Department of Chemistry, Trent University, Peterborough, Ontario, Canada K9J 7B8

A.N.Frumkin Institute of Electrochemistry, Russian Academy of Sciences, 117071 Moscow, Russia

Department of Mechanics, Royal Institute of Technology, 100 44 Stockholm, Sweden

Polarization curves are analyzed for parallel electrochemical reactions of two different types with no supporting electrolyte in the solution. Consideration of the problem is based on an analytic solution, in parametric form, of the Nernst - Planck equations for electrodiffusion with boundary conditions of the Butler - Volmer type. It is found that the current - voltage curves for such systems clearly display the interaction of ionic components due to the migration- current exaltation phenomena.

Problem statement

1) One-dimensional transport:

- spherical micro-electrodes;

- membranes.

2) Nernst-Plank equation for each ion species:

- activity coefficients are unity.

- thickness of diffusion layer is much greater than Debye length.

- can be solved together with the system of Nernst-Plank equations for simple electrode reaction such as Az+ (aq) + z e- ÷ ö A(s).

- For more complicated parallel reactions only the limiting current as

*V6 7*
could be found. The Butler-Volmer equations reduce to
*C*ks =0.

5) Boundary conditions:

*C*k(*L*)=*C*kb, f(*L*)=0

(+) Arbitrary values of diffusion coefficients;

(+) Multi-charged ions;

(+) Possible homogeneous reactions in transport layer;

but

(-) the system is at steady-state;

(-) no convection;

(-) no temperature effects

(-) one electrode problem

"To solve problem" - means to find a solution to the system of differential equations and boundary conditions in a closed analytical form.

Concentration bulk concentration

Potential *RT/F*~25 mV

Current density *L / FD*kCb
diffusion limiting current density

Dimensionless Nernst-Plank equation:

Single electrochemical reaction:

W. Nernst, Z. Phys. Chem. -1904, B.17, S.52

A+ + e- ÷ ö A9 .

Limiting diffusion-migration current:

*j*lim = 2

Binary zA-zB electrolyte:

*j*lim = 1+zA/zB

A. Euken, Z. Phys. Chem. -1907, B.59, S.72

Limiting diffusion-migration current:

*j*lim =

*j*lim 6
2 at 6 1 (binary
electrolyte)

*j*lim 6
1 at 6 0 (large excess of
supporting electrolyte)

zA-zB electrolyte with zF-zB buffer:

L.Hsieh and J. Newman -Ind. Eng. Chem. Fund 1971, V.10, P.615.

Yu.Gurevich and Ju. Kharkats -. Sov. Electrochemistry 1979, V.15, P.94.

R. Schl` gl, Z. Phys. Chem.(Frankfurt) -1954, B.1, S.305

For simplicity: *
*z*k* =1

*j*1 is the current of the reaction in
question: A+(aq) + e- ÷
ö A(s)

*j*0 is the current of the
parallel reaction, for example:

O2(aq) + 2H2O + 4e- 6
4OH-(aq)
*or*

O2(aq) + 2H2O + 2e- 6 H2O2(aq) + 4OH-(aq)

Transport of O2 occurs by ordinary diffusion,
therefore *j*0 does not
depend on *j*1, but the
opposite it not true due to a migration interaction of
A+ with OH-.

Voltamperometric curve for a=1/2:

Limiting diffusion-migration current in conventional variables:

*i*lim = *i*0 + 2
*i*1dif +
(D1/DOH)
*i*0

For a…OQ 1/2:

where *h*
= *V* - *F*s is the
reaction overvoltage. The last two formulas give voltamperometric
curve *j*1(*V*) in
the parametric form {*j*1(*h*)*,V(**h*)}.

A1+(aq)+ e- ÷ ö A1(s).

A2+(aq)+ e- ÷ ö A2(s)

The system of equations include three Nernst-Plank equations for A1+, A2+ and B-, as well as two Butler-Volmer equations for A1+ and A2+. This problem could be reduced to one transcendental equation.

compare with
*Isaak Rubinstein "Electro-Diffusion of Ions" - SIAM
1990*

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no transport-limited current?

There are some systems with
three parallel electrode reactions for which the condition of
zero concentrations of all reagents *C*ks=0 and fast (compared to the transport) electrode kinetics
are not sufficient to give a value of the current. In such
situations the current depends on the relationships between the
equilibrium potentials for all values of the applied potential.
The whole concept of transport-limited current should be
re-defined for such systems.

A+(aq)+ e- ÷ ö A(s).

O2 + 2H2O + 4e- 6 4OH-

OH- + H+ 6 H2O

Diffusive double-layer split into two parts:

One remains near the electrode and is responsible for the reaction kinetics - the "normal" double layer. The other, much stronger, layer moves into the region where the homogeneous reaction of water recombination takes place. It is called the "reaction double layer" and is responsible for transport limitation of the entire system.