Chemical Modeling of Aqueous Systems II - American Chemical Society

Chapter 2

Activity Coefficients in Aqueous Salt Solutions Hydration Theory Equations Thomas J. Wolery and Kenneth J. Jackson

Downloaded by CORNELL UNIV on July 21, 2012 | http://pubs.acs.org Publication Date: December 7, 1990 | doi: 10.1021/bk-1990-0416.ch002

Lawrence Livermore National Laboratory, L-219, P.O. Box 808, Livermore, CA 94550

A new set of phenomenological equations for describing the activity coefficients of aqueous electrolytes has been derived, based on the hydration theory concept of Stokes and Robinson (1), but using the "differentiate down" approach in which an expression is first defined for the excess Gibbs energy. Separate equations are given for the activity of water and the activity coefficients of ionic solutes. The new equations incorporate an empirical but thermodynamically consistent scheme for using an average ion size parameter in the Debye-Hückel part of the model. This permits the new equations to be applied to mixtures of aqueous electrolytes. The new equations, applied to the case of a pure aqueous electrolyte, do not reduce to the familiar Stokes-Robinson equation owing to a minor difference in how the DebyeHückel model is presumed to apply to formally hydrated solutes. As a first step in evaluating the usefulness of the equations, we have fit a "two-parameter" (ion size plus hydration number) model to data for a number of pure aqueous electrolytes. The quality of the fits is excellent in many cases, but there are indications that more satisfactory results would be obtained by fixing reasonable values for the ion sizes and compensating for the loss of a fitting parameter by including ion pairs in the models, by including virial coefficient terms in the equations, or both. Thermodynamic modeling of aqueous geochemical systems depends on functions which approximate the activity of water and the activity coefficients of the solutes. Ten years ago, the state of the art for addressing this problem in geochemical modeling codes was the use of simple extensions of the Debye-Hiickel equation, such as the Davies equation or the "B-dot" equation (see ref. 2). These equations were useful up to concentrations approaching sea water (I ~ 0.7 m, I = ionic strength), although they can be shown to depart from precise physical measurements in much less concentrated solutions. These approximations were used in the modeling codes in conjunction with a large number of ion-pairing and complexation constants to compute models of speciation and degree of saturation with respect to various minerals and other solids. Ideally, a model for approximating activity coefficients in geochemical or engineering modeling codes would have the following properties: consistency with the laws of thermodynamics; compact mathematical form; high accuracy over wide ranges of temperature, pressure, and concentration; applicability to systems including most of the elements in the periodic table; consistency with generally all types of physical measurements bearing on activity coefficients, phase equilibria, and speciation; and existence of useful predictive means to estimate the model parameters, as opposed to obtaining them exclusively by data fitting. These predictive means might include estimation via empirical or semi-empirical correlation methods, which have been shown to exist for some models Q, 4). The best predictive means would be obtained if the parameters could be expressed as functions of the intermolecular forces (5-7). Some kind of predictive means is probably NOTE: This chapter is Part II in a series. 0097-6156/90/0416-0016S06.00/0 o 1990 American Chemical Society

In Chemical Modeling of Aqueous Systems II; Melchior, D., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

2.

WOLERY & JACKSON

Activity Coefficients in Aqueous Salt Solutions 17


essential to making substantial progress in modeling the behavior of aqueous systems containing many chemical elements, such as transition metals and actinides, that tend to exist as aqueous complexes more so than as dissociated ions. No model currently exists which satisfies all of the above requirements. For modeling geochemical systems, the requirement of mathematical compactness has focused attention on phenomenological models guided to some degree by theoretical considerations. The most successful such example is Pitzer's equations (8-10). which are thermodynamically consistent and generally produce highly accurate results. The virial coefficients employed in this phenomenology have been obtained by fitting experimental data, originally only of vapor pressure equilibrium and electromotive force (emf) data (see ref. £), and more recently also of solubility data (see ref. 10). Model development has been largely restricted to systems in which the solutes tend to exist in mostly dissociated form. Ion pair species are formally included in the models only when the data can not be adequately fit otherwise. Application of the Pitzer phenomenology has ignored some kinds of data. An important example is electrical conductance, the interpretation of which is generally ascribed to ion-pairing (see ref. 11). The picture obtained from such data is one in which there are many more ion pair species than are included in the present models based on Pitzer's equations. No generally useful predictive relations governing the Pitzer's equations parameters have been shown to exist; consequently, the usefulness of the existing models is largely restricted to systems that are part of the larger "sea-salt" system. Some other kinds of models have shown parameters that seem to follow useful correlation relationships. Among these are the virial coefficient model of Burns Q), the interaction coefficient model of Helgeson, Kirkham, and Flowers (4), and the hydration theory model of Stokes and Robinson (1). The problem shared by all three of these models is that they employ individual ion size parameters in the Debye-Huckel submodel. This led to restricted applicability to solutions of pure aqueous electrolytes, or thermodynamic inconsistencies in applications to electrolyte mixtures. Wolery and Jackson (in prep.) discuss empirical modification of the Debye-Huckel model to allow ion-size mixing without introducing thermodynamic inconsistencies. It appears worthwhile to examine what might be gained by modifying these other models. This paper looks at the hydration theory approach. PREVIOUS DEVELOPMENT Hydration theory was first applied to aqueous electrolytes by Stokes and Robinson (1) in 1948. The approach is to correct for the fact that the actual solute species do not exist as bare ions as they are normally written, but as hydrated aqueous complexes. In the case of a solution of an aqueous nonelectrolyte, this reduces to the assumption that the solution is ideal, or nearly so, if one writes the formula of the solute to include the waters of hydration and computes its concentration and that of the solvent on this basis. In the case of an aqueous electrolyte, it reduces to the assumption that the Debye-Huckel model becomes adequate (or at least more adequate) to describe the activity coefficients if this treatment is applied. Hydration theory deals with aqueous solutions in terms of two parallel definitions of the set of components, the usual one in which the solutes are considered formally unhydrated and the amount of solvent is the nominal amount, and a second in which the solutes are formally hydrated and the amount of water is reduced from the nominal amount. The objective is to correct for hydration effects and obtain a model giving activity coefficients in terms of the usual set of components. Quantities pertaining to the second set of components will be denoted by an asterisk. The number of moles of water in the first set is given by n

w

=

i

i

w

w

(1)

where Q is a constant, the number of moles of water having a mass of one kilogram t55.51), and w is the number of kilograms of nominal water. The number of moles of water in the second set is given by w


18

CHEMICAL MODELING O F AQUEOUS SYSTEMS II

n = w

O - Z h.m (2)

where hi is the hydration number (number of bound water molecules per molecule of solute) and mi is the molality of the i-th solute. The number of moles of a solute is the same in both component sets (ni* = ni). The molality of a solute in the usual component set is given by ftn . m . = ——w 1

n

(3)


The molality of a solute in the second component set is given by the same equation, but with asterisks attached to the m and n quantities. It can be related to the nominal molality by m. m* = - J i D

(4)

where D= 1 -

I i

h.m. 1

1

Q

(5)

Thus, the amount of "free" water is less than the nominal amount, and the molality of a solute, corrected for hydration, is greater than the nominal molality. The model encounters a singularity when all the "free" water has been used up. Using an argument based on the Gibbs-Duhem equation, Stokes and Robinson Q) derived the following equation for a solution of a single aqueous electrolyte (A), presumed to be fully dissociated:\ (

X) m

\

h

I z z l

A

A

(6)

(Note: the original equation has been translated into our notation). Here y± is the mean activity coefficient of the electrolyte, D A is the number of ions produced by its dissociation, ITIA is its molality, hA is its hydration number, ZA+ is the charge on the cation, and ZA- is the charge on the anion. The last term contains f = df/dl, where f(I, a/0 is a Debye-Huckel function which depends on I, the ionic strength, and aA, an ion size parameter characteristic of the electrolyte. The DebyeHuckel function used by Stokes and Robinson is what Pitzer (&, 2) refers to as the DHC (DebyeHuckel-charging) form. Pitzer in his equations uses the alternative DHO (Debye-Hiickel-osmotic) form. Equations for these forms are summarized in refs. & and £. The ionic strength is given by tA

T 1=

1

2

v m .z. i 1

2

1

(7)

In such solutions, D = 1 - (h/^m/JQ). The hydration number of the electrolyte is easily identified as the stoichiometric sum of the hydration numbers of the individual ions. The relationship of the characteristic ion size parameter to the sizes of the individual ions was not addressed. Stokes and Robinson (T) fit their equation to data reported for 35 pure aqueous electrolytes of the 1:1 and 2:1 types and obtained good results to fairly high concentrations. For example, for NaCl, the reported average difference in ln y± was only 0.002 for over a concentration range of 0.15.0 m (I = 0.1-5.0 m); for CaCl2, 0.001 from 0.01-1.4 m (I = 0.06 to 8.4 m). These statistics can not be directly compared with those reported for fitting with Pitzer's equations (e.g., ref. 9) as the


2.

WOLERY & JACKSON


types of statistics employed are not the same; nevertheless, it is clear that the accuracy of this twoparameter model is close to that of Pitzer's model, which employs three virial coefficient parameters to fit a single aqueous electrolyte. Values of the characteristic ion size ranged from 3.48 to 6.18 Angstroms, values which appear reasonable. The hydration numbers ranged from 0.6 to 20.0 and also seem reasonable, although hydration numbers in the higher end of this range would require water molecules to be bound in other than the first hydration shell. Stokes and Robinson further proposed a correlation between the ion size parameter and the hydration number, and examined a "oneparameter" model in which the ion size was taken as a function of the hydration number. This was surprisingly successful, though less accurate.


The Stokes-Robinson model was later modified by its creators for use in more concentrated solutions by accounting for dehydration equilibria (12). The original model was also modified by Nesbitt (13). who made the hydration number a decreasing function of the ionic strength. Other workers too numerous to mention here have also attempted to do something with hydration theory. Given that the Stokes-Robinson model was proposed well before the development of modern computers, and that the fitting accuracy of the "two-parameter" form is quite good, one might ask why this model was not used in geochemical modeling codes as soon as such codes were developed. One reason is that the mean activity coefficient can only be obtained if one knows the activity of water in the solution. Stokes and Robinson did not give a separate equation for this, as it was not necessary for what they were trying to do. In the majority of cases, a solution of a pure aqueous electrolyte is studied by the is< ^iestic method, which yields the osmotic coefficient (), which is related to the activity of water ( $ = - (Q/Ximj) ln a ). The mean activity coefficient can be obtained by numerically integrating the following form of the Gibbs-Duhem equation: w

ain -urn A A

din a

7 + A

3m . A

= -1>

A

-

Q.

dm

w

4

A

/

C

n

W

(sometimes the route is reversed, as when emf methods are used). In essence, Stokes and Robinson used their equation as an alternative to this numerical integration. Thus, in addressing the problem of interest to them, they already knew the activity of water. Other problems are associated with the ion size parameter. The model could not be extended to electrolyte mixtures without introducing some means of allowing mixing of characteristic ion size parameters or losing accuracy by using a single value for all electrolytes in a given mixture. Many empirical schemes that have been proposed for ion size mixing can be shown to run afoul of thermodynamic consistency relations (see Wolery and Jackson, in prep.). Furthermore, the use of an ion size parameter characteristic to an electrolyte makes it difficult to see how the Stokes-Robinson equation can be adapted to a form giving single ion activity coefficients, which is generally useful and even necessary for some purposes in geochemical modeling codes. It appears worthwhile to attempt to overcome these deficiencies, as the hydration concept has a powerful physical appeal. This does not mean that one must hold that the Debye-Huckel model is really adequate if one simply corrects for hydration. It has a number of deficiencies, as has been pointed out for example by Pitzer (&, 9). Like Pitzer's equations, the kind of model that we are addressing here incorporates some degree of theory as a guide but in essence remains phenomenological. We only expect that the hydration correction will turn out to be a useful part of such a model. THERMODYNAMIC FRAMEWORK A thermodynamic model that satisfies all of the necessary consistency relations can be devised by writing an expression for the excess Gibbs energy and applying appropriate partial differentiation with respect to the number of moles of the components (see Wolery, in prep.). If the excess Gibbs energy is defined with reference to mole fraction ideality, the differential equations are


20

CHEMICAL MODELING OF AQUEOUS SYSTEMS II

ln a

•

= - ln

j

i

1 RT

EXx 3G 3n. (9)

ln y . = - ln

1+

J

J

1 RT

a )

EXx 3G 3n. (10)

(Note- the solute components here are taken to be individual species; i.e., not neutral electrolytes.) Here R is the gas constant, T is the absolute temperature, and G is the mole-fraction based excess Gibbs energy. The first term on the right hand side of the equation for the activity of water is equivalent to the logarithm of the mole fraction of that substance; the second term, the logarithm of its mole fraction activity coefficient. Alternative differential equations apply if one uses the excess Gibbs energy defined with reference to ideality based on molalities ( G ; see Wolery, in prep.), and such were used by Pitzer (&, £) to develop his equations. There is a tendency to use the same expressions for the excess Gibbs energy regardless of the specific reference ideality employed. Therefore, it makes a difference in the resulting equations which set of differential equations one uses. As far as the development of phenomenological models is concerned, however, the choice is probably best viewed as a matter of taste. In this paper, we will use the set of differential equations given above. The work of Stokes and Robinson (T) was also consistent with mole fraction ideality.


E X x

E X m

EXCESS GIBBS ENERGY AND THE DEBYE-HUCKEL FUNCTION The excess Gibbs Energy for a pure Debye-Huckel model is given by (see Wolery and Jackson, in prep.) G

E X x

=w

RT f(I)

w w

(11)

(Here the ion size parameter is taken as a fixed constant.) Partial differentiation gives ( ln a = - ln

i ,

J

j

•i- [

w

tf'(I)-f(I)] (12)

ln y . = - ln

1

'

J +

i

-

f

'

( I )

(B) ION SIZE AVERAGING A simple, empirical, but thermodynamically consistent approach to ion size averaging has been recently proposed by Wolery and Jackson (in prep.). The excess Gibbs energy is written as EXx G

=w RTf(I.a) w

(14) where a is the average ion size defined by


2.


WOLERY & JACKSON

I i

m.w.a. 1

I

1

1

m.W.

(15)

Here 2l\ denotes the individual size of the i-th solute species. Wj is an empirical charge-based 2

weighting cofactor. One might choose Wj = 1, lzi,l or z\ / 2 . Partial differentiation then gives X m 1 ln a

w

= - In

-i- [ r a , s ) - f a , a ) ]

1 +


(16) I ln y . = - ln

1 +

m.

W.f

i a 5,a)

2

J

Q

1

X W.m J

J

a

J

y (17)

r

rhe equation for the activity of water is essentially unchanged, but a new term, the "averaging term," has been introduced into the equation for the solute activity coefficient. Here f is a derivative DebyeHiickel function, defined as a

f -a

it

a

Ua

(18)

For both the DHC and DHO models, it may be shown that (Wolery and Jackson, in prep.) f =3If(I,a)-2f(I,a)

(19)

a

The mean activity coefficient of electrolyte A (assumed to be completely dissociated) is given by X m. Z

In Y

±

A

=-ln

1+ -

w

f A

n J g

a

q- )

A

A

Z

A+ A -

a

A

- a

I W.m. J

J

(20)

J

where WA is a characteristic weighting factor for the electrolyte defined by t) . W +u W A+,A A+ A-,A AA

W

A

A =

(21)

and a A is a characteristic average ion size for the electrolyte defined by %

+

W

, A %

+

A , A

a +

W

A A

+

^ A - , A +

+

W

% - , A

A W

a

A -

A -


(22)

22


In a solution of a pure electrolyte, the averaging term vanishes. PROPOSED NEW HYDRATION THEORY MODEL If we assumed that the Debye-Huckel model worked for a set of components in which the solutes were considered formally hydrated, we could write ..EXx*

= w ; RT f(I*,a) (23)

Note that the excess Gibbs energy carries an asterisk to denote that it pertains to the set of components with formally hydrated solutes. Thus, w * is the number of kilograms of "free" water (w* = w D), and I* is the molal ionic strength, calculated in the usual way but with the nominal molalities replaced by molalities corrected for hydration (thus I* = I/D). If it were desired to place less reliance on the Debye-Huckel model, other terms could be added to the right hand side in the usual way. We will presently ignore this as we are immediately interested in the hydration correction itself. The objective is to obtain an expression for G from that assumed for G * and obtain expressions for the activity coefficients of the components in the usual component set by partial differentiation. First, however, we are motivated to make a change in the expression for G * . In the early stages of developing Debye-Huckel theory, the ionic strength is defined on a volumetric basis, and the molal ionic strength is later substituted as a convenient approximation. In general, we should expect the nominal molal ionic strength I to be closer to the volumetric ionic strength than is the molal ionic strength corrected for hydration. As did Stokes and Robinson (1), we will take the ionic strength to be the nominal molal quantity. We therefore change our starting equation to w


w

w

E X x

E X x

E X x

EXx*

:w*

RT f(I,a)

(24)

The total Gibbs energy is independent of how one chooses to define the components; therefore, G

EXx

IDx* =G

G

IDx

EXx* + G

(25)

The ideal chemical potential of the a-th solution component in the nominal set of components is given by IDx =^ |i° a + RT ln xa (26) o

(the subscript a may denote either w or i). Here n° is the standard state chemical potential (for a standard state mole fraction of unity) and x is the mole fraction of the component. This can be substituted into the first-order sum rule (see Wolery, in prep.) to obtain a

a

.IDx

= 1 n (|i° a a

A similar expression may be constructed for G

G

IDx *

I D x

IDx -G = RT n l n w

a

+ RT l n x ) fl

(27)

* . It can then be shown that

r * \ X.

+ X n . ln i

h

1

*

I

V w where the mole fractions can be related to the nominal molalities by


(28)

2.


WOLERY & JACKSON

X

Q Q+ £ m .

w

(29)

x.

:

J

j

Q - X h.m. j £2- £ h.m. + I m. j j

(3D

n-Yh.m. j

(32)

J

w


J

i

(30)

J

J

J

J

J

+ ym. j J

The expression for the excess Gibbs energy can therefore be written as

G

EXx

f

x = RT w Df(I,a) + n ln w ' w v

J

+

X

I n.ln i

*

Y

X.

1

(33) The actual process of partial differentiation of this equation is quite tedious and will not be presented in detail. We note the following key intermediate results:

G

IDx*

IDx - G RT

dn

^ _EDx*

G 3n.

= ln

( * \ X w X

)

(34)

rox^i

- G RT

l

ln

h *

i (35)

The final results are:

ln a

w

[ DIf(I,a)-f(I,a) ]

= - ln

(36) ( In 7 . = - ln D + ( h . - l ) l n

I J

1

QD J

+ D - y - f'a.a)


24


h.

1

in

)

f(I,a)

+

D

W .1f

B

)

QD J

1%

f(I,a)

A

D

+

I W m J J j :

:

A " V a

a

(38)

Note that, compared with the counterpart equations without the hydration correction, the factor D appears in various terms. In the equations for molal activity coefficients, there are two additional terms, one containing ln D, the other containing the Debye-Huckel function f. Combination of the equations for the activity of water and the mean activity coefficient for the case of a single aqueous electrolyte gives f h In a.,

+ D

A

A

h

A

I^ f'(I,a)

A

(39)

This is not the Stokes-Robinson equation, though some common elements may be discerned. Our result is different because we substituted I for I* in the Debye-Huckel equation for the excess Gibbs energy, whereas Stokes and Robinson (1) made the substitution in the Debye-Huckel equation for the activity coefficient of a formally hydrated solute (ln Y*±,A)- Differentiation of our equation for the excess Gibbs energy would yield a somewhat different equation for y*±,APRELIMINARY MODEL DEVELOPMENT The equations derived above constitute a "phenomenology" upon which specific models might be constructed. Using the same phenomenological equations, one can develop models that are distinct with regard to speciation. It is also possible to develop models that are distinct according to the observational data (e.g., isopiestic, emf, solubility, electrical conductivity, spectroscopic) that are to be explained. Within the framework of the equations given above, one may choose from among different Debye-Huckel submodels, which will also lead to distinct models (see ref. 14 for a review of advances in Debye-Huckel models). In addition, one might modify the equations, for example by including higher-order electrostatic terms (15) or adding empirical virial coefficient terms. Such modifications can be accomplished easily and safely by adding the requisite terms to the equation for the excess Gibbs energy and applying the partial differential equations given earlier in this paper. A complete model development is beyond the scope of the present paper. We present here only some results of fitting our hydration theory equations to data for several pure aqueous electrolytes, ignoring ion pairing. The data fit are values for the osmotic coefficient as tabulated by Robinson and Stokes (11). The Debye-Huckel model employed was the DHC, and the Debye-Huckel parameters used were taken from ref. 16. The fitting was accomplished by unweighted least squares minimization, using as fitting parameters the characteristic average ion size and total hydration number of each electrolyte. The mean molal activity coefficient was then computed from equation 38


2.

WOLERY & JACKSON


and compared with the values tabulated by Robinson and Stokes (H); the latter are related to the corresponding tabulated values for the osmotic coefficient via numerical integration of the GibbsDuhem equation.


Figures 1 and 2 show the results of fitting the osmotic coefficient of the aqueous electrolytes sodium perchlorate and potassium chloride, respectively. Analysis of the variance in fitting the osmotic coefficient indicates that the fits are about as good as those obtained using Pitzer's equations, despite the fact that our equations have one less fitting parameter. For sodium perchlorate, the standard deviation in our fit is 0.0011, whereas Pitzer (2) reports 0.001 using his equation. For potassium chloride, the standard deviation in our fit is 0.00036, that in Pitzer's, 0.0005 (for a maximum molality of 4.8 opposed to our 4.5). Figure 3 shows the predicted mean molal activity coefficients evaluated from these fits. The fit for aqueous potassium chloride is essentially perfect in terms of both deviation in fitting the osmotic coefficient and the deviation between the activity coefficient predicted from our model and the same quantity obtained by numerical integration. The fit for aqueous sodium perchlorate, however, is not quite as good and is actually much more typical of the results obtained fitting data for 1:1 electrolytes. Not only are the residuals for fitting the osmotic coefficient greater, but the activity coefficient computed from our model rises noticeably (though not greatly) above the values obtained by numerical integration for an ionic strength greater than about three or four molal. The results of fitting a number of 1:1 aqueous electrolytes are given in Table 1. The fits are noticeably less good for data likely to have been obtained in part or all by emf as opposed to isopiestic means. The data themselves are probably not as good. These cases include the hydroxides, significantly volatile acids, and salts of lithium. Values for the fitting parameters are physically reasonable in most cases. Exceptions are the negative hydration numbers of aqueous potassium, rubidium, and cesium nitrate. Except for these electrolytes and those mentioned above for which the data are probably not of highest quality, fairly regular and expected patterns are apparent. In the chloride series, for example, the characteristic ion sizes and the hydration numbers decrease going to alkali cations farther down the periodic table. If single ion values are desired, a thermodynamic convention can be applied. For example, following the Maclnnes convention, one might take the sizes of the potassium and chloride ions to be equal to the characteristic ion size of aqueous potassium chloride, and the hydration number of each of these ions to be equal to one-half the hydration number of the same aqueous electrolyte. A thermodynamically valid convention need not require physical (i.e., positive values) for ion sizes and hydration numbers of single ions. The meaning of obtaining negative hydration numbers requires some discussion. The role of the hydration number is to cause the activity coefficient to increase at high ionic strength (assuming of course a positive value). The Debye-Huckel term only causes the activity coefficient to tend to decrease. The data for the potassium, rubidium, and cesium nitrate salts are consistent with an activity coefficient that is below that predicted by the Debye-Huckel model, hence the fit produces a negative hydration number. Ion pairing, which is ignored in the model, also tends to decrease the activity coefficient, and may be the real factor involved. Reported ion pairing constants suggest that this is possible. Smith and Martell Q7) report a log K of -0.15 for association of potassium with nitrate, in contrast to one of -0.7 for association of the same cation with chloride. One should keep in mind that the hydration numbers obtained for all of the aqueous electrolytes given in the table would be larger if ion pairing were explicitly treated in the model. In general, attempts to fit the simple hydration theory model to 2:1 and 3:1 electrolytes were markedly less successful. In many cases, the fitting residuals were extremely large. Reducing the upper end of the concentration range fit in some cases resulted in acceptable fits. In many of these cases, however, we obtained physically unacceptable parameter values, typically negative characteristic ion sizes. In the case of 1:1 electrolytes, we found that varying the upper limit of the concentration range in the fitting produced little change in the fitting parameters. In the case of 2:1 and 3:1 electrolytes, we expected to see convergence of the fitting parameters as the upper


26


NaCI0

4

1


1 —'—

r

Ionic strength Figure 1. The fit of our hydration theory model to osmotic coefficient data for sodium perchlorate tabulated by Robinson and Stokes (11).

KCI -r-ri—i—|—i— —i—|—10.0006 9

Ionic strength Figure 2. The fit of our hydration theory model to osmotic coefficient data for potassium chloride tabulated by Robinson and Stokes (11).


WOLERY & JACKSON



2.

Figure 3. The mean molal activity coefficient of sodium perchlorate (left) and potassium chloride (right). The curves show the predictions of our model, fit to data for the osmotic coefficient. The squares represent the data tabulated by Robinson and Stokes (11).


28


Table 1. Results of fitting several 1:1 aqueous electrolytes max.m Electrolyte aA h A

NaC104

12.11 18.26 7.74

5.40 5.74 1.56

6.0 4.0 6.0

0.062 0.028 0.0028

HC1 LiCl NaCl KC1 RbCl CsCl

15.40 11.25 7.15 6.45 5.79 4.43

4.88 4.53 2.51 1.44 1.32 1.21

6.0 6.0 6.0 4.5 5.0 6.0

0.076 0.0033 0.0025 0.00059 0.0017 0.0058

HBr LiBr NaBr KBr RbBr CsBr

11.67 12.03 7.97 7.14 5.89 4.20

6.30 5.15 3.05 1.53 1.13 1.14

3.0 6.0 4.0 5.5 5.0 5.0

0.012 0.068 0.0033 0.00053 0.0031 0.0038

HNO3

CSNO3

9.01 14.51 5.45 2.98 2.43 2.72

3.47 3.38 0.29 -0.94 -0.66 -1.27

3.0 6.0 6.0 3.5 4.5 1.4

0.0025 0.041 0.0056 0.0075 0.0083 0.0040

LiOH NaOH KOH CsOH

4.47 7.20 10.72 4.88

1.24 3.11 4.14 6.24

4.0 6.0 6.0 1.0

0.023 0.018 0.038 0.040

HCIO4 UCIO4


max IA0I

LiN03 NaN03 KNO3

RbN03

concentration limit was decreased. However, such behavior was generally not observed. We did note that for some 2:1 electrolytes, it was possible at selected upper limits of concentration to obtain reasonably good fits and physically realistic parameter values. However, at greater upper limits of concentration, either the deviation in the activity coefficient would increase and/or one or both of the parameter values would become unrealistic. Lack of perfection is apparent even in the results for the 1:1 salts. Although the parameters show significant regularity, they do not satisfy additivity relations within the fitting error. Letting q stand for the characteristic ion size or hydration number of an electrolyte, relations such as q(KBr) = q(KCl) + q(NaBr) - q(NaCl) should be satisfied. For KBr, we predict values of 7.27 and 1.98 for the ion size and hydration number; the fitted values are 7.14 and 1.53. Although this is not too bad, if we attempt to predict the same parameters for CsBr from those of CsCl, NaBr, and NaCl, we obtain 5.25 and 1.75, compared to fitted values of 4.20 and 1.14. One could go further and test the ability of the models for pure aqueous electrolytes to be extrapolated to predict the properties of mixtures, such as the system NaCl-KCl-H20, as no additional fit parameters are required by the model. However, because the additivity relations are not precisely satisfied, there seems little point in doing so. It seems likely that a major problem with the simple hydration theory model is the lack of explicit consideration of ion pairing, which is known to be stronger when the ions present have higher electrical charges. There may also be greater problems in the electrostatic part of the model. A more sophisticated electrostatic model than the DHC may required. Another factor may be kinetic repulsion, which is not addressed in electrostatic models such as Debye-Huckel. This, as well as other factors, might be addressed by including virial coefficient terms (hopefully to only second order) in the phenomenological equations.


2. WOLERY & JACKSON


CONCLUSIONS


We have developed the basic equations for applying a thermodynamically consistent hydration correction for both pure aqueous electrolytes and mixtures of aqueous electrolytes. A thermodynamically consistent method of using a compositionally averaged ion size parameter is included in this treatment. A simple hydration theory model using the DHC version of DebyeHuckel theory and ignoring ion pairing works rather well for pure 1:1 aqueous electrolytes and offers encouragement to the development of more sophisticated models that might be useful in geochemical modeling calculations. The simple model does not work well for 2:1 and higher pure aqueous electrolytes. Future efforts should be directed to develop more advanced hydration theory models than the simple form examined here. Ion pairing should be treated in an explicit manner, even for the 1:1 electrolytes. Other possible changes that might be explored include the use of a more sophisticated electrostatic model and the usage of virial coefficient terms in the phenomenological equations. These models must be tested against not only the ability to fit data, but also the abilities to satisfy additivity relationships and to predict the properties of mixtures of aqueous electrolytes.

ACKNOWLEDGMENTS Thanks are offered for encouragement at various stages in the development of this work to Judith Moody, Terry Steinborn, Paul Cloke, and Larry Brush. This work was begun with support from the Salt Repository Project Office and was continued with support from the Waste Isolation Pilot Plant. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405-Eng48. We thank Paul Cloke, Richard Knapp, and three anonymous reviewers for their helpful comments. Literature Cited 1. 2.

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Stokes, R.H.; Robinson, R.A. J. Amer. Chem. Soc. 1948, 70, 1870-1878. Nordstrom, D.K.; Plummer, L.N.; Wigley, T.M.L.; Wolery, T.J.; Ball, J.W.; Jenne, E.A.; Bassett, R.L.: Crerar, D.A.; Florence, T.M.; Fritz, B.; Hoffman, M.; Holdren, G.R., Jr.; Lafon, G.M.; Mattigod, S.V.; McDuff, R.E.; Morel, F.; Reddy, M.M.; Sposito, G.; and Thrailkill, J. In Chemical Modeling in Aqueous Systems, Jenne, E.A., ed. Amer. Chem. Soc. Symp. Series 1979, vol 79; pp. 857-892. Burns, D.T. Electrochim. Acta 1964, 9, 1545-1547. Helgeson, H.C.; Kirkham,D.H.,; Flowers, G.C. Amer. J. Sci. 1981, 281, 1249-1516. Friedman, H. L. Ionic Solution Theory 1962; Intersicience Publishing, New York. Friedman, H.L. Ann. Rev. Phys. Chem. 1981, 32, 179-204. Dickson, A.G.; Friedman, H.L.; Millero, F.J. Appl. Geochem. 1988, 3, 27-35. Pitzer, K.S. J. Phys. Chem. 1973, 77, 268-277. Pitzer, K.S. In Activity Coefficients in Electrolyte Solutions 1979; Pytkowicz, R.M., ed., CRC Press, New York; pp. 157-208. Harvie, C.E.; Moller, N.; Weare, J.H. Geochim. Cosmochim. Acta 1984, 48, 723-758. Robinson, R.A.; Stokes, R.H. Electrolyte Solutions 1965; second edition, Butterworths, London. Stokes, R.H.; Robinson, R.A. J. Soln. Chem. 1973, 2, 173-184. Nesbitt, H.W. J. Soln. Chem. 1982, 11, 415-422. Pitzer, K.S. Acc. Chem. Res. 1977, 10, 371-377. Pitzer, K.S. J. Soln. Chem. 1975, 4, 249-265. Helgeson, H.C.; Kirkham, D.H. Am. J. Sci. 1974, 274, 1199-1261. Smith, R.M.; Martell, A.E. Critical Stability Constants, Volume 4, Inorganic Complexes 1976; Plenum Press, New York.

RECEIVED July 20, 1989


Chemical Modeling of Aqueous Systems II - American Chemical Society

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