Activity Coefficient Models

Easily solve activity coefficient models for a missing term.

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Activity coefficient models consist of two experimental terms (f_i and I) and several fitting parameters, where
f_i is the activity coefficient of ion i in a solution.
I is the solution ionic strength, typically given in molar (M) or molal (m) units.

and where

A is a fitting parameter equal to 1.82x10⁶·(ε_rT)^-3/2.
B is a fitting parameter equal to 50.3·(ε_rT)^-1/2.
C is a model fitting parameter, in L/mol so molar ionic strength units cancel out. It is usually set between 0.2 and 0.5. Our tool uses 0.3 by default.
a_i is the ion size (not radius) in nanometers, nm (1 nm = 10 angstrom).
a°_i is a model fitting parameter, in nm.
b_i is a model fitting parameter, in L/mol, and usually set between 0.1 and 0.3. Our tool uses 0.2 by default.
ε_r is the solvent relative permittivity (dielectric constant), which is the ratio of the electric field strength in vacuum to that in a given medium. For pure water at 25° C (298.15 kelvins) Malmberg & Maryoff (1956) reported a value of 78.3, which is not that far from the IUPAC accepted value of 78.36 ±0.05. (Barthel, Krienke, & Werner). NIST reports a value of 78.4 (Fernandez, Mulev, Goodwin, & Levelt Sengers, 1995), which is our tool default value.
T is the temperature in kelvins at standard conditions.
z_i is the valence (charge) of ion i, which can be treated as an integer parameter because computing a fractional z_i has no chemical meaning in the context of activity coefficient models.

The molar ionic strength of a solution, I, is a function of the sum of products of all ion concentrations, c_i, present in solution times the square of their ionic charge (valence). The sum is divided by 2 as both cations and anions are included; i.e.

I = 1/2*Σc_iz_i²

I can be molar (mol/L) or molal (mol/kg water). To avoid confusion the units should be stated explicitly, or be used unitless (Solomon, 2001). In this work we explicitly assume that I is in mol/L.

Debye-Hückel theory of electrolytes was initially based on several assumptions: among others, that ions in a diluted solution dissociate completely, are symmetric, have a negligible ionic radius, and do not interact with other ions or the solvent. Several extended models have been proposed since then, each applicable up to an upper bound of ionic strength. These are given in Figure 1.

Figure 1. Activity coefficient models with estimated upper bound values of ionic strength, I, where activity coefficients, f_i, steadily decrease.

According to Figure 1, Debye-Hückel Limiting Law model is applicable to ions in highly diluted solutions (I < 0.005 M). By contrast, Debye-Hückel Extended Limiting Law and Güntelberg are used for ions in moderately diluted solutions (I < 0.1 M), while Davis's applies to solutions with relatively high ionic strength (I < 0.5 M) and Truesdell-Jones's to even more concentrated solutions (I < 1 M); for instance, sea water whose major components are Na⁺, K⁺, Mg²⁺, Ca²⁺, Cl^-, and SO₄^2- (Ryan, 2004).

Finally, for highly saline solutions (1 < I < 20 M), the Pitzer Equation, a more sophisticated model is recommended. This model, however, requires a lot of additional parameters (virial coefficients) and is not included in our tool.

From the above discussion we may conclude, as Henry Freiser (1992) asserted that

"The limiting law is truly limiting and not a very useful law, however. Some chemists say it is only useful in 'slightly dirty water'."

In general as I decreases, the models converge to Debye-Hückel Limiting Law. See Figure 2, inspired by the one given at Aqion.

Figure 2. Parameterized relationships between several activity coefficient models. For highly diluted solutions (i.e., at low I) they all converge to the Debye-Hückel Limiting Law. I is assumed in mol/L.

Thus, arbitrarily applying Debye-Hückel Limiting Law, or any ionic strength model, is contraindicated as one may end computing meaningless activity coefficients, or the wrong ones.

Interpretation of results
A ≃ 0.51 M^-1/2 and B ≃ 3.3 M^-1/2nm^-1 for diluted aqueous solutions at 25° C; i.e., T = 298.15 in the Kelvin scale. C doesn't have a generally accepted value and is usually set between 0.2 < C < 0.5 (Chembuddy.com, 2017). Actually Davis used 0.2 and 0.3 (Davis, 1962). MINEQL program uses 0.2 and there is an option for its use in PHREEQE with C = 0.3. (Pearson & Berner, 1991).

The ion size parameter, a_i is not the ionic radius, but a measure of the diameter of a hydrated ion and has to be known in order to calculate its activity coefficient. By contrast, a°_i and b_i are ion-specific parameters fitted from mean salt activity coefficient data.

In many applications, a_i = 0.3 nm; therefore, Ba_i ≃ 1 so we can simplify the several models by setting a°_i = a_i = 1/B, as can be seen from Figure 2. One can also simplify the models by setting a_i = b_i = 0. Notice that the prevailing factor affecting activity coefficients and the selection of a model is the solution ionic strength.

Another reasons for using simplified activity coefficient models comes from the fact that the ion size parameter is not well known for all ions, particularly the complex ones, and because for the most commonly known ions this parameter is between 0.3 to 0.9 nm, as can be seen from Table 1, derived from Kielland (1937).

Table 1. Common *a_i* values in water
a_i (nm)	ion
0.9	H₃O⁺, Al³⁺, Fe³⁺, La³⁺, Ce³⁺
0.8	Mg²⁺, Be²⁺
0.7	COO^-
0.6	Ca²⁺, Zn²⁺, Cu²⁺, Fe²⁺, Sn²⁺, Mn²⁺
0.5	Ba²⁺, Sr²⁺, Pb²⁺, CO₃^2-
0.4	Na⁺, HCO₃^-, H₂PO₄^-, HPO₄^2-, PO₄^3-, SO₄^2-, CH₃COO^-,
0.3	K⁺, Ag⁺, NH₄⁺, OH^-, NO₃^-, Cl^-, Br^-, I^-, HS^-,

If using Truesdell-Jones model, one would also need to know the ion-specific parameters, determine these experimentally, or assume some values. Table 2 lists these for some common ions (Hellevang, 2006).

Table 2. Truesdell-Jones *a°_i* and *b_i* values for some ions in water
ion	a°_i (nm)	b_i (L/mol)
Na⁺	0.43	0.06
K⁺	0.37	0.01
H⁺	0.48	0.24
Mg²⁺	0.55	0.22
Fe²⁺	0.51	0.16
Ca²⁺	0.49	0.15
Al³⁺	0.67	0.19
Cl^-	0.37	0.01
OH^-	1.07	0.21
HCO₃^-	0.54	0
CO₃^2-	0.54	0

Drawbacks
If we define x = I^1/2, it can be demonstrated that Davis and Truesdell-Jones models lead to third degree polynomial expressions in x so the nonlinearity of these models can be understood on mathematical terms.

This explains why mapping I to f_i with these models is faster than mapping f_i to I. Since a close form function for I does not exist for these models, one would need to solve for I using successive approximations or residual analysis.

An important drawback, common to most activity models, is that at high I they all break down on chemical and mathematical grounds. Let see why.

At high I, ions can no longer be treated as point charges as they can interact with the solvent or other ions (e.g. by forming ion pairs). The ionic radius, negligible in diluted solutions, can then be comparable to the radius of the ionic atmosphere. As a result of this, f_i increases for relatively high ionic strengths and the models predict that some values of f_i can be mapped to more than one I. Thus, the models break down completely.

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