## Molar Conductivity

Molar and Equivalent Conductivities

The electrical conductivity EC is an easy-to-measure parameter; its exact calculation, however, is rather non-trivial. Today exist a variety of approaches 1, but all of them are no more than approximations (especially for waters of arbitrary composition). In any case, physical-based approaches to EC start always from the concept of molar or equivalent conductivities:

 (1a) electrical conductivity (specific conductance) EC 2,3 in S/m  (or µS/cm) (1b) molar conductivity Λm = EC / c in S cm2 mol-1 (1c) equivalent conductivity Λeq = Λm / |z| in S cm2 eq-1

Here c symbolizes the molar concentration of the electrolyte (in mol/L) and z refers to the electrical charge. The molar conductivity Λm 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 (rather than molarity). It accounts for the obvious fact that ions with higher z are able to transport more charge. Introducing the

 (2) equivalent concentration:     ceq = |z| c

the equivalent conductivity in Eq.(1c) becomes

 (3) Λeq = EC / ceq

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 Λm acts as proportionality constant. Unfortunately, nature is not so simple: Λm is not constant and diminishes when c raises. About 100 years ago F. Kohlrausch deduced from experimental data the “Square-Root 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 electrolytes4 at low concentrations, c ≤ 10 mM. The Kohlrausch parameter K depends on the type of electrolyte. A theoretical explanation of the square-root dependence of c was provided by Debey, 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 cm2 eq-1 (5b) molar limiting conductivity $\Lambda_{m}^{0}$ in S cm2 mol-1

These are the only experimentally accessible, basic 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 $\nu_-$ label the stoichiometric coefficients. Some typical values of limiting molar conductivities5 at 25°C are:

cation $\Lambda_m^+$ [S cm2 mol-1]   anion $\Lambda_m^-$ [S cm2 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 aqueos solution, Eq.(6) allows us to compute 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, which bases on diffusion coefficients. The corresponding formula is derived here.

Remarks & References

1. An overview of up-to-date approaches is given in: R.B. McCleskey, D.K. Nordstrom, and J.N. Ryan: Comparison of electrical conductivity calculation methods for natural waters, Limnol. Oceanogr.: Methods 10, 952–967 (2012)

2. 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.

3. To recapitulate: Electrical conductivity σ is a material-specific constant with units S/m. It should not be confused with electrical conductance G that has units S (= Ω-1).

4. In contrast, weak elektrolytes – 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.

5. These values are taken from the larger table here