Improved Thermodynamic Model for Aqueous NaCl Solutions from

Improved Thermodynamic Model for Aqueous NaCl Solutions from 350 to 400 °C. Bing Liu, John L. Oscarson,* Craig J. Peterson, and Reed M. Izatt...
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Ind. Eng. Chem. Res. 2006, 45, 2929-2939

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Improved Thermodynamic Model for Aqueous NaCl Solutions from 350 to 400 °C Bing Liu, John L. Oscarson,* Craig J. Peterson, and Reed M. Izatt Departments of Chemistry and Biochemistry and Chemical Engineering, Brigham Young UniVersity, ProVo, Utah 84602

A model, based on Helmholtz energy, that can be used to describe thermodynamics of NaCl solutions in the water near-critical region has been developed and tested as a function of temperature (350-400 °C), pressure (18-41 MPa), and solute concentration (0-5 molal) using literature densities, apparent NaCl (aq) molar volumes, heats of dilution, and apparent isobaric molar heat capacities. Densities, heats of dilution, apparent molar isobaric heat capacities, and apparent NaCl (aq) molar volumes calculated using the model agree well with literature values over the conditions studied. Calculated apparent molar isobaric heat capacities agree with literature data up to 450 °C, even though the data used in model regression were valid only at temperatures from 350 to 400 °C. Since calculation of apparent molar isobaric heat capacities involves the second temperature derivative of Helmholtz energy, extrapolation to temperatures > 400 °C is expected to give reasonable results. 1. Introduction High-temperature (T) aqueous solutions are of increasing interest in many fields. Corrosion and the resultant failure of steam generators in nuclear power plants have caused an increased interest in corrosion processes occurring in aqueous systems at high T values. Knowledge of the chemistry in aqueous solutions from 280 to 320 °C as a function of solute concentrations was included in a code developed by us called MULTEQ1 that allowed operators to adjust conditions at nuclear plants so that corrosion was minimized. A code similar to MULTEQ is needed that is applicable at higher T and pressure (P) values. The model reported in this work is a starting point in the development of such a code that would be useful as a tool to calculate thermodynamic properties including speciation at high T and P values. Power generation from geothermal wells and the discovery of ocean vents and the biological life associated with them2 have led workers to realize the need to know the chemistry in these environments. The knowledge required includes the effect of aqueous chemistry on corrosion, precipitation, life processes, and biochemical reactions at high T values. The discovery that oxidation of organic wastes in supercritical water is extremely rapid and efficient has led workers to propose supercritical water oxidation as an alternative to conventional methods of waste disposal.3-8 Aqueous-solution chemistry under high T and P conditions has been studied using solubility measurements,9,10 potentiometry,11,12 electrical conductivity,13-15 calorimetry,16-22 isopiestic measurements,23 spectroscopy,24-26 and density (F) measurements.27-29 Making experimental measurements and interpretation of results involving aqueous solutions at these T and P conditions are challenging. Major problems encountered in the experimental measurements include the potential corrosion of any materials in contact with the solution, plugging of flow and sample lines due to formation of precipitates, and safe containment of hot, pressurized fluids. Interpretation of the results obtained at high T and P values is complicated by the tendency for all ions to associate and by the significant deviation of activity coefficients from unity. Difficulties in both measurement and interpretation of results increase markedly as the critical temperature (Tc) and critical pressure (Pc) of water are approached. Water becomes * To whom correspondence should be addressed. Fax: (801) 422-0151. Phone: (801) 422-6243. E-mail: [email protected].

very compressible as its Tc and Pc are approached. The result is that small fluctuations in T, P, or molality (m) in the dilute region cause large changes in F, enthalpies (H), and degree of solute dissociation of the constituent ions of the solute.20 As m increases, Tc and Pc also increase.30 Hence, as m increases at constant T and P values near or above the water critical point, the solution becomes a subcritical fluid. As a result, changes in F, H values, and equilibrium constants for any reaction involving ionic species become small with fluctuations in T, P, and/or m. Interpretation of experimental results is further complicated because traditional models based on excess Gibbs energy (Gex) become increasingly less effective for correlating the thermodynamic properties of aqueous solutions above ∼300 °C.31,32 In addition, it is difficult to separate effects due to dilution from those due to reactions. Models based on residual Helmholtz energy (Ares) as a function of T, F, and m are more effective in describing aqueous solution thermodynamic properties than those based on Gex as a function of T, P, and m at T > 300 °C.19,20,33 In our previous papers,19,20 we modeled Ares as a function of T, F, and m. The starting point of these models was the Ares model developed by Anderko and Pitzer (AP).33 The first model developed by us (RI model)19 consisted of a term for the change in A due to the ionization of NaCl0 (aq) to Na+ + Cl- at infinite dilution (∆ionA) plus the unmodified AP model. The ∆ionA term was obtained using the equation of Gruszkiewicz and Wood34 for the ionization of NaCl0 (aq). Generally, better agreement between calculated and measured heats of dilution (∆dilH) was found using the RI model19 as compared to those found using the AP model. However, calculation of ∆dilH values at high solute concentrations using the RI model resulted in values significantly different than the measured values. These differences were not unexpected, since the ∆ionA term is valid only at infinite dilution. Significant improvement in the model was achieved by adding a term that accounted for the difference in A between solutions at ionic strength, µ f 0, and those where µ > 0.20 This additional term was based on the work of Myers et al.35 where a mean spherical approximation (MSA) approach was used. The improved model thus created was termed the RII model.20 To improve the agreement with measured ∆dilH values, the parameters in the term for the change in A with concentration, ∆AMSA, and the ∆ionA terms were adjusted in the RII model.

10.1021/ie0511579 CCC: $33.50 © 2006 American Chemical Society Published on Web 03/30/2006

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Table 1. Parameters for Water Used in the Perturbation Term in the AP33 and RIII Models parameter

AP model33

aH2O (MPa‚cm3‚mol-1) cH2O dH2O eH2O Tr

(1.718248 + 1.828379/Tr + 1.546648/Tr + 0.107189/ Tr4) × 105 2.953548 - 8.874823/Tr + 3.179334/Tr2 - 0.168698/ Tr4 2.139339 + 9.442203/Tr - 3.144017/ Tr + 0.149539/Tr42 -9.0 T/Tc ) T/647.067 K

RIII model

The RII model fits ∆dilH values over the entire concentration range much better than did the AP model33 or the RI model.19 The model reported here (RIII model) is an improved version of the RII model in several respects. In the RIII model, an improved equation for the ∆ionA term is used. In the earlier equation,19 obviously incorrect results were found at certain values of F and T. Second, in the RII and RIII models, the AP term is used to describe the behavior of the neutral species. However, in the AP model, there is no differentiation between NaCl0 (aq) and NaCl (aq). In the RIII model, the perturbation term in the AP model was modified to describe only NaCl0 (aq). Third, the parameter regression of the RIII model involved fitting apparent molar volumes (Vφ), apparent molar isobaric heat capacities (Cp,φ), F, and ∆dilH values. The agreement of the above measured quantities with those calculated using the RIII model demonstrates the validity and wide applicability of this model. 2. Model Development In the RIII model, a term is incorporated for the neutral NaCl0 (aq) species, Aneu, and two terms are incorporated which account for the ions interacting with each other, ∆ionA and ∆AMSA. The Aneu term is similar to that used by Anderko and Pitzer.33 The ∆ionA term is based on the approach used by us,19,20 which was taken from the results of Gruszkiewicz and Wood.34 The MSA term is a modified version of that reported by Myers et al.35 The Ares term is a sum of several other terms, as shown in eq 1,

Ares ) Aneu + nNa+∆ionA + ∆AMSA

(1.698183 + 1.801095/Tr + 1.523088/Tr2 + 0.107041/ Tr4) × 105 2.830988 - 8.476943/Tr + 2.970851/Tr2 - 0.169394/ Tr4 2.137675 + 9.315539/Tr - 3.156785/ Tr2 + 0.149472/Tr4 -9.0 T/Tc ) T/647.067 K

2

(1)

where nNa+ is the number of moles of Na+ in the solution. The Ares term, used in the AP model33 to describe aqueous NaCl solutions, was fitted to phase equilibria and F over wide T (300-927 °C) and P (up to 500 MPa) ranges. The AP model was based on the assumption that none of the NaCl (aq) is dissociated. The Ares term contains a hard sphere term, a dipolar interaction term, and a perturbation term. The hard sphere and dipolar terms are based on theory and, thus, are incorporated unchanged into the Aneu term in eq 1. The perturbation term is included in a modified form in the Aneu term. There were three reasons for modifying the perturbation term in the RIII model. First, in the AP model, dissociation effects were implicitly incorporated in the perturbation term, since the parameters in this term were fitted to data where NaCl ionization is known to occur. Second, there was a need to improve the accuracy of calculated property values in the near-critical region, where small changes in T, F, and m result in large changes in these values. Third, the parameters in the AP model were fitted using only F and phase equilibrium data, and in this study, the RIII model was fitted to H and Vφ data as well. Without these changes, large errors may occur when the model is used to calculate such properties as H, molar isobaric heat capacity (Cp), or Vφ, which require differentiation. The a, acb, adb2, and aeb3 quantities in the perturbation term of the AP model are related to the second, third, fourth, and fifth virial coefficients.33 These quantities can be calculated using

Table 2. Comparison of Water Densities Reported by NIST37 in the Critical Region of Water with Those Calculated Using the AP33 and RIII Models AP model33 (g‚cm-3) T (°C)

P (MPa)

NIST (g‚cm-3)

350 360 365 370 375 380 390 395 400

18 20 22 22 24 25 28 30 30

0.5872 0.5480 0.5381 0.4930 0.4856 0.4508 0.4187 0.4205 0.3574

a

RIII model (g‚cm-3)

value

% deviation

value

% deviation

0.5823 0.5399 0.5294 0.4771 0.4682 0.4273 0.3924 0.3754 0.3350 Σ|% deva|

-0.834 -1.478 -1.617 -3.225 -3.583 -5.213 -6.281 -10.73 -6.267 39.2

0.5777 0.5424 0.5343 0.4934 0.4854 0.4606 0.4161 0.4150 0.3492 Σ|% deva|

-1.618 -1.022 -0.706 0.081 -0.041 2.174 -0.621 -1.308 -2.294 9.8

dev ) deviation.

the pure-fluid parameters a, b, c, d, and e and the interaction parameters, R0ij, γ0ijk, δ0ijkl, 0ijklm, and τ as described by Anderko and Pitzer.36 In the present work, the parameter b is the same as that used in the AP model, because it is the classical covolume term used in the repulsive and dipolar terms. Anderko and Pitzer33 argued that the parameter e is similar in different compounds; therefore, the value of e was kept constant in the present work. The other three pure-component parameters and the five interaction parameters were modified using available FH2O, solution F, and ∆dilH values. Accurate calculation of ∆dilH, Cp,φ, and Vφ requires accurate FH2O values. However, the disagreement between FH2O values calculated using the AP model and those determined experimentally increases rapidly as the Tc and Pc of water are approached and water becomes more compressible. In the RIII model, the coefficients of the water parameters aH2O, cH2O, and dH2O were refitted using FH2O values provided by the National Institute of Standards and Technology (NIST) online database37 at the T and P values of interest. In Table 1, the parameters used for water in the perturbation term of the AP33 and RIII models are shown. In Table 2, FH2O values calculated using the AP and RIII models are compared with those reported by NIST.37 At the given T and P values, FH2O values calculated using both models agree well with the NIST data, but the values found using the RIII model agree better with the NIST data than do those values calculated using the AP model, except in the case of FH2O at 350 °C and 18 MPa. This improvement in fitting the experimental FH2O values results in significantly better agreement with measured Vφ values, since small changes in FH2O cause large changes in calculated Vφ and ∆dilH values as the critical point of water is approached. Anderko and Pitzer33 argued that parameters a, c, d, and e for pure components should be used for both water and NaCl (liq) at the T and P values of interest. In the present work, the T range of interest is well below the melting point of NaCl. Therefore, pure liquid solute does not exist. Properties of this hypothetical solute, NaCl (liq), are expected to be solvent dependent and are those of NaCl0 (aq) where the solute mole

Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 2931 Table 3. Parameters for the Hypothetical NaCl (liq) Used in the AP33 and RIII Modelsa parameterb

AP model33

aNaCl

{14.412 + 5.644 exp[-0.4817(θ 10 exp[-0.6154(θ - 5.403)3.26]} × 105 -2.7501 8.0969 -9.0 θ ) T/(100 K)

cNaCl dNaCl eNaCl θ

RIII model 8.959)2]

{11.185 + 5.6464 exp[-0.4934(θ - 9.854)2] 9.835 exp[-0.5682(θ - 5.483)1.8613]} × 105 -2.5215 7.7331 -9.0 θ ) T/(100 K)

a Coefficients in the RIII model were regressed using measured solution densities and apparent molar volumes of NaCl (aq),27,38-40 heats of dilution,22,41-43 and apparent molar isobaric heat capacities.44 b Units of aNaCl are MPa‚cm3‚mol-1. The remaining parameters are dimensionless.

Table 4. Interaction Parameters Used in the AP33 and RIII Modelsa parameter

AP model33, b

RIII modelb

R012

-3.0873 + 1.2688θ + + 0.4213 exp[-1.3614(θ - 5.491)2] 2.057 + 0.06552(θ - 9.7315) + 0.2058 exp[-1.604(θ - 5.667)2] 0.7865 + 0.1487θ + 0.2537 exp[-(1.162 × 10-6)(θ - 0.57315)8] 1.306 + 0.091 exp[-1.720(θ - 5.7315)2] 1.389 1.293 (0.10119 + 6.4313 exp[-0.9961(θ - 5.7315)0.79026]) × (1 + 6.1005q - 27.151q2 + 141.3q3 - 136.43q4) (θ - 5.7315)/10

-3.0072 + 1.1539θ - 0.10736θ2 + 0.0037464θ3 + 0.4143 exp[-1.3459(θ - 5.485)2] 1.9567 + 0.07083(θ - 5.7315) + 0.1923 exp[-1.546(θ - 5.667)2] 0.8422 + 0.1244θ + 0.2210 exp[-(1.205 × 10-6)(θ - 0.57315)8] 1.316 + 0.091 exp[-1.5880(θ - 5.7315)2] 1.3602 1.269 (0.12169 + 6.9321 exp[-1.0427(θ - 5.7315)0.70177]) × (1.1025 + 5.5249q - 24.7741q2 + 167.1q3 - 131.89q4) (θ - 5.7315)/10

γ0112 γ0122 δ01112 δ01122 δ01222 τ θ

0.11850θ2

0.0036853θ3

a

Coefficients in the RIII model were regressed using measured solution densities and apparent molar volumes of NaCl (aq),27,38-40 heats of dilution,41-43 and apparent molar isobaric heat capacities.44 b In both models: R011 ) a022 ) γ0111 ) γ0222 ) δ01111 ) δ02222 ) 011111 ) 011112 ) 011122 ) 011222 ) 012222 ) 022222 ) 1

Figure 1. Plot of FNaCl(liq) as a function of T at 28 MPa calculated using the AP33 and RIII models.

fraction (x) is unity. Since the water parameters have been modified in this work, the parameters for NaCl (liq) should also be modified. To better fit the experimental data, the purecomponent NaCl (liq) parameters and the interaction parameters in the perturbation term of the AP model were re-regressed in development of the RIII model. The Aneu term, which includes the perturbation term, in the RIII model represents NaCl0 (aq) and water and their interaction. The effects of NaCl0 (aq), Na+, Cl-, and H2O are implicitly accounted for in the perturbation term of the AP model. The parameters found in the perturbation term regressed in this study are listed in Tables 3 and 4 along with those used by Anderko and Pitzer.33 The parameters in Tables 1 and 3 are those found in the portion of the RIII equation that can be used to calculate the properties of H2O and NaCl (liq), respectively. The interaction between NaCl (liq) and H2O in the mixture can be found using the parameters listed in Table 4. The parameters reported in Tables 3 and 4 were regressed using measured solution F/Vφ,27,38-40 ∆dilH,41-43 and Cp,φ 44 values. In Figure 1, the F values of the hypothetical NaCl (liq) predicted using the AP model are shown together with those calculated using the RIII model as a function of T at 28 MPa. The FNaCl(liq) values calculated using the RIII model are slightly lower than those found using the AP model. In the RIII model, the FNaCl(liq) values are based only on the associated NaCl0 (aq),

Figure 2. Plot of log Km, calculated using Gruszkiewicz and Wood’s equation,34 for the reaction Na+ + Cl- ) NaCl0 (aq) as a function of FH2O at 350, 360, 370, 380, 390, and 400 °C.

whereas in the AP model, the F values are based on the NaCl (aq) in solution, which is composed of both ionized and associated NaCl. The FNaCl(liq) values in the RIII model decrease as T increases, as would be expected for a liquid, whereas FNaCl(liq) values calculated using the AP model increase with increasing T. The ∆ionA term in eq 1 is based on the equation for the logarithm of the equilibrium constant, log Km, written as a function of T and F. In the RI model,19 the ∆ionA term was derived using the same log Km equation as that reported by Gruszkiewicz and Wood,34 who claimed that their equation predicted log Km values more accurately than did the equation developed by Zimmerman et al.15 However, the equation and parameters adopted by Gruszkiewicz and Wood34 were fitted mainly to log Km values at high T and low F values or low T and high F values. At medium or low F and low T values, the log Km values calculated using Gruszkiewicz and Wood’s equation fail to follow reasonable trends. An example of this is shown in Figure 2. At T > Tc, the log Km values calculated using the equation developed by Gruszkiewicz and Wood follow the same trends as those predicted by Chialvo and coworkers.14,45 At T < Tc, unexpected trends in log Km values with F are seen at F values between 0.26 and 0.35 g‚cm-3. The

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trends predicted in this region using the equation of Gruszkiewicz and Wood are not consistent with the general observation that ion association increases as F decreases at constant T.20,46 Also, the derivatives in the equation of Gruszkiewicz and Wood with respect to T or F are not accurate, since H and Vφ values calculated using this equation34 are incorrect. An expression for log KVm (KVm is Km at constant molar volume (V) and at standard state conditions) is given in eq 2:

log Kvm ) -1.7563 +

(

[(

3.2372 -

) )]

753 3.313 × 104 T T2

4.3007 - 1.1839 log Fhyp +

383 2291 + Fhyp F 2 hyp

(2)

Figure 3. Comparison of literature log Km 15,34 values for the reaction Na+ + Cl- ) NaCl0 (aq) as a function of FH2O at 347.28, 379.5, and 400 °C with those predicted using Gruszkiewicz and Wood’s equation and eq 2.

Equation 2 can be used to calculate ∆ionA. The Fhyp term in eq 2 represents the hypothetical density of water in the solution as defined in eq 3,

Fhyp ) (1 - xNaCl(aq))MH2O/Vcalc

(3)

where MH2O is the molecular mass of water and Vcalc is the V value calculated using the RIII model. The moles of water per mole of solution is equal to 1 - xNaCl(aq), because the ∆ionA term is calculated based on 1 mol of solution, assuming that all the NaCl is associated. The ∆ionA term in eq 1 can be obtained using eq 4:

∆ionA ) ln(10) RT log KVm

(4)

Equation 2 gives equilibrium constants at isochoric conditions. These values are also valid for an isobaric process for infinitely dilute solutions at the same T and FH2O values. The equivalence of the two equilibrium constants is illustrated by eqs 5-9 for an ion dissociation reaction at constant P.

(∆ionG)V ) (∆ionG)P +

∫PP VproddP f

(5)

i

In eq 5, the subscript “ion” refers to an ionization process, (∆ionG)V is the change in Gibbs energy at constant V and T, (∆ionG)P is the change in Gibbs energy at constant P and T, Pi is the P value at the start of the reaction, Pf is the P value at the end of the reaction that will make the V of the products equal to that of the reactants, and Vprod is the V of the products of the reaction.

(∆ionA)V ) (∆ionG)V - V(Pf - Pi) ) (∆ionG)P +

∫PP Vprod dP - v(Pf - Pi) f

(6)

i

In eq 6, (∆ionA)V is the change in A at constant V. At infinite dilution in the liquid phase,

∫PP Vprod dP ≈ V(Pf - Pi)

(7)

(∆ionA)V ≈ (∆ionG)P

(8)

log KVm ≈ log KPm

(9)

f

i

Therefore,

and

The value of log Km is dependent only on T and F whether the reaction is isochoric or isobaric. In the following discussion,

Figure 4. Plot of ((∂ log Km)/(∂T))FH2O, calculated using the equation of Gruszkiewicz and Wood34 (- - -) and that used in the RIII model (s), as a function of T at FH2O ) 0.6 g‚cm-3.

log Km will be used without indicating whether it refers to an isochoric or isobaric process. In Figure 3, literature log Km values as a function of FH2O at given T values are compared with those calculated using the equation of Gruszkiewicz and Wood34 and eq 2. At F > 0.4 g‚cm-3, log Km values calculated using both equations agree with the literature data. At F < 0.4 g‚cm-3, log Km values calculated using the equation of Gruszkiewicz and Wood do not follow the expected trend at 347.28 °C. The values calculated using the Gruszkiewicz and Wood equation agree with the literature data slightly better than those calculated using eq 2 at higher T values (e.g., at 379.5 °C). However, the experimental uncertainties at low F values increase as T approaches Tc.34 To avoid the two-phase region or the region where small disturbances in P and/or T cause large changes in fluid properties, Gruszkiewicz and Wood34 measured the equivalent conductances of the solutions at high concentrations where a significant fraction of the ions were associated. As a result, the conductances extrapolated to infinite dilution are not as accurate as those at low T and high F values. For example, at 347.28 °C, 0.6 g‚cm-3, and 1.6118 × 10-5 mol‚kg-1, 99.95% of the NaCl (aq) is dissociated.34 However, at 379.32 °C, 0.2523 g‚cm-3, and 2.0407 × 10-6 mol‚kg-1, only 98.3% of the NaCl (aq) is dissociated.33 Thus, extrapolations based on incomplete dissociation of NaCl underestimate the actual log Km values. Therefore, the slightly higher log Km values predicted using eq 2 are reasonable. The main advantage of eq 2 over that of Gruszkiewicz and Wood34 is that differentiation of eq 2 with respect to T and F gives reasonable derivative values over a wider T and P range. Use of Vφ27,38-40 and ∆dilH values22,41-43 in the development of eq 2 ensures that differentiation of the equation with respect to T and F gives the correct values. In Figures 4 and 5, plots of (∂logKm/∂T)F vs T and (∂logKm/∂FH2O)T vs FH2O, respectively,

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Figure 5. Plot of ((∂ log Km)/((∂FH2O))T, calculated using the equation of Gruszkiewicz and Wood34 and the equation used in the RIII model, as a function of FH2O at 380 °C. Table 5. Parameters Used in Calculating the Average Diameter, σ, in the Myers et al.35 and RIII Models parameter (10-10

σ1 m) σ2 (10-10 m‚K)

Myers et al. model35

RIII model

5.695 -551.3

5.935 -581.7

are given. Derivatives in both cases were calculated using eq 2 and the equation of Gruszkiewicz and Wood.34 The derivatives at any T or FΗ2Ο value are different for the two equations. Thus, ∆dilH values, which depend on the T derivatives, vary according to which equation is used. As seen in Figure 5, (∂logKm/∂FH2O)T values calculated using the two equations agree well at both low and high FH2O values. At medium FH2O values (0.25 g‚cm-3 < F < 0.45 g‚cm-3), however, the derivative values using the equation of Gruszkiewicz and Wood34 change more abruptly than do those found using eq 2. Thus, the Vφ values, which depend on the FH2O derivative, calculated using the equation of Gruszkiewicz and Wood are in error in this region. This result is understandable since few conductance values have been measured in this region. The parameters in eq 2 were regressed using several measured Vφ values of NaCl (aq) at these T and FH2O values, and thus, (∂logKm/∂FH2O)T values found using eq 2 are more accurate. To obtain correct Ares values at finite values of µ, the difference between Ares values for solutions containing ions at µ f 0 and µ > 0 must be taken into account. This difference is calculated using the ∆AMSA term in eq 1 as developed by Myers et al.35 Their original development involved fitting the parameters in this term to data collected at T < 300 °C, where NaCl (aq) can be assumed to be completely dissociated and solution V values are close to those calculated by adding solute and solvent V values. As T increases, the solution becomes more compressible and ion association becomes significant. The ∆AMSA term is a function of T, F, the mean ion diameter (σ), and the dielectric constant () of the medium between the ions. The latter two terms must be estimated, since they cannot be measured directly. Myers et al.35 used eq 10 to estimate the dependence of σ on T.

σ ) σ1 +

σ2 T

(10)

The values of σ1 and σ2 were found by fitting literature activity coefficients for NaCl solutions below 300 °C. In the present study, the parameters, σ1 and σ2, were refitted using measured ∆dilH22,41-43 and Vφ 27,38-40 values valid from 350 to 400 °C. In Table 5, the σ1 and σ2 values used by Myers et al.35 are compared to those regressed in this study. In Figure 6, the average diameters used in the model developed by Myers et

Figure 6. Plot of the mean ion diameter, σ, used in the ∆AMSA term calculated using the model of Myers et al.35 and the RIII model versus T.

al.34 when extrapolated to higher T values and those used in the RIII model are compared. As seen in Table 5, the σ1 and σ2 values from the two studies are similar. The σ values for both cases as shown in Figure 6 increase as T increases. This increase is reasonable since σ refers to the diameters of the solvent cavities around the bare ions and not to the bare ion diameters.35 As T increases, the distance between the bare ions and the neighbor water molecules increases, and thus, the size of the cavities increases with T. The mathematical functions of  are usually formulated as functions of T and F, especially for compressible fluids. Myers et al.35 used FH2O at the T and P values of interest in calculating . At the conditions of the present study, the F value of the solution increases dramatically with solute concentration at any set of T and P values. For example, at 400 °C and 28 MPa, FH2O is 0.259 g‚cm-3, and the calculated solution F value at m ) 0.61 mol‚kg-1, assuming V values are additive, is 0.267 g‚cm-3. The measured solution F at m ) 0.61 mol‚kg-1 is 0.487 g‚cm-3.41 In this work, eq 11, developed by Archer and Wang,47 was used to calculate .

() (

)

B1 B2 + B3 + B4T* F*2 + F* + T* T* B9 B5 B8 + + B6T* + B7T*2 F*3 + + B10 F*4 (11) 2 T* T* T*

)1+

(

) (

)

The F value used in eq 11 is the F of the solution found using the RIII model as explained in the following section. In eq 11, B1-B10 are fitting parameters.47 The T* and F* terms are dimensionless T and F, respectively, and can be calculated using eqs 12 and 13.

T* ) T/(298.15 K) -3

F* ) F/(10 kg‚m ) 3

(12) (13)

3. Model Application 3.1. General Development. Summing the Ares (eq 1) and Helmholtz energy ideal gas (Aig) terms gives the following equation for A,

A ) Ares + Aig

(14)

The Aig term must include the ideal-mixing contribution of the species, ∆mixA. Expressing eq 14 as a sum of the previously discussed terms results in eq 15,

()

V0 + Aneu + V nNa+∆ionA + ∆AMSA + ∆mixA (15)

A ) xH2OAHig02O + xNaCl(aq)Aig0 NaCl + RT ln

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where A is the Helmholtz energy per mole of [H2O + NaCl (aq)]; xH2O and xNaCl(aq) are the x values for water and NaCl (aq), respectively, assuming all NaCl is associated; AHig02O and Aig0 NaCl are the ideal gas A values of water and NaCl, respectively, at the reference V value, V0 (V0 ) 1 m3‚mol-1); and ∆mixA is defined by eq 16,

∆mixA ) RT

ni

∑ni ln1 + nNa

(16) +

where ni is the number of moles of component i [i ) H2O, Na+, Cl-, or NaCl0 (aq)]. The first two terms on the right-hand side of eq 15 can be found using eq 17,

()

T - R(T - T0) + 0 T ig T ig T Cp (T) dT (17) C (T) dT T T0 p T0 T

xH2OAHig02O + xNaClAig0 NaCl ) RT ln

∫

∫

where T0 is the standard state reference T (273.15 K) and Cig p (T) is the molar ideal gas heat capacity of the H2O + NaCl mixture as defined by eq 18, ig ig Cig p (T) ) xNaCl(aq)Cp,NaCl(T) + (1 - xNaCl(aq))Cp,H2O(T) (18) ig ig where Cp,NaCl and Cp,H are the molar isobaric ideal gas heat 2O capacity values for NaCl and H2O, respectively, and can be found using the DIPPR database.48 The computation was simplified by using quadratic expressions to fit the experimental ig ig and Cp,H values over the range 570-680 K. The Cp,NaCl 2O resulting regressed equations are

ig (T) Cp,NaCl

J‚mol-1‚K-1 ig Cp,H (T) 2O

J‚mol-1‚K-1

) 34.4138 + (6.86566 × 10-3)T (3.43249 × 10-6)T2 (19) ) 30.419 + (8.3037 × 10-3)T + (2.59597 × 10-6)T2 (20)

The remaining terms in eq 15 have been defined earlier. All thermodynamic properties of the solution can now be found using eq 15 and its derivatives with respect to T and F. Examples are presented below showing the use of eq 15 to calculate values for several thermodynamic properties and the agreement of these values with literature values. 3.2. Density and Speciation. To find the F value of the solution at a given T, P, and m, an iterative scheme was used. More details of this iteration scheme are available.49 The iteration scheme is based on determining the compressibility factor (Z ) PV/RT) instead of F. Equations 21-23 are used in the iteration scheme.

P)

F2 ∂A MH2O ∂F T

( )

(21)

F ∂A RT ∂F T

(22)

Z)

( )

A(Na+) ) minimum

(23)

Expressions for calculating the derivatives are available.49 It is necessary to determine the amount of NaCl (aq) that is

Figure 7. Comparison of literature F values27,38-40 with those calculated using the AP33 and RIII models. Some measured F values were converted from measured Vφ values27,38 and measured FH2O values.37

dissociated when determining F, since eqs 21-23 must be solved simultaneously. In Figure 7, F values found using the AP33 and RIII models are compared with literature values.27,38-40 Perfect agreement between the model predictions and the literature values is represented by the 45° line. Good agreement is found for both models at high and low F values. It is expected that good agreement between the F values calculated using the AP model and the literature values will be found at low F values for either of two reasons. First, if there is appreciable NaCl (aq) present, the fraction associated is high. Second, most of the low F values are for low NaCl (aq) concentrations, and the F value is close to that of water. It is also expected that the F values calculated using the AP model will give good agreement with literature values at high F values. All of the data at high F values were collected at high solute concentrations, where a large fraction of the NaCl (aq) is associated. In intermediate regions, agreement between results calculated using the AP model and literature values is poorer, since an appreciable fraction of the NaCl (aq) is dissociated. Good agreement is found between the F values calculated using the RIII model and those from the literature over the entire range of F values, since the RIII model contains terms for the Na+ and Cl- ions. Once the F value and fraction NaCl (aq) dissociated are known, incorporation of these values allows the calculation of other thermodynamic properties as shown below. The extent of the agreement between calculated and measured values is an excellent test of the validity of the RIII model. 3.3. Apparent Molar Volume of NaCl (aq). A quantity related to F is Vφ. This quantity is defined by eq 24,

Vφ )

1 + mMNaCl 1 F F H2 O m

(24)

where MNaCl is the molecular mass of NaCl. The Vφ term describes the effect of solute on the solution V. In Figure 8, Vφ values as a function of P measured by Majer et al. at 378 °C38 are compared with those calculated using the AP, RI, and RIII models. All Vφ values are negative, as would be expected since the charged ions attract water molecules. The Vφ values become more negative as P decreases, suggesting that the effect of solute is more significant at low P values. This is reasonable since the water molecules are already relatively close together at high P values and, thus, an increase of ion concentration in the solution does not result in as large a contraction in V as would be the case at low P values where the water molecules are relatively far apart. Values calculated using the RIII model are in excellent agreement with literature values38 at both m ) 3.1 mol‚kg-1 and m ) 0.005 mol‚kg-1. The RI model contains a

Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 2935

Figure 8. Plot of calculated (using the indicated models) and literature38 Vφ values as a function of P at 378 °C and at (a) m ) 3.1 mol‚kg-1 and (b) m ) 0.005 mol‚kg-1.

Figure 10. Comparison of literature ∆dilH values43 as a function of m at 370 °C and 24.7 MPa with those calculated using the AP,33 RI,19 and RIII models.

Figure 9. Comparison of literature Vφ values38 as a function of m0.5 at 400.03 °C and 28 MPa with those calculated using the AP33 and RIII models.

Figure 11. Comparison of literature ∆dilH values43 as a function of m at 380 °C and 24.7 MPa with those calculated using the AP,33 RI,19 and RIII models.

term for ion association but does not contain a term for the effect of ion concentration. As a result, the agreement between calculated and experimental values38 using the RI model is not as good as that found using the RIII model. At 3.1 mol‚kg-1, the values found using the AP model33 are in good agreement with the literature values.38 At 0.005 mol‚kg-1, the -Vφ values calculated using the AP model are much smaller than the literature values.38 The reason for this result is that the AP model does not incorporate NaCl (aq) dissociation. At 0.005 mol‚kg-1, a larger fraction of the NaCl (aq) is dissociated than at 3.1 mol‚kg-1; therefore, the assumption in the AP model33 that all the NaCl (aq) is associated is more accurate at the higher concentration. In Figure 9, literature Vφ values are compared with those calculated using the AP33 and RIII models at T and P values where water has a high isothermal compressibility. The Vφ term becomes more negative as m decreases, since the compressibility of the solution is greater at low than at high m values. At high solute concentrations, Vφ values calculated using both models agree with literature Vφ values, but at low solute concentrations, those calculated using the RIII model are in better agreement with the experimental data. 3.4. Heats of Dilution. Solution H values must be known in order to calculate ∆dilH values. These H values can be obtained by first calculating molar internal energy (U) using eq 25.

U)

( ( )) ∂ A ∂(1/T) T

F

(25)

Once U is obtained, H can be found from eq 26.

H ) U + PV

(26)

Details of the differentiation of A (eq 15) with respect to T are available.49

Agreement between calculated and measured ∆dilH values provides a stringent test for model validity. Values of ∆dilH can be calculated from solution H values determined using the RIII model and eq 27

∆dilH )

(n + 1)Hcalc - Hcalc - nHHcalc f i 2O xNaCl(aq),i

(27)

where the subscripts f and i refer to the quantities after and before dilution, respectively. The value of n (number of moles of water added) can be calculated using eq 28.

n)

xNaCl(aq),i xNaCl(aq),f

-1

(28)

Substitution of eq 28 into eq 27 and rearranging gives

∆dilH )

Hcalc - HHcalc f 2O xNaCl(aq),f

-

Hcalc - HHcalc i 2O xNaCl(aq),i

(29)

Comparison of ∆dilH values calculated using the RIII model, the RI model, and the AP33 model with those measured calorimetrically is shown in Figures 10-12. As shown in Figures 10 and 11, the ∆dilH values calculated using the RIII model agree better with the literature values43 than those calculated with either the AP33 or RI model at the given conditions. Similar plots are available49 that show that, at some conditions, the values calculated using the AP model33 are in better agreement with literature values than are those calculated using the RI model.19 However, at all conditions, values calculated using the RIII model are in better agreement with the literature values22,41-43 than are those calculated using either the RI19 or AP33 model. This is shown in Figure 12. A plot

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Figure 12. Plot of ∆dilH values calculated using AP,33 RI,19 and RIII models vs literature ∆dilH values22,41-43 for aqueous NaCl solutions. The 45° line represents perfect agreement.

Figure 14. Comparison of literature Cp,φ values of NaCl (aq)49 as a function of m0.5 at 350 °C and 20 MPa with those predicted using the Archer model51 and the RIII model.

Figure 13. Comparison of literature Cp,φ values of NaCl (aq)50 as a function of m0.5 at 350 °C and 30.3 MPa with those predicted using the Archer model51 and the RIII model.

Figure 15. Plot of literature44 Cp,φ values and those calculated using the RIII model as a function of T at 32 MPa and 0.015 mol‚kg-1.

similar to that in Figure 12 was reported earlier by us.20 However, additional data and values calculated using the RIII model are shown in Figure 12. The good agreement between calculated and literature ∆dilH values indicates that the first derivative of the RIII model with respect to T is essentially correct. 3.5. Apparent Isobaric Molar Heat Capacities. The apparent isobaric molar heat capacity, Cp,φ, is defined by eq 30,

Cp,φ )

ˆ p,H2O C ˆ p(m) - C m

(30)

where C ˆ p,H2O is the heat capacity of 1 kg of water and C ˆ p(m) is the heat capacity of the solution per kilogram of water at the same T and P. It follows that calculation of Cp,φ requires calculation of the second derivative of A with respect to T. Values of Cp,φ calculated by Hnedkovsky´ et al.50 using Archer’s model51 and by us using the RIII model are compared with literature50 values as a function of m at 350 °C and at 30.3 MPa (Figure 13) and 20 MPa (Figure 14). At 30.3 MPa, the agreement of the values calculated using the RIII model with the literature data is excellent while the agreement of those calculated using the Archer model51 is poor. However, this T value is 23 °C above the upper T limit of Archer’s model and indicates that his model does not yield reliable results when extrapolated to higher T and P values. Hnedkovsky´ et al.50 proposed that the poor agreement between the values calculated using Archer’s model51 and literature values at 350 °C was a result of Archer basing his calculations primarily on Cp,φ and other values measured at T values C ˆ p(m) and is positive in the region where C ˆ p(m) > C ˆ p,H2O. The general agreement between the literature and calculated Cp,φ values in the region of rapid transition indicates that the equations for water and for the solution are valid in the range of 350-450 °C. Values of Cp,φ are accurately predicted up to at least 450 °C, even though the model was correlated using data valid only up to 400 °C. This is impressive, especially since the second derivatives of A with respect to T are used to calculate Cp,φ. The difference between the predicted and literature Cp,φ values close to 410 °C are larger than those at other T values. This is not unexpected since solution properties change dramatically in this region with small changes in T, P, and/or m. Both literature values and model predictions are likely to have relatively large uncertainties in this region. In Figures 17-19 are shown plots of literature52 and calculated Cp,φ values as a function of m0.5 at various T and P values. At medium and high m values, the agreement between

AP ) Anderko Pitzer MSA ) mean spherical approximation35 MULTEQ ) multiple equilibrium (code to calculate equilibrium distribution of impurities in crevices of steam generators) NIST ) National Institute of Standards and Technology DIPPR ) Design Institute for Physical Properties NaCl Species NaCl (aq) ) total NaCl in solution whether associated or dissociated NaCl0 (aq) ) associated NaCl in solution NaCl (liq) ) hypothetical pure liquid NaCl at the temperature and pressure of the system Roman A ) Helmholtz energy, J‚mol-1 A(Na+) ) A as a function of Na+, J‚mol-1 Aig ) ideal gas A, J‚mol-1 AHig02O ) Aig for H2O at standard state, J‚mol-1 ig -1 Aig0 NaCl ) A for NaCl at standard state, J‚mol neu -1 A ) A for neutral species, J‚mol Ares ) residual A, J‚mol-1 a ) pure-fluid parameter in perturbation term, MPa‚cm6‚mol-2 aH2O ) a for H2O, MPa‚cm6‚mol-2

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aNaCl ) a for NaCl, MPa‚cm6‚mol-2 b ) pure-fluid parameter in perturbation term, cm3‚mol-1 B1-B10 ) fitting parameters used in eq 11 c ) pure-fluid parameter in perturbation term cH2O ) c for H2O cNaCl ) c for NaCl Cp ) molar isobaric heat capacity, J‚mol-1‚K-1 Cig p ) Cp for ideal gas mixture as defined in eq 18, J‚mol-1‚K-1 ig ig -1 -1 Cp,H O ) Cp for H2O, J‚mol ‚K ig 2 -1‚K-1 Cp,NaCl ) Cig for NaCl, J‚mol p Cp,φ ) apparent Cp of NaCl (aq), defined in eq 30, J‚(mol of NaCl (aq))-1‚K-1 C ˆ p ) heat capacity per kg of water, J‚(kg of H2O)-1‚K-1 C ˆ p(m) ) C ˆ p of solution as a function of m, J‚(kg of H2O)-1‚K-1 C ˆ p,H2O ) C ˆ p of water, J‚kg-1‚K-1 d ) pure-fluid parameter in perturbation term dH2O ) d for H2O dNaCl ) d for NaCl e ) pure-fluid parameter in perturbation term eH2O ) e for H2O eNaCl ) e for NaCl G ) Gibbs energy, J‚mol-1 Gex ) excess G, J‚mol-1 H ) enthalpy, J‚mol-1 Hcalc ) H calculated for solution after dilution, J‚mol-1 f calc HH2O ) H calculated for H2O, J‚mol-1 Hcalc ) H calculated for solution before dilution, J‚mol-1 i log Km ) logarithm of the equilibrium constant for the reaction Na+ + Cl- ) NaCl0 (aq) at standard state log KPm ) logarithm of the equilibrium constant for the reaction Na+ + Cl- ) NaCl0 (aq) at constant P at standard state log Kvm ) logarithm of the equilibrium constant for the reaction Na+ + Cl- ) NaCl0 (aq) at constant V at standard state m ) molality, mol‚kg-1 MH2O ) molecular mass of H2O, kg‚mol-1 MNaCl ) molecular mass of NaCl, kg‚mol-1 n ) moles of water added during dilution, mol ni ) moles of component i, mol nNa+ ) moles of Na+ in solution, mol P ) pressure, MPa Pc ) critical pressure, MPa Pi ) P prior to reaction, MPa Pf ) P at end of reaction, MPa q ) parameter used in equation for τ as defined in Table 4 R ) ideal gas constant, J‚mol-1‚K-1 RI ) name for model from ref 19 RII ) name for model from ref 20 RIII ) name for model from this work T ) temperature, K or °C Tc ) critical temperature, K T0 ) standard state reference T, K Tr ) reduced temperature, T‚Tc-1 T* ) dimensionless T, defined by eq 12 U ) molar internal energy, J‚mol-1 V ) molar volume, cm3‚mol-1 V0 ) standard state V of 1 m3‚mol-1, m3‚mol-1 Vcalc ) V value calculated using RIII model, cm3‚mol-1 Vprod ) V of the products of reaction, cm3‚mol-1 Vφ ) apparent V of NaCl (aq), cm3‚(mol of NaCl(aq))-1 x ) mole fraction xH2O ) x of H2O

xNaCl(aq) ) x of NaCl (aq) xNaCl(aq),f ) xNaCl after dilution xNaCl(aq),i ) xNaCl before dilution Z ) compressibility factor, PV/RT Greek R0ij ) interaction parameter in perturbation term γ0ijk ) interaction parameter in perturbation term δ0ijkl ) interaction parameter in perturbation term ∆AMSA ) change in A due to changes in ion concentration as predicted by the MSA equation,35 J‚mol-1 ∆dilH ) change of H upon dilution of NaCl, J‚(mol of NaCl(aq))-1 ∆ionA ) change in A upon NaCl0 (aq) ionization at infinite dilution, J‚mol-1 (∆ionA)V ) change in A for ionization reaction at constant V and T, J‚mol-1 (∆ionG)P ) change in G for ionization reaction at constant P and T, J‚mol-1 (∆ionG)V ) change in G for ionization reaction at constant V and T, J‚mol-1 ∆mixA ) change in A upon mixing, J‚mol-1  ) dielectric constant 0ijklm ) interaction parameter in perturbation term Λ ) description of shape of curve µ ) ionic strength, (mol of ions)‚(kg of H2O)-1 F ) density, g‚cm-3 FH2O ) F for H2O, g‚cm-3 Fhyp ) hypothetical F for liquid NaCl, g‚cm-3 FNaCl(liq) ) F for NaCl, g‚cm-3 F* ) dimensionless F defined by eq 13 σ ) mean ion diameter, m σ1 ) parameter in eq 10, m σ2 ) parameter in eq 10, m‚K θ ) dimensionless T‚(100 K)-1 τ ) interaction parameter in perturbation term Literature Cited (1) Oscarson, J. L.; Christensen, J. J.; Izatt, R. M. Development and Uses of the EPRI Program (MULTEQ) for Predicting Concentrations and Partitioning of Chemical Species in High-Temperature Aqueous Solutions. In Proceedings: Symposium on Chemistry in High-Temperature Water, Provo, UT, August 1987; EPRI Report NP-6005; Electric Power Research Institute: Palo Alto, CA, March 1990; Paper 4b. (2) Mid-Ocean Ridges: Hydrothermal Interactions between the Lithosphere and Oceans; German, C. R., Lin, J., Parson, L. M., Eds.; Geophysical Monograph 148; American Geophysical Union: Washington, DC, 2004. (3) Shaw, R. W.; Brill, T. B.; Clifford, A. A.; Eckert, C. A.; Franck, E. U. Supercritical Water, A Medium for Chemistry. Chem. Eng. News 1991, 69, 26-39. (4) Tester, J. W.; Holgate, H. R.; Armellini, F. J.; Webley, P. A.; Killilea, W. R.; Hong, G. T.; Barner, H. E. Supercritical Water Oxidation Technology. Process Development and Fundamental Research. In Emerging Technologies in Hazardous Waste Management III; Tedder, D. W., Pohland, F. G., Eds.; American Chemical Society: Washington, DC, 1993. (5) Gloyna, E. F.; Li, L. Supercritical Water Oxidation Research and Development Update. EnViron. Prog. 1995, 14, 182-192. (6) Modell, M. Supercritical-Water Oxidation. In Standard Handbook of Hazardous Waste Treatment and Disposal; Freeman, H. M., Ed.; McGraw-Hill Book Co.: New York, 1989; pp 8.153-8.168. (7) Griffith, J. W. Design and Operation of the First Supercritical Wet Oxidation Industrial Waste Destruction Facility, 1995. Chem. Oxid. 1997, 5, 22-38. (8) Minett, S.; Fenwick, K. Supercritical Water OxidationsThe Environmental Answer to Organic Waste Disposal? Eur. Water Manage. 2001, 4, 54-56. (9) Moore, R. C.; Mesmer, R. E.; Simonson, J. M. Solubility of Potassium Carbonate in Water between 384 and 829 K Measured Using the Synthetic Method. J. Chem. Eng. Data 1997, 42, 1078-1081.

Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 2939 (10) Armellini, F. J.; Tester, J. W. Solubility of Sodium Chloride and Sulfate in Sub- and Supercritical Water Vapor from 450 to 550 °C and 100-250 Bar. Fluid Phase Equilib. 1993, 84, 123-142. (11) Kettler, R. M.; Wesolowski, D. J.; Palmer, D. A. Dissociation Quotient of Benzoic Acid in Aqueous Sodium Chloride Media to 250 °C. J. Solution Chem. 1995, 24, 385-407. (12) Lvov, S. N.; Zhou, X. Y.; Ulyanov, S. M.; Macdonald, D. D. Potentiometric Measurements of Association Constants and pH in HighTemperature HCl(aq) Solutions. Power Plant Chem. 2000, 2, 5-8. (13) Ho, P. C.; Palmer, D. A.; Gruszkiewicz, M. S. Conductivity Measurements of Dilute Aqueous HCl Solutions to High Temperatures and Pressures Using a Flow-through Cell. J. Phys. Chem. B 2001, 105, 12601266. (14) Chialvo, A. A.; Cummings, P. T.; Simonson, J. M. H3O+/Cl- Ion Pairing in Hydrothermal Solutions by Simulation and Electrical Conductance. A Review. J. Mol. Liq. 2003, 103-104, 235-248. (15) Zimmerman, G. H.; Wood, R.H. Conductance of Dilute Sodium Acetate Solutions to 469 K and of Acetic Acid and Sodium Acetate/Acetic Acid Mixtures to 548 K and 20 MPa. J. Solution Chem. 2002, 31, 9951017. (16) Holmes, H. F.; Busey, R. H.; Simonson, J. M.; Mesmer, R. E.; Archer, D. G.; Wood, R. H. The Enthalpy of Dilution of HCl(aq) to 648 K and 40 MPa. J. Chem. Thermodyn. 1987, 19, 863-890. (17) Inglese, A.; Sedlbauer, J.; Wood, R. H. Apparent Molar Heat Capacities of Aqueous Solutions of Acetic, Propanoic and Succinic Acids, Sodium Acetate and Sodium Propanoate from 300 to 525 K and a Pressure of 28 MPa. J. Solution Chem. 1996, 25, 849-864. (18) Oscarson, J. L.; Gillespie, S. E.; Chen, X.; Schuck, P. C.; Izatt, R. M. Enthalpies of Dilution of Aqueous Solutions of HCl, MgCl2, CaCl2, and BaCl2 at 300, 325, and 350 °C. J. Solution Chem. 2001, 30, 31-53. (19) Oscarson, J. L.; Palmer, B. A.; Fuangswasdi, S.; Izatt, R. M. A New Model Incorporating Ion Dissociation for Sodium Chloride Solutions Near the Critical Point of Water. Ind. Eng. Chem. Res. 2001, 40, 21762182. (20) Oscarson, J. L.; Liu, B.; Izatt, R. M. A Model Incorporating Ion Dissociation, Solute Concentration, and Solution Density Effects to Describe the Thermodynamics of Aqueous Sodium Chloride Solutions in the Critical Region of Water. Ind. Eng. Chem. Res. 2004, 43, 7635-7646. (21) Gillespie, S. E.; Chen, X.; Oscarson, J. L.; Izatt, R. M. Enthalpies of Dilution of Aqueous Solutions of LiCl, KCl, and CsCl at 300, 325, and 350 °C. J. Solution Chem. 1997, 26, 47-61. (22) Busey, R. H.; Holmes, H. F.; Mesmer, R.E. The Enthalpy of Dilution of Aqueous Sodium Chloride to 673 K Using a New Heat-Flow and Liquid-Flow Microcalorimeter. Excess Thermodynamic Properties and Their Pressure Coefficients. J. Chem. Thermodyn. 1984, 16, 343-372. (23) Holmes, H. F.; Baes, C. F., Jr. Isopiestic Studies of Aqueous Solutions at Elevated Temperatures. I. KCl, CaCl2, and MgCl2. J. Chem. Thermodyn. 1978, 19, 983-996. (24) Bennett, G. E.; Johnston, K. P. UV-Visible Spectroscopy of Organic Probes in Supercritical Water. J. Phys. Chem. 1994, 98, 441447. (25) Xiang, T.; Johnston, K. P. Spectroscopic Measurement of pH in Aqueous Sulfuric Acid and Ammonia from Sub- to Supercritical Conditions. Ind. Eng. Chem. Res. 1996, 35, 4788-4795. (26) Ratcliffe, C. I.; Irish, D. E. Vibrational Spectral Studies of Solutions at Elevated Temperatures and Pressures. VII. Raman Spectra and Dissociation of Nitric Acid. Can. J. Chem. 1985, 63, 3521-3525. (27) Crovetto, R.; Lvov, S. N.; Wood, R. H. Vapor Pressures and Densities of NaCl (aq) and KCl (aq) at the Temperature 623 K and CaCl2 (aq) at the Temperatures 623 and 643 K. J. Chem. Thermodyn. 1993, 25, 127-138. (28) Gates, J. A.; Wood, R. H. Density and Apparent Molar Volume of Aqueous CaCl2 at 323-600 K. J. Chem. Eng. Data 1989, 34, 53-56. (29) Sharygin, A. V.; Wood, R. H. Densities of Aqueous Solutions of Sodium Carbonate and Sodium Bicarbonate at Temperatures from 298 to 623 K and Pressures to 28 MPa. J. Chem. Thermodyn. 1998, 30, 15551570. (30) Knight, C. L.; Bodnar, R. J. Synthetic Fluid Inclusions. IX. Critical PVTX Properties of NaCl-H2O Solutions. Geochim. Cosmochim. Acta 1989, 53, 3-8. (31) Levelt Sengers, J. M. H.; Harvey, A. H.; Crovetto, R.; Gallagher, J. S. Standard States and Finite-Concentration Effects in Near Critical Mixtures with Applications to Aqueous Solutions. Fluid Phase Equilib. 1992, 84, 85-107.

(32) Gallagher, J. S.; Friend, D. G.; Given, J. A.; Levelt Sengers, J. M. H. Critical Lines for Type III Aqueous Mixtures by Generalized Corresponding-States Models. Int. J. Thermophys. 1994, 6, 1271-1278. (33) Anderko, A.; Pitzer, K. S. Equation-of-State Representation of Phase-Equilibria and Volumetric Properties of the System NaCl-H2O above 573-K. Geochim. Cosmochim. Acta 1993, 57, 1657-1680. (34) Gruszkiewicz, M. S.; Wood, R. H. Conductance of Dilute LiCl, NaCl, NaBr, and CsBr Solutions in Supercritical Water Using a Flow Conductance Cell. J. Phys. Chem. B 1997, 101, 6549-6559. (35) Myers, J. A.; Sandler, S. I.; Wood, R. H. An Equation of State for Electrolyte Solutions Covering Wide Ranges of Temperature, Pressure, and Composition. Ind. Eng. Chem. Res. 2002, 41, 3282-3297. (36) Anderko, A.; Pitzer, K. S. Equation of State for Pure Fluids and Mixtures Based on a Truncated Virial Expansion. AIChE J. 1991, 37, 13791391. (37) Lemmon, E. W.; McLinden, M. O.; Friend, D. G. Thermophysical Properties of Fluid Systems. In NIST Chemistry WebBook, NIST Standard Reference Database Number 69; Linstrom, P. J., Mallard, W. G., Eds.; National Institute of Standards and Technology: Gaithersburg, MD, June 2005; http://webbook.nist.gov. (38) Majer, V.; Hui, L.; Crovetto, R.; Wood, R. H. Volumetric Properties of Aqueous 1-1 Electrolyte-Solutions near and above the Critical-Temperature of Water. 1. Densities and Apparent Molar Volumes of NaCl (aq) from 0.0025 mol‚kg-1 to 3.1 mol‚kg-1, 604.4 K to 725.5 K, and 18.5 MPa to 38.0 MPa. J. Chem. Thermodyn. 1991, 23, 213-229. (39) Bischoff, J. L. Densities of Liquids and Vapors in Boiling NaClH2O SolutionssA PVTx Summary from 300 °C to 500 °C. Am. J. Sci. 1991, 291, 309-338. (40) Urusova, M. A. Volumetric Properties of Sodium Chloride Aqueous Solutions at Elevated Temperatures and Pressures. Zh. Neorg. Khim. 1975, 20, 3103-10. (41) Oscarson, J. L.; Peterson, C. J.; Izatt, R. M. Unpublished results, this laboratory. (42) Chen, X.; Oscarson, J. L.; Cao, H.; Gillespie, S. E.; Izatt, R. M. A New Flow Calorimeter Designed for Operation to 450 °C and 50 MPa. Thermochim. Acta 1996, 285, 11-23. (43) Fuangswasdi, S.; Oscarson, J. L.; Zhou, L.; Izatt, R. M. A New Flow Calorimeter Using a Eutectic Molten Salt as the Temperature Control Medium. Thermochim. Acta 2001, 373, 13-22. (44) White, D. E.; Wood, R. H.; Biggerstaff, D. R. Heat Capacities of 0.0150 mol‚kg-1 NaCl(aq) from 604 to 718 K at 32 MPa. J. Chem. Thermodyn. 1988, 20, 159-168. (45) Chialvo, A. A.; Cummings, P. T.; Cochran, H. D.; Simonson, J. M.; Mesmer, R. E. Na+-Cl- Ion Pair Association in Supercritical Water. J. Chem. Phys. 1995, 103, 9379-9387. (46) Marshall, W. L. Complete Equilibrium Constants, Electrolyte Equilibria and Reaction Rates. J. Phys. Chem. 1970, 74, 346-355. (47) Archer, D. G.; Wang, P. The Dielectric Constant of Water and Debye-Hu¨ckel Limiting Law Slopes. J. Phys. Chem. Ref. Data 1990, 19, 371-411. (48) Rowley, R. L.; Wilding, W. V.; Oscarson, J. L.; Yang, Y. DIPPR Data Compilation of Pure Chemical Properties; Design Institute for Physical Properties, Brigham Young University: Provo, UT, 2005; http:// dippr.byu.edu. (49) Liu, B. Properties Model for Aqueous Sodium Chloride Solutions near the Critical Point of Water. Ph.D. Dissertation, Brigham Young University, Provo, UT, 2005. (50) Hnedkovsky´, L.; Hynek, V.; Majer, V.; Wood, R. H. A New Version of Differential Flow Heat Capacity Calorimeter; Tests of Heat Loss Corrections and Heat Capacities of Aqueous NaCl from T ) 300 K to T ) 623 K. J. Chem. Thermodyn. 2002, 34, 755-782. (51) Archer, D. G. Thermodynamic Properties of the NaCl + H2O System. II. Thermodynamic Properties of NaCl(aq), NaCl‚2H2O(cr), and Phase Equilibria. J. Phys. Chem. Ref. Data 1992, 793-829. (52) Carter, R. W. Apparent Molar Heat Capacities at Constant Pressure of 1:1 Aqueous Electrolytes from 598 to 691 K and 28 to 38 MPa. Ph.D. Dissertation, University of Delaware, Newark, DE, 1992.

ReceiVed for reView October 17, 2005 ReVised manuscript receiVed February 8, 2006 Accepted March 8, 2006 IE0511579