# Activity Coefficients (Activity Models)

Activity Models

The step from molar concentrations (measured data) to activities (used in mass-action law calculations) requires the calculation of activity coefficients $$\gamma_i$$. 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 charge of ion i.

All quantities carrying the subscript i are ion-specific parameters (ai, ai0 and bi). On the other hand, the prefactors A and B depend on the temperature T and the dielectric constant εr:

 (5a) A = [1.82∙106 K3/2 M-1/2]   ⋅ (εrT)-3/2 (5b) B = [50.3 nm-1 K1/2 M-1/2] ⋅ (εrT)-1/2

For water at 25 (T = 298 K) and with εr = 78.54, we obtain:

 (6a) A = 0.5085 M-1/2 (6b) B = 3.281 M-1/2 nm-1

Note the length unit in the last equation: 1 nm = 10-9 m = 10 Ångström.

Pitfalls. The prefactors A and B depend on the units chosen (and other conventions). So be careful when comparing them with other prefactors given in the literature.

Example 1. If the activity corrections are expressed in terms of ln $$\gamma_i$$ (instead of lg $$\gamma_i$$), then A should be multiplied by (ln 10):

 A $$\quad\Longrightarrow\quad$$ A · (ln 10) = 1.172 M-1/2

Example 2. If the prefactor B in 6b is expressed using the non-SI unit Ångström (instead of nm), then B should be divided by 10:

 B $$\quad\Longrightarrow\quad$$ B/10 = 0.3286 M-1/2 Å-1

Model Hierarchy. The relationship between the various 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 $$\lg\gamma_i^0$$, the above equations 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, the Davies and the Truesdell-Jones equations contain an additive term that causes a re-increase when I approaches 1 mol/L — see diagrams below.

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 absence 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: $$\lg\gamma_i = \textrm{const}\cdot I$$.

Truesdell-Jones (WATEQ Debye-Hückel)

The empirical approach of Truesdell and Jones3 was proposed for the hydrochem 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 $$\gamma_i$$ 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

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 $$\gamma_i$$, 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 thdyn 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