Conductivity (electrolytic)

Strong electrolytes are hypothesized to dissociate completely in solution. The conductivity of a solution of a strong electrolyte at low concentration follows Kohlrausch's Law

${\displaystyle \Lambda _{m}=\Lambda _{m}^{0}-K{\sqrt {c}}}$

where ${\displaystyle \Lambda _{m}^{0}}$ is known as the limiting molar conductivity, K is an empirical constant and c is the electrolyte concentration. (Limiting here means "at the limit of the infinite dilution".) In effect, the observed conductivity of a strong electrolyte becomes directly proportional to concentration, at sufficiently low concentrations i.e. when

${\displaystyle \Lambda _{m}^{0}\gg K{\sqrt {c}}}$

As the concentration is increased however, the conductivity no longer rises in proportion. Moreover, Kohlrausch also found that the limiting conductivity of an electrolyte;

• ${\displaystyle \lambda _{+}^{0}}$ and ${\displaystyle \lambda _{-}^{0}}$ are the limiting molar conductivities of the individual ions.

The following table gives values for the limiting molar conductivities for some selected ions.[11]

Table of limiting ion conductivity in water at 298 K (approx. 25 °C)[11]
Cations	${\displaystyle \lambda }$+o / mS${\displaystyle \cdot }$m2${\displaystyle \cdot }$mol−1	Cations	${\displaystyle \lambda }$+o / mS${\displaystyle \cdot }$m2${\displaystyle \cdot }$mol−1	Anions	${\displaystyle \lambda }$−o / mS${\displaystyle \cdot }$m2${\displaystyle \cdot }$mol−1	Anions	${\displaystyle \lambda }$−o / mS${\displaystyle \cdot }$m2${\displaystyle \cdot }$mol−1${\displaystyle \lambda }$
H+	34.982	Ba2+	12.728	−OH	19.8	SO42−	15.96
Li+	3.869	Mg2+	10.612	Cl−	7.634	C2O42−	7.4
Na+	5.011	La3+	20.88	Br−	7.84	HC2O4−	40.2 ???
K+	7.352	Rb+	7.64	I−	7.68	HCOO−	5.6
NH4+	7.34	Cs+	7.68	NO3−	7.144	CO32−	7.2
Ag+	6.192	Be2+	4.50	CH3COO−	4.09	HSO32−	5.0
Ca2+	11.90			ClO4−	6.80	SO32−	7.2
Co(NH3)63+	10.2			F−	5.50

An interpretation of these results was based on the theory of Debye and Hückel, yielding the Debye-Hückel-Onsager theory:[12]

${\displaystyle \Lambda _{m}=\Lambda _{m}^{0}-(A+B\Lambda _{m}^{0}){\sqrt {c}}}$

where A and B are constants that depend only on known quantities such as temperature, the charges on the ions and the dielectric constant and viscosity of the solvent. As the name suggests, this is an extension of the Debye–Hückel theory, due to Onsager. It is very successful for solutions at low concentration.

A weak electrolyte is one that is never fully dissociated (i.e. there are a mixture of ions and complete molecules in equilibrium). In this case there is no limit of dilution below which the relationship between conductivity and concentration becomes linear. Instead, the solution becomes ever more fully dissociated at weaker concentrations, and for low concentrations of "well behaved" weak electrolytes, the degree of dissociation of the weak electrolyte becomes proportional to the inverse square root of the concentration.

Typical weak electrolytes are weak acids and weak bases. The concentration of ions in a solution of a weak electrolyte is less than the concentration of the electrolyte itself. For acids and bases the concentrations can be calculated when the value(s) of the acid dissociation constant(s) is (are) known.

For a monoprotic acid, HA, obeying the inverse square root law, with a dissociation constant K_a, an explicit expression for the conductivity as a function of concentration, c, known as Ostwald's dilution law, can be obtained.

${\displaystyle {\frac {1}{\Lambda _{m}}}={\frac {1}{\Lambda _{m}^{0}}}+{\frac {\Lambda _{m}c}{K_{a}(\Lambda _{m}^{0})^{2}}}}$

Various solvents exhibit the same dissociation if the ratio of relative permittivities equals the ratio cubic roots of concentrations of the electrolytes (Walden's rule).

Both Kohlrausch's law and the Debye-Hückel-Onsager equation break down as the concentration of the electrolyte increases above a certain value. The reason for this is that as concentration increases the average distance between cation and anion decreases, so that there is more inter-ionic interaction. Whether this constitutes ion association is a moot point. However, it has often been assumed that cation and anion interact to form an ion pair. Thus the electrolyte is treated as if it were like a weak acid and a constant, K, can be derived for the equilibrium

A⁺ + B⁻ ⇌ A⁺B⁻; K=[A⁺] [B⁻]/[A⁺B⁻]

Davies describes the results of such calculations in great detail, but states that K should not necessarily be thought of as a true equilibrium constant, rather, the inclusion of an "ion-association" term is useful in extending the range of good agreement between theory and experimental conductivity data.[13] Various attempts have been made to extend Onsager's treatment to more concentrated solutions.[14]

The existence of a so-called conductance minimum in solvents having the relative permittivity under 60 has proved to be a controversial subject as regards interpretation. Fuoss and Kraus suggested that it is caused by the formation of ion triplets,[15] and this suggestion has received some support recently.[16][17]

Other developments on this topic have been done by Theodore Shedlovsky,[18] E. Pitts,[19] R. M. Fuoss,[20][21] Fuoss and Shedlovsky,[22] Fuoss and Onsager.[23][24]

The limiting equivalent conductivity of solutions based on mixed solvents like water alcohol has minima depending on the nature of alcohol. For methanol the minimum is at 15 molar % water,[18][25][26] and for the ethanol at 6 molar % water.[27]

Generally the conductivity of a solution increases with temperature, as the mobility of the ions increases. For comparison purposes reference values are reported at an agreed temperature, usually 298 K (≈ 25 °C), although occasionally 20 °C is used. So called 'compensated' measurements are made at a convenient temperature but the value reported is a calculated value of the expected value of conductivity of the solution, as if it had been measured at the reference temperature. Basic compensation is normally done by assuming a linear increase of conductivity versus temperature of typically 2% per Kelvin. This value is broadly applicable for most salts at room temperature. Determination of the precise temperature coefficient for a specific solution is simple and instruments are typically capable of applying the derived coefficient (i.e. other than 2%).

The change in conductivity due to the isotope effect for deuterated electrolytes is sizable.[28]