SPECIFIC CONDUCTANCE: how to calculate, to use, and the pitfalls

The specific conductance or electrical conductivity (SC, μS/cm) of a solution is easy to measure and very useful for checking parameters of the aqueous model or defining association constants for new species.
PHREEQC calculates the SC of a solution from the concentration, the activity coefficient, and the diffusion coefficient of all the charged species. Results obtained with the traditional ion-association and the Pitzer model agree very well with measurements of a large variety of solutions, spanning a range of conductances from 10 to 100 000 μS/cm. The figure compares data provided by P.J. Stuyfzand with calculated values.


The molar conductivity of a solute species and its diffusion coefficient are related by:
where Λ0m is the molar conductivity (S/m / (mol/m3)), z the charge number (-), F is Faraday's constant (Coulomb/mol), R the gas constant (J/oK/mol), T the absolute temperature (K), and Dw the diffusion coefficient (m2/s). Multiplying the molar conductivity with the concentration m and summing up for all the solutes, gives an estimate of the specific electrical conductance of the solution:
SC = Σ (Λ0m m)
The only problem is that the molar conductivity changes with the concentration.
Kohlrausch's law states that the equivalent conductivity decreases with the square root of the concentration:

Λeq = Λ0eq - Κ (|z| m)0.5
where Κ is Kohlrausch's constant. We could use Λeq (= Λm / |z|) instead of Λ0m to calculate SC. Alternatively, we can stick with Λ0m, but correct the molar concentration with an electrochemical activity coefficient. Typically, this is a job for PHREEQC since the program calculates concentrations and activities of the various species in solution.
Of course, the two methods should produce the same result:
SC = Σ ((Λ0eq - Κ (|z| m)0.5) |z| m) = Σ (Λ0m γsc m)
where γsc is the activity coefficient that corrects the molar concentration. This is not simply the Debye-Hückel activity coefficient, but it is related to it.
From the equation above, and using exponentiation since γsc is close to 1:
γsc = 1 - Κ/Λ0m |z|1.5 m0.5 ≈ exp(-Κ/Λ0m |z|1.5 m0.5)
It can be compared with the limiting Debye-Hückel activity coefficient for low concentrations:
γDH = 10^(-0.5 |z|2 m0.5) = exp(-ln(10) 0.5 |z|2 m0.5)
Thus, multiplying the logarithm of the activity coefficient with ff = Κ / (Λ0m ln(10) 0.5 |z|0.5), gives the electrochemical activity coefficient. The factor used by PHREEQC is ff = 0.6 / |z|0.5 for ionic strength I < 0.36 |z|, and ff = I0.5 / |z| otherwise.

PHREEQC's calculations were checked with data from the Handbook of Chemistry and Physiscs, which lists equivalent conductivities at 25oC, and SC's for salt solutions at 20oC. PHREEQC prints SC in the output file and the special BASIC variable SC provides it in keywords USER_PUNCH or _GRAPH. Results calculated with PHREEQC input file salt_sc.phrq are graphed below. Note that the calculations need diffusion coefficients given in PHREEQC.DAT that goes with version 2.17.

Except for NaCl, some deviations are evident for all the salts when SC > 60 000 μS/cm. At lower conductivity the fit is quite good except for MgSO4. Mg-solutions have relatively high viscosities compared with equimolar other solutions, but this should result in a decrease of the measured SC (relative to the calculated value) instead of increasing it as shown in the graph.
Using the Pitzer database, aimed at improving calculations at high salt concentrations, deteriorates the fit even more (Dw's were added to PITZER.DAT). The Pitzer model, as implemented, is firmly based on emf measurements done in SO4 solutions and on the solubilities of SO4-minerals. Thus, be suspicious of PHREEQC's calculated SC of SO4-rich solutions (more than about 0.1 M).

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