Activity Coefficients (Activity Models)

Activity Models

The step from molar concentrations (analytical data) to activities (that enter law-of-mass-action 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

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	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

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Here, z_i is the valence of ion i. All quantities carrying the subscript i are ion-specific parameters (a_i, a_i⁰ 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 ∙ 106 (εT)-3/2
(5b)	B = 50.3 (εT)-1/2

For standard conditions (water at 25°C) 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 Debye-Hückel formula in Eq.(1). Denoting the “Debye-Hückel building block” by lg γ_i⁰ the equations above can be rewritten as:

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	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

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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 renewed increase when I approaches 1 mol/L – see diagrams below.

Model Hierarchy. The empirical model of Tuesdell-Jones in Eq.(4) and Eq.(4b), with its two ion-specific parameters a_i⁰ and b_i, 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 Eq.(1), this approach was extended in Eq.(2) by an additional term in the denominator containing the parameters B and a_i. The extended formula considers the fact that the central ion has a finite radius (rather than 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+, HCO3-, SO4-2
a = 0.3 nm	for	K+, NH4+, OH-, Cl-, NO3-

Note. The ion size parameter a_i is an empirical fit parameter; it is larger than the ionic radius because it includes some aspect of hydrated shell.

Davies Equation

The Davies formula in Eq.(3) is an empirical approach which differs in two respects from the extended Debye-Hückel Eq.(2):

• it get rid of the ion-size 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 strength up to I ≈ 0.5 M. Because of its mathematical simplicity and lack of free 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.

Truesdell-Jones (WATEQ Debye-Hückel)

The empirical approach of Truesdell and Jones³ was proposed for the hydrochemistry program WATEQ in 1974 with the aim to describe NaCl-containing solutions. Due to the additional fit parameter b_i in Eq.(4) the range of application was enlarged to sea waters (I = 0.72 M and above).

Equation (4) is based on two fit parameters: a_i⁰ and b_i, whereas the effective ion radius differs from the extended Debye-Hückel model (a_i⁰ ≠ 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 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 γ_i on the y-axis (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 Truesdell-Jones caused by the additive terms in Eqs.(3) and (4).

Which Activation 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 a_i⁰ and b_i 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 activation model is set in the thermodynamic database wateq4f.dat for each aqueous species i separately.

High-Saline 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, however, a lot of additional parameters (virial coefficients). The Pitzer model is not contained in aqion.

References & Remarks