Equations for calculating hydrogeochemical reactions of ...leo/Modelagem_Hidrogeoquimica/... · of the pressure-dependent speciation in a solution. If the CO 2 pressure is maintained

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Geochimica et Cosmochimica Acta 125 (2014) 49–67

Equations for calculating hydrogeochemical reactions ofminerals and gases such as CO2 at high pressures and temperatures

C.A.J. Appelo a,⇑, D.L. Parkhurst b, V.E.A. Post c

a Valeriusstraat 11, 1071 MB Amsterdam, The Netherlandsb U.S. Geological Survey, P.O. Box 25046, MS 413, Denver, CO 80225, United States

c School of the Environment/National Centre for Groundwater Research and Training, Flinders University, GPO Box 2100,

Adelaide, SA 5001, Australia

Received 3 October 2012; accepted in revised form 1 October 2013; available online 19 October 2013

Abstract

Calculating the solubility of gases and minerals at the high pressures of carbon capture and storage in geological reservoirsrequires an accurate description of the molar volumes of aqueous species and the fugacity coefficients of gases. Existing meth-ods for calculating the molar volumes of aqueous species are limited to a specific concentration matrix (often seawater), havebeen fit for a limited temperature (below 60 �C) or pressure range, apply only at infinite dilution, or are defined for saltsinstead of individual ions. A more general and reliable calculation of apparent molar volumes of single ions is presented,based on a modified Redlich–Rosenfeld equation. The modifications consist of (1) using the Born equation to calculatethe temperature dependence of the intrinsic volumes, following Helgeson–Kirkham–Flowers (HKF), but with Bradley andPitzer’s expression for the dielectric permittivity of water, (2) using the pressure dependence of the extended Debye–Huckelequation to constrain the limiting slope of the molar volume with ionic strength, and (3) adopting the convention that theproton has zero volume at all ionic strengths, temperatures and pressures. The modifications substantially reduce the numberof fitting parameters, while maintaining or even extending the range of temperature and pressure over which molar volumescan be accurately estimated. The coefficients in the HKF-modified-Redlich–Rosenfeld equation were fitted by least-squares onmeasured solution densities.

The limiting volume and attraction factor in the Van der Waals equation of state can be estimated with the Peng–Robinsonapproach from the critical temperature, pressure, and acentric factor of a gas. The Van der Waals equation can then be usedto determine the fugacity coefficients for pure gases and gases in a mixture, and the solubility of the gas can be calculated fromthe fugacity, the molar volume in aqueous solution, and the equilibrium constant. The coefficients for the Peng–Robinsonequations are readily available in the literature.

The required equations have been implemented in PHREEQC, version 3, and the parameters for calculating the partialmolar volumes and fugacity coefficients have been added to the databases that are distributed with PHREEQC. The easeof use and power of the formulation are illustrated by calculating the solubility of CO2 at high pressures and temperatures,and comparing with well-known examples from the geochemical literature. The equations and parameterizations are suitablefor wide application in hydrogeochemical systems, especially in the field of carbon capture and storage.� 2013 Elsevier Ltd. All rights reserved.

0016-7037/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.gca.2013.10.003

⇑ Corresponding author. Tel.: +31 206716366.E-mail addresses: [email protected] (C.A.J. Appelo), [email protected] (D.L. Parkhurst), [email protected]

(V.E.A. Post).


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50 C.A.J. Appelo et al. / Geochimica et Cosmochimica Acta 125 (2014) 49–67

1. INTRODUCTION

Carbon capture and storage (CCS) is currently beinginvestigated and tested as a mitigation strategy to reducethe release of CO2 produced by the combustion of fossilfuels (Benson and Cole, 2008; Kharaka and Cole, 2011;Wilcox, 2012). For calculating the geochemical reactionsin CCS systems in deep aquifers, the solubility of CO2

and minerals in the reservoir must be known as a functionof the temperature and the gas pressure (Gaus et al., 2005;Cantucci et al., 2009). The pressure dependence is a func-tion of the volume change of the reaction, and thus, the mo-lar volumes of aqueous species are required for itscalculation. In addition, at pressures above 20 atm, the sol-ubility of CO2 deviates from Henry’s law because the fugac-ity coefficient of CO2 gas decreases markedly. It is ourpurpose here to obtain equations that allow an efficientcomputation of the pressure dependence of hydrogeochem-ical reactions up to 200 �C and 1000 atm, and perhapsbeyond.

The high pressures in CCS change the solubility of min-erals and acid dissociation constants substantially. Forexample, going from 1 to 500 bars (at 25 �C), the solubilitiesof sulfates like gypsum and celestite, and of carbonates likecalcite increase by a factor of about 1.7 (Blount and Dick-son, 1973; Macdonald and North, 1974; Howell et al.,1992), and the log Ks of dissociation reactions for H2CO3

and HCO3� increase by 0.2 and 0.4 units, respectively

(Ellis, 1959; Millero, 1983). However, the effect of pressureon hydrochemical reactions is usually assumed to be smallin groundwater systems, and in CCS models it is either ne-glected, or possibly accounted for by adapting equilibriumconstants to a higher, but constant, pressure (Møller et al.,1998; Xu et al., 2006; Kharaka et al., 2009).

The pressure dependence of solubility is also a functionof the pressure-dependent speciation in a solution. If theCO2 pressure is maintained at 1 atm, the calcite solubilityincreases only by 1.2 (instead of 1.7) when the total pressureincreases from 1 to 500 bars, because HCO3

� then becomesthe dominant species, with an aqueous volume that is muchlarger than of CO3

2�. Thus, pressure should be included asa variable in speciation and solubility calculations just liketemperature effects. Pressure effects on hydrogeochemicalreactions are often explicitly accounted for in oceanography(Millero, 1983; Monnin, 1999; Monnin et al., 2003) and inhydrothermal systems (Ellis and Mahon, 1964; Helgesonet al., 1981), and always depend on the molar volumes ofthe solute species.

The apparent molar volume of a dissolved salt can becalculated from the measured densities of the pure solventand the solution (Ellis, 1966; Millero, 1971). Conventionalvolumes for single ions are defined relative to a standard,by setting the volume of one of the ions (most commonlyH+) to zero (Millero, 1971, 1972). The resulting volumesof aqueous species are a function of temperature, ionicstrength, and pressure (Millero, 1983). Millero (1983,2000) presented polynomials for calculating the molar vol-umes of ions in water as a function of temperature and ionicstrength. The polynomials used by Millero (1983, 2000) givethe temperature dependence of the terms in the

Redlich–Rosenfeld equation (Redlich and Rosenfeld,1931; Redlich and Meyer, 1964), which expresses the molarvolume as a tripartite function of the intrinsic molar vol-ume (the molar volume at infinite dilution), and the changesin volume when going from low to high concentrations. Atlow concentrations, the molar volume changes with thesquare root of the ionic strength, according to the“Debye–Huckel limiting slope”, which results from thepressure derivative of the electrostatic excess energy fromthe Debye–Huckel equation. The major variables thatdetermine the limiting slope are the dielectric permittivity,its pressure dependence, and the compressibility of purewater. Because these are known as a function of pressureand temperature (Bradley and Pitzer, 1979; Fernandezet al., 1997; Wagner and Pruß, 2002), the Redlich–Rosen-feld equation is extremely useful for first estimates andextrapolation, and moreover, it has been validated timeand again (Redlich, 1940; Redlich and Meyer, 1964; Dunn,1966, 1968a,b; Millero, 1971, and many others). A modifiedRedlich–Rosenfeld equation is proposed in this paper to ex-tend the Millero (1983, 2000) calculations across a greaterrange of pressures and temperatures.

Alternatively, the conventional molar volumes can beextracted from equations for the density or the volume ofsalt solutions (Monnin, 1994, 1999; Krumgalz et al., 2000;Laliberte, 2009). Laliberte (2009) devised an equation with5 parameters to fit the densities of virtually all the measuredsalt solutions up to about 125 �C at 1 atm pressure or at thesaturation pressure of water, with standard deviationsmostly less than 1 g/L. For concentrations of the salt below50 mM, however, the fit diverges from observations, andthe intrinsic molar volumes differ from the accepted or rec-ommended values (Millero, 1971; Hunenberger and Reif,2011) because Laliberte’s equation does not contain thesquare root dependence on the ionic strength that followsfrom the Debye–Huckel theory. Monnin (1994) andKrumgalz et al. (2000) used the equations developed byPitzer and coworkers (Pitzer, 1973, 1986; Rogers and Pitzer,1982; Pitzer et al., 1984; Phutela et al., 1987) with parame-ters fit to measured densities for 1 atm at 25 �C (Monnin,1994), up to 95 �C (Krumgalz et al., 2000) and to 200 �C(Monnin, 1999). The parameters are obtained from solu-tions containing a single salt, but the applicability tomulticomponent solutions can be tested, for example, bycalculating the density of mixed (NaKMgCa)Cl-solutionsmeasured by Krumgalz and Millero (1982) and Krumgalzet al. (1992). The absolute difference (measured � calculated)is less than 0.5 g/L with Monnin’s (1994) model. Even smallerdifferences are reported by Krumgalz et al. (2000).

Similar to the Redlich–Rosenfeld equation, Pitzer’sequation contains the intrinsic molar volume and thechange of the molar volume (Vm) with ionic strength (I).In addition, the Pitzer equation contains a virial expansionof the excess (electrostatic) energy in the solution. If theexpansion provides the Gibbs free energy as a function oftemperature and pressure, many chemical and physicalproperties can be derived, for example, for aqueous NaCl(Pitzer et al., 1984). However, in general, the coefficientsfor molar volumes cannot be obtained from the activityparameters in existing Pitzer models because pressure

Fig. 1. The apparent molar volumes of HCl, NaCl, and CaCl2 inaqueous solutions of the salt as a function of the square root of theionic strength. Symbols give the measured data (sources listed inTable 1). Thin lines indicate the Debye–Huckel limiting slopes,dashed and full lines give the calculated molar volumes accordingto Millero (1983, 2000) and PHREEQC, respectively. The limitingslope for HCl is drawn with the convention that V m;Hþ ¼ 0 for all I,the thin line overlaps with the full line calculated by PHREEQC.

C.A.J. Appelo et al. / Geochimica et Cosmochimica Acta 125 (2014) 49–67 51

dependence is not included in the parameterization. Also,the molar volumes of salts in water expand, in a ratherstraightforward manner, with the square root of the ionicstrength (Masson, 1929; cf. Fig. 1 of this paper), so thatthe many fitting parameters that are used for calculatingactivities in the Pitzer formulas may be unnecessary for

Fig. 2. Intrinsic molar volumes of Na+, K+, Mg2+, Ca2+, Cl�, HCO3� a

pressure, calculated by PHREEQC (lines) or by using coefficients fromfrom Millero (1971, 1983, symbols).

the molar volumes. Furthermore, the polynomials used inthe Pitzer expressions are computer intensive and probablyinaccurate outside the fitting range.

The concave downward variation of the intrinsic molarvolume with temperature (Millero, 1972; cf. Fig. 2 of thispaper) is fit with up to 6 coefficients by Krumgalz et al.(2000) and up to 9 coefficients by Monnin (1999). If anunderlying, physical explanation for the variation and theassociated equation can be identified, fewer fitting parame-ters will be necessary, and extrapolation will be more accu-rate. The obvious candidate is the pressure derivative of theBorn equation for the solvation energy of an ion (Helgesonet al., 1981; Myers et al., 2002). This derivative contains thepressure dependence of the inverse dielectric permittivity,which, for water, shows the same concave downward shapeas the intrinsic molar volumes with temperature. Based onthe work of Helgeson and coworkers (Helgeson and Kirk-ham, 1976; Helgeson et al., 1981; Shock and Helgeson,1988; Tanger and Helgeson, 1988; Shock et al., 1992), theSUPCRT code (Johnson et al., 1992) requires only 5 coef-ficients to give the temperature and pressure dependenceof the intrinsic molar volumes of single ions over a widetemperature and pressure range.

The parameterization for intrinsic molar volumes devel-oped by Helgeson et al. (1981) provides an excellentapproximation for experimental data (e.g., Tanger and Hel-geson, 1988; Shock et al., 1992; Sharygin and Wood, 1997),but in its practical application (SUPCRT), the volumes aredefined only for infinite dilution. Thus, the large depen-dence of the volumes on solution concentrations must stillbe incorporated for applications in natural systems. If com-bined with the modified Redlich–Rosenfeld equation pro-posed in this paper, the measured densities of saltsolutions, up to 10 M, can be fit with 5 additionalparameters.

nd SO42� as a function of temperature at 1 atm or water saturation

SUPCRT (Johnson et al., 1992, dashed lines) and tabulated values


For CCS, in addition to pressure dependence of molarvolumes, it is necessary to account for the fugacity coeffi-cient of CO2 and other gases. The fugacity coefficient ofCO2 can be calculated by using the polynomial function gi-ven by Duan et al. (2006), which yields values for a widerange of temperatures and pressures, but does not accountfor multicomponent interactions with other gases that willbe present in a CCS system, like CH4, H2S, N2, and ofcourse, H2O. More general approaches for calculating thethermodynamic properties of gases at high pressure arethe extensions of the Van der Waals equation by Redlichand Kwong (1949), and further by Soave (1972) and Pengand Robinson (1976). The coefficients in the Redlich–Kwong equation for CO2 and H2O have been fitted by Spy-cher et al. (2003) and Springer et al. (2012). Peng and Rob-inson (1976) estimated the limiting volume and attractionfactors in the Van der Waals equation from the critical tem-perature and pressure and the acentric factor of the gas,which are readily available numbers. With the parametersin the Van der Waals equation known, the fugacity coeffi-cient of a gas can be calculated (Redlich and Kwong,1949; Peng and Robinson, 1976). The Peng–Robinsonequation is widely applied in chemical engineering, and itwill be shown in this paper that it provides, without any fit-ting necessary, fugacity coefficients for CO2 that are closelysimilar to Duan’s polynomial function. Moreover, thePeng–Robinson equation accounts for mixing in the gasphase, and it can give the mole-fraction of water in thegas phase in a CO2–H2O system as calculated by Spycheret al. (2003) and the aqueous solubility of CO2 in a ternaryCO2–CH4–H2O mixture as measured by Qin et al. (2008),again without the need for any fitting parameters.

The equations for calculating the conventional aqueousvolumes for single ions and the fugacity coefficients of gasesin gas mixtures have been included in PHREEQC, version 3(Parkhurst and Appelo, 2013). Using the least-squaresparameter-estimation program PEST (Doherty, 2003), thecoefficients in the modified Redlich–Rosenfeld equationwere optimized from measured densities of salt solutionsor measured molar volumes and included in two of the dat-abases that accompany PHREEQC. The parameters forcalculating fugacity coefficients were also entered in the dat-abases. This paper presents the logic for selecting the formof the modified Redlich–Rosenfeld equation, and providesexamples showing that solubilities of gases and minerals(in pure water) can be calculated reliably up to 200 �Cand 1000 atm. The atmosphere was selected as the pressureunit because PHREEQC’s thermodynamic databases arebased on it, and equations from the literature are convertedto conform to this unit. However, in this paper the bar orthe Pascal may be used when data from the literature arecompared with model results.

2. APPARENT MOLAR VOLUMES OF AQUEOUS

SPECIES

The apparent molar volume of a dissolved salt can beobtained from the measured difference between the densityof the solution and of pure water:

V m ¼1

m1000þ mMW

q� 1000

q0

� �; ð1Þ

where Vm is the molar volume of the salt (cm3/mol), m is themolality (mol/kg H2O), MW is the molecular weight of thesalt (g/mol), and q and q0 are the densities of the solutionand of pure water at the same pressure and temperature,respectively (g/cm3). Fig. 1 shows the apparent molar vol-umes of HCl, NaCl and CaCl2 as a function of the squareroot of the ionic strength (I, mol/kg H2O) at 25 �C.

There are at least three aspects to be noted in Fig. 1. First,the molar volumes increase substantially with ionic strength,which needs to be taken into account in the calculation of thepressure dependence of geochemical reactions. Second, theincrease is almost linear with the square root of the ionicstrength. Above I � 0.5, the slopes are 0.8, 1.9 and 3.3 forHCl, NaCl, and CaCl2, respectively. For other alkali-chlo-rides the slopes are nearly equal to the slope for NaCl, andsimilarly, for other alkaline-earth chlorides (except SrCl2),the slopes are close to the slope for CaCl2. The similarity ofslopes means that the volumes of the individual ions in a mix-ture simply can be added to obtain the total volume of thesolutes, and hence the density of the solution (Harned andOwen, 1958; Ellis, 1968; Millero, 1972). It also means thatthe molar volumes of salts may be added or subtracted to ob-tain the volume of the desired ions in a salt (Millero, 1972;Monnin, 1999). The third point is that the intrinsic molar vol-ume of NaCl is smaller than that of HCl, indicating that Na+

has a smaller molar volume than H+ if the molar volume ofCl� is the same in both solutions. Also at I = 0, the volume ofCaCl2 is only slightly larger than of NaCl, and thus, Ca2+

must have even smaller volume than Na+. It may be notedthat in Eq. (1), the density change is wholly attributed tothe dissolved salt, for which an apparent volume is calcu-lated. Physically, the density change is due to compactionof water molecules around the ions by electrostatic attractionof the dipoles on H2O to the ions.

The increase of the molar volume with the square rootof the ionic strength can be linked with the pressurederivative of the excess free energy in the solution,DVm = oGE/oP = �RT Ro(ln c)/oP, where GE is excessGibbs free energy, P is pressure (atm), R is the gas constant(82 cm3�atm/K/mol), T is the absolute temperature (K),and c is the activity coefficient. At low concentrations, thedependence of the excess energy on I can be calculated withthe Debye–Huckel theory (Redlich and Rosenfeld, 1931;Redlich and Meyer, 1964):

V m ¼ V 0m þ AvI0:5

X0:5miz2

i þ bI ; ð2Þ

where V 0m is the intrinsic molar volume, Av is the Debye–

Huckel limiting slope (cm3/mol/(mol/kg H2O)0.5), mi is thestoichiometric coefficient of element i in the salt, zi is thecharge number of i, and b is a coefficient for fitting thechange of the molar volume at higher concentration. Eq.(2) is referred to as the Redlich–Rosenfeld equation.

The Debye–Huckel limiting slope is (Redlich andRosenfeld, 1931):

Av ¼ RT � Ac �2

3� 2:303

3@ ln er

@P� j0

� �; ð3Þ


where Ac is the Debye–Huckel A parameter (0.51 (mol/kgH2O)�0.5 at 25 �C), er is the relative dielectric constant ofpure water (�), and j0 is the compressibility of pure water(1/atm). Using the (P, T) dependent equation for er fromBradley and Pitzer (1979), which gives, with bar-to-atmconversion, o(ln er)/oP = 4.79e�5/atm, and j0 = 4.52e�5/atm (Kell, 1975), Av = 1.89 cm3/mol/(mol/kg H2O)0.5 at25 �C and 1 atm.

The Debye–Huckel theory thus gives slopes dVm/dI0.5

of 1.89, 1.89 and 5.67 for HCl, NaCl and CaCl2, respec-tively, clearly different from the observed slopes atI > 0.5 mol/kg H2O except for NaCl (Fig. 1). However,the slopes are in close agreement when I is less thanabout 0.02, also at temperatures other than 25 �C (Red-lich, 1940; Redlich and Meyer, 1964; Dunn, 1966,1968a,b; Millero, 1970, 1971; and many others). When I

increases beyond 0.2, the limiting slope deviates markedlyfrom the measured slope in HCl and CaCl2 solutions.Thus, if Eq. (2) is used to fit data for these solutions,b in the third term must be negative. If at higher I, thelinear third term of Eq. (2) takes precedence over thesquare root term, the model gives increasingly smallervolumes, as illustrated by the lines calculated with theequations proposed by Millero (1983, 2000) in Fig. 1.

2.1. Modified Redlich–Rosenfeld equation for apparent molar

volumes of single ions

The Redlich–Rosenfeld equation offers a thermody-namic way to obtain intrinsic molar volumes, but, fromthe foregoing discussion, it is clear that its effective rangeis limited to low ionic strengths. The challenge is to findan equation that lets the limiting slope change smoothlyto the slope at higher concentrations.

One option is to follow Pitzer and coworkers and use alogarithmic constraint on the slope, replacing I0.5 in Eq. (2)with (Pitzer, 1973, 1986; Rogers and Pitzer, 1982):

lnð1þ bAvI0:5Þ=bAv; ð4Þ

where bAv is fixed to 1.2 by Rogers and Pitzer (1982).Another option is, to differentiate the extended Debye–

Huckel equation with respect to pressure (Harned andOwen, 1958; Millero, 1970; Helgeson and Kirkham,1976). Assuming that the ion-size parameter is constant,the I0.5 term in Eq. (2) is replaced with:

I0:5

1þ�aBcI0:5; ð5Þ

where a is the ion-size parameter (A) and Bc is the Debye–length parameter (1/A). The pressure derivative of Bc ismore than 50 times smaller than Av and can safely beneglected.

Eqs. (4) and (5) give almost the same results when bAv isfixed to 1.2 in Eq. (4), and a � Bc is set to 0.48 in Eq. (5).Because the volumetric slopes are different for alkali- andalkaline earth-chlorides (Fig. 1), and for other halides aswell, bAv or a must vary for different salts and for individualions. Different a values for the single ions are available andwere introduced in the extended form of Debye–Huckelequation. When calculated and measured molar volumes

were compared, the fit obtained with the extended De-bye–Huckel option was better in all cases (about half theroot of the mean sum of squared differences, RMSD) thanthe Pitzer logarithmic constraint with a fixed bAv. Thus, theextended Debye–Huckel form was selected for inclusion inPHREEQC.

The third term of the Redlich–Rosenfeld equation mod-ifies the slope of the molar volume at high ionic strength.This slope is often nearly constant for temperatures up to100 �C, but becomes temperature-dependent at higher tem-peratures. The slope may change with pressure, but it is dif-ficult to find conclusive data because measurements at highpressures also are usually at high temperatures, where theDebye–Huckel slope increases substantially and becomesmore dominant in the equation. To account for the temper-ature effect, the coefficient b was replaced by a three-param-eter temperature-dependent factor. Moreover, anadjustable exponent for I in the third term was added to al-low for data fitting when necessary.

The resulting modified Redlich–Rosenfeld equation forsingle ions is

V m;i ¼ V 0m;i þ Av0:5z2

i

I0:5

ð1þ�aiBcI0:5Þ

þ b1;i þb2;i

T � 228þ b3;iðT � 228Þ

� �Ib4;i ; ð6Þ

where b1..b4 are adjustable coefficients for fitting experi-mental data.

The intrinsic molar volume is calculated with the equa-tion developed by Helgeson and coworkers (Helgeson andKirkham, 1976, and the final form in Helgeson et al.,1981; cf. also Shock and Helgeson, 1988; Tanger and Hel-geson, 1988; Johnson et al., 1992; Shock et al., 1992):

V 0m;i ¼ 41:84 0:1a1;i þ

100a2;i

2600þ P bar

þ a3;i

ðT � 228Þ

�

þ 104a4;i

ð2600þ P barÞðT � 228Þ � xi@e�1

r

@P bar

�; ð7Þ

where a1..a4 and x are parameters for individual solutes,and 41.84 converts cal/bar/mol to cm3/mol. The combina-tion of Eqs. (6) and (7) is referred to as the Helgeson–Kirk-ham–Flowers-modified-Redlich–Rosenfeld (HKFmoRR)equation.

In principle, x is based on the Born equation (Born,1920; Helgeson et al., 1981), for calculating the energy ofsolvation,

DGs ¼ �N Avogadroz2i

q2e

8pe0ri1� 1

er

� �¼ �xabs

i 1� 1

er

� �; ð8Þ

where DGs is the solvation energy (cal/mol), NAvogadro isAvogadro’s number (6.022e23/mol), zi is the charge num-ber, qe is the electron charge (1.602e�19 C), e0 is thedielectric permittivity of vacuum (3.704e�11 C2/cal/m),ri is the ion radius (m), xi

abs is the Born coefficient(cal/mol), and er is the relative dielectric constant of purewater (�). The pressure derivative of the solvation energygives the volume change associated with transferring anion from vacuum into water, and the change therein withtemperature allows estimating how this volume changewill vary with temperature. The pre-multiplier x has been


estimated for individual ions by Helgeson et al. (1981)from crystal radii, but is used here as a fitting parameterfor single ions (the differences between the two values willbe discussed). In Eq. (6), the pressure derivative of Bc isneglected, but the variation with temperature and pres-sure is accounted for with Bradley and Pitzer’s (1979)equation for the dielectric constant of water.

The molar volume of a salt becomes

V m ¼X

miV m;i: ð9Þ

With Eqs. (6), (7), and (9), conventional molar volumes ofsingle ions can be defined relative to V m;Hþ ¼ 0, startingwith V m;Cl� from the densities of HCl solutions, then of cat-ions using the densities of cation-Cl solutions, and of otheranions using cation–anion solutions (Millero, 1971, 1972).The actual molar volume of the proton may be �5.6 cm3/mol (Conway, 1981) or �4.5 ± 5 cm3/mol (Hunenbergerand Reif, 2011), but its value cancels in the calculationof the pressure-dependent solubilities, because for asolid, dissolving in water, the aqueous volume isVm,real = Rmi(Vm,i � zi V m;Hþ ) = Vm, because Rmizi = 0.

Usually, V m;Hþ is set to 0 for all T and P (Millero, 1983,2000; Hunenberger and Reif, 2011), without discussing thatit also must be decided whether this volume is zero at all io-nic strengths. The consequences of the choice are far-reaching. If, in the set of equations that is used to calculatethe system, the proton contributes to the calculated molarvolume of HCl at ionic strengths greater than zero, the con-tribution of Cl� will change. Because the volumes of thecations are based on the volumes for Cl�, the calculatedcation contributions to the volumes of the salts will changeas well. And, continuing along this line, the contribution ofthe anions other than Cl� also will change.

If V m;Hþ ¼ 0 for all T, P, and I, the proton does not con-tribute to the calculated molar volume of HCl at any ionicstrength. Accordingly, the increase of HCl’s molar volumewith the square root of ionic strength is wholly assignedto Cl�. In the ideal Redlich–Rosenfeld equation, i.e., witha = 0 and bI = 0, the slope is Av/2 = 0.94 at 25 �C forHCl with this convention. The measured data (Fig. 1) showa slope of 0.8. For NaCl, the slope is 1.9 (Fig. 1), to whichCl� contributes 0.8, and Na+ the remainder of 1.1. If theproton is accounted for in the molar volume of HCl, Cl�

would contribute less than 0.8, and Na+ more than 1.1,and both contributions would be deviating more from theideal of 0.94. Thus, if the volume of the proton is zero, inde-pendent of ionic strength, the system of equations behavesmore ideally, meaning that it is simpler and probably re-quires fewer fitting parameters than if the proton volumeis assumed to vary with ionic strength. Accordingly, thevolume of the proton is set to 0, for all T, P and I.

2.2. Strategy for optimizing the coefficients in the HKFmoRR

equation

The coefficients in Eqs. (6) and (7) were optimized forthe major cations and anions in natural waters by usingdata compiled by Laliberte (2009), omitting obvious outli-ers and supplementing with data at lower concentrationsand at higher temperatures and pressures. The referencesare listed in Table 1. The coefficients were obtained by using

non-linear least-squares fitting techniques provided by theprogram PEST (Doherty, 2003), following three rules: (1)if the standard deviation for the value of a parameter islarge, redo the optimization while omitting that parameter,setting its value equal to zero; (2) if parameters are highlycorrelated, try to omit them one-by-one, while keepingRMSD within some limit; (3) reduce the number of digitsused for the final value for a parameter, without affectingthe RMSD.

Some of the coefficients were found to be highly cor-related, or were given a wide probability range by PEST.By trial and error, when inclusion of a variable in theoptimization did not reduce the RMSD by more than20%, the coefficient was fixed to zero, or, for the pressurecoefficients (a2 and a4) to the SUPCRT number. By thesame criterion, the ion-size parameter ai was set to zerofor anions, and to the number in PHREEQC.DAT forcations. [PHREEQC.DAT is a database with parametersfor calculating complexes and the activity coefficient withthe extended Debye–Huckel equation. PHREEQC.DATand PITZER.DAT are two of the databases distributedwith PHREEQC (Parkhurst and Appelo, 2013).]

The parameters optimized for PHREEQC.DAT can beentered simply in PITZER.DAT (which contains theparameters for the Pitzer interaction model) when com-plexes (e.g., CaCl+) have a negligible contribution to the to-tal concentration. For sulfate and carbonate, chargedcomplexes with Na+ are important in PHREEQC.DAT,but absent in PITZER.DAT, and the optimization was per-formed separately for the two databases.

Generally, the molar volumes, calculated with Eq. (1)from the measured densities, are modeled with the sameaccuracy as for the salts shown in Fig. 1 for the full con-centration range of the major cations and anions in nat-ural waters for temperatures up to 150 �C. At highertemperatures the estimates may be less accurate, althoughthey remain mostly within 1 cm3/mol. The standard devi-ations of the fit of calculated and measured molar vol-umes are listed in Table 2; for MgCl2 and CaCl2, theyare only slightly higher than those noted by Phutelaet al. (1987), who used 11 parameters over a smaller tem-perature range.

The intrinsic molar volumes are compared with num-bers estimated by Millero (1971, 1983) and from SUP-CRT (Johnson et al., 1992) in Fig. 2. The agreement isexcellent at temperatures below 100 �C. The molar vol-umes of anions calculated by PHREEQC are smaller athigh temperatures, because the data fitting equationsare different and may extrapolate differently to the intrin-sic volumes (this is discussed later). For example, at200 �C, V 0

m;Cl� is smaller than in the previous estimates.Consequently, V 0

m;Naþ must be larger because the volumeof NaCl is the same. The larger V 0

m;Naþ results in smallerapparent volumes for HCO3

� and SO42�, which are cal-

culated from densities of Na solutions.

2.3. Values for the coefficients in the HKFmoRR equation

When the apparent molar volume of a dissolved salt iscalculated from the density of the solution with Eq. (1),

Table 1References for the molar-volume and density model used in PHREEQC.

Property or ion References

Density of pure H2O At saturation: Wagner and Pruß (2002); at higher pressures (up to 1000 atm), 0–300 �C interpolationfunctions using the IAPWS tables

Dielectric constant er Bradley and Pitzer (1979)OH� (pKw) Allred and Woolley (1981), Herrington et al. (1986), Hershey et al. (1984), Huckel and Schaaf (1959),

Magalhaes et al. (2002), Patterson et al. (2001), Perron et al. (1975a), Roux et al. (1984), Singh et al. (1976),Sipos et al. (2000), Vazquez et al. (1996), Bandura and Lvov (2006) for pKw (for calculating the (T, P)dependent OH� concentration)

Cl� (HCl) Allred and Woolley (1981), Fortier et al. (1974), Goldsack and Franchetto (1977), Herrington et al. (1985),Hershey et al. (1984), Pogue and Atkinson (1988), Redlich and Bigeleisen (1942), Rizzo et al. (1997), Salujaet al. (1986), Sharygin and Wood (1997), Singh et al. (1976), Torok and Berecz (1989)

Na+ (NaCl) Allred and Woolley (1981), Chen et al. (1980), Connaughton et al. (1986), Dedick et al. (1990), Dunn(1968ab), Ellis (1966), Fabuss et al. (1966), Fortier et al. (1974), Gates and Wood (1985), Gonc�alves andKestin (1977), Isono (1980), Korosi and Fabuss (1968), Kumar (1988), Lengyel et al. (1964), Majer et al.(1988), Manohar et al. (1994), Millero (1970), Motin (2004), Olofsson (1979), Out and Los (1980), Perronet al. (1975b, 1981), Romankiw and Chou (1983), Singh et al. (1976), Vaslow (1966), Zhang and Han (1996)

K+ (KCl) Dedick et al. (1990), Dunn (1968a, 1968b), Ellis (1966), Fortier et al. (1974), Gates and Wood (1985),Gonc�alves and Kestin (1977), Gucker et al. (1975), Isono (1980), Korosi and Fabuss (1968), Kumar (1988),Lengyel et al. (1964), Nickels and Allmand (1937), Out and Los (1980), Patterson et al. (2001), Romankiwand Chou (1983), Saluja et al. (1992), Singh et al. (1976), Zhang and Han (1996)

Mg2+ (MgCl2) Call et al. (2000), Chen et al. (1980), Connaughton et al. (1986), Ellis (1967), Gates and Wood (1985), Isono(1984), Millero and Knox (1973), Perron and Desnoyers (1974), Perron et al. (1981), Romankiw and Chou(1983), Saluja and LeBlanc (1987)

Ca2+ (CaCl2) Dunn (1966, 1968a, 1968b), Ellis (1967), Gates and Wood (1989), Isono (1984), Kumar et al. (1982), Oakeset al. (1990, 1995), Perron and Desnoyers (1974), Perron et al. (1981), Romankiw and Chou (1983), Salujaand LeBlanc (1987), Wimby and Berntsson (1994), Zhang et al. (1997)

Sr2+ (SrCl2) Ellis (1967), Isono (1984), Perron and Desnoyers (1974), Pilar Pena et al. (1997), Saluja and LeBlanc (1987).Ba2+ (BaCl2) Isono (1984), Manohar et al. (1994), Perron and Desnoyers (1974), Puchalska and Atkinson (1991)Fe2+ (FeCl2) Kaminsky (1956), Pogue and Atkinson (1989).HCO3

� + CO32� (NaHCO3,

Na2CO3)Barbero et al. (1983), Ellis and McFadden (1972), Hershey et al. (1983), Magalhaes et al. (2002), Perronet al. (1975a), Sharygin and Wood, (1998)

SO42� (Na2SO4) Chen et al. (1980), Connaughton et al. (1986), Dedick et al. (1990), Ellis (1968), Fabuss et al. (1966), Glass

and Madgin (1934), Isono (1984), Korosi and Fabuss (1968), Magalhaes et al. (2002), Millero and Knox(1973), Pabalan and Pitzer (1988), Perron et al. (1975a), Phutela and Pitzer (1986), Saluja et al. (1992),Sanchez et al. (1994)


the density difference between the solution and pure wateris wholly attributed to the dissolved electrolytes; the aque-ous solvent keeps the density of pure water. Thus, the con-ventional molar volumes of Na+, Mg2+ and Ca2+ arenegative (Fig. 2) because the solution density increasesmore than the weight increase resulting from replacementof H+ by Na+, Mg2+, or Ca2+. Physically, the density in-crease is due to compression of the water molecules in thehydration shell around the ion by electrostatic attractionof the oppositely charged dipole-ends of the water mole-cules (Benson and Copeland, 1963; Conway, 1981; Marcus,2011). In the Born-model adopted by these authors, thedielectric property of water is taken as a continuum, andthe different orientation of the water dipoles near cationsand anions is neglected (Hunenberger and Reif, 2011).However, Table 2 shows that the parameters in the HKF-moRR model that are associated with the dielectric con-stant of water are different for cations and anions.Notably, for the anions, the simple Debye–Huckel equationcan be used (ai = 0), while for the cations, the extended De-bye–Huckel equation gives a much better fit. Except forNa+, Sr2+, and SO4

2�, the exponent b4 for the ionicstrength in the third term of the Redlich–Rosenfeld equa-tion is 1. It is greater than 1 for Sr2+, probably because

complexes form at higher concentration. For SO42�, the

exponent b4 depends on the speciation model; b4 is smallerthan 1 for SO4

2� in the Pitzer interaction model (Table 2),but the third term cancels altogether for SO4

2� when theNaSO4

� complex is included (Table 3). The exponent issmaller than 1 for Na+, and becomes even smaller for ionicstrengths greater than 3.

2.4. Additivity of individual molar volumes

The additivity of molar volumes at infinite dilution hasbeen proven (Harned and Owen, 1958; Millero, 1972), butthe principle applies with a remarkable precision for mix-tures of salts at high concentrations. Fig. 3 shows excellentagreement between calculated and measured densities ofmixtures of NaCl, MgCl2, and CaCl2 solutions. The largestdifference is 3.1 g/L at a measured density of 1154.1 g/L, fora solution containing 0.93 M NaCl and 1.36 M CaCl2 at5 �C (Fig. 3B). Using Eq. (1), the calculated molar volumeof the mixed salt (Na0.26Ca0.37Cl) is 12.9 cm3/mol, com-pared to 12.2 cm3/mol from the measured density. Themixed (NaKMgCa)Cl data by Krumgalz et al. (2000),quoted in the introduction, can be modeled with less than0.4 g/L difference for the 25–45 �C temperature range.

Table 2Coefficients in the HKFmoRR equations [Eqs. (6) and (7)] for calculating the (P, T, I) dependent conventional molar volumes of the major cations and anions in water, relative to V m;Hþ ¼ 0, forthe PITZER.DAT database (i.e., no aqueous complexes). Also listed are the resulting intrinsic molar volumes at 25 �C.

Na+ K+ Mg2+ Ca2+ Sr2+ Ba2+ Fe2+ OH� Cl� HCO3� CO3

2� CO2 SO42�

a1/(10 cal/bar/mol) 2.28 3.322 �1.410 �0.3456 �1.57e�2 2.063 �0.3255 �9.66 4.465 8.54 4.91 20.85 �7.77a2/(10�2 cal/mol) �4.38 �1.473c �8.60c �7.252c �10.15c �10.06c �9.687c 28.5 4.801c 0 0 �46.93 43.17a3/(cal kelvin/bar/mol) �4.10 6.534 11.13 6.149 10.18 1.9534 1.536 80.0 4.325 �11.7 0 �79.0 141.1a4/(10�4 cal kelvin/mol) �0.586 �2.712c �2.39c �2.479c �2.36c �2.36c �2.379c �22.9 �2.847c 0 �5.41 27.9 �42.45x/(10�5 cal/mol) 0.09 0.0906 1.332 1.239 0.860 0.4218 0.3033 1.89 1.748 1.60 4.76 �0.193f 3.794a/(10�10 m) 4.0d1 3.5d1 5.5d1 5.0d1 5.26d2 5.0d3 6.0d3 0 0 0 0 0b1

a 0.30 0 1.29 1.60 0.859 1.58 �0.0421 1.09 �0.331 0 0.386 4.97b2 52 29.7 �32.9 �57.1 �27.0 �12.03 39.7 0 20.16 116 89.7 26.5b3 �3.33e�3 0 �5.86e�3 �6.12e�3 �4.1e�3 �8.35e�3 0 0 0 0 �1.57e�2 �5.77e�2b4 0.566e 1 1 1 1.97 1 1 1 1 1 1 0.45RMSDb 0.22 0.20 0.45 0.66 0.59 0.57 1.6 0.24 0.12 0.30 0.025 0.56V 0

m;i/(cm3/mol), 25 �C �1.52 8.98 �21.9 �18.3 �17.9 �12.8 �22.2 �3.91 18.0 24.7 �3.87 29.1 14.2

a The units of b1..3 are such, that cm3/mol results after division or multiplying with temperature and Ib4.b The root of the mean squared deviations of the fit of calculated and measured (references in Table 1) molar volumes (cm3/mol).c Shock and Helgeson (1988).

d1 Truesdell and Jones (1974).d2 Busenberg et al. (1984).d3 Kielland (1937).

e For Na+ and I > 3, use b4 = 0.45.f Schulte et al. (2001).

56C

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Table 3Coefficients in the HKFmoRR equations [Eqs. (6) and (7)] for calculating the (P, T, I) dependent conventional molar volumes of the majorcations and anions in water, relative to V m;Hþ ¼ 0, in the PHREEQC.DAT database. Numbers for cations, OH�, and Cl� are the same as inTable 2 (PITZER.DAT).

HCO3� CO3

2� NaCO3� NaHCO3 SO4

2� NaSO4�

a1/(10 cal/bar/mol) 8.472 5.95 3.89 0.431 8.0 1e�5a2/(10�2 cal/mol) 0 0 �8.23e�4 0 2.30 16.4a3/(cal kelvin/bar/mol) �11.5 0 20.0 0 �46.04 �0.0678a4/(10�4 cal kelvin/mol) 0 �5.67 �9.44 0 6.245 �1.05x/(10�5 cal/mol) 1.56 6.85 3.02 0 3.82 4.14a/(10�10 m) 0 0 9.05e�3 0 0 0b1

a 0 1.37 3.07 0 0 6.86b2 146 106 0 0 0 0b3 3.16e�3 �0.0343 0.0233 0 0 0.0242b4 1 1 1 0 1 0.53

RMSDb 0.31 0.35

V 0m;i/(cm3/mol), 25 �C 24.6 �5.39 �1.06 1.8 13.9 18.2

Log Kc 10.33 0.65 0.4 0.7DHr

c/(kcal/mol) (poly) 0.7 �0.4 1.12

a The units of b1..3 are such, that cm3/mol results after division or multiplying with temperature and Ib4.b The root of the mean squared deviations of the fit of calculated and measured (references in Table 1) molar volumes (cm3/mol).c Log K and DHr at 25 �C for the association reaction. (poly) indicates that a polynomial is used for log K(T).

Fig. 3. Calculated and measured densities of mixed solutions of (A) NaCl + MgCl2 (measured by Saluja et al., 1995), (B) NaCl + CaCl2(Kumar and Atkinson, 1983; Oakes et al., 1990; Saluja et al., 1995 and Zhang et al., 1997), and (C) MgCl2 + CaCl2 (Saluja et al., 1995).


3. EXAMPLE: GYPSUM AND ANHYDRITE

SOLUBILITY AS A FUNCTION TEMPERATURE AND

PRESSURE

Blount and Dickson (1973) summarized the measuredsolubilities of gypsum and anhydrite as a function of tem-perature and pressure, and their data (from Fig. 2 in Blountand Dickson, 1973) are compared with PHREEQC’s calcu-lation (PITZER.DAT) in Fig. 4. The agreement is excellent.

Fig. 4 shows that the transition temperature of gypsuminto anhydrite (intersections of red and green lines) in-creases from 55 �C at 1 bar, to 61 �C at 500 bar, and to66 �C at 1000 bar. Thus, the stability of gypsum relativeto anhydrite increases with pressure, which is because waterin the gypsum crystal has a smaller volume than water insolution:

CaSO4 � 2H2O ¼ CaSO4 þ 2H2O; DV r

¼ 8:3 cm3=mol at 25 �C; ð10Þ

and the equilibrium constant changes with pressure as:

log KP ;T ¼ log KP¼1;T � DV rP � 1

2:303RT; ð11Þ

where KP=1,T is the equilibrium constant at 1 atm and tem-perature T (the temperature dependence is calculated with apolynomial in PHREEQC).

Fig. 4 also illustrates that the solubilities of gypsum andanhydrite increase with pressure because the sum of theaqueous molar volumes of the solute species is smaller thanthe molar volume of the minerals and water. This pressureeffect is observed for the solubility of all inorganic minerals.

The pressure effect on anhydrite solubility in NaCl solu-tions described by Blount and Dickson (1969), also ex-plored by Monnin (1990), can be modeled quite well at100 �C with the database with Pitzer coefficients developedby Plummer et al. (1988). However, the higher temperaturesthat usually go hand-in-hand with high pressure solubilityexperiments may require adaptation of the interaction

Fig. 4. Solubility of gypsum (red symbols and lines) and anhydrite(green symbols and lines) in pure water up to 1000 bar and 160 �C.Data (symbols) from Blount and Dickson (1973), lines calculatedwith PHREEQC. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of thisarticle.)


coefficients. For example, barite solubility in NaCl solu-tions according to Blount (1977, Table 5) can be calculatedwell, but only if the Na+–SO4

�2 interaction coefficientsfrom Pabalan and Pitzer (1987) are used, and when the Pit-zer coefficients for Ba2+–Cl� interactions (B0, B1 and C0)are revised.

4. PENG–ROBINSON EQUATION OF STATE FOR

GASES

The solubility of gas i is given by

mi ¼P iP 0

uicim0

KH; ð12Þ

where KH (Henry’s constant) is the temperature- and pres-sure-dependent equilibrium constant, P is the pressure, u isthe fugacity coefficient, m is the molality, and c is the activ-ity coefficient in water. Subscript 0 indicates the standardstate, 1 atm for gases, and 1 mol/kg H2O for aqueoussolutes.

The fugacity coefficient is determined by the excess freeenergy, which can be solved from an equation of state suchas the Van der Waals equation (Redlich and Kwong, 1949;Soave, 1972):

P ¼ RTV m � b

� aa

V 2m þ 2bV m � b2

; ð13Þ

where b is the gas’ minimal volume, a is the Van der Waalsattraction, and a is an additional correction obtained fromthe acentric factor.

Peng and Robinson (1976) derived the limiting volumeand the attraction factors for both paraffins and nonhydro-carbon gases from the critical temperatures and pressures,and the acentric factors:

a¼ 0:457235RT cð Þ2

P c

b¼ 0:077796RT c

P c

a¼ 1þð0:37464þ1:54226x�0:26992x2Þ 1�

ffiffiffiffiffiffiffiffiffiffiffiffiTT c

� �s ! !2

ð14Þ

where Tc and Pc are the temperature and pressure at thecritical point, and x is the acentric factor. Using thePeng–Robinson definitions in the Van der Waals equationis referred to as the Peng–Robinson equation of state orsimply Peng–Robinson.

Using Peng–Robinson, the pressure P can be calculatedwhen Vm is given, or conversely, Vm can be solved from thecubic equation when P is known. Fig. 5 shows a plot of P

vs. Vm, with data from Michels and Michels (1935) andMichels et al. (1935, 1937) and lines calculated withPeng–Robinson [Eq. (13)]. Michels et al. (1935) measuredthe system with pressures up to 3117 atm at 150 �C, andfor these very high pressures the calculated molar volumesare smaller than measured (Peng and Robinson, 1976).However, in Fig. 5, the pressure is limited to 500 atm todemonstrate the particular behavior of a gas below its crit-ical temperature (31.2 �C for CO2). At 25 �C the curve flat-tens out when CO2 is compressed below 0.17 L/mol. Theconstant pressure for a decreasing volume indicates thatCO2 gas transforms into liquid. When, at 0.065 L/mol, allthe gas has been liquefied, a further decrease of the molarvolume results again in pressure increase. Thus, the gaspressure may remain constant (or, possibly even slightly de-crease) even as the molar volume decreases. Above the crit-ical temperature, liquid and gas are indiscernible, and thepressure–volume relationship is continuous, with no dis-crete phase change.

The abrupt gas–liquid transition below the critical pointcomes with three roots in the cubic Van der Waals equa-tion, of which only one is valid. The correct pressure forthe range of molar volumes, where a mixture of liquidand gas is present, is the one where liquefaction starts whenthe molar volume is decreased. Thus, when the pressure iscalculated from the molar volume of the gas or the gas mix-ture, it must be determined whether the equation has threeroots. If so, it must be tested further whether a larger molarvolume can be found with a larger root, and the largest root(i.e., the highest pressure) is then the physically correct solu-tion of the equation. The maximum can be obtained bycombining interval-halving and Newton iteration on theVm derivative of Peng–Robinson.

From the equation of state, the fugacity coefficient canbe solved (Redlich and Kwong, 1949; Peng and Robinson,1976):

lnðuÞ ¼ PV m

RT� 1

� �� ln

P ðV m � bÞRT

� �þ aa

2:828bRT

� ln V m þ 2:414bV m � 0:414b

� �ð15Þ

Fig. 5. Plot of CO2 gas pressure vs. molar volume at 25, 50, 100, and 150 �C. Symbols are measured data from Michels and Michels (1935)and Michels et al. (1935, 1937). Lines are calculated with PHREEQC. Also shown is the trace of liquefaction at 25 �C, where the Van derWaals equation solves to 3 roots (black line at 0.06 < Vm < 0.17 L/mol).


Fig. 6A compares the fugacity coefficients from Peng–Rob-inson and the Duan polynomial (Duan et al., 2006), show-ing that they give essentially the same results. The fugacitycoefficient is close to 1 when the total pressure of the gasphase is smaller than about 10 atm, but it decreases to lessthan 0.4 at 25 �C and above 130 atm.

Other gases at high pressures may interact with CO2 andmay change the fugacity coefficient markedly (formulas aregiven in Section 4.1) as illustrated in Fig. 6B, where the ini-tial CH4 pressure is set to 400 atm in a fixed volume. In thiscase, the fugacity coefficient of CO2 is less than 0.5 for all

Fig. 6. Fugacity coefficients of CO2. (A) Coefficients calculated with Peng(symbols) for 25, 50, and 75 �C. The total pressure is equal to the CO2 prCO2/CH4 mixture, when the initial CH4 pressure is 400 atm (lines), and ththe system.

CO2 pressures for all three temperatures. The fugacity coef-ficient slightly decreases, and then increases again as theCO2 pressure increases. Also note that the total pressurein the system is higher than the sum of the initial pressureof the system (equal to P CH4

¼ 400 atm) and the pressureof CO2. For example, at 50 �C and 500 atm CO2 pressure,the total pressure has increased to 1200 atm, because thepressure of CH4 changes from 400 to 700 atm when CO2

is added to the fixed gas volume.Using the fugacity coefficient for CO2, the temperature-

dependent Henry’s constant from the PITZER.DAT

–Robinson (lines) compared to the polynomial of Duan et al. (2006)essure. (B) Calculated fugacity coefficients for 25, 50, and 75 �C in ae change in total pressure (thin lines with circles) as CO2 is added to


database (Plummer and Busenberg, 1982), and the volumeof aqueous CO2 calculated with parameters listed in Table 2,the solubility of CO2 gas can be calculated with Eq. (12),which becomes:

mCO2¼

P CO2uCO2

cCO2

KH exp�V m;CO2

ðP � 1ÞRT

� �1 mol

kg H2O

1 atm: ð16Þ

The gaseous molar volume is not an explicit part of Eq. (16)because it is already included in the Peng–Robinson equa-tion of state and thus, is contained in P CO2

uCO2.

Results are compared with experimental data in Fig. 7.Water vapor is included in the model calculations by usingbinary interaction coefficients of CO2 and H2O from Sør-eide and Whitson (1992). The effect of H2O is to lowerslightly the fugacity coefficient of CO2, but the effect is min-or for the temperatures used in Fig. 7. Other experimentaldata at higher temperatures and in CaCl2, NaCl, or Na2SO4

solutions (Takenouchi and Kennedy, 1964; Nighswanderet al., 1989; Rumpf and Maurer, 1993; Rumpf et al.,1994) and data compiled by Springer et al. (2012) can bemodeled equally well.

4.1. Gas mixtures

For gas-mixtures, Peng and Robinson (1976) took theweighted sum of a, b and a in Eq. (13):

b ¼ bsum ¼X

yibi ð17Þ

and

aa ¼ aasum ¼X

i

Xj

yiyj aiai � ajaj

� �0:5� � !

ð1� kijÞ; ð18Þ

where yi is the mole-fraction of gas i in the mixture, and kij

is a binary interaction parameter for gases i and j.The fugacity coefficient of gas i in a mixture becomes

Fig. 7. The solubility of CO2 in pure water as a function of gaspressure at 25, 50, 75, and 100 �C, calculated with PHREEQC(lines) and measured data compiled by Duan et al. (2006) andSpycher et al. (2003).

lnðuiÞ ¼ Brðz� 1Þ � lnðz� BÞ

þ A

2:828B

Br � 2aasum;i

aasum

� �ln

zþ 2:414B

z� 0:414B

� �; ð19Þ

where Br = bi/bsum, z is the compressibility factor, z

= PVm/(RT), B = bsumP/(RT), A = a asum/(RT)2, andaasum;i ¼

Pj yiyj aiai � ajaj

� �0:5� �

ð1� kijÞ.For example, the fraction of H2O in the gas phase of

CO2–H2O mixtures can be calculated and compared withdata compiled by Spycher et al. (2003). The results areshown in Fig. 8. At 25 and 50 �C, the calculations matchthe data very well. At 75 �C, the calculated fractions arehigher than measured, but there is scatter in the data,and, because the solubility of CO2 in H2O is calculatedaccurately at this temperature (Fig. 7), by mass balance,the modeled line in Fig. 8 is probably correct. The fractionscalculated by Spycher et al. (2003), using the Redlich–Kwong equation with optimized a and b for H2O andCO2, are in almost exact agreement with the Peng–Robin-son results for this temperature. Similarly for higher tem-peratures, the comparison with data compiled by Springeret al. (2012), shows excellent agreement at pressures below100 atm, but shows some deviation at higher pressures.

Fig. 8 also shows the effect of liquefaction of CO2 whenthe pressure increases above 60 bar at 25 �C, which is belowthe critical temperature for CO2. The curve for the fractionof water in the gas phase jumps up when liquefaction startsbecause CO2 is lost from the gas phase. In a more gradualform, the effect is present at temperatures above the criticalpoint.

Qin et al. (2008) measured the solubility of CO2 in waterat 102 �C in hydrothermal bombs filled with H2O, CO2 andCH4 at total pressures of 10.7, 20.5, 30.3, 40.2 and49.9 MPa. The apparent Henry’s constant (KH in Eq.(12), with uCO2

¼ 1, and mCO2expressed as mole-fraction in-

stead of molality) is compared with data (Table 3 in Qinet al., 2008) in Fig. 9. The results show the same effects as

Fig. 8. The mole-fraction of H2O in the CO2-rich gas phase in amixture of CO2 and H2O as a function of the total pressure. Linescalculated by PHREEQC; symbols from Spycher et al. (2003).


were noted with Figs. 6 and 7. With increasing gas pressure,the fugacity coefficient decreases, resulting in a lower solu-bility of CO2, and thus, a notable increase of the apparentHenry’s constant. At a given pressure, there is a slight in-crease of the apparent Henry’s constant when the fractionof CO2 in the CO2 + CH4 mixture increases from 0.5 to0.9 because the Van der Waals attraction factor for CO2 in-creases slightly. In the calculations, the binary interactioncoefficients of CO2 and CH4 (kij) were set to 0, in goodagreement with the data.

5. DISCUSSION

The Peng–Robinson equation of state has been claimedto be “. . . one of the greatest success stories in the history ofapplied chemical thermodynamics. Somehow, Peng andRobinson constructed the optimum balance of repulsiveand attractive terms such that the sum gives a remarkablyclose representation of the configurational thermodynamicproperties of a large number of pure fluids and their mix-tures” (Wu and Prausnitz, 1998). This equation works,not only for the examples presented in this paper forCO2, but for other gases and gas mixtures as well. The accu-racy for hydrogeochemical calculations of CCS systemsshould be more than satisfactory for most practicalapplications.

The modified equation for molar volumes of aqueousspecies that we have proposed, may not gain the impactof Peng–Robinson’s additions (Peng and Robinson, 1976)to the Soave–Redlich–Kwong equation (Soave, 1972), butit does offer improved and simpler applicability for a widerrange of temperatures than equations employed previously.In a sense, we tried to honor the arguments of Redlich andMeyer (1964) that an equation based on first principles will

Fig. 9. Apparent Henry’s constant of CO2, determined in amixture of CO2 + CH4 + H2O, as a function of the CO2 fractionof total C in the system, for five different total gas pressures.Symbols indicate the data from Qin et al. (2008), lines are modeledwith PHREEQC; temperature is 102 �C. In the apparent Henryconstant, CO2,aq is expressed as a fraction and the fugacitycoefficient is set to 1 for gaseous CO2.

reduce the number of variables that have to be allocated,and will always be superior to fitting coefficients of polyno-mials in T, P, and I. The improvements are the result of (1)using the Born equation for calculating the temperaturedependence of the intrinsic volumes following Helgesonand coworkers (Helgeson and Kirkham, 1976; Helgesonet al., 1981) (2) taking the pressure derivative of the ex-tended Debye–Huckel equation, which allows the limitingslope to apply only at low concentrations when necessary,and (3) setting the reference molar volume of the protonto zero at all ionic strengths, in addition to all T and P.

The Born equation is not the perfect solution for obtain-ing solvation energies (Conway, 1981; Fawcett, 2004), andhence, for calculating the temperature dependence of theintrinsic, aqueous molar volumes. Its pressure derivativedecreases continuously with temperature, which then re-quires an additional, positive temperature factor to fit theobserved increase of the intrinsic volumes with temperatureat lower temperatures (cf. Fig. 2). Helgeson et al. (1981) de-rived an effective electrostatic radius for ions from crystalradii, and these are compared in Table 4 with values calcu-lated from our optimized values of xi by using reff,i = 1.66zi

2/(xi + 0.5387zi) (Helgeson et al., 1981). The optimizedeffective radius is, for cations, somewhat larger than the va-lue from the SUPCRT database (Johnson et al., 1992), and,for anions, considerably smaller. Accordingly, the effect ofthe Born term [x in Eq. (7)] is larger for anions, and smallerfor cations than calculated by Helgeson et al. (1981). Thedifference can be attributed partly to the way the molar vol-umes are extrapolated to infinite dilution in our study, andpartly to the fact that Helgeson et al. (1981) use the Bornequation not only for molar volumes, but also for calculat-ing the heat capacity of electrolytes. Helgeson et al. (1981)took the arithmetic average of the effective radii as the ion-size parameter for both anion and cation in the extendedDebye–Huckel equation, whereas in this study, differentvalues are optimized for anions (a = 0) and cations (a closeto the value in the PHREEQC.DAT database, and, becausethe inclusion as an adjustable parameter did not improveRMSD, fixed to that value). The slope of the molar volumewith ionic strength is substantially higher for the simple De-bye–Huckel equation (applicable to anions) than for the ex-tended Debye–Huckel equation (applicable to cations) forionic strengths above 0.1, as illustrated in Fig. 10. Because0.1 is about the smallest ionic strength for which solutiondensities have been measured at high temperatures (forexample, HCl, Sharygin and Wood, 1997), the higher slopefor anions translates to a smaller intrinsic volume, andhence, to a smaller reff,optimized.

The smaller effective radius for anions is the major rea-son that the intrinsic molar volume is smaller than the SUP-CRT value above 100 �C (Fig. 2). At lower temperaturesthe intrinsic volumes are determined mainly by the a1..4

parameters in Eq. (7) because the Born term is small dueto the (relatively) small derivative oe�1/oP.

Also shown in Fig. 10, is the effect of the logarithmicconstraint used by Pitzer and coworkers, which falls be-tween the simple and extended Debye–Huckel slope. Inthe Pitzer equation, the logarithmic control is added tothe extended Debye–Huckel term to become a multiplier

Table 4Ion-size parameters (10�10 m) in Eq. (7) for calculating the molar volume of aqueous species. reff,optimized is calculated from x in Table 2; reff,SUPCRT is from Johnson et al. (1992); a is the ion-size parameter used in the extended Debye–Huckel equation (numbers for cations are fromPHREEQC.DAT).

Na+ K+ Mg2+ Ca2+ Sr2+ Ba2+ Fe2+ OH� Cl� HCO3� CO3

2� SO42�

reff,optimized 2.64 2.64 2.76 2.87 3.43 4.43 4.81 1.23 1.37 1.56 1.80 2.44reff, SUPCRT 1.91 2.27 2.54 2.87 3.00 3.22 2.62 1.40 1.81 2.26 2.87 3.21a 4.0 3.5 5.5 5.0 5.26 5.0 6.0 0 0 0 0 0

Fig. 10. The limiting slope resulting from the simple (Av I0.5, lines)and extended (Av I0.5/(1 + a Bc I0.5), dashed lines) Debye–Huckelequation, and the logarithmic constraint (ln(1 + bAv I0.5)/bAv,dotted lines) as a function of temperature for I = 1 (red) and 0.1(blue). a = 5 A. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of thisarticle.)


for the activity coefficient of a salt. However, the extendedDebye–Huckel term is left out in the pressure derivative forthe molar volume. Pitzer (1986) notes that this is based on aredundancy that appears in the second virial coefficient ofhis equation. In the early applications (Rogers and Pitzer,1982; Phutela and Pitzer, 1986), the value of bAv was fixedto 1.2. However, the combination of b1..3 terms in Eq. (6),give a value for b that goes from slightly positive to slightlynegative, from 25 to 200 �C. With the smaller correction ofthe limiting slope by the logarithmic control, b would shiftto more negative values, and, perhaps, this is the reasonwhy RMSD in the optimization decreases about twofoldwhen the extended Debye–Huckel is used.

The third improvement, obtained by setting V m;Hþ ¼ 0at all T, P, and I, may seem to be built on an inconsistencyin that the proton contributes to the ionic strength, while itsvolume remains zero, even though the electrostatic excessenergy that results from increased ionic strength affectsthe absolute volume of the proton as well as of the otherions. However, the convention simply selects a suitable ref-erence level for calculating the volumes of ions. One may setthe proton’s volume to zero at I = 0 and let it increase withthe Debye–Huckel limiting slope, or let it remain zero andassign the actual volume change to the anion that

accompanies the proton when I increases. The choice de-pends on what is most convenient in practical applications,and, because ions were shown to behave more ideally, V m;Hþ

is chosen to be 0 at all T, P, and I.

6. CONCLUSIONS

For calculating pressure effects on hydrochemical reac-tions, the molar volumes of single ions can be derived froma combination of the parametrization developed by Helge-son et al. (1981) for the intrinsic volume, and the Redlich–Rosenfeld equation for the influence of ionic strength. Thepressure derivative of the Born equation, which is part ofthe equation developed by Helgeson and coworkers, pro-vides the concave downward change of the intrinsic molarvolumes with temperature, and avoids numerous polyno-mial terms for fitting the temperature dependence. Theapplicability of the Redlich–Rosenfeld equation can be im-proved substantially if the pressure derivative of the ex-tended Debye–Huckel equation is used instead of thesimple form. Furthermore, by defining the volume of theproton to be 0 for all T, P, and I, the molar volumes tendto conform better to Debye–Huckel limiting slopes. Thesetwo modifications are also helpful for reducing the numberof parameters needed to fit molar volumes as a function oftemperature, pressure, and ionic strength. The Peng–Rob-inson equation of state uses estimates for the Van der Waalslimiting volume and attraction factor that are derived fromthe critical temperature, the critical pressure, and the acen-tric factor, and allows calculation of the fugacity coeffi-cients of pure gases and mixtures of gases. Withoutfurther adjustable parameters, the Peng–Robinson modelis able to calculate the solubility of CO2 and other gasesin water at the high pressures encountered in CCS systemsquite well.

The equations described above, comprising a hybrid ofpublished models to account for volume as a function oftemperature, pressure and ionic strength, have been imple-mented in PHREEQC, version 3. The parameters for calcu-lating the molar volumes of aqueous species and thefugacity coefficients of gases have been added to two ofthe databases distributed with PHREEQC, PHRE-EQC.DAT and PITZER.DAT. The program can be usedto calculate the solubility of gases and minerals from 0 to200 �C, and 1 to 1000 atm.

ACKNOWLEDGMENTS

We thank Jean-Franc�ois Boily for handling this manuscript forGCA, and Dennis Bird, Andy Felmy, Christophe Monnin and 2


more reviewers, as well as Alex Blum for taking time to read thepaper carefully and offering suggestions and ideas, which have beenused to clarify the text and to improve the fits of coefficients in thePHREEQC databases. Special thanks go to Lingli Wei for inform-ing one of us (CAJA) about Peng–Robinson.

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