Mobility
of multi-valent ions – as used in MIFE flux
calculations
- Ian
Newman, 14 June 2002, with modifications by Peter Barry
The flux of an ion depends on the ion’s mobility u (see Newman 2001, Eqn 2), where mobility is defined as the ion’s speed per unit force acting on it. The force may be more conveniently expressed as that acting per mole of ions. Thus explicit SI units for u are
(m s^{-1}) per (N mol^{-1}),
so u relates the limiting speed v to the magnitude of the force f per mole: -
v = u f. (1)
Robinson and Stokes (2^{nd} edition 1965, pages 42-43) refer to u as the “absolute mobility”, but they are not clear in their specification of its units.
If the force is solely of electric origin, due to the presence of an electric field of magnitude E, we can express f in terms of E and the electric charge per mole of ions zF, where z is the magnitude of the valence and F is the Faraday number (96500 coulomb mol^{-1}).
Thus, f = z F E,
so v = u z F E. (2)
As Robinson and Stokes note in the case where there are no other forces, it may be convenient to consider the electric field E as representing the force and hence to relate v directly to E as:-
v = u’ E, (3)
where u’, defined through Eqn (3), is the “electrical mobility” (Robinson & Stokes, page 43) and is what is generally used by physical chemists.
Thus u and u’ are dimensionally different, through the Faraday number. They also differ according to the ionic valency.
Thus u’ = u z F. (4)
Neither u nor u’ can be conveniently measured but they can be related to l, the limiting equivalent ionic conductivity, which can be measured.
To find this relationship, consider a column in the solution of length l and of uniform cross sectional area A. If there is a potential difference V between the ends of the column, it will drive a current I of one ion species through the column. If G is the conductance of the column for that ion species, the current will be given by Ohm’s law as: -
I = G V. (5)
The conductance G depends on length and area and is proportional to the ionic conductivity s, so
G = s A/l. (6)
The limiting equivalent conductivity l is obtained by dividing the conductivity s by the ionic concentration in equivalents (z ´ concentration c): -
l = s/(zc). (7)
For full consistency with SI units, c should be expressed in units of mol m^{-3}.
The electric field driving the current is the voltage gradient,
E = V/l. (8)
From these four equations (5) to (8), we can derive an expression for the current, namely
I = l z c A E. (9)
We can also relate the current to the ionic speed v. Since the electric charge per unit volume is c z F, the quantity of charge in the column considered is c z F A l. This charge moves out of the column in the time l/v. Thus the rate of passage of electric charge, which is just the electric current, is
I = c z F A l/(l/v) = c z F v A. (10)
Equating the two expressions for the current, equations (9) and (10),
c z F v A = l z c A E,
or v = E l/F. (11)
This, with Eqn (3) gives the electric mobility u’ in terms of l as
u’ = l/F, (12)
whereas from Eqn (2) the absolute mobility u is related to l also through the valency: -
u = l/(zF^{2}). (13)