Overview Activity Chem Equilibrium Acids Carbonate System Mineral Phases

Activity Coefficients (Activity Models)

Activity Models

The step from molar concentrations (analytical data) to activities (used in mass-action law calculations) requires the calculation of activity coefficients γ_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

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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 water at 25 and with ε = ε_rε₀ = 78.54 · 8.854 10^-12 C²/(J m), 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 clear when they are all traced back to the simple Debye-Hückel formula in 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 rise again when I approaches 1 mol/L – see diagrams below.

Model Hierarchy. The empirical model of Truesdell-Jones in 4 and 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 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+, HCO3-, SO4-2
a = 0.3 nm	for	K+, NH4+, OH-, Cl-, NO3-

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 Debye-Hückel 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 strengths up to I ≈ 0.5 M. Due to its mathematical simplicity and the 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.

Truesdell-Jones (WATEQ Debye-Hückel)

The empirical approach of Truesdell and Jones³ was proposed for the hydrochemistry program WATEQ in 1974 with the aim of describing NaCl-containing 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_i⁰ and b_i, whereas the effective ionic 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 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 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 the activation model is set in the thermodynamic 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