Electrical Conductivity (EC)
Three Calculation Methods
The electrical conductivity (EC) or specific conductance is a useful waterquality parameter. There are several methods available to calculate EC.^{1} Three of them are used by the program:
 Linear approach (proportional to ionic strength)
 Pseudolinear approach (Inverse Marion & Babcock)
 Diffusion coefficientbased approach (Appelo 2010)^{2} ⇐ default method
The first two approaches are simple empirical methods based on the ionic strength. The third approach is more advanced and relies on diffusion coefficients; it is the default algorithm of the program.^{3}
For any given aqueous solution two values are displayed: (i) the calculated EC at the actual water temperature and (ii) EC_{25} after conversion to the reference temperature of 25. The latter allows measurements or sampling performed at different T to be compared.
Method 1: Linear Approach based on Ionic Strength
The simplest empirical method relies on a linear relationship between electrical conductivity and the ionic strength I:
(1)  EC (µS/cm) = 6.2 10^{4} × I (mol/L) 
This equation is equivalent to the common approximation (i.e. the inverse of 1):
(2)  I (mol/L) = 1.6 10^{5} × EC (µS/cm) 
In this way, based on the easily measured EC values, a rough estimate for the ionic strength is obtained. But we take the opposite way: The ionic strength I, which enters 1, is strictly determined by the actual water composition/speciation as:
(3)  \(\large I = \frac{1}{2} \, \sum\limits_{i}z_{i}^{2} \, c_{i}\) 
where the sum runs over all ions i with molar concentration c_{i} and charge number z_{i}. In hydrochemistry, the ionic strength I is calculated anyway because it enters the activity model to account for ionion interactions in nonideal solutions.^{4}
TDS. There is also a simple linear relationship between EC value and TDS – see here.
Method 2: Pseudolinear Approach (Inverse Marion & Babcock)
The pseudolinear approach is also based on the ionic strength. According to Sposito^{5}, who adopts the results of Marion & Babcock^{6}, the relationship between EC and I is nonlinear:^{7}
(4)  lg I = 1.159 + 1.009 lg EC  for I ≤ 0.3 mol/L 
In this equation the units of I are mmol/L (= mM) and the units of EC are dS/m – which differ significantly from the units in 1. The rearrangement of 4 into a form similar to 2 can be done step by step:
lg (EC / dS∙m^{1})  =  0.991 lg (I / mM) – 1.149  
lg (10^{3} EC / µS∙cm^{1})  =  0.991 lg (10^{3} I/M) – 1.149  
lg (EC / µS∙cm^{1}) – 3  =  0.991 [ 3 + lg (I/M) ] – 1.149  
lg (EC / µS∙cm^{1})  =  4.824 + lg (I/M) 
which yields
(5)  EC (µS/cm) = 6.67 10^{4} × [ I (mol/L) ]^{ 0.991} 
Because I^{0.991} ≈ I, this equation is very similar to 1. Thus, this approach is called ‘pseudolinear’. Due to a slightly larger prefactor in 5, the EC of the pseudolinear approach is a little higher than the EC of the linear approach.
[The alternative name ‘inverse Marion & Babcock’ arises from the fact that we have rearranged the original equation from I = I(EC) to EC = EC(I).]
Method 3: Approach based on Diffusion Coefficients
The NernstEinstein equation establishes a physical relationship between the molar limiting conductivity \(\Lambda_{m,i}^{0}\) and the diffusion coefficient D_{i} for a given ion i. Appelo^{2} adopted this idea for practical use:
(6)  \(EC \ = \ \left( \dfrac {F^2}{RT} \right) \ \sum\limits_i \, D_i z_i^2 \, (\gamma_i)^{\alpha} \, c_i\) 
The meaning of the symbols and the mathematical derivation of this formula is described here. This approach is used in aqion as the default algorithm.
Temperature Compensation: EC ⇒ EC_{25}
The EC of most natural waters, including seawater, increases with temperature 13% per degree Celsius. Measured EC values are usually referred to 25°C – indicated by EC_{25}. For this purpose the program converts the calculated EC (valid for the given water temperature T) to EC_{25} at 25°C. Principally, there are two main approaches: (i) a nonlinear model and (ii) its linear approximation.
Nonlinear T Compensation. The nonlinear model is an outcome of the physical relationship between electrical conductivity, diffusion coefficients, and the viscosity of water. The equation is given by:
(7)  EC_{25} = 1.125 · 10^{A/B} · EC 
with the two parameters taken from Atkins:^{8}
(7a)  A = 1.37023 (T – 20) + 8.36·10^{4} (T – 20)^{2} 
(7b)  B = 109 + T 
and T in °C. The nonlinear compensation model is the standard method used in aqion.
Linear Approximation. Instead of the general approach in 7, linear formulas are in widespread use. The most common type of a linear expression is obtained from 7 by Taylorseries expansion (as shown here):
(8)  EC_{25} = EC / [ 1 + a (T – 25) ] 
with a = 0.020 °C^{1} and T in °C.
Program Output. The program displays both values, the calculated EC (based on diffusion coefficients) and the compensated value EC_{25}. This is done in the output tables:
EC_25  if checkbox mol is on 
EC (T)  if checkbox mol is off 
Conversion of Units
EC’s physical units take some getting used to. The conversions between µS/cm (micro Siemens per centimeter) and other EC units are:
1 mS/m  =  10 µS/cm 
1 dS/m  =  1000 µS/cm 
1 dS/m  =  1 mS/cm 
1 µmho/cm  =  1 µS/cm 
where 1 S = 1 Siemens = 1 ohm^{1} = 1 mho. The program uses µS/cm as the default unit for EC.
Typical Conductivities of Aqueous Solutions
absolute pure water  0.055  µS/cm 
distilled water  0.5  µS/cm 
rain water  5 – 30  µS/cm 
potable water  500 – 1000  µS/cm 
groundwater  30 – 2000  µS/cm 
industrial wastewater  ≥ 5000  µS/cm 
seawater  54 000  µS/cm 
concentrated acids and bases  up to 1 000 000  µS/cm 
Pure Water. Due to the self ionization of water into H^{+} and OH^{} ions, the electrical conductivity of pure water is nonzero: EC = 0.055 µS/cm at 25.
References

RB McCleskey, DK Nordstrom, JN Ryan: Comparison of electrical conductivity calculation methods for natural waters, Limnol. Oceanogr.: Methods 10, 952–967 (2012) ↩

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

To select the calculation method for EC, click on Settings→EC in the upper menu bar. ↩

The ionic strength I, calculated by 3, is displayed in the upper part of the output table. ↩

Garrison Sposito: The Chemistry of Soils, 2nd Edition, Oxford University Press, 2008, (see Eq.(4.23) p.111) ↩

GM Marion, KL Babcock: Predicting specific conductance and salt concentration in dilute aqueous solutions. Soil Sci. 122, 181–187 (1976) ↩

“lg a” abbreviates the decadic logarithm “log_{10} a”. ↩

P Atkins and J de Paula: Physical Chemistry, 8th Edition, W.H. Freeman and Company New York, 2006, Table 21.4, p. 1019 ↩