## Temperature Compensation for Conductivity

The electrical conductivity (EC) of an aqueous solution increases with temperature significantly: about 2% per degree Celsius. In practice, there are several empirical formulas in use to convert measured EC values to the reference temperature at 25°C. But what is the physical idea behind all these equations?

The answer to this question lies in the fundamental relationship between electrical conductivity, diffusion coefficients, and the viscosity of water.

Electrical Conductivity, Diffusion Coefficient(s), and Viscosity of Water

The two equations that interrelate the three physical quantities (EC, diffusion coefficient(s) D, and viscosity η of water) are:

 Nernst-Einstein equation: EC ⇔ D Stokes-Einstein equation: D ⇔ η

The Nernst-Einstein equation states the proportionality between EC and the diffusion coefficient(s) Di of dissolved ions:1

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

where F denotes Faraday’s constant, R the gas constant, and T the temperature in Kelvin. The sum runs over all dissolved ions with charge zi and molar concentration ci. To keep the notation simple (and without loss of generality), we focus on just one single ion:

 (1b) $EC \ = \left( \dfrac {F^2}{RT} \right) \, D z^2 \, c \ = \ const \cdot \dfrac{D}{T}$

The Stokes-Einstein equation describes the relation between the diffusion coefficient D and the viscosity η

 (2) $D \ = \ \dfrac {k_B T}{6 \pi \, \eta \, r}$

Here, kB denotes Boltzmann’s constant and r is the ‘hydraulic radius’ of the diffusing particle. But we can get rid of the last parameter when considering D and η at two temperatures T1 and T2 which implies the general relation:

 (3) $\dfrac {D_1 / T_1}{D_2 / T_2} \ = \ \dfrac {\eta_2}{\eta_1}$

On the other hand, according to Eq.(1b), EC is directly proportional to D/T which yields:

 (4) $\dfrac {EC_1}{EC_2} \ = \ \dfrac {D_1 / T_1}{D_2 / T_2} \ = \ \dfrac {\eta_2}{\eta_1}$

If T1 refers to the water temperature T, and T2 refers to the reference temperature 25°C, then – after rearranging of Eq.(4) – the temperature compensation formula is expressed as a ratio of the viscosity of water at T and 25°C:

 (5) $EC_{25} \ = \ \left( \dfrac {\eta}{\eta_{25}} \right) \, EC$

where EC und η refer to the water temperature T.

Viscosity of Water as a Function of Temperature

The viscosity of pure water decreases with temperature. Typical value are:

 η20  =  1.003·10-3   kg m-1 s-1 at 20 °C η25  =  0.891·10-3   kg m-1 s-1 at 25 °C

A nonlinear parameterization of the dynamical viscosity is presented in the classical textbook of Atkin’s Physical Chemistry:2

 (6) $\log \left( \dfrac {\eta_{20}}{\eta} \right) \, = \, \dfrac {A}{B}$ or $\left( \dfrac {\eta_{20}}{\eta} \right) \, = \, 10^{A/B}$

with the two parameters:

 (6a) A = 1.37023 (t-20) + 8.36·10-4 (t-20)2 (6b) B = 109 + t     and    t in °C

It describes the viscosity of water over its entire liquid range (0 to 100°C) with less than 1% error. [Please note that this equation refers to 20°C and not to 25°C.]

Nonlinear Temperature Compensation of EC

Inserting Eq.(6) into Eq.(5) yields:

 (7) $EC_{25} \ = \ \left( \dfrac {\eta_{20}}{\eta_{25}} \right) \,\left( \dfrac {\eta}{\eta_{20}} \right) \, \cdot EC \ = \ \left( \dfrac {\eta_{20}}{\eta_{25}} \right) \, 10^{-A/B} \,\cdot EC$

The numerical value of the conversion from 20°C to 25°C can be easily calculated: (η2025) = 1.125. Thus we obtain the final formula:

 (8) $EC_{25} \ = \ 1.125 \cdot \, 10^{-A/B} \,\cdot EC$

This nonliner equation is used by the program aqion to convert between the EC value at the given water temperature and EC25. Whereas the EC value is an outcome of the general approach based on diffusion coefficients and molar concentrations of the dissolved ions.

[Note: Although it is not obvious at first sight, Eq.(8) provides the rigth normalization, i.e., the ratio EC25/EC becomes exactly 1 for 25°C .]

Linearization

Instead of the general approach in Eq.(8), linear formulas are in widespread use. The most common type of a linear expression is obtained from Eq.(8) by Taylor series expansion (in which higher-order terms in (t - 25) are neglected):

 (9) EC25   =   EC / [ 1 + a (t - 25) ] with   a = 0.020

The mathematical derivation is presented in the appendix.

Empirical values of the parameter range between a = 0.01 and 0.03. For example, Hayashi3 deduced a compensation factor of 0.019 from the examination of natural waters. This is in close agreement with the theoretical value:

 theoretical value: a = 0.020 °C-1 Hayashi: 3 a = 0.019 °C-1

Comparison of the Nonlinear and Linear Model

It is quite instructive to compare the general and linear compensation formulas. The following diagram presents EC25/EC in the temperature range between 0 and 100°C:

 nonlinear model –   general approach in Eq.(8) linear approx. (a=0.020) –   Eq.(9) with a = 0.020 linear approx. (a=0.019) –   Eq.(9) with a = 0.019  [Hayashi]

Obviously, at t = 25°C the correction is exactly 1. This is valid for all methods.

Appendix: Linearization of the General Equation

We start with Eq.(8) and use the Taylor series expansion, ex = 1 + x + … . It yields:

 (A1) $\dfrac {EC_{25}}{EC} \ = \ \dfrac {1.125}{10^{A/B}} \ = \ \dfrac {1.125}{e^{A/B\cdot \ln 10}} \ \approx \ \dfrac {1.125}{1+ A/B\cdot \ln 10}$

Now we simplify the ratio A/B, that is based on Eqs.(6a) and (6b). Ignoring all quadratic and higher terms in θ = t - 25 leads to:4

 (A2) $\dfrac {A}{B} \ \approx \ \dfrac {1.37\cdot (t-20)}{109+t} \, = \, \dfrac {1.37\cdot (\theta +5)}{134+\theta} \, = \, \dfrac {1.37}{134} \cdot\dfrac {\theta +5}{1+\theta / 134} \, \approx \, 0.010 \cdot (\theta +5)$

That is, we get  A/B · ln 10 = 2.302 · A/B ≈ 0.023 (θ + 5). Inserting this into Eq.(A1) yields

 (A3) $\dfrac {1.125}{1+ 0.023 \, (\theta +5)} = \dfrac {1}{0.889 + 0.020 \, (\theta +5)} = \dfrac {1}{1 + 0.020 \, (\theta -0.5)}$

Returning from θ = t - 25 to t and ignoring the small temperature offset of 0.5, gives the final expression:

 (A4) $\dfrac {EC_{25}}{EC} \ = \ \dfrac {1}{1 + 0.020 \, (t - 25)}$

This is the linear approximation used in Eq.(9).

Remarks and References

1. The general equation for non-ideal, real solutions is given here

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

3. M. Hayashi: Temperature-electrical conductivity relation of water for environmental monitoring and geophysical data inversion, Environmental Monitoring and Assessment 96, 121-130, 2004  2

4. Theoretisch könnte man sogar noch weiter gehen, und am Ende noch den Nenner (1+θ/134) approximieren: (1+θ/134)-11 - θ/134 + … Das bringt aber kaum Zugewinn an Genauigkeit, so dass sich der Aufwand nicht lohnt.