Appendix I
The Ion-Interaction Model of Pitzer for Multielectrolyte Solutions

Section 9.14 gives a brief discussion of the ion-interaction model of Pitzer and expressions for the activity coefficient and the osmotic coefficient of binary (single-electrolyte) solutions. This Appendix gives the genera! expressions for these coefficients for multielectrolyte solutions. Also given are tables that list the ion-interaction parameters and their temperature dependence for several common aqueous electrolytes.

For an electrolyte solution containing w_s kg of solvent s and solutes i, j,… with molalities m_i, m_j,…, Pitzer (1995) assumed that the excess Gibbs energy is

(I-1)¹

¹ Equation (1-1) is the same as Eq. (9-58).

Function f(I) depends on ionic strength I, temperature and solvent properties; it is a modified Debye-Hückel term

(I-2)

with

(I-3)²

² Equation (1-3) is the same as Eq. (9-43).

The empirical universal constant b = 1.2 kg^1/2 mol^-1/2 is used for all electrolytes. A_φ is the Debye-Hückel parameter for the osmotic coefficient given by

(I-4)

where N_A is Avogadro’s constant, d_s is the solvent density in g cm^-3, e is the electronic charge (e = 1.60218×l0^-19 C), ε_o the permittivity of free space (ε_o = 8.85419×l0^-12 C² N^-1 m^-2), ε_r is the dielectric constant or relative permittivity of the solvent, k is Boltzmann’s constant and T is absolute temperature.³

³ If the solvent is water (ε_r = 78.41), A_φ is 0.392 kg^1/2 mol^-1/2 at 25°C. A_φ increases with temperature to 0.462 at 100°C and to 0.962 at 300°C.

Interaction parameters λ_ij(I) and Λijk. are analogous to second and third virial coefficients; they represent the effects of short-range forces between, respectively, two and three ions. For highly concentrated solutions, fourth or higher-order interactions may be required in Eq. (1-1). The λ_ij and Λ_ijk matrices are symmetric, with λ_ij = λ_ijk, Λ_ijk = Λ_ijk = Λ_ijk etc. Parameters λ_ij and Λ_ijk are also used for ion-neutral and neutral-neutral interactions in the solvent. For the ion-neutral case, theory suggests an ionic-strength dependence when the neutral species has a significant dipole moment. However, these parameters are usually considered to be constants at a given temperature, independent of ionic strength.

For electrolytes with cations c, c′,… and anions a, a′,… only some combinations of λ’s and Λ’s are measurable. The following definitions simplify the final equations for activity coefficients:

(I-5)

(1-6)

(1-7)

(1-8)

with corresponding expressions for Φ_aa, and Ψaa’c- Because parameters λ_ij, are functions of ionic strength, B_ca, Φ_cc. and Φ_aa are also functions of I. For brevity, however, the explicit notation B(I), etc. is omitted except for f(I).

The excess Gibbs energy becomes

(1-9)

The double-summation indices, c < c’, a < a’, and n < n’ denote sums over all distinguishable pairs of different cations, anions, and neutral solutes, respectively.

Applying Eqs. (9-42) and (9-43) to Eq. (1-9), gives expressions for the osmotic coefficient and the mean ionic activity coefficients of the various electrolytes in a multielectrolyte mixture. To obtain the mean ionic activity coefficient y±x for an electrolyte M_v+ X_v–, it is preferable to obtain initially the expressions for the activity coefficients y_+,M and y_-x of the individual ions ” and ; the expression for y_+,M is then obtained in a subsequent step. The results are:

(1-10)

(1-11)

(1-12)

(1-13)

where second-virial terms for neutral species have been added but third-virial terms for neutrals are omitted. Terms B and C can be evaluated empirically from data for binary (single electrolyte + solvent) systems. Terms Φ and Ψ arise only for multi-electrolyte solutions; they can best be determined from data for common-ion mixtures. Quantity F includes the Debye-Huckel term and other terms as follows:

(1-14)

Also,

(1-15)

(1-16)

(1-17)

A corresponding expression is used for .B’ and Φ’ are the ionic-strength derivatives of B and Φ. The sums over i include all solute species; uncharged species do not contribute to I or Z. Parameter C_MX is related to the commonly tabulated by

(1-18)

The ionic-strength dependence of B terms is taken into account by

(1-19)

(1-19a)

(1-19b)

where functions g and g’ are given by

(1-20)

(1-20a)

, and are solute-specific parameters fitted to isothermal or isobaric and data for single-electrolyte solutions, is important only for 2-2 or higher-valence electrolytes that show a tendency toward electrostatic ion pairing. For solutions of electrolytes containing at least one univalent ion, α₁ is taken to be 2.0 kg^1/2 mol^-1/2 and a₂ = 0. For 2-2 electrolytes at 25°C, the optimized values of α₁ and α₂ are 1.4 and 12 kg1’2 mol”s/2, respectively. For many applications these values can be assumed independent of temperature and pressure, but there are theoretical reasons (Pitzer, 1991) for setting α₂ proportional to the Debye-Hücke! parameter A.

The Φ terms have a strong ionic-strength dependence for unsymmetric cases (e.g. Na+ with Mg2+ or Cl^- with ”) arising from long-range electrostatic forces. The expressions for Φ_ij are

(I-21)

(I-21a)

(I-21b)

where ^Eθ(I) and ^Eθ’(I) account for electrostatic unsymmetric mixing effects and depend only on charges of ions i and j, the total ionic strength, and on the density and dielectric constant of the solvent (i.e. on the temperature and pressure). Equations for calculating these terms have been developed; they are given elsewhere (Pitzer, 1991). The remaining term θ_ij, arising from short-range forces, is taken as a constant for any particular c, c’ or a, a’ at a given temperature and pressure. Terms ^Eθ(I) and ^Eθ’(I) are often omitted for solubility calculations.

Pitzer’s ion-interactive model gives expressions for the osmotic and activity coefficients of multi-electrolyte mixtures in terms of six types of empirical parameters, viz.,,, ,θ_ij and φ_ijk. Provided that the temperature and pressure dependencies of these parameters are known, solubilities in binary, ternary and higher mixtures can be calculated.

The excess Gibbs energy given by Eq. (I-1) yields other thermodynamic properties such as excess enthalpies and heat capacities by appropriate differentiation. These other excess properties can be measured directly. In the literature there is a wide array of experimental data for obtaining ion-interaction parameters and their dependence on temperature and pressure. Tables published elsewhere (Pitzer, 1991, 1995; Clegg and Whitfield, 1991; Zemaitis et al., 1986) give ion-interaction parameters for numerous aqueous solutions of electrolytes. Table I-1 gives parameters at 25°C for a few common aqueous electrolytes and Table I-2 gives the temperature dependence of parameters for those electrolytes used in Sec. 9.14 for model calculations. The mixed-electrolyte parameters for variable-temperature calculations are shown in Table I-3.

With model parameters determined as discussed above, and with experimental solubility products, Pitzer’s model can be used to predict solid solubilities in aqueous mixed-salt systems. It is relatively easy to calculate solubilities in two-salt solutions with a common ion. If the two salts do not have a common ion or, if there are more than two salts, the calculations become complex; they require solution of numerous simultaneous equations.

Section 9.14 presents two examples to illustrate how Pitzer’s equation, coupled with solubility products, can be used to calculate solid-liquid equilibria in aqueous systems containing two salts. Table I-4 gives standard-state chemical potentials, enthalpies of formation, and entropies at 25°C of the ions and solids considered there and for some other species of common interest. It is an important characteristic of the equations of the ion-interaction model of Pitzer that, for calculating solid-liquid equilibria, all parameters can be evaluated from measurements for single-electrolyte solutions and solutions containing two electrolytes with a comrnon-ion. No new parameters are required. Therefore, calculations for multi-salt systems are predictions.

Table I-1 Ion-interaction parameters at 25°C. Parameters only apply to a maximum molaity of 4-6 mol kg”1 (from Pitzer, 1995).

Table I-2 Standard-state heat capacity and temperature dependence of ion-interaction parameters for aqueous NaCl, KCl, and Na₂SO₄ (from Pitzer, 1995).

Table I-3 Mixed-electrolyte parameters for variable-temperature calculations (T in kelvin) (Pabalan and Pitzer, 1991).

Table I-4 Standard-state chemical potentials, enthalpies of formation, and entropies of aqueous species and solid salts at 25°C (Pitzer, 1995). Values in parentheses have large uncertainties.

The equations presented in this appendix are complex but they are all analytical. For application, a suitable computer program is required.

The practical applications of Pitzer’s method are not due to mathematical complexity but follow from the need for a large experimental-data base to fix parameters.

References

Clegg, S. L. and M. Whitfield, 1991. In Activity Coefficients of Electrolyte Solutions, (K. S. Pitzer, Ed.), 2^nd Ed., Chap. 6. Boca Raton: CRC Press.

Pabalan, R. T. and K. S. Pitzer, 1991. In Activity Coefficients of Electrolyte Solutions, (K. S. Pitzer, Ed.), 2^nd Ed., Chap. 7. Boca Raton: CRC Press.

Pitzer, K. S., 1991. In Activity Coefficients in Electrolyte Solutions, (K. S. Pitzer, Ed.), 2^nd Ed., Chap. 3. Boca Raton: CRC Press.

Pitzer, K. S., 1995, Thermodynamics, 3^rd Ed. New York: McGraw Hill.

Zemaitis, R. M., Jr., D. M. Clark, M. Rafal, and N. C. Scrivner, 1986, Handbook of Aqueous Electrolyte Thermodynamics. New York: A.I.Ch.E.