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Ind. Eng. Chem. Res. 2002, 41, 3298-3306
Predictive Model for Equilibrium Constants of Aqueous Inorganic Species at Subcritical and Supercritical Conditions Kiwamu Sue, Tadafumi Adschiri, and Kunio Arai* Department of Chemical Engineering, Tohoku University, Aoba-07, Aramaki-Aza, Aoba-ku, Sendai 980-8579, Japan
A simple estimation model is proposed for describing dissociation constants of alkali-metal salts and dissolution constants of metal oxides in subcritical and supercritical water. The required parameters of the model are the Gibbs free energy, enthalpy, heat capacity, and conventional Born coefficient of the dissolved species at a given reference state. The g function needed for calculation of the temperature- and pressure-dependent conventional Born coefficient was reevaluated by regression with new experimental data of NaCl and KCl dissociation constants. The model shows a good agreement with the recently reported metal salt dissociation constants at near-critical and supercritical regions compared with the revised Helgeson-Kirkham-Flowers model. Introduction Production of metal oxide fine particles is greatly affected by the hydrothermal synthesis conditions.1 In high-temperature water, the reaction equilibrium of metal nitrates shifts toward metal hydroxides or metal oxides, which subsequently precipitate from solution according to the degree of supersaturation.1 Hydrothermal syntheses at supercritical conditions have been used to produce new functional materials2,3 and to separate metals from simulated high-level radioactive wastewater.4 The thermodynamic properties and the solubility of inorganic species at these conditions are essential, not only for these applications but also for geochemistry, inorganic synthesis, corrosion, and other fields. In this work, we focus on the evaluation of the molal equilibrium constant, K, of aqueous inorganic species at infinite dilution in supercritical water. There are many methods for estimating the reaction equilibrium but very few predictive models especially for the supercritical region. Marshall and Franck (1981)5 proposed a simple empirical model, which is a function of temperature and density of water and has been widely applied for data correlation. Anderson et al. (1991)6 simplified this model, made a thermodynamic analysis, and related the terms to constants having physical meaning. The three constant parameters for the model can be determined with the equilibrium constant, enthalpy change of a reaction, heat capacity change of a reaction, water density, and thermal expansion of water at the reference state. Although these constant parameters at a given reference state are available in most cases, the densitydependent term in the model is basically empirical. Therefore, prediction of equilibrium constants over a wide range of temperatures and pressures, especially around the critical point, can be unreliable. The Helgeson-Kirkham-Flowers (HKF) model7 has been successfully applied to the correlation of the equilibrium constant for hundreds of inorganic aqueous species of interest to geochemistry over a wide range of * To whom correspondence should be addressed. E-mail:
[email protected]. Phone/Fax: +81-22-217-7245/ 7246.
conditions (25-1000 °C and 1-5000 bar). In this model, the standard partial molal Gibbs free energy of formation, ∆G°i, at given temperatures and pressures is described by the following relation:
∆G°i ) ∆G°r,i + ∆G°s,i + ∆G°n,i
(1)
where ∆G°r,i is the Gibbs free energy of formation at the reference state (298.15 K and 1 bar). ∆G°s,i and ∆G°n,i are the solvation Gibbs free energy change and the nonsolvation Gibbs free energy change of aqueous species, i, from the reference state to given temperatures and pressures, respectively, which are defined as
[ ()
∆G°n,i ) -S°Pr,Tr(T - Tr) - c1 T ln
{[(
) (
)](
]
T - T + Tr Tr
1 1 Θ-T T-Θ Tr - Θ Θ Tr(T - Θ) Ψ+P T ln + a1(P - Pr) + a2 ln + 2 Ψ + Pr T(Tr - Θ) Θ 1 Ψ+P a (P - Pr) + a4 ln (2) T-Θ 3 Ψ + Pr
[
c2
]}
( )[
∆G°s,i ) ωP,T
(
)
(
)
(
)
(
) )]
1 1 - 1 - ωPr,Tr -1 + P,T Pr,Tr ωPr,TrYPr,Tr(T - Tr) (3)
where a1, a2, a3, a4, c1, c2, and ω represent speciesdependent parameters and Ψ, Θ, and YPr,Tr designate solvent-dependent parameters. For water, Ψ, Θ, and YPr,Tr are equal to 2600 bar, 228 K, and -5.8 × 10-5 K-1, respectively. Sverjensky et al. (1997)8 reported that each parameter (a1, a2, a3, a4, c1, and c2) in eq 2 could be estimated with heat capacity, partial molal volume, and a conventional Born coefficient at a given reference state. Tanger and Helgeson (1988)9 and Shock et al. (1992)10 proposed a revised HKF model by introducing temperature and pressure dependence into the Born coefficient,
10.1021/ie010956y CCC: $22.00 © 2002 American Chemical Society Published on Web 05/31/2002
Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002 3299
ωP,T. The Born coefficients, ωP,T, for ionic species were given by
(
ωP,T ) η
)
Z2 Z re,Pr,Tr + |Z|g 3.082 + g
(4)
where η ) 6.946 57 × 105 J mol-1 and Z refers to the charge on ionic species, respectively, and g is a function of the solvent temperature, pressure, and density given by Shock et al. (1992).10 The re,Pr,Tr term in eq 4 denotes the effective electrostatic radius of the species at the reference pressure and temperature. For monatomic ions, re,Pr,Tr is given by
re,Pr,Tr ) rx + |Z|kZ
(5)
where rx designates the crystallographic radius of the ion. In eq 5 kZ is set equal to zero for anions and 0.94 for cations based on the theory of the HKF model.7 For nonionic species, the value of the Born coefficient ωP,T is constant (-12552 J mol-1) as in the revised HKF model. The revised HKF model provides a substantial improvement over previous models in predicting equilibrium constants. In its application, the empirical part of the revised HKF model has seven parameters for each reaction. These parameters can be estimated through correlation with reference state thermodynamic properties for each species.8 Thus, in use of the revised HKF model, ∆G°Tr,Pr, S°Tr,Pr, CP°Tr,Pr, V°Tr,Pr, and ωTr,Pr are required for each species. Shock et al. (1997)11 provided a database for these parameters. However, the number of experimentally determined V°Tr,Pr data for inorganics are generally less than that of experimentally determined other data. For this reason, in the viewpoint of the prediction for equilibrium constant, K, at supercritical conditions, a more simple and predictive model is needed. Furthermore, the revised HKF model cannot apply to the nearcritical region because the applicability range of the g function is not included in the region.10 Recently, we demonstrated that the equilibrium constant for dissociation of water and phenol and for dissolution of metal oxides in subcritical and supercritical water could be well described with a simple relation equation:12,13
ln KT,F ) ln KTr,Fr -
(
)
∆H°Tr,Fr 1 1 R T Tr ∆ωTr,Fr RT
(
)
1 1 (6) T,F Tr,Fr
where is the dielectric constant of water. In eq 6, KTr,Fr is the equilibrium constant and ∆H°Tr,Fr is the enthalpy change at the reference state conditions of pure water at 25 °C and 0.997 g cm-3. ∆ωTr,Fr in eq 6 represents the change of the Born coefficient at a given reference state. In the work, ∆ωTr,Fr was determined from the experimental data. At saturation conditions and temperatures ranging from 25 to 350 °C, the calculated dissociation constants for water were in good agreement with the literature data. Equation 6 was applied to
correlate newly measured solubilities of copper oxide and lead oxide in the critical region, and it was found that retrograde behavior could be accurately described.12 In this study, we constructed the simple estimation model for the equilibrium constant by simplifying the empirical part of the revised HKF model from the function of temperature and pressure to that of temperature and water density, to eliminate the parameter needed, V°Tr,Pr and to reevaluate the g function to expand the applicability range to the near-critical region. 2. Development and Analysis of the Model 2.1. Theory. To calculate the equilibrium constant, K, at subcritical and supercritical aqueous systems, we begin with the following thermodynamic framework. The equilibrium constant in high-temperature water can be described as follows:
∑i ni∆G°i
ln K ) -
)-
∆G°Tr,Fr + ∆G°
RT
(7)
RT
where the reference state is 25 °C (Tr) and 0.997 g cm-3 (Fr). In this state, the pressure Pr is 1 bar. ∆G°Tr,Fr is the Gibbs free energy change of reaction at the reference state, and ∆G° is the Gibbs free energy change of reaction from the reference state to given temperature and density. The terms in eq 7 can be expressed as follows. The electrostatic interaction between the solute and the solvent is determined by the following Borntype equation based on the Born14 or Amis and Hinton theory.15
∆G°s,i ) ωT,F
(
)
(
)
1 1 - 1 - ωTr,Fr -1 T,F Tr,Fr
(8)
where ωT,F is estimated by eq 4. Thus, by using eq 8, ∆G° can be separated into two terms as follows:
∆G° ) ∆G°s(Tr,FrfT,F) + ∆G°res(Tr,FrfT,F) (9) where ∆G°res is the other contributions (residual term) including nonelectrostatic solute-solvent or solutesolute interactions and the rearrangement of solvent molecules. 2.2. Evaluation of ∆G°res. At this section, we modified and simplified the empirical part of the revised HKF model by taking two steps: (i) changing the temperature from Tr to T at reference density Fr and (ii) changing the density from Fr to F at a given temperature. (i) ln K at Constant Density Fr. At a constant density, the relation between ln KT,F and T can be expressed by the van’t Hoff relation over both subcritical and supercritical regions despite the divergence of various properties around the critical point.5,16,17 For this case, the change in K with temperature can be written as
ln KT,Fr ) -
∆GT,Fr RT
)-
∆G°Tr,Fr RTr
-
(
∆H°T,Fr 1 1 R T Tr
)
(10) where ∆G°Tr,Fr is estimated from thermochemical data at the reference state. ∆H°T,Fr is a constant and is
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Figure 1. Temperature dependence of the dielectric factor at constant density Fr. Dashed line: Johnson and Norton (1991). Solid line: eq 15.
Figure 2. Comparison of the correlation results and literature data for the dissociation constant of NaCl at saturation conditions.
evaluated from the reference state Tr to an arbitrary liquid state (T ′) less than about 100 °C by the following relationship:
-
(
)
∆H°Tr,Fr 1 KT ′ 1 ) ln R T ′ Tr KTr
(11)
where ln KTr is the equilibrium constant at 25 °C and is calculated with the standard state Gibbs free energy, ∆G°Tr,Fr. The equilibrium constant at T ′ is evaluated as follows:
ln KT ′ ) ln KTr -
(
) (
∆H°Tr,Fr 1 1 + R T Tr ∆CP°Tr,Fr T Tr - 1 (12) ln + R Tr T
where, in this work, we selected 50 °C as T ′.
)
Figure 3. Comparison of the correlation and literature data for the dissolution constant of ZnO at saturation conditions.
A comparison of eqs 7 and 9 with eq 10 gives
-
∆G°Tr,Fr RTr
-
(
)
∆H°T,Fr 1 1 ) R T Tr ∆G°Tr,Fr ∆G°s ∆G°res (13) RT RT RT
At the constant density of Fr and over a range of temperatures from 25 to 600 °C, the g function in eq 4 is negligible.10 Therefore, the value of ωT,F in eq 4 can be regarded as constant10 and thus equal to ωTr,Fr, which gives
(
∆G°s ) ∆ωTr,Fr
)
(
)
1 1 - 1 - ∆ωTr,Fr -1 ) T,Fr Tr,Fr
(
∆ωTr,Fr
)
1 1 (14) T,Fr Tr,Fr
At these conditions, the relationship between the value of the dielectric factor, 1/T,Fr - 1/Tr,Fr, in eq 14 and
Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002 3301 Table 1. Thermodynamic Data at the Reference State and Coefficient β Fitted to the Revised HKF Model at Saturation Conditions ∆G (kJ/mol) LiCl NaCL KCl RbCl CsCl LiOH NaOH KOH RbOH CsOH NaBr KBr RbBr CsBr NaI KI RbI CsI NaF RbF Zn2+ ZnOH+ ZnOaq HZnO2Cu2+ CuOH+ CuOaq HCuO2Ca2+ CaOH+ Fe2+ FeOH+ FeOaq HFeO2Mg2+ MgOH+ Mn2+ MnOH+ MnOaq HMnO2Ni2+ NiOH+ NiOaq HNiO2Pb2+ PbOH+ PbOaq HPbO2Sn2+
∆H (kJ/mol)
-8.6259 -3.3681 -4.4357 -5.0459 -9.9960 -11.7278 -5.4804 -13.1922 -0.7935 -2.7322 1.9752 -0.1213 -1.1988 -0.4602 -2.5290 20.7234 -1.2331 -8.3554 -9.6477 -18.1878 -7.7467 -6.8743 -9.9160 -12.5102 -6.9475 -13.9327 0.1256 -9.2885 -8.7914 -8.0835 -9.1225 -8.5395 -5.4804 -7.9496 5.6060 10.6692 -5.6973 -7.2090 5.4804 -1.9246 -147.277 -153.385 -339.699 -363.966 -282.085 -327.565 -463.252 -595.676 65.584 65.689 -126.357 -174.473 -87.027 -143.093 -251.458 -361.079 -552.790 -543.083 -716.719 -751.446 -91.504 -92.257 -275.516 -326.687 -212.212 -263.383 -399.154 -525.929 -453.985 -465.960 -624.483 -639.964 -230.538 -221.334 -407.103 -446.851 -340.996 -414.634 -506.264 -627.182 -45.606 -53.974 -21.124 -283.048 -164.599 -235.266 -343.004 -496.892 -23.891 0.920 -225.727 -288.278 -164.640 -187.025 -338.904 -435.973 -27.489 -8.786
∆ω (kJ/mol)
∆CP (J/mol‚K)
β (kJ/mol)
828.52 763.41 702.37 676.22 566.26 937.55 872.45 814.75 796.97 774.88 747.43 662.54 646.85 620.99 679.90 623.88 604.42 540.07 901.90 810.94 609.78 136.40 -12.55 781.11 617.93 270.79 308.95 228.61 517.39 188.11 601.74 293.01 -12.55 776.72 643.16 353.51 586.01 228.61 -12.55 739.94 630.40 344.01 -12.55 906.42 451.37 293.01 -12.55 567.60 469.28
-147.700 -120.080 -160.670 -51.463 -48.534 -105.020 -84.935 -42.258 -33.890 -29.288 -121.750 -84.935 -58.994 -41.840 -121.750 -84.935 -58.994 -41.840 -121.750 -58.576 -22.594 41.840 -41.840 84.517 -8.786 67.362 -5.439 -102.926 -30.962 5.858 -32.635 63.178 0.000 92.885 -21.757 129.286 -16.736 36.819 -53.137 -12.970 -48.116 130.122 41.840 201.669 -52.718 -83.262 -175.728 -292.462 -64.434
-5.0 -1.0 -2.5 4.0 4.5 6.0 10.0 11.0 13.0 16.0 -1.0 0.1 4.0 8.0 -3.0 -2.0 1.0 5.05 4.0 8.0 10.0 10.5 -1.5 25.0 15.0 16.0 8.0 -5.0 5.0 5.0 80. 15.0 3.4 25.0 12.0 24.0 8.0 11.0 -3.0 12.0 8.0 23.0 10.0 41.0 3.0 -2.0 -18.0 -24.0 2.0
temperature can be expressed as a linear function with temperature as shown in Figure 1. In this work, we approximated this by
1 1 ) R(T - Tr) T,Fr Tr,Fr
(15)
where the slope R is constant at 6.385 × 10-5 K-1 and is calculated from the equation of Johnson and Norton (1991).18 When eqs 13-15 are compared, the residual term can be described as
-
(
)
∆H°T,Fr 1 ∆Gres 1 + )RT R T Tr R∆ωTr,Fr(T - Tr) ∆G°Tr,Fr 1 1 + ) R T Tr RT ∆H°T,Fr - ∆G°Tr,Fr + R∆ωTr,FrTr 1 1 (16) R T Tr
(
)
(
)
Therefore, at a given density Fr, equilibrium constant
SnOH+ SnOaq HSnO2Al3+ AlOH2+ AlO+ HAlO2 AlO2Bi3+ BiOH2+ BiO+ HBiO2 BiO2Cr3+ CrOH2+ CrO+ HCrO2 CrO2Fe3+ FeOH2+ FeO+ HFeO2 FeO2Ga3+ GaOH2+ GaO+ HGaO2 GaO2In3+ InOH2+ InO+ HInO2 InO2Sc+ ScOH2+ ScO+ HScO2 ScO2Tl3+ TlOH2+ TlO+ HTlO2 TlO2Y3+ YOH2+ YO+ HYO2 YO2-
∆G (kJ/mol)
∆H (kJ/mol)
∆ω (kJ/mol)
∆CP (J/mol‚K)
β (kJ/mol)
-245.182 -224.262 -407.103 -483.708 -692.347 -661.859 -869.017 -831.332 95.730 -135.143 -122.591 -331.791 -258.153 -206.271 -420.492 -388.275 -577.810 -524.255 -17.238 -241.835 -222.170 -423.002 -368.192 -158.992 -381.162 -362.334 -574.463 -538.481 -97.906 -312.126 -290.370 -501.243 -446.433 -586.597 -79.914 -768.182 -969.014 -912.530 214.639 -18.828 -13.389 -240.998 -174.054 -685.339 -878.640 -828.850 -1011.273 -951.442
-266.939 -251.458 -510.866 -530.676 -766.927 -715.046 -951.860 -929.371 81.002 -187.443 -126.775 -361.079 -299.574 -251.040 -496.222 -439.320 -658.143 -620.487 -49.580 -292.880 -255.224 -503.335 -463.169 -211.710 -488.273 -422.166 -663.582 -643.918 -104.600 -396.225 -321.750 -565.258 -524.255 -614.211 -839.729 -794.123 -1022.151 -979.474 196.648 -78.659 -21.548 -274.470 -219.660 -715.046 -924.246 -856.046 -1065.246 -1019.222
108.57 -12.55 622.33 1151.86 721.61 400.41 -12.55 728.77 949.72 567.02 110.21 -12.55 395.64 1146.54 736.68 394.84 -12.55 725.13 1079.97 601.74 300.87 -12.55 613.46 1162.61 866.09 417.81 -18.37 743.79 1060.48 726.59 247.15 -12.55 581.62 1046.13 541.58 253.68 -12.55 559.36 951.94 593.79 118.49 -12.55 405.81 1041.40 552.33 243.97 -12.55 549.99
-53.974 -135.562 -189.117 -128.867 56.066 -125.102 -209.200 -48.953 4.184 -62.342 -345.598 -621.324 -532.623 -94.558 65.270 -128.449 -157.737 -85.772 -76.567 -33.890 -200.832 -312.126 -234.722 -109.621 166.105 -114.642 -128.030 -57.321 -64.015 58.576 -241.417 -359.824 -281.165 -56.484 -80.333 -236.814 -384.510 -309.616 -0.837 -42.258 -338.904 -607.098 -518.816 -52.718 -73.220 -243.509 -403.756 -323.005
-2.0 -13.0 -11.0 0.1 20.0 0.1 -20.0 0.1 17.0 6.0 -35.0 -65.0 -50.0 5.0 15.0 0.1 -7.0 0.1 10.0 10.0 -12.0 -25.0 -10.0 10.0 30.0 -4.0 -9.0 10.0 10.0 17.0 -17.0 -30.0 -18.0 12.0 2.0 -20.0 -27.0 -20.0 15.0 5.0 -36.0 -70.0 -53.0 15.0 0.1 -15.0 -30.0 -20.0
K can be expressed with the solvation and the residual terms as follows:
∆GT,Fr
)-
∆G°Tr,Fr
(
)
RT RTr ∆ωT,Fr 1 ∆H°T,Fr + R∆ωTr,FrTr 1 1 -1 + R T Tr RT T,Fr
ln KT,Fr ) -
(
∆ωTr,Fr RT
(
)
)
1 - 1 (17) Tr,Fr
(ii) Effect of Density F on ln K. In eq 17, the density dependence of the solvation term, eq 8, is clearly related to the density dependence by replacing ∆ωT,Fr and T,Fr with ∆ωT,F and T,F, respectively. The density dependence of the residual term was determined by comparing eq 17 with the literature data.7 To analyze the density dependence of the residual term, the dissociation reaction of sodium chloride at
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be developed. In general, the database for ωTr,Fr and CP°Tr,Fr are sufficient compared with that of V°Tr,Fr. Thus, in the latter part, correlation of β with the available parameters will be examined.
Figure 4. Calculated results of the NaCl dissociation constant. Circles are data calculated by the revised HKF model.
Figure 5. Parity plot of the g values for NaCl and KCl dissociation constants.
saturation conditions was considered because many literature data, including the results from the revised HKF model, were reported over a wide range of conditions including the supercritical region. When the density was changed from Fr to F, the density dependence of the residual term was found to correlate as
ln KT,F,lit. - ln KT,F ) -
β(1 - F*) R
(
2/3
1 1 T Tr
)
(18)
ln KT,F ) ln KTr,Fr -
)
∆H°T,Fr + β(1 - F*)2/3 + R∆ωTr,FrTr 1 1 R T Tr
(
)
For predictions over a wide range of inorganics, it would be preferable to correlate β with ∆ωTr,Fr and ∆CP°Tr,Fr. This is because the literature data for conventional Born coefficients ∆ωTr,Fr and the standard heat capacities ∆CP°Tr,Fr are more extensive than ∆V°Tr,Fr tabulated for the revised HKF model. Considering that ∆G°res,F,dif is the perturbation term from the second term of eq 19, we analyzed and found that a linear relation existed between β and ∆CP°Tr,Fr + ∆ωTr,Fr/5000. Therefore, in this study, the following linear function was employed:
β ) λ1∆CP°Tr,Fr + λ2∆ωTr,Fr + λ3
where ln KT,F,lit. shows literature data. When eq 18 was substituted into eq 17, the following equation was obtained:
( (
The equilibrium constant for the dissociation of alkalimetal salts and the Gibbs energy of formation for the divalent and trivalent cations and metal hydroxide complexes as shown in Table 1 were calculated with eq 19 at temperatures ranging from 25 to 350 °C and at saturation conditions to use β as a correlation parameter for the results of the revised HKF model. In the calculations, ∆G°Tr,Fr, ∆H°Tr,Fr, ∆CP°Tr,Fr, and ∆ωTr,Fr were evaluated using the database of the standard state parameters of the revised HKF model.11 The water density was calculated from the equation of Haar et al. (1984),19 and the dielectric constant was calculated from the equation of Johnson and Norton (1991).18 The standard deviations in log K in the case of comparison to the revised HKF model are less than 0.126. Correlation results of zinc ions are shown in Figure 3 as ZnO dissolution reactions with the values calculated by the revised HKF model and literature data. The Gibbs energy of formation for zinc oxide was taken from Shock et al. (1998).11
)
∆ωTr,Fr 1 ∆ωT,F 1 -1 + - 1 (19) RT T,F RT Tr,Fr The calculation results of eq 19 are shown in Figure 2 for the NaCl dissociation reaction. The coefficient β fitted with the value from eq 19 was -1.0 kJ mol-1. From the results described above, it was considered that the Gibbs energy of formation of the dissolved species could be estimated with the theory of eq 19. 2.3. Estimation Method of β. In this section, a method to estimate β from the available information will
(20)
where λ1, λ2, and λ3 were determined to be 9.737 × 101 K, 1.947 × 10-2, and -5.96 × 102 J mol-1, respectively. The standard deviation of error between the calculated value of β from eq 20 and the fitted value of β is 4.097 kJ mol-1. 2.4. Comparison of the Estimated Dissociation Constants with Those of the Revised HKF Model at Supercritical Conditions. With the parameters in Table 1, the equilibrium constants of sodium chloride were estimated by eq 19 at supercritical conditions (25600 °C and 500-1000 bar) as shown in Figure 4. Even though β in eq 19 was obtained by the fitting with the data at saturation conditions, the calculation result showed a good agreement with that of the revised HKF model. Using the correlation function, eq 20, we considered that equilibrium constants of inorganics not only for saturation conditions but also for supercritical conditions could be reliably estimated and predicted. 2.5. Reevaluation of the g Function. Within the framework of the revised HKF model, the range of application of the g function is at pressures from 500 to 5000 bar and densities from 0.35 to 1 g cm-3 in the supercritical region and is not included in the near-
Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002 3303
critical region. The g function in the supercritical region is expressed by regression with experimental data of dissociation constants for NaCl by Quist and Marshall (1968).20 Since the development of the revised HKF model, new NaCl and KCl experimental data have been reported in the near-critical and supercritical regions by Gruszkiewicz and Wood (1997)21 and Ho et al. (1994, 2000, 1997).22-24 In the reevaluation of the applicability region of our model, we assumed that the g function in
the electrostatic term should be reevaluated because the residual term is not sensitive to changing temperature and density at the supercritical region compared with the electrostatic term.7-11 In this part, the g function was evaluated with these experimental data. Data used in the regression were at conditions of temperatures ranging from 250 to 600 °C, pressures ranging from 223 to 998 bar, and densities ranging from 0.20 to 0.81 g cm-3. In the framework of
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Figure 6. Comparison of the dissociation constant between the literature data and calculated result with the conditions of data. Closed circles: this work. Open circles: the revised HKF model.
the revised HKF model, the g function was given by Shock et al. (1992)10 as
eq 21 and found that the behavior could be described by the following function:
g ) ag(1 - F*)bg - f(P,T)
(21)
f(P,T) ) cg1(F*) + cg2(F*)2 + cg3 ln(F*)
ag ) ag1 + ag2T + ag3T 2
(22)
bg ) bg1 + bg2T + bg3T 2
(23)
[(T -300155) + (T -300155) ][c (1000 - P) + c 4.8
f(P,T) ) cg1
16
3
g2
g3(1000
- P)4] (24)
where ag1, ag2, ag3, bg1, bg2, bg3, cg1, cg2, and cg3 are constants. In eq 21, F* is a density ratio formed by dividing the pure water density by its density at a reference condition that was chosen to be that of pure water at atmospheric pressure at 25 °C. The function f(P,T) is used at temperatures above 150 °C and pressures under 1000 bar. We analyzed the term f(T,P) in
(25)
In eq 25, cg1, cg2, and cg3 are constants that were determined by Marquardt’s method correlating to experimental data of dissociation constants of NaCl and KCl. The values of cg1, cg2, and cg3 were estimated to have values of 0.18359, -0.18632, and 0.11531, respectively. A comparison between gmodel in eq 21 and gfit in eq 25 is shown in Figure 5 with the g values from the revised HKF model. Calculated values, gmodel, with eq 25 are in good agreement with the values, gfit, fitted with the experimental data. The deviation of the revised HKF model from gfit became larger as the g value became smaller because of the lower densities. With the new g function defined by eqs 21 and 25, the equilibrium constants of some metal salts (LiCl,25 NaCl,21-23 NaOH,26 KCl,24 KOH24), which were recently determined by electric conductance measurement at supercritical conditions, were estimated. The results of this evaluation are summarized in Figure 6. The results
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showed a good agreement with experimental data compared to the results from the revised HKF model. 3. Conclusions A simple predictive model was developed for describing equilibrium constants for metal salt dissociation and metal oxide dissolution in subcritical and supercritical water. The required parameters of the model are the Gibbs free energy, enthalpy, heat capacity, and conventional Born coefficient of dissolved species at the reference state. The proposed model given by eq 19 with the coefficient β given by eq 20 can be applied to estimate the equilibrium constant of many inorganic species not only for saturation conditions but also for the supercritical region. Literature Cited (1) Adschiri, T.; Hakuta, Y.; Arai, K. Hydrothermal synthesis of metal oxide fine particles at supercritical conditions. Ind. Eng. Chem. Res. 2000, 39, 4901-4907. (2) Kanamura, K.; Goto, A.; Ho, R. Y.; Umegaki, T.; Toyoshima, K.; Okada, K.; Hakuta, Y.; Adschiri, T.; Arai, K. Preparetion and electrochemical characterization of LiCoO2 particles prepared by supercritical water synthesis. Electrochem. Solid-State Lett. 2000, 3 (8), 256-258. (3) Hakuta, Y.; Seino, K.; Ura, H.; Adschiri, T.; Takizawa, H.; Arai, K. Production of phosphor (YAG:Tb) fine particles by hydrothermal synthesis in supercritical water. J. Mater. Chem. 1999, 9 (10), 2671-2674. (4) Smith, R. L., Jr.; Atmaji, P.; Hakuta, Y.; Kawaguchi, M.; Adschiri, T.; Arai, K. Recovery of metals from simulated high-level liquid waste with hydrothermal crystallization. J. Supercrit. Fluids 1997, 11, 103-114. (5) Marshall, W. L.; Franck, E. U. Ion product of water substance, 0-1000 °C, 1-10 000 bar; New international formulation and its background. J. Phys. Chem. Ref. Data 1981, 10, 295304. (6) Anderson, G. M.; Castet, S.; Schott, J.; Mesmer, R. E. The density model for estimation of thermodynamic parameters of reactions at high temperature and pressures. Geochim. Cosmochim. Acta 1991, 55, 1769-1779. (7) Helgeson, H. C.; Kirkham, D. H.; Flowers, G. C. Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: IV. Calculation of activity coefficients, osmotic coefficients, and apparent molal and standard and relative partial molal properties to 600 °C and 5 kb. Am. J. Sci. 1981, 281, 1249-1516. (8) Sverjensky, D. A.; Shock, E. L.; Helgeson, H. C. Prediction of the thermodynamic properties of aqueous metal complexes to 1000 °C and 5 kb. Geochim. Cosmochim. Acta 1997, 61, 13591412. (9) Tanger, J. C.; Helgeson, H. C. Calculation of the thermodynamic and transport properties of aqueous species at high pressures and temperatures: Revised equations of state for the standard partial molal properties of ions and electrolytes. Am. J. Sci. 1988, 288, 19-98. (10) Shock, E. L.; Oelkers, E. H.; Johnson, J. W.; Sverjensky, D. A.; Helgeson, H. C. Calculation of the thermodynamic and transport properties of aqueous species at high pressures and temperatures: Effective electrostatic radii to 1000 °C and 5 kb. J. Chem. Soc., Faraday Trans. 1992, 88, 803-826. (11) Shock, E. L.; Sassani, D. C.; Willis, M.; Sverjensky, D. A. Inorganic species in geologic fluids: Correlations among standard molal thermodynamic properties of aqueous ions and hydroxide complexes. Geochim. Cosmochim. Acta 1997, 61, 907-950. (12) Sue, K.; Aida, M. T.; Hakuta, Y.; Smith, R. L., Jr.; Adschiri, T.; Arai, K. Measurement and correlation of metal oxide solubility in sub- and supercritical water. In Physical and Chemistry: Meeting the Needs of Industry; Tremaine, P. R., et al., Eds.; NRC Research Press: Ottawa, Ontario, Canada, 2000; pp 782-789. (13) Sue, K.; Murata, K.; Matsuura, Y.; Tsukagoshi, M.; Adschiri, T.; Arai, K. Flow-through electrochemical cell for pH
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Received for review November 26, 2001 Revised manuscript received April 5, 2002 Accepted April 15, 2002 IE010956Y
