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Molar Conductivity

Molar and Equivalent Conductivities

The electrical conductivity EC is an easy-to-measure parameter; however, its exact calculation is non-trivial. Today exist a variety of semi-empirical 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)	electr. 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 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:     ceq = |z| c

the equivalent conductivity in 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 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 “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 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 cm2 eq-1
(5b)	molar limiting conductivity	\(\Lambda_{m}^{0}\)	in S cm2 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 conductivities⁵ (at 25):

	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 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