PII S (99)

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1 ergamon II S6-737(99)3- Geochimica et Cosmochimica Acta, Vol. 64, No. 3, pp , Copyright Elsevier Science Ltd rinted in the USA. All rights reserved 6-737/ $.. Infinite dilution partial molar properties of aqueous solutions of nonelectrolytes. I. Equations for partial molar volumes at infinite dilution and standard thermodynamic functions of hydration of volatile nonelectrolytes over wide ranges of conditions ANDEY V. LYASUNOV,, *JOHN. O CONNELL, and OBET H. WOOD 3, Institute of Experimental Mineralogy, ussian Academy of Sciences, Chernogolovka, 443, ussia Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 93, USA 3 Department of Chemistry and Biochemistry, and Center for Molecular and Engineering Thermodynamics, University of Delaware, Newark, Delaware 976, USA (eceived October, 998; accepted in revised form July 9, 999) Abstract A semitheoretical expression for partial molar volumes at infinite dilution of aqueous nonelectrolyte solutes has been developed employing the collection of properties from fluctuation solution theory for use over wide ranges of temperature and pressure. The form of the solution expression was suggested by a comparison of solute/solvent and solvent/solvent direct correlation function integrals (DCFI). The selection of solvent density and compressibility as model variables provides a correct description in the critical region while second virial coefficients have been used to give a rigorous expression in the low density region. The formulation has been integrated to obtain analytic expressions for thermodynamic properties of hydration at supercritical temperatures. The equation is limited to solutes for which B (the second cross virial coefficient between water and a solute molecule) is known or can be estimated. egression of the three remaining parameters gives good correlations of the available experimental data. A strategy for estimating these parameters allows prediction from readily available data. Copyright Elsevier Science Ltd. INTODUCTION * Author to whom correspondence should be addressed resent address: Department of Earth and lanetary Sciences, Washington University, St. Louis, MO 633, USA. 495 The development of an equation of state (EoS) for the infinite dilution partial molar properties of aqueous nonionic solutes for very wide ranges of temperatures and pressures/ densities is an important task with many practical applications in geochemistry and technology. Despite progress towards this goal, most noticeably due to Helgeson, Shock and their coworkers (Tanger and Helgeson, 988; Shock et al., 989; Shock and Helgeson, 99; Shock et al., 99; 997), recent experimental (Hnedkovsky, 994; Hnedkovsky et al., 995; 996; Hnedkovsky and Wood, 997), theoretical (Levelt Sengers, 99a; 99b; O Connell, 994; 995), and data correlation (O Connell et al., 996) developments show that there is much to be done to improve both the quality of correlations and the reliability of predictions, especially at near-critical and supercritical conditions. A successful EoS must provide an accurate description of experimental data and allow a reasonable extrapolation to temperature/pressure ranges where there are no experimental results. For practical geochemical applications, it is important to have some methods for reliable estimation of the EoS parameters where few experimental data are available such as at 98 K. There are several justifications for developing an EoS from the infinite dilution partial molar volume of solutes in water, V. First, rigorous fluctuation solution theory (Kirkwood and Buff, 95), establishes a direct relationship between V, the macroscopic thermodynamic properties of the pure solvent and integrals of statistical mechanical correlation functions. Second, a volumetric equation alone, if valid at all densities, allows calculation of all thermodynamic functions of a solute at supercritical temperatures, if they are known for the solute in an ideal gas state. Finally, such an equation gives the pressure dependence of all thermodynamic properties of the solutes.. EQUATION OF STATE FO INFINITE DILUTION ATIAL MOLA VOLUMES OF VOLATILE NONELECTOLYTES IN WATE.. eview of revious Equations Several correlations have been proposed for infinite dilution partial molar volumes of aqueous nonelectrolytes over wide ranges of T and along with numerous specific equations of state for a limited number of water gas supercritical mixtures (typically involving CO,CH 4,H and some other nonpolar gases). The best known and most widely used by geochemists and chemists is the revised equation of Helgeson, Kirkham, and Flowers (Tanger and Helgeson, 988; Shock et al., 99, etc), the revised HKF, incorporated into a comprehensive database and the SUCT software package (Johnson et al., 99). The HKF equation was originally proposed for aqueous ions (Tanger and Helgeson, 988) and then later extended by Shock et al. (989) to include neutral aqueous species. In the HKF equation, V is divided into a solute-specific part for the internal or nonsolvation volume and another solvent-specific term for solvation based on the Born equation. The revised HKF equation for the infinite dilution partial molar volume of neutral species is V a a / a 3 /T a 4 / T / /T, () where a to a 4 are adjustable parameters, stands for pressure

2 496 A. V. lyasunov, J.. O Connell, and. H. Wood in bars, 6 bar, 8 K, is the solvent dielectric constant and is the conventional Born coefficient for ions but which becomes an additional adjustable parameter for nonelectrolyte solutes. Consistent treatment of thermodynamic properties of neutral solutes in this framework has resulted in a valuable supplement to the existing database. For many substances the predicted values of V up to 5 K, are in fair agreement with experiments performed after publication of the predictions (Criss and Wood, 996). However, recent results have revealed some limitations of this approach. recise volumetric data (Hnedkovsky, 994) show that the revised HKFmodel does not provide the correct behavior for nonelectrolyte solutes in the near-critical and supercritical regions. Further, predictions of standard chemical potentials of dissolved gases are inaccurate at supercritical temperatures, as shown by comparison with experimental fugacity/concentration ratios for dissolved H (lyasunov, 99) and with computer simulations and equations of state for aqueous CH 4 (Lin and Wood, 996). The solvent critical region causes the infinite-dilution standard state properties of solutes to behave in a complex manner (Harvey et al., 99). There have been several suggestions to minimize the consequent difficulties, especially in the pressure and temperature derivatives of Henry s law constant (Harvey et al., 99; O Connell, 994; Akinfiev, 997) but these do not give quantitative agreement with V and Cp (the infinite dilution partial molar heat capacity) near the critical point of pure water and at low temperatures. Another, and more accurate (at least at near critical conditions) correlating equation for V of nonelectrolytes in water has been proposed by O Connell, Sharygin, and Wood (996). These authors correlated the dimensionless quantity A at the limiting condition of infinite dilution of a solute () in the water () A V lim T N 3 V/T N, () T,V,N as a simple function of water properties. Here, is the isothermal compressibility of pure water, is the gas constant, T is temperature, and V stand for pressure and total volume of a system, and the derivative is taken with respect to the number of moles of solute at constant T, V and the number of moles of solvent. The parameter A is well-behaved, i.e., it is a smooth, continuous, and finite function at all conditions, even at the critical point of the solvent, where both V and diverge, but their ratio is finite. The property A ij (i, j, ) arises in two different analyses. Levelt Sengers connects A in the critical region with the useful Krichevskii parameter (Levelt Sengers, 99b). It also arises naturally in the statistical mechanics of fluctuation solution theory FST (Kirkwood and Buff, 95; O Connell, 97; 99; 994) by being related to integrals of microscopic correlation functions. For example, A V T C, (3) where C is the dimensionless integral of the infinite dilution solute solvent direct correlation function conceived by Ornstein and Zernike (McQuarrie, 976). reviously such an approach had been used for nonelectrolytes by Brelvi and O Connell (97) and by Crovetto et al. (99). O Connell et al. (996) used recent precise V data for five nonelectrolytes (CH 4,CO,H S, NH 3,H 3 BO 3 ) in water, measured at temperatures between 98 and 75 K and pressures 8 and 35 Ma to develop the model V V a b exp T, (4) where V is the molar volume of pure water, cm 3 mol,.5 m 3 kg is an universal constant for aqueous solutes, a and b are adjustable parameters. This relation was found to be superior to the five-parameter Eq. () over all the range of experimental data from 98 to 75 K and at 8 and 35 Ma... Development of a New Equation A successful model is expected to capture the essential physics of the system of interest and also be able to be manipulated conveniently to obtain all thermodynamic properties. Many models have been based on the Born concept of spherical charges in a continuum (structureless) incompressible dielectric. An attractive feature of the Born model is its capability to give directly qualitative estimates of the Gibbs energy of hydration of an ion at different temperatures and pressures, when the solute radius and solvent dielectric constant are available. However, ions in water involve specific hydration features in addition to the charge-dielectric effect and to account for this the so-called electrostatic radius of a solute must become an empirical function of temperature and pressure/density, to obtain adequate agreement for experimental thermodynamic functions of hydration. However, such improvement is achieved by considerable erosion of the theoretical basis of the Born model and when this concept is applied to nonelectrolytes in water, it can be argued that Born-type relations lose all theoretical basis. An appeal of Eq. (3) is its formal relationship between the property of interest, V, and a well-behaved statistical mechanical quantity, the direct correlation function, c (r, T, ). Thus the collection of quantities in A ij (i, j, ) has a rigorous foundation. A possible approach could be to use models of the microscopic direct correlation function, such as from integral equation or other theoretical methods (McQuarrie, 976). However, the results would not be quantitatively reliable except in the low- to moderate-density second virial region. Our approach has been to utilize the benefits of FST s unique collection of variables and the insights that theory provides while avoiding the inaccuracies of microscopic modeling. Thus, similar to previous work (Brelvi and O Connell, 97; Cooney and O Connell, 987; Huang and O Connell, 987; O Connell et al., 996), we correlate the particularly simple density behavior of the FST integrals directly from macroscopic V data. It is important to recognize the density and temperature behavior of the direct correlation function, c ij (r, T, ), that constrains variations of its integral, C ij, and therefore A ij through Eq. (3). O Connell (994) shows the general behavior of pure water and ammonia direct correlation functions based on experimental neutron and x-ray scattering data. All pairs of species have the same general characteristics. In terms of A ij behavior there are short-range positive values from exclude volume effects that increase significantly with density. These are often modeled with rigid body approximation (McQuarrie,

3 Equation of state for V ). There are also longer-range negative values from attractive forces that change little with density. Theory, for example, the random-phase approximation (McQuarrie, 976), suggest that the long-range limit of the direct correlation function, c ij (r, T, ), is related to the intermolecular pair potential function, though this is not rigorous except for the ideal gas and at densities where the EoS truncated at the second virial coefficient is accurate. Under these conditions A ij B ij (T), where B ij (T) stands for the second virial coefficient between particles i and j. These considerations allow us to say that all pairs of A ij (i, j, ) will have the similar characteristics: () equal to one at zero density with identifiable (usually negative) linear density dependence beginning with ; () large, positive and increasing rapidly at high densities; (3) well-behaved in the solvent critical region. The last is true even though V and the pure solvent compressibility are both infinite at the critical point of a pure solvent; their ratio is finite. For a pure solvent component pair at the critical point, A is zero. For the present case, A involves only water molecules, whereas A replaces one of them with a solute molecule. Thus, all multibody interactions, such as excluded volume effects, would be similar for both A and A giving large positive values at high densities. From zero density through the critical region, A A if the effects of attraction are less those for pairs (as for most of the hydrophobic species considered in this work) while A A if the attractive direct correlations are greater (solutes with a strong affinity to water). In fact, A can be negative (giving V ) for solutes with strong attractive interactions with water, like salts. At high densities A A if the solute is larger in size than the solvent, which is common for many solutes. Thus, there can be a crossover of A and A at moderately high densities for such systems. All of these expectations are fulfilled by the available data for both aqueous and nonaqueous systems. We test the expected similarity of the temperature and density variations of A and A beyond that of previous workers by analyzing experimental data for some systems in Figs. a c. First, we start with consideration of the T dependence of the reduced bulk modulus of pure water A, which can be defined as A V T C. (5) A can also be thought of as A for a solute water molecule in water. The Hill (99) equation of state for pure water was used to calculate A for a number of isotherms between 373 and 73 K at different densities as shown in Fig. a. While this function is both temperature and density dependent, the temperature variations become weaker at supercritical conditions. This behavior has been found for all substances (Huang and O Connell, 987). At zero density, the ideal gas value of unity is found. The limiting slope is related to the second virial coefficient. At the critical point, A. At high densities, A, the water water interactions are dominated by repulsive forces and increasing T decreases the distance of closest approach, resulting in a decrease of A with temperature at these densities. At low densities the water water correlations are attractive. They become less attractive relative to k B T (where k B is the Boltzmann constant) when T increases, resulting in the increase of A at lower densities. In Figs. b and c we investigated the two solute water systems. First, estimates of A for dissolved silica SiO (aq) were made from the very extensive experimental information on the solubility of SiO (c) at different T and as correlated by Fournier and otter (98) and by Manning (994) for K and up to and Ma, respectively. Both correlations give the value of the molality of dissolved silica as a function of temperature and water density, i.e., the value of the equilibrium constant, K, for the reaction SiO (c)asio (aq). For this reaction the volume change V T( ln K/) T, and since the molalities of the dissolved silica are small, the volume change can be assumed to be essentially at infinite dilution: V V V V m (SiO (c)), where the last term stands for molar volume of quartz. The difference in A for the dissolved silica calculated from both the Fournier and otter (98) and Manning (994) correlating equations is less than.5 in the overlapping density range. In addition, the A values for dissolved methane were estimated from the Jacobs Kerrick EoS for the CH 4 H O system (Jacobs and Kerrick, 974) at K at different densities. These A values are expected to have uncertainties of less than at densities below 3 kg m 3 and less than 3 4 at densities above 8 kg m 3. As expected, Fig. shows the similarity in T dependence of A for water and A for SiO (aq) and CH 4 (aq). The principal difference is that the value of A at the solvent critical point is not zero; for SiO (aq) it is negative whereas for CH 4 it is positive. The isotherms are only qualitatively the same but a correlation depending strongly on density and weakly on temperature is expected to hold. Thus we propose the leading term in A is proportional to A with some differences: A N A additional terms. (6) At high densities the approximation that A A when species is water is expected to hold if a solute were the same size as water. Since it generally is not, we use the scaling factor N to correct the consequent excluded volume effect. This approximation would be rigorous if the direct correlation functions for water water and water solute pairs were identical in reduced coordinates and N would be a measure of the excluded volume ratio of unlike to pure water pairs. In general it is expected than the additional terms are likely to be required. We find that N can be correlated with various experimental measures of solute size (see below).... The limiting form of the EoS at low densities To be theoretically correct, a successful EoS must go to the ideal gas law at zero-density limit, and at low densities be consistent with the virial equation of state. For the present case of a binary mixture the virial series is written as follows: T N V N V N V B T N V B T N N V B T, where N and N stand for a number of moles of solvent (water) and solute species, respectively, V is the total volume of the mixture, B (T), B (T), and B (T) are the second virial

4 498 A. V. lyasunov, J.. O Connell, and. H. Wood coefficients related to interactions of two molecules of water, two molecules of a solute, and one molecule of water with one molecule of solute, respectively. The virial coefficients are (only) temperature dependent. Using the definition of the A parameter, see Eq. (), one obtains the relation A B (T)..., and A B (T)... (O Connell, 99, Crovetto et al., 99; O Connell et al., 996). Therefore, at the limit of low densities, Eq. (6) should have the form A N A N B T N B T, (7) where ( 3 /M w ) mol kg is the conversion factor needed if the values of the second virial coefficients are given in units cm 3 mol and values of are in kg m 3 ; M w stands for the molar mass of pure water in g mol. The values of the second virial coefficient for pure water are well known over 73 5 K (LeFevre et al., 975; Eubank et al., 988; Kell et al., 989), and the values of the second cross virial coefficients between gases and water can be reliably estimated using literature correlations (Tsonopoulos, 974; Hayden and O Connell, 975). We chose the simple correlation of Tsonopoulos (Tsonopoulos, 974; Tsonopoulos and Heidman, 99), which requires critical properties, the itzer acentric factor, the dipole moments of pure substances, and one interaction parameter specific for each binary pair, k ij. The use of this correlation is described in Appendix A together with an empirical method to estimate k ij when necessary. Comparison of the few available experimental and predicted values of B (T) using the Tsonopoulos correlation for nonpolar and weakly polar solutes allows us to conclude that the uncertainty in B (T) is often less than to 5 cm 3 mol at temperatures above 6 65 K, decreasing with increasing temperature. Unfortunately, no data for relatively strongly polar solutes are available for comparison, so, the errors in predictions could be much larger below 4 K. For polar solutes with strong attraction with water (hydrogen bonding, dimerization, etc.) uncertainties could be extremely large with values not being negative enough. However, these effects are not expected to significantly affect the accuracy of our predictions. First at high temperatures, where B values are not large, even an apparently large error will not be very significant. Second, as shown below, the final equation contains a term to compensate the unrealistically large contribution of the second cross virial term at high densities and low temperatures, so, errors in B (T) values will be minimized under these conditions. In general, we used predictions from the Tsonopoulos model at temperatures above 4 K. To further simplify our correlation, we used the analytical relation for the square-well potential (itzer, 995) as a way to express the temperature dependence of B ij : B ij T N 3 A ij 3 3 exp ij, (8) k B T Fig.. The values of A (the reduced bulk modulus) and A for some representative isotherms as a function of pure water density: (a) A for water; (b) A for aqueous silica; and (c) aqueous methane. where N A is the Avogadro number; ij is the collision diameter, in Å; ij /k B is depth of the potential well; is the width of the potential well in molecular diameters. From published values of B, which cover a temperature range K

5 Equation of state for V 499 Table. The values of the Gibbs energies of hydration h G, molar volumes at normal boiling point V b, hard-sphere diameters d, the Tsonopoulos interaction parameters k ij, as well as parameters, /k B, N, a, b of Eq. () for some well-studied aqueous nonelectrolytes. In the last three columns the values of / are given for Eq. (), for the O Connell Sharygin Wood model (column g) and for the HKF model (column h), respectively ( is the difference between experimental and calculated values of V for a solute and is the estimated uncertainty of the experimental point). h G a V b b d c k ij c /k B N 9 a d 3 b e / f / g / h CH 4,3V points i,j, 98 T, K 75,., Ma k 37.7 l 3.7 m.34 n CO,3V points i, 98 T, K 75,, Ma o 33.3 p 3.94 m.6 q H S, 3 V points i,r, 83 T, K 75,., Ma o 34.3 l 3.6 l.5 s NH 3,3V points i,t, 83 T, K 75,., Ma 35.5 o 5. l.9 l. s Ar, 6 V points u,j, 98 T, K 687,., Ma k 9. l 3.4 m.3 v C H 4,6V points u, 98 T, K 687,, Ma k 48.6 l 4.7 m.5 w Morpholine OC 4 H 8 NH, 9 V points x, 83 T, K 573,., Ma 9.8. y 96 z 5.4 aa.6 bb Methanol CH 3 OH, 3 V points dd, 73 T, K 573,., Ma y 4.5 l 3.69 m. bb propanol CH 3 (CH ) OH, 35 V points ee,ff, 73 T, K 53,., Ma 8.7 y 8.8 l 4.55 l.7 bb henol C 6 H 5 OH, 6 V points ff, 98 T, K 598,., Ma y 3 z 5.54 aa.9 bb yridine C 5 H 5 N, 3 V points ff, 98 T, K 598,., Ma 8.74 y 93 z 5.33 aa.5 bb N-propylamine CH 3 (CH ) NH,3V points ff,gg, 78 T, K 53,., Ma 8.45 y 8.4 l 5. aa.4 bb ropionamide CH 3 CH CONH,8V points ff, 98 T, K 53,., Ma ii 88 z 5.3 aa. bb Succinic acid HOOC(CH ) COOH, 8 V points ff, 98 T, K 53, 8, Ma 6 jj 9 z 6. aa.8 bb Adipic acid HOOC(CH ) 4 COOH, 9 V points ff, 98 T, K 53, 8, Ma jj 74 z 6.69 aa.6 bb ropanoic acid CH 3 CH COOH, V points ff,kk, 78 T, K 53,., Ma y 9 z 5.8 aa.7 bb ,4-butanediamine, H N(CH ) 4 NH,V points ff, 98 T, K 53,., Ma 8.4 ii 5 z 5.93 aa. bb ,6-hexanediamine, H N(CH ) 6 NH,V points ff, 98 T, K 53,., Ma 8.9 ii 69 z 6.6 aa. bb ,4-butanediol, HO(CH ) 4 OH, 6 V points ff,ll, 78 T, K 53,., Ma 8.6 ii z 5.69 aa.8 bb ,6-hexanediol, HO(CH ) 6 OH, 4 V points ff,ll, 78 T, K 53,., Ma 8. ii 55 z 6.43 aa.7 bb a kj mol ; b cm 3 mol ; c Å; d m 3 kg K 5 ; e m 3 kg ; f the present model, Eq. (); g the O Connell et al. (996) model; h the HKF model; i (Hnedkovsky et al., 996); j (Tiepel and Gubbins, 97); k (Wilhelm et al., 977); l (eid et al., 987); m (Wilhelm and Battino, 97); n (Tsonopoulos, 974); o (Wagman et al., 98); p estimated using the Tyn and Calus method, recommended for inorganic gases in (eid et al., 987, p. 53); q our estimate from experimental values of B (T) (Coan and King, 97; atel et al., 987; Wormald and Lancaster, 988); r (Tsonopoulos, 978); s (Barbero et al., 98); t (Allred and Woolley, 98); u (Biggerstaff and Wood, 988); v our estimate from experimental values of B (T) (igby and rausnitz, 968; Wormald and Lancaster, 988); w our estimate from experimental values of B (T) (Wormald and Lancaster, 988); x (Tremaine et al., 997); y (Cabani et al., 98); z estimated using the LeBas method (eid et al., 987, p. 53); aa estimated using Eq. (A7); bb estimated using Eq. (A6); cc (Makhatadze et al., 99); dd (Alexander, 959; Friedman and Scheraga, 965; Hoiland, 98; Makhatadze et al., 99; Sakurai et al., 994); ee (Alexander, 959; Friedman and Scheraga, 965; Hoiland, 98; Makhatadze and rivalov, 989; Sakurai et al., 994); ff (Criss and Wood, 997); gg (Kaulgud et al., 98); hh (Makhatadze and rivalov, 988; Sakurai et al., 994); ii estimated using group contribution values (Cabani et al., 98); jj calculated from the standard Gibbs energies of solution at 98 K [saturation molalities are taken from solubility compilations (Seidel, 94; Seidel and Linke, 95)], combined with the Gibbs energies of formation of compounds in crystalline and gaseous state (Domalsky and Hearing, 993); kk (Makhatadze et al., 99); ll (Hoiland, 98). (Eubank et al., 988; Kell et al., 989; LeFevre et al., 975), we determined the parameters of the square-well potential for water water interactions:.56.6 Å; /k B 55 5 K;.. [these numbers compare well with previous estimates given by Hirschfelder et al. (954):.66, 6, and.99 respectively]. The weighted leastsquares procedure was used with the weights equal to /, where is the estimated uncertainty. The same procedure was used to determine parameters for B (T) for different water solute interactions. In principle, all three parameters could be fitted for each solute, but it was found that the value of. can be used for all solutes with little sacrifice of accuracy, particularly taking into account large uncertainties of available B (T) data/estimates. The values of and /k B are given in Table together with other parameters for the solutes we have considered.

6 5 A. V. lyasunov, J.. O Connell, and. H. Wood... The EoS for infinite dilution partial molar volumes of nonpolar and weakly polar solutes There are many ways to build the correlation equation for infinite dilution partial molar volumes consistent with the low density limiting form. One of the simplest is the form A N A N B T N B T exp (c) ft,. (9) The challenge to any correlation that covers the whole density range is to include the linear in density and temperature dependent term important at low to moderate densities but eliminate it at high densities where the pair interactions are overwhelmed by excluded volume effects. This task is accomplished by exponential reduction of the linear density term and by introduction of the f(t, ) term to account properly for higher order contributions at high densities. Equation (9) has the right second virial coefficient provided f(t, ) does not have any n terms in its expansion, where n. This approach differs from that of another paper from this laboratory which is not as rigorous at low densities, but is able to treat very polar and ionic substances (Sedlbauer et al., 999). The final form of the correlating equation was developed from Eq. (9) by trial and error, keeping in mind two main requirements: () to provide an accurate description of the available experimental results using a minimum of fitted parameters; () to provide a fair prediction of the infinite dilution partial molar volumes and related thermodynamic functions of solutes in T regions outside the fitted region. The basic volumetric data set included recent precise experimental partial molar volumes of inorganic nonelectrolytes CH 4, CO, H S, NH 3, Ar, and C H 4, taken from two principal sources (Hnedkovsky et al., 996; Biggerstaff and Wood, 988). In all about 3 experimental points for V were used. The data range over 83.5 K T 75 K and. Ma 35 Ma. In this range the V values show sharp extrema, the accurate description of which is one of the important goals of this study. Only results at T 69 K were used from the earlier work (Biggerstaff and Wood, 988) for Ar and C H 4. Also, a few data points at low temperatures were added from other recent sources for NH 3 (Allred and Woolley, 98), and H S (Barbero et al., 98). In addition, we included values of standard Gibbs energy of hydration h G, the standard enthalpy of hydration h H, and standard heat capacity of hydration h Cp at K and 8 Ma for CH 4,CO,H S, and NH 3. Experimental values of these quantities were evaluated using the tabulated thermodynamics of these solutes at 98.5 K,. Ma (Cox et al., 989; Wagman et al., 98) together with the values of V (T, 8 Ma) and Cp (T, 8 Ma) from Hnedkovsky et al. (996) and Hnedkovsky and Wood (997). The final form of the correlating equation was chosen to be A N A N B T N B T exp (c ) a T 5 bexp c, () or for partial molar volumes V N V T N T B T N B T exp (c a T 5 bexp c () where N, a, and b are adjustable parameters to be fitted to volumetric data; c.33 m 3 kg and c. m 3 kg are universal constants. At the limit of low densities, Eq. () correctly transforms into Eq. (7). The very strong temperature dependent term involving a corrects for the unrealistically large contribution of the N B (T) exp(c ) term at low temperatures. The rapid decrease with temperature is compensated by the a/t 5 term. The term may also empirically account for the strongly temperature dependent hydrogenbonding effects in lower-temperature aqueous solutions. To obtain the values of N, a, and b we regressed the data with a weighted least-squares procedure, with the weights equal to /, where is the estimated uncertainty of a given experimental result. The uncertainty of each V data point was assumed to consist of two contributions: experimental uncertainty of measurement at finite (low) concentration of solute (typically from % to 4% depending on temperature) and an additional % uncertainty due to extrapolation to infinite dilution, in accordance with recommendations of O Connell et al. (996). Table summarizes the results for many aqueous nonelectrolytes together with the average value of /, where is the absolute value of the difference between calculated and experimental V. For comparison, the last columns of Table give the values of / for the model of O Connell et al. (996) and for the revised HKF model for the same data sets. Besides six gases (CH 4,CO,H S, NH 3, Ar, and C H 4 ), studied up to K, we give in Table the values of the corresponding parameters for some additional nonelectrolytes, where the experimental values of V were measured up to 53 6 K, below the temperature of near-critical extrema of infinite dilution partial molar volumes (Criss and Wood, 997; Tremaine et al., 997; Xiao et al., 997). Typical estimates of V uncertainties for these nonelectrolytes are within.5 to 4. cm 3 mol depending on temperature. Table shows that in practically all cases Eq. () describes well the experimental data for both inorganic and organic solutes, including ones of very large size. For subcritical V the quality of fit of Eq. () and the revised HKF-model is about the same. Generally, they are both better than the model of O Connell et al. (996). However, as found by O Connell et al. (996) the accuracy of the HKF at supercritical temperatures is lower despite its five fitting parameters. The reasons for this are that the HKF model does not have the correct low-density limits, i.e., it does not yield either the second virial or the ideal gas EoS and further, as shown by Hnedkovsky (994), its Born-type relations cannot provide an accurate description of the divergence of V of nonelectrolytes near the critical point. The O Connell et al. (996) equation describes the data very well, especially in the near-critical region, although as found in other work from this laboratory (Sedlbauer et al., 999) this two-parameter model does not reproduce low-temperature and low density V results within experimental accuracy. No attempt was made to describe infinite dilution partial molar isothermal compressibilities,, of nonelectrolytes, mainly because these data are available only for some of the

7 Equation of state for V 5 3 Fig.. redictions and experimental values of h V G. (a) The predicted (lines), experiment [open squares for Kishima (989) and triangle for lyasunova and Ivanov (99) results], and SUEFLUID (open circles) values of h V G for dissolved H S at 673 and 73 K as a function of pure water density; (b) redictions using the present model (solid lines) and the O Connell Sharygin Wood (996) equation (dashed lines), compared to SUEFLUID results (open circles) and the Lin and Wood (996) MD-based (filled circles) values of h V G for the dissolved CH 4 at 873 and 473 K as a function of pure water density; (c) redicted (lines) and SUEFLUID (open circles) values of h V G for dissolved CO at 673 and 73 K as a function of pure water density. organic solutes at ambient temperatures. For most solutes the calculated values of infinite dilution partial molar volumes monotonically increase with temperature at ambient conditions (73 33 K,. Ma). However, for a number of nonelectrolytes, there is a very flat (about..6 cm 3 mol ) minimum in V at temperatures between 8 to 3 K. Although the model is successful for correlation of the data considered, it has limitations. It requires an independent estimation of the second cross virial coefficient for water solute pairs. This is reliable for volatile nonelectrolytes where values of the critical temperature, pressure, itzer acentric factors, and dipole moments are available or can be estimated. It was found that some predictions of V are sensitive to the values of B, so care is required in its estimation. We suggest using the procedure outlined in Appendix A which is reliable even though no attempt was made to optimize B values from fitting V data. However, Eq. () may not be reliable for very polar solutes because of their strong interactions with water which influence low density descriptions. Another paper from this laboratory (Seldbauer et al., 999) successfully uses an alternative based on high density results for correlations and predictions for both ions and very polar substances, but it has less accuracy at the lowest densities and highest temperatures..3. Accuracy of the roposed Equation Beyond the T ange Used for arameterization It is expected that Eq. () will be accurate at least up to 3 K at low water densities due to its correct limiting form. However, this is yet to be verified since there are few possibilities to check the accuracy of such high temperature high pressure predictions. Values of the infinite dilution partial molar chemical potential are known for a number of solutes at supercritical temperatures from experimental fugacity/concentration ratios, predictions from different EoS and from computer simulations. Equation () can be examined by comparing the results from experimental and calculated fugacity coefficients,, and therefore the chemical potential of infinitely dilute solutes at supercritical temperatures. Thus ln A d ln V () T and from Eqs. () and () it follows that

8 5 A. V. lyasunov, J.. O Connell, and. H. Wood Fig. 3. redicted (lines) and experimental (Hnedkovsky and Wood, 997) values of the standard partial molar heat capacity of aqueous CH 4 (circles) and NH 3 (triangles) at 8 Ma and supercritical temperatures. Fig. 4. Correlation of the a parameters and B (43 K) N B (43 K). The solid line is Eq. (4). ln N ln c B TN B T exp [c ] ) a T 5 b expc c N ln V T, () where is the fugacity coefficient of pure water. A convenient function to compare the experimental and predicted values of the chemical potential of infinite dilute solute at supercritical temperatures is h V G, the Gibbs energy of transfer of a solute from the ideal gas to an equal volume of solution: V h G T ln V (3) T. The zero-density value of this function is zero, the initial departure is linear in water density and its sign at low densities is determined by the sign of the second cross virial coefficient between water and a solute. Thus, it is a sensitive check of the accuracy of any EoS for mixtures at low densities. The use of Eq. (3) for V h G and other hydration properties assumes Eq. () for V is valid over a whole density range down to zero density. This is no problem for temperatures above or equal the critical temperature of water (i.e., for systems in the fluid state). However, because we do not expect accurate integration using Eq. () through the two-phase region, Eq. (3) cannot be used at lower temperatures. Sedlbauer et al. (999) have suggested one way to overcome this difficulty but that approach was not implemented here. The set of equations for calculation of the Gibbs energy, enthalpy, entropy, and heat capacity of hydration of a solute at infinite dilution at temperatures above the critical temperature of water consistent with Eq. () is given in Appendix B. The set of equations for the calculation of the pressure dependence of the standard chemical potential (the Gibbs energy), enthalpy and heat capacity of a solute is given in Appendix C. There are several sources of experimental values of V h G. For H S it can be calculated from gaseous fugacity/solution concentration ratios of studies of buffer mineral assemblages in equilibrium with water (Kishima and Sakai, 984; Kishima, 989; lyasunova and Ivanov, 99) at temperatures up to K and pressures up to Ma. In the former work the authors determined the fugacity/concentration ratio Y (bar mol kg), while in the latter work the related function K H /Y was considered. These results can be converted into V h G values as follows: V h G T ln 55.5 V T ln Y. These results are probably accurate to kjmol. For CH 4 values of V h G at K and pressures corresponding to densities of pure water up to kg m 3 are available from Lin and Wood (996), who performed molecular dynamics (MD) simulations to obtain the Gibbs energy of hydration of infinite dilute methane, ethane and propane in water, confirming their results by comparison with several equations of state for the CH 4 H O system. Finally fugacity coefficients of dissolved gases at infinite dilution (actually at mole fraction X 6 ) are calculated using the SUEFLUID computer program (Belonoshko et al., 99), which provides VT properties and Gibbs energies for any mixture of C H O N S Ar composition up to very Table. Correlations useful for estimating the parameter N of Eq. (): V (98) is the partial molar volume of a solute in water at 98 K; V b is the molar volume of pure substance at its normal boiling point; V c stands for the critical volume of pure substance. atio For polar solutes For nonpolar solutes V (98)/N V b /N V c /N

9 Equation of state for V 53 Table 3. Estimated parameters of Eq. () for a number of inorganic nonelectrolytes. Solute h G a V b b V c k ij / B d N 9 a e 3 b f He 9.47 g 3.5 h 6 i Ne 9. g 6.8 h i Kr 4.86 g 34.6 h 33 i Xe 3.5 g 4.9 h 43 k n.6 g 5 h 47 i H 7.73 l 8.4 h 6 l D 7.57 l 4 h 7 i N 8.9 g 34.8 h 35 j O 6.5 g 7.9 h 3 m O 3.8 g 35 h 35 i CO 7.4 g 34.9 h 37 j NO 5.49 g 3 h 7 i N O 9. g 36 h 36 i COS 9.54 g 47 h 45 i NF g 46 h 44 i N F g 56 i 5 i SO.48 n 44 h 39 o SF 6.6 g 79.8 h 65 i HCN 6. n,p 39 h 38 i H 3.9 g 46 i 44 i AsH 3.69 g 49 h 46 i a Gibbs energy of hydration at 98.5 K,. Ma, kj mol ; b molar volume of pure substance at its normal boiling point, cm 3 mol ; c at 98.5 K,. Ma, cm 3 mol ; d Å; e m 3 kg K 5 ; f m 3 kg ; g ((Wilhelm et al., 977); h (eid et al., 987); i estimation, see text; j (Moore et al., 98); k (Biggerstaff and Wood, 988); l (Muccitelli and Wen, 978); m (Tiepel and Gubbins, 97; Moore et al., 98); n (Wagman et al., 98); o (Barbero et al., 983; Sharygin et al., 997); p (Edwards et al., 978). high T and. Note that the value to be used in Eq. () is equal to the product of the fugacity coefficient of the pure gas and the activity coefficient of the solute at infinite dilution from the SUEFLUID program. In SUEFLUID the properties of pure components are based on experimental VT-data and MD calculations, whereas the properties of mixtures depend on the mixing rules employed. Although it was claimed that the integral VT properties of mixtures are not sensitive to the choice of mixing rules (Belonoshko and Saxena, 99), it is not clear that this remains true for partial (derivative) properties close to infinite dilution. Comparison of V h G values for methane and H S from different sources (Kishima, 989; lyasunova and Ivanov, 99; Lin and Wood, 996; Belonoshko et al., 99) suggests that the accuracy of V h G values from SUEFLUID may be better than 5 kj mol. Figure shows predictions of properties at high temperatures using the parameters (listed in Table ) obtained from fitting data in the database described above. In Fig. a we compare predicted and experimental values of V h G for infinitely dilute H S at 673 and 73 K. At 673 K there are accurate (presumably within kjmol ) determinations related to experiment (Kishima, 989; lyasunova and Ivanov, 99), but at 73 K only calculated estimates from SUEFLUID can be used. Figure a shows good agreement of our model with the experimental data, better than SUEFLUID, while consistency with SUEFLUID is maintained at 73 K. In Fig. b we compare predicted values of V h G with MD-based calculations (Belonoshko et al., 99; Lin and Wood, 996) for dissolved methane at 873 and 473 K up to pressures corresponding a density of pure water kg m 3. Also shown are predictions from the model of O Connell, Sharygin, and Wood (996). At 873 K, all results are within 4 kj mol while at 473 K the O Connell Sharygin Wood (996) model is too high, the present model is close to, and at the higher densities SUEFLUID deviates from the accurate results of Lin and Wood (996). Figure c shows results for CO at 673 and 73 K compared to SUEFLUID (Belonoshko et al., 99). Though Fig. c shows no experimental data at these conditions, the comparison is made because of the system s exceptional importance for geochemistry and its extensive database of several properties at other conditions that were utilized in the development of SUEFLUID. The agreement between the models for this system suggests that the present approach is quite reliable. We conclude from these comparisons that Eq. () can be used for reliable estimation of infinite dilution partial molar volumes (and chemical potentials above the critical temperature) without large errors at least up to 3 K and pressures corresponding pure water density kg m 3, even when parameters are obtained from lower temperatures and pressures. Of course, in the absence of direct experimental determinations any equation at high T and should be used with caution. To our knowledge the only relevant data for other standard thermodynamic functions of hydration at supercritical temperatures are the infinite dilution partial molar heat capacities reported by Hnedkovsky (994) and Hnedkovsky and Wood (997). Model and experimental results are plotted in Fig. 3 for CH 4 and NH 3. The values of Cp were calculated from Cp h Cp Cp(ideal gas), where the standard heat capacity of hydration h Cp can be predicted from the volumetric Eq. (B3), with Cp(ideal gas) from thermodynamic compilations (Stull and rophet, 97; Cox et al., 989). Using parameters obtained only from V data, the agreement between experiment and predictions is quite satisfactory in view of its enormous sensitivity at near-critical conditions to even very small variations in temperature and pressure, see Levelt Sengers, 99a. It

10 54 A. V. lyasunov, J.. O Connell, and. H. Wood Table 4. Estimated parameters of Eq. () for representative groups of organic solutes. Solute h G a V b b V c k ij /k B d N 9 a e 3 b f n-alkanes Ethane C H g 54.9 h 53 i ropane C 3 H 8 6. g 75.8 h 7 i n-butane C 4 H 6.63 g h 84 k n-pentane C 5 H 7.68 g 6 h 98 k n-hexane C 6 H g 3 h k alkenes -propene C 3 H g 69 h 57 i butene C 4 H g 94 h 7 k pentene C 5 H 4.88 g h 86 k hexene C 6 H 4.94 g 6 h k alkynes Ethyne C H 7.87 g 4.3 h 43 i propyne C 3 H g 6.9 h 53 k butyne C 4 H g 83. h 68 k pentyne C 5 H g 98.7 h 8 k n-alkylbenzene Benzene C 6 H g 96.5 h 83 g Toluene C 7 H 8 4. g 3 k 98 g Ethylbenzene C 8 H 4.59 g 3 k 5 l n-propylbenzene C 9 H 5.69 g 5 k 3 k n-butylbenzene C H g 7 k 46 k alcohols Ethanol C H 5 OH 3.6 g 6 k 55 g butanol C 4 H 9 OH.8 g 4 k 87 g pentanol C 5 H OH.8 g 6 k 3 g ketones Acetone C 3 H 6 O 8. g 77.5 h 67 g butanone C 4 H 8 O 7.3 g 97 k 83 g pentanone C 5 H O 6.36 g 8 k 98 g n-carboxylic acids Acetic CH 3 COOH.3 g 64. h 5 g Butanoic C 3 H 7 COOH 8.67 g 9 k 85 g amines Methanamine CH 3 NH.7 g 44. h 4 g Ethanamine C H 5 NH.9 g 66 h 58 g propanamine C 3 H 7 NH.45 g 83 h 74 g butanamine C 4 H 9 NH.5 g 99 h 9 g a Gibbs energy of hydration at 98.5 K,. Ma, kj mol ; b molar volume of pure substance at its normal boiling point, cm 3 mol ; c at 98.5 K,. Ma, cm 3 mol ; d Å; e m 3 kg K5; f m 3 kg ; g (Cabani et al., 98); h (eid et al., 987); i (Moore et al., 98); k estimated; l (Sakurai, 99). should be noted that at the lowest temperatures and at the extrema the model yields Cp values that are about 5 % different from experimental ones, perhaps exceeding the combined experimental error. 3. ESTIMATION OF THE AAMETES OF EQ. () FO VOLATILE NONELECTOLYTES Because of the limited V data for aqueous nonelectrolytes, estimations and predictions must be made of the parameters of Eq. (). Besides the parameters of the square-well potential for solute water interactions, Eq. () contains three adjustable coefficients, a, b, and N. As discussed above, the term a/t 5 is mainly to compensate for the unrealistically large contribution of the second cross virial coefficient at low temperatures and high densities. This suggests a correlation of a based on {B (T) N B (T)}. In Fig. 4 the a parameter is plotted versus {B N B } at 43 K (for many solutes the lowest temperature for the Tsonopoulos correlation). Within the uncertainties of most of the data, there is the linear correlation a B 43 K N B 43 K, (4) where numerical values of the second virial coefficients are in cm 3 mol. The parameter N is expected to reflect the size of solutes. Convenient measures of solute size include the molar volume of the liquid at the normal boiling point V b, the critical volume V c, and V at 98 K. Table gives average values of V b /N, V c /N, and V (98)/N. Any of these can be used since the variations are similar. We found that there seem to be two classes of solutes, nonpolar (CH 4, CO, Ar, C H 4 ), and polar (H S, NH 3 and many organic solutes). Obviously this classification is crude, and some solutes, like benzene, with zero dipole moment in gas phase, will show special effects from polarization in aqueous solutions. Finally, b can be fitted to a single V value, either experimental or estimated where necessary. Thus, we suggest the following strategy for estimating the required parameters:

11 Equation of state for V 55 Fig. 5. redicted (solid line) V values for Xe at 35 Ma and data ones (circles) from (Biggerstaff and Wood, 988) at Ma at T 6 K and at 33 Ma at lower temperatures versus pure water densities. The temperature interval is from 98 to 7 K. () /k B and. When experimental values of B are not available, values of /k B and in Eq. (8) can be found by fitting estimations of B from the Tsonopoulos correlation over 4 3 K. The necessary properties can be found in eid et al. (987) and the binary interaction parameter k ij can be obtained from Eq. (A). () N. A value of N for Eq. () can be found using one of the ratios of Table. The value of V b, V c, V (98) are either available or can be estimated (eid et al., 987). (3) a. Values of a in Eq. () can be estimated with Eq. (4) while values of b require either a single experimental value, such as at 98.5 K in Moore et al. (98), Tiepel and Gubbins (97), Cabani et al. (98), or an estimated one. redictive methods are available in Brelvi and O Connell (97) and in Moore et al. (98). The above strategy was employed to estimate the parameters of Eq. () for a number of inorganic and organic solutes. Making predictions, we used as a criterion of the polarity the value of the Gibbs energy of hydration at 98.5,. Ma, h G (98.5 K,. Ma). Solutes with h G (98.5 K,. Ma) kj mol were treated as nonpolar, other as polar. We understand the arbitrariness of this decision and the necessity to develop a more objective correlating scheme as soon as parameters of the model become available for a larger number of solutes. Tables 3 and 4 show results when the above strategy was used. For a few solutes, V data are available at temperatures above 33 K [benzene, toluene (Makhatadze and rivalov, 988), ethanol, butanol, pentanol (Makhatadze and rivalov, 989), acetic acid, n-butanamine (Makhatadze et al., 99)]; they were fitted to obtain a and b. The test of the results obtained can be made for Xe, where experimental values of V are available up to 7 K (Biggerstaff and Wood, 988), but the data were not as numerous as for other solutes, so they were not used to derive Eq. (). In Fig. 5, predicted values of infinite dilution partial molar volumes of Xe at 35 Ma (solid line) are compared with experiment. Fig. 6. redicted and literature values of h V G for the aqueous H at 673 and 73 K as a function of pure water density. Data are filled circles (Kishima and Sakai, 984), open circles are from SUE- FLUID. The agreement is good. In Fig. 6, predicted values of h V G for dissolved H at 673 and 73 K are compared with accurate experimental data from (Kishima and Sakai, 984) and values from SUEFLUID (Belonoshko et al., 99). As in Fig. the agreement with the lower temperature experimental values is excellent and better than SUEFLUID while there are significant differences from SUEFLUID at the higher temperatures where no data exist. For N, a comparison can be made with results from an EoS [Gallagher et al., 993(b)] for the nitrogen-water system valid at T 44 K and Ma. Figures 7(a) and 7(b) show good agreement along the saturation line for V and at 7 K for h V G. For ethane and propane one can compare predicted and MD-based (Lin and Wood, 996) values of h V G at very high temperatures. Figures 8a and b shows that predictions are usually within the estimated uncertainty of the simulation. Unfortunately, all of these encouraging comparisons are for nonpolar solutes. For strongly polar substances tests are possible only in limited temperature ranges for very few solutes. For example, Fig. 9 shows predicted and experimental values of V for acetic acid along the saturation curve. The data are estimates based on V for the acetic acid dissociation obtained from the pressure dependence of the dissociation reaction (Lown et al., 97; Mesmer et al., 989) and partial molar volumes of the acetate ion using the revised HKF data base (Johnson et al., 99). Note, that in the Mesmer et al. (989) work, there are two sets of values depending on the model employed to describe the pressure dependence of the reaction, and the difference between them may be an estimate of the uncertainty. The predictions could be considered satisfactory. Obviously, the availability of high-temperature experimental measurements is important both to obtain model parameters and correlations. For solutes where V values are not known even at a single temperature it is not possible to know how accurate the estimation method is. Extension of the set of