# Molar conductivity

Molar conductivity is defined as the conductivity of an electrolyte solution divided by the molar concentration of the electrolyte, and so measures the efficiency with which a given electrolyte conducts electricity in solution. It is the conducting power of all the ions produced by dissolving one mole of an electrolyte in solution. Its units are siemens per meter per molarity, or siemens meter-squared per mole. The usual symbol is a capital lambda, Λ, or Λm. It can also be defined as follows: Molar conductivity of a solution at a given concentration is the conductance of the volume (V) of the solution containing one mole of electrolyte kept between two electrodes with area of cross section (A) and at a distance of unit length.

## History

Friedrich Kohlrausch's researches in 1875-79 established that to a high accuracy in dilute solutions, molar conductivity is composed of individual contributions of ions. This is known as the law of independent migration of ions.[1]

## Description

From its definition, the molar conductivity is given by:[2]

${\displaystyle \Lambda _{\mathrm {m} }={\frac {\kappa }{c}}}$

where:

• κ is the measured conductivity (formerly known as specific conductance)[3]
• c is the electrolyte concentration.

Two cases should be distinguished: strong electrolytes and weak electrolytes.

For strong electrolytes, such as salts, strong acids and strong bases, the molar conductivity depends only weakly on concentration. Based on experimental data Friedrich Kohlrausch (around the year 1900) proposed the non-linear law for strong electrolytes:

${\displaystyle \Lambda _{\mathrm {m} }=\Lambda _{\mathrm {m} }^{\circ }-K{\sqrt {c}}}$

where

• ${\displaystyle \Lambda _{\mathrm {m} }^{\circ }}$ is the molar conductivity at infinite dilution (or limiting molar conductivity), which can be determined by extrapolation of ${\displaystyle \Lambda _{\mathrm {m} }}$ as a function of ${\displaystyle {\sqrt {c}}}$
• K is the Kohlrausch coefficient, which depends mainly on the stoichiometry of the specific salt in solution.

This law is valid for low electrolyte concentrations only; it fits into the Debye-Hückel-Onsager equation :.[4]

For weak electrolytes (i.e. incompletely dissociated electrolytes), however, the molar conductivity strongly depends on concentration: The more dilute a solution, the greater its molar conductivity, due to increased ionic dissociation. For example, acetic acid has a higher molar conductivity in dilute aqueous acetic acid than in concentrated acetic acid. This is also the case of SDS-coated proteins in the stacking gel of an SDS-PAGE.

## Molar ionic conductivity

The limiting molar conductivity can be decomposed into contributions from the different ions (Kohlrausch's law of independent migration of ions):

${\displaystyle \Lambda _{\mathrm {m} }^{\circ }=\Sigma _{i}\nu _{i}\lambda _{i}}$

where:

• ${\displaystyle \lambda _{i}}$ is the limiting molar ionic conductivity of ion i.
• ${\displaystyle \nu _{i}}$ is the number of ions i in the formula unit of the electrolyte (e.g. 2 and 1 for Na+ and SO42− in Na2SO4)

The molar ionic conductivity of each ionic species is proportional to its electrical mobility (${\displaystyle \mu }$), or drift velocity per unit electric field, according to the equation ${\displaystyle \lambda =z\mu F}$, where z is the ionic charge and F is the Faraday constant.[5]

The limiting molar conductivity of a weak electrolyte cannot be determined reliably by extrapolation. Instead it can be expressed as a sum of ionic contributions which can be evaluated from the limiting molar conductivities of strong electrolytes containing the same ions. For aqueous acetic acid as an example,[6]

${\displaystyle \Lambda _{\mathrm {m} }^{\circ }(CH_{3}COOH)=\lambda (CH_{3}COO^{-})+\lambda (H^{+})=\Lambda _{\mathrm {m} }^{\circ }(CH_{3}COONa)+\Lambda _{\mathrm {m} }^{\circ }(HCl)-\Lambda _{\mathrm {m} }^{\circ }(NaCl)}$

Values for each ion may be determined using measured ion transport numbers. For the cation

• ${\displaystyle \lambda ^{+}=t_{+}\cdot {\frac {\Lambda _{0}}{\nu ^{+}}}}$ and for the anion ${\displaystyle \lambda ^{-}=t_{-}\cdot {\frac {\Lambda _{0}}{\nu ^{-}}}}$

Most monovalent ions in water have limiting molar ionic conductivities in the range of 40–80 S cm2 mol–1, for example cations Na+ 50.1, K+ 73.5 and Ag+ 61.9, and anions F 55.4, Br 78.1 and CH3COO 40.9.[6] Exceptionally high values are found for H+ (349.8) and OH (198.6), which are explained by the Grotthuss proton-hopping mechanism for the movement of these ions.[6] The H+ also has a larger conductivity than other ions in alcohols which have a hydroxyl group, but behaves more normally in other solvents including liquid ammonia and nitrobenzene.[6]

For multivalent ions, it is usual to consider the conductivity divided by the equivalent ion concentration in terms of equivalents per litre, where 1 equivalent is the quantity of ions which have the same amount of electric charge as 1 mol of a monovalent ion: 1/2 mol Ca2+, 1/2 mol SO42–, 1/3 mol Al3+, 1/4 mol Fe(CN)64–, etc. This quotient can be called the equivalent conductivity, although IUPAC has recommended that use of this term be discontinued and the term molar conductivity be used for the values of conductivity divided by equivalent concentration.[7] If this convention is used, then the values are in the same range as monovalent ions, e.g. 59.5 S cm2 mol–1 for 1/2 Ca2+ and 80.0 for 1/2 SO42–.[6]

From the ionic molar conductivities of cations and anions, ionic radii can be calculated using the concept of Stokes radius. The values obtained for an ionic radius in solution calculated this way can be quite different from the ionic radius for the same ion in crystals, due to the effect of hydration in solution.

## Applications

Ostwald's law of dilution, which gives the dissociation constant of a weak electrolyte as a function of concentration, can be written in terms of molar conductivity. Thus, the pKa values of acids can be calculated by measuring the molar conductivity and extrapolating to zero concentration. Namely, pKa = p(K/(1 mol dm−3)) at the zero-concentration limit, where K is the dissociation constant from Ostwald's law.

## References

1. ^ Castellan, G.W. Physical Chemistry. Benjamin/Cummings, 1983.
2. ^ The best test preparation for the GRE Graduate Record Examination Chemistry Test. Published by the Research and Education Association, 2000, ISBN 0-87891-600-8. p. 149.
3. ^ Conductivity IUPAC Gold Book
4. ^ Atkins, P. W. (2001). The Elements of Physical Chemistry. Oxford University Press. ISBN 0-19-879290-5.
5. ^ Atkins, P. W.; de Paula, J. (2006). Physical Chemistry (8th ed.). Oxford University Press. p. 766. ISBN 0198700725.
6. Laidler K.J. and Meiser J.H., Physical Chemistry (Benjamin/Cummings 1982) p.281-3 ISBN 0-8053-5682-7
7. ^ Low Electrolytic Conductivity Standards Yung Chi Wu and Paula A. Berezansky, J. Res. Natl. Inst. Stand. Technol. 100, 521 (1995)]

[1]