Ind. Eng. Chem. Res. 2000, 39, 3625-3630
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Aqueous Nonionic Solutes at Infinite Dilution: Thermodynamic Description, Including the Near-critical Region† Jorge L. Alvarez,‡ Roberto Fernandez-Prini,*,‡,§,⊥ and M. Laura Japas‡,| Unidad Actividad Quı´mica, Comisio´ n Nacional de Energı´a Ato´ mica, Av. Libertador 8250, 1429 Buenos Aires, Argentina, INQUIMAE, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina, and Escuela de Ciencia y Tecnologı´a, Universidad Nacional de General San Martı´n, 1651 San Andre´ s, Prov. Buenos Aires, Argentina
The possibility of describing the thermodynamic behavior of solutes at infinite dilution using a procedure based upon the Krichevskii function J ) (∂p/∂x)∞T,V has been explored using properties of N2, CO2, NH3, and B(OH)3 dissolved in water. The expressions obtained by fitting one property of the binary systems over a wide range of temperature and fluid density could be used to predict successfully other properties of the same solution over the same range of thermodynamic states, including the near-critical region. Alternative routes to obtain J are discussed. Introduction There is an increasing number of chemical processes that rely on the use of supercritical fluids; hence, the possibility of describing the behavior of solutions, mainly dilute solutions, over the complete density range of fluid existence is of great practical importance. Such a general descriptive tool should also be applicable to the particularly difficult near-critical region where moderate changes in temperature or in pressure cause appreciable variations in the fluid’s density, a thermodynamic region where the majority of the processes using supercritical fluids takes place. Many of these chemical processes are optimized by fine-tuning the intermolecular interactions between solute and solvent molecules. This is accomplished by a careful adjustment of the thermodynamic variables and/or by addition of adequate cosolvents; therefore, it is important to have a good knowledge of the thermodynamic behavior of the solutes and how this is affected by the intermolecular interactions which prevail in the solution. The behavior of solutes at infinite dilution is a consequence of solvation phenomena; solute solvation under equilibrium conditions is characterized by the values of its standard thermodynamic properties which refer to the Henry activity scale. From the scientific point of view, supercritical near-critical fluids are the best systems to explore the effect of the solvent number density on intermolecular interactions, a variable which is known to be of paramount importance in characterizing solvation phenomena. To understand at a molecular level those processes which contribute to the behavior of infinitely diluted solutes in near-critical fluids, it is essential to separate the effects produced by the high solvent susceptibility / ) from the contribu(i.e., isothermal compressibility κT1 † Paper contributed in honor of Professor Reed Izatt. * To whom correspondence should be addressed. Tel: +5411-4754-7175.Fax: +54-11-4724-7886.E-mail:
[email protected]. ‡ Comisio ´ n Nacional de Energı´a Ato´mica. § Universidad de Buenos Aires. | Universidad Nacional de General San Martı ´n. ⊥ Member of Carrera del Investigador (CONICET).
tions of short-range intermolecular interactions.1 This is clearly illustrated by the procedure normally followed to evaluate solvation properties in liquid solvents which relies on the determination of the solute’s partial molar quantities; in the near-critical region partial molar quantities are strongly affected by the large susceptibility exhibited by the solvent.1 For an extensive property Y, the solute’s partial molar property at infinite dilution Y∞2 is given by
Y∞2 ) Y/1 -
∂p (∂Y ∂p ) ( ∂x ) T
∞ T,Y
(1)
According to Griffiths and Wheeler,2 the quantity (∂Y/ ∂p)/T, a response function of the pure solvent, will diverge strongly at the solvent’s critical point, causing the partial molar quantity Y∞2 to diverge. The desirable strategy to circumvent this problem is to describe the solute’s thermodynamic behavior in terms of the factor (∂p/∂x)∞T,Y, which will diverge weakly at the solvent’s critical point.2 As an example, let us consider that Y is the system’s volume V and then / J V∞2 F/1 ) 1 + κT1
(2)
where F/1 is the density of water, J ≡ (∂p/∂x)∞T,V ) -(∂2A/ ∂V ∂x)∞ is the Krichevskii function of the solution, and / is the solvent isothermal compressibility. While J κT1 is considered to be well-behaved in the near-critical / diverges strongly at the solvent critical region, κT1 point. The divergence of V∞2 is due to the solvent compressibility, but J determines its sign and amplitude. The advantage of separating the two contributions is illustrated in Figure 1, which shows in the upper panel plots for V∞2 for the system B(OH)3-H2O at two pressures3 and in the lower panel plots for the solvent compressibility and J/T, calculated with the partial molar volume data according to eq 2. It is clear that V∞2 / peaks at the same density as κT1 does and that J/T, which contributes the negative sign to V∞2 , is wellbehaved. Recently, this feature has been exploited to describe the behavior of aqueous4-6 and nonaqueous7 dilute
10.1021/ie000003k CCC: $19.00 © 2000 American Chemical Society Published on Web 09/13/2000
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Figure 2. J(CO2) against water density. Points were calculated with V∞2 from ref 13 for isotherms between 400 and 750 K.
Figure 1. B(OH)3 in water at infinite dilution. Upper panel: V∞2 reported in ref 3; (b) 28 MPa and (O) 35 MPa. Lower panel: (b) / values of J/T calculated with the V∞2 values in ref 3. κT1 : (solid curve) 28 MPa; (dashed curve) 35 MPa. The vertical dashed line gives the critical density of water.
binary solutions. Because of the existence of ample experimental information for aqueous systems, they are convenient systems to test this approach, but it must be remarked that the procedure is also applicable to nonaqueous systems. A massive overall adjustment of several properties of binary aqueous solutions has been successfully used to describe the behavior of solutes at infinite dilution on the basis of J.4,5 The extension of that approach to systems, either aqueous or nonaqueous, where there are not so many data, would be important. However, we have reported recently that basing the thermodynamic description upon J is not completly free of difficulties because of the density gap of coexisting phases at subcritical temperatures;8 information complementary to J may be required to enable a reasonably precise prediction of other properties over wide density ranges. In the present work, we shall analyze the application of this strategy to dilute nonionic aqueous binary solutions, with the final objective of using as the main input the values of a single property of the solution over wide ranges of (T, F/1) in order to predict other properties of the solute over the same range of (T, F/1) including the near-critical region. We shall summarize our findings and discuss alternative procedures for a general thermodynamic description of dilute binary solutions. Molecular Features and Thermodynamic Relations Solvation phenomena in near-critical fluids are often described in terms of the local density of solvent in the vicinity of a solute particle, i.e., the number of solvent molecules per unit volume in the region which is close to the solute particle. At low and intermediate bulk solvent densities F/1, the local density around the solute molecule is larger than F/1 because of the effect of
attractive solvent-solute interactions occurring in the expanded fluid.1,9 Many nondivergent equilibrium properties of solutes in near-critical fluids, as well as their spectroscopic and dynamic behavior in that region, are observed to depend nonlinearly on the fluid density; this is rationalized on the basis of the local density.1,10 From the equations which relate these properties and J to the solute-solvent radial distribution function g12(r), it is observed that all of them depend on the value of g12(r) in the vicinity of the solute molecule. Hence, none of these properties will diverge at the solvent’s critical point; they are all short-ranged functions. The nondivergence of J at the solvent’s critical point, illustrated for aqueous B(OH)3 in Figure 1, is also evident in the expression which relates J to the direct correlation functions cij, which are known to be short-ranged:11
J ) RT(F/1)2(cˆ 11 - cˆ 12)
(3)
where cˆ ij is the integral of cij over the system’s volume and F/1 is the density of the solvent. Another advantage of Krichevskii’s function J is that its limiting behavior for very low and very high densities is well-established. For very low density, the limit can be conveniently expressed in terms of second (Bij) and third (Cijk) virial coefficients
lim
F/1f0
J ) 2(B12 - B11) + 3(C112 - C111)F/1 / 2 RT(F1)
(4)
At high fluid density repulsion between molecule rules, the observed behavior and the limiting value of J is welldescribed by the hard-sphere model.12 A feature frequently noted is the strong dependence of J on F/1 and its weak dependence on temperature. From a study of the solubility of CHI3 in several nearcritical supercritical fluids,7 we have shown that a change of 40 K does not affect the value of J beyond the effect of the experimental uncertainty of the solubility measurements. This characteristic is also illustrated in Figure 2 for CO2 at infinite dilution in water where we have plotted values of J calculated with the V∞2 reported by Gallagher et al.13 for isotherms between 400 and 750 K. It may be seen that all of the values of J collapse into a single curve although the data span a temperature range of 350 K. J is related to other equilibrium properties of the solution so that, in principle, all of them can be described in terms of J. However, it is convenient to distinguish the thermodynamic properties which are related to J or to its derivatives (type I) from those
Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000 3627
of type III, is given by
Table 1. Classification of Different Thermodynamic Properties of Solutes at Infinite Dilution According to Their Relation with J type I
II
III
relation to J directly with J or with its V and/or T derivatives through the integral I
()
R/pT ∂J 2 / F1 ∂T
partial molar volume, compressibility, and expansivity chemical potential, solubility, enhancement factor, fugacity coefficient, and distribution constant enthalpy, entropy, and heat capacities
mixed
which require an isothermal integration of J (type II).
I(0,F/1) )
∫0F
/ 1
[
J dF/1 / 2 (F1)
]
(5)
[
]
1 1 1 J+ / - / RT(F/1)2 κT1 F1
(6)
However, this approach shows the same difficulties displayed by J in the two-phase region;8 hence, the present paper is mainly centered on using J, but we shall also consider the one based upon -cˆ 12 when solvent-solute interactions are very weak. The properties of the solute at infinite dilution to which we shall make direct reference in this work are the partial molar volume and heat capacity C∞p2, the fugacity coefficient Φ∞2 , and the distribution constant of the solute between water and coexisting steam KD. The relation between V∞2 , a property of type I, and J is given by eq 2. Both ln Φ∞2 and ln KD are properties of type II related to the integral I of the Krichevskii function by the equations
ln Φ∞2 ) ln Φ/1 +
1 I(0,F/1) RT
(7)
and
RT ln KD ) I[F/1(v),F/1(l)]
) [( ) ] ( )
F
+ T(R/p)2
∂2I(0,F/1)
∂J / ∂F1
∂T2 -
T
-
F
J TJ ∂Rp - / / F1 F1 ∂T
/
(9)
p
Q denotes the ideal gas heat caIn this expression Cvi pacity at constant volume of component i at temperature T. For each system the polynomial which is used to express J or I fixes the values of all of the V and T derivatives of these quantities. This probably ignores details in the functional temperature dependence of J and I; hence, a small approximation may exist in the calculation of C∞p2.
Systems Studied and Adopted Procedure
Table 1 lists thermodynamic properties which belong to each category including type III, which denotes to properties related to both J and I; the individual expressions for these properties can be derived easily from the well-known relationships between thermodynamic quantities. We have shown8 that the greatest problem to implement the procedure based on the calculation of J occurs at subcritical temperatures, where two fluid phases coexist. This feature generates a major problem to describe properties related to I(0, F/1) because it is necessary to perform an isothermal integration of J across the two-phase region. An alternative strategy to describe the equilibrium properties of dilute binary systems was followed by O’Connell, Wood, and co-workers,4,5 to avoid the problems of the near-critical region. They made use of -cˆ 12 to describe the solute’s behavior; this quantity depends only on solute-solvent interactions and is related to J by
-cˆ 12 )
(
C∞p2 ) C/p1 + (CQv2 - CQv1) - T
properties of solutes at infinite dilution
(8)
The relation between J and -cˆ 12 with ln KD is given by eqs 5, 6, and 8. The equation for C∞p2, a mixed property
In the present work we have analyzed binary aqueous solutions of N2, CO2, and NH3. For the first two solutes we have used the critically evaluated data reported by Gallagher et al.,13,14 which contain values of V∞2 and of ln Φ∞2 for isotherms covering a wide range of pressure (density). For NH3 we have used as input the values of V∞2 reported by Hnedkovsky et al.15 for the isobars of 28 and 35 MPa. To test the capacity of this procedure to predict values for some other properties of the binary systems, we have used the experimental information available for ln KD for the three volatile solutes,16,17 the values of C∞p2 for NH3 in water reported by Sedlbauer et al.,5 and for aqueous CO2 the values of C∞p2 determined by Hnedkovsky and Wood18 and of V∞2 determined by Hnedkovsky et al.15 Recently, we have obtained experimental values of the distribution constant of B(OH)3 between steam and coexisting water over a wide range of temperatures.6 Using a procedure which involved the calculation of J, we have shown that those values were consistent with the V∞2 data for aqueous B(OH)3 reported by Hnedkovsky and Wood.18 As a further example of the predictive capacity of this approach, we shall use in the present work the expression obtained by fitting both sets of data in order to calculate the values of C∞p2 of B(OH)3 and compare them with those determined experimentally for the same binary system.3 For the properties of water in different (T, F/1) states, we used the formulation of the scientific properties of the water substance produced by the International Association for the Properties of Water and Steam.19 Equation 2 suggests that the most direct route to describe the behavior of dilute solutions is the calculation of J for the binary systems using V∞2 data as input. However, to determine from J the values of properties of types II and III, it is necessary to integrate [J/(F/1)2] as a function of the fluid density at every temperature. Obviously, at subcritical temperatures there is an experimentally inaccessible range of densities, that between the densities of the two coexisting phases. For weak solute-H2O interactions, the integrand in eq 5 at subcritical temperatures is not easily described by a function of F/1 valid over the entire fluid density range. In a preliminary paper,8 we showed that for weakly interacting binary systems, e.g., N2-H2O, it is better to fit -cˆ 12 instead of J; we shall return to this point
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Table 2. Property F in Eq 10 for the Binary Systems and Input Data Used N
system
F
type of input
L
1 2 3 4
N2-H2O CO2-H2O CO2-H2O NH3-H2O
-cˆ 12 J/(F/1)2 (I/RT) J/(F/1)2
I I II I
3 4 4 5
below. For the other binary systems we have studied, it was found possible to obtain a reasonable function for the density dependence of [J/(F/1)2]. In the particular case of CO2-H2O, for which there is abundant experimental information, we have also evaluated the performance of the procedure which uses as input values of a type II property in order to calculate I(0,F/1) and then by differentiation obtain J and predict with it properties of types I and III. The following general polynomial was used to fit the values of a property F L
F)
ai(F/1)i ∑ i)0
(10)
This expression will be used to describe thermodynamic properties of the system which are related to F. It should be noted that for some of the adjusted properties the first ai coefficients were temperature dependent. Tables 2 and 3 contain all of the details of the above equation for the binary systems studied in the present work; in Table 2, N gives the number assigned to each F calculated. Table 3 gives the values of the coefficients for eq 10; units for F/1 are mol‚dm-3, those for -cˆ 12 dm3‚mol-1, and those for J MPa. Results and Discussion The main objective of this work is to explore possible ways of describing the behavior of binary dilute solutions over wide ranges of thermodynamic variables, including the near-critical region, when sufficient experimental information exists only for a single property of the system. Consequently, we have not tried to obtain general formulations which could be applied to all of the systems that we have studied; we preferred to determine the best expressions capable of describing the behavior of each individual system, expressions which then could be used to predict other properties of the systems as well. For the N2-H2O system having weak solute-solvent interactions, it was not possible to fit J to a single function of F/1 valid at all fluid densities; hence, for this binary system, we used the expression obtained by fitting -cˆ 12, in which the strong water-water molecular interactions are absent.8 The information available for N2 dissolved in water at infinite dilution over a wide density range, apart from that reported in ref 14, is the equilibrium constant for the distribution of nitrogen between water and steam.16 Figure 3 shows the comparison between ln KD calculated with the adjusted expression for -cˆ 12 and that given by a formulation based on the critical evaluation of experimental data. It may be observed that the prediction was poor except close to the solvent critical point, and this is attributed to the already noted lack of information about -cˆ 12 in the two-phase region. To circumvent this difficulty, we have used the value of ln KD (373.2 K) to guide the adjustment of -cˆ 12, forcing it to pass through that point.
The resulting expression is given in Tables 2 and 3, and the values of ln KD calculated with it are also plotted in Figure 3; in this case the agreement with the experimental formulation of ln KD is very good. For the systems CO2-H2O and NH3-H2O, the best fit of the J values calculated with V∞2 was obtained with polynomials in the density, i.e., having no temperature dependence; their coefficients are given in Table 3. These polynomials were then used to calculate the distribution of the solutes between steam and water. For the solute CO2, a very good fit was obtained when the fitting equation was forced to pass through the experimental value of ln KD at 373.2 K. For ammonia it was not necessary to complement the V∞2 data with any additional information to obtain a good prediction of the distribution constant. For both solutes the calculated values of ln KD are compared in Figure 3 with the values obtained using for CO2 the formulation based upon experimental data for the distribution constant16 and for ammonia the experimental results of Jones.17 For the case of the CO2-H2O system, we have also used an alternative procedure based on the use of values of the fugacity coefficients of the solute and the solvent to calculate I(0,F/1) with eq 7. Values for ln(Φ∞2 /Φ/1) were used which covered the same temperature range as was the case with V∞2 to calculate J; the values of the coefficients of the equation for I given in Table 2 are reported in Table 3. The logarithm of the fugacity coefficient ratio is a property of type II (Table 1), i.e., of the same type as the distribution constant; in fact, these two quantities are related by
(
ln KD ) ln
)
Φ∞2 (F/1(gas)) Φ∞2 (F/1(liq))
Consequently, predicting the distribution constant from the fugacity coefficients is trivial and does not afford a test of the procedure’s predictive capacity. So, we have employed the expression obtained for I(0,F/1) in order to predict properties of types I and III. We have calculated C∞p2 for the systems where experimental information is available and compared the calculated values with the experimental ones. Figure 4 shows the calculated curves and the experimental points at 28 MPa for the solutes carbon dioxide, ammonia,18 and boric acid;3 the experimental uncertainties of the data are given in the figure. In the case of CO2, the figure shows two curves for the heat capacity: one was calculated with the expression for J obtained using V∞2 as input (N ) 2 in Tables 2 and 3); the other curve corresponds to the equation for I which resulted from the adjustment of the fugacity coefficients of the solute (N ) 3 in Tables 2 and 3). It should be remembered that in the first case the fitted expression was forced to pass through the value of ln KD (373.2 K), while in the second case no complementary information was necessary. A caveat is necessary when looking at Figure 4, which includes the near-critical region because the magnitude of the heat capacity is very different for different solvent densities; the same is true of its experimental uncertainty. For this reason it is convenient to analyze separately the C∞p2 data points corresponding to F/1 below 35 mol‚dm-3 from those above that value. At high fluid density C∞p2 has values in the region of 100-200 J‚(K‚mol)-1, and typically their uncertainties are a few
Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000 3629 Table 3. Coefficients of F According to Eq 10 (Units: mol‚dm-3 for F/1; dm3‚mol-1 for -cˆ 12; MPa for J) N a0 a1 a2 a3 a4 a5 b1 b2 b3 b4
1
2
3
4
b1 exp(b2/T) b3 exp(b2/T) 2.403 58 × 10-6 -1.288 235 × 10-8
0.576 554 5 -2.373 889 × 10-2 1.405 964 × 10-3 -3.990 46 × 10-5 4.544 531 × 10-7
0 b1 + (b2 + b3T) exp(b4/T) 8.675 364 × 10-4 -5.502 192 × 10-5 7.743 703 7 × 10-7
5.531 2868 × 10-4 -5.524 520 × 10-5 3.610 079 4 × 10-6 -1.236 932 6 × 10-7 2.145 342 7 × 10-9 -1.308 360 9 × 10-11
1.889 346 × 10-3 -549.12 K -9.914 721 × 10-4
0.185 018 8 0.039 206 80 -1.127 655 × 10-4/K 697.78 K
Figure 3. ln KD against F/1: (9) N2-H2O, ref 16; (2) CO2-H2O, ref 16; (b) NH3, ref 17. Solid curves were calculated with the expressions given in this work. The dashed curve was calculated with the expression for N2-H2O when fitting is not forced through a KD point for low temperature.
percent. Heat capacities for F/1 e 35 mol‚dm-3 are seen in Figure 4 to have values 100 times greater than those at high fluid density; moreover, their absolute experimental uncertainties are also quite large. They even can be of the same magnitude of C∞p2; both effects are consequences of the near-critical behavior of the heat capacities. This dramatic change in the value of C∞p2 hampers the description of the data over a wide density range, employing a simple expression for its density dependence. This is due in part to the already noted approximations in its calculation with eq 9; to improve the description, it would be necessary to have supplementary information at low temperature.5 Although our method is able to reproduce quite well the extremely different behavior observed experimentally when F/1 changes, it cannot reproduce the high density values of the partial molar heat capacity within experimental uncertainty. On the other hand, the partial molar heat capacity is described within experimental uncertainty in the near-critical region. According to the results obtained for CO2-H2O, it may be said that the heat capacities may be predicted successfully using the expressions originated in adjusting either J or I. Finally, we have compared the V∞2 for CO2-H2O calculated with the equation obtained in this work using as input the fugacity coefficients of CO2 reported by Gallagher et al.13 with the experimental values determined by Hnedkovsky et al.15 from density measurements. Gallagher et al. also report V∞2 , but we have preferred to compare our calculated values with a source of partial molar volumes which is completely independent from the data bank which was used as input. In Figure 5, we compare the calculated V∞2 for the isobars of 28 and 35 MPa with the experimental data; the agreement is very satisfactory, and in this case the
Figure 4. C∞p2 against F/1 at 28 MPa. Symbols are experimental points, and curves were calculated with the expressions derived in this work. For CO2-H2O, the dotted curve was obtained with the polynomial derived using a property of type II as input and the solid curve that when a property of type I was used. Bars on each data point give its experimental uncertainty.
calculated V∞2 are within the experimental uncertainty of the data points. It may be seen that the strategy of using either J or its integral I(0,F/1) to base the description of the thermodynamic behavior of dilute binary solutions is very adequate, except for solutes interacting very weakly with the H2O molecules. It circumvents successfully the difficulties of describing the near-critical region. With regard to its predictive capacity, it may be said that it works well even when a single property is known over a sufficiently wide (T, F/1) range to be used as input. If this property is of type II, it appears that the information provided is self-sufficient. However, if the property
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Literature Cited
Figure 5. V∞2 (CO2) against F/1: (b and dashed curve) 28 MPa; (9 and full curve) 35 MPa.
is of type I, the description is also good, but some systems require that one supplement the input with values, usually just one, of a type II property corresponding to the subcritical region. This is not a severe limitation considering the benefit which the formulation yields. From the analysis made above, it would appear that solubilities could provide the best route to calculate I(0, F/1), just as V∞2 gives direct access to J. This is conditioned to the fact that the solubility must be sufficiently low so that the saturated binary solution may be considered to be within the Henrian range. For many nonionic solutes, this is the case; deviations from Henrian behavior are only expected at high concentrations. Solubility is related to ln Φ∞2 , which was one of the properties we have used as input to adjust eq 10 in order to describe successfully the behavior of the system. Conclusions The procedure used in this work is able to describe the behavior of the solutes over all solvent densities, including the near-critical region. Using as input values of a single property, either of type I or type II, it is possible to predict other properties of the binary systems. In the first case, the input property of type I, to predict properties of types II or III, it is necessary to use some, usually only one, experimental values of the calculated property in the subcritical region to circumvent the problems posed by the isothermal integration necessary to calculate from J properties of types II or III. The route which makes use of properties of type II to describe the behavior of the solutions does not require any complementary information. The expressions obtained by fitting the input property have between five and seven coefficients. These expressions are able to describe the behavior of the systems over wide (T, F/1) ranges, including the near-critical region. It is, however, unable to give C∞p2 within experimental uncertainties for liquidlike densities at subcritical temperatures. Acknowledgment The authors are grateful for partial economic support from CONICET and ANPCyT.
(1) Ferna´ndez-Prini, R.; Japas, M. L. Chemistry in Near-critical Fluids. Chem. Soc. Rev. 1994, 23, 155. (2) Griffiths, R. B.; Wheeler, J. C. Critical Points in Multicomponent Systems. Phys. Rev. A 1970, 2, 1047. (3) Hnedkovsky, L.; Majer, V.; Wood, R. H. Volumes and Heat Capacities of H3BO3(aq) at Temperatures from 298.15 to 705 K and at Pressures to 35 MPa. J. Chem. Thermodyn. 1995, 27, 801. (4) O’Connell, J. P.; Sharygin, A. V.; Wood, R. H. Infinite Dilution PartialMolar Volumes of Aqueous Solutes over Wide Ranges of Conditions. Ind. Eng. Chem. Res. 1996, 35, 2808. (5) Sedlbauer, J.; O’Connell, J. P.; Wood, R. H. Equation of State for Correlation and Prediction of Thermodynamic Properties at Infinite Dilution of Aqueous Electrolytes and Nonelectrolytes. Chem. Geol. 2000, 163, 43. (6) Kukuljan, J.; Alvarez, J. L.; Ferna´ndez-Prini, R. Distribution of B(OH)3 between Water and Steam at High Temperature. J. Chem. Thermodyn. 1999, 31, 1511. (7) Japas, M. L.; Alvarez, J. L.; Gutkowski, K.; Ferna´ndez-Prini, R. Determination of the Krichevskii Function in Near-critical Dilute Solutions of I2(s) and CHI3(s). J. Chem. Thermodyn. 1998, 30, 1603. (8) Japas, M. L.; Alvarez, J. L.; Kukuljan, J.; Gutkowski, K.; Ferna´ndez-Prini, R. The Krichevskii Function for Binary Aqueous Systems. In Steam, Water and Hydrothermal Systems; Tremaine, P., Hill, P. G., Irish, D., Balakrishnan, P. V., Eds.; NRC Press: Ottawa, Canada, 2000. (9) Tom, J. W.; Debenedetti, P. G. Integral Equation Study of Microstructure in Model Attractive and Repulsive Supercritical Mixtures. Ind. Eng. Chem. Res. 1993, 32, 4573. (10) Tucker, S. C.; Maddox, M. W. The Effect of Solvent Density Inhomogeneities on Solute Dynamics in Supercritical Fluids: A Theoretical Perspective. J. Phys. Chem. 1998, 102, 2437. (11) Stanley, H. E. Introduction to Phase Transitions and Critical Phenomena; Clarendon Press: Oxford, U.K., 1971. (12) Carnahan, N. F.; Starling, K. E. Equation of State for Nonattractive RigidSpheres. J. Chem. Phys. 1969, 51, 635. (13) Gallagher, J. S.; Crovetto, R.; Levelt Sengers, J. M. H. The Thermodynamic behavior of the CO2-H2O System from 400 to 1000 K, up to 100 MPa and 30% Mole Fraction of CO2. J. Phys. Chem. Ref. Data 1993, 22, 431. (14) Gallagher, J. S.; Levelt Sengers, J. M. H.; Abdulagotov, I. M.; Watson, J. T. R.; Fenghour, A. Thermodynamic Properties of Homogeneous Mixtures of Nitrogen and Water from 440 to 1000 K, up to 100 MPa and 0.8 mole fraction N2; NIST Technical Note 1404; NIST: Gaithersburg, MD, 1993. (15) Hnedkovsky, L.; Wood, R. H.; Majer, V. Volumes of Aqueous Solutions of CH4, CO2, H2S and NH3 at Temperatures from 298.15 to 705 K and Pressures to 35 MPa. J. Chem. Thermodyn. 1996, 28, 125. (16) Alvarez, J. L.; Corti, H. R.; Ferna´ndez-Prini, R.; Japas, M. L. Distribution of Solutes between Coexisting Steam and Water. Geochim. Cosmochim. Acta 1994, 58, 2789. (17) Jones, M. E. Ammonia Equilibrium between Vapor and Liquid Aqueous Phases at Elevated Temperatures. J. Phys. Chem. 1963, 67, 1113. (18) Hnedkovsky, L.; Wood, R. H. Apparent Molar Heat Capacities of Aqueous Solutions of CH4, CO2, H2S and NH3 at Temperatures from 304 to 704 K and Pressures to 28 MPa. J. Chem. Thermodyn. 1997, 29, 731. (19) NIST Standard Reference Database. NIST/ASME Steam Properties; NIST: Gaithersburg, MD, 1997.
Received for review January 11, 2000 Accepted August 21, 2000 IE000003K
