Activity Coefficients (Activity Models)
Activity Models
The step from molar concentrations (analytical data) to activities (used in mass-action law calculations) requires the calculation of activity coefficients γi. For this task several semi-empirical approaches are available, whereas each activity model has its own range of validity defined by the ionic strength I as shown here:1
Model | Equation | Validity | ||
(1) | Debye-Hückel | \(\lg \gamma_{i} = -Az^{2}_{i} \ \sqrt{I}\) | I < 10-2.3 M | |
(2) | Extended Debye-Hückel | \(\lg \gamma_{i} = -Az^{2}_{i} \ \left ( \dfrac{\sqrt{I}}{1+Ba_i\sqrt{I}} \right )\) | I < 0.1 M | |
(3) | Davies | \(\lg \gamma_{i} = -Az^{2}_{i} \ \left ( \dfrac{\sqrt{I}}{1+\sqrt{I}} - 0.3 \!\cdot\! I\right )\) | I ≤ 0.5 M | |
(4) | Truesdell-Jones (WATEQ Debye-Hückel) | \(\lg \gamma_{i} = -Az^{2}_{i} \ \left ( \dfrac{\sqrt{I}}{1+B a_i^0 \sqrt{I}}\right ) + b_i \!\cdot\! I\) | I < 1 M |
Here, zi is the valence of ion i. All quantities carrying the subscript i are ion-specific parameters (ai, ai0 and bi). On the other hand, the parameters A and B depend on temperature T and the dielectric constant ε:
(5a) | A = 1.82 ∙ 106 (εT)-3/2 |
(5b) | B = 50.3 (εT)-1/2 |
For water at 25 and with ε = εrε0 = 78.54 · 8.854 10-12 C2/(J m), we get:
(6a) | A = 0.5085 M-1/2 |
(6b) | B = 3.281 M-1/2 nm-1 |
Please note the length unit: 1 nm = 10-9 m = 10 Ångström.
The relationship between the activity models becomes most clear when they are all traced back to the simple Debye-Hückel formula in 1. Denoting the “Debye-Hückel building block” by lg γi0 the equations above can be rewritten as:
Model | Equation | Validity | ||
(1b) | Debye-Hückel | \(\lg \gamma_{i}^{0} \ =\ -Az^{2}_{i} \ \sqrt{I}\) | I < 10-2.3 M | |
(2b) | Extended Debye-Hückel | \(\lg \gamma_{i} \ =\ \dfrac{\lg \gamma_{i}^{0}}{1+Ba_i\sqrt{I}}\) | I < 0.1 M | |
(3b) | Davies | \(\lg \gamma_{i} \ =\ \dfrac{\lg \gamma_{i}^{0}}{\ 1+\sqrt{I}\ } \ + \ 0.3 Az^{2}_{i} \!\cdot\! I\) | I ≤ 0.5 M | |
(4b) | Truesdell-Jones | \(\lg \gamma_{i} \ =\ \dfrac{\lg \gamma_{i}^{0}}{1+B a_i^0 \sqrt{I}} \ + \ b_i \!\cdot\! I\) | I < 1 M |
The activity coefficients decrease steadily when the ionic strength I rises. However, both the Davies and Truesdell-Jones equations obey an additive term that causes a rise again when I approaches 1 mol/L – see diagrams below.
Model Hierarchy. The empirical model of Truesdell-Jones in 4 and 4b, with its two ion-specific parameters ai0 and bi, represents the most general approach. By specifying and/or ignoring these two parameters we obtain the other three activity models.
Extended Debye-Hückel Equation
Due to the narrow validity range of the Debye-Hückel formula in 1, this approach was extended in 2 by an additional term in the denominator containing the parameters B and ai. The extended formula takes into account the fact that the central ion has a finite radius (instead of a point charge). The parameter ai represents the effective size of the corresponding ion, for example:2
a = 0.9 nm | for | H+, Fe+3, Al+3 |
a = 0.8 nm | for | Mg+2 |
a = 0.6 nm | for | Ca+2, Fe+2 |
a = 0.4 nm | for | Na+, HCO3-, SO4-2 |
a = 0.3 nm | for | K+, NH4+, OH-, Cl-, NO3- |
Note. The ion size parameter ai is an empirical fitting parameter; it is larger than the ionic radius because it contains a part of the hydrate shell.
Davies Equation
The Davies formula in 3 is an empirical approach which differs in two respects from the extended Debye-Hückel 2:
- it get rid of the ion-size parameter ai (which is not well known for complex ions)
- the additional term 0.3⋅I, which is linear in the ionic strength I
Irrespective of the fact that there is no strict theoretical justification for the additional term, it improves the empirical fit to higher ionic strengths up to I ≈ 0.5 M. Due to its mathematical simplicity and the lack of free parameters, the Davies equation is a preferred choice in hydrochemistry modeling (see below).
For neutral species (zi = 0) the Davies formula collapses to the Setchenow equation: lg γi = const⋅I.
Truesdell-Jones (WATEQ Debye-Hückel)
The empirical approach of Truesdell and Jones3 was proposed for the hydrochemistry program WATEQ in 1974 with the aim of describing NaCl-containing solutions. The additional fit parameter bi in 4 extended the scope to seawater (i.e. I = 0.72 M and above).
Equation (4) is based on two fit parameters: ai0 and bi, whereas the effective ionic radius differs from the extended Debye-Hückel model (ai0 ≠ ai). Typical values for bi are in the order of 0.1.
Example: Activity Coefficient for Mg
The subsequent two diagrams plot the activity coefficient γi of the ion Mg+2 as a function of the ionic strength:
• upper diagram: | I = 0 … 2 M | (linear) |
• bottom diagram: | I = 0.0001 … 5 M | (logarithmic) |
The calculations are based on the following ion-specific parameters:
zi = 2 | ||
ai = 0.80 nm | for Extended Debye-Hückel | |
ai0 = 0.55 nm | for Truesdell-Jones 4 | |
bi = 0.2 | for Truesdell-Jones 4 |
[Please note: The diagrams display γi on the y-axis (and not lg γi).]
At I=0 (ideal solution) the activity coefficient is 1. It decreases with increasing ionic strength I. At high ionic strength (I ≈ 1 M) there is again an increase of γi, but only for Davis and Truesdell-Jones caused by the additive terms in 3 and (4).
What Activity Model is used in the Program?
The program aqion, which is based on PhreeqC, uses
- either the Davies equation (3) – as default setup
- or the Truesdell-Jones equation (4) – if the parameters ai0 and bi are provided5
Model | Parameters | Name | Used in aqion? |
---|---|---|---|
Davies | 0 | – | yes (default) |
Extended DH | 1 | ai | no |
Truesdell-Jones | 2 | ai0, bi | yes |
The type of the activation model is set in the thermodynamic database (e.g. wateq4f.dat) for each aqueous species separately.
High-Saline Solutions (I > 1 M)
The Pitzer approach6 is a sophisticated ion-interaction model used for high-saline solutions up to I = 20 M. Unfortunately, a lot of additional parameters (virial coefficients) are required for this. The Pitzer model is not implemented in aqion.
References & Remarks
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lg (= log10) denotes the decadic logarithm. ↩
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The effective ionic radii are selected from the classical paper: J. Kielland, J. Am. Chem. Soc., 59, 1675 (1937). Unit conversions: 1 Å = 0.1 nm = 10-8 cm. ↩
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AH Truesdell, BF Jones: WATEQ – A computer program for calculating chemical equilibria of natural waters; Journal of Research, U.S.G.S. v.2, p.233-274, 1974 ↩
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These data are taken from the thermodynamic database wateq4f. They are defined in the data block for Mg+2 in the line
-gamma 5.5 0.200
, where the first and second parameters represent ai0 and bi. Note that ai0 is in units of Ångström, i.e. 5.5 Å = 0.55 nm. ↩ ↩2 -
If no value for bi is provided, then the bi = 0.1 is used in the calculations. ↩
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KS Pitzer: Activity coefficients in electrolyte solutions (2nd ed.), Boca Raton, CRC Press, 1991 ↩