Activity Coefficients (Activity Models)
Activity Models
The step from molar concentrations (measured data) to activities (used in massaction law calculations) requires the calculation of activity coefficients \(\gamma_i\). For this task several semiempirical 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)  DebyeHückel  \(\lg \gamma_{i} = Az^{2}_{i} \ \sqrt{I}\)  I < 10^{2.3} M  
(2)  Extended DebyeHückel  \(\lg \gamma_{i} = Az^{2}_{i} \ \left ( \dfrac{\sqrt{I}}{1+Ba_i\sqrt{I}} \right )\)  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)  TruesdellJones (WATEQ DebyeHückel)  \(\lg \gamma_{i} = Az^{2}_{i} \ \left ( \dfrac{\sqrt{I}}{1+B a_i^0 \sqrt{I}}\right ) + b_i \!\cdot\! I\)  I < 1 M 
Here, z_{i} is the charge of ion i.
All quantities carrying the subscript i are ionspecific parameters (a_{i}, a_{i}^{0} and b_{i}). On the other hand, the prefactors A and B depend on the temperature T and the dielectric constant ε_{r}:
(5a)  A = [1.82∙10^{6} K^{3/2} M^{1/2}] ⋅ (ε_{r}T)^{3/2} 
(5b)  B = [50.3 nm^{1} K^{1/2} M^{1/2}] ⋅ (ε_{r}T)^{1/2} 
For water at 25 (T = 298 K) and with ε_{r} = 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 nonSI 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 DebyeHückel formula in 1. Denoting the “DebyeHückel building block” by \(\lg\gamma_i^0\), the above equations can be rewritten as:
Model  Equation  Validity  
(1b)  DebyeHückel  \(\lg \gamma_{i}^{0} \ =\ Az^{2}_{i} \ \sqrt{I}\)  I < 10^{2.3} M  
(2b)  Extended DebyeHückel  \(\lg \gamma_{i} \ =\ \dfrac{\lg \gamma_{i}^{0}}{1+Ba_i\sqrt{I}}\)  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)  TruesdellJones  \(\lg \gamma_{i} \ =\ \dfrac{\lg \gamma_{i}^{0}}{1+B a_i^0 \sqrt{I}} \ + \ b_i \!\cdot\! I\)  I < 1 M 
The activity coefficients decrease steadily when the ionic strength I rises. However, the Davies and the TruesdellJones equations contain an additive term that causes a reincrease when I approaches 1 mol/L — see diagrams below.
The empirical model of TruesdellJones in 4 and 4b, with its two ionspecific parameters a_{i}^{0} and b_{i}, represents the most general approach. By specifying and/or ignoring these two parameters we obtain the other three activity models.
Extended DebyeHückel Equation
Due to the narrow validity range of the DebyeHückel formula in 1, this approach was extended in 2 by an additional term in the denominator containing the parameters B and a_{i}. The extended formula takes into account the fact that the central ion has a finite radius (instead of a point charge). The parameter a_{i} 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^{+}, HCO_{3}^{}, SO_{4}^{2} 
a = 0.3 nm  for  K^{+}, NH_{4}^{+}, OH^{}, Cl^{}, NO_{3}^{} 
Note. The ion size parameter a_{i} 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 DebyeHückel 2:
 it get rid of the ionsize parameter a_{i} (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 (z_{i} = 0) the Davies formula collapses to the Setchenow equation: \(\lg\gamma_i = \textrm{const}\cdot I\).
TruesdellJones (WATEQ DebyeHückel)
The empirical approach of Truesdell and Jones^{3} was proposed for the hydrochem program WATEQ in 1974 with the aim of describing NaClcontaining solutions. The additional fit parameter b_{i} in 4 extended the scope to seawater (i.e. I = 0.72 M and above).
Equation (4) is based on two fit parameters: a_{i}^{0} and b_{i}, whereas the effective ionic radius differs from the extended DebyeHückel model (a_{i}^{0} ≠ a_{i}). Typical values for b_{i} 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 ionspecific parameters:
z_{i} = 2  
a_{i} = 0.80 nm  for Extended DebyeHückel  
a_{i}^{0} = 0.55 nm  for TruesdellJones ^{4}  
b_{i} = 0.2  for TruesdellJones ^{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 TruesdellJones 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 TruesdellJones equation (4) — if the parameters a_{i}^{0} and b_{i} are provided^{5}
model  parameters  name  used in aqion  

Davies  0  —  yes (default)  
Extended DH  1  a_{i}^{}  no  
TruesdellJones  2  a_{i}^{0}, b_{i}  yes 
The type of the activation model is set in the thdyn database (e.g. wateq4f.dat) for each aqueous species separately.
HighSaline Solutions (I > 1 M)
The Pitzer approach^{6} is a sophisticated ioninteraction model used for highsaline 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

lg (= log_{10}) denotes the decadic logarithm. ↩

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. ↩

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.233274, 1974 ↩

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 a_{i}^{0} and b_{i}. Note that a_{i}^{0} is in units of Ångström, i.e. 5.5 Å = 0.55 nm. ↩ ↩^{2} 
If no value for b_{i} is provided, then the b_{i} = 0.1 is used in the calculations. ↩

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