Ind. Eng. Chem. Res. 2002, 41, 1057-1063
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Nonrandom Distribution of Free Volume in Fluids and Their Mixtures: Hydrogen-Bonded Systems Theano Vlachou,† Ioannis Prinos,† Juan H. Vera,‡ and Costas G. Panayiotou*,† Department of Chemical Engineering, University of Thessaloniki, 54006 Thessaloniki, Greece, and Department of Chemical Engineering, McGill University, Montreal, PQ, H3A 2B2 Canada
A new equation-of-state model is presented that accounts for the nonrandom distribution of the free volume, as well as of the molecular segments, in systems of fluids interacting with strong specific forces. Expressions for the basic thermodynamic quantities are derived for pure fluids and for mixtures. The new formalism is applied to the description of the volumetric behavior of water at near-critical and supercritical conditions. The vapor pressures of subcritical water are also calculated. The degree of hydrogen bonding at these conditions is estimated and compared with the results of NPT molecular dynamics calculations and with available experimental data. Introduction In a recent monograph,1 Sandler, Orbey, and Lee discussed the status of equations of state as applied to various systems including those containing associated fluids. They pointed out both the problems and the merits associated with the equations of state. One of the problems that remains unsolved in equation-of-state models and, in general, in thermodynamic models of mixtures is the proper consideration of nonrandom arrangements of the various molecular entities in the system. The nonrandom distribution of molecules or segments of macromolecules in mixtures is a subject of considerable importance and has been repeatedly addressed by numerous scientists over the past few decades. One classical treatment is Guggenheim’s quasichemical theory.2 The essential point of this method is that molecules, or segments of molecules, with favorable intermolecular interactions tend to be close neighbors. By consistently applying this approach, one can derive expressions for the local compositions in mixtures.3 When the focus is on the interactions between molecules, the quasi-chemical theory is useful for the tratment of systems with intermolecular interaction energies that are relatively large in comparison with the thermal energy. For work in systems containing associated fluids, i.e., in systems with strong specific interactions such as hydrogen bonds, a variety of different treatments have been used.4 The majority of these approaches can be classified as either association models or combinatorial models.4,5 Association models invoke the existence of multimers or association complexes and seek expressions for their populations. Combinatorial models do not invoke the existence of association complexes but, instead, focus on donor-acceptor contacts and seek combinatorial expressions for the number of ways of forming hydrogen bonds in systems containing protondonor and proton-acceptor groups. Both types of models * Author for correspondence. Phone: +3031-996223. Fax: +3031-996232. E-mail:
[email protected]. † University of Thessaloniki. ‡ McGill University.
imply that the molecules tend to be nonrandomly distributed because of hydrogen-bonding interactions. The quasi-chemical theory and the hydrogen-bonding approaches address the same problem, and in a sense, they are not complementary. Yet, the two methods can be combined in the treatment of the distribution of empty space between strongly interacting molecules. In this case, the hydrogen-bonding approach can be used to account for the nonrandom distribution of molecules and the quasi-chemical theory for the redistribution of the free volume. A quasi-chemical equation-of-state model to account for the nonrandom distribution of free volume in pure nonpolar fluids as well as in their mixtures has been recently proposed.6 The model proved to be successful in describing the phase equilibria of these systems, especially in the near-critical region. This model is combined in the present work with the combinatorial approach5 for hydrogen bonding. The resulting equationof-state model for hydrogen-bonded systems also accounts for the nonrandom distribution of free volume. The framework used in this work is based on a treatment proposed almost 20 years ago7 for a more general case of r-mer fluids having q external contacts for each of their r segments. However, for the case of mixtures, that treatment7 assumed the free volume to be randomly distributed throughout the system. Here, we confine ourselves to the Flory approximation of the partition function of the system, which forms the basis of the well-known lattice fluid (LF) model.8,9 In addition, we remove the assumption of randomly distributed free volume as pseviously proposed.6 In the first part of the next section, we summarize the basic formalism for the nonrandom distribution of free volume in fluids that do not necessarily interact with strong specific forces. Theory A. The Quasi-Chemical Approach. Let us consider first a system of N molecules of a pure r-mer fluid at temperature T and external pressure P. The molecules are considered to be arranged on a quasi-lattice of Nr sites, N0 of which are empty. The empty sites, however, are not considered to be distributed randomly through-
10.1021/ie0103660 CCC: $22.00 © 2002 American Chemical Society Published on Web 09/22/2001
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out the volume of the system. We assume here that the partition function of the system can be factorized as follows
Q(N,T,P) ) QRQNR
(1)
where QR is the partition function for a hypothetical system with a random distribution of the empty sites and QNR is a correction term for the actual nonrandom distribution of the empty sites. For the first factor, we use the simple lattice fluid (LF) expression,8,9 and for the second factor, we use Guggenheim’s quasi-chemical theory2 as proposed previously7
[( ) ]
N0r0 ! 2 QNR ) Nr0 Nrr!N00! ! 2 N0rr!N000!
The reduced volume is defined as
v˜ )
V 1 1 ) ) V* F˜ f
where F˜ is the reduced density. In the absence of empty sites, the average interaction energy per segment is
s * ) 2
* ) RT* ) P*v*
[( ) ]
(2)
(9)
The scaling temperature T* and scaling pressure P* are related by
2
2
(8)
(10)
The reduced temperature and pressure are defined as
T ˜ )
T P , P) T* P*
(11)
In eq 2, Nrr is the number of external contacts between the segments of the molecules, N00 is the number of contacts between the empty sites, and Nr0 is the number of contacts between a molecular segment and an empty site. The superscript 0 refers to the case of randomly distributed empty sites. The molecules of the pure fluid are considered to be divided into r segments, each of volume v*. The same volume v* is assigned to an empty site. It is assumed that two neighboring empty sites do not coalesce (i.e., they remain discrete). Thus, the total volume of the system is given by
In analogy to eq 7, in the random case, the number of contacts between empty sites is given by the equation
V ) Nrv* + N0v* ) Nrv* ) V* + N0v*
For the number of contacts in the system in the nonrandom case, following the previous treatment,7 we write
(3)
According to the LF model, the partition function QR can be written as
QR )
()() ( 1 f0
N0
ω f
N
exp -
)
E + PV RT
N0 1 1 1 N000 ) N0s ) N0sf0 ) N0s(1 - F˜ ) 2 Nr 2 2
while the number of contacts between a segment and an empty site is given by
N0r0 ) rNs
N0 rN ) N0 s ) rNs(1 - F˜ ) ) N0sF˜ (13) Nr Nr
Nrr ) N0rrΓrr
(4)
N00 ) N000Γ00
where the site fractions f0 and f for the empty sites and the molecular segments, respectively, are related by
Nr0 ) N0r0Γr0
N0 Nr - rN ) )1-f f0 ) Nr Nr
(5)
In eq 4, ω is a characteristic quantity for the fluid that takes into account the flexibility and symmetry of the molecules. This parameter cancels out in all applications of interest and will not be retained further. The potential energy E arises from the intermolecular interactions in the system. In this work, we assume that only first-neighbor segment-segment contacts contribute to E, whereas the contacts of an empty site with a segment or with another empty site are assigned a zero interaction energy. If s is the average number of external contacts per segment (either molecular or empty sites) and is the segment-segment interaction energy, the potential energy of the system is given by
-E ) Nrr
(6)
In the random case, Nrr takes the form8,9
1 Nr 1 ) Nrsf N0rr ) Nrs 2 Nr 2
(7)
(12)
(14)
The nonrandom Γ factors in eq 14 are equal to unity in the random case. These numbers must satisfy the material balance equations3
2N00 + N0r ) N0s
(15)
2Nrr + N0r ) rsN By combining the last three equations, we obtain
(1 - F˜ )Γ00 + F˜ Γr0 ) 1
(16)
F˜ Γrr + (1 - F˜ )Γr0 ) 1 Thus, the three nonrandom factors Γ are not independent. When one is known, the other two can be obtained from eqs 16. The reduced density needed in eqs 16 can be obtained from the equation of state of the system. From statistical thermodynamics, the Gibbs free energy of the system is obtained from the partition function as
G ) - RT ln Q
(17)
At equilibrium, the number of empty sites in the system
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or, equivalently, the reduced density is obtained from the minimization condition
(∂G ∂F˜ )
T,P,N,Nr0
)0
(18)
sk is the average number of contacts per segment k (a surface-to-volume ratio characteristic of molecule k). In the one-fluid approach, the following combining and mixing rules are assumed for the mixture
while the number of contacts Nr0 or, equivalently, the nonrandom factor Γr0 is obtained from the minimization condition
( ) ∂G ∂Nr0
)0
(19)
T,P,N,F˜
Equation 18 leads directly to the equation of state
[
(
P ˜ +T ˜ ln(1 - F˜ ) + F˜ 1 -
]
In the random case (Γrr ) Γ00 ) 1), this equation reduces to the familiar LF equation of state.8 Equation 19 leads to the quasi-chemical relation
4NrrN00 Nr02
)
4ΓrrΓ00 Γr02
) exp )A RT
( )
(21)
By substituting eq 16 into eq 21, we obtain a quadratic equation for Γr0, with the physically meaningful solution
Γr0 )
2 1 + [1 - 4F˜ (1 - F˜ )(1 - A)]1/2
∂G (∂N )
T,P,Nr0,F˜
s s * ) ) (θ1211 + 2θ1θ212 + θ2222) 2 2
(28)
θ1 )
r1s1N1 φ1s1 φ1s1 ) ) ) 1 - θ2 r1s1N1 + r2s2N2 φ1s1 + φ2s2 s (30)
A Berthelot-type combining rule is adopted for the cross term 12, namely
12 ) ξ12x1122
* ) φ11* + φ22* - RTφ1θ2X12
(32)
where
()
This set of equations suffices for performing the basic thermodynamic calculations for pure fluids. We now consider a binary system of N1 and N2 molecules of components 1 and 2, with r1 and r2 segments of segmental volumes v1* and v2*, respectively. The total number, Nr, of lattice sites is now
(25)
where N ) N1 + N2 is the total number of molecules in the system and x1 and x2 are the mole fractions of components 1 and 2, respectively. The average interaction energy per segment of molecule k (k ) 1 or 2) is given by
k* ) (sk/2)kk
(31)
The dimensionless parameter ξ12 is expected to have values close to unity, its value in the classical Berthelot’s rule. Equation 10 can be written in an alternative form as
F˜ µ rF˜ P ˜ v˜ ) r(v˜ - 1) ln(1 - F˜ ) + ln - Γrr + r + RT ω T ˜ T ˜ 1 - F˜ Γrr rs [ln Γrr + (v˜ - 1) ln Γ00] - r (24) 2 T ˜
) N(x1r1 + x2r2) + N0
(29)
and
(23)
Taking into account eqs 18 and 19, eq 23 gives
Nr ) N1r1 + N2r2 + N0 ) rN + N0
r1N1 x1r1 ) ) 1 - φ2 rN r
φ1 )
(22)
Equations 16, 20, and 21 are coupled equations and must be solved simultaneously for the reduced density and the nonrandom factors. The chemical potential is obtained from
µ)
(27)
where the segment fractions φi and surface (contact) fractions θi are defined by
1 s + ln Γ00 ) 0 (20) r 2
)
v* ) φ1v1* + φ2v2*
(26)
where kk is the interaction energy per k-k contact and
* + X12 )
x
(33)
) ξ12x1*2*
(34)
s1 *-2 s2 2 RT
s1 * s2 12
and
12* )
xs1s2 2
12
The total volume of the system is given by eq 3, where v* is now given by eq 27. Because only segmentsegment interactions contribute to the potential energy E, we can write
E ) N1111 + N1212 + N2222
(35)
The number of intersegmental contacts Nij will be obtained by applying the quasi-chemical theory. The following key assumption is now adopted: the empty sites are distributed nonrandomly and make no distinction between their neighbor molecular segments. Thus,
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for the nonrandom distribution, the number of intesegmental contacts satisfies the equations
N11 ) N110Γrr N22 ) N022Γrr N12 ) N012Γrr Nrr ) N11 + N12 + N22 s Nrr ) N0rrΓrr ) NrF˜ Γrr 2 N00 ) N000Γ00 Nr0 ) N0r0Γr0
(36)
QR )
()( )( ) ( 1 f0
ω1 f1
N1
ω2 f2
N2
exp -
)
E + PV RT
(37)
The equation of state for the mixture is obtained by the same minimization procedure discussed previously. The result is an equation of state identical in form to eq 20. Similarly, the quasi-chemical condition for the mixture is identical in form to eq 22, and the material balance equations are formally identical to eqs 16. On the other hand, the variables r, s, , and F˜ in these equations now represent quantities pertinent to the mixture. The chemical potential of each component in the mixture is obtained by a procedure analogous to eq 23, which must now read
( )
∂G µi ) ∂Ni
[
(
P ˜ +T ˜ ln(1 - F˜ ) + F˜ 1 -
The above assumption implies that, with the Nij values obtained from eq 36, eq 2 is also valid for the case of a mixture. For a mixture, the QR term in eq 1 is similar to eq 4 and is given by N0
potential energy, the enthalpy, the volume, and the chemical potential of each component in the mixture consist of two contributions: one physical (denoted by the subscript P) and one chemical or hydrogen bonding (denoted by the subscript H). As proposed previously,5 we consider that there are m types of proton donors and n types of proton acceptors in the mixture. Let dik be the number of donor groups of type i in each molecule of type k and ajk be the number of acceptor groups of type j in each molecule of type k. Let Nij be the total number of hydrogen bonds between donors of type i and acceptors of type j in the system. Using the LFHB procedure,5 we obtain the following expression for the equation of state of the mixture
1 1 ) - νH rj r
(41)
and the reduced total number of hydrogen bonds in the system is given by m n
νH )
m n
Nij
∑i ∑j νij ) ∑i ∑j rN
(42)
The hydrogen-bonding contribution to the chemical potential of component 1 takes the form
µ1,H RT
m
) r1νH -
∑ i)1
d1i
ln
νid
n
-
νi0
∑ j)1
a1j ln
νja
ν0j
(43)
where
(38)
2
νid
By making the appropriate substitutions, eq 38 gives
( )
]
)
where
T,P,Nj,Nr0,F˜
r1 µ1 ) ln φ1 + 1 - φ2 + r1F˜ X12θ22Γrr + RT r2 r1F˜ P ˜ v˜ v1* F˜ r1(v˜ - 1) ln(1 - F˜ ) + ln + Γrr + r1 ω1 T ˜1 T ˜ v* θ1 1 - F˜ Γrr r1s1 [ln Γrr + (v˜ - 1) ln Γ00] - r1 (39) 2 φ1 T ˜
1 s + ln Γ00 ) 0 (40) rj 2
)
Nid
)
∑ dki Nk k)1
rN
rN
Nja
∑ akj Nk k)1
(44)
and 2
νja
)
)
rN
rN
νi0 ) νid -
νij ∑ j)1
ν0j ) νja -
νij ∑ i)1
(45)
while B. Hydrogen-Bonded Systems. Let us now consider the case of a mixture where, in addition to the nonrandom distribution of empty sites, there are strong specific interactions between the segments (e.g., hydrogen bonds). The distribution of segments is no longer random, and favorable distributions are those congruent with the formation of hydrogen bonds. For the description of this nonrandom distribution, we adopt the LFHB procedure,5 while for the nonrandom distribution of empty sites we adopt the quasi-chemical theory. For simplicity, we restrict the discussion to the case of binary mixtures. The extension to multicomponent mixtures is straightforward.5 The key assumption of this treatment is that the intermolecular forces can be divided into physical and chemical (or hydrogen bonding). This implies that the
n
(46)
and similarly m
(47)
As discussed previously,5 the νij’s satisfy the minimization conditions
( )
-GH νij ij ) F˜ exp νi0ν0j RT
for all i, j
(48)
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In eq 48, GH ij is the free enthalpy of formation of the hydrogen bond of type i-j, and it is given in terms of the energy (E), volume (V), and entropy (S) of hydrogenbond formation by the equation H H H GH ij ) Eij + PVij - TSij
(49)
To obtain the full expression of the chemical potential of component 1 in the mixture, eq 43 must be added to eq 39. Applications In this section, the model is applied to the description of the volumetric behavior of water at moderate to high temperatures and pressures. The well-known picture of icebergs, present at lower temperatures, is not expected to be of any importance in this case. In each water molecule, there are two proton donors and two proton acceptor sites and only one type of hydrogen bond. In this case, we have only one minimization condition (eq 48), which, when solved for the number of hydrogen bonds, gives the physically meaningful solution
ν11 )
4 + B - xB(B + 8) 2
Figure 1. Experimental12 (symbols) and calculated (line) vapor pressures of water.
(50)
where
B ) rv˜ exp
( ) GH RT
(51)
To perform further calculations, we must first determine the parameters of the model. There are two types of constants that must be determined: the equation-ofstate scaling constants (T*, P*, and F* or, equivalently, *, v*, and r) and the hydrogen-bonding constants (EH, VH, and SH), i.e., six parameters in total for pure water. If all of these parameters were obtained by a leastsquares fit to the experimental data, a multitude of sets that describe the experimental data equally well would be obtained. To reduce the number of parameters, we have set r ) 1, i.e., we have considered water to be a monosegmental molecule. Because VH and F* are interrelated, we have set F*)1.00 g/cm3, and we have fixed v*)18.02 cm3/mol. EH and SH are also interrelated. Hence, we have fixed SH ) 26.5 J/(mol K) as for hydroxyl interactions. In this way, from the value of EH, it is possible to have a direct comparison of the strength of the water-water hydrogen bond with that of the OHOH interaction. The estimated values for the remaining parameters are EH ) -19.9 kJ/mol and * ) RT* ) P*v* ) 3359 J/mol. In this oversimplified picture, the remaining parameter VH must necessarily vary with both temperature and pressure in order to give an adequate description of the volumetric behavior of water. For this purpose, we have adopted a simple relation analogous to that proposed by Marcus,10 namely
(
)
(T - 383.15)3 VH ) (VH 0 - βP) 1 + R |T - 373.15|
(52)
3 -9 m3/MPa, with VH 0 ) -0.51 cm /mol, β ) 2.03 × 10 and R ) -5.66 × 10-5 K-2.
Figure 2. Experimental12 (symbols) and calculated (line) orthobaric densities of water.
Some comments are in order regarding the values of the above parameters. The energy change upon formation of one water-water hydrogen bond is significantly weaker than the corresponding energy for the OH-OH interaction in alkanols (-25.6 kJ/mol).5,11 It is important to keep in mind that each water oxygen can participate in two hydrogen bonds with protons from other water molecules. The estimated value for VH is negative and almost 1 order of magnitude smaller than the corresponding value for the OH-OH interaction in alkanols (-5.6 cm3/mol).5 If the focus were on the densities at the lower range of T-P conditions, we could even expect a positive value for VH. At high temperatures, however, the degree of hydrogen bonding per water molecule is expected to be significantly lower than 2, and thus, the lack of persistence of long-range order and the geometrical constraints associated with it will no longer lead to positive volume changes upon hydrogen-bond formation. Thus, it is essential to have an estimation of the degree of hydrogen bonding of water in the range of temperature and pressure studied. Figure 1 presents the experimental12 and calculated vapor pressures of water up to the critical point. Figure 2 compares the experimental12 and calculated orthobaric densities of water. In both cases, the agreement is satisfactory. Figure 3 compares the calculated densities of water over an extended range of supercritical conditions with the results of calculations made by Saul and Wagner’s expressions,13 which are essentially experimental data. In view of the significant variation in temperature and pressure, the agreement is again considered satisfactory.
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Figure 3. Experimental13 and calculated densities of supercritical water, as calculated by the equation-of-state model. Figure 5. Average number of hydrogen bonds per oxygen as a function of temperature.
Figure 4. Degree of hydrogen bonding in supercritical water as a function of temperature and pressure as calculated by the equation-of-state model.
As mentioned above, it is essential to have an estimation of the degree of hydrogen bonding of water at supercritical conditions. The maximum number of hydrogen bonds per molecule as calculated by eq 50 is 2, which is the expected value. Thus, the percent degree of hydrogen bonding, NHB, in the system is equal to 100ν11/2. Figure 4 shows the calculated degree of hydrogen bonding of water over an extended range of supercritical conditions. Experimental data at these conditions are sparse. To have an alternative estimation of this degree, we conducted molecular dynamics (MD) calculations for water with the MSI Cerius2 suite of Molecular Simulations Inc. and using the Dreiding 2.11 force field. The energetic/geometric criteria14-15 for the hydrogen-bonding cutoff parameters of 2.5 Å for the O-H distance and 90° for the minimum D-H-A angle were adopted. A sample of 216 water molecules was used in the calculations. The simple picture of three point charges was adopted with the typical charge values of -1.04 for oxygen and +0.52 for hydrogen, neglecting polarizability.14 The picture remains practically unaltered by using different sets of reasonable charge values. For the estimation of the degree of hydrogen bonding, we first calculated the radial distribution function gOH(r), as well as the corresponding functions for O-O and H-H pairs. The calculated radial distribution functions compare favorably with recent experimental data for liquid water at different pressures and temperatures obtained by X-ray and neutron scattering.16,17 Figure 5 compares the calculations for the degree of hydrogen bonding of water
Figure 6. Percentage (%) of hydrogen-bonded water molecules as a function of temperature at 50 and 80 MPa calculated by MD simulations and the equation-of-state (EOS) model.
with experimental data and the results of MD calculations reported by Mountain.18 As observed, the MD calculations of this work are closer to the experimental values, but as the temperature rises, they produce values that are too low. Figure 6 compares the results of our MD calculations of the degree of hydrogen bonding in supercritical water with the corresponding results obtained with the equation-of-state model developed in this work. As observed, there is only qualitative agreement between the results obtained with the two methods. The equation-of-state calculations and the MD calculations have similar trends, but the former are systematically higher than the MD calculations. As discussed previously, the available experimental data indicate that this result is in the right trend, but further experimental verification is required. Discussion and Conclusions The physical picture adopted in this work for the hydrogen bonding in water is a rather oversimplified one. We have considered a one-state hydrogen bonding model and attributed to it a small negative value for the volume change upon its formation. This picture might be justifiable for the relatively high temperatures of interest in this work, but it is not expected to be appropriate for the low-temperature region. In the latter case, the picture of cooperative hydrogen bonding19 with a positive volume change is more appropriate. However,
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1063
this would introduce three additional parameters in the model and would add more complexity to the formalism. Although the nature of water might justify such additional complexity, the simpler version adopted in this work seems appropriate for work in the region of interest here. For this reason, we preferred to arbitrarily fix three of the six model parameters rather than attempting to describe the thermodynamic properties of water at low temperatures with additional parameters. The model presented in this work takes into account two types of nonrandom distributions in pure compounds as well as in multicomponent fluids: the nonrandom distribution of free volume and the nonrandom distribution and orientation of molecules forming hydrogen bonds. The approach that has been adopted leads to compact analytical expressions for all basic thermodynamic quantities of systems of fluids. When applied to water, the model is able to describe satisfactorily the volumetric behavior over an extended range of subcritical and supercritical conditions and the vapor pressures up to the critical point. The degree of hydrogen bonding calculated under supercritical conditions is in qualitative agreement with the limited experimental data available and with the results of molecular dynamics calculations. There has been much debate about the structure of water under high-temperature and -pressure conditions as obtained from experimental and simulated data of radial distribution functions.16-18,20-22 The recent experimental work by Yamaguchi,22 however, has clarified at least the different effect that pressure has on the degree of hydrogen bonding below and above the critical temperature of water. These measurements showed that, at 400 °C, an increase in pressure increases the degree of hydrogen bonding, in agreement with our present calculations. Acknowledgment Financial support from EU/INTAS 96-1989 is gratefully acknowledged. This work is dedicated to the memory of Dr. Hasan Orbey. Literature Cited (1) Sandler, S. I.; Orbey, H.; Lee, B.-I. In Models for Thermodynamic and Phase Equilibria Calculations; Sandler, S., Ed.; Marcel Dekker: New York, 1994. (2) Guggenheim, E. A. Mixtures; Clarendon Press: Oxford, U.K., 1952. (3) Panayiotou, C.; Vera, J. H. The Quasichemical Approach for Nonrandomness in Liquid Mixtures. Expressions for Local Surfaces and Local Compositions with an Application to Polymer Solutions. Fluid Phase Equilib. 1980, 5, 55.
(4) Economou, I. G.; Donohue, M. D. Chemical, Quasi-Chemical and Perturbation Theories for Associating Fluids. AIChE J. 1991, 37, 1875. (5) Sanchez, I. C.; Panayiotou, C. In Models for Thermodynamic and Phase Equilibria Calculations; Sandler, S., Ed.; Marcel Dekker: New York, 1994. (6) Taimoori, M.; Panayiotou, C. The Nonrandom Distribution of Free Volume in Fluids: Nonpolar Systems. Fluid Phase Equilib., in press. (7) Panayiotou, C.; Vera, J. H. Thermodynamics of r-mer Fluids and Their Mixtures. Polym. J. 1982, 14, 681. (8) Sanchez, I. C.; Lacombe, R. Elementary Molecular Theory of Classical Fluids. Pure Fluids. J. Phys. Chem. 1976, 80, 2352. (9) Panayiotou, C. Lattice-Fluid Theory of Polymer Solutions. Macromolecules 1987, 20, 861. (10) Marcus, Y. Supercritical Water: Relationships of Certain Measured Properties to the Extent of Hydrogen Bonding Obtained from a Semiempirical Model. Phys. Chem. Chem. Phys. 2000, 2, 1465. (11) Panayiotou, C. Thermodynamics of Alkanol-Alkane Mixtures. J. Phys. Chem. 1988, 92, 2960. (12) Perry, R. H.; Green, D. W. Chemical Engineers’ Handbook [CD-ROM]; McGraw-Hill: New York, 1999. (13) Saul, A.; Wagner, W. A Fundamental Equation for Water Covering the Range from the Melting Line to 1273 K at Pressures up to 25 000 MPa. J. Phys. Chem. Ref. Data 1989, 18, 1537. (14) Bertolini, D.; Cassettari, M.; Ferrario, M.; Grigolini, P.; Salvetti, G. Dynamical Properties of Hydrogen-Bonded Liquids. Adv. Chem. Phys. 1985, 62, 277. (15) Marti, J.; Padro, J. A.; Guardia, E. Molecular Dynamics Simulation of Liquid Water Along the Coexistence Curve: Hydrogen Bonds and Vibrational Spectra. J. Chem. Phys. 1996, 105, 639. (16) Yamanaka, K.; Yamaguchi, T.; Wakita, H. Structure of Water in the Liquid and Supercritical States by Rapid X-ray Diffractometry Using an Imaging Plate Detector. J. Chem. Phys. 1994, 101, 9830. (17) Okhulkow, A. V.; Demianets, Y. N.; Gorbaty, Y. E. X-ray Scattering in Liquid Water at Pressures of up to 7.7 kbar: Test of a Fluctuation Model. J. Chem. Phys. 1994, 100, 1578. (18) Mountain, R. D. Length Scales for Fragile Glass-Forming Liquids. J. Chem. Phys. 1989, 90, 1866; Comparison of a Fixed Charge and a Polarizable Water Model. J. Chem. Phys. 1995, 103, 3084. (19) Missopolinou, D.; Panayiotou, C. Hydrogen-Bonding Cooperativity and Competing Inter- and Intramolecular Associations: A Unified Approach. J. Phys. Chem. A 1998, 102 (20), 3574. (20) Chialvo, A. A.; Cummings, P.; Simomson, J. M.; Mesmer, R. E.; Cochran, H. D. Interplay Between Molecular Simulation and Neutron Scattering in Developing New Insights into the Structure of Water. Ind. Eng. Chem. Res. 1998, 37, 3021. (21) Touba, H.; Mansoori, G. A. Structure and Property Prediction of Sub- and Supercritical Water. Fluid Phase Equilib. 1998, 150-151, 459. (22) Yamaguchi, T. Structure of Subcritical and Supercritical Hydrogen-Bonded Liquids and Solutions. J. Mol. Liq. 1998, 78, 43.
Received for review April 24, 2001 Revised manuscript received July 9, 2001 Accepted July 14, 2001 IE0103660
