Practical Exercises in Physical Chemistry
Advanced Level
Jan/2001
Institute of Physical and Theoretical Chemistry, Freie Universität Berlin
4: Diffusion overpotential
1. Basic principles
Considering an electrochemical cell composed of a working electrode and a nonpolarizable
reference electrode (f.e. standard calomel electrode – SCE) the potential of the working
electrode referring to SCE at open circuit (i=0) is the equilibrium value Eeq. By applying an
external voltage of magnitude Eappl a current is forced through the cell and the potential of the
working electrode will shift to a new value E. Then:
Eappl = E + IRS = Eeq +  + IRS
(1)
IRS – ohmic potential drop in the solution
 - overpotential
In order to minimize IRS a three-electrode arrangement is preferable. In this arrangement the
current passes between the working electrode and a counter electrode (f.e. Pt – electrode).
The potential of the working electrode is measured relative to a seperate reference electrode
(f. e. SCE). If the ohmic resistance is neglectable the difference between the applied (or
measured) potential and the equilibrium potential Eeq is the overpotential . In other words:
The overpotential is the difference of the electrode potential if a current I passes between the
electrodes from the equilibrium potential at open circuit.
E(I0) - Eeq(I=0) = 
(2)
An overpotential is generally caused by a kinetic inhibition of one reaction step of the
electrochemical process. There are different overpotential contributions associated with
different reaction steps:
1) charge transfer overpotential CT
The charge transfer through the Helmholtz layer is the rate determing step.
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2) mass transport overpotential MT = D + M + C
; D – Diffusion
M – Migration
C - Convection
Mass transport is the rate determing step.
3) reaction overpotential R
Reactions procceeding or following the electrode reaction are rate determing.
The total overpotential can be considered as a sum of different overpotential contributions.
=CT+MT+R
(3)
In this experiment the electrodes are dipped into the same solution of following redox couple:
[Fe(II)(CN)6]4- [Fe(III)(CN)6]3- + e-
(4)
Since the electron transfer is fast compared to mass transport and migration and convection
are supressed in the setup the rate of the electrochemical reaction depends only on the
overpotential due to diffusion.
D >>CT + R
The rate determining process of the electrochemical reaction is the diffusion of the ions to the
electrodes. In order to achieve well-defined diffusion conditions the working electrode (here:
glassy carbon electrode) is rotating. Thus a constant concentration gradient within layer of a
stagnant thickness  near the electrode surface can be assumed (Nernst diffusion layer).
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The thickness  is proportional to the inverse of the square root of the angular velocity  ( =
2) and depends also on the kinematic viscosity of the solvent  ( = viscosity/density) and
the diffusion coefficient D of the substance:
 = 1.75 1/2 1/6 D1/3
(5)
Within this layer mass transport occurs only by diffusion. Because of the constant
concentration gradient we can apply Fick’s first law:
dN/Adt = -D NA dc/dx
(6)
NA – Avogadro number
If we consider the reduction of Fe3+ to Fe2+ at the cathode the corresponding current is
regarded as being positiv. By multiplying both sides of equation (6) with ze the current
density i is obtained:
i = z F D (cO* – cO(x=0))/ O
(7)
z- number of exchanged electrons
(cO* – cO(x=0))/ O = dc/dx
cO* - Concentration of the oxidized species (Fe3+) in the bulk
cO(x=0) - Concentration of the oxidized species (Fe3+) at the electrode surface
O - thickness of Nernst diffusion layer
The value of cO(x=0) depends on the electrode potential E. At a certain value the
corresponding current reaches its maximum – the limiting current density il. When the
limiting current flows, the electrode process is occuring at the maximum rate, because the
oxidized species is being reduced as soon as it arrives at the electrode surface. As a
consequence cO(x=0) = 0 under this conditions and the according current density is given by
:
id = z F D cO* / O
(8)
Plotting id against 1/2 (see equation 5) yields a straight line from whose slope D can be
calculated. Substituting  O in (7) by using (8) the concentration at the electrode surface can
be written as:
cO(x=0) = cO*(1-i/ i d, O)
(9)
cR(x=0) = cR*(1-i/ i d, R)
cO(x=0) - concentration of the oxidized species (Fe3+) at the electrode surface
cO* - Concentration of the oxidized species (Fe3+) in the bulk
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cR(x=0) - concentration of the reduced species (Fe2+) at the electrode surface
cR* - Concentration of the reduced species (Fe2+) in the bulk
i d, O – limiting current density for oxidized species in case of cathodic current (i positiv)
id, R – limiting current density for reduced species in case of anodic current (i negativ)
If the electron transfer is supposed to be comparably fast the electrode potential be can
expressed by the Nernst equation:
E = E° + (RT/zF)lncO(x=0)/cR(x=0)
(10)
E° - standard potential
In case of no current flowing cO(x=0) = cO* (equation 8) so the equilibrium potential can be
written as (9):
Eeq(I=0) = E° + (RT/zF)lncO*/cR*
(11)
In case of current flowing the electrode potential results from (9) and (10):
E = E° + (RT/zF)ln cO*(1-i/ id,O) - (RT/zF)ln cR*(1-i/ id,R)
(12)
With equation (2) the overpotential becomes:
 = (RT/zF) ln(1-i/ id,O) - (RT/zF)ln(1-i/ id,R)
(13)
2. Tasks:
1.) Before starting the experiment you are asked to prepare respectively 100 ml of a 0.1 molar
K4[Fe(II)(CN)6] solution and a 0.1 molar K3[Fe(III)(CN)6] solution.
Molar masses: M(K4[Fe(II)(CN)6]) = 422.42 g/mol
M(K3[Fe(III)(CN)6]) = 329.26 g/mol
2.) Record voltage-current curves of redoxsystem (4) at a rotating disk electrode at 100, 200,
600, 400 and 1000 revolutions /min in a potentiostatic circuit.
c(K4[Fe(II)(CN)6]) = c(K3[Fe(III)(CN)6]) = 10-3 mol/l
in 1n KCl
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3.) Plot the measured limiting current densities id,O and id,R against the square root of the
angular velocity  of the disk electrode. Determine from the slope the diffusion coefficient D1
and D2 of both Hexacyanoferrate ions.
Viscosity = 0.009 Poise
Density = 1.0 g/ cm3
4.) Record voltage-current curves at 600 revolution/min with different concentrations than in
exercise (2):
c(K4[Fe(II)(CN)6]) = 2 *10-3 mol/l
c(K3[Fe(III)(CN)6]) = 10-3 mol/l
in 1n KCl
5) Calculate the theoretical curve of D using equation (13) with the measured limiting current
densities for 600 revolutions/min obtained in task 4). Draw the theoretical curve on the
measuring sheet.
Compare the experimental curve with the curve calculated.
Electrode diameter: 5.8 mm
6.) Calculate the diffusion layer thickness for D1 and 1000 revolutions/min.
7.) Determine the equlibrium potential of the redoxsytem (4) for both concentrations used.
3.) Experiment

measuring cell

glassy carbon disk electrode (working electrode) with driving motor

standard calomel electrode SCE (reference electrode), saturated, EB = +0.24 V against
normal hydrogen electrode (Don’t overturn! Before and after use rinse with destilled
water. After use store the SCE in the prepared KCl solution)

Pt-electrode (counter electrode)

potentiostat

x, y – plotter
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The voltage-current curves are measured in a potentiostatic circuit. In a three-electrode
arrangement the potentiostat controls the potential difference between the working electrode
WE and the reference electrode RE, which serves as the potential basis for the working
electrode, to a predetermined value. In this experiment the difference potential UWE-RE is
varied continuously from –500mV to +300mV. A current flows from the working electrode to
the counter electrode when the redox species are converted in each other. This current (Y
entry of the plotter) is plotted against UWE-RE (X entry of the plotter).
4.) Proceeding
The cell is filled with 50 ml 1n KCl solution. Then 0.5 ml of 0.1 molar K4[Fe(II)(CN)6
solution and 0.5 ml of 0.1 molar K3[Fe(III)(CN)6 solution are added. Afterwards the
electrodes are connected (Rinse the SCE before and after use with destilled water!) Now the
solution is degased for 20 min with nitrogen. The disk electrode is set on the requested
angular velocity (Start with 1000 revolutions/min). The potentiostat is switched on and the
scan speed is choosen to be 2mV/s. The scan range is set between –300mV and +800mV. An
appropiate current range has to be determined (usually 1mA). Thereafter the cell is connected
(cell off/on button). Don’t forget to disconnect the cell whenever manipulations are made
at the cell or the potentiostat!! The plotter is switched on and the starting point is marked on
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the paper. The plotter settings for the X-axis and the Y-axis at the plotter are given in V/cm.
To transform the Y values into a current note the choosen current range at the potentiostat (the
maximum value there corresponds to 1V). For starting the measurement press the start button
at the potentiostat. Check if the current range is reasonable. Degase 5 min between the
measurements. Repeat the procedure with 600, 400 and 200 revolutions/min. Don’t forget to
label the curves. Before starting a new measurement the potential has to be reset to its initial
value.
In order to work on task 4 add additionally 0.5 ml of 0.1 molar K4[Fe(II)(CN)6. The voltagecurrent curve is recorded at 600 revolutions/min on the same paper.
In this arrangement voltage-current curves depending on the angular velocity of the disk
electrode are obtained. The recorded voltage corresponds to UWE-RE. Referring to equation (2)
the diffusion overpotential D arises from:
D = EWE(i)- EWE(i=0)
D = UWE-RE(i) + ERE - UWE-RE(i=0) - ERE
Since UWE-RE = EWE - ERE
= UWE-RE(i) - UWE-RE(i=0)
UWE-RE corresponds to the X – values and i to the Y – values of the plotter.
5.) Additional questions
1.) Sketch the graph of  as a function of i, where x=  and y=log i n case a) of pure transfer
potential and b) there is also diffusion overpotential.
2.) At a sufficient deviation of the recorded potential UWE-RE from its equilibrium value a
current increase beyond the limiting diffusion current is observed. Explain this effect!
3.) Supposing a diffusion controlled reaction occurs at the anode:
Red  Ox +eWith  = 10-3 cm
cR* = 10-3 mol/l
cO* = 2*10-3 mol/l
DR = DO
Draw the concentration for the oxidized and the reduced species as a function of the distance
from the electrode ( as multiple of ) if
a) i=0
b) i=0.5 id,R
c) i= id,R
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4) According to Brintzinger (Remy, Bd. 2, S. 332) the [Fe(II)(CN)6]4- - ion is in aqueous
solution stronger hydrated than [Fe(III)(CN)6]3- due to different diffusion velocities. Is this
in agreement with your measurements?
6.) Literatur
Faulkner, L., Bard, A. Electrochemical methods, New York 1980
Lehrwerk Chemie, Elektrolytgleichgewichte und Elektrochemie Lehrbuch 5, Leipzig 1988
Wedler, G., Lehrbuch der Physikalischen Chemie, 4th edition, Weinheim 1997
Gileadi, E., Electrode kinetics for chemists, engineers and material scientists, Weinheim 1993
Remy, H., Lehrbuch der anorganischen Chemie, Leipzig 1973
Dohrmann, J., lecture notes, FU Berlin
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