Ind. Eng. Chem. Res. 1994,33,2819-2829
2819
Ion Hydration in Supercritical Water Ram B. Gupta and Keith P.Johnston' Department of Chemical Engineering, The University of Texas at Austin, Austin, Texas 78712-1062
A molecular thermodynamic model is presented for the prediction of the chemical potential of a n ion in supercritical water, and also the standard free energy, internal energy, and entropy of hydration. The reference state consists of ion-water clusters in the gas phase at low pressure, containing integer numbers of water molecules from 1to 6. The chemical potentials of the ionwater clusters are examined as a function of t h e relevant intermolecular forces, i.e., van der Waals repulsive and attractive forces, hydrogen bonding between the clusters and additional water molecules in the bulk, and Born electrostatic solvation. The hydrogen bonding stabilizes an ion-water cluster to a much greater extent than it is destabilized by repulsive forces. Consequently, hydrogen bonding increases the mean hydration number and the effective Born
radius. Introduction Molecular models of the thermodynamic properties of ions in water at high temperatures and pressures are of interest in hydrothermal reactions, for example, supercritical water oxidation (Tester et al., 1993) and in steam power generation and geochemistry (Tanger and Pitzer, 1989a,b;Tanger and Helgeson, 1988; LeveltSengers, 1991). Ion solvation in supercritical water plays a n important role in solubility phenomena, ion mobility, equilibrium behavior for ion association reactions, and chemical kinetics. For example, specific interactions between water and transition states have large effects on energy barriers for reactions such as the water-gas shift reaction (Melius et al., 1990) and the decomposition of nitrobenzene (Melius, 1992). A better understanding of molecular interactions between water and solutes of varying polarity is greatly needed for describing solvent effects on chemical reactions in supercritical water, particularly ion association reactions (Mesmer et al., 1991). The number of water molecules about an ion can be measured precisely in the gas phase at low pressures with mass spectrometry. For example, hydration numbers from 1 t o 6 have been measured for alkali halide ions (Kebarle, 1977). We will refer to these hydrated ions as clusters. The concentrations of clusters with various stoichiometries have been measured over a wide range of temperatures to determine standard state (ideal gas) free energies, enthalpies, and entropies of hydration. Cluster properties lie between those of the gas and the bulk condensed states as a function of the number of water molecules in the cluster (Castleman and Keesee, 1986). Pitzer and co-workers have developed various models for the standard Gibbs free energy of hydration of ions at high temperatures and pressures (Pitzer, 1983; Pitzer and Pabalan, 1986). Recently, Tanger and Pitzer (1989a) developed a semicontinuum model for temperatures to 1000 "C and pressures to 5 kbar. The above thermodynamic data for gas phase clusters from mass spectrometry served as a convenient reference in the model. The effect of the excluded volume of the cluster on its chemical potential was treated by using an empirical effective volume increment per HzO of hydration. In the critical region, the unusual properties present a challenge for thermodynamic models. The density varies markedly with temperature and pressure, and the isothermal compressibility approaches infinity. Con-
sequently, the magnitudes of partial molar properties of ions, such as volumes, enthalpies, entropies and heat capacities, can be enormous compared with those in conventional liquids (Levelt Sengers, 1991; Wood et al., 1981). Large partial molar heat capacities of ions are observed in the range 700-1000 K, arising from the enthalpy of dissociation of water. Not only have macroscopic partial molar properties been studied but also the structure of supercritical fluid solutions has been characterized spectroscopically and with computer simulation, for fluids such as COz with critical temperatures near ambient (Kim and Johnston, 1987; Knutson et al., 1992; Munoz and Chimowitz, 1992; Sun et ai., 1992; Carlier and Randolph, 1993). FTIR spectroscopy is particularly useful for investigating how supercritical fluids influence hydrogen bonding, e.g., between methanol and triethylamine (Gupta et al., 1993) and perfluorotert-butyl alcohol and dimethyl ether (Kazarian et al., 1993). These data have been modeled successfully with a lattice fluid hydrogen-bonding model (LFHB) (Gupta and Johnston, 1994). Very few spectroscopic studies have examined solvation in supercritical water. Hydrogen bonding of water about acetone has been measured with W-visible spectroscopy, on the basis of spectral shifts in the n n* band of acetone (Bennett and Johnston, 1994). At 380 "C, hydrogen bonding persists a t a density of only 0.1 g/mL but rapidly disappears a t lower densities. Future spectroscopic studies of the structure of supercritical aqueous solutions are needed t o further test theoretical models; presently the models must be tested primarily with computer simulation. Because partial molar volumes are pronounced in supercritical fluids, activation volumes and volume changes for chemical reactions can be large. The effect of pressure on the redox potential of the I d - couple was measured in water from 230 to 300 bar at 385 "C. The resulting partial molar volume change for the reduction of 12 to I-, ADrxn, is pronounced because of the large isothermal compressibility of the solvent multiplied by the change in solute-solvent interactions upon reaction from a molecule to an ion (Flarsheim et al., 1989). A better understanding of ion solvation would aid the interpretation of these results. Another important consideration in modeling supercritical solutions is the effect of Lewis acid-base interactions, in particular hydrogen bonding. These interactions have been measured spectroscopically for many ions in ambient water, e.g., CH&OCH2-, Gd3+,
OSSS-5885l94l2633-2819~Q4.50l~0 1994 American Chemical Society
2820 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994
and F- (Meot-Ner, 1988; HeRer, 1991). Specific interactions have also been studied by molecular dynamics simulation for pure supercritical water (Mountain, 1989; Kalinichev and Heinzinger, 1992) and aqueous solutions containing ions (Cummings et al., 1991; Cochran et al., 1992; Cui and Harris, 1994). For TIP4P water, hydrogen bonding persists to supercritical temperatures from 1t o 0.1 g/mL, as evidenced by the maximum in &?OH a t 0.18 nm (Mountain, 1989; Kalinichev and Heinzinger, 1992). The hydrogen bonding persists despite the fact that the water structure resembles that of a simple liquid such as argon, on the basis ofgoo (Kalinichev and Heinzinger, 1992). The persistence of hydrogen bonding is also observed in predictions of a lattice fluid hydrogen bonding model (Gupta and Johnston, 19941, based on energy, entropy, and volume of hydrogen bonding parameters obtained from spectroscopic data a t subcritical conditions. These predictions are in good agreement with computer simulation (Mountain, 1989). The results for the vapor pressure of water are considerably better for an improved version of the LFHB model (Gupta and Johnston, 1994)than for the earlier version (Gupta et al., 1992). The improved version is mathematically very similar to the statistical association fluid theory (SAFT) model (Chapman et al., 19901, although there are large differences in the derivations of the models and subtle differences in the hydrogen-bonding parameters. An important advancement is the determination of the equilibrium structure of water molecules about Na+ and C1- ions a t two states near the critical point (Cummings et al., 1991; Cochran et al., 1992). The local density of water in the first solvation shell approaches that of liquid water a t ambient temperature. Very recently, Cui and Harris (1994) studied this system over a wide range of temperatures and densities and also considered ion association, which was found to be driven by the entropic contribution to the potential of mean force. Balbuena et al. (1994a,b) examined the solvation of species of varying polarity along the reaction coordinate for the Sp~2reaction of C1- and CH&1 from ambient water to SCW. The behavior of the free energy, energy, and entropy changes along the reaction coordinate was influenced by the hydrogen-bonding interactions with the various species. These simulation studies of pure water and aqueous solutions suggest that hydrogen bonding can play an important role even at supercritical temperatures. However, hydrogen bonding interactions have not been considered in thermodynamic free energy models for aqueous ionic solutions at supercritical temperatures. Our objective is to develop a molecular thermodynamic model to calculate the chemical potential of ions and the thermodynamic properties for ion hydration in supercritical water. Again, the reference state consists of ion-water clusters in an ideal gas for which experimental data are available. We wish to investigate the transition between an ion in a gas phase cluster and an ion in dense liquid-like water. We view the cluster as a single solute species and determine how it interacts with additional water as the bulk water density is varied from zero to “liquid-like’’densities. The physical part of our model includes van der Waals repulsive (excluded volume) and attractive forces. In addition, the hydrogen bonding is described with a recently developed lattice model (Gupta, 1993). The effects of each of these types of intermolecular forces will be examined in detail. The thermodynamic formalism of Tanger and Pitzer has
vdw r e p u l s i o n t a t t r a c t i o n hydrogen bonding Born s o l v a t i o n
I
Figure 1. Schematic representation of solvation of an ion-cluster solute in water.
been adopted to define the standard free energy of hydration. The calculated results for the standard Gibbs free energy of hydration, mean hydration number, and Born radius will be compared with those of Tanger and Pitzer, and the importance of the various molecular interactions will be quantified. The Born radius will be regressed by forcing the model to agree with experimental data for the standard Gibbs free energy of hydration (Tanger and Helgeson, 1988; Tanger and Pitzer, 1989a,b). The mean hydration number will be compared with values from computer simulation.
Theory Chemical Potential of an Ion in a Cluster. The molecular thermodynamic model for the chemical potential of an ion in a cluster is described in Figure l . In this example, the solute is a cluster containing four water molecules about an ion. van der Waals repulsive forces destabilize the cluster due to excluded volume. van der Waals attractive forces stabilize the cluster since it is more polarizable than the anhydrous ion. In addition the cluster forms hydrogen bonds with additional water molecules. The purpose of this section is to write a free energy expression to describe these interactions. The molecular thermodynamic model is expressed in terms of the residual Helmholtz energy per mole, ares, defined as
areS(T,V,N) = a(T,VN - aig(T,V,N)
(1)
where a ( T , V N and a i g ( T , V N are the total and ideal gas values of the Helmholtz energy per mole at the same temperature and density. I t is assumed that the residual Helmholtz energy may be described by four terms ares
= ahs
+
adis + achain + aassoc
(2)
The expressions for the first three terms are the physical part of SAFT (Huang and Radosz, 1990,1991). The first term is given by a well-known equation for the repulsive forces for a mixture of hard spheres (Mansoori et al., 1971). The adisterm describes van der Waals mean field dispersion interactions between segments in the molecules. The achain term, which describes covalent chainforming bonds among segments, is not very important for small molecules such as water but is included anyway. The association term aassoc, which describes donor-acceptor interactions (e.g., hydrogen bonding) between segments, is derived from a modified version
Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 2821 of the hydrogen-bondingpartition function presented by Panayiotou and Sanchez (1991) in the LFHB model. The hydrogen-bonding contribution to the Helmholtz free energy is obtained from a partition function, by determining the number of ways of distributing the hydrogen bonds among the donor and acceptor groups in the system (Panayiotou and Sanchez, 1991). The donor and acceptor must be in close spatial proximity as described by a mean field probability. The probability that a specific acceptor will be near a given donor depends upon the system volume, the segment-segment correlation function, and the entropy loss (intrinsically negative) associated with the formation of a hydrogen bond. The general formalism is applicable for multicomponent systems of molecules having any number of hydrogen bond donor and acceptor groups and is applicable over a wide range in density. Further details concerning the free energy model are in the Appendix. The composition derivative of the free energy is used to determine the chemical potential of the ion, Pn, for the various values of hydration number n, analytically (Gupta, 1993). Given the chemical potential of the ion in the gas phase kg)and in a cluster, the ratio of fugacity coefficients (4) may immediately be written as
(3)
AG,, = AhG"
Ag
+ n H 2 0 = A(H20),
(4)
(7)
where AhGis is the inner-shell contribution, which accounts for the first six HzO molecules of hydration, and AhGosis the outer-shell contribution, which accounts for long range Born electrostatic solvation. The AG,, term simply changes the standard state from a hypothetical ideal gas a t 1 bar to a 1 m solution as described elsewhere (Tanger and Pitzer, 1989a) and is given by
)::(
AG,, = RT In where the gas constant R is in bar cm3K-l, T i n K, and e in g / ~ m - ~ . The subdivision of hydration free energy into innershell and outer-shell contributions is consistent with the widely used conceptual models proposed by Frank and Wen (1957) and Gurney (1953). The limit in inner-shell water molecules at six is consistent with experiment and simulation for alkali-metal ions (Chandrasekhar et al., 1984; Marcus, 1985). The inner-shell contribution is defined as (Tanger and Pitzer, 1989a,b)
--
RT where xn and xg represent the mole fractions of the hydrated ion-cluster and anhydrous ion, respectively. Chemical Reaction Equilibria. Having described the nonidealities of a cluster in a condensed phase in terms of fugacity coefficients, the next step is to write a set of chemical reaction equilibria expressions for an ion, A, as follows
+ A,Gos + AG,,
- -In(x,/xg)
n=6
= -In
where xt is the total mole fraction of ions, either as ions in clusters or as pure anhydrous gaseous ions. Equation 9 is simply the free energy per mole for the transfer of a solute from the gas phase to a cluster (Lewis et al., 1961). Here the clusters are sufficiently dilute such that the activity coefficients are unity. The mole fractions of the clusters with various hydration numbers may be obtained from the molecular model according to eqs 6 and 3 and substituted into eq 9 to yield
The equilibrium constant, Kn,is related to the fugacities of reaction species as In K, = fg f i 2 0
where the standard state fugacity is 1bar. The fugacities cfi) can be rewritten in terms of mole fractions and fugacity coefficients, 4i, where 4i = filyiP. The ratio of the mole fraction of a cluster of hydration number n, xn, to that of a gas phase ion, xg, is
The fugacity coefficient ratio may be calculated with eq 3. Equations 2 and 6 offer a complete molecular thermodynamic model for the nonidealities of the clusters and the chemical reaction equilibria (excluding the Born solvation). All of the calculations for chemical potentials in this work are done a t infinite dilution which results in each p, being independent of xm, where m f n. Standard Free Energy of Hydration. The standard free energy of hydration may be calculated given a molecular model for the mole fractions of the various clusters. We utilize the thermodynamic framework of Tanger and Pitzer (1989a) as described in the following summary. The standard free energy of hydration may be expressed as
Experimental values of Kn for alkali-metal, halide, and hydroxide ions are listed in literature (Tanger and Pitzer, 1989). The temperature dependence of Kn may be written as
As; AH; R RT
In K, = -- where ASK and AH; are literature values of the standard entropy and enthalpy of the nth hydration reaction, as given in Table 1, from measurements of Kn over a wide range of temperatures from 300 to 700 K (Dzidic and Kebarle, 1970;Arshadi and Kebarle, 1970; Arshadi et al., 1970). The outer-shell interaction is given by the Born equation (12) where the universal constant 77 is 83549 A K (7 = e2/ 8 n D a where e is the charge on an electron and Dois the permittivity in vacuum), E denotes the dielectric constant of water, Zi is the ionic charge, and Ri* represents the effective Born radius of the spherical
2822 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 Table 1. Standard Enthalpy and Entropy Changes for Hydration of Na+ and C1- Ions in the Gas Phase (Dzidic and Kebarle, 1970; Arshadi et al., 1970)
nHzO
+ Na+ = Na+(HzO), - As;
nHzO
m;
+ C1- = Cl-(HZO),
n
(kcalimol)
(calimol/K)
- AH; (kcalimol)
1 2 3 4 5 6
24.0 43.8 59.6 73.4 85.7 96.4
21.5 43.7 65.6 90.6 118.7 144.7
13.1 25.8 37.5 48.6 59.3 69.7
-
- As; (cal/moYK) 16.5 37.3 60.5 86.3 112.1 137.9
cavity for the inner-shell region in the dielectric continuum. For the kth electrolyte, the outer-shell contribution based on eq 12 may be written as
C
al
0
i
1
(13) where the effective Born radius, &*, is defined as (14)
species
where vi is the stoichiometric coefficient of each ion. The complete equation of Tanger and Pitzer for the AhGk of the kth electrolyte is
where V k is the total number of ions in the electrolyte k, Le., Y k = 1 i o n s v ~ . Parameter Evaluation. The parameters required in the free energy model are as follows. Each pure molecule is described by three physical segment parameters (segment length, r; segment-segment interaction energy, u"/k;and segment volume, uoo). Each type of hydrogen bond is characterized by three association parameters: energy, E"; entropy, s";and volume, v". Two of the physical parameters, the segment length (r) and the segment interaction energy (uo/k),for the pure salts NaCl and KC1 were obtained from the pure liquid density and the normal boiling point (Kirshenbaum et al., 1962). The segment volume (v"") was set equal to 10 cc/mol, the value for water. A parameter in the SAFT model, e/k, was chosen as 1 for all species in the present study, as was recommended for water (Huang and Radosz, 1990). The other parameters are for NaC1, r = 1 and u"/k = 1667 K; for KC1, r = 1.05 and u"/k = 1667 K, and for water, r = 1.33 and u"/k = 218.8 K. The calculated and experimental values of the liquid density for NaCl and KC1 are in good agreement, as shown in Figure 2. The physical parameters for an ion, i, are estimated from corresponding electrolyte k parameters as follows rk =
Cri
(16)
2
and (17)
molecular weight
r
58.5 74.6 23.0 35.5 41.0 59.0 77.0 95.1 113.1 131.1 53.5 71.5 89.5 107.5 125.5 143.6
1.00 1.05 0.50 0.50 0.89 1.28 1.67 2.07 2.46 2.85 0.75 1.01 1.26 1.52 1.77 2.02
To avoid adding any adjustable parameters, the value of r for Na' has been set equal to half of the r for NaCl and u"/k of Na+ has been set equal to that of NaC1. The parameters for the remaining of the ions have been evaluated by using eqs 14 and 15. The segment length parameter, r, is assumed to scale as the molecular weight ( M w )of the ion clusters as follows
'Na-(H20),
rC1-(H20),
(MWNa+ - rNa'
-
- 'C1-
(MWCl-
+ nMWNa+(H20)n)
(18)
MWNa+
+ nMWCl-(H20~,)
(19)
MWCl-
The segment length parameters for electrolytes, ions, and ion clusters are listed in Table 2. The trends in r have physical meaning, although the values for the smaller Na+ clusters are low relative to that of pure water. This minor discrepancy would not be expected to have a large influence on the results. An anion-cluster with n water molecules is considered to have only 2n proton acceptor sites (two sites per oxygen atom of each water molecule), and a cationcluster is considered to have only 2n proton donor sites (one site per hydrogen atom of each water molecule). The hydrogen-bonding parameters (E", So, and VO) between a water molecule in a cluster and a bulk water molecule are considered to be the same as those for bulk
Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 2823 water, i.e., E" = -15466 J/mol; Sa = -15.65 J/mol/K; and V' = -0.97 cm3/mol (Gupta and Johnston, 1994). There is some justification for this assumption. Cochran et al. found that there is an absence of any significant difference in the hydrogen bond structure around quite different solutes (Cochran et al., 1992). To evaluate the chemical potentials of ions in the clusters, the fraction of hydrogen bonds for various donor-acceptor pairs in the solution is required, which can be obtained from the fundamental equation of hydrogen-bonding equilibria (see eq A7). In the case of a dilute solution of an anion-cluster with n water molecules, this equation may be written as (see eq A l l )
exp(-
&n-w
r
%)
(20)
where rvn-, is the fraction of cluster-water hydrogen bonds, rvw-w is the fraction of water-water hydrogen bonds, g n - w is hard sphere pair correlation function for the cluster-water interaction, and is the reduced density of the mixture. In the case of a very dilute solution of clusters in water, rvn-w is negligible in comparison to rvw-w. Hence, the water-water hydrogen-bonding equilibria can be considered as independent of cluster-water hydrogen bonding. This simplifies the evaluation of the fraction of water-water hydrogen bonds as
e
This quadratic equation can be solved for rvw-wto yield
rvw-w-
(43cw
+
-
~ 4 - w ( A w -+w b,) 2
(22)
where Aw-wis defined as
-)RT
Aw-w= r exp(Gw-w egw-w
(23)
In the case of a dilute solution of a cation-cluster with n water molecules, an equation similar to eq 20 may be written as
rvw-ll (2xw- rvW-,)(2mn- rvW-J @w-n
exp(-
m)(24) Gv-n
r where clusters have proton donor sites only. The effective Born radius was determined by equating values of AhGk from eq 15 with those provided elsewhere (Tanger and Pitzer, 1989a), which came from an empirical thermodynamic model (Tanger and Helgeson, 1988). The required densities and fugacities of H20 were calculated from the equation of state of Haar et al. (1984). The dielectric constants were obtained from Uematsu and Franck (1980).
Results and Discussion The key objective of the molecular thermodynamic model is to calculate the chemical potential of an ion in
21
"
1
"
'
'
'
"
I
'
Na' P i 30 MPa
1 .E
1.6
.
--
B
Be 1 . 4
1.2
n-6
- _--._ ..
4\
----:::1 I:------
1 '
. . _' . f l . 5
----_--_- - - - _ _ _ _
------- - --
"
400
"
550
"
700
"
'
850
n=2
"
1000
T ("1 Figure 3. Fugacity coefficient ratio of Na+ in a cluster of hydration number n to anhydrous gaseous Na+ for the Tanger and Pitzer (1989a) model.
a cluster. A closely related property is the fugacity coefficient ratio for the ion in the cluster to that of the anhydrous ion, c#d4g. To better understand the solvation phenomena, we examine &/dg, the mean hydration number, and A&'* for three cases: (1) the model of Tanger and Pitzer, (2) the physical part of eq 2 based on the repulsive and attractive part of the free energy model (excluding association), and (3) the total free energy model including association, eq 2. We are particularly interested in the effects of association (hydrogen bonding), since they have not been modeled previously. All the following calculations were performed above T, and P, at a constant pressure of 30 MPa. Calculations were not performed closer to the critical point due to large uncertainties in the equation of state in this region. In order to test the model, calculations were performed for Na+ in liquid water from 25 to 75 "C at 1 bar. The mean hydration number was 6.0 over this temperature range, which is consistent with experiment (Burgess, 1988). The number of water molecules hydrogen bonded to each of these waters was 2.0, which is in agreement with molecular dynamics computer simulation for pure water (Mountain, 1989). As shown by Tanger and Pitzer (19891, this type of model can be applied to aqueous solutions at both ambient conditions and supercritical conditions. Given the agreement of these results with the literature, we now examine ions in supercritical water. The results for the model of Tanger and Pitzer are given in Figure 3. Here, the q5n/#g ratio is obtained from a Poynting correction integral as a function of the volume increment for each water molecule in the cluster. The fugacity of the ion in the cluster increases as the size of the cluster (excluded volume) increases, as expected. As kT increases, repulsive forces become less important and rprJ& decreases toward unity. The new results for the physical model (case 2 above) for &/4 are shown in Figure 4. Clearly, the repulsive term must be dominant, since the attractive forces would lead to a fugacity coefficient ratio below unity. There are some differences between the new model and the model of Tanger and Pitzer. In the earlier model, the repulsive forces are treated in terms of empirical effective volume increments, while they are treated with a molecular free energy model in this work. The trends
2824 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994
1
n=6
1
10
NE*
\
P
30 MPa
'.\
350
500
Tyc,
800
950
Figure 6. Predicted number of hydrogen bonds per ion-cluster for Na+ clusters for the total free energy model. I 1
,
,
,
,
,
500
350
, 650
,
1
,
800
5 . 5 1 . '
+
5
physlcal
s
- -1anger
z
Po
\
3 ' 5.5
C
P 5 L
I
0.01
'
350
'
'
' 500
'
"
'
1
+ hydrogen bondlng and Pltzer (1989)
'
"
1
'
'
"
"
'
'
"
'
'
1
- - --Ideal
\
gas
-physical + hydrogen bondlng --physlcal - -Tanger and Pitrer (1989)
\',
I
'
"
- -physical $ n
.o
n-6
"
- - - Ideal . gas
950
T ("1 Figure 4. Fugacity coefficient ratio for the physical part (repulsive attractive terms) of the free energy model.
1
"
1
'
650
800
950
T ("c) Figure 5. Fugacity coeficient ratio for the total free energy model.
are similar to those of Tanger and Pitzer, but the new physical model predicts much larger destabilization (a greater especially for the larger clusters. The difference in the models becomes more apparent at lower temperatures where densities and thus repulsive forces are largest. At the highest temperatures, the clusters sense an environment approaching an ideal gas. The &/& ratio changes dramatically when the hydrogen bonding term is added (Figure 5). This ratio changes from above unity to below unity in all cases studied. Likewise, the chemical potential of the ion in the cluster changes from above to below that of the gas phase ion. The stabilization of the clusters due to hydrogen bonding dominates the destabilization due to excluded volume effects. Again, these effects are greatest at lower temperatures where densities are largest, and potential energy is largest relative to kT. As the clusters become larger, the stabilization actually increases due to the added hydrogen bonds (see Figure 6). Because more hydrogen-bonding sites become available on the cluster, the probability of a donor-acceptor interaction increases. For example, approximately two water molecules hydrogen bond t o a cluster for n = 6 a t 400 " C . As temperature increases, thermal energy liberates these waters from the cluster and &J& goes toward unity.
= I
8
2.5
1.5 400
700
550
850
1000
T ("C)
Figure 7. Comparison of the calculated mean hydration number for Na+ and C1- clusters from various models.
The mean hydration number (average cluster size) is shown for the above three cases of interest and also for the ideal gas case where & = 4gin Figure 7 for sodium and chloride ion clusters. The mean hydration number, (n),has been defined previously as n=6
Cn(xn/xg) 1 + C(xn/xg) n=l
Tanger and Pitzer's model predicts that the clusters are smaller than those in an ideal gas for which @n = @g. The physical model predicts that the clusters are even smaller than those of Tanger and Pitzer's model. This progression is consistent with the increasing strength of repulsive forces which raises +nl#g.In contrast, the mean cluster size for the total free energy model
Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 2825 including hydrogen bonding exceeds that of the ideal gas clusters. Here the larger clusters are favored due to hydrogen bonding, i.e., $nl$g< 1. In all cases, the mean hydration number decreases as the temperature increases, since hydration is exothermic. At near critical temperatures, the calculated hydration number approaches 6, the value also observed in ambient water. This observation is in general accordance with molecular dynamics simulations (Cummings et al., 1991; Cochran et al., 19921, although the simulations were performed below the critical temperature of water (374 "C), whereas our calculations were performed above T,. The strong hydration of ions in SCW was also observed in the MD simulations of the sN2 reaction of C1- and CH&1 (Balbuena et al., 1994a,b). The results for the mean hydration number are in general agreement with the molecular dynamics simulation data of Gui and Harris (1994)for the Na+ and C1- ions. It is not possible to make an exact comparison for the following reasons. The two studies cover similar ranges in temperature, although the simulations were done at constant density, whereas our calculations were done at constant pressure. More importantly, the definition of the first hydration shell is not exact. For the simulations, the size of the first hydration shell is fixed at a desired value. In the model, it is not specified explicitly but is obtained from eqs 6 and 25. Therefore, the comparison between the simulations and the model is qualitative. In order to place the results in perspective, we refer to the results in ambient water at 25 "C discussed above. At 25 "C, the number of hydrogen bonds per cluster is 12, a much larger value than for SCW in Figure 6. In SCW, kT is sufficiently high and the density is sufficiently low to break a large number of the hydrogen bonds between water and the water molecules which hydrate the ion. In contrast, the mean hydration number a t 425 "C is not much smaller than the value of 6 in ambient water. The different temperature effects on water-water hydrogen bonds and on the hydration number has been explained on the basis of simulation studies (Balbuena et al., 199413). The difference is due t o the fact that the donor-acceptor bonds between the oxygen in water and the Na+ ion are much stronger than the water-water hydrogen bonds (Balbuena et al., 1994b). In the simulation study, two properties were calculated, the excess hydration number Ne, and the total hydration number N h . For Na+, the simulation results are similar for Ne, and N h ; however N h is significantly larger than Ne, for C1-. For C1-, our results are closer to Ne,, as we focus on inner-shell water molecules quite close to the ion. At 427 "C, the hydration numbers are 3.5 and 4.8 for the simulation and calculation, respectively. At 527 "C, the values are 4.0 and 3.9. Similar agreement is found for Na+. For both simulation and theory, the hydration number decreases with temperature and the decrease is larger for C1-. The inner-shell contribution to the Gibbs free energy of hydration for Na+ and C1- ions is presented in Figure 8, again for the three cases of interest. The results are consistent with those for $,& and the mean hydration number. The total free energy model predicts the most negative values for AhGis because of the stabilization of the clusters due to the hydrogen bonding. For example, the magnitude of A h P S is about 20 kJlmol greater than for Tanger and Pitzer's model a t 425 "C and 30 MPa. As temperature increases, this difference diminishes as the hydrogen bonds are broken.
100 Na*
i
- --anger
phy8kGal
-2001
'
-401
-55
'
'
"
"
'
"
"
i
and Pltzer (1989)
+ '
'
hydrogen bondlng
"
" # '
"
'
I
'
I ----physical
-85
- -Tanger
and Piirer (1989)
physical - 1 00 400
+
hydrogen bonding
700
550
850
1000
T ("C)
Figure 8. Gibbs free energy of hydration due to the inner-shell for Naf and C1- clusters. 4.25
,
1
NaCl
P=M 0-
U
4.00
UP1
~
physical
Y
t
hydrogen bondlng
E l
4 3.50
I
400
550
700
T
850
1000
("1
Figure 9. Effective Born radius for NaC1.
The results for the total Gibbs free energy of hydration are the same as predicted by Tanger and Pitzer's model, since the Born radius was fit to experimental values (Tanger and Pitzer, 1989a) of A&' by using eq 15 at each temperature. The effective Born radius of NaCl (sum of Born radius for Na+ and C1-) for the Tanger and Pitzer model and the new model are given in Figure 9 at 30 MPa. The new model predicts a higher effective Born radius, which is equivalent to a smaller magnitude for &,GOs. Notice that less of the solvation is included in the outer-shell term because of the greater solvation in the inner shell due to hydrogen bonding. The larger effective Born radius is consistent with the larger mean hydration number. The effective Born
2826 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 1.0,
200
t
NBCl
P
0
= 30
r
MPa
.....................................
-.-1 . o
-
A
0
2.
0
-
.
E
E
-200
7 Y
v
---2.0 -
:
m= a
-400
-3.0 AnCP'
-
' '
NaCl P = 30 MPa ,standard state
(Na.]
*--'innsr.shell
- -inner-shell (CI') - -outershell -total
il
-A,G
-600
350
500
800
650
950
T ("c) Figure 12. Standard entropy of hydration and its various components for NaCl. 1.5
!I
Na' P = 30 MPa
__-----
* - -
_ _ - -..........-.e............... /
NaCl P = 30 MPa .standard state
.- ' ___.
inner-shell (Na*)
- -inner-shell - -ouler.shell
(CI.1
i
i
-3500 350
1 '
500
650
800
950
T ("1 Figure 11. Standard enthalpy of hydration and its various components for NaCl.
radius of NaCl is larger than its crystal radius because of the volume increase due to the inner-shell water molecules. The new model could also be used to predict solvation properties without fitting the Born radius, since it takes into account all of the key molecular interactions. The Born radius could be estimated as the radius of a cluster with the mean hydration number. The various contributions to the Gibbs free energy of hydration for NaCl at 30 MPa are given in Figure 10 for the new model. The inner- and outer-shell terms are of similar magnitude. Indeed the physical and hydrogen-bonding interactions are significant compared with the electrostatic interactions. Whereas both the inner- and outer-shell terms become more negative as temperature decreases, the change is somewhat larger for the outer-shell term. Additional thermodynamic properties of solvation, i.e., entropy, enthalpy, and heat capacity, have been calculated by taking the appropriate derivatives of AhG numerically. For the new model, the enthalpy of hydration and its various components are presented in Figure 11. Energetically, the ion clusters are stabilized by both hydrogen bonding and electrostatic solvation.
400
'
'
"
550
"
700
"
I
'
850
'
1000
T ("C) Figure 13. Polydispersity of the Na+ ion-clusters.
The latter is influenced much more strongly by temperature, particularly below 500 "C, where density changes are largest. Here large changes in the dielectric constant have a large effect on the Born term. The pressure and temperature derivatives of the dielectric constant diverge at the critical point, which explains the large variations in the Born term. The entropy of hydration is shown in Figure 12. The clusters become more ordered by hydrogen bonds and electrostatic forces. There is some compensation between the enthalpy and entropy of hydration, such that the changes in the free energy with temperature are less pronounced. The polydispersity in the hydration number is shown in Figure 13 in terms of the standard deviation over the mean. Two compensating effects are apparent. At temperatures well above T,, the polydispersity increases with T as thermal fluctuations grow, due to an increase in TS. Near the critical point, there is a large increase in polydispersity with a small decrease in T. Here the large increase in the isothermal compressibility produces large concentration fluctuations which increase the polydispersity. This near critical effect was also observed in CFBHas measured with a fluorescent probe (Betts et al., 1992).
Ind. Eng. Chem. Res., Vol. 33,No. 11, 1994 2827
Conclusions The molecular thermodynamic model for the chemical potential of ion-water clusters provides a means to calculate a wide variety of properties including the distribution of hydration numbers and the standard free energy, internal energy, and entropy of hydration. Because all of the parameters in the model are based on molecular properties, the model has predictive capabilities. Hydrogen bonding stabilizes an ion-cluster to a much greater extent than it is destabilized by repulsive forces. Consequently, hydrogen bonding increases the mean hydration number and the effective Born radius. The existence of clusters with high local densities of water about ions is consistent with molecular dynamics simulation (Cochran et al., 1992 ; Balbuena et al., 1994a,b; Cui and Harris, 1994) and electrochemical measurements of diffusion coefficients of ions (Flarsheim et al., 1986). In the new model, the nonelectrostatic forces stabilize the ion-water clusters, which is the opposite result of the model of Tanger and Pitzer and our purely physical model (which excludes association). At 425 "C, about two water molecules form hydrogen bonds with a cluster containing six water molecules about Na+. The free energy of hydration is stabilized by about 20 kJ/mol, and the mean hydration number increases by a full unit from 4.5 t o 5.5. The hydration numbers, which are in good agreement with simulation (Cui and Harris,1994), decrease with temperature. The stabilization due to hydrogen bonding is comparable to that by Born solvation. As the temperature is lowered toward the critical temperature, the density of water increases significantly at 30 MPa. Therefore, hydration becomes much more exothermic, and the clusters become more ordered primarily due t o hydrogen bonding and electrostatic solvation. The solvation of ion-water clusters in supercritical water may be expected t o have a large influence on chemical reactions in which the polarity changes upon activation.
Acknowledgment Acknowledgmentis made t o the US.Army Research Office for University Research Initiative Grant No. DAAL 03-92-6-0174,to the Separations Research Program at the University of Texas, and to the Camille and Henry Dreyfus Foundation for a Teacher-Scholar Grant (to K.P.J.). We are grateful to Drs. S. T. Cui and J. G. Harris for providing a preprint containing simulation data and to Dr. A. Anderko of Simulation Science Inc. for helpful comments.
and similarly the total number of acceptor groups of type j is given by
There are Nv such bonds in the system, and the total number of hydrogen bonds may be written
The number of donors of type i, Nio, and acceptors of type j , NO,,that are not hydrogen bonded are given by
and i
The number of ways of distributing the Nu bonds among the functional groups of the system is the same as derived by Panayiotou and Sanchez (1991) by generalizing the argument of Veytsman (1990) to the case of multigroup molecules. The mean field probability that a specific acceptor will be proximate t o a given donor is proportional to the volume of the acceptor group divided by the total segment volume, i.e., 1N (or GlrN). We modify this probability to include the donoracceptor pair correlation function, gv, which is assumed to be same as the segment-segment pair correlation function on which donor and acceptor groups reside. The formation of the hydrogen bond is also accompanied by a loss of rotational degrees of freedom. In general, for the donor-acceptor i-j pair, the probability is given by
where Sv" is the entropy loss (intrinsically negative) associated with hydrogen bond formation of an (ij ) pair. The Helmholtz free energy due to hydrogen bonding (aassoc) can be obtained from the canonical partition function.
m
Appendix The expressions for the first three terms in the free energy model (eq 2) are the same as in SAFT presented by Huang and Radosz (1990). Only the fourth term will be described here which is based on a modified form of the hydrogen-bonding partition function presented by Panayiotou and Sanchez (1991) in the LFHB model. Let d? be the number of proton donor groups of type i in each molecule of type k and af be the number of proton acceptor groups of typej in each molecule of type k. The total number of donor groups of type i in the system is given by
"io In -
+ Ed: In
I
where
Here rN is the total number of molecular segments in the system, and the standard Gibbs free energy and Helmholtz free energy of hydrogen bond formation are
and
2828 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994
(A101 respectively. Here Elio is the favorable energy change upon hydrogen bond formation between a donor group, i, and an acceptor group,j . (This energy is in excess of any physical interaction energy.) Finally Vgo is the volume change accompanying hydrogen bond formation. Minimizing the Gibbs free energy with respect to the number of (ij)hydrogen bonds (keeping the number of all other types of hydrogen bonds constant) yields
(All) and the equation of state can be obtained by minimizing the total Gibbs free energy with respect t o the system volume. A detailed methodology for obtaining the equation of state and the chemical potentials can be found elsewhere (Gupta et al., 1992,1993; Gupta and Johnston, 1994; Huang and Radosz, 1990, 1991, Panayiotou and Sanchez, 1991).
Literature Cited Arshadi, M.; Kebarle, P. Hydration of OH- and 0 2 - in thr Gas Phase. Comparative Solvation of OH- by Water and the Hydrogen Halides. Effects of Acidity. J . Phys. Chem. 1970, 74, 1483- 1485. Arshadi, M.; Yamdagni, R.; Kebarle, P. Hydration of the Halide Negative Ions in the Gas Phase. 11. Comparison of Hydration Energies for the Alkali Positive and Halide Negative Ions. J . Phys. Chem. 1970, 74, 1475. Balbuena, P. B.; Johnston, K. P.; Rossky, P. J. Molecular Simulation of a Chemical Reaction in Supercritical Water. J. Am. Chem. SOC.1994a, 116,2689-2690. Balbuena, P. B.; Johnston, K. P.; Rossky, P. J. Theoretical study of an sN2 reaction in Supercritical Water. J . Phys. Chem. 1994b, submitted for publication. Bennett, G. E.; Johnston, K. P. W-visible Absorbance Spectroscopy of Organic Probes in Supercritical Water. J . Phys. Chem. 1994, 98, 441. Betts, T. A.; Zagrobelny, J.; Bright, F. V. Supercritical Fluid Technology, Theoretical and Applied Approaches i n Analytical Chemistry; Bright, F. V., McNally, M. E. P., Eds.; ACS Symposium Series 488; American Chemical Society: Washington, DC, 1992. Burgess, J. Ions i n Solution: Basic Principples of Chemical Interactions; John Wiley: New York, 1988. Carlier, C.; Randolph, T. W. Dense-Gas Solvent-Solute Clusters a t Near-Infinite Dilution: EPR Spectroscopic Evidence. AIChE J . 1993, 39, 876. Castleman, A. W., Jr.; Keesee, R. G. Clusters: Bridging the Gas and Condensed Phases. Acc. Chem. Res. 1986,19,413-419. Chandrasekhar, J.; Spellmeyer, D. C.; Jorgensen, W. L. Energy component analysis for dilute aqueous solutions of Li+, Na+, F-, and C1- ions. J . Am. Chem. SOC.1984,106,903-910. Chapman, W.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Liquids. Ind. Eng. Chem. Res. 1990,29, 1709. Chateauneuf, J. E.; Roberts, C. R.; Brennecke, J. F. ACS Symp. Ser. 1992, 488 106. Cochran, H. D.; Cummings, P. T.; Karaborni, S. Solvation in Supercritical Water. Fluid Phase Equilibr. 1992, 71, 1-16. Cui, S. T.; Harris, J. G. Chem. Eng. Sci. 1994, in press. Cummings, P. T.; Cochran, H. D.; Simonson, J. M.; Mesmer, R. E.; Karaborni, S. Simulation of supercritical water and of supercritical aqueous solutions. J . Chem. Phys. 1991,94,56065621. Dzidic, I.; Kebarle, P. Hydration of the Alkali Ions in the Gas Phase. Enthalpies and Entropies of Reactions M+(H20),-1 + HzO = M+(HzO),. J . Phys. Chem. 1970, 74, 1466. Flarsheim, W. M.; Tsou, Y. M.; Trachtenberg, I.; Johnston, K. P.; Bard, A. J. Electrochemistry in Nearcritical and Supercritical
Fluids. 3. Studies of Br-, I-, and Hydroquinone in Aqueous Solutions. J . Phys. Chem. 1986, 90, 3857. Flarsheim, W. M.; Bard, A. J.; Johnston, K. P. Pronounced Pressure Effects on Reaction Equilibria in Supercritical Fluids. J. Phys. Chem. 1989,93, 4234-4242. Frank, H. S.; Wen, W. Y. Structureal Aspectes of Ion-solvent Interaction in Aqueous Solutions: Water Structure. Discuss. Faraday SOC.1957, 24, 133. Gupta, R. B. Hydrogen Bonding in Supercritical Fluids. Ph.D. Dissertation, The University of Texas a t Austin, TX, 1993. Gupta, R. B.; Johnston, K. P. Hydrogen Bonding Lattice Fluid Model with a Local Segment Density. Fluid Phase Equilibr. 1994, 99, 135. Gupta, R. B.; Panayiotou, C.; Sanchez, I. C.; Johnston, K. P. Theory of Hydrogen Bonding in Supercritical fluids. AIChE J . 1992, 38, 1243-1253. Gupta, R. B.; Combes, J. R.; Johnston, K. P. Solvent Effect on Hydrogen Bonding in Supercritical Fluids. J . Phys. Chem. 1993, 97, 707-715. Gurney, R. W. Ionic Processes i n Solution; Dover: New York, 1953. Haar, L.; Gallagher, J.; Kell, G. N B S I N R C Steam Tables; Hemisphere: Washington, DC, 1984. Hefter, G. T. Fluoride solvation: the case of missing ion. Pure Appl. Chem. 1991, 63, 1749-1758. Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydispersed, and Associating Molecules. Ind. Eng. Chem. Res. 1990,29, 2284-2294. Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydispersed, and Associating Molecules: Extension to Fluid Mixtures. Ind. Eng. Chem. Res. 1991,30, 1994-2005. Kalinichev, A. G.; Heinzinger, K. In Advances in Physical Chemistry: Thermodynamic Data, Systematics and Estimation; Saxena, S., Ed.; Springer-Verlag: New York, 1992. Kazarian, S. G.; Gupta, R. B.; Clarke, M. J.; Johnston, K. P.; Poliakoff, M. How is Hydrogen-Bonding Influenced by Solvent Density? The Spectroscopic Study and Modeling of the Interaction between a Proton Donor and Acceptor from the Gas to Supercritical Fluid States. J . Am. Chem. SOC.1993,115, 11099. Kebarle, P. Ion thermochemistry and solvation from gas phase ion equilibria. Ann. Reu. Phys. Chem. 1977,28, 445-476. Kim, S.; Johnston, K. P. Molecular Interactions in Supercritical Fluid Extraction and Reaction Processes. Ind. Eng. Chem. Res. 1987,26, 1206-1213. Kirshenbaum, A. D.; Cahill, J. A.; McGonigal, P. J.; Grosse, A. V. The density of NaCl and KC1 and an estimate of their critical constants togather with those of the other alkali halides. J . Inorg. Nucl. Chem. 1962,24, 1287-1296. Knutson, B. L.; Tomasko, D. L.; Eckert, C. A,; Debenedetti, P. G.; Chialvo, A. A. Local Density Augumentation in Supercritical Solutions: A Comparision between Fluorescence Spectroscopy and Molecular Dynamics Results. Supercritical Fluid Technology; Bright, F. V., McNally, M. E. P., Eds.; ACS Symposium Series 488; American Chemical Society: Washington, DC, 1992; p 60. Levelt Sengers, J. M. H. In Supercritical Fluid Technology; Ely, J., Bruno, T. J., Eds.; CRC Press: Boca Raton, FL, 1991; p 1. Lewis, G. N.; Randall, M.; Pitzer, K. S.;Brewer, L. Thermodynamics; McGraw-Hill: New York, 1961; p 314. Mansoori, G. A.; Carnahan, N. F.; Starling, K. E.; Leland, T. W. Equilibrium Thermodynamic Properties of the Mixture of Hard Sphere. J . Chem. Phys. 1971, 54, 1523. Marcus, Y. Ion Soluation; Wiley: Chichester, U.K., 1985. Melius, C. F. The thermochemistry and reaction pathways of energetic material decomposition and combustion. Philos. Trans. R . SOC.London A 1992, 365. Melius, C. F.; Bergen, N. E.; Sheperd, J. E. Effects of Water on Combustion Kinetics a t High Pressure. Twenty-Third Symposium (International) on Combustion, The Combustion Institute, 1990; pp 217-223. Meot-Ner, M. The Ionic Hydration Bond and Ion Solvation. 7. Interaction Energies of Carbanions with Solvent Molecules. J . A m . Chem. SOC.1988,110, 3858-3862. Mesmer, R. E.; Palmer, D. A,; Simonson, J. M. Ion Association a t High Temperatures and Pressures. In Activity Coefficient in Electrolyte Solutions, 2nd ed.; Pitzer, K. S., Ed.; CRC Press: Boca Raton, FL, 1991; p 491. Mountain, R. D. Molecular dynamics investigation of expanded water a t elevated temperatures. J. Chem. Phys. 1989,90,18661870.
Ind. Eng. Chem. Res., Vol. 33, No. 11,1994 2829 Munoz, F.; Chimowitz, E. H. Integral equation calculations of the solute chemical potential in a near-critical fluid environment. Fluid Phase Equilibr. 1992,71,237. Panayiotou, C.; Sanchez, I. C. Hydrogen Bonding in Fluid: an Equation-of-State Approach. J . Phys. Chem. 1991,95, 10090. Pitzer, K. S. Thermodynamics of Sodium Chloride Solutions in Steam. J . Phys. Chem. 1983,87,1120-1125. Pitzer, K.S.;Pabalan, R. T. Thermodynamics of NaCl in Steam. Geochim. Cosmochim. Acta 1986,50,1445-1454. Sun, Y.-P.; Bennett, G.; Johnston, K. P.; Fox, M. A. Studies of solute-solvent interactions in mixtures of supercritical fluids using fluorescence spectroscopy.J . Phys. Chern. 1992,96,10001. Tanger, J. C., N; Helgeson, H. C. Calculations of the Thermodynamic and Transport Properties of Aqueous Species a t High Pressures and Temperatures: Revised Equations of State for the Standard Partial Molal Properties of Ions and Electrolyts. Am. J . Sei. 1988,288,19. Tanger, J. C., N,Pitzer, K. S. Calculation of the Thermodynamic Properties of Aqueous Electrolytes to 1000 "C and 5000 bar from a Semicontinuum Model for Ion Hydration. J. Phys. Chem.
Tester, J. W.; Holgate, H. R.; h e l l i n i , F. J.; Webley, P. A.; Killilea, W. R.; Hong, G. T.; Barnes, H. E. Supercritical Water Oxidation Technology. Energy Technologies in Hazardous Waste Management; Tedder, D. W., Phland, F. G., Eds.; ACS Symposium Series 518 American Chemical Society: Washington, DC,
1993. Uematsu, M.;Franck, E. U. An Equation of State for the Static Dielectric Constant of Water from 0 to 550 "C and up to 5 kbar. J . Phys. Chem. Ref. Data 1980,9,1291. Veytsman, B. A. Are Lattice Models Valid for Fluids with Hydrogen Bonds? J. Phys. Chem. 1990,94,8499. Wood, R. H.; Quint, J. R.; Grolier, J.-P. E. Thermodynamics of a Charged Hard Sphere in a Compressible Dielectric Fluid: A Modification of the Born Equation To Include the Compressibility of the Solvent. J. Phys. Chem. 1981,85,3944.
Received for review February 7, 1994 Revised manuscript received July 19,1994 Accepted July 27, 1994@
1989a,93,4941-4951. Tanger, J. C., Iv,Pitzer, K. S. Calculation of the Ionization Constant of HzO to 2,273K and 500 MPa. AIChE J. 1989b,35,
1631-1638.
@
Abstract published i n Advance ACS Abstracts, October 1,
1994.
