Practical Exercises in Physical Chemistry    Jan/2001

Advanced Level

 

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 + h + IRS                                   (1)

 

IRS – ohmic potential drop in the solution

h - 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 h. 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(I¹0) - Eeq(I=0) = h                                                             (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 hCT

            The charge transfer through the Helmholtz layer is the rate determing step.

 

2)      mass transport overpotential hMT = hD + hM + hC ; D – Diffusion

 M – Migration

 C - Convection

Mass transport is the rate determing step.

3)      reaction overpotential hR

Reactions procceeding or following the electrode reaction are rate determing.

The total overpotential can be considered as a sum of different overpotential contributions.

h=hCT+hMT+hR                                                                             (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.

hD >>hCT + hR


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 d near the electrode surface can be assumed (Nernst diffusion layer).

 

 

 


The thickness d is proportional to the inverse of the square root of the angular velocity w (w = 2pn) and depends also on the kinematic viscosity of the solvent g (g = viscosity/density) and the diffusion coefficient D of the substance:

d = 1.75 w1/2 g1/6 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))/d O                                                                  (7)

z- number of exchanged electrons

(cO* – cO(x=0))/d 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

dO - 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* /d O                                                                  (8)

Plotting id against w1/2 (see equation 5) yields a straight line from whose slope D can be calculated. Substituting d 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

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:  

h = (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

 

3.) Plot the measured limiting current densities id,O  and id,R against the square root of the angular velocity w 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 hD 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

 

 

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 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 voltage-current 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 hD arises from:

hD = EWE(i)- EWE(i=0)

hD =  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 h  as a function of i, where x= h 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 à Ox +e-

            With  d = 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 d) if

a) i=0                                      b) i=0.5 id,R                             c) i= id,R

 

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