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Electrical Conductivity (EC)

Three Calculation Methods

The electrical conductivity (EC) or specific conductance is a useful water-quality parameter. There are several methods available to calculate EC.¹ Three of them are used by the program:

• Linear approach (proportional to ionic strength)
• Pseudo-linear approach (Inverse Marion & Babcock)
• Diffusion coefficient-based 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) EC₂₅ 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)   =   6.2 104 × 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 ion-ion interactions in non-ideal solutions.⁴

TDS. There is also a simple linear relationship between EC value and TDS – see here.

Method 2:  Pseudo-linear Approach (Inverse Marion & Babcock)

The pseudo-linear approach is also based on the ionic strength. According to Sposito⁵, who adopts the results of Marion & Babcock⁶, 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 (103 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)  =  6.67 104 × [ I (mol/L) ] 0.991

Because I^0.991 ≈ I, this equation is very similar to 1. Thus, this approach is called ‘pseudo-linear’. Due to a slightly larger prefactor in 5, the EC of the pseudo-linear 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 Nernst-Einstein 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. Appelo² 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 ⇒ EC₂₅

The EC of most natural waters, including seawater, increases with temperature 1-3% per degree Celsius. Measured EC values are usually referred to 25°C – indicated by EC₂₅. For this purpose the program converts the calculated EC (valid for the given water temperature T) to EC₂₅ 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)	EC25   =   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 Taylor-series expansion (as shown here):

(8)	EC25  =  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 EC₂₅. 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 non-zero: EC = 0.055 µS/cm at 25.

References