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Electrical Conductivity based on Diffusion Coefficients

Nernst-Einstein Equation

The Nernst-Einstein equation establishes the relationship between the molar limiting conductivity \(\Lambda_{m,i}^{0}\) and the diffusion coefficient D_i for any given ion i:

(1)

\(D_i = \dfrac{RT}{z_i^2 F^2} \, \Lambda_{m,i}^0\)

or

\(\Lambda_{m,i}^0 = z_i^2 D_i \, \left( \dfrac{F^2}{RT} \right)\)

with

zi		charge of ion i
T	in K	absolute temperature
F	= 9.6485·104 Coulomb/mol	Faraday’s constant
R	= 8.31446 J/(K mol)	gas constant
F2/(RT)	= 3.7554·106 s·S mol-1	proportionality constant at 25°C
Di	in m2/s	diffusion coefficient of ion i
\(\Lambda_{m,i}^{0}\)	in S m2/mol (= 104 S cm2/mol)	molar limiting conductivity of ion i

Example. If we enter some well-known values of D_i (taken from literature) into 1 the following molar limiting conductivities are obtained:

	ion	Di [m2 s-1]	\(\Lambda_{m,i}^0\) [S cm2 mol-1]
	H+	9.31·10-9	349.6
	Na+	1.33·10-9	50.0
	K+	1.96·10-9	73.6
	OH-	5.27·10-9	197.9
	Cl-	2.03·10-9	76.2
	Br-	2.01·10-9	75.5

The goal is now to exploit this method for the calculation of electrical conductivities (EC) of aqueous solutions of arbitrary composition.

EC of Ideal Aqueous Solutions

In the limit of infinite dilution (non-interacting ions) we obtain from Eq.(7a) a simple formula that relies on diffusion coefficients:

(2)	\(EC^{(0)} \ = \ \sum\limits_i \, \Lambda_{m,i}^0 \, c_i \ = \ \left( \dfrac {F^2}{RT} \right) \ \sum\limits_i \, D_i z_i^2 \, c_i\)

In the realistic case of non-ideal solutions, however, the method becomes somewhat more elaborate.

EC of Real (Non-Ideal) Aqueous Solutions

The Nernst-Einstein equation is restricted and valid only for molar limiting conductivities \(\Lambda_{m,i}^{0}\). In contrast, the EC of real aqueous solutions rest upon molar conductivities \(\Lambda_{m,i}\):

(3)	\(EC \ = \ \sum\limits_i \, \Lambda_{m,i} \, c_i\)

Both quantities are related by Kohlrausch’s Square-Root Law, but this law requires the knowledge of an extra parameter K which depends non-trivially on the type of electrolyte (and which is hardly available in tables or literature). An alternative was proposed by Appelo¹ who rearranged 3 into

(4)

\(EC \ = \ \sum\limits_i \, \Lambda_{m,i}^0 \, \gamma_{corr} \, c_i\)

where all aspects regarding the ion-ion interaction are put into the correction factor

(5)

\(\ln \gamma_{corr} \ \simeq \ - (K /\Lambda_{m,i}^0) \, \mid\! z_i\!\mid^{1.5} \! \sqrt{I}\)

with I as ionic strength. It is no surprise that this ion-ion correction factor closely resembles the activity model of Debye-Hückel:

(6)

\(\ln \gamma_i \ =\, - (\ln 10) \ Az^{2}_{i} \ \sqrt{I}\)

(Debye-Hückel)

with A = 0.5085 M^-1/2. This strategy looks promising because we replaced the non-trivial Kohlrausch parameter K by the activity constant γ – a quantity that belongs to the standard repertoire of hydrochemical models (available for each ion and aqueous species).

Formally, 4 can be converted into

(7)

\(EC \ = \ \sum\limits_i \, \Lambda_{m,i}^0 \, (\gamma_i)^{\alpha} \, c_i\)

with

(8)

\(\alpha \ = \ \dfrac{\ln \, \gamma_{corr}}{\ln \, \gamma_i} \ = \ \dfrac{K} {\Lambda_{m,i}^0 \, (\ln 10) \ A \, \mid\! z_i\mid^{0.5}}\)

The only thing we need is a clever parameterization of α as a fairly constant quantity.

Parameterization à la Appelo (PhreeqC 3)

Appelo¹ proposed the following parameterization for 8:

(9)	\(\alpha \ = \ \begin{cases} \ 0.6 \,/ \mid\! z_i\!\mid^{0.5} = const & \ \ \ \text{if } \ \ I\leq 0.36 \mid\! z_i\mid \\ \ \sqrt{I} \, / \mid\! z_i\mid & \ \ \ \text{otherwise } \end{cases}\)

Just this parameterization is used in PhreeqC and aqion. The equation behind the EC calculation is:

(10)	\(EC \ = \ \left( \dfrac {F^2}{RT} \right) \ \sum\limits_i \, D_i z_i^2 \, (\gamma_i)^{\alpha} \, c_i\)

This equation is used as the default method in aqion. The diffusion coefficients D_i are taken from here.

[Remark to γ_i: While 6 represents the most simplest form of an activity model (in order to demonstrate the main ideas in the present article), in the program, however, more sophisticated approaches for γ_i are used.]

Appendix: Rearrangement of the EC Equation

The aim is to convert 3 into a form similar to the ideal-solution formula:

(A1)

\(EC \ = \ \sum\limits_i \, \Lambda_{m,i} \, c_i\)

\(\Large \Rightarrow\)

\(EC \ = \ \sum\limits_i \, \Lambda_{m,i}^0 \, \gamma_{corr} \, c_i\)

We start from Eq.(7b) and get from Kohlrausch’s Square-Root Law:

(A2)

\(\begin{align*} EC \ &= \ \sum\limits_i \, \Lambda_{eq,i} \mid\! z_i\!\mid c_i \\ &= \ \sum\limits_i \ \left\{ \Lambda_{eq,i}^0 - K \sqrt{\mid z_i\!\mid c} \right\} \mid\!z_i\!\mid c_i \\ &= \ \sum\limits_i \ \Lambda_{eq,i}^0 \ \left\{1 - (K /\Lambda_{eq,i}^0) \, \sqrt{\mid z_i\!\mid c} \right\} \mid\!z_i\!\mid c_i \end{align*}\)

Using \(\Lambda_{eq}^{0} = \Lambda_{m}^{0} \, / \!\mid\! z_i\mid\), the last line yields:

(A3)	\(EC \ = \ \sum\limits_i \ \Lambda_{m,i}^0 \ \left\{1 - (K /\Lambda_{m,i}^0) \, \mid\! z_i\!\mid^{1.5} \! \sqrt{c} \, \right\} \, c_i\)

Note that c, and not c_i, enters the square root. This is because c refers to the ‘medium effect’ of the electrolyte (as a composition of several ion types). In other words, c appears as a sibling of the ionic strength I = ½ Σ z_i² c_i.² Thus, replacing c by I in A3 yields the desired equation:

(A4)

\(EC \ = \ \sum\limits_i \, \Lambda_{m,i}^0 \, \gamma_{corr} \, c_i\)

with

\(\gamma_{corr} \ = \ 1 - (K /\Lambda_{m,i}^0) \, \mid\! z_i\!\mid^{1.5} \! \sqrt{I}\)

The value of \(\gamma_{corr}\) is near to 1, which permits the expansion exp(-a) = 1–a + … :

(A5)

\(\gamma_{corr} \ \simeq \ \exp \,\left\{ - (K /\Lambda_{m,i}^0) \, \mid\! z_i\!\mid^{1.5} \! \sqrt{I} \,\right\}\)

References & Remarks