SED & PREDOM / Ionic Strength / Activity Coefficient Models

The simplified Helgeson-Kirkham-Flowers (HKF) model

The equation to calculate the activity coefficient, γ_i, of an ion “i” with electric charge Z_i is:

log γi =

−Aγ Zi2 (√I  /  (1 + åBγ √I )) − log(1+0.018 ∑mk) + bγ,MX I

where I is the ionic strength, A_γ and B_γ are Debye-Hückel parameters which are temperature- and pressure dependent (see Table), å is the effective ionic radius (discussed below), and b_γ,MX is a solute-specific parameter. The sum (∑m_k) is made over all solute species “k” in the aqeuous solution, both charged and neutral. The model is further described in Helgeson, Kirkham and Flowers (1981) (eqs. 121, 165-167, 297, 298), Oelkers and Helgeson (1990) (eqs. 22 & 23) and in Oelkers and Helgeson (1991) (eqs. 3 & 9).

In SPANA (and SED/PREDOM) the HKF model is simplified by using the NaCl parameters:

• the parameter “å” is the effective ionic distance for NaCl: å = (1.91+1.81) Å = 3.72 Å; see Table 9 in Shock et al. (1992) and Table 3 in Helgeson et al. (1981). This results in åB_γ = 1.22 (mol/kg)^−½ at 25°C. This effective ionic distance changes at high temperature or pressure as described in Shock et al. (1992) (eqs. 4-5, 24-26, Table 3).
• b_γ,MX for NaCl solutions is used, see Table. b_γ,NaCl= 0.064 kg/mol at 25°C.
• the calculated ionic strength (I) is used instead of ∑m_k. This avoids problems when making diagrams where the redox potential, Eh, is outside the stability range of H₂O. Convergence problems in the calculations occur if large amounts of either O₂(aq) or H₂(aq) are calculated when Eh is too high or too low, respectively, because then ∑m_k becomes unrealistically large.

The activity of water (H₂O) is given by

log aH2O =

− Φ ∑mk / (ln(10) 55.508)

where “Φ” is the osmotic coefficient, calculated using the HKF model as follows:

Φ =

ln(1+0.018 ∑mk)/(0.018 ∑mk) − (2 AΦ (1/∑mk) I3/2 σ(åBγ√I)) + (1/∑mk) (1/2) ln(10) bγ,NaCl I

where A_Φ = (1/3) ln(10) A_γ, and

σ(x) =

(3/x3) ((1+x) − 1/(1+x) − 2 ln(1+x))

In SPANA (and SED/PREDOM) these equations are simplified by using the ionic strength value (I) instead of ∑m_k. This avoids the convergence problems explained above in diagrams where the redox potential, Eh, is outside the stability range of H₂O.

References

• H.C. Helgeson, D.H. Kirkham and G.C. Flowers, Amer. Jour. Sci., 281 (1981) 1249-1516. doi: 10.2475/ajs.281.10.1249
• E.H. Oelkers and H.C. Helgeson, Geochim. Cosmochim. Acta, 54 (1990) 727-738. doi: 10.1016/0016-7037(90)90368-U
• E.H. Oelkers and H.C. Helgeson, Geochim. Cosmochim. Acta, 55 (1991) 1235–1251. doi: 10.1016/0016-7037(91)90303-M
• E.L. Shock, E.H.Oelkers, J.W. Johnson, D.A. Sverjensky, H.C.Helgeson J. Chem. Soc. Faraday Trans., 88 (1992) 803-826. doi: 10.1039/FT9928800803

 Models for activity coefficients

 Comparison between experimental data and models