The simplified Helgeson-Kirkham-Flowers (HKF) model:
|
log γi = |
−A Zi2
(√I /
(1+B √I
))
− log(1+0.018 ∑mk)
+ b I
|
where I is the ionic strength,
γi is the activity coefficient of an
ion i with electric charge Zi;
A is the Debye-Hückel
slope (see the table below);
B (= åBγ in the
Debye-Hückel eqn.)
includes the distance of closest approach (effective ionic radii);
and b is a solute specific parameter. The sum
is made over all solute species k in the aqeuous solution.
See the list of references.
In SPANA (and SED/PREDOM)
the HKF model is simplified by using the NaCl parameters:
-
the B parameter
(= åBγ),
which includes the effective ionic radii å,
is obtained by multyplying
the Debye-Hückel constant Bγ (from
Table 1, p.1295, in Helgeson et al. 1981)
with the effective ionic distance for NaCl
(å = 1.91+1.81 = 3.72;
see Tables 2 or 3 in Helgeson et al. 1981
or Table 9 in Shock et al. 1992).
This results in B = 1.22 at 25°C (see table below).
- the b parameter for NaCl solutions = 0.064 at
25°C, see table below (from Table 5, p.1327, in
Helgeson et al. 1981,
and Table A-2 in
Oelkers and Helgeson 1990).
- for the sum (∑mk):
if a constant value for the ionic strength is given, then the value of I
is used instead of ∑mk. However, if the ionic strength is calculated
iteratively at each point, then ∑mk is calculated as well.
The HKF parameters A, B, and b
for this simplified model are (Helgeson et al. 1981,
Oelkers and Helgeson 1990):
t (°C) |
A |
B |
b |
0 | 0.491 | 1.21 | 0.041 |
25 | 0.509 | 1.22 | 0.064 |
50 | 0.534 | 1.24 | 0.074 |
100 | 0.600 | 1.27 | 0.076 |
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 simplified HKF model as follows:
|
Φ = |
ln(1+0.018 ∑mk)/(0.018 ∑mk)
− (2/3) (ln(10)/∑mk) A I3/2
σ(B√I)
+ (ln(10)/∑mk)
(b/2) I
|
where
|
σ(x) = |
(3/x3) ((1+x) − 1/(1+x) − 2 ln(1+x))
|
References
- H.C.Helgeson, D.H.Kirkham and G.C.Flowers, Amer. Jour. Sci., 281 (1981)
1249-1516; eqns. 165-167, 121, 297, 298
- E.H.Oelkers and H.C.Helgeson, Geochim. Cosmochim. Acta, 54 (1990)
727-738; eqns.22 and 23.
- E.L. Shock, E.H.Oelkers, J.W. Johnson, D.A. Sverjensky, H.C.Helgeson
J. Chem. Soc. Faraday Trans., 88 (1992)
803-826
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