(T,V,N), where a(T,V,N)and aided(T,V,N) are the total Helmholtz energy per mole and the ideal gas Helmholtz energy per mole a t the same temperature and density. The residual Helmholtz energy aresis a sum of three terms representing contributions from different intermolecular forces. The first term aseg accounts for the part of aresthat represents segment-segment interactions, i.e., hard sphere and mean-field (dispersion) interactions. The second term acheinis due to the presence of covalent chain-forming bonds among the segments. The third term am accounts for the increment of am due to the presence of site-site specific interactions among the segments, for example, hydrogen-bonding interactions. The general expression for the Helmholtz energy is given by ares = aseg + a h a h + aassoc (11) The segment Helmholtz energy a w , per mole of molecules, can be calculated from aseg
= ma
seg
(12)
where sow, per mole of segments, is the residual Helmholtz energy of nonassociated spherical segments and m is the segment number. Let us allow ao- to be composed of hard sphere and dispersion parts, as follows aoseg
=
aohs
+ a disp
(13)
As usual, the hard sphere term can be calculated as proposed by Carnahan and Starling (1969) (14)
where 7 is the reduced density given by eqs 1 and 2. The dispersion term is a power series initially fitted by Alder et al. (1972) to molecular dynamics data for a square-well fluid. This equation, which also provided the basis for the Perturbed Hard Chain Theory of Beret and Prausnitz (1975), is given by
Dij are universal constants. In this work, we use Dij's that have been refitted to accurate PVT, internal energy, and second viral coefficient data for argon, by Chen and Kreglewski (1977). Both chain and association terms are the same as those described by Chapman et al. (1990). The chain term, the
Ind. Eng. Chem. Res., Vol. 29, No. 11, 1990 2287 16
r /
12
/
O
' O .
t
/
-
m = 0 05096 MM
E 10
0
20 Molar Mars. kgimole
40
30
Figure 2. Segment number m a~ a linear function of molar mass for n-alkanes and long-chain polymers; PP = polypropylene, P E = polyethylene, PIB = polyisobutene.
Helmoltz energy increment due to bonding, can be determined from 4
1
.-.chain
1-
,'I
Equation 16 is derived based on the associating fluid theory, as explained by Chapman et al. (1990), where the association bonds are replaced by covalent, chain-forming bonds. The Helmholtz energy change due to association is calculated for pure components from
where M is the number of association sites on each molecule, X A is the mole fraction of molecules NOT bonded at site A, and LA represents a sum over all associating sites on the molecule. The mole fraction of molecules NOT bonded a t site A can be determined as follows
0 1 0
1
I
I
I
100
I
200
1
300
Molar Mass, g/mole
Figure 3. Segment numbers m for n-alkanes and polynuclear aromatics, as linear functions of molar mass, set boundaries for other hybrid classes of hydrocarbons. The branches of diamonds correspond to n-alkyl derivatives of the polynuclear aromatics, e.g., methyl-, ethyl-, tz-propyl-, n-butylnaphthalene.
Our final equation of state can be presented as a sum of the compressibility factor terms 2, analogous to the Helmholtz energy terms in eq 11, as given below - 1 = Zseg + Zchain + z a s s o c (22) where
X A = [ 1 + N A ~ C PBAAB]-l X (summation over ALL B
sites: A, B, C, ...) (18) where NAv is Avogadro's number and p is the molar density of molecules. Am in eq 18 is the association strength that is approximated as AAB = g(d)"g[exp(tAB/kT) -
5
,9 - T 2
(19)
AAB is our key property characterizing the association bonds that depends on the segmental radial distribution function g(d)seg. Since we approximate our segments as hard spheres, we approximate g(dIsegas the hard sphere radial distribution function (Carnahan and Starling, 1969): 1 1 - -q 2 g(d)"g = g(d)h" = (20) (1 - 7d3 The only density dependence in Am is given by g(d)w,and the only explicit temperature dependence is given by the tAB/kT,in eq 19. However, we should also note the temperature dependence of d and hence, implicitly, 9. The temperature-independent segment diameter u, used in eq 19 to make KAB dimensionless, can be calculated based on eq 4 as follows (21)
The equation of state presented above has been used to correlate vapor-liquid equilibria of over 100 real fluids. For each fluid, the pure component parameters (om, m, and u"/k for non associating plus tAAand K~~ for associating fluids) have been regressed from the least-squares minimization using differences in calculated and experimental vapor pressure and liquid density data. Correlation results for both nonassociating and associating fluids are presented in the following two sections.
Segment Parameters for Real Nonassociating Fluids For each nonassociating compound, Table I1 lists the molar mass (MM), temperature range, uW, m, uo/k, percent
2288 Ind. Eng. Chem. Res., Vol. 29, No. 11, 1990 Table 11. Segment Parameters for Real Nonassociating Fluids AAD%
MM
Trange, K
uw, mL/mol
m 1.0 1.0 1.221 1.417 1.147 1.463 1.133
uoJk,K 123.53 150.86 111.97 216.08 367.44 396.05 335.84
PMt
diq
0.68 (6Ia 2.8 (8) 1.9 (12) 4.5 (12) 1.1 (8)
0.26 (6)O 0.86 (8) 0.76 (12) 1.6 (12) 2.1 (6)
190.29 191.44 193.03 195.11 200.02 202.72 204.61 206.03 203.56 205.46 205.93 209.40 210.65 211.25 209.96 208.74 207.73
1.4 (10) 1.8 (15) 2.1 (IO) 2.3 (12) 1.9 (12) 2.3 (14) 1.8 (14) 1.6 (13) 0.86 (11) 2.2 (14) 0.71 (12) 0.82 (12) 2.1 (14) 2.1 (10) 1.3 (11) 2.2 (11) 4.8 (10)
0.35 (10) 1.6 (15) 1.8 (IO) 2.6 (12) 3.0 (12) 3.5 (14) 3.4 (14) 3.4 (13) 1.9 (7) 3.5 (14) 3.3 (12) 2.8 (11) 3.1 (12) 2.6 (10)
nitrogen argon carbon monoxide carbon dioxide chlorine carbon disulfide sulfur dioxide
28.013 39.948 28.010 44.010 70.906 76.131 64.063
72-121 218-288 180-400 278-533 283-413
19.457 16.29 15.776 13.578 22.755 23.622 22.611
methane ethane propane butane pentane hexane heptane octane nonane decane dodecane tetradecane hexadecane eicosane octacosane hexatriacontane tetratetracontane
16.043 30.070 44.097 58.124 72.151 86.178 100.205 114.232 128.259 142.276 170.340 198.394 226.432 282.556 394.77 506.99 619.21
92-180 160-300 190-360 220-420 233-450 243-493 273-523 303-543 303-503 313-573 313-523 313-533 333-593 393-573 449-704 497-768 534-725
n-Alkanes 21.576 1.000 14.460 1.941 13.457 2.696 12.599 3.458 12.533 4.091 12.475 4.724 12.282 5.391 12.234 6.045 12.240 6.883 11.723 7.527 11.864 8.921 12.389 9.978 12.300 11.209 12.000 13.940 12.0 19.287 12.0 24.443 12.0 29.252
data source* 1 1 2 3 4 5 2 1
6 7 8 2, 9 2 2 2 2 2, 10 2, 10 2, 10 2, 10 2, 10 10 10 10
Polymers Poly propylene ethylene isobutylene
15 700 25 000 36 000
263-303 4 13-473 333-383
12.0 12.0 12.0
822.68 1274.08 1823.95
209.96 216.15 267.44
1.0 (30) 2.1 (24) 0.77 (30)
11 12 12
1.1 (5)
2, 10 2, 10 2, 10 10
cyclopentane methyl ethyl propyl butyl pentyl
70.135 84.162 98.189 112.216 126.244 140.272
253-483 263-503 273-5 13 293-423 314-458 333-4 83
n-Alkylcyclopentanes 12.469 3.670 13.201 4.142 13.766 4.578 14.251 5.037 14.148 5.657 13.460 6.503
226.70 223.25 229.04 232.18 230.61 225.56
1.7 (13) 1.5 (13) 1.5 (13) 0.39 (8) 0.53 (9) 0.45 (9)
1.3 (5) 1.6 (6) 1.4 (6) 1.1 (5) 0.94 (4)
cyclohexane methyl ethyl propyl butyl pentyl
84.162 98.189 112.216 126.243 140.270 154.297
283-513 273-533 273-453 313-453 333-484 353-503
n-Alkylcyclohexanes 13.502 3.970 15.651 3.954 15.503 4.656 15.037 5.326 14.450 6.060 14.034 6.804
236.41 248.44 243.16 238.51 234.30 230.91
0.68 (13) 1.3 (14) 0.88 (10) 0.41 (8) 0.44 (10) 0.43 (11)
1.0 (5) 3.1 (14) 1.4 (6) 0.99 (4) 0.90 (4) 0.53 (4)
2, 10 2, 10 2, 10 10
benzene methyl (toluene) ethyl n-propyl n-butyl m-xylene tetralin biphenyl
78.114 92.141 106.168 120.195 134.212 106.168 132.205 154.213
300-540 293-533 293-573 323-573 293-523 309-573 293-673 433-653
Benzene Derivatives 11.421 3.749 11.789 4.373 12.681 4.719 12.421 5.521 12.894 6.058 12.184 4.886 13.196 5.163 12.068 6.136
250.19 245.27 248.79 238.66 238.19 245.88 279.17 280.54
1.4 (13) 2.6 (13) 1.2 (15) 1.5 (14) 0.83 (13) 2.1 (14) 2.1 (20) 2.5 (12)
2.1 (13) 2.9 (13) 3.0 (15) 2.0 (9) 2.1 (9) 2.9 (14) 3.9 (20) 2.8 (12)
2 2, 10 2, 10 10 2, 10 10 13 2
naphthalene I-methyl 1-ethyl 1-n-propyl 1-n-butyl phenanthrene anthracene pyrene triphenylene
128.174 142.201 156.228 170.255 184.282 178.234 178.234 202.255 228.293
373-693 383-551 393-563 403-546 413-566 373-633 493-673 553-673 573-773
Polynuclear Aromatics 13.704 4.671 13.684 5.418 12.835 6.292 13.304 6.882 13.140 7.766 16.518 5.327 16.297 5.344 18.212 5.615 21.271 6.016
304.80 293.45 276.18 266.82 252.11 352.00 352.65 369.38 379.12
1.8 (17) 0.43 (IO) 0.29 (11) 4.2 (6) 11.5 (9) 0.43 (14) 0.31 (10) 4.9 (7) 2.2 (11)
2.6 (17) 0.50 (3) 0.53 (3) 2.6 (1) 7.0 (1) 1.6 (11) 1.1 (6)
14 10 10 10 10 15 15 16 16
dimethyl methyl ethyl methyl n-propyl diethyl phenyl
46.069 60.096 74.123 74.123 170.212
179-265 266-299 267-335 273-453 523-633
207.83 203.54 208.13 191.92 276.13
1.9 (8) 2.4 (4) 2.7 (7) 2.4 (10) 0.37 (7)
10 10
10
10
Ethers 11.536 10.065 10.224 10.220 12.100
2.799 3.540 4.069 4.430 6.358
2.2 (10) 1.5 (7)
17 17 17 2 2
Ind. Eng. Chem. Res., Vol. 29, No. 11, 1990 2289 Table I1 (Continued) AAD 90 MM dimethyl (acetone) methyl ethyl methyl n-propyl diethyl trimethylamine triethylamine
Trange, K
uoo, mL/mol
m
58.080 72.107 86.134 86.134
273-492 257-376 274-399 275-399
Ketones 7.765 4.504 11.871 4.193 11.653 4.644 10.510 4.569
59.111 101.192
193-277 323-368
Tertiary Amines 14.102 3.459 11.288 5.363
u o / k ,K
PUt
210.92 229.99 230.40 235.24
2.3 (10) 1.3 (8) 2.0 (8) 1.4 ( 8 )
196.09 201.31
0.90 (16) 0.70 (5)
uiiq
1.0 (5)
data sourceb 2, 18 17 17 17 19 19
Esters methanoate methyl ethyl n-propyl n-butyl ethanoate methyl ethyl n-propyl n-butyl propanoate methyl ethyl n-propyl butanoate methyl ethyl n-propyl ethene propene 1-butene I-hexene chloromethane dichloromethane trichloromethane tetrachloromethane chloroethane 1-chloropropane 1-chlorobutane I-chlorohexane chlorobenzene
60.046 74.073 88.100 102.127
225-324 242-348 262-377 279-403
7.548 7.954 8.671 10.100
3.886 4.727 5.276 5.460
215.48 202.72 203.98 212.34
0.61 (8) 0.34 (8) 0.71 (8) 2.1 (8)
0.53 (5) 1.0 (6) 1.5 (7) 1.5 (6)
17 17 17 17
74.073 88.100 102.127 116.154
244-498 273-493 278-542 293-413
7.366 7.897 8.637 10.829
4.918 5.566 6.106 5.918
200.58 196.19 197.94 212.11
3.1 (12) 1.5 (12) 1.9 (11) 1.5 (7)
2.9 (10) 3.0 (12) 2.5 (11) 0.82 (5)
17, 20 2 17, 20 17
88.100 102.127 116.154
261-512 277-538 293-413
7.991 8.621 10.332
5.493 6.064 6.090
199.61 197.45 207.13
1.7 (11) 1.9 (11) 1.3 (7)
3.2 (10) 3.1 (11) 1.4 (6)
17, 20 17, 20 17
102.127 116.154 130.181
285-533 298-423 316-445
8.634 11.273 10.575
6.176 5.675 6.713
196.83 214.04 205.67
1.3 (15) 2.9 (8) 2.6 (8)
1.4 (5) 1.2 (5) 1.2 (5)
17 17 17
28.054 42.081 56.108 84.156
133-263 140-320 203-383 213-403
Alkenes 18.157 1.464 15.648 2.223 13.154 3.162 12.999 4.508
212.06 213.90 202.49 204.71
2.0 (8) 3.6 (10) 1.8 (10) 2.3 (11)
0.68 (8) 1.5 (10) 2.4 (10) 2.8 (11)
21 22 2, 10 2, 10
50.488 84.933 119.378 153.823 64.515 78.542 92.569 120.623 112.559
213-333 230-333 244-357 273-523 212-440 238-341 262-375 306-435 273-543
Chlorinated Hydrocarbons 10.765 2.377 238.37 10.341 3.114 253.03 10.971 3.661 240.31 13.730 3.458 257.46 11.074 3.034 229.58 11.946 3.600 229.14 12.236 4.207 227.88 225.82 12.422 5.458 276.72 13.093 3.962
0.75 (7) 2.2 (7) 0.25 (7) 1.1 (14) 3.1 (13) 0.50 (8) 0.40 (8) 0.33 (8) 1.5 (15)
0.82 (7) 0.99 (7) 1.1 (6) 2.5 (14) 3.4 (13) 0.94 (6) 1.2 (6) 1.0 (4) 2.7 (15)
2 17 17 2 17, 20 17 17 17 2
"Numbers in parentheses indicate number of data points used in the correlation. b ( l )Parameter values are taken from Kreglewski (1984) without fitting. (2) Vargaftik, N. B. Tables on the Thermophysical Properties of Liquids and Gases; John Wiley & Sons: New York, 1975. (3) IUPAC. Carbon Dioxide. International Tables of Fluid State; Pergamon Press: Oxford, 1976; Vol. 3. (4) IUPAC. Chlorine. International Tables of the Fluid State; Pergamon Press: Oxford, 1985; Vol. 8 (tentative tables). (5) O'Brien, L. J.; Alford, W. J. Ind. Eng. Chem. 1951,43,506. (6) Goodwin, R. D.; Roger, H. M.; Straty, G. C. Thermodynamic Properties of Ethane from 90 to 600 K and Pressures to 700 bar. NBS Technical Note 687; NBS: Boulder, CO, 1976. (7) Goodwin, R. D.; Haynes, W. M. Thermodynamic Properties of Propane from 85 to 700 K and Pressures to 70 Mpa; NBS Monograph 170; NBS: Boulder, CO, 1982. (8) Haynes, W. M.; Goodwin, R. D. Thermodynamic Properties of n-Butane from 135 to 700 K and Pressures to 70 Mpa; NBS Monograph 169; NBS: Boulder, CO, 1982. (9) Das, T. R., Reed, C. O., Jr. J . Chem. Eng. Data 1977,22,3. (10) Selected Values of Properties of Hydrocarbons and Related Compounds. Research Project 44 of the American Petroleum Institute and the Thermodynamic Research Center, Texas A&M University: College Station, TX, Loose-leaf data supplements to 1989. (11) Passaglia, E.; Martin, G. M. J . Res. Natl. Bur. Stand., Sect. A 1964, 68A, 273. (12) Beret, S.; Prausnitz, J. M. Macromolecules 1975, 8, 536. (13) Kudchadker, A. P.; Kudchadker, S. A.; Wilhoit, R. C. Tetralin; API Monograph 705; API: Washington, DC, 1978. (14) Kudchadker, A. P.; Kudchadker, S. A.; Wilhoit, R. C. Naphthalene; API Monograph 707; API: Washington, DC, 1978. (15) Kudchadker, A. P.; Kudchadker, S. A.; Wilhoit, R. C. Anthracene and Phenanthrene; API Monograph 708; API: Washington, DC, 1979. (16) Kudchadker, A. P.; Kudchadker, S. A.; Wilhoit, R. C. Four-Ring Condensed Aromatic Compounds; API Monograph 709; API: Washington, DC, 1979. (17) Thermodynamic Tables for non-Hydrocarbons; Thermodynamic Research Center, Texas A&M University: College Station, loose pages to 1988. (18) Ambrose, D.; Sprake, C. H. S.; Townsend, R. J. Chem. Thermodyn. 1974, 6, 693-700. (19) Boublik, T.; Fired, V.; Hala, E. The Vapor Pressure of Pure Substances, 2nd ed.; Elsevier, Amsterdam, 1984. (20) Perry, R. H.; Chilton, C. H. Chemical Engineer's Handbook, 5th ed.; McGraw-Hill: New York, 1973. (21) IUPAC. Ethylene. International Tables of the Fluid State; Pergamon Press: Oxford, 1976; Vol. 2. (22) IUPAC. Propylene (Propene). International Tables of the Fluid State; Pergamon Press: Oxford, 1980; Vol. 7.
average absolute deviation (AAD%)in vapor pressure (pt, behavior. This is important because the key future except for polymers) and liquid molar volume (u"q), inchallenge lies in estimating the equation of state paramcluding the number of data points used in regression, and eters for polydispersed, poorly defined pseudocomponents data source. The quality of fit for both vapor pressure and of real fluid mixtures, rather than in fine-tuning precise liquid density is as good as can be usually expected for a predictions (however important) for well-defined pure reasonable, three-parameter equation of state. However, components. Apart from the expected scatter due to inour focus is not on the quality of fit but on the parameter accuracies in experimental data and fitting itself, the pa-
2290 Ind. Eng. Chem. Res., Vol. 29, No. 11, 1990
r
Table 111. Correlation of t h e Segment Number m for Hydrocarbons m = A(1) + A(2'MM
L
/O'
A(') 0.70402 2.673 3 0.823 60 -0.010038 0.51928 -2.3190
n-alkanes polynuclear aromatics n-alkylcyclopentanes n-alkylcyclohexanes n-alkylbenzenes 1-n-alkylnaphthalenes 0' 0
I
I
I
100
I
200
I
I
300
Molar Mass, g/mOle
Figure 4. Close-packed molar volumes muM for n-alkanes and polynuclear aromatics, as linear functions of molar mass, set boundaries for other hybrid classes of hydrocarbons. The branches of diamonds correspond to rz-alkyl derivatives of the polynuclear aromatics, e.g., methyl-, ethyl-, n-propyl-, n-butylnaphthalene. 400
MM range in fittine 16-619 78-202 70-140 84-154 78-134 128-184
A(2'
0.046647 0.014 781 0.039 044 0.043 096 0.041 112 0.054 566
Table IV. Correlation of t h e Close-Packed Molar Volume mvw for Hydrocarbons mum = A") + A(2'MM
MM range A(1)
11.888 5.0117 4.2053 3.6438 -6.1400 -21.619
n-alkanes polynuclear aromatics n-alkylcyclopentanes n-alkylcyclohexanes n-alkylbenzenes 1-n-alkylnaphthalenes
in fittini 16-619 78-202 70-140 84-154 78-134 128-184
~ 1 2 '
0.551 87 0.469 42 0.598 17 0.59961 0.62468 0.66647
Table V. Correlation of the Segment Energy u o / kfor n -Alkanes a n d Polynuclear Aromatics
u o / k = A")
300
-
A'2) e ~ p [ - A ( ~ ) A l M l
MM range
.-E>
-u
A")
X
A(2)
in fitting
A'3'
n-alkanes 210.0 26.886 0.013341 polynuclear aromatics 472.84 357.02 0.0060129
$3
200
16-619 78-228
Table VI. Correlation of the Segment Energy u o l k for Other Hydrocarbons u o / k = A('' - A(2)MM
MM range 100
0
100 200 Molar Mass, gtmole
300
Figure 5. Segment energies u o / k for n-alkanes and polynuclear aromatics as smooth but nonlinear functions of molar mass, set boundaries for other hybrid classes of hydrocarbons. The branches of diamonds correspond to n-alkyl derivatives of the polynuclear aromatics, e.g., methyl-, ethyl-, n-propyl-, n-butylnaphthalene,
rameter values reported in Table I1 are well-behaved and suggest predictable trends upon increasing the molar mass of similar compounds. The segment number m increases with increasing molar mass within each homologous series practically linearly. For example, m for n-alkanes is plotted versus molar mass in Figure 2. It is reassuring to find that a single linear relationship holds not only for all the small n-alkanes, which is shown in the insert, but also for macromolecular chains of varying degree of branchiness, such as polypropylene, polyethylene, and polyisobutylene. It is also reassuring to fmd out that m is essentially a linear function of molar mass for different homologous series of aromatic molecules, i.e., upon increasing the side chain length for alkylbenzenes, alkylnaphthalenes, etc., as shown in Figure 3. The plots in Figure 3 also suggest that the segment numbers m for all hydrocarbons fall between two boundaries set by n-alkanes and plain polynuclear aromatics (PNA's) and that for a given molar mass m will decrease with increasing aromaticity. This means that, if there are no accurate PVT data available, which often is the case, m can be estimated from molar mass only.
n-alkylcyclopentane n-alkylcyclohexanes n-alkylbenzenes 1-n-alkylnaphthalenes
A'1)
A(2)
239.56 278.59 267.39 425.70
0.085 618 0.313 11 0.218 25 0.941 11
in fittine 98-140 98-154 78-134 128-184
We note that the effective segment numbers m reported for hydrocarbons in Table I1 are systematically smaller than the corresponding carbon numbers. A physical picture of a n-alkane therefore is that of a chain of overlapping spherical segments. Hence, the segment volume u" should correspond to the volume occupied by such segments. Expectedly, the segment volume uoo for methane is the largest among alkanes because it corresponds to a single ( m = 1) CH, unit, and it gradually decreases upon increasing the chain length, reaching an asymptotic value of 12 (set 12.0 for C,,) for long chains. Since uoo does not vary much with chain length and remains constant for long chains and since m is a linear function of the molar mass, the product mum (volume occupied by a mole of molecules in a close-packed arrangement) is also a linear function of the molar mass, as shown in Figure 4. As for m alone (Figure 3), n-alkanes and plain polynuclear aromatics set the boundaries for the mu" domain for all the hydrocarbons. Apart from testing these reassuring trends, a practical reason for developing a correlation for mu- is to provide the basis for estimating L)", if there are no accurate PVT data available, for a given m and molar mass. A similar molar mass correlation can be developed for the segment energy uo/k, which is shown in Figure 5. As for m and mu", n-alkanes and plain polynuclear aromatics
Ind. Eng. Chem. Res., Vol. 29, No. 11, 1990 2291 Table VII. Monomer Fractions XAfor Different Bonding Types type A approximations 1
XA approximations
XA -1
A"#O
+ (1 +
PA)'/^
2PA -l'+ (1
+ 8pA)'12
4pA -1
+ (1 +
PA)'/^
PA -1
+ (1 + 1 2 ~ A ) ' / ~ PA
-(I - pA)
+ ((1 +
+ 4pA)'12
4pA -1
+ (1 + 1 6 ~ A ) ' / ~ 8PA
XA = XB = XC XD = 3 X A - 2
Table VIII. Types of Bonding in Real Associating Fluids" rigorous species formula type assigned type
alkanol
A
-o:B
3B
2B
4C
3B
1
non-self-associating
2B
2B
3B
3B
4B
3B
CH A
water
B:ij:Hc H O
amines tertiary
- N-
A
I
secondary
-..N- A HB
primary
-NL
S
A
":NOHe HC
set the boundaries for the uo/k domain for all the hydrocarbons. Unlike m and mu", however, although smooth, uo/k is nonlinear with respect to the molar mass. For the ease of estimating, m, mu", and uof k have been regressed as simple functions of the molar mass (MM) for many homologous series. For example,
m = A(')
+ A(2)MM
muw = A(') + A(2)MM
for all hydrocarbons (26) for all hydrocarbons
(27)
u o / k = A(') - A(2) ~ x ~ ( - A ( ~ ) M M ) for n-alkanes and PNA's (28) u o / k = A(') - A(2)MM
for other hydrocarbons
+ ((1 + 2pA)' + 4pA)'I' 6PA
justed when accurate data become available for hybrid hydrocarbons, such as aromatic rings with long side chains. The solid lines and curves shown in Figures 3-5 have been predicted from eqs 26-29. Specific values of the regression coefficients A(i) are reported in Tables III-VI. This way we have a useful method for estimating all the equation of state parameters for nonassociating pure fluids where no accurate data are available and, especially, for poorly defined pseudocomponents where only average molar mass and average aromaticity are available. Although this method is primarily applied to predicting phase equilibria in mixtures (Huang and Radosz, 19901, this method can also be used to estimate pure component properties, such as vapor pressures, densities, and critical properties. Segment a n d Site-Site P a r a m e t e r s for Real Self-Associating Fluids
HA
ammonia
-(1 - 2pA)
(29)
We note that eq 29 is only a linear approximation that is valid up to the MM of a corresponding n-alkane; for higher MM values, we set u o / k to be the same as that for a corresponding n-alkane. This can easily be tested and ad-
Vapor pressure and liquid density correlations for self-associating fluids require the use of the am term (eq 17; zero for the nonassociating fluids). Before we can calculate am, however, we have to define and specify all the non-zero site-site interactions and hence non-zero A's (eq 19) needed to determine the monomer fractions X A (eq 18). While eq 18 is not X A explicit in general, it can be made X A explicit in many specific cases. Examples of one-, two-, three-, and four-site models and A approximations are provided in Table VI1 along with analytical X A expressions for each model that can be used in place of' eq 18. In order to select a proper expression for X Afrom Table VII, one has to assign association sites and non-zero sitesite interactions. Ideally, one should have detailed, independent data from spectroscopy on the association strength for each site-site interaction. Since such data are scarce and usually qualitative (we are addressing this problem in a separate project), one has to make simplifying approximations aimed at reducing the number of parameters that have to be fitted. For example, different hydrogen bonds are assumed to be equivalent, e.g., AAC= ABC, which means that different sites A and B form equivalent bonds with C , or double hydrogen bonds are
2292 Ind. Eng. Chem. Res., Vol. 29, No. 11, 1990 Table IX. Segment and Site-Site Parameters for Real Self-Associating Fluids" AAD 70
UM,
Trange, K
mL/mol
17.032 18.015 34.080
200-380 283-613 284-369
10.0 10.0 10.0
methanol ethanol 1-propanol 1-butanol 1-pentanol 1-hexanol 1-heptanol 1-octanol 1-nonanol 1-decanol "propanol 2- butanol 2-methyl-1-propanol 2-methyl-2-propanol phenol
32.042 46.069 60.096 74.123 88.150 102.177 116.204 130.231 144.258 158.285 60.096 74.123 74.123 74.123 94.114
273-487 302-483 293-493 313-493 333-513 343-573 353-573 373-593 383-613 393-633 273-373 293-393 293-403 303-373 346-482
12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0
methanoic ethanoic n-propanoic n-butanoic n-pentanoic n-hexanoic n-heptanoic n-octanoic n-nonanoic n-decanoic benzoic
46.025 60.052 74.080 88.107 102.134 116.161 130.187 144.215 158.242 172.269 122.124
293-393 313-553 313-463 333-493 353-483 372-504 385-522 399-540 411-557 423-572 405-523
15.5 14.5 13.5 13.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0
methanamine ethanamine 1-propanamine 1-butanamine 1-pentamine 1-hexanamine 1-heptanamine aniline
31.057 45.084 59.111 73.138 87.165 101.192 115.219 93.129
204-413 221-433 243-473 265-503 285-401 307-429 325-457 350-673
Primary Amines (Model 3B) 12.0 2.026 226.86 1045 12.0 2.781 208.61 940.3 12.0 i3.450 211.36 779.1 12.0 4.062 210.91 895.4 12.0 4.773 215.81 746.1 12.0 5.476 215.73 722.0 12.0 6.153 218.62 553.9 12.0 4.034 288.10 1269
dimethylamine diethylamine
45.084 73.138
208-436 240-483
Secondary Amines (Model 2B) 12.0 2.637 208.23 1064 12.0 3.996 212.62 581.3
ammonia water hydrogen sulfide
m ua/k, K (Model 3B) 1.503 283.18 1.179 528.17 1.935 226.38
elk, K
MM
10*~
Put
dig
data sourceb
893.1 1809 804.1
3.270 1.593 0.911
1.6 1.3 (13) 0.69 (14)
3.2 (10)' 3.2 (13) 1.4 (14)
Alkanols (Model 2B) 1.776 216.13 2.457 213.48 3.240 225.68 3.971 225.96 4.642 225.15 5.315 222.88 6.028 221.62 6.642 220.69 7.322 219.18 8.024 217.14 3.249 202.94 3.933 216.41 4.028 232.90 3.966 202.35 4.103 290.08
2714 2759 2619 2605 2587 2556 2579 2532 2516 2448 2670 2457 2698 2515 1894
4.856 2.920 1.968 1.639 1.637 1.873 1.654 2.001 2.249 2.892 2.095 1.634 0.4032 1.118 4.315
0.83 (12) 0.86 (10) 0.16 (11) 0.23 (10) 0.32 (IO) 0.77 (13) 0.61 (12) 1.0 (13) 1.1 (14) 2.1 (14) 0.27 (11) 0.32 (11) 0.58 (12) 0.21 (8) 0.24 (9)
0.88 (12) 0.83 (10) 1.2 (111 1.0 (6) 1.1 (5) 1.22 (51 0.96 (41 1.0 (4) 0.47 ( 4 , 0.57 (41 0.96 (11) 1.2 (11, 1.4 (121 0.79 (81 0.02 (1I
Acids (Model 1) 1.341 333.28 2.132 290.73 3.084 296.03 3.800 268.93 4.719 248.63 5.482 243.39 6.059 241.50 6.628 240.41 7.274 238.97 7.847 237.10 4.608 272.66
7522 3941 3400 4155 4322 4683 4734 4745 4798 4906 5930
0.1625 3.926 0.8193 0.3700 0.5103 0.2352 0.2309 0.2430 0.2005 0.1844 0.3149
0.62 (6) 1.6 (13) 0.25 (9) 0.30 (9) 0.21 (8) 0.44 (7) 0.46 (7) 0.60 (7) 0.44 (7) 0.60 ( 7 ) 0.66 (8)
0.49 (5) 0.69 ( 1 3 ) 0.10 (9) 0.84 (91 1.0 (8)
0.43 14)
3 1 3 3 3 3 3 3 3 3 3
6.310 10.65 17.94 15.08 15.15 16.63 16.57 11.27
0.22 (13) 0.27 (13) 1.3 (13) 2.2 (13) 2.3 (7) 0.84 (7) 1.5 (7) 2.6 (16)
0.38 (13) 0.22 (12) 1.5 (10) 2.6 (10) 2.0 (7) 1.7 (6) 0.99 (3) 3.5 (16)
3 3 3 3 3 3 3 3
15.61 15.36
0.73 (12) 3.2 (11)
1 1 2
1,3 1 1, 3
3 3 3 3 3 3 3 1. 3 3 3 3 3
4 4
"Numbers in parentheses indicate number of data points used in the correlation. b ( l )Vargaftik, N. B. Tables on the Thermophysical Properties of Liquids and Gases; John Wiley & Sons: New York, 1975. (2) Sage, B. H.; Lacey, W. N. Some Properties of the Light Hydrocarbons, Hydrogen Sulfide, and Carbon Dioxide; API Monograph 37; API: New York, 1955. (3) Thermodynamic Tables for nonHydrocarbons; Thermodynamic Research Center, Texas A&M University: College Station, TX, loose pages to 1988. 14) Perry, R. H.; Chilton, C. H. Chemical Engineer's Handbook, 5th ed.; McGraw-Hill: New York, 1973.
assumed to be represented as strong single bonds, e.g., between carboxylic groups. Other examples of rigorous and approximated sets of site-site interactions for selfassociating molecules that are used in this work are given in Table VIII. The sets defined as rigorous in Table VI11 include all the electron-donor and -acceptor sites on each molecule. An example for alkanols will illustrate the use of Tables VI1 and VIII. Each hydroxylic group (OH) in alkanols, in principle, has three association sites, labeled A and B on oxygen and C on hydrogen, as shown in Table VIII. The association strength A due to the like, oxygen-oxygen or hydrogen-hydrogen (AA, AB, BB, and CC) interactions is assumed to be equal to zero. The only non-zero A is due to the unlike (AC and BC) interactions, which moreover are considered to be equivalent. Hence, the rigorous type selected for alkanols in Table VI1 is 3B. Another approximation is to allow only one site on oxygen (A) and one site on hydrogen (B). In this case, the assigned type for alkanols (Table VIII) is 2B, where the only non-zero
interactions AB and BA are equivalent. Although we tested both types of site assignment for alkanols, 3B and 2B, we felt that the limited experimental data used for fitting did not justify the use of the rigorous type 3B. Therefore, we report results for the alkanols of type 2B in Table IX. For each self-associating compound, Table IX lists the temperature range, um, m, uo/k,P l k , KAA, AAD% in vapor pressure, AAD% in liquid density, including the number of data points used in regression, and data source. Only m, u o / k ,c"/k, and K" have been allowed to be adjustable. The segment volumes uoo have been set equal to 12.0. except for ammonia, water, and hydrogen sulfide where it has been set equal to 10.0 and except for small carboxylic acids where uoo has been made similar to those of corresponding n-alkanes. Similar to the trends observed for the nonassociating fluids, the segment number m increases with increasing molar mass within each homologous series, for example, for chain acids and alkanols.
Ind. Eng. Chem. Res., Vol. 29, No. 11, 1990 2293
E
*Oo0
8
alkanoic acids
. m u m 8 8 '
sW
2t
400
too0
Figure 7. Critical temperatures for n-alkanes predicted by our equation of state falling between those predicted by others.
amines
1000
10.000
Molar Mars. glmole
** 0
o 0
-
0 100
2
3
Experimental Huang el 01. (19a8) Mohin 6 R n p (1987) 99.5 pb Confldence EOS Pndlclion
1000
2
3
10,000
Molar Mars, glmole
0
100
200
Molar Mass, glmole
Figure 6. Plot of the associating energies t / k and segment energies u o / k versus molar masses of organic compounds in homologous series. The ratio of the former to the latter is roughly 10 to 1.
The segment energies uo/k are close to 200 K and, hence, are similar to those derived for nonassociating chain molecules, as shown in Figure 6. Especially, uo/k for amines and alkanols essentially matches uo/k for n-alkanes. Somewhat larger values for carboxylic acids are due to the one-site approximation of the double hydrogen bond. We tested this by allowing two and three sites on acid molecules and, as a result, obtaining uo/k values that were very close to 200 K. Figure 6 also shows the association energies eAA for acids (one-site approximation), alkanols, and amines. As expected, the association energies decrease upon going from acids to amines. The ratios of P / u 0 are also in the reasonable range (of the order of 10).
Figure 8. Critical pressures for n-alkanes predicted by OUT equation of state agreeing with those of Mohan and Peng (1987).
Mohan and Peng used the Perturbed Hard Chain equation of state (Cotterman et al., 1986) to extrapolate the critical properties for long chain alkanes up to polyethylene. As a result, they had to extrapolate all three parameters C, V*, and T* (Mohan and Peng verified that the polyethylene density was correctly predicted). By contrast, the only parameter in our equation of state that has to be extrapolated is m, which is linear with respect to MM. Therefore, we found our equation of state to be particularly suitable for estimating the critical properties of large molecules. Like all the analytical equations of state, this one overpredicts the critical temperature and critical pressure. However, since the differences between calculated and predicted values have been very consistent for all the n-alkanes for which there are experimental data, simple correlations have been developed as follows: In T, = 1.0169 In T p S- 0.14703
Example: Critical Constants Can Be Estimated for Large Molecules While the critical temperature and pressure are commonly used in numerous engineering correlations, such as those based on cubic equations of state and corresponding states, they cannot be measured for high molar mass compounds because of thermal decomposition. Therefore, there have been various useful empirical methods proposed, such as that of Lee and Kesler (1975), that allow for extrapolating the critical temperature and pressure to high molar mass. An alternative, proposed by Mohan and Peng (1987), is to use a reliable equation of state for such extrapolation, which at least can be tested on other properties in the high molar mass range.
In P, = 0.89858 In P,""
+ 0.18496
(30) (31)
where T, is the critical temperature in kelvin, P, is the critical pressure in bar, and the superscript eos means predicted from the equation of state, using a simplified set of parameters that are specialized for large molecules: m = 0.05096MM, uoo = 12.0, and uo/k = 210. Equations 30 and 31 are applicable to all the n-alkanes, including macromolecules, down to about Clo. Sample results are shown in Figure 7 for the critical temperature and in Figure 8 for the critical pressure. The pairs of solid curves shown in Figures 7 and 8 have been calculated from eqs 30 and 31, respectively. Each pair of curves corresponds to a 99.5% confidence band. Our
2294 Ind. Eng. Chem. Res., Vol. 29, No. 11, 1990
predicted critical temperatures agree well with the experimental data and with those predicted by Huang et al. (1988) but deviate somewhat from those predicted by Mohan and Peng (1987). Our predicted critical pressures on the other hand agree well with the experimental data and with those predicted by Mohan and Peng but deviate from those predicted by Huang et al. It is worth noting that, unlike Huang et al.'s, Mohan's and our critical pressures asymptotically reach zero as the molar mass increases to infinity, as should be expected (Tsonopoulos, 1987).
Conclusions The equation of state developed in this work is applicable to small, large, polydisperse, and associating molecules over the whole density range. The equation of state parameters, segment number, segment volume, and segment energy, derived for over 100 real fluids, can be readily extended to other hydrocarbon molecules based on molecular structure and molar mass only. The association parameters, also derived in this work from bulk experimental data, characterize the strength of specific site-site interactions that lead to molecular association. This way one can separate and quantify the effects of nonspecific segment properties and specific site-site interactions on bulk fluid properties. Our future work on this equation of state will focus on the phase behavior of binary mixtures and polymer solutions and on the use of spectroscopy for probing molecular interactions leading to association.
Acknowledgment This work is an extension of a joint project on associating fluid theory with the Cornell University group of Professor Keith E. Gubbins. Helpful comments and stimulating discussions with Professor Gubbins, John Walsh, and Karl Johnson are gratefully acknowledged.
Nomenclature A") = regression constants, each use defined in the text a = molar Helmholtz energy (total, res, seg, bond, assoc. etc.), per mole of molecules a. = segment molar Helmholtz energy (seg), per mole of
segments C = integration constant in eqs 5 and 6 d = temperature-dependent segment diameter, A e l k = constant in eq 8 k = Boltzmann's constant = 1.381 X J/K m = effective number of segments within the molecule (segment number) muo" = volume occupied by 1 mole of molecules in a closepacked arrangement, mL/mol M = number of association sites on the molecule MM = molar mass, g/mol molar = molar with respect to molecules N = total number of molecules NA, = Avogadro's number = 6.02 X molecules/mol P = pressure P, = critical pressure, bar psst = saturated vapor pressure R = gas constant segment molar = molar with respect to segments T = temperature, K T , = critical temperature, K ulk = temperature-dependent dispersion energy of interaction between segments, K
u o /k = temperature-independent dispersion energy of inter-
action between segments, K V = total volume u = molar volume, uliq = liquid molar volume, mL/mol of bulk fluid u0 = temperature-dependent segment volume, mL/mol of segments uw = temperature-independent segment volume, mL/mol of segments X = mole fraction XA = monomer mole fraction (mole fraction of molecules NOT bonded at site A) 2 = P u / ( R T ) ,compressibility factor K~~ = volume of interaction between sites A and B AAB = "strength of interaction" between sites A and B, A3 cAB/k = association energy of interaction between sites A and B, K 7 = (r/6)pnmd3= (aNA,/6)pmd3,pure component reduced density, the same for segments AND molecules p = pn/NAv,molar density, mOl/A3 pn = number density (number of molecules in unit volume), l-3
Lennard-Jones segment diameter (temperature independent), A EA= summation over all the sites (starting with A) a =
Superscripts
A, B, C, D, ... = association sites res = residual seg = segment assoc = associating, or due to association hs = hard sphere ideal = ideal gas
Literature Cited Alder, B. J.; Young, D. A.; Mark, M. A. J . Chem. Phys. 1972, 56, 3013. Barker, J. A.; Henderson, D. J . Chem. Phys. 1967, 47, 4714. Beret, S.; Prausnitz, J. M. AZChE J. 1975, 21, 1123. Carnahan, N. F.; Starling, K. E. J . Chem. Phys. 1969,51,635. Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. Fluid Phase Equilib. 1989,52, 31-38. Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. Ind. Eng. Chem. Res. 1990,29, 1709. Chen, S. S.; Kreglewski, A. Ber. Bunsen-Ges. Phys. Chem. 1977,81, 1048. Cotterman, R. L.; Schwartz,B. J.; Prausnitz, J. M. AIChE J . 1986, 82, 1787. Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953; pp 318-325. Huang, S.H.; Radosz, M. Equation of State for Mixtures Containing Small, Large, Polydisperse, and Associating Molecules. Presented at the Annual AIChE Meeting, Chicago, Nov 1990. Huang, S.H.; Lin, H. M.; Tsai, F. N.; Chao, K. C. Ind. Eng. Chem. Res. 1988, 27, 162. Kreglewski, A. Equilibrium Properties of Fluids and Fluid Mixtures; The Texas Engineering Experiment Station (TEES)Monograph Series; Texas A & M University Press: College Station. 1984. Lee, B. I.; Kesler, M. G. AIChE J. 1975, 21, 510. Mohan, R.; Peng, D. Y. Correlation for T,, P, and Acentric Factor of High Molecular Weight n-Paraffins. Presented at the Annual AIChE Meeting, New York, Nov 1987. Simnick, J. J.; Lin, H. M.; Chao, K. C. Ado. Chem. Ser. 1979, 182,
209. Tsonopoulos, C. AIChE J . 1987,33, 2080. Wertheim, M. S. J . Stat. Phys. 1984, 35, 19, 35. Wertheim, M. S. J. Stat. Phys. 1986a, 42, 459, 477. Wertheim, M. S. J . Chem. Phys. 1986b,85, 2929, 7323.
Received for review April 3, 1990 Accepted June 20, 1990
