Subject: Mobility paradox, summary

From: [email-masked] (Eberhard von Kitzing)
Date: Tue, 2 Apr 1996 12:44:26 +0200

Dear colleges on the Computational Chemistry List,

this morning I thought about an apparently simple problem where I do not
know a solution jet. Consider a symmetric 1:1 electrolyte system with a
single dielectric constant which is divided by a planar surface into a
right hand and a left hand side. The cationic and anionic concentrations,
cC and cA, on both sides are the same; to obtain global electro neutrality
cC == cA is required. Exceptions from this rule are allowed only locally,
e.g. within a few Debye length around the boundary. The ionic mobilities
are mu1 > mu2 on the right hand side and mu2 and mu1 on the other for
cations and anions, respectively. At equilibrium there is no problem.

Now an external electric field E is applied and this creates the problem.
On the first view one would write the individual ionic fluxes jC and jA far
away from the central boundary:

             left hand side       and      right hand side

cations  jC(left) = + F mu1 E cC  and  jC(right) = + F mu2 E cC

anions   jA(left) = - F mu2 E cA and   jA(right) = - F mu1 E cA

where F is Faraday's constant. But this implies jC(left) != jC(right) and
jA(left) != jA(right) which violates the particle conservation law. I would
assume that such a problem has been discussed in literature. Has anybody
some ideas?

=========================================================================

> >I think  that the current density at the cathode is
> >  j(K)=(mu+  +  mu- )F E |Z+|(n+  - (mu-/mu+)dn-  - dn-(diff))
> >and at the anode:
> >  j(A)=(mu+  +  mu- )F E |Z-|(n-  -  dn-          + dn-(diff))
> >and for kinetic equilibrium j(K)= j(A) obtains, because the diffusion
> >current dn-(diff) will balance the unequal changes in particle density
> >in the left and right parts of the cell which stem from the inequality
> >of mobility values.
>
> This may be the situation close to the boundary. But one cannot have
> a non-zero concentration gradient over macroscopic distances.

I did not realize that by 'planar surface' you meant a material division
which separates the cell into two compartments (that is, a membrane).In
this case there will of course be no diffusion gradient over macroscopic
distances, but instead a diffusion poential which influences the field
strength in the left and right compartment.The faster ions will be slowed
down and vice versa until the current densities are equal again. I do not
think that under these conditions concentration differences between left an
right part can be stationary. But if the concentrations are equal again,the
diffusion potential should vanish, which gives rise to concentration
differences,this again to diffusion potential, ... . I suppose therefore an
oscillatory behaviour.

Best Wishes, M. Mauksch, CCC uni-erlangen

-------------------------------------------------------------------------

Eberhard von Kitzing
Max-Planck-Institut fuer Medizinische Forschung
Jahnstr. 29, D69120 Heidelberg, FRG

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