J . Phys. Chem. 1992, 96, 10995-11004
10995
Physical Theory for Fluids of Small Associating Molecules John M. Walsh: Henrique J. R. Cuedes,$ and Keith E.Cubbms* School of Chemical Engineering, Cornell University, Zthaca, New York 14853 (Received: January 10, 1992; Zn Final Form: August 4, 1992)
The physical approach is used to derive a theory for the properties and phase equilibria of hydrogen-bonding and associating fluids. The theory accounts for nonpolar and multipolar interactions and for the strong and highly directional short-range attractive interactions which lead to association. For nonpolar and multipolar interactions, the u expansion of perturbation theory is used, and Wertheim's theory is used for association interactions. The theory is used to predict the properties of a variety of real fluids, with emphasis on the prediction of mixture phase behavior. The agreement with experiment is generally good.
I. Introduction The distinguishing feature of associating fluids is the presence of strong, highly directional attractive forces between the molecules. These forces, which are mostly electrostatic, cause aggregation and result in associated species such as dimers, trimers, and larger chainlike species and in some cases branched, network, or ring structures. Because of thermal motion, the associated species are continually forming and breaking apart. Their concentrations and the lifetime of any particular aggregate are strong functions of the temperature and, to a lesser extent, of the density; typical lifetimes are on the order of picoseconds. Thus, the aggregates are not nearly as stable as the molecular species but are nevertheless observable by a variety of spectroscopic techniques (infrared, ultraviolet, Raman, NMR).'-3 In fluids such as the alcohols, acids, and water, the intermolecular OH*-0 interaction is known as hydrogen bonding. The hydrogen atom is small and is therefore capable of forming a compact electrostatic link between the oxygen atoms on two different molecules. In other cases, the association interaction may involve a weak form of chemical bonding (e.g., nitric 0xide~9~) or be dominated by a charge-transfer effect (e.g., trinitrobenzene and mesitylene6). As discussed by Walsh et al.,' most association interactions involve both electrostatic and charge-transfer contributions and cooperative (multimolecular) effects as well. The formation of associated species has a profound effect on fluid properties. Relative to fluids of small nonpolar molecules (e.g., methane, argon, oxygen), fluids of small associating molecules (e.g., nitric oxide, methanol, water) have high heats of vaporization, low vapor pressures, and high critical temperatures and pressures. It has been estimated8 that if water did not hydrogen bond, ice would melt at -90 OC and liquid water would have a normal boiling point of -80 "C. At moderate temperatures and pressures, associating and nonpolar fluids mix poorly. For conditions at which they mix more readily (higher temperatures and pressures), they have large molar heats of mixing. For predicting the properties of associatingfluids, one approach that has been partially successful is the chemical approach. In this approach, the existence of various associated species is postulated and a chemical equilibrium expression is written for the formation of each associated species (e.g., A + B AB). Values of the equilibrium constants ( K ) are then assigned based on experimental measurements9J0 (spectroscopy or calorimetry) or based on equation of state calculations of activities"J2 or are treated as adjustable parameter^.^' These theories are accurate for predicting phase behavior provided sufficient data are available to determine the K values. The main drawback of the chemical approach, however, is that for compounds such as water or the alcohols, many chainlike and branched structures are formed and
'
Permanent address: Separations/Data Department, Shell Development Company, Westhollow Research Center, P.O. Box 1380, Houston, TX 7725 1-1 380. Permanent address: Departamento de Quimica, Universidade Nova de Lisboa, Lisboa, Portugal.
many equilibrium constants are required. For these cases, there are seldom enough experimental data available to avoid the ad hoc assignment of K values. In contrast to the chemical approach, the physical approach relies only on a specified intermolecular potential. Statistical mechanics and computer simulation are used to relate the potential to the liquid structure, to the equation of state, to concentrations of monomeric and associated species, and to other thermodynamic properties. The use of intermolecular potentials eliminates the need to devise ad hoc chemical equilibrium expressions and to assign equilibrium constants to them. Instead, equilibrium constants together with fractions of monomers and associated species are predicted. Further, computer simulation can be used to test unambiguously the approximations in the theory. In such a test, the same intermolecular potential is used as input to both simulation and theory, and the output is compared without adjustment of parameters. For a theory based on the chemical approach, such an unambiguous test cannot be carried out unless the theory gives an unambiguous way of calculating the equilibrium constants from the intermolecular potential. To our knowledge, such a theory, based on the chemical approach, has not yet been developed. The main disadvantageof the physical approach is that for mast associating molecules the intermolecular potential, which is a strong function of distance, orientation, and multibody effects, is not precisely k n ~ w n . ~ Quantum ~J~ mechanics is exact in principle, but in practice, the amount of computer time required to determine the exact intermolecular potential is prohibitive. Quantum mechanics is useful, however, for estimates of the potential and for estimates of the relative contribution of various componentsI6-l8 (overlap or exchange repulsion, dispersion, electrostatic, polarization, charge transfer, etc.). Most successful intermolecular potential models for computer simulation of associating fluids are empirical two-body potent i a l ~ . ' ~They . ~ ~ are composed of a nonpolar core plus various fractional point charges which are placed at positions within the core. A molecular dipole moment can be calculated from the point charge distribution. However, the magnitude of this calculated dipole moment is usually higher than the experimentallymeasured dipole moment since the point charges are adjusted to account for a number of effects which are not otherwise explicitly accounted for (polarization, charge-transfer interactions, and multibody cooperative effects21,22). Several theories, based on the physical approach, have been developed recently for the properties of fluids of associating molecule^.^^-^^ Using graphical resummation techniques, Wertheim has derived an approximation to the partition function for molecules with highly directional, strong, short-range attractive force^.^'-^^ The Wertheim theory agrees well with simulation result^^^-^* but does not account for nonassociating interactions such as dispersion or multipolar interactions. Therefore, in addition to Wertheim's theory for association, an auxiliary treatment must be used to account for the dispersion and longer-range multipole electrostatic interactions which occur in real associating fluids.
0022-3654/92/2096-10995.$03.00/0 0 1992 American Chemical Society
10996 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
In this paper, Wertheim's theory for association is combined with the u expansion for dispersion and multipolar interactions. The theory is derived in section 11. In section 111, theontical results are compared with measured data of pure fluid properties (PVT and vapor pressure) and phase behavior for mixtures. The parameters in the theory are intermolecular potential parameters which are adjusted to correlate the properties of various assoCiating and nonassociating fluids. The theory is developed for and applied to the properties of small associating molecules, but not to long-chain associating molecules.
II. Theory
Walsh et al. In a general perturbation expansion of the configurational Helmholtz energy (A,), there is a series of terms A , = A0 Ai .f A2 A3 .,. (6)
+
in which A. is the Helmholtz energy of the reference fluid and expressions for the AI, A*, etc., tmns are generated by performing a Taylor series expansion of the canonical ensemble partition function in powers of X (e.g., AI = -kT(d In Q/aX),.,). For the reference fluid chosen here, we use Wertheim's theory and calculate & as the sum of a nonspherical overla contribution (AO,) plus a contribution due to association (Aoh%) A. = A$
The thermodynamic theory presented here is based on Wer-
thei"s theory for association and nonspherical overlap (repulsive) forces, plus the u expansion for multipolar, dispersion, and induction forces.43 The intermolecular potential is written as the sum of a reference (uo) and a perturbation (up) potential: uA( 1 2 ) = #O( 1 2 ) + Xu,( 1 2 ) (1) in which uA(with X = 1 ) is the full intermolecular potential. To obtain a rapidly convergent perturbation expansion, the reference potential is chosen such that the reference fluid structure (as measured by the pair correlation function) is similar to that of the real fluid. For the real fluids considered in this paper, the molecules possess nonspherical mes,association and longer-range multipolar forces, and other anisotropic interactions (dispersion and induction). Previous i n v e ~ t i g a t i o n shave ~ ~ ~indicated ~ that the repulsive core and short-range forces which lead to d a t i o n have the strongest effect on the fluid structure. We therefore include these two kinds of forces in the reference potential: UO(l2) uov(12) + Uhb(12) (2) in which ,u is the nonspherical (repulsive) overlap potential and Uhb is the association or hydrogen-bonding potential; here (1 2) represents the center of mass and orientational coordinates for molecules 1 and 2, Le., (12) = (rI2,wI,w2) where r12= r2 - rl, with r, being the center of molecule i, and ai= 4,,t9,,Xi is the set of orientational coordinates for molecule i. The nonspherical overlap potential is given by an extension of the united atom model of Hsu et In this model, various chemical functional groups (CH3, 0, etc.) are treated as hard spherical segments which overlap with each other (the bond lengths are generally less than the segment diameters). In our model, each spherical segment has a soft repulsive (WCA) potential which is obtained by cutting the LJ potential at its minimum and shifting the repulsive branch, at its end, to zero energy. The association potential is given by short-range, off-center square-well attraction sites:
in which r A B is the distance between site centers, A and B are indica which range over the set of site labels denoted by r, and uAB is the square-well potential which is defined as ~ A B ( ~ A B ) -CAB
A TAB
(3) in which A is the diameter of the association site and eAB is the depth of the square well. For most cases, the intermolecular forces in eq 2 will have the greatest effect on fluid structure, so the other forces are included in the perturbation potential up(12) = Ump(12) + udis(l2) + Uind(l2) (4) U A B ( ~ A B )=
0
in which ump,Udbr and uindare the contributions from multipolar, dispersion, and induction interactions, respectively, and can be written as a sum of spherical harmonic terms.43 The multipolar forces obey the relations (ump(12))ui = ( U m p ( 1 2 ) ) q q 0 (5) indicates the unweighted average over the oriin which entations of moiecules i and j .
+ +
+ Aohb
(7)
The explicit equation for AoWfrom Wertheim's theory is referred to as the modified-TPT equationMand is given by
-AOov - - Y-4y - 3y2 NkT
(1 - y ) 2
(8)
in which Y is calculated from Y = 2a - 1 , where a is the shape parameter of scaled particle theory and y is the packing fraction (y = V d ,in which V, is the hard core volume of a molecule). The modified TPT has been proven accurate for hard molecules of arbitrary shape for values of a less than roughly 8 (this includes all of the molecules in the present study). To extend the modified TPT from hard to soft molecules, the hard core volume (V,) is used to defme an effective hard core diameter (4V, = d / 6 ) which is temperature and density dependent and is calculated by the procedure of Verlet and Weis."* This is similar to the WCA procedure used to extend the Camahan-Starling equation from hard to soft sphere molecules. As described in ref 46,a,and hence Y, can be estimated from a specified molecular geometry. However, in this work, accurate results were obtained when Y is treated as an adjustable parameter as described in the section on results. For mixtures, Y is given by Y -- ~ x r vini which xi is the mole fraction of component i. The Helmholtz energy due to association is given in Wertheim's theorySoby
-NAohb -k T -
C AEr
(
lnXA-? - ) + M2
(9)
in which X Ais the fraction of molecules not bonded at site A and M is the number of attraction sites on each molecule. The fraction not bonded is calculated from
The quantity A is given by the general expression AAB
S g ( l 2 ) f A B ( l 2 ) d(12)
(11)
in which f A B is the Mayer f function for interaction sites A and B and g" is the pair correlation function for the association reference potential. The association reference potential is given by the reference potential of the u expansion without association interactions (the nonspherical soft repulsive potential). To calculate A, the number, position, and geometry of the attraction sites must be specified, and g must be determined either by computer simulation or by calculations with a suitable integral equation. In general, the evaluation of A could be tedious. However, Jackson et have evaluated A for the simple case of spherical hard core molecules with off-center square-well attraction sites. The following expressions were obtained: AAB = 4rghsKABFAB (12) and 1 -u/2
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 10997
Theory for Fluids of Small Associating Molecules and
in which the bonding volume parameter ( K A B ) is a constant with respect to temperature and density and depends only on the geometry of the interaction between association sites A and B?9 Both K A B and CAB are treated as adjustable parameters. To extend eq 12 from hard to soft core molecules, y is calculated using the temperatura and densitydependent diameter described previously for soft repulsive forces. For each AB site-site interaction, there are two association parameters ( K A B and C A B ) . For those associating fluids for which all sites are equivalent, the association parameter values are equal and CAB = eHB and K A B = K H B . The first-order term in the perturbation expansion is given by
Upon substituting up into this expression, separate terms result from each separate contribution to the anisotropic potential (A, = Almp AIdis+ A,'"); for example,
+
The general expressions for A2 and A3 have been given elsewhere (see eqs 4.20, 4.21, and 4.13 and Appendix 4A of ref 43). At present, no satisfactory theory exists for the reference fluid pair correlation function go(12), and essentially nothing is known about the higher-order correlation functions (e.g., go(123)) for such fluids. In this work, we therefore introduce a further approximation; we replace the unknown reference fluid pair correlation functions by the corresponding functions for a fluid of spherical Lennard-Jones molecules. This greatly simplifies the expraions for AI, A2, and A3. Thus, AI becomes
For multipolar forces, A2 becomes
A3 is given in ref 43 (eqs 4.30-4.35 for general potentials and eq 4.39 for multipolar ones). The multipolar potential is often the most important part of the perturbing potential. For these forces, the AlmPterm vanishes, the A2mPand A3"'pterms are calculated explicitly, and the A4"'P and higher-order contributions are treated by way of the Pade approximant of Stell et al.$'
+
+
A2"'P A3"'P A4mP+ ...
= A2"'P[1 - A3"'P/A2"P]-'
(19)
The exact equations which are explicit in terms of density and temperature are given by Gubbins et al.52 The approximation used here of replacing go for the nonspherical reference potential (uo) with gLJ for the spherical Lennard-Jones potential is similar to that used successfully by Martina et aLS3 In their study of hard diatomic molecules with a point quadrupole, g for the hard diatomic fluid is replaced with g for hard spheres. In considering the accuracy of our approximation, we note that (a) the perturbation forces are relatively small compared to the hydrogen-bonding forces and (b) the range of the hydrogen-bonding forces is small and restricted to a narrow range of angles. We therefore would expect the approximation to be crude over the short range of the hydrogen-bonding forces and would expect it to be good outside of this range. However, the accuracy of this approximation has not yet been tested by computer simulation studies. For dispersion interactions, go is replaced with g for a WCA fluid of soft spheres, and ( t 4 d i ~ ( 1 2 ) )is~ ,approximated ~~ by the Lennard-Jonespotential. Thus, the expression for A,& is the same
TABLE I: Parameter Values for Pure Fluids compd elk, K u, A3 v eHB/k.K argon 116.3 3.404 1.00 0.00 methane 149.9 3.657 1.00 0.00 nitrogen 130.3 3.401 1.910 0.00 oxygen 142.4 3.227 1.491 0.00 ethane 339.9 3.963 2.133 0.00 propane 446.6 4.306 2.420 0.00 nitric oxide 134.2 3.208 1.12 1842 acetone 475.8 4.463 2.380 0.0 chloroform 700.9 4.492 2.829 0.0 sulfur dioxide 406.0 3.975 2.695 0.00 dimethyl ether 495.6 4.021 2.916 0.00 methanol 391.0 3.669 2.37 1474 water 257.11 3.259 1.00 249.8 TABLE 11: Multipole Parameter Values for Pure Fluid@ compd lo2%,cm3 P, D Qxx,DA Qyy, DA acetone 6.37 2.88 2.149 1.447 chloroform 9.00 1.04 -0.942 -0,942 sulfur dioxide 4.28 1.63 -5.300 4.000 dimethyl ether 5.16 1.30 3.300 -1.300 3.28 methanol 1.70 -1.328 3.660 water 1.50 1.88 1.970 -1.890
10SKHe,A3 0.00 0.00 0.00 0.00
0.00 0.00 0.002 0.00 0.00 0.00 0.00 9.8 24.2
Qz,
DA
-3.596 1.884 1.300 -2.000 -2.332 -0,080
LI Multipole parameter values were obtained from Appendix D of ref 43, except those for dimethyl ether and sulfur dioxide, which were obtained from ref 52 and the quadrupole moment values for acetone and chloroform which were calculated as described in the text. The units for Q are DA = esu.
as that given by the Weeks-Chandler-Ander~en~~~~ theory for spherically symmetric molecules AIdis= 2rpNJuLJ(r)go(r)$ dr
in which uLJ = 4c[(~/r)-'~ - (u/r)+]. For dispersion interactions, the higher-order terms (AZdis,Ajdis, etc.) are ignored. The LJ diameter (a)and energy (e) are constants with respect to temperature and density, and u is related to the temperature- and density-dependent hard core diameter (d)through the prescription of Verlet and Weis. For mixtures, the Lorentz-Berthelot combining rules are used. Some mixtures require the use of the binary dispersion parameter (kij). This is included in the usual way, eij (eii~,j)'/~(l - k,,). Altogether then, the intermolecular potential model is composed of a nonspherical repulsive core with a LJ attractive interaction, nonaxial multipole moments (dipole and quadrupole), and short-range square-well attraction sites for interactions leading to molecular association. The parameters in the theory (Tables I and 11) are intermolecular potential parameters which, as described below, are adjusted to correlate the properties of various associating and nonassociating fluids. The short-range interactions which lead to association could arise from the short-range portion of an electrostatic interaction (not already accounted for by a multipole expansion) or from a charge-transfer interaction. The contribution of the attraction sites to the thermodynamic properties is given by Wertheim's theory of association. The contribution of a nonspherical repulsive core is given by a modified TPT equation based on Wertheim's theory for hard-chain molecules. The contribution of longer-range electrostaticinteractions is given by the u expansion for the nonaxial multipole interactions of Gra~-Gubbins,4~ which has been implementedS2and tested prev i o u ~ l yfor ~ ~fluids of nonassociating molecules. 111. Results
Before discussing several specific compounds and the theoretical results obtained, a few general comments are in order. In comparisons of theory and experiment, an ever present problem is that the true intermolecular potential of most molecules is not precisely known. The parameters in a theory, which represent the magnitudes of various intermolecularforces, must therefore be adjusted to obtain agreement with experiment. It is most desirable to use pure component data (vapor pressure and liquid density) for
10998 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 TABLE III: Average Errm in Pure Fluid Results liquid density vapor pressure av av error, error, T,K % ' T,K P, bar 9% "pd argon 110-201 418-3038 0.5 84-150 1.5 methane 250-423 30-483 2.7 105-188 2.6 1.8 nitrogen 90-120 5-626 0.8 75-125 70-145 1.9 oxygen 200-400 70-500 1.5 ethane 160400 34-741 2.3 170-290 1.7 propane 311-360 14-344 1.7 183-351 2.7 nitric oxide 120-170 10-700 3.1 111-170 1.5 1.2 acetone 293-398 10-588 0.8 259-390 254-518 3.1 chloroform 293-600 10-202 8 sulfur dioxide 323-523 8-288 16 237-405 2.5 dimethyl ether 283-333 3-15 0.9 283-363 0.5 methanol 298-473 5-1000 2.5 288-463 2.5 water -273-423 25-500 2.9 293-500 1.1 TABLE I V Parameter Vdues for Mixture Cross Interactions mixture cHB/k,K 10sKHB,A' k, nitric oxide/methane 0.0 0.0 0.05 0.0 0.05 nitric oxide/nitrogcn 0.0 3.401 0.0 acetone/chloroform 130.3 339.9 3.963 0.0 sulfur dioxide/dimethyl ether 0.0 -0.14 methanol/propane 0.0 methanol/water 750.0 13.0 -0.09
parameter adjustment and to predict mixture properties, without further adjustment of parameters, as a test of theory. Where possible, this is the procedure, For some of the mixtures considered, however, this procedure is not possible due to the presence of cross association interactions between the two mixture components. Such cross association requires the use of mixture-adjustable association parameters (eHBand KHB).For such cases, a minimum of .mixture data is used (two isotherms of phase equilibria) for parameter adjustment, and the remaining isotherms as well as enthalpy of mixing are predicted and compared with experiment where these data are available for comparison. Depending on the intermolecular form present, a molecule may have several parameters. Not all of the parameters in the theory are adjusted, however. The two Lennard-Jones parameters (e and a) and Y are adjusted. For our purposes, most literaturereported values of multipole moments are sufficiently accurate to use without adjustment. In specifying the a d a t i o n potential, it is necessary to specify the number of association sites and to adjust the depth (eAB) and bonding volume (KAB)parameters for all possible sittsite pair interactions. To specify the number of association sites requires, in some cases, a review of quantum mechanical, computer simulation, and experimental results. For each of the associating fluids studied, therefore, there is presented a discussion of the appropriate number of association sites to use in the assumed intermolecular potential. While the parameters represent the magnitudes of various intermolecular interactions, it is not possible to use parameter values to determine a unique intermolecular potential. This is particularly true of parameters which depend on molecular geometry (v, KHB,multipole values). To illustrate the point, several different distributions of fractional point charges can give the same value of dipole moment. A measured value of the dipole moment therefore does not allow calculation of a unique distribution of fractional point charges. Similarly, a v value obtained by regression of data cannot be used to specify a unique repulsive core geometry (bond lengths, bond angles, and segment diameters). Consequently, the intermolecular potentials for the molecules treated in this work are not uniquely specified but are instead characterized by the parameter values and the number of association sites. Fluids of Nonpolar Molecules. All nonpolar compounds are treated with at least two adjustable parameters (c and a), and in the case of nonspherical molecules, v is also adjustable. These parameter values were determined by the correlation (nonlinear regression) of liquid density and vapor pressure data. The pa-
Walsh et al.
,
I
30
-
T = 122.04 K -A experiment theory
-
20-
B \
a
0
0.50
0.25
I.o
0.75
XN>YN
Figure 1. Comparison of theory and experiment9' for the P j y diagram of the mixture, nitrogen/methane. T = 283.15K
*.
experiment - t h y
20
0
0.25
0.5
0.75
1.0
XE~YE
Figure 2. Comparison of theory and experiment9*for the P j y diagram of the mixture, ethane/propane.
rameter values are given in Table I, and the percent errors (pacent difference between measured and calculated) in pure component properties are given in Table 111. Argon and methane are both treated as spherical with Y = 1. Nitrogen, oxygen, ethane, and propane are treated as nonpolar, nonassociating, and nonspherical molecules; the multipole parameters are zero, the association parameters are zero, and v # 1. For mixtures, the unlike pair parameters au and tu were obtained . from the Lorentz-Berthelot combining rules, and v = ~ , x , v t In Figure 1 is shown a phase envelope for the nitrogen/methane mixture. The theoretical results show good agreement with experiment as shown, without the use of mixture-adjustable parameters. The ethane/propane mixture is shown in Figure 2 for two isotherms. Again the agreement is good without mixtureadjustable parameters. Pure Nitric Oxide md Mhvblres with Nitric Oxide. Nitric oxide plays a major role in photochemical smog. It is a component of vehicle emissions and eventually diffuses into the upper atmosphere where it reacts with sunlight and atmospheric oxygen to produce nitrogen dioxide. Nitrogen dioxide, in turn, undergoes photochemical reactions to produce activated oxygen radicals and a cascade of higher molecular weight organic compound^.^^ A theory for the thermodynamic properties of nitric oxide would be useful in understanding the mechanism of its production and in predicting its fate in the atmosphere. Since it is one of the simplest high explosives,56it has been studied extensively in shock-initiated detonation studies to gain a better understanding of the microscopic features of explosive reactions. It is ideally suited for the study of rotational dispersion on surfacess7 because it absorbs infrared radiation at energies easily accessible by lasers. It has been studied in several other connections as well.58ds Nitric oxide has a strong tendency to dimerize due to the presence of one unpaired ?r electron in each monomer. In the dimer, the two r electrons from the monomers are shared and all orbitals are then filled. This explains why only dimers are formed and not oligomers of several associating molecules. The structure of gas-phase nitric oxide dimers has been determined from measurements of the microwave rotational transition frequencies for various isotopes66and is shown in Figure 3.
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 10999
Theory for Fluids of Small Associating Molecules
0
J
Figure 3. Structure of the gaseous nitric oxide dimer determined by measurements of the microwave rotational transition frequencies for various isotopes.&
I 100
= 43.7 I20
140
160
I
180
T/ K I
I
I
A
-B
I
I
argon oxyen nitrogen
f
a
-3 -4
-
Figure 5. Comparison of theory and experiment for the fraction of nitric oxide molecules in the dimer state. The fraction of dimers was determined by magnetic susceptibility measurements?
W + omciation NO
-5
1
I
1.0
1.2
I
1.4
b,N2,02
I
1.6
1.8
2.0
Tc/T
Figure 4. Comparison of theory and experiment for the vapor pressures of nitric oxide and other small molecules. The vapor pressures for the three compounds argon, nitrogen, and oxygen obey roughly a corresponding states relation, while the vapor pressure of nitric oxide deviates significantly from corresponding states.
AHvapt J/mol 5 582 6814 13 587 40 673
0.75
I.o
1.25
1.5
T I Tc
Figure 6. Calculated results for the fraction of nitric oxide dimers over
TABLE V Pure Fluid Vaporization Properties "pd nitrogen oxygen nitric oxide water
I
I
0.5
ASvlptJ/Wmol)
72.2 75.6 111.9 109.1
Though nitric oxide has a strong tendency to form dimers, it has only a small dipole moment ( p = 0.16 D). Thus, it is one of the few strongly associating fluids which does not also have strong multipolar interactions. For this reason, it is an interesting compound for the application of the thermodynamic theory since the hydrogen-bonding contribution can be evaluated without the added complication of the longer-range electrostatic interactions due to multipole moments. In applying the theory, nitric oxide is modeled as a diatomic molecule with a single, self-associating, square-well attraction site. Due to the small value of the dipole moment, multipolar interactions are ignored. The parameter values (e, u, v, eHB, K H B ) were determined by regression of pure component vapor pressure and liquid densities. All five parameters were adjusted until the error was minimized. These parameter values, given in Table I, were then used without further adjustment to calculate the fraction of dimers and the phase behavior of various mixtures (see below). The average error in pure nitric oxide liquid density and vapor pressure is given in Table 111. Shown in Table V are AHva and ASvspat the normal boiling point for 02,N2,NO, and H26.Note that the values for nitric oxide are significantly different from those of either nitrogen or oxygen. Nitrogen and oxygen have no tendency to associate. They both have even numbers of electrons with all electrons paired. If NO did not associate,its properties would be roughly intermediate between O2and N2.The calculated values of AHvspand ASvap for nitric oxide arc 15 OOO J mol-' and 130 J K-'mol-', respectively. In Figure 4, theory and experiment are compared for the vapor pressure of pure nitric oxide. Excellent agreement is found. As shown in Figure 4, the vapor pressures for the three compounds argon, nitrogen, and oxygen obey roughly a corresponding states relation, while the vapor pressure of nitric oxide deviates significantly from corresponding states. The slopes of these vapor pressure curves follow the trends in AH shown in Table V, as would be expected from the Clausius-Clapeyron equation. For
a range of temperatures and densities.
the three nonpolar fluids (N2,02,Ar),a single theoretical result is shown which is actually the result for Ar. In Figure 5 is shown a quantitative comparison of theory and experiment for the fraction of dimers. The theoretical results shown here are predicted without adjustment of parameter values. Nitric oxide dimers have a weak (if not negligible) magnetic moment due to an even number of electrons, while monomers have an odd number of electrons and a fairly strong magnetic dipole moment. These magnetic properties were used in the determination of the fraction of dimers from measurements of magnetic susceptibility! As shown, the theory compares reasonably well with measurements over nearly the entire saturation curve which ranges from the triple point of 109 K to the critical point of 180 K. The bonding energy parameter corresponds to roughly 15.3 kJ/mol of dimers. For molecules with a single site, such as NO, the fraction of sites not associated equals the fraction of molecules not associated for which eq 10 can be solved analytically: X=
-1
+ (1 + 4pA)'I2 2PA
(21)
Further, the ideal mixture equilibrium constant (K,)is related directly to A:
in which CHB is the concentration of associated molecules and CM = Xp is the concentration of unassociated (monomeric) molecules. As shown in Figure 6, the theory can be used to calculate the fraction of dimers over a wide range of temperatures and densities. As expected, the fraction of dimers decreases with increasing temperature and increases with increasing density. At constant density, increasing temperature has a very strong effect such that the fraction of dimers is significantly reduced even for the highly compressed densities that are shown. Along the saturation curve, the combined effect of increasing temperature and decreasing density dramatically reducts the fraction of dimers such that at the critical point few dimers are formed.
llO00 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
Walsh et al.
1-1 1.2
= 323.15K
T
a
0.8
experiment
I \\
I
T = 111.2K 0
'
.
.
'
I
"
'
'
0.5
0
I .o
T = 298.15K
0,3
XNO'YNO
Figure 7. Comparison of theory and experiment for the P , x y diagram of the mixture, nitric ~xide/methane.~~
0.20
0.25
0.5
XCncIs
I
0.75
IO
YCnCl,
Figure 9. Comparison of theory and experimentg5for the P,XJ diagram of the mixture, acetone/chloroform.
L
I
T = 2 9 8 .I5 K
I
P
I bar
experimmt
- theory
0..
- theory ea
experiment
n
-2500 0.25
0
0 0
025
0.5
0.75
10
xNo* YNO
Figure 8. Comparison of theory and experiment for the P,x,y diagram of the mixture, nitric ~xide/nitrogen?~
Shown in Figure 7 is the P j y diagram for the mixture, nitric oxide/methane. A binary dispersion parameter (k,,),given in Table IV, was required for accurate results. Good agreement is found between experiment and theory. Shown in Figure 8 is the P j y diagram for the mixture, nitric oxidelnitrogen. Again, with the use of a binary dispersion parameter (k,,),given in Table IV, there is good agreement between experiment and theory. Pure Acetone md ChlOrorMm and Their Mixtures. Both pure acetone and chloroform are modeled as nonassociating fluids. Literature reported values of the dipole moment were used. S i n c e measured values of the quadrupole moments could not be found in the literature, they were estimated by way of a multipole expansion (eq 2.58 of ref 43), using point charge values from literaturereported computer simulation potential models.67@The three nonpolar parameters (e, u, and v ) for each pure component were determined by regresoion of vapor pressure and liquid density data. All of the parameters used for these compounds are given in Tables I and 11. The regression results are given in Table 111. The acetone/chloroform mixture represents an interesting test case for the theory since for most conditions it has a negative azeotrope. Empirical eviden~e~~.'O indicates that neither pure fluid has appreciable hydrogen bonding but the mixture has a cross association interaction. To account for the mixture cross association, Wertheim's theory is used. Chloroform is modeled as having one hydrogen-bond donor site, and acetone is modeled as having two hydrogen-bond acceptor sites. The two sites of acetone correspond to the two pairs of unbonded electrons on the oxygen. Placing two sites on the oxygen is consistent with spectroscopic evidence which indicates that in an acetone/chloroform mixture, some acetone molecules have two chloroform molecules bonded to tbem.1° Accounting for the hydrogen bonding in this mixture introduces two mixtwadjustable parameters (eHB and KHB).The values of these parameters were adjusted so that the theory gives the correct azeotropic composition and pressure. The values of
0.50
0.75
I .o
Xcnci,
Figure 10. Comparison of experimentg6and theory for the enthalpy of mixing for the mixture, acetone/chloroform.
-2m
L
298.15
323.15
348 15
373 15
T/ K
Figure 11. E~perimentally~~ measured second virial coefficients for components in the sulfur dioxide/dimethyl ether mixture.
the mixture association parameters are given in Table IV. The theoretical results shown in Figure 9 agree well with the measured data,but since mixture assoCiation parameters were required, these results are not a rigorous test of the theory. As a test of theory, excess enthalpy was calculated. The calculations were carried out using Wx= H-R - x l H I R- xzHzR in which HR denotes residual enthalpy and H,R is the residual enthalpy of pure component i at the system temperature. Residual enthalpy is calculated using HR = ER + PV- RT, and the residual energy ( E R ) is calculated using E R = t3(AR/T)/t3(l/n. The calculated results are compared with measured data in Figure 10. Agreement with experiment is very good. Pure Sulfur Dioxide a d Dimethyl Ether pad Tbeir Mixtures. The sulfur dioxide/dimethyl ether mixture shows similar features to the acetone/chloroform mixture. Although both sulfur dioxide and dimethyl ether have signiiicant multipolar interactions,neither has appreciable self-amciation. In Figure 11 are shown the second virial coefficients7'for the 1:l and 2:2 (likelike) and 1:2 (cross) interactions. As shown, the BIZcross virial coefficient follows a
The Jouinul of Physical Chemistry, Vol. 96, No. 26, 1992 11001
Theory for Fluids of Small Associating Molecules
m
4 experiment ...... N P + M P
24
1-
..,? ,...:..
- NP + MPt as?,,...:::.':,'"' ..." ,..:::...". .. . .. .... .... ....."'." ,..'
, ,
,
,,
20
. n
T 363.15 K " " " " '
-8
I
1.0
1.2
I
1.4
1.6
I
I .e
2.0
Tc/T
/1
,I
Figure 13. Comparison of theory and experiment9' for the vapor prcssure of methanol and water.
I " " " " ' 1
NP + MP + assoc 0
50
I .o
0.5
T = 373.1
XW>Y,
40-
Figure 12. Comparison of theory and for the P , X j diagram for the mixture, dimethyl ether/sulfur dioxide.
curve which sharply deviates from that of either Bll or BZ2.This indica* that there is a strong cross interaction between a dimethyl ether and a sulfur dioxide molecule which does not occur when either molecule interacts with a like molecule. Though both compounds Nve si@icant electrostaticinteractions, presumably the charge distributions and molecular shapes give rise to steric hindrance and do not allow self-association. The association interaction which occufs in the mixture is most likely due to charge transfer. This could be experimentally verified by ultraviolet SP-VcoPY * In Figure 12 is shown the result of two theoretical approaches for this mixture. For the dotted curves, nonpolar and multipolar (NP MP) interactions are accounted for without accounting for d a t i o n . This approach uses three adjustable parameters (e, u, and v ) for each pure component and no mixture-adjustable parameters. The multipole parameter values reported in Table I1 were obtained from the literature (see ref 43, Appendix D) and used without adjustment. Since the NP MP approach does not account for association, there is poor agreement with the experimentaldata, as expected. In the second theoretical approach (NP MP Assoc), association is accounted for between the unlike species. The results, showing good agreement with experiment, are given by the solid m es of Figure 12. In this approach, there are three adjustable parameters for each pure component. The pure component parameters are the same as in the NP MP approach. Also, there are two association parameters (eHB, K H B ) to account for the 1:2 cross association interaction which occurs between the unlike species in the mixture. These two association parameters were adjusted using the lower temperature data ( T = 283.15 and 323.59 K), and the higher temperature (T= 363.15 K)results were predicted without further adjustment of the parameters. Good agreement with experiment was obtained for all three temperatures. The mixture association parameter values are given in Table IV. The higher temperature predictions give a measure of the accuracy of the theory when it is used to extrapolate from measurements. This mixture is an example of a case in which the full multipole treatment does not give accurate results; chemical association must be accounted for. Fwe Mahrnd In applying the theory to methanol, the number of association sites must be specified. Methanol is modeled as having three hydrogen-bonding sites (a, b, and c):
+
+
+
+
+
c b CH3 -0,
Ha
Site (1 is centered on the hydrogen atom of the OH group; sites 6 and c are centmd within the Lennard-Jones core of the oxygen atom. The model used here, having a separate site for each of the two pairs of unbonded electrons on the oxygen atom, is similar to the so-called primitive model of Kolafa and Nezbeda.'* The
30
s
-
20 -
n
2
10-
XMEOH
I
YMEOH
Figure 14. Comparison of theory and experiment9*for the P j y diagram of the mixture, propane/methanol.
predominantly electrostatic nature of the hydrogen-bonding interactions in methanol eliminates the attractive H:H and 0:O interactions and leaves only the O:Hinteractions. In terms of the bond energies, E,, = Ebb = ecc = 0 and tab = sac # 0. The symmetry in this model reduces the number of independent Xs to one (X,= X, = I/&). The values of five potential parameters are needed for pure methanol, the three nonpolar parameters, E, u, Y, and two association parameters, EHB and KHB. To determine the adjustable parameter values in the model, pure component vapor pressure and liquid densities were regressed. These parameter values were used without further adjustment. The theoretical results for pure methanol vapor pressures are shown in Figure 13. The parameters used in this model are given in Tables I and 11, and the regression results are given in Table 111. Mixtures of Metbaod and Propane. Mixtures of methanol and propane show large deviation from ideality. Methanol has strong hydrogen-bonding forces, while propane is nonpolar, and no hydrogen bonding occur^ between methanol and propane. To dissolve even a small amount of propane in methanol requiresa high partial pressure. That such high pressures are required is a consequence of the strong self-association of methanol. Pure methanol is almost entirely hydrogen bonded so that when propane dissolves, it breaks bonds. The theoretical results for this mixture are shown in Figure 14. As reported in Table IV,a rather large value of a dispersion binary parameter (ko)was required to obtain agreement with experiment. Rue Water. In applying the theory to water, as with the previously studied fluids, the number of association sites must be specified. Empirical evidence indicates that in condensed phases, water molecules form a tetrahedral hydrogen-bonded structure. Water molecules in ice (polymorph IC, the most common), for example, form a hexagonal structure in which each molecule is hydrogen bonded to four others in a tetrahedral The
11002 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
tetrahedral bonding structure would suggest the use of four association sites; however, the potentials developed in computer simulation studies use only three sites and still give rise to a tetrahedral bonding structure. Although there have been several intermolecular potentials investigated for water, there is no single potential that is clearly better than all There is little debate that positive fractional point charges should be placed on each of the hydrogen atoms, but the placement of negative fractional point charges is less certain. For our purposes, the potential models for water can be organized into three categories according to their source (ab initio or empirical) and the number of charge sites used (three or four). The early effective pair potentials (Rowlinson,B0J1 BNS,82,83,ST284)use four charge sites. The ab initio isolated dimer potentials (Popkie et al.,85MCY-CIa6) use three charge sites. The more recent effective pair potentials (SPC,8’ TIPS,B8J9) use three charge sites. As discussed previously, the ab initio isolated dimer potentials are most likely to have realistic electrostatic charge distributions and the recent effective potentials use the quantum results as their starting point. The early effective pair potentials (BNS and ST2), incorporated four charge sites in a tetrahedral geometry. The hydrogen-bonding structure observed by simulation was tetrahedral, in agreement with experimental observations. The more recent potential models (Popkie et al., MCY-CI, SPC, TIPS) employ only three charge sites, one each for the two hydrogen atoms and one for the unpaired electrons of the oxygen atom. With these potentials, the oxygen charge site has, on average, two hydrogen sites bonded to it such that the observed hydrogen-bonding structure is tetrahedral. Thus, the oxygen site is relatively large so that it can accommodate the two smaller hydrogen sites. In the Wertheim theory, however, this type of bonding topology is forbidden. In the Wertheim theory, it is assumed that no two sites will be simultaneously bonded to a third. This restriction is easily incorporated in models for water by assigning four sites to water. This allows a tetrahedral geometry in which two hydrogen atoms, each having one site, can bond to a single oxygen atom which has two sites. Water, therefore, is modeled as having four hydrogen-bonding sites (a, b, c, and d): a b
/ O\
H c
H d
As with methanol, the predominantly electrostatic nature of the hydrogen-bonding interactions in water eliminates the attractive H:H and 0:Ointeractions and leaves only the O:H interactions. In terms of the bond energies, e,, = ebb = e,, = edd = 0,tab = 0 and e,d = 0. The symmetry in this model reduces the number of independent X’s to one (x,= xb= X,= Xd = x). Calculated vapor pressures are compared with literature-reported vapor pressures in Figure 13. The average errors in regressing liquid density and vapor pressure are given in Table 111. Mixtures of Water and Methanol. The most challenging application of the thermodynamic model to date is in predicting the phase behavior and properties of methanol/water mixtures. An understanding of their phase behavior is important in the gas industry, which uses methanol to inhibit the formation of gas hydrates in pipelines and well bores. Water/methanol mixtures are class I mixtures and show only slightly nonideal phase behavior. At moderate temperature and pressures, no azeotrope is formed, and the two components are readily miscible in all proportions. Although the mixture phase behavior is fairly ideal, the intermolecular forces are complex. Both fluids have strong association interactions in their pure states, there is an interaction between the two unlike pairs in a mixture of methanol and water, and both methanol and water have substantial nonaxial multipolar interactions. The ideality of the mixture phase behavior therefore arises from strong intermolecular forces which happen to be of roughly similar strength. An accurate theoretical treatment must account for not only the strong self-associationforces of the pure fluids but also the cross interactions which occur in the mixture as well.
Walsh et al.
r---l - theory
o.:m -
1
T = 338.15K
T = 323.15 K
0.51
T
:
1
308.15K
l%zeE?I
0
0.5
0
XMEW
IO
TYMEOH
Figure 15. Comparison of theory and experiment99for the P j y diagram of the mixture, water/methanol. 0
I
I
-500-
.
P =
T = 298.15
I bar
..
z -1oooW
I
- Ixx)
I
I
**
0
0.25
experiment
--
I theory
I
= 0.5
0.75
I .o
XMEW
Figure 16. Comparison of theory and experimentlwfor the excess enthalpy of the water/methanol mixture.
In predictive theories for mixtures, it is desirable to use combining rules which relate the parameters for cross interactions to the parameters for self-interactions. For example, the LorentzBethelot combining rules are often used for the nonpolar cross interactions of simple molecules. For the short-range interactions which lead to association (charge transfer, short-range electrostatics, cooperative effects), accurate combining rules have not yet been developed. Therefore, the association parameters (eWM and KWM)accounting for methanol/water cross association are treated as independent of the parameters for self-association and must therefore be adjusted using mixture data. Unlike macroscopic equation of state parameters, however, the cross association parameters correspond to intermolecular potential parameters and are therefore treated as independent of temperature and pressure. As given in Table IV, an additional mixture-adjustable parameter (k,)was used for the unlike pair dispersion interaction. The two association parameters and k, were adjusted using the lowertemperature data (T = 308.15 and 323.15 K), and the highertemperature ( T = 338.15 K)results were predicted, as a test of theory, without further adjustment of parameters. Excellent agreement is obtained for the phase behavior over the entire temperature range as shown in Figure 15. As a further test of theory, excess enthalpy was calculated. The calculated result is compared with measured data in Figure 16. These results were predicted without adjustment of parameters, and the agreement with experiment is rather poor. The magnitude of the predicted minimum is inaccurate and the composition at the minimum does not agree with the experimental data. One possible reason why this poor agreement is obtained is that the structure of the reference fluid is very much different from the assumed structure (see eqs 15 and 17). In the theory, go for the hydrogen-bonding fluid is replaced by go for a fluid of LJ atoms, and hence, the assumed fluid structure is independent of the
Theory for Fluids of Small Associating Molecules fraction of molecules associated. Further, a liquid of LJ atoms has a coordination number of roughly 9, whereas spectroscopic studies have shown that liquid water has a coordination number closer to 4.w Nevertheless, considering the simplicity of the theory used here, the results of Figures 15 and 16 are encouraging.
IV. Summary and Conclusions In this paper, we have presented theory and application for the properties of associating fluids. The theory is based on the physical approach and accounts for the effect of repulsion, dispersion, longer-range electrostatic (multipolar) interactions, and the short-range chargetransfer and electrostatic interactions that lead to association. Unlike the many chemical equilibrium models for associating fluids, the theory is based on the use of well-defined intermolecular potentials. To our knowledge, this is the first attempt to combine Wertheim’s theory of association with the u expansion for nonpolar and multipolar interactions and thus to account for all of the components in a realistic two-body potential for associating fluids. Among the associating fluids considered are nitric oxide, methanol, and water; the pure fluid properties, enthalpy of mixing, and mixture phase behavior are calculated. Also, mixtures in which a cross association (solvation) occurs between the unlike species are treated; these include acetone/chloroform and dimethyl ether/sulfur dioxide. While promising results are obtained for various properties, still much more work needs to be done. In comparisons of theory and experiment, an ever present problem is that the true intermolecular potential is not known and the parameters in the theory must be adjusted to obtain agreement. When poor agreement occurs, as with the enthalpy of mixing for the water/methanol mixture, it is difficult to know if it is due to poor theoretical approximations or due to a poor choice of the assumed intermolecular potential. Spectroscopic determinations of the fractions of molecules associated would be useful as an independent check of the theory and as a source of information on the association interactions. An important weakness in the theory is the approximation used for the structure of the associating reference fluid. A more accurate theory for associating liquid structure is needed to improve this situation.
Acknowledgment. Support of this research by a contract from the Gas Research Institute (Contract No. 5086-260-1254) is gratefully acknowledged. We thank Professor John A. Zollweg and Dr. Joe A. Noles for their helpful discussions and advice. References and Notes (1) Stillinger, F. H. In Advances in Chemical Physics; higogine, I., Rice, S.A., Eds.; John Wiley and Sons: New York, 1975; p 1. (2) Schuster, P.; Zundel, G.; Sandorfy, C. The Hydrogen Bond. Recent Developments in Theory and Experiments; North-Holland Publishing Co.: Amsterdam, 1976. (3) Nesbit, D. J. Chem. Rev. 1988, 88, 843. (4) Smith, A. L.; Johnston, H. L. J . Am. Chem. Soc. 1952, 74, 4696. (5) Guedes, H. J. R. Propriedada Termodinamicas de Sistemas Liquidos Simples. Ph.D. Thesis, Universidade Nova de Lisboa, Lisbon, 1988. (6) Thompson, C. C.; deMaine, P. A. D. J . Phys. Chem. 1965,69, 2766. (7) Walsh, J. M.; Koh, C. A.; Gubbins, K. E. Fluid Phase Equilib. 1992, 76, 49. (8) Luck, W. A. P. Hydrogen Bonds in Liquid Water. In The Hydrogen Bond. Recent Developmenrs in Theory and Experiments; Schuster, P., Zundel, G., Sandorfy, C., Eds.; North-Holland Publishing Co.: Amsterdam, 1976; Chapter 28, p 1369. (9) Eckert, C. A.; McNiel, M. M.; Scott, 8. A,; Halas, L. A. AIChE J . 1986, 32, 820. (10) Walsh, J. M.; Greenfield, M. L.; Ikonomou, G. D.; Donohue, M. D. Inr. J. Thermophys. 1990, 1 1 , 119. ( 1 1 ) Heidemann, R. A.; Prausnitz, J. M. Proc. Narl. Acad. Sci. U.S.A. 1976, 73, 1773. (12) Ikonomou, G. D.; Donohue, M. D. AIChE J . 1986, 32, 1716. (1 3) Noles, J. Vapor-Liquid Equilibria of Solvating Binary Mixtures. Ph.D. Thesis, Cornell University, Ithaca, NY, 1991. (14) Murrell, J. N.; Carter, S.;Farantos, S.C.; Huxley, P.; Varandas, A. J. C. Molecular Potential Energy Functions; John Wiley & Sons: New York, d 1984. (15) Maitland, G. C.; Rigby, M.; Smith, E. B.; Wakeham, W. A. Intermolecular Forces; Clarendon Press: Oxford, 1987. (16) Kitaura, K.; Morokuma, K. Int. J . Quantum Chem. 1976, 10, 325. (17) Morokuma, K. Acc. Chem. Res. 1977, I O , 294.
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~
Local Composition Model for SquareWell Chains Using the Generalized Fiory Dimer Theory Costas P. Bokis, Marc D. Donohue,* Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 21 218
and Carol K. Hall Department of Chemical Engineering, North Carolina State Uniuersity, Raleigh, North Carolina 27695 (Received: March 20, 1992)
The generalized Flory dimer (GFD) theory of Hall and co-workersprovides a highly accurate equation of state for hard-chain molecules. It has been extended here to treat chain molecules that interact with a square-well site-site potential by using closed-form expressions for the equations of state for squarewell monomer and dimer fluids based on local composition theory and using the analytic solution of the Percus-Yevick theory for monomers and RISM/MSA calculations for dimas. Comparison with Monte Carlo calculations for square-well molecules of lengths 4, 8, and 16 shows that there is very good agreement between this theory and simulation data.
Introduction In recent years, Hall and co-workersI4 have derived continuous-space analogues of the Florys lattice theory. They developed the generalized Floryl (GF)equation of state for freely-jointed tangent hard chains (n-mers). The GF compressibility factor can be written in the form ZHC(v,n)= 1
+ aZcp(q,l)
(1)
where ZnP(v,l) is the compressibility factor for a system of hard spheres (monomers), which can be calculated by the CarnahanStarling6 equation, v is the volume fraction (9 = n ~ p d / 6where u is the segment diameter), and a is the ratio of the excluded volume of the n-mer to the excluded volume of the monomer (a = zc(n)/ue(l)). Their derivation was based on the realization that the nary lattice estimate for the probability of ihpertinga segment in a chain fluid is too large for a continuous-space fluid. Hence, they replaced the Flory insertion probability with the insertion probabilitycalculated from the Carnahan-Starling equation. They made the assumption that the probability of inserting a monomer into a chain fluid is the same as the probability of inserting a monomer into a monomer fluid, at the same volume fraction. The ratio of the excluded volumes, a,appears in eq 1 because the insertion of the second and subsequent segments of the chain Author to whom correspondence should be addressed.
0022-3654/92/2096-11004$03.00/0
requires a smaller hole in the fluid than the insertion of the first segment. As noted by Vimalchand and Donohue,' the GF equation has the same functional form as the perturbed-hardchain theory (PHCT).* Honnell and Hall4 extended and improved the GF theory by accounting for chain connectivity through the use of a dimer equation of state. They developed the generalized Flory dimer (GFD)theory for a hard-chain n-mer, which has the form ZHC(v,n)= Z(a2)
+ Y,(Z(v,2)- Z(v,1))
(2)
where Z(q,l) is the hard monomer compressibility factor, calculated from the Carnahan-Starling equation, Z(tj.2) is the hard dimer compressibility factor, calculated from the Tild~aley-Streett~ equation of state, and Y,is a function of the excluded volumes, given by
(3) where ue(l), ue(2), and u,(n) are the excluded volumes of the monomer, dimer, and n-mer, respectively. The GF'D theory assumes that the probability of inserting the whole chain into the chain fluid is the product of the probability of inserting the first site times the product of the conditional probabilities of inserting each subsequent site along the chain, given that the previous site has already been inserted. Q 1992 American Chemical Society
