Calculation of activity coefficients and degrees of formation of neutral ion pairs in supercritical electrolyte solutions

0016.7037/91/$3.00 + .OO

Calculation of activity coefficients and degrees of formation of neutral ion pairs in supercritical electrolyte solutions ERIC H. OELKERSand HAROLDC. HELGESON Department of Geology and Geophysics, University of California, Berkeley,CA 94720, USA (Rclceived January

I I, 1990: accepted in revisedjbrm Fehruury 22, I99 1 )

Abstract-Although hydrothermal solutions at high temperatures and low pressures where the solution density is low are widely regarded as highly associated alkali chloride solutions in which neutral ion pairs are the predominant solute species, reconsideration in the present study of experimental conductance data reported by QUIS~ and MARSHALL( 1968a,b,c), DUNN and MARSHALL( 1969) and FRANTZ and MARSHALL( 1984) for 1:1 electrolytes suggests that this may not be the case.’ Quist, Marshall, and their coworkers interpreted their high temperature/pressure conductance data in part by considering the activity coefficients of the neutral species to be negligibly different from unity in dilute solutions. However, these data can also be interpreted by regarding the logarithm of the activity coefficients of neutral complexes (7 ,,) to be a linear function of the effective ionic strength (f), which is consistent with both the SETCH~~NOW ( 1892) and HOCKEL ( 1925 ) equations ( HELGESONet al., 198 1). Accordingly, supercritical conductance data for five electrolytes were regressed in the present study using the H~~CKEL( l925), SETCH~NOW ( 1892). and SHEDLOVSKY( 1938) equations, together with the law of mass action and values of the limiting equivalent conductances of the electrolytes taken from OELKERSand HELGESON( 1989) for the temperatures and pressures of the experiments to generate simultaneously the logarithms of dissociation constants (log K, ) and SetchCnow coefficients (b,,,) for neutral 1: 1 complexes. The resulting values of log K, are in close agreement with corresponding values originally reported by Marshall and coworkers. However, the regression calculations indicate that h,, and thus the activity coefficients of neutral species in concentrated electrolyte solutions increase dramatically with increasing temperature and decreasing pressure, which opposes the effect on ion pair formation of the accompanying decrease in log K, . Distribution of species calculations using these values of log K, and h,., with and without provision for the formation of triple ions ( OELKERSand HELGESON,1990) indicate that the degrees of formation of neutral 1:1 ion pairs maximize at stoichiometric ionic strengths of -0.2 molal and become negligible at stoichiometric ionic strengths greater than -0.8 molal in the temperature range 400 to 800°C at pressures to 4 kb. Consequently, it appears that these neutral species may account for only a minor part of the solute in concentrated hydrothermal solutions at supercritical temperatures and pressures. invalid in concentrated electrolyte solutions. To test whether this is indeed the case, activity coefficients were computed in the present study by regressing the supercritical conductance data reported in the literature for dilute solutions of NaCl, NaBr, NaI, HCl, and HBr using limiting equivalent conductances and the law of mass action, together with the HOCKEL (1925), SHEDLOVSKY(1938), and SETCH~NOW ( 1892 ) equations. By assuming that the Hiickel equation and retrieved SetchCnow coefficients determined in dilute solutions at supercritical temperatures and pressures are valid at higher concentrations, the degree of formation of neutral ion pairs in 1: 1 electrolyte solutions were calculated at concentrations to 1 molal. The purpose of this communication is to present the results of these calculations and consider their effect on predicted degrees of formation of neutral ion pairs in hydrothermal solutions at supercritical temperatures and pressures.

INTRODUCTION MANY MEASUREMENTS OFTHE conductances of supercritical electrolyte solutions have been reported in the literature over the past 30 years (FOGO et al., 1954; FRANCK, 1956a,b,c; PEARSON et al., 1963a,b; QUIST et al., 1963, 1965; QUIST and MARSHALL, 1966, 1968a,b,c,d, 1969; RITZERT and FRANCK, 1968; MANGOLD and FRANCK, 1969; DUNN and MARSHALL, 1969; HWANC et al., 1970; FRANTZ and MARSHALL, 1982, 1984). Because the goal of most of these experimental studies was to retrieve reliable values of the thermodynamic dissociation constants for neutral ion pairs from the conductance data obtained for dilute solutions (~0.1 m), activity coefficients of the neutral ion pairs (7.) were assumed to be unity. Although it has been demonstrated that this procedure introduces negligible uncertainty in the derived dissociation constants (OELKERS and HELGESON, 1988), the distribution of the conductance data with increasing concentration indicates that the assumption that 7n = 1 may be

CALCULATION OF SETCHfiNOW COEFFICIENTS AND DISSOCIATION CONSTANTS FROM CONDUCTANCE DATA

’ This observation is supported also by electrostatic considerations, which indicate that higher order complexes may predominate in concentrated electrolyte solutions at supercritical temperatures and pressures (PITZER and SCHREIBER, 1987; OELKERS and HELGESON, 1990).

The standard state for aqueous species adopted in the present study is one of unit activity of the solute in a hypothetical one molal solution referenced to infinite dilution at any pres1235

1236

E. H. Oelkers and H. C. Helgeson

sure and temperature. The activity coefficient of the ion pair, MX”(yM,e) in a single 1: 1 electrolyte solution in which the solute is designated by MX can be computed from conductance data by first taking account of ( HELGESON et al., 198 1) log

MP%M++Xcan be expressed

?M,@’

=

b,,.uxo~+

r7,

~0.1 m. K’,

electrolyte solutions with solute concentrations for the dissociation of MX” in accord with

(4)

as

(1)

where br,hl,p represents the short range interaction parameter for the ion pair MX’, which depends only on temperature, pressure, and the identity of the electrolyte, fstands for the effective ionic strength of the solution in molality units of concentration, which can be expressed as I=

t C Zfm,,

I and I’, denotes the mole fraction to molality conversion given by Tr = -log

(1 + O.O180153m*),

(2) factor

(3)

where m, and Z, designate the molality and the charge of the ith aqueous species, respectively, and m* refers to the sum of the molalities of all solute species in solution. Equation ( 1) represents a form of the SETCH~NOW ( 1892) equation for the molality scale of concentration. The Setchenow equation has been used with success to describe the activity coefficients of neutral species (including dissolved gases and neutral ions pairs) in aqueous electrolyte solutions at 25°C by RANDALL and FAILEY (1927a,b,c), COHN and EDSALL (1943), SCHNEIDER (1969), CAMPBELL et al. (1970), CONWAY and NOVAK (1975), KONNIK (1977), YASUNISHI and YOSHIDA (1979), ARMENATE and KARLSSON (1982), and CLEVER ( 1983), and at elevated temperatures to 350°C by ELLIS and GOLDING ( 1963),CLEVER and HOLLAND ( 1968 ). HELGESON (1969), MARSHALL (1980), MASON and KAo (1980), HELGESON et al. (1981), CHEN and MARSHALL (1982), MARSHALL and CHEN (1982) and others.2 Accordingly, Eqn. ( 1) was adopted in the present study to describe the activity coefficients of neutral species in supercritical aqueous electrolyte solutions. br,Me in Eqn. ( 1) is commonly referred to as the SetchCnow coefficient. Attempts to characterize Setchenow coefficients for a large number of neutral aqueous species at various temperatures have been made using electrostatic theory (DEBYE~~~ MACAULAY, 1925; LONG and MCDEVIT, 1952), scaled particle theory ( MASTERDON and LEE, 1975 ), and semi-empirical correlations (VAN KREVELEN and HOFTIJZER, 1948 ). Setchtnow coefficients were calculated in the present study from the concentration dependence of the effective dissociation constant (K’, ) of the neutral species MX” in single

(5) where K1 refers to the molal standard state dissociation constant, a,+,+and ax- correspond to the activities of the cation (M+) and anion (X-), respectively, mM+, mx- and mM,@ designate the molalities of the subscripted species, -i;*,+,* and m stand for the mean molal ionic activity coefficient and total molality of the electrolyte, and LY,,,~o represents the degree of association of the neutral species MX” defined by

(6) which can be expressed for solutions in which the only predominant solute species are W, X-, and/or MX” as I-i a,+,,@ = -

(7)

’

where I denotes the stoichiometric ionic strength, which is equal to m for 1: 1 electrolytes.3 Combining the logarithmic analog of the first equality in Eqn. (5) with Eqn. ( 1) leads to log K; = log K, -t by,,,,,&+

r7.

(8)

Because standard state dissociation constants (K, ) and Sechtnow coefficients (b 7,Mxo)depend only on pressure, temperature, and the identity of the electrolyte, and because the absolute value of T7 is ~0.00 1 for m 5 0.1 m, it follows from Eqn. (8) that log K’, can be regarded as a linear function of F for these molalities with a slope and intercept equal to b?,MXO and log K, , respectively. This linear relation was used to generate graphically values of log K, and br,MXousing effective dissociation constants calculated from Eqn. (5) and values of + k,Mx and (Y,+,x~computed in the manner described below. Values of the mean molal ionic activity coefficient of the single electrolyte represented by MX in Eqn. (5) can be calculated from the Hiickel equation, which can be written for the molality scale of concentration as (HELGESON, et al., 1981)

log Yt,Mx = * Although MARSHALL( 1980) argued on empirical grounds that the Setchtnow equation should be used only in conjunction with the molarity concentration scale, consideration of the effects of ion pair formation on r and taking account of r7 indicates that his experimental data can be described within the uncertainty of the data with Eqns. (l)-(3). 3 Although triple ions and higher order complexes may form in supercritical single electrolyte solutions, they are apparently present in negligible concentrations if the total molality of the solute is 50. I m at the pressures and temperatures considered in this study (OELKERS and HELGESON,1990).

I

A,I&+Zx-

Ii”? + r +

1 + 6B,P2

y

b

J

Y,MA 1

(9)

where Z,+ and Z,- refer to the charge on the cation and the anion, respectively, br,MXand d represent the extended-term and ion size parameters for the electrolyte, which are functions only of temperature and pressure, and A, and B, represent the Debye-Htickel coefficients in units of kg’j2 mol-‘I’ and kg’/’ mol-“* cm-‘, respectively, defined by A, =

1.8248 X 106P”2 (cT)~‘*

Activity coefficients in supercritical electrolytes

1237

Table 1. Molal solute concentrations for which equivalent conductances were considered in the present study.

and

(11) where T designates temperature in K, and t and p stand for the dielectric constant and density in g cmm3 of H20. (~~~0 in Eqn. ( 5 ) can be calculated from conductance data for dilute single I: 1 electrolyte solutions by taking account of (SHERILL and NOYES, 1926; MACINNES, 1926; MACINNES and SHEDLOVKSY, 1932; HARNED and OWEN, 1956; ROBINSON and STOKES, 1968 ) (1 -CY,@)=-_

(A,A”

A,

+ B,,)f”*,

(13)

(14) and = 82.48~“~ A

am”

’

(15)

where t) refers to the viscosity of H20 in poises. Equations (7) and ( 12) to ( 15) were used in the present study, together with experimental equivalent conductances reported in the literature ( QUIST and MARSHALL, 1968a,b,c; DUNN and MARSHALL, 1969; FRANTZ and MARSHALL, 1982 ) and limiting equivalent conductances evaluated from these data by OELKERS and HELGESON ( 1989) to compute values of ?and 01~~0for single electrolyte solutions at several dilute concentrations. These values were then used together with Eqns. (3), (5), and (9) to ( 11) to calculate corresponding values of log K’, , which were regressed graphically with Eqn. (8) to generate simultaneously values of b,,,o and log K, from the concentration dependence of log K\ . Regression

NaBr*

Na13

0.001000 0.005023 0.01000 0.01492

0.002010 0.005000 0.007000 0.01000

0.0009996 0.0050010 0.010000 0.05000

0.0 1994

0.01496

0.10000

HC14

HBr’

0.002 0.005 0.01

0.001957 0.004937 0.01000 0.01471

( 12)

A,

where Ah and B,, for a 1: 1 electrolyte in units of kg”* mol-I’* and kg”* mall”* cm2 ohm-’ equiv-‘, respectively, can be computed from

B

NaCl’

0.04942 0.1000

where A, stands for the experimentally determined equivalent conductance of the electrolyte solution and A, corresponds to the equivalent conductance of the hypothetical completely dissociated electrolyte in a solution of the same effective ionic strength (f). Values of A, can be computed from the limiting equivalent conductance of the electrolyte (A’) using the SHEDLOVSKY ( 1938) equation, which is given by A<>= A0 - $

Electrolyte

Calculations

Experimental measurements of the equivalent conductances of aqueous NaCl, NaBr, NaI, HBr, and HCl at supercritical pressures and temperatures have been reported by QUIST and MARSHALL ( 1968a,b,c), DUNN and MARSHALL ( 1969), and FRANTZ and MARSHALL ( 1984) for the concentrations given in Table 1. Regression of these data with Eqns. (3),(5),(7), and(9)to( 15)resultedin thevaluesof log K’, and rshown in Figs. 1 through 5. The diagrams in these figures are arranged from left to right and bottom to

1) Quist and Marshall (1968a) 2) Quist and Marshall (1968b) 3) Dunn and Marshall (1969) 4) Fran& and Marshall (1984) 5) Quist and Marshall (1968~)

top in ascending order of temperature and pressure, respectively. The curves in Figs. 1 through 5 are consistent with both the linear relation between log K’, and irepresented by Eqn. (8) and a smooth variation of the intercepts (log K,) and slopes (h+~ 0) with increasing or decreasing pressure and temperature. The values of log K, for NaCl, NaBr, NaI, HCl, and HBr corresponding to the intercepts of the curves shown in Figs. 1 through 5 are listed in Table 2 and plotted as functions of pressure at constant temperature in Fig. 6. The slopes of the curves in Figs. 1 through 5, which correspond to the Setchenow coefficients of the neutral ion pairs, are depicted as symbols in Fig. 7. The values of the density and viscosity of Hz0 used in the present study were generated from equations and parameters reported by HAAR et al. ( 1984) and WATSON et al. ( 1980), respectively. Corresponding values of the dielectric constant of H20 were computed from equations and parameters given by HELGESON and KIRKHAM (1974) at temperatures c 500°C and those adopted by PITZER ( 1983) for higher temperatures. Values of d in Eqn. (9) were calculated from algorithms reported by HELGESON et al. ( 198 1)) which have been corrected by HELGESON ( 1982), together with equations describing the temperature and pressure dependence of the effective electrostatic radii of ionic species given by SHOCK et al. ( 199 1). As a first approximation, b,,, in Eqn. (9) was taken to be equal to by,NaCI at all pressures and temperatures, which is consistent with the similarity in magnitude of Born coefficients and short-range interaction parameters, respectively, for 1: 1 electrolytes ( HELGESON et al., 198 1; SHOCK et al., 1991). The values of b,,Nacl were generated from equations and parameters given in Appendix B of OELKERS and HELGESON ( 1990), which are consistent with the revised HKF equations of state for the standard partial molal thermodynamic properties of aqueous species (TANGER and HELGESON, 1988; SHOCK and HELGESON, 1988; SHOCK et al., 1989). Although considerable scatter is apparent in the distribution of the symbols shown in Figs. 1 through 5, it can be seen that in most cases the experimental data represented by the

TEMPERATURE

TEMPERATURE

FIG. I. Logarithm of the effective dissociation constant (log K’, ) of NaCl’ as a function of effective ionic strength (f) in NaCl solutions at temperatures from 400 to 800°C and pressures to -4 kb (see text).

500 *c

550 *c

TEMPERATURE

450 T

6OOOC

TEMPERATURE

FIG. 2. Logarithm of the effective dissociation constant (log K’,) of NaBr” as a function of effective ionic strength (1) in NaBr solutions at temperatures from 400 to 800°C and pressures to -4 kb (see text).

400 T

1240

E. H. Oelkers and H. C. Helgeson

W

a 3 cn cn

-22 0

002RMoiaeRm30)0

I mole kgm-’

1

mole

kgm-’

0

Ro2004om008om f

W (II CL

f

mde

kgm-’

mole

kg&’

1 mde kgm-’

?i mde kg&x

IO*

65O*C 7OO*C 75O*C 8OO*C TEMPERATURE

TEMPERATURE FIG. 3. Logarithm of the effective dissociation constant (log K;) of NaI’ as a function in NaI solutions at temperatures from 400 to 800°C and pressures to -4 kb (see text).

of effective

ionic

strength(F)

800°C and pressures

constant

450 OC

the effective dissociation to -4 kb (see text).

FIG. 4.Logarithm of

400 OC

550 Oc

(log K’I ) of HCl” as a function

OC

from 400 to

700

at temperatures

650 OC

(I) in HCl solutions

600 OC

of effective ionic strength

TEMPERATURE

500 OC

1242

E.

-

H. Oelkers and

H. C. Helgeson

Activity coefficients in supercritical electrolytes symbols are consistent with the linear curves shown in the figures. In certain cases, the interpretation of the data represented by the linear curves shown in Figs. 1 through 5 is permissive, rather than compelling; i.e., a non-linear curve or a linear curve with a different slope could be chosen to represent equally well the data shown in some of the diagrams. However, the fact that a linear curve represents the simplest interpretation of the data and the constraint that log K, and be smooth functions of temperature and pressure genhY,M_Xe erally precludes such freedom of choice. It should be emphasized in this regard that the commonly made but untenable assumption that neutral ion pairs have unit activity coefficients in supercritical electrolyte solutions would require linear curves with zero slopes in Figs. 1 through 5, which would be inconsistent with the systematic increase in log K’, with increasing Iexhibited by the bulk of the data represented by the symbols in these figures. It can be seen in Fig. 6 that the log K, values shown in the figure (which correspond to those given in Table 2) decrease with increasing temperature and decreasing pressure, which favors neutral ion pair formation in dilute high-temperature/ low-pressure solutions. These log K, values are consistent with those generated by OELKERS and HELGESON(1988) from conductance data reported by QUISS and MARSHALL ( 1969), who studied many electrolytes over the equivalent temperature and pressure range, but at different pressures for the same temperatures. Comparison of the log K, values in Table 2 with those obtained originally from experiments by QUIST and MARSHALL( 1968a,b,c), DUNN and MARSHALL ( 1969), and FRANTZ and MARSHALL( 1984) can be made in Fig. 8. Note that in every case the symbols fall within 0.2 log units of the linear curves representing equal values of log K, at a given pressure and temperature, which justifies the assumption made by Quist, Marshal and their coworkers that the activity coefficients of neutral ion pairs can be regarded as unity in the dilute solutions (~0. I m) considered in their experiments without intr~ucing significant unce~ainty in the values of log K, calculated from the data. However, Eqn. ( 1f and the curves shown in Figs. 1 through 5 indicate that this assumption is not applicable to concentrated solutions. To permit interpolation of the values of h,,,+,,Y~ shown in Fig. 7 over the whole range of pressure and temperature represented by the array of experimental data, the h,,,,,,-o values were regressed with a power function of the form bY.,,l‘to= i : a;,,n,yo( T - 273.15)‘$, i-0 ,=-,

(16)

where uli.>4j,uo denotes regression coefficients for the neutral species. Values of a,,, a,y0for NaCl’, NaBr*, NaI*. HCl’, and HBr” are given in Table 3. Equation ( 16) was used together with the regression parameters shown in Table 3 to generate the curves depicted in Fig. 7. The dashed parts of the curves in this figure represent computed extrapolations of the experimental data depicted by the symbols4 It can be seen in Fig. 7 that the curves represent closely 4 Although it has been argued that the temperature

and pressure dependence of /?.+e can be described solely as a function of the dielectric constant of Hz0 (K. DING, 1990. pers. comnr. elaborating on DING and SEYFRIED, 1990). the values of h-r,b,A~generated in the present study indicate that this is not the case,

1243

Table 2. Logarithms of the association constants of neutral ion pairs (log KI) retrieved from the experimental data depicted in Figs. l-5 (see text). for NaCl”

LogK,,

Density, g cmm3

0.35 0.40 0.45 0.50 0.55 0.60 0.65 285 :2:62 250 :2:26 214 :1:90 Ill :3:38 314 :2:98 180 :I:55 146 :1:20

‘1

-3.50

-3.14

-2.71

-2.35

-1.97

-1.61

-1.33

-3.63 -3.93 -4.03 -4.15 -4.23 -4.27

-3.28 -3.36 -3.41 -3.55 -3.73 -3.15

-2.80 -2.87 -2.98 -3.03 -3.12 -3.26

-2.41 -2.41 -2.55 -2.60 -2.67 -2.71

-2.05 -2.10 -2.16 -2.20 -2.24 -2.24

-1.71 -1.75 -1.78 -1.84 -1.90 -1.87

-1.41 -1.48 -1.50 -1.52 -1.51 -1.52

0.70

-1.17 -1.21

-

1

Lo@,, for NaBr”

rT, “C

Density, g crnT3 0.40

0.35 400 450 500 550 600 650 700 750 800

0.45

-2.58 -2.83 -3.08 -3.25 -3.32 -3.47 -3.56 -3.62

-3.09 -3.32 -3.60 -3.72 -3.90

1

0.50

-2.23 -2.39 -2.50 -2.64 -2.70 -2.82 -2.96 -3.12

-1.94 -2.01 -2.14 -2.22 -2.30 -2.38 -2.45 -2.61

T, “C 0.45 -.201 -2.21 -2.27 -2.36 -2.45 -2.52 -2.62 -2.70 -2.76

T, “C

0.45

0.50

0.55

-5.69

-5.15

-4.61 -4.91 -5.29 -5.63 -5.84

-4.02 -4.31 -4.65 -4.95 -5.23

“(

450 500 550 600 650 700 750 800

IO:96 -1.10 -1.15 -1.21 -1.28 -1.38 -1.42 -1.53

-1.02 -1.08

0.50 -. -1.80 -1.86 -1.93 -2.01 -2.12 -2.16 -2.25 -2.32

0.60 -_ -1.15 -1.28 -1.40 -1.43 -1.56 -1.60 -1.72 -1.80

0.55 -1.44 -1.53 -1.59 -1.65 -1.76 -1.86 -f .94 -2.00

0.65

-1.00 -1.08 -1.24 -1.31 -1.52

for HCl”

Density, g cm-3 0.40

Lo@,, r.

0.70

Density, g cmw3 0.40 -.2 16 -2.47 -2.60 -2.72 -2.85 -2.97 -3.02 -3.12 -3.22

1 0.35 I -.246 -2.89 -3.03 -3.16 -3.28 -3.40 -3.49 -3.53 -3.64

LogK,,

400 450 500 550 600 650 700

-1.64 -1.74 -1.79 -1.86 -1.91 -2.01 -2.13 -2.22

0.65

0 90

for NaI”

Lo@,,

400 450 500 550 600 650 700 750 800

0.60 -1.35 -1.39 -1.45 -1.52 -1.60 -1.65 -1.73 -1.81

0.55

0.60 2 94 :3:21 -3.48 -3.17 -4.10 -4.37 -4.58

0.65

0.70

0.75

0.80

-2.71 -3.02 -3.32 -3.61 -3.85 -4.04

-2.25 -2.60 -2.87 -3.13 -3.36 -3.54

-1.84 -2.17 -2.44 -2.61

-1.40 -1.74

for HBr”

Density, g cmm3 0.35

0.40

0.45

4.60 5.17

-3.95 -4.26 -5.02

-3.48 -3.86 -4.26 -4.60 -4.94

0.50 241 :2:88 -3.27 -3.60 -3.93 -4.23 -4.54 -4.72 -5.03

0.55 208 :2:41 -2.72 -3.04 -3.32 -3.62 -3.91 -4.05 -4.31

0.60 170 :1:89 -2.15 -2.44 -2.77 -3.08 -3.35 -3.52 -3.68

0.65 124 :I:50 -1.75 -2.03 -2.32 -2.57 -2.80 -2.99 -3.23

0.70

0.75

-1.03 -1.33 -1.63 -1.92 -2.15 -2.37

-0.98 -1.23 -1.51 -1.73 -1.97

0.80

-0.95 -1.23

E. H. Oelkers and H. C. Helgeson

1244

x-

- 2.5

3 -35

J 5

- 4.5 p

PRESSURE. KB

4

-,5w

5

5 PRESSURE. KE

PRESSURE. KB

I - 2.5

J

5

- 3.5 - 4.5

- 5.5b

I

-550

2

FIG. 6. Logarithms of the dissociation constants of NaCl’, NaBr’, NaI”, HCI’, intercepts ofthe curves depicted in Figs. 1 through 5 (symbols) as functions ofpressure in “C). The curves represent graphic interpolation of the intercepts.

PRESSURE. KB

3

4

5

PRESSURE, KB

PRESSURE. KB

and HBr’ corresponding to the at constant temperature (labeled

PRESSURE, KB

PRESSURE. KE

300

200 o:: A 100

0

I.0

21)

3.0

4.0

PRESSURE, KB FIG. 7. Setchenow coefficients for NaCI’, NaBr’, depicted in Figs. 1 through 5 (symbols) as functions were generated from Eqn. ( 16) using the parameters

5.0

0

I

2

3

4

5

PRESSURE, KB

NaI’, HCI’, and HBr’ corresponding to the slopes of the curves of pressure at constant temperature (labeled in “C). The curves given in Table 3 (see text)

Activity coefficients in supercritical electrolytes

1245

- 0.5 f

- 1.5

i -2.5 UJ - 3.5 9 P -4.5

3

-5.5

-4.5

-35

-2.5

-1.5

-0.5

J

-5.5 STuDY, LoG Kl.NoBrfPREsENT

LOG K, (PRESENTSTUDY)

I Q

-2.5

r a

-35

5

-4.5

3

- 5.5

I r’-’

-45

-35

‘OG Kt.NoI

-2.5

-1.5

-a5

@‘RESENT STWY)

- 2.5

-4.5 LOG Kl

-3.5 ,+.I

-2.5

- I .5

-a5

P+tESEMsNoY)

LOGKIHBr wREsEMsNDy)

FIG. 8. Logarithms of dissociation constants for NaCl “, NaBr”, NaI”, HCI’, and HBr’ reported by QUIST and MARSHALL( 1968a,b,c), DUNN and MARSHALL( 1969), FEUNTZand MARSHALL.( 1984) plotted against those generated in the present study (see text).

the pressure and temperature dependence of by,MXoand that calculated independently by HELGESON the values of by,NaC,~ et ai. ( 198 1) for 250,275, and 300°C at PSAT5and POKROVSKII and HELGESON ( 1991) at 400,500, and 600°C from 0.5 to 2 kb are consistent with those generated at supercritical pressures and temperatures in the present study, which strongly supports the interpretation of the data represented by the curves in Figs. 1 through 5. The values of br,NaCIo in ’ PsATrepresents pressures corresponding to liquid-vapor equilibrium for the system HzO, except at temperatures < 100°C where it refers to the reference pressure of 1 bar.

Fig. 7 are also similar in magnitude and behavior to the Setchenow coefficients for aqueous HZ and HlS computed by DING and SEYFRIED ( 1990) from the solubility of H2 gas and H2S gas in NaCi solutions coexisting with pyrite, pyrrhotite, and magnetite at tem~ratu~s from 300 to 425°C and pressures from 300 to 500 bars. Setchenow coefficients computed from Eqn. ( 16) and the parameters listed in Table 3 are given in Table 4 for temperatures and pressures from 400-800°C and 0.5-5 kb in increments of 50”and 0.5 kb. It can be seen in Fig. 7 that b +,,p increases in response to increasing temperature and decreasing pressure, which is opposite to the behavior exhib-

Table 3. Parameters for use in Eqn. (16) to compute Setchkmv coefficienu for 5 neutral ion pairs.

i,j o,-1 ~1,-I z-1

3.-l 4.-l 0.0 I.0 2.0 3,O 0,l j Ll / 2.1 / 0,2 / 1.2 I 0,3

NaCI”

NaBP

1.064859 x100

1.598052 Xl02

8.249023 xl@

9.522827 xtW2 -2.022128 x10-* 1.173785 x10-7 1.439940 x10-” -7.738881 x10’ -1.137903 x10-1 3.541372 x10-’ -2.432701 x10-’ 2.309298 x 10’ -1.263004 x10-’ 3.351660 x IO-’ -2.182633 xtb 5.366976 x IO-’ 8.009462 x10’

-4.638729 x10-’ 7.599159 x10-4 -6.394614 x10-’ 2.285512 x10-” -5.604571 x10* 6.430498 x10-’ -3.178222 x10-’ 1.720241 x10-* 1.075740 x103

-5.193119 x10-’

-7.846173 x10-’ 2.050769 x lo4 -9.200614 xld 3.440019 x10-1 2.765662 x10’

NSP

2.385547 x IO-’ -9.851578 x10-’

HCI” 1.951585 Xl@ -1.726522 x10’ 3.728887 xIO-~

HBP -5.080346 x10’ -3.565666 x10’ 1.074747 x10-2

-2.672885 x~O-~

-2.172120 x10-’

- 1.022092 x lo-‘0

1.336374 x10-*

-8.502564 x10-”

-1.053926 x10’

4.746038 x10’ 1.072841 x10’

-1.246493 x100

-1.689762 x10-’ -5.188445 xlOd

3.301488 x103

-3.894041 x10-* -3.393355 x10-6

-2.972109 x10-*

6.396254 x10-’

1.186646 xld 7.542943 x10-’ -6.007195 ~10~ -3.624652 xl@ 4.514899 x10-* 1.113304 xl02

-1.795720 X104 2.032616 x10’ 2.806229 x lo-’ 1.403626 x104 -2549746 x10’ -6.198890 xi0’

-6.751774 x103 2.378356 x10’ 1.846489x10-2 4.915252 xld -2.062484 x10’ 3.754412 x103

2.776214 x~O-~

1246

E. H. Oelkers and H. C. Helgeson

Table 4. Setchehow coefficients computed from Eqn. (16) and the parameters listed in Table 3.

1

b -,,NsCI’ P, kb 400 07 0:3 0.2

-ox1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

450 30 0:6 0.3 0.2

500 1.2 0.6 0.4 0.3

Temperature, 550 600 2.2 1.0 0.7 0.5 0.5

“C 650

1.6 1.0 0.8 0.7 0.6

2.3 1.4 1.1 0.9 0.8 0.8

700

2.1 1.5 1.2 1.0 0.9 0.9

750

800

2.1 1.6 1.3 1.1 0.9 0.9

3.2 2.3 1.7 1.3 1.1 0.9

750

800

the assumption that Y,, = I and the behavior of log K, as a function of temperature for neutral ion pairs in the supercritical region (Fig. 6), the curves in Fig. 9 indicate that increasing temperature at constant pressure in this region leads to increasing association of ion pairs only in dilute solutions. At higher concentrations increasing temperature at constant pressure in the supercritical region causes increasing dissociation of neutral ion pairs.

b 7,NBBP P.kb

1 1 400

450

500

1.7 0.6

2.1 1.1 0.3

2.7 1.7 1.0 0.5

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

.

Temuerature. 550 - 600

’

‘C 650

700

I

4.2 2.4 1.8 1.2 0.8 0.4 0.2

3.2 2.4 2.0 1.6 1.2 1.0 0.8 0.7

3.2 2.6 2.2 1.9 1.7 1.5 1.4

4.1 3.2 2.7 2.4 2.2 2.1 2.0

4.0 3.3 2.9 2.6 2.5 2.4

5.0 3.9 3.3 3.0 2.8 2.7

Temperature, 550 600

“C 650

700

750

800

4.2 3.3 2.8 2.4 1.9 1.5 1.1 0.6 0.3

5.1 3.7 3.2 2.8 2.5 2.2 1.9 1.6

4.6 3.1 3.3 2.9 2.6 2.3 2.0

4.4 3.8 3.4 3.0 2.7 2.4

5.0 4.4 4.0 3.6 3.3 3.0

b r,NaI’

1 P, kb 400 26 0:3

05 1:o 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

450 36 2:6 1.5 0.4

500 3.3 2.8 2.1 1.4 0.8

3.8 3.2 2.8 2.5 2.1 1.8 1.5 1.2

The different configurations of the curves in Figs. 9a and b have a profound effect on the degree of association of neutral ion pairs ( aMe) with increasing concentration. The curves in Fig. 9 indicate that LY~~Oat high temperatures and low pressures is large in dilute solutions, maximizes at intermediate values of m, and becomes small in concentrated solutions. Hence, contrary to what is commonly deduced from

DEGREE OF FORMATION OF ION PAIRS AS A FUNCTION OF CONCENTRATION It follows from Eqn. (6) that ol,+,e can be calculated for a given total molality if mMXOis known. However, calculation of accurate values of mMXo(and therefore 01~~0) requires explicit consideration of all the solute species in solution. It has been demonstrated ( OELKERS and HELGESON, 1990) that

b r.HCI’ P, kb 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

L

I

400 50.0 13.1 8.4

450

500

29.7 13.2 7.7 6.3

63.8 31.4 19.0 12.1

Temperature, 550

65.7 39.6 26.5 17.4 9.7 2.6

‘C 600

78.5 50.5 34.3 22.3 11.8

650

700

98.0 64.9 44.1 28.3 14.7

126.8 86.7 59.9 39.5

b y,HBr’ I‘,

ilT 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 -5.0

400 @O -8:5 7.9

450 498 11:6 5.5 4.0

500 24.9 9.2 4.6 2.7 2.3

Temperature, 550 600 100.5 22.8 10.0 5.8 3.8 2.8 2.5

70.7 25.2 13.2 9.0 7.2 6.2

‘C 65Cl

700

750

65.3 29.7 18.1 14.0 12.6 12.1

65.6 35.0 24.0 20.4 19.7

67.4 40.4 30.3 27.6

800

68.5 44.8 36.2 35.0

P = 1.0 molol

-2.0

ited by log K, in Fig. 6. Hence, in dilute solutions log K’, decreases with increasing temperature and decreasing pressure, but in concentrated solutions log K; increases with increasing temperature and decreasing pressure. This behavior is apparent in Fig. 9, where log K’, for NaCl is plotted as a function of temperature and pressure for I = 0.1 and 1.O m.

400

800 TEMPEEk”RE.

co

FIG. 9. Logarithm of the effective dissociation constant (log K’,) of NaCl’ as a function of temperature at constant pressure (labeled in kilobars) for f = 0.1 (A) and 1.O molal ( B) computed from Eqn. (8) using values of 6, respectively.

b7,NaCI~ and log K, taken from Table 4 and Fig.

1247

Activity coefficients in supercritical electrolytes substantial triple ion ( M2X+, MX :) formation may occur in moderately concentrated single electrolyte solutions at supercritical temperatures and pressures where the dielectric constant of Hz0 is 520. Taking account of single ions, neutral ion pairs, and triple ions, the total molality of the electrolyte can be expressed as m = mM+ + mMXOf 2mM,.Y+ f mMXr.

NaCl 500°C

(17)

The molalities of the triple ions in Eqn. ( 17) can be computed from the law of mass action expressions for the dissociation reactions represented by MIX+

4MX”+M+

(18)

MX;

%MX’+X-

(19)

and

which can be combined

to give aM+aMXO

=

ax-

=

mM+?

001

aMP

ahfxi

aM2X+

M+ mMX6

MX”

mM2Xt?M2Xf

=

mX-TX-mMX”?MXo (20) rnMX$iMX$

where ( FUOSS and -AUS,

’

1

0.5

CONCENTRATION,

1.0

MOLAL

FIG. 10. Computed degrees of ion pair formation in NaCl solutions (o~n~c,o) as a function of the total solute concentration at 500, 600, and 700°C and 1, 2, 3, and 4 kilobars. The curves were generated fromEqns.(l),(2),(3),(5),(6),(9),(10),(11),(17),(2O),and (22) using parameters given in Fig. 6 and Table 4 together with log K2 values taken from OELKERSand HELGESON( 1990).

1933)

K2 z KM2x+ = KMX, 3

(21)

where K,,,Y + and KMxi refer to the dissociation constants of the subscripted species. Electrical neutrality in solution requires that mM+ + m&x+ = mx- + mMXr.

(22)

Equations ( I), (21, (3), (5), (6), (9), ( lo), ( 111, ( 17), (20), and (22) were used in the present study, together with values of log K, shown in Fig. 6 and those of log K2 taken from OELKERS and HELGESON ( 1990) to compute the degree of neutral ion pair formation in NaCl solutions ((~~~~~0)at various temperatures and pressures. Because the activity coefficients of monovalent aqueous species are similar in magnitude to one another ( HELGESON et al., 198 1; OELKERS and HELGESON, 1990), Y,+,+, rx-, Yh,2X+, and TMX~ were taken in the calculations to be equal to r +,.,=c, and computed from Eqn. (9).6 The activity coefficients of the neutral aqueous species represented by MX” were calculated from Eqns. ( 1) and ( 16) using the ail,,+,p values given in Table 3. The results of the calculations are depicted in Figs. 10 and 11, where CY,+~,O is plotted as a function of the total molality of NaCl at 1, 2, and 3 kb for temperatures from 400 to 800°C. It can be seen in these figures that the curves maximize with increasing NaCl concentration, which can be attributed in part to the relatively low dielectric constant of HZ0 at supercritical temperatures and pressures. The existence of maxima in

’ It has been shown (OELKERS and HELGESON,1990) that Eqn. (9) applies to single electrolyte solutions in which the predominant species include triple ions as well as single ions and neutral

ion pairs.

curves representing (Y.&,X0 as a function of electrolyte concentration in low dielectric constant media was anticipated by KRAUS ( 1954, 1956 ). Note in Figs. IO and 11 that formation of NaCl’ in supercritical NaCl solutions increases with increasing concentration in dilute solutions, maximizes at molalities of -0.1 to -0.5, depending on the temperature and pressure, and then decreases with increasing concentration at higher molalities. For example, at 700°C and 2 kb, (~~~c.0in Fig. 10 reaches a maximum of -0.7 at a total NaCl concentration of -0.1 molal and decreases to ~0.1 at concentrations >, 0.8 m. It can be seen in Fig. 10 that the higher the temperature at a given pressure, the greater the maximum and the lower the concentration at which the maximum occurs. This is a consequence of the fact that the largest SetchCnow coefficients occur at low pressures and high temperatures, which are the pressures and temperatures at which log K, is the most negative (Fig. 6). The maxima in a,,,@ as a function of concentration in Figs. 10 and 1 1 are consistent with the results of statistical mechanical calculations, which also indicate that the degree of neutral ion pair formation with increasing solute concentration in single 1: 1 electrolyte solutions at supercritical temperatures and pressures maximizes at low solute concentrations ( PITZER and SCHREIBER, 1987). The latter observation is apparently independent of the definition of ion clusters. In accord with the BJERRUM (1926) model of ion pair formation, VALLEAU et al. (1980), GILLIAN ( 1983), and PITZER and SCHREIBER ( 1987) adopted a primitive model in which a group of ions forms a cluster if the distance between each ion is less than a specific “associating” distance. In this model, sequentially larger ion clusters dominate the solute as the solution becomes more concentrated. Thus, cy,+,,ro declines as

E. H. Oelkers and H. C. Helgeson

1248

IO

NaCl

00

3

kb

“,pd 0.5

IO

CONCENTRATION. MOLAL FIG. 11.Computed degrees of ion pair formation in NaCl solutions ( aNaclo) as a function of total solute concentration at 400 to 800°C and 2, 3, and 4 kb-see

caption of Fig. 10.

the neutral ion pairs combine with free ions and other ion pairs to form larger complexes. For example, the calculations reported by PITZER and SCHREIBER( 1987) indicate that triple ions, neutral ion quadruples, quintuple ions, and neutral ion sextuples account for more than 95% of the solute in - 1 molal NaCl solutions at - 1 kb and 550°C. The remaining 5% consists of single ions and neutral ion pairs. However, at a concentration of 0.1 m, the calculations of PITZER and SCHREIBER ( 1987) indicate that neutral ion pairs account for -3O-50% of the species in solution at this pressure and temperature. In contrast to the approach taken by PITZER and SCHREIBER ( 1987), CORTI and FERNANDEZ-PRINI ( 1986) and LARIA et al. ( 1990) consider ion clusters to exist only when a set of two or more ions have interparticle interactions well differentiated from those acting on the average ions in the system. In this model, clusters have physical meaning only if the cluster sizes are significantly smaller than the mean interparticle distance, and therefore distinct clusters do not exist in concentrated solutions. As a consequence of this definition, the calculations of LARIA et al. ( 1990) indicate that neutral ion pairs at supercritical temperatures and pressures redissociate into single ions as concentration increases and approaches 1 molal. Hence, although the cluster definitions adopted by PITZER and SCHREIBER ( 1987) and LARIA et al. ( 1990) lead to quite different results with respect to the relative predominance of single, triple, quadruple, quintuple, and sextuple species with increasing solute concentration in 1: 1 electrolyte solutions, both sets of calculations support the conclusion reached above from consideration of conductance data that the degrees of formation of neutral ion pairs maximize in dilute solution with increasing concentration at supercritical temperatures and pressures. The significance of this observation is underscored by the fact that the SetchCnow coefficients used to calculate the values of aMXoreported above were extrapolated from conductance data for dilute solutions

((0.1 m) to 1 m. It thus appears that statistical mechanical calculations could be used to calculate directly the magnitude of Setchenow coefficients in concentrated supercritical electrolyte solutions. To determine the extent to which triple ion formation is responsible for the configurations of the curves in Figs. 10 and I 1, analogous calculations were carried out omitting provision for the possible formation of triple ions. The results indicate that under these conditions, aNaCIoexhibits behavior similar to that shown in Figs. 10 and 11, but the isotherms and isobars maximize at slightly higher total concentrations. Hence, the conclusions reached in the present study with respect to the extent to which SetchCnow coefficients affect the formation of ion pairs in supercritical electrolyte solutions are independent of whether or not triple ions actually form in appreciable degrees in these solutions. It can be deduced from Figs. 10 and 11 that the calculations described above indicate that neutral ion pair formation becomes small in concentrated sodium chloride solutions at supercritical pressures and temperatures. The values of log K, are similar in magnitude for all of the alkali metal halides considered in the present study, which is also true of the values of log K2 and b,,,,p, respectively. Hence, the ~(~~c,o values depicted in Figs. 10 and 11 are typical of those for solutions of other alkali metal halides. In contrast, it can be seen in Fig. 7 that b+,p for HCl’ and HBr’ are approximately two orders of magnitude greater than those of the alkali metal halides at a given temperature and pressure. It seems likely that this large difference can be attributed to enhanced hydrogen bonding resulting from much stronger short-range interaction among HZ0 dipoles and the solute species in HCl and HBr solutions. The transport of H+ in these solutions is greatly enhanced by the prototropic mechanism corresponding to transfer of protons down a potential gradient from Hz0 dipole to Hz0 dipole (OELKERS and HELGESON, 1989). As a consequence, the cvMXocurves for HCl and HBr solutions exhibit higher maxima which occur at lower concentrations than those shown for NaCl’ in Figs. 10 and 11. The concentrations of HCl, HBr, and other acids are generally much lower than those of the alkali metal halides in natural hydrothermal solutions. Consequently, the Setchknow coefficients for such species as HCl’ and HBr’ in these solutions are probably also much lower than they are in HCl and HBr solutions. Because NaCl is commonly the dominant solute component in natural hydrothermal solutions, b+,aCjOis probably more representative of the Setchenow coefficients of other neutral aqueous species in such solutions than the corresponding coefficients for the species in single electrolyte solutions with the same stoichiometry as the species. COMPUTATIONAL

UNCERTAINTIES

Uncertainties in the values of b,,,o and CQ,~O(which increase with increasing concentration) generated above are difficult to estimate, but they may be large. For example, omission of provision for quadruple, quintuple, and other higher order complexes from the calculations responsible for the values of cu,,,e shown in Figs. 10 and 11 introduce additional uncertainty in computed degrees of formation of

1249

Activity coefficients in supercritical electrolytes A&X*as concentration increases above -0.5 m. In addition, estimated uncertainties in experimental data and computed limiting equivalent conductances indicate that in the cases where the experimental data in Figs. 1 through 5 exhibit a scattered distribution, uncertainties in the b,,,o values listed in Table 4 may be as great as 100%. Nevertheless, the absence of such scatter and the systematic distribution of the data in most of the diagrams in these figures, as well as the smooth variation of the retrieved values of log Ki and h7,*fXoas functions of tem~rature and pressure in Figs. 6 and 7 lend considerable support to the inte~retation of the conductance data described above. This observation is supported by the similarity in magnitude of the Setchenow coefficients computed above for NaCl’ in NaCl solutions and those reported by DING and SEYFRIED( 1990) for aqueous Hz and H2S in NaCl solutions at 300 to 425°C and 300 to 500 bars. Similar Setchenow coefficients for different neutral aqueous species in a given electrolyte solution at 25°C and 1 bar were noted 35 years ago by HARNED and OWEN (1956). Although the values of br,MX~generated in the present study should be regarded as provisional approximations, at the very least it appears to us that they afford considerable improvement over the commonly made (and unfound~) assumption that y,WXo= 1 at high temperatures and pressures. In the absence of more accurate and extensive experimental conductance and/or spectrographic data for supercritical aqueous electrolyte solutions, the values of b,,,o in Table 4 should afford a far more realistic assessment of the distribution of neutral ion pairs as a function of concentration, temperature, and pressure than could otherwise be achieved. It should be emphasized in this regard that interpretation of supercritical mineral solubility data in the present state of knowledge can neither confirm nor deny the calculations summarized above. Analyses of this kind currently require assumptions regarding the identities of the predominant aqueous species in solution. In addition, several conflicting experimental values of the dissociation constants for these species may be available in the literature. Under these circumstances, no unique interpretation of mineral solubility data can be made without additional information that is diagnostic with respect to the speciation in solution. As emphasized above, the values of log K, generated in the present study are relatively insensitive to the magnitude ofb,,n,.w0, owing to the low concentrations at which the conductance measurements were made. Consequently. the uncertainties in the computed values of log k; in the present study are comparabIe to those assessed by OELKERS and HELC~ESON f 1988) for supercritical log 1yi values computed from the experimental conductance data reported by QUIST and MARSHALL( 1968a,b,c), DUNN and MARSHALL( 1969), and FRANTZ and MARSHALL(1982). CONCLUDING REMARKS The calculations described above indicate that neutral ion pairs in 1:1 electrolyte solutions are major species only in dilute to moderately concentrated (SO.8 molal) solutions at supercritical pressures and temperatures. This ~~~~1~~~~~is

~~iid~~r all vahres ofb l,,tfXnti 0, regardkss qf rhedegree to ~~~~ichtriple ions or other hither

order c~)~n~le~esform

with

increasing

concentration. It is also valid for small values of br,Mp if triple ions and/or other higher order complexes form to appreciable degrees (OELKERS and HELGESON, 1990, 199 1) , Hence, both of these phenomena favor redissociation of ion pairs with increasing concentration, Because the solute in supercritical electrolyte solutions in nature consists primarily of NaCl, it appears likely that neutral ion pair formation in these solutions is restricted in a similar manner to that described above for NaCl solutions. Conseo values generated above should afford close quently, the bT,Naci approximation of the activity coefficients of neutral 1: 1 ion pairs in hydrothermal solutions at supercritical pressures and temperatures. Although the generality of the conclusions reached in the present study have yet to be demonstrated, it appears on the basis of the calculations described above that assuming unit activity coefficients for neutral aqueous species in supercritical electrolyte solutions may lead to erroneous interpretation of experimental solubility data at supercritical temperatures and pressures, as well as unrealistic prediction of mineral solubilities and phase relations in hydrothermal systems. In any event, it is clear that more experimental data and inte~retative theory are required to achieve a better understanding of the thermodynamic behavior of electrolyte solutions at high temperatures and pressures in the Earth’s crust. Ackno~~ledgments-The research described above was supported by the Department of Energy (DOE Grant DE-FG03-85ER- I34 19), the National Science Foundation (NSF Grant EAR 8606052), and the Committee on Research at the University of California, Berkeley. We thank William L. Marshall and John V. Walther for providing insightful reviews of this manuscript. We are indebted to Vitalii Pokrovskii, Peter C. Lichtner, Everett L. Shock, Barbara L. Ransom, William M. Murphy, James W. Johnson, and Dimitri Sverjensky for helpful discussions, encouragement, and assistance during ;he c&se of this study. Technical assistance from Jefferv DeBraal. Alice Wolff. and LaurieStenberg is gratefully acknowledged. Thanks are also due Kim Suck-kyu, Lillian Mitchell, and Rebecca Arington for drafting figures, Joachim Hampel for photographic assistance, and Tony Wong and Mike Stewart for their computer expertise. Editorial bundling: T. S. Bowers

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