Page 614 - Handbook of Battery Materials

588 17 Liquid Nonaqueous Electrolytes

17.4.5.3 The Walden Rule and the Haven Ratio
In 1906, P. Walden published his empirically discovered relation known as the
Walden rule [407]. It states that the product of the limiting molar conductivity Λ 0
m
and the viscosity of the pure solvent, η, is constant for infinitely diluted electrolyte
solutions, as expressed by Equation 17.49
0
·η = C = constant (17.49)
m
−1
0
log( ) = log(C) + log(η ) (17.50)
m
It has been empirically found that the Walden rule is often well obeyed, especially
by solutions of large and only weakly coordinating ions in solvents with nonspecific
ion–solvent interactions. In this case, ion mobility and hence conductivity is
solely governed by ion migration, according to the Stokes–Einstein equation, see
Equation 17.51 [408],
0
D = u i k B T (17.51)
i
0
where D i is the diffusion coefficient of an ion i and u I its mobility, k B is Boltzmann’s
constant, and T the absolute temperature. The mobility is linked to the single ion
0
conductivity λ i via
0
λ = z i Fu i (17.52)
i
yielding
e o z i F 1
0
λ η = (17.53)
i
f R i
and for a 1 : 1 salt
e o F 1
0
η = = C (17.54)
m
f (R + + R − )
where e 0 is the elementary charge, z i the charge of the ion, and F Faraday’s
constant. f is a factor based on Stokes law (f = 6π for perfect stick or f = 4π for
perfect slip) and R i is the Stokes radius of the ion, that is, the ionic radius plus
a contribution due to solvation of the ion by solvent molecules. R + and R − are
the Stokes radii (ionic radii and solvation contribution) of the cation and anion
respectively. To avoid size variation effects of ions dissolved in different solvents,
these should preferably interact only weakly and/or at least in a similar manner
with the electrolytes’ ions. This assists comparable ion coordination behavior, and
by eliminating specific solvent–solute interactions, the viscosity of the solvent is
then and only then the key property determining the conductivity of the solution.
The molar conductivity Λ 0 , calculated from self-diffusion coefficients, es-
NMR
timated by pulsed field gradient nuclear magnetic resonance (pfg-NMR) (see
Section 17.4.7.2), is described by the Nernst–Einstein equation [409]
|z|eF
Λ 0 = (D + + D − ) (17.55)
NMR
k B T
where z is the charge number, e the elementary charge, F the Faraday constant,
k B the Boltzmann constant, T the temperature, and D + and D − the self-diffusion