Application of fluctuation solution theory to strong electrolyte solutions

83 (1993) 233-242 Elsevier SciencePublishersB.V., Amsterdam

233

Fluid Phase Equilibria,

Application of fluctuation solution theory to strong electrolyte solutions

John P. O’Connell University

of Virginia,

Charlottesville,

VA 22903

USA

Keywords: theory, statistical mechanics, activity, density, critical, electrolytes

ABSTRACT Fluctuation solution theory via direct correlation integrals for correlating and predicting the densities and activities of aqueous strong electrolytes up to saturation is described theoretically and from experiment. The subtlety, but importance, of the McMillan-Mayer (MM) treatment is contrasted with other thermodynamic developments. Also, formal results are shown which distinguish the effects of solvent compressibility (fluctuations) and electrostatics on property divergences in systems near the solvent’s critical point.

INTRODUCTION Electrolyte solutions are of considerable practical and theoretical importance. The behavior of thermodynamic and transport properties of aqueous ionic systems is complex because of the combination of long range ionic and short range excluded volume effects. The result is that models for these properties have become quite complicated. Thus, while the original approach to activity coefficients of Debye and Hiickel is rigorous in the limit of infinite dilution, corrections due to short range interactions of the ions has been handled in a great variety of ways. (See, e.g., Pytkowicz, 1979 and Pitzer, 1991.) Fluctuation solution theory, from the statistical mechanics of the grand canonical ensemble (Kirkwood and Buff, 1951; Matteoli and Mansoori, 1990) yields expressions for concentration derivatives of the chemical potential and pressure in terms of integrals of molecular correlation functions (O’Connell, 1971,199O). Previous applications were on liquid volumes (Huang and O’Connell, 1987) and gas solubility (Campanella, et al., 1987); theoretical work has also treated electrolytes (Perry and O’Connell, 1984; Perry, et al., 1988). The purpose here is to describe the current state of development of this method. In particular, use of different thermodynamic variables within fluctuation solution theory explicitly shows 1) the proper quantities for converting

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234

from the continuous dielectric solvent of Debye and Hiickel to the actual molecular system of solutions and 2) the roles of electrostatic and compressibility divergences near the solvent critical point. Also shown are data generated from a careful analysis of NaCl data over wide ranges of conditions. FUNDAMENTALS Kirkwood and Buff (1951) used fluctuation statistical thermodynamics in the grand ensemble, which has independent variables of T, V’, ,g, to show that

This was reformulated (O’Connell, 1971) via cij and the DCFI Cij.

(2) Extension of the method was made to electrolyte solutions (Perry and O’Connell, 1984), with a direct correlation function for ions cr and /? which has a convergent integral in eq. (2). Th e result then gives ionically-additive “component” DCFI, for salt (2) in solvent (1) for the derivatives

s

x1(1

-

Cl2)

+

v2z2(1

-

C$2)

(4)

Derivatives with respect to pi are obtained by interchanging the subscripts 1 and 2 in eq. (3) and (4). The desired properties of yi and p at a given T, P, 22 are obtained by isothermal integration of a model for the DCFI (expressed in terms of T and _p)from a set of reference properties, P”,p”. Results include infinite dilution limits of the salt (pure solvent). 1 - C& = pT/n$RT

(5)

which is zero at the solvent’s critical point. 1 - Cz = r

jv2t+RT

(6)

235

Cooney and O’Connell (1987) gave a simple function of Cg with T and p? for a variety of salts from ambient to near-critical conditions where 7: is extremely negative. Crovetto (1990) found similar behavior for various gases dissolved in water where the partial molar volume is positively divergent. The Debye-Hiickel Limiting Law (DHLL) applies at infinite dilution

(7) where the sums are over all ions in salt 2. The last term diverges. THERMODYNAMIC VARIABLES A major theme in thermodynamics is the relations among partial derivatives where identifying variables held constant is essential. In solution theories from statistical mechanics, there are three principal sets. The extensive Kirkwood-Buff (1951) (KB), or canonical, variables are T, Vt , Nit,Ni, yielding eq. (3). McMillan and Mayer (1945)(MM) analyzed dilute solutions; they held the chemical potentials of the solvents constant while adding solute, i.e., the variables are T, Vt, Ni, &. The MM analog to eq. (3) is

(8) where the extra term arises from matrix manipulations (O’Connell, 1971). Finally, the common practical variables of T, I’, Ni, Ni, are attributed to Lewis and Randall (LR) ( see e.g., Pitzer, et al., 1984) leading to

~(~)?r4=~(~)~,~=l-c~2-u~~~T =

1 - ci2 -

[Sl(l

$1

-

Gl)

+

62)

+

2v23w2(1

v222(1 -

Cl2)

-

C22)12 +

Z$(l - C$)

(9a)

Eq. (9) are complex when intensive variables such as pi and m are used. Fig. 1 shows the different thermodynamic paths associated with the above sets (Friedman, 1972). Its relevance to electrolytes is that many models assume the Debye-Hiickel Limiting Law (DHLL) and its extensions (EXTDH) for ionic effects have LR variables (ri is from an excess gibbs energy) or KB variables (ri is from an excess helmholtz energy) though the proper set of variables is MM (Friedman, 1962; Pailthorpe, et al., 1984). Some of the complexity of the models may be due to trying to describe the last term in

236

eq. (9b). The alternative suggested here is to model the various C$ and then integrate (4) to obtain p and then integrate one of (3),(8) or (9) for yi.

I

T Constant

SOLUTE MOLE FRACTION

1

x2

Figure 1. Thermodynamic paths for solute addition.

DCFI MODEL EXPRESSIONS The Rushbrooke and Scoins (1953) treatment suggests that DCFI can be written in a virial series. Thus, a nonelectrolyte (1) has the form 1 -

Cl1

=

1 -

P[Jil(T,

PT) + P14111(q

+ P295112(q

(10)

For the salt-solvent DCFI there is an added electrostatic term (Redlich and Meyer, 1964; Perry, et. al., 1988) so the form is 1 - Cl2 = 1 -

GXTDH

- PPl2(T,

/8> + Ph2(~)

+ P24122(~)1

(11)

where to the first order in p2 3 alne~

= -_ GXTDH 4 ap?

L ln $$TD* >T v2

(12)

For the salt-salt DCFI, the virial form with an MM electrostatic term is 1 -

ci2

= 1 -

GXTDH

- &?22(~,

I> + w#J122(77

+ /324222(T)]

(13)

where

(14) In each of eq. (10) - (14), the virial coefficients and electrostatic terms are

237

consistent with a rigorous higher order DCFI relation (Perry, et al., 1988). (A small electrostatic term in (10) has been ignored here.) The theory (Perry and O’Connell, 1984) shows that the coefficients .Z$ and &k are ionically additive. DATA ANALYSIS Though sets of DCFI values for nonelectrolytes exist (e.g., Wooley and O’Connell, 1991), there are none for electrolytes. A thorough fitting of activity, volumetric and thermal data is available over a wide range of conditions (e.g., Archer, 1991; 1992). Archer’s equations have been used to generate fig. 2 - 4 for NaCl at fixed pressures just above the water saturation pressure at temperat~es from ambient to near-critical.

25

010

6

4

2

Salt Concentration,

mol/L

Figure 2. Solvent-solvent DCFI for aqueous N&l. 20 15 oz I

10

-

5 .

5

.

t

..

.

.

-5 -10

0

2

Salt Concentration,

4

6

mol/L

Figure3. Salt-solvent DCFI for aqueous NaCl. The behavior of 1 - Cl1 and 1 - Ci, up to 200 “C is regular and their slopes are weak functions of 2”. The values at 300 “C are different perhaps

238

due to uncertainties

in ICTnear the upper bound of the fitting range. The low concentration variation of 1 - C 12 s h ows the electrostatic effect of eq. (11) and (12). The divergence of 1 - C$ at infmite dilution in fig. 4 is from eq. (7).

0

4

2

Salt Concentration,

6

mol/L

Figure 4. Salt-salt DCFI for aqueous NaCl.

Fig. 5 shows the derivatives of eq. (3), (8) and (9) and the electrostatic term of the DHLL expression for NaCl at 25 “C. Subtracting the DHLL contribution and plotting the remaining “nonelectrostatic” contributions is shown in fig. 6. The MM and LR variations are similar to each other but somewhat different from KB. However, in all cases the large and rapidly varying contribution at low concentrations must be either residual from the approximations

in the model EXTDH

or contained

Values from Equations

0

in Fzz (Piker,

of Arc her(1992)

1984).

1

I

1

3

2

4

5

6

7

m, Molality Figure 5. Salt activity coefficient derivatives for aqueous NaCl.

In any case, model elsewhere,

equations

have been developed,

to fit all the Cij values (except

and will be published

at 300 “C) to within l%, probably

239

better

than experimental

error and equivalent

Values

01

from

Equations

to the lines in the figures.

of Arched1

992)

I

0

1

2

3

4

5

6

m. Molality

Figure 6. Nonelectrostatic

contributions

to salt activity

coefficient derivatives

for aqueous

NaCl at 25 “C.

NEAR-CRITICAL

BEHAVIOR

Griffiths and Wheeler (1970) and Leung and Griffiths (1973), as most recently discussed by Levelt Sengers (1991ab), d iscuss the relationship of critical point divergences to sets of intensive variables. There are “field” variables (f) such as T, P, pi which have the same value in the phases and “density” variables (p) which are derivatives of one field with respect to another (e.g., a pure fluid V is 6’p/aP),.) are different

and density variables, Near-critical diverge

D ensity variables usually include properties

in the phases such as V, xi, m. There are conjugate denoted

behavior

to infinity

which

pairs of field

with ‘,“, . . . .

of thermodynamic

derivatives,

or go to zero, depends

especially

on the variables,

those which

e.g., derivatives

such as dp/df)f~,f~ diverge strongly in both pure and mixed systems. quantitative variation is different for classical or nonclassical treatments. set of derivatives other

of the form dp/df)p’;~,,f,,,,...

nonconjugate

fields,

(f”, f”‘, etc.),

with one density, p’, and all the

held constant,

for pure and mixed systems but can become becomes

an infinitely

dilute

to the critical

and often can be ignored near the solvent’s

is weakly

divergent “Weakly

divergent

as a mixture divergent”

only in a very small range of conditions

with two or more densities constant, All solutions

strongly

solute in a pure solvent.

means that the effect is important

The The

for practical

purposes.

close

Derivatives

dp/8f)~,~,;~~~,,..., are always nondivergent. critical

point have some properties

which

diverges strongly as 22 is decreased along any path to the diverge, e.g., c solvent’s critical(though the value depends on the path). This is due to the divergence

of the IETwhich is easily seen in DCFI

terms for an electrolyte

as

Note that (1 - Cg) is finite for both point and real particles. The DCFI analysis of eq. (3), (8) and (9) g ives insights about the nature of critical divergences. Thus, in nonelectrolytes the KB derivative of eq. (lo), (1 - Cg), goes weakly to zero since it is reciprocal to the form ap/6’f)~;f~~,f~~~,... which is weakly divergent. On the other hand, the limiting derivatives of eq. (8) (MM) and eq. (9) (LR) both diverge since their second terms vary as n$ (the inverse of 1 - Ci’r) at 222= 0. For dilute electrolytes, the DHLL is eq. (8) with (7) for Ci2 and (5) for Crr. The two terms show that the DHLL has both electrostatic (first term) and compressibility (second term) critical divergences. This may be significant for theory. The LR derivative (fixed T, P) of (9) also always diverges from both contributions. With 1 - Ci2 varying as (l/~)r/~ and a term in l/(1 - Cri), as in the second one of (9a), varying as (1/~2)~/~ for classical systems and as (1/~2)~/~ for nonclassical systems, the composition variation of ri at fixed T and P will be complex. Ultimately, the compressibility effects having higher powers make the dominant contribution as the solvent critical is approached. This may be significant for experiment. While there are no explicit DCFI formulae for variations along the phase envelope, this case should also be complicated. The data of Busey, et al. (1984) show that at temperatures slightly above the water critical, the 22 variation of P is very sharp and of varying sign. CONCLUSIONS This work shows some applications to electrolyte solutions of fluctuation solution theory via DCFI. Recognizing the connection of electrostatic effects with MM variables and the corrections of the theory to variables for KB and LR systems yields a more rigorous unified model for activities and densities. The formulation also infers the roles of solvent fluctuations relative to electrostatics in activity coefficient divergences near the solvent critical. LIST OF SYMBOLS c = direct correlation function, eq. (2) C = Direct Correlation Function Integral (DCFI), eq. (2) e = electronic charge f = field variable having same value in all phases in equilibrium Fij = virial coefficient in correlation for Cij, eq. (lo), (ll), (13). gij = radial distribution function for components i and j

241 I = molal ionic strength, c k ~u,sz%ps k = Boltzmann constant tm* m = molality, moles of salt per kg of solvent hf, = molecular weight of component i N’ = number of moles N4 = Avogadro’s number P = total pressure r = separation between ions, molecules, etc. R = universal gas constant 2re6NA ‘Ia $1 = Debye-Hiickel Limiting Law Property, = (e;kT)3 [ T = absolute temperature V = volume Va = partial molar volume of component 2 5i = mole fraction of component i Z, = charge on ion cy ^/fz= mean ionic activity coefficient on molarity scale Sij = Kronecker delta, = 1 for i = j, = 0 for i # j ET = solvent dielectric constant nT = isothermal compressibility pi = chemical potential of component i uOz= number of CYions in salt 2 va = c, va2 p = molar density, density variable p; = molar concentration of component i virial coefficient in correlation for Cij, eq. (lo), (ll), (13).

1

bijk

=

o = spatially convergent direct correlation function DHLL,EXTDH = limiting law, extended Debye- Hiickel form ’ = reference state, pure solvent ’ = total amount oI)= infinite dilution ‘,” ,“‘, etc. = different field or density variables f~ = mean ionic value for salt 2 components a~ = ions i,j

=

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