Electrical Conductivity based on Diffusion Coefficients
NernstEinstein Equation
The NernstEinstein 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
z_{i}  charge of ion i  
T  in K  absolute temperature 
F  = 9.6485·10^{4} Coulomb/mol  Faraday’s constant 
R  = 8.31446 J/(K mol)  gas constant 
F^{2}/(RT)  = 3.7554·10^{6} s·S mol^{1}  proportionality constant at 25°C 
D_{i}  in m^{2}/s  diffusion coefficient of ion i 
\(\Lambda_{m,i}^{0}\)  in S m^{2}/mol (= 10^{4} S cm^{2}/mol)  molar limiting conductivity of ion i 
Example. If we enter some wellknown values of D_{i} (taken from literature) into 1 the following molar limiting conductivities are obtained:
ion  D_{i} [m^{2} s^{1}]  \(\Lambda_{m,i}^0\) [S cm^{2} 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 (noninteracting 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 nonideal solutions, however, the method becomes somewhat more elaborate.
EC of Real (NonIdeal) Aqueous Solutions
The NernstEinstein 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 SquareRoot Law, but this law requires the knowledge of an extra parameter K which depends nontrivially on the type of electrolyte (and which is hardly available in tables or literature). An alternative was proposed by Appelo^{1} who rearranged 3 into
(4)  \(EC \ = \ \sum\limits_i \, \Lambda_{m,i}^0 \, \gamma_{corr} \, c_i\) 
where all aspects regarding the ionion 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 ionion correction factor closely resembles the activity model of DebyeHückel:
(6)  \(\ln \gamma_i \ =\,  (\ln 10) \ Az^{2}_{i} \ \sqrt{I}\)  (DebyeHückel) 
with A = 0.5085 M^{1/2}. This strategy looks promising because we replaced the nontrivial 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)
Appelo^{1} 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 idealsolution 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 SquareRoot 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_{i}^{2} 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

C.A.J. Appelo: Specific conductance – how to calculate the specific conductance with PhreeqC (2010), http://www.hydrochemistry.eu/exmpls/sc.html ↩ ↩^{2}

The following example for NaCl illustrates the close relationship between the electrolyte concentration c and ionic strength: I = ½ (c_{Na} + c_{Cl}) = c_{NaCl}. ↩