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.

hierarchy of activity models

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:

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).]

activity coefficient for Mg+2 as function of ionic strength

activity coefficient for Mg+2 as function of ionic strength (log scale)

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

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

  1. lg (= log10) denotes the decadic logarithm. 

  2. 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. 

  3. 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 

  4. 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

  5. If no value for bi is provided, then the bi = 0.1 is used in the calculations. 

  6. KS Pitzer: Activity coefficients in electrolyte solutions (2nd ed.), Boca Raton, CRC Press, 1991 

[last modified: 2022-05-29]