3. Transport in electrolyte solutions

3.1. Conductivity of an electrolyte solution

Let’s consider the situation in Figure 3.1: a particle (ion) with the charge q = ze feels a coulombic force F_c in an electric field E:

F_c=qe=zeE . (3.4)

Forces acting on a charged particle.

Figure 3.1. Forces acting on a charged particle.

The coulombic force makes the particle move with the velocity v (no vector notation):

v=uE , (3.5)

where u is the mobility of the particle. Friction resists the movement:

F_f=-fv (3.6)

In Equation (3.6) Ff  is the frictional force and f  the friction coefficient for which Einstein derived an expression

\displaystyle f= \frac{kT}{D} (3.7)


At equilibrium, the friction force and the coulombic force cancel each other out, and we get:

\displaystyle zeE= \frac{kT}{D}v \Leftrightarrow v=\frac{zeD}{kT}E=\frac{zFD}{RT}E . (3.8)

Comparing Equation (3.5) with (3.8) it is immediately seen that

\displaystyle u= \frac{zFD}{RT} . (3.9)

It is easy to prove that a particle reaches constant velocity immediately after switching on the electric field. Newton’s second law can be written in the form

\displaystyle m \frac{dv}{dt} =qE-fv (3.10)


With the initial condition v(t = 0) = 0 the solution is


\displaystyle v(t)= \frac{qE}{f}(1-e^{-(f/m)t}) (3.11)


Inserting typical values of the quantities into the above equation, we see that the exponential (f/m) is of the order of 109 s-1, i.e. the relaxation time is of the order of nanoseconds.

Electric current density i is defined using the Ohm’s law:
i= \kappa E (3.12)

where \kappa is the conductivity of the solution. It can be written as the sum of the product of molar conductivities, \lambda k, and concentrations, ck, of all ions:

\displaystyle\kappa =\sum\limits_{k}\lambda_kc_k (3.13)


In the solution, electric current is the sum of all ionic fluxes:
\displaystyle i=F\sum\limits_{k}z_kJ_k=F\sum\limits_{k}z_kv_kc_k (3.14)

Inserting Equation (3.8) into (3.14), the following is obtained:
\displaystyle i= \frac{F^2}{RT}\sum\limits_kz_k^2D_kc_k \cdot E (3.15)


Comparing Equations (3.12) and (3.13) with (3.14) and (3.15), the relation between \lambda k and Dk is found to be
\displaystyle\lambda_k= \frac{F^2z_k^2D_k}{RT} (3.16)

The relation between uk and \lambda k is

\lambda_k=z_ku_kF (3.17)

The above equation means that the mechanical and electrical mobility are assumed to be equal. Stokes' law defines the friction coefficient of a spherical object as

f=6\pi\eta a (3.18)

where η is the viscosity of the solution and a is the particle radius. Quite surprisingly, Stokes' law applies also to ions although they are of the same order of magnitude as solvent molecules, and although the law was derived via hydrodynamic considerations for macroscopic objects moving in a homogeneous medium. Assuming a as the ion radius, it follows from Equation (3.7) that

\displaystyle D= \frac{kT}{6\pi\eta a} (3.19)

Multiplying this equation by η, only solvent independent constants are left on the right hand side: we have derived Walden's rule:

D\eta= constant (3.20)

Walden's rule can be used to estimate the values of diffusion coefficients in solvents where there is no measured data. The rule does not, however, apply particularly well to ions in aqueous solutions due to their strong hydration (see Figure 3.2).

Walden rule

Figure 3.2. The product \eta\ \lambda_i for K+ (●) and Cs+ (■) cations in selected solvent: DMSO = dimethyl sulphoxide, NMF = n-methyl formamide, DMF = dimethyl formamide, THF = tetrahydrofurane, DMOE = dimethoxy ethane, ACN = acetonitrile. Note water! A.K.Kontturi et al., Ber. Bunsenges. Phys. Chem. 99 (1995) 1131. (P = poise = 0.1 Ns/m2).