Molar Conductivity
Molar and Equivalent Conductivities
The electrical conductivity EC is an easytomeasure parameter; however, its exact calculation is nontrivial. Today exist a variety of semiempirical approaches ^{1}, but all of them are no more than approximations (especially for waters of arbitrary composition). In any case, physicalbased approaches to EC start always from the concept of molar or equivalent conductivities:
(1a)  electr. conductivity (specific conductance)  EC ^{2}^{,}^{3}  in S/m (or µS/cm) 
(1b)  molar conductivity  Λ_{m} = EC / c  in S cm^{2} mol^{1} 
(1c)  equivalent conductivity  Λ_{eq} = Λ_{m} / z  in S cm^{2} eq^{1} 
Here c symbolizes the molar concentration of the electrolyte (in mol/L) and z refers to the electrical charge. The molar conductivity Λ_{m} is defined as the conductivity of a 1 molar aqueous solution placed between two plates (electrodes) 1 cm apart.
The equivalent conductivity refers to the normality of the solution (and not to the molarity). It accounts for the obvious fact that ions with higher z are able to transport more charge. Introducing the
(2)  equivalent concentration: c_{eq} = z c 
the equivalent conductivity in 1c becomes
(3)  Λ_{eq} = EC / c_{eq} 
Kohlrausch’s Law for Strong Electrolytes (Limiting Conductivities)
Strong electrolytes (in contrast to weak electrolytes) are salts, acids and bases that dissociate completely. For strong electrolytes one might expect a linear relationship between EC and the concentration, i.e. EC = const · c, where the molar conductivity Λ_{m} acts as proportionality constant. Unfortunately, nature is not so simple: Λ_{m} is not constant and diminishes when c raises. About 100 years ago F. Kohlrausch deduced from experimental data the “SquareRoot Law”:
(4a)  \(\Lambda_{eq} \ = \ \Lambda_{eq}^{0}  K \sqrt{c_{eq}}\) 
or, equivalently,
(4b)  \(\Lambda_{m} \ = \ \Lambda_{m}^{0}  K' \sqrt{c}\)  with K’ = K / z^{1.5} 
It is valid for strong electrolytes^{4} at low concentrations, c ≤ 10 mM. The Kohlrausch parameter K depends on the type of electrolyte. A theoretical explanation of the squareroot dependence of c was provided by Debye, Hückel and Onsager about 50 years later.
Limiting Conductivities. In the very special case of zero concentration, c → 0 (infinite dilution), the above equations collapse to the
(5a)  equivalent limiting conductivity  \(\Lambda_{eq}^{0}\)  in S cm^{2} eq^{1} 
(5b)  molar limiting conductivity  \(\Lambda_{m}^{0}\)  in S cm^{2} mol^{1} 
These are the only experimentally accessible electrotransport properties of a given ion.
Kohlrausch’s Law of the Independent Migration of Ions
According to the Law of independent migration the limiting molar conductivity can be expressed as a sum of cation and anion contributions:
(6)  \(\Lambda_{m}^0 \ = \ \nu_+ \Lambda_m^+ + \nu_ \Lambda_m^\) 
where \(\nu_+\) and \(\nu_\) are stoichiometric coefficients. Some typical values of limiting molar conductivities^{5} (at 25):
cation  \(\Lambda_m^+\) [S cm^{2} mol^{1}]  anion  \(\Lambda_m^\) [S cm^{2} mol^{1}]  

H^{+}  349.6  OH^{}  197.9  
Na^{+}  50.0  Cl^{}  76.2  
K^{+}  73.6  Br^{}  75.5 
Given the composition of an aqueous solution, 6 predicts its electrical conductivity EC as a sum over all dissolved ions i:
(7a)  ideal solution (c → 0):  \(EC^{(0)} \ = \ \sum\limits_i \, \Lambda_{m,i}^0 \, c_i \ = \ \sum\limits_i \, \Lambda_{eq,i}^0 \mid\! z_i\!\mid c_i\) 
(7b)  real solution:  \(EC \ \ \ = \ \sum\limits_i \, \Lambda_{m,i} \, c_i \ = \ \sum\limits_i \, \Lambda_{eq,i} \mid\! z_i\!\mid c_i\) 
Equation (7b) constitutes the background for the third calculation method used by aqion that is based on diffusion coefficients. The corresponding formula is derived here.
Remarks & References

An overview of uptodate approaches is given in: RB McCleskey, DK Nordstrom, and JN Ryan: Comparison of electrical conductivity calculation methods for natural waters, Limnol. Oceanogr.: Methods 10, 952–967 (2012) ↩

We abbreviate the electrical conductivity by “EC” to be in accordance with the name used by aqion. However, it’s very common to abbreviate this quantity by the Greek letter σ. ↩

To recapitulate: Electrical conductivity σ is a materialspecific constant with units S/m. It should not be confused with electrical conductance G that has units S (= Ω^{1}). ↩

In contrast, weak electrolytes — i.e. chemicals with incomplete dissolution — are described by Ostwald’s Dilution Law. This law requires additional parameters: the equilibrium constants of the corresponding weak acid or weak base. ↩