Activity Coefficients (Activity Models)
Activity Models
The step from molar concentrations (analytical data) to activities (used in massaction law calculations) requires the calculation of activity coefficients γ_{i}. For this task several 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 valence 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 parameters A and B depend on temperature T and the dielectric constant ε:
(5a)  A = 1.82 ∙ 10^{6} (εT)^{3/2} 
(5b)  B = 50.3 (εT)^{1/2} 
For standard conditions (water at 25) 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 evident when they are all traced back to the simple DebyeHückel formula in 1. Denoting the “DebyeHückel building block” by lg γ_{i}^{0} the equations above 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, both the Davies and TruesdellJones 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 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 ion radius because it includes some aspect of hydrated 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. Because of its mathematical simplicity and lack 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 γ_{i} = const⋅I.
TruesdellJones (WATEQ DebyeHückel)
The empirical approach of Truesdell and Jones^{3} was proposed for the hydrochemistry 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 ion 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 γ_{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} 
[Please note: The diagrams display γ_{i} on the yaxis (and not lg γ_{i}).]
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 γ_{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 thermodynamic database wateq4f.dat for each aqueous species separately.
HighSaline Solutions (I > 1 M)
The Pitzer equation is a much more sophisticated ion interaction model that has been used in very high strength solutions up to I = 20 M. It requires a lot of additional parameters (virial coefficients). 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. ↩

A.H. Truesdell, B.F. 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. ↩