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ABSTRACT

AN EXPERIMENTAL EVALUATION OF THEORY FOR DIFFUSION
CONTROLLED COULOSTATIC RELAXATION

Previously, eXperimental applications of the coulostatic
technique have been limited to perturbations from equilibrium
of less than 4/n mV, because of assumptions in the derivation
of the theoretical equations. Recent theoretical studies,
however, have indicated that perturbations as large as 25/n mv
should yield the same results as the previous small perturbation
theory when the normalized relaxation curves are compared. In
this work an attempt was made to test these predictions experi-
mentally for the case of diffusion controlled relaxation. Thus,
the ferrous-ferric oxalate couple, which is known to have a large
electron transfer rate constant, was investigated by the
coulostatic method. Normalized relaxation curves for perturba-
tions ranging from 7 to 43 mV from equilibrium were recorded and
compared with the small perturbation theory. A profound lack
of agreement was found between the theory and experimental results
even for small perturbations from equilibrium. In an effort to
determine the source of this disagreement between theory and

experiment, each of the assumptions of the theoretical model was

tested experimentally, but no discrepancies were found which

could explain the magnitude of the deviation observed.

AN EXPERIMENTAL EVALUATION OF THEORY FOR DIFFUSION

CONTROLLED COULOSTATIC RELAXATION

BY

Frank Matthew Michaels

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Chemistry

1970

VITA

Name: Frank Matthew Michaels
Born: March 12, 1943, in Cleveland, Ohio

Academic Career: Benedictine High School
Cleveland, Ohio 1957 - 1961

Case Institute of Technology
Cleveland, Ohio 1961 - 1965

Degree Held: 3.5. Case Institute of Technology (1965)

ii

ACKNOWLEDGEMENT

The author wishes to convey his sincere appreciation to
Professor Richard S. Nicholson for his guidance and encouragement
in the course of this work.

The author also wishes to thank Professor C. G. Enke for

permitting the use of some of his research equipment during this

study.

iii

TABLE OF CONTENTS

INTRODUCTION 0 O O O O O O O O O O O 0
THEORY O O O O O O O O O O O O O O O 0

Model. .
Results 0 O O O O O O O O O O O 0

EXPERIMENTAL 0 O O O O O O O O O O O 0

Selection of a Chemical System .

Instrumentation for Coulostatic Experiments
Circuit of Delahay. . . . . . . . . .
Circuit of Lauer. . . . . . . . . . .
Circuit of Daum and Enke. . . . . . .
Measurement of Double Layer Capacitanc

Cell and Electrodes. . . . . . . . . . . .

Chemicals. 0 O O O O O O O O O O O O O O 0

Procedures . . . . . . . . . . . . . . . .

RESULTS AND DISCUSSION . . . . . . . . . . . . .

Low Overpotential Experiments. . . . . . .
Effect of Electron Transfer Kinetics.
Errors in To and I o o o o o o o o 0
Effect of Finite Cfirrent Injectio Tim

Errors in AEi . . . . . . .
Other Sources of Error. . .
Effect of Large Overpotentials .

iv

0
e

O

O

Page

12

12
14
14
16
19
28
28
29
29

31

31
32
35
39
40
41
42

TABLE

TABLE Page

I. Comparison of Relaxation Curves for Large and
Small overPOtentialS o o o o o o o o o o o o o 11

FIGURE

LIST OF FIGURES

Diagram of the circuit due to Delahay. . . . . .
Circuit Diagram of the semiconductor switch. . .
Diagram of the circuit due to Lauer. . . . . . .
Equivalent circuit for the cell. . . . . . . . .
IR compensator and final circuit configuration .

Comparison of theoretical and experimental
relaxation curves. . . . . . . . . . . . . . . .

Stationary electrode polarogram for supporting
electrolyte without depolarizer. . . . . . . . .

Relaxation curves for larger perturbations from
equilibrium on the ferrous-ferric oxalate system

vi

Page

18
21
24

27

34

38

44

INTRODUCTION

The use of relaxation methods to study the rates of fast
homogeneous chemical reactions is fairly commonplace. General-
ly, with these methods the equilibrium state of a chemical
system is perturbed by abruptly changing one of the variables
defining that equilibrium state. The rate of return (relaxa-
tion) of the system to equilibrium is determined by chemical
reaction rates, and therefore can be used to measure rate
constants.

The coulostatic technique“2 can be thought of as an
application of these ideas to the study of heterogeneous
electrochemical reactions. In the initial equilibrium state
the electrode potential is determined by the concentration of
depolarizer at the electrode surface through the Nernst
equation. This equilibrium potential is then rapidly per-
turbed by altering the charge in the electrical double layer
of the electrode. This rapid injection of charge can be
accomplished in several ways, the simplest being the circuit
of Figure 1 -'(this circuit was first used by Delahay3 in his
initial investigations of the coulostatic technique).
Initially, capacitor, C, is charged by battery, B. Switch,

8, is then thrown to charge the electrical double layer. By

having C small with respect to the double-layer capacitance,
the discharge of C can take place in a very short time. The
electrode potential reached at the end of the coulostatic

impulse is

E=Eeq+g (1)

Figure 1. Diagram of the Circuit Due to Delahay.3
A: Cell
B: Battery

C: Capacitor

 

 

 

 

_— A
C —‘— '
0
'_'F—

 

 

 

 

Figure 1

where Ee is the initial equilibrium potential, q is the

q
coulombic charge injected, and C is the double layer
capacitance.

Since the new electrode potential, E, does not corres-
pond to the equilibrium concentration of depolarizer,
Faradaic reaction takes place, discharging the double-layer.
As the double layer is discharged, the electrode potential
relaxes back toward the initial potential, Eeq- Initially,
the extent of Faradaic reaction is governed by the rate of
the electron transfer reaction, and therefore the relaxation
is kinetically controlled.' As the Faradaic reaction proceeds,
however, concentration gradients develop near the electrode
surface and the mass transport process (usually diffusion)
also determines the rate of relaxation. Thus, theory for
the coulostatic method must consider both diffusion and
electron transfer kinetics. For a first order electrode
reaction, the rate of electron transfer is given by the follow-

ing well known equation“:

Co CR
i = nFAks (C *)1'9 (C *)° "' exp (-anF ) - -- exp|}l-o)nF]
‘5 O R Co* ‘1?!" CR* if"

(2)

where if is the faradaic current

a = transmission coefficient

kS = standard electrochemical rate constant
n = overpotential
CR,CO = surface concentrations of reduced and
oxidized species
CR*,CO* = bulk concentrations of reduced and

oxidized species
The other parameters have their usual meaning.

When diffusion also must be considered, Equation 2
constitutes a boundary condition for solution of the Fick
diffusion equations. Because the relationship between
potential and concentration in Equation 2 is nonlinear, in
general the Pick boundary value problem cannot be solved
analytically. Thus it is common practice to linearize
Equation 2 by expanding the exponentials in a Taylor series
and retaining only the first two terms. For this approxima-
tion to be valid at the 1% error level the argument of the
exponentials must be less than 0.15. In terms of Equation 2
this condition corresponds to potential excursions less than

3/N mV. From an experimental point of view the use of such

small overpotentials is difficult when signal to noise ratios
are considered. Actually, it is conceivable that the ex-
ponential terms are of minor importance in the final result,
so that larger overpotentials could be used without increasing
the above error level. In the case of purely diffusional
relaxation this appears to be so. Nicholson5 has solved the
boundary value problem without introducing linearization, and
found that results agree with linearized theory for over-
potentials as large as 25/n mV. Since this fact has important
implications in terms of electrochemical relaxation methods,
it was decided to test experimentally the theory published by
Nicholson. The results are described in the remainder of this

thesis.

THEORY

For an experimental evaluation of coulostatic theory to
be meaningful it is important to carefully define the theoreti-
cal model. The model assumed by Nicholson and in most other
treatments of coulostatic relaxation is therefore described
next.

£2931. The following major assumptions were used by
Nicholson in develOping a theoretical description of coulostatic
relaxation:

1. Mass transport occurs by linear diffusion. This
implies the absence of convection or electrical
migration affecting the electroactive species, and
assumes that a planar electrode is employed.

2. The concentration of depolarizer in the bulk of
solution remains unchanged within the time scale of
the experiment.

3. No adsorption or other accumulation occurs at the
electrode surface.

4. The injection of charge is instantaneous.

5. The double layer capacitance is independent of
potential over the potential range of the experiment.

6. The potential relaxation is diffusion controlled at
all times. This means that the electrochemical rate
constants are large enough that the Nernst equation
concentrations are always maintained at the elect-
rode surface even immediately after the initial
impulse. The rate of potential decay will then be
determined by the rate of diffusion of electroactive
species to the surface.

Results. The above model can be expressed mathematically

as the following nonlinear integral equation first derived by

Nicholsons.

 

 

 

t/I t/TO
‘1 2&7}? - f 0 Mend: =5 exp mm -1 dz; (3)
O
lt/to - E 1 + JIR7IO exp[Y(£)]
The following definitions apply to Equation (3)
Y(t) = E AE(t) (4)
RT
‘l’i = nF AEi (5)

RT'

10

 

 

10% = RT _c_ 1 (6)
nZDZ A
555730"
I]: = RT C l (7)
nzeX _"'
YnDR CR*

There AE(t) is the electrode potential as a function of
time; AEi is the initial potential displacement from
equilibrium, C is the differential double layer capacitance,

A is the electrode area, D and DR are diffusion coeffici-

O

ents for the oxidized and reduced forms of the depolarizer,

C6 and CE are initial bulk concentrations of depolarizer,

and other terms have their usual meaning. Thus, the function
?(t), which is the solution of Equation 3, directly gives the
potential - time relaxation for any initial perturbation, Vi.
The solution of Equation 3, and hence the rate of relaxation,
is determined by the two diffusional relaxation constants, To

and TR.

Equation 3 is nonlinear because of the exponential terms,
and therefore Equation 3 cannot be solved analytically. If
potential excursions are restricted to less than 3/n mV, however,

the exponential terms can be expanded as a truncated Taylor

11

series in which only first order terms are retained. This
results in the following solution obtained first by Reinmuth.6
W(t) 8 Vi exp [t/u(/?;’+ /?;)2] erch/EY/?1/?;'+ /?;)] (8)
all terms in Equation 8 have been defined previously.

Equation 3 was solved numerically by Nicholson for
various perturbations from equilibrium. The results of solv-
ing Equation 3 for a 25/n mV, perturbation are listed in
Table I along with data calculated from Equation 8. A
comparison of the two sets of data shows that they agree with-
in 2% . Thus these theoretical data indicate that some of the
errors introduced by the linearization tend to cancel, so that
larger perturbations can be used in the coulostatic technique

while the data can sti11 be interpreted in a straightforward

manner using Equation 8.

11

Table I. Comparison of Relaxation Curves for

Large and Small Overpotentials

a

 

t/rO w/wi 41/13
.01 .969 .977
.05 .933 .940
.10 .907 .915
.50 .809 .819

1.0 .747 .758

5.0 .554 .563

10. .458 .466
50. .256 .259

 

a

Linearized theory from Equation 8.

b

Nonlinearized theory from Equation 3; data

for Vi = 25/n mV.

are

12

EXPERIMENTAL

Selection of a Chemical System

Success in comparing theory with experiment is depend-
ent on how faithfully the theoretical model describes the
experimental conditions. Some assumptions of the model are
satisfied without difficulty, while others are impossible to
achieve vigorously. Thus, the discussion that follows con-
siders each point of the model in terms of selection of
experimental conditions and a depolarizer.

The linear diffusion and constant bulk concentration
requirements - the so called semi-infinite linear diffusion
assumption - is a common constraint in electrochemistry, parti-
cularly polarography. This assumption is realized experi-
mentally through use of excess supporting electrolyte to insure
mass transport by diffusion, and measurement times less than 1
second to insure linear diffusion.

Adsorption effects would make it impossible to determine
the initial concentrations for the experiment. Also, preferent-
ial accumulations of the electroactive species at the surface

during the experiment would affect the potential - time curves

13

to an indeterminate degree. These problems can be averted by
selecting a depolarizer for which adsorption phenomena are
known to be absent as indicated by other electrochemical
techniques. Since only small overpotentials are used in
coulostatic experiments, the variation of double layer capaci-
tance with potential is usually negligible over these relative-
ly small potential ranges. The double layer itself is almost
entirely composed of ions of the supporting electrolyte,
because of the large excess of electrolyte with respect to the
concentration of depolarizer. The double layer capacitance
therefore, can be measured as a function of potential by per-
forming cyclic voltammetry experiments on the inert electrolyte.
The model assumes that Nernstian conditions are maintained
during the entire experiment, a condition which is clearly
impossible to achieve vigorously since it would require an in-
finitely large electron transfer rate constant. Nevertheless,
by using a depolarizer for which the electron transfer rate
constant is very large (greater than 1 cm/sec.), deviations
from Nernstian behavior can be limited to the first few micro-
seconds of the relaxation. In addition, by employing relatively
low concentrations of depolarizer, onset of diffusion control

can be hastened.

14

To summarize, in terms of the model the major require-
ments for a chemical system are; large electron transfer rate
constant, absence of adsorption, and both forms of the redox
couple soluble in the solution phase. An examination of the
electrochemical literature revealed that the ferrous - ferric
oxalate couple in 0.15 M binoxalate satisfies these criteria
[e.g. ks > 1 cm/sec 7], and therefore this system was selected
for evaluation of the theoretical calculations.

The only point of the model not yet discussed is Point 4,
the assumption that the coulombic charge injection can be
accomplished in an infinitesimally small time. This condition,
of course, can never be realized experimentally, since it would
require an infinite current from the potential source. Never-
theless, with suitable instrumentation the charge injection can
be accomplished in a very short time. A number of different
approaches are possible, and several of these were evaluated as
a part of this investigation. The results of this study, as
well as a description of the circuit actually used to collect
data reported in this thesis, are presented next.
Instrumentation for Coulostatic Experiments

Circuit of Delahay.3 The circuit of Figure 1, due to

Delahay, although simple in principle, cannot be used directly

15

since the mechanical switching operations introduce large
spikes and spurious noise which take up to 100 u Sec. to
decay. It would seem that the switching could be done
electronically, however, and this possibility was tested
using a commercial SPDT electronic switch (Crystallonics
Model CAG7) to duplicate the function of the switch in
Delahay's circuit. Since contacts are not formed or broken
in this type of switch, the noise problems are eliminated.

The CAG7 has a 1.5 u sec. turn on time with a 69 on
resistance and a lOOma current capability. The circuit for
the switch is shown in Figure 2. Briefly, its Operation is

as follows. With a logic level of + 5 V, transistor T1 is
maintained in the off state. Point P is then negative, diode
D1 conducts but the field effect transistor (FET) F1 is
reverse biased and is in the off state, so that the cell is
effectively out of the circuit. Transistor T2 is also off,
keeping point Q at + 15 V. Since diode D2 is reversed biased,
FET F2 is on and the capacitor is charged by the battery. When
the "logic in” level is switched to 0 (ground), both transis-
tors are on and now F1 is on while F2 is off so that the

capacitor discharges across the cell. The resistor across the

16

capacitor is used to provide proper bias for the FET's. The
resistor was chosen to be small enough to bias the FET's, yet
large enough not to contribute significantly to the discharge
of the capacitor across the cell.

Unfortunately it was found with the above circuit that
the cell was always biased by a small voltage during the
experiment, corresponding to a net current flowing through the
cell during the decay. The source of this current was the
fact that the diode D1, though reversed biased, still had a
finite resistance. Thus, the +15V from point P is divided
through the diode, the drain to gate resistance, and the cell
resistance giving the resulting bias during the decay. No
immediate solution to this problem was apparent.

Circuit of Lauer.8 A circuit published by Lauer was
designed to eliminate the switching noise of Delahay's design
by putting the cell in the control loop of an Operational
amplifier. The circuit is given in Figure 3. A square wave
is input to the amplifier as shown. Current flows through the
feedback loop until capacitor CA reaches a potential equal in
magnitude, though opposite in sign to the input source. Point

S is then at virtual ground so no further net current flows.

17

Figure 2. Circuit Diagram of the semiconductor switch.
T1,T2: Transistor
D1,D2: Diodes
F1,F2: Field Effect Transistors
A: Cell
B: Battery

C: Capacitor

18

 

 

 

 

+15

Figure 2

 

 

 

 

 

19

Since all the current which flows through the capacitor also
flows through the cell, by knowing the capacitance of CA, the
number of coulombs input to the cell can be calculated. The
initial potential can then be determined if the double layer
capacitance is known.

Although Lauer claims to have obtained good results with
this circuit, when it was constructed in this laboratory con-
siderable difficulty was encountered in terms of maintaining
stability. Thus, in order to insure stability, response time
had to be sacrificed so that under the best conditions 50 u
sec. were required to charge the cell, which is at least an
order of magnitude too large to be useful.

Circuit of Daum and Enke.g A commercial current pulse
generator (Intercontinental Instruments Incorporated model
PG-33) has been used by Daum and Enke to give fast charging
without the noise problems obtained with Delahay's circuit.
With this instrument the pulse duration can be controlled from
30 n sec. to 2 sec., and the magnitude of the current from
zero to 200 ma. The use of a constant current source has a
further advantage in that the double layer capacitance can be

measured during the changing pulse as described by Wier and

20

Figure 3. Diagram of the circuit due to Lauer.8
G: Square wave pulse generator
R's: Resistors of equal value
oA: Operational amplifier
A: Cell

C: Capacitor

21

 

 

 

 

 

 

Figure 3

 

22

Enke.10 The equivalent circuit of the cell for the charging
process is shown in Figure 4. The magnitude of the constant

current, ip, is given by the differential equation

ip = C dAE + AB (9)
HF“ ‘RE

where AB is the potential of the cell without the iR drop due

to solution resistance.- The solution of Equation 9 is

-t
AB = ipRC (1 - e /Rcc)

During charging the potential measured, 5, includes the

(10)

solution resistance,

:1: ipRe + AB (11)
so that the slope of the observed charging curve is
. ’t/R c

‘HE' '3?
For measurement times much smaller than the time constant
RcC, the 310pe will equal ip/C. A typical time constant for
these experiments is approximately 5 u sec.

During the charging pulse relatively large currents are
flowing through the cell and and appreciable IR drop, as large
as 8V, can be developed across the cell during the current
pulse. With the oscilloscope at the high sensitivity required

to observe the relaxation curve (e.g. 5mV/div.), saturation of

23

Figure 4. Equivalent circuit for the cell
Re: Electrolyte resistance
RC: Charge transfer resistance

C : Double layer capacitance

Re

24

 

 

Figure 4

 

25

the oscilloscope's input amplifier occurs. These amplifiers
recover from saturation fairly slowly, thereby obscuring the
initial portion of the relaxation curve. To avoid this
problem Daum11 devised an IR compensator, in which a variable
resistor simulates the cell resistance. The current is
divided into two branches, one path through this dummy cell,
the other through the real cell. By measuring the potential
between the resistor and the cell differentially, IR drOp
across the cell is cancelled, and saturation of the oscillo-
scope amplifiers prevented. This approach is illustrated in
Figure 5. In this figure CPG denotes the current pulse
generator, the output of which is divided by matched diodes.
Approximately half of the current flows through a resistor
which simulates the cell resistance, and whose value can be
adjusted with potentiometer P. The other half of the current
flows through the cell. The potential is measured different-
ially between the cell and dummy cell via the oscilloscope.
The oscillosc0pe used was a Tektronix Model 535A with Type W
plug in high gain differential amplifier and delayed trigger
(T in Figure 5). The net current through the cell was measured

between points M and Q across the precision resistor R. All

26

Figure 5. IR compensator and final circuit configuration
CPG: Current Pulse Generator
R's: Precision resistors of equal value

A Cell

P Potentiometer

w : Differential amplifier plug in unit

T : Delayed trigger

 

 

 

CPG

27

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5

 

 

28

oscilloscope traces were photographed using a Tektronix
Type C-12 camera and Model 100 projected graticule.

Measurement of Double Layer Capacitance. The double
layer capacitance as a function of potential was determined
with cyclic voltammetry on a solution of supporting electro-
lyte without depolarizer. Double layer capacitance was
calculated from the equation

c = i (13)
3173?

where C is the capacitance at potential E, dE/dt is the scan
rate and i is the observed current. These curves were recorded
with a three electrode potentiostat (Wenking Potentiostat
Model 61 RS) and a commercial function generator (exact, Type
255 s). The same equipment was used_in a polargraphic con-
figuration to measure the concentrations of depolarizer.
Cell and Electrodes

For the coulostatic work a two electrode cell configuration
was used. The working electrode was a hanging mercury drop.
The counter - reference electrode was a mercury plated platinum
gauge with area much greater than the working electrode. The

geometry of the cell is similar to that described by Wier and

29

Enke12 with the gauze surrounding and concentric with the
working electrode to minimize noise.

The cell and electrode system used for the cyclic volt-
ammetry and polarographic work were of conventional design and
are described elsewhere.12
Chemicals.

All chemicals were reagent grade and used without further
purification. Argon was saturated over water and passed into
the cell to purge the system of dissolved oxygen.

Procedures.

Experiments were performed for various perturbations from
equilibrium ranging from 7 mV to 50 mV. For each experiment
the charging curve was measured to calculate the double layer
capacitance. The current through the cell and charging pulse
duration measured via the 100 a precision resistor in series
with the cell were also recorded for each experiment.

Ammonium sulfate salts were used to introduce the ferrous
and ferric ions in approximately equimolar concentrations
(~2x10‘4M). The salts were weighed out on an analytical

balance and the concentration checked polarographically.

30

A 0.15M oxalate solution, made from potassium bin-
oxalate acted as a complexing agent, buffer (pH~2.5), and

supporting electrolyte.

31

RESULTS AND DISCUSSION

Although the original objectives of this research were
to evaluate the use of large overpotential relaxation, it
seemed wise initially to evaluate the iron oxalate system
using standard low overpotential experiments. As will be seen
below, the results of these experiments were such that in fact
it was impossible to obtain a meaningful evaluation of the
effect of large overpotentials.
Low overpotential experiments

Many experiments were performed with over potentials less
than SmV. These data can be conveniently summarized with a
single figure by plotting AE/AEi XE} time, which effectively
normalizes the experimental data. A plot of this type is shown
as Curve A in Figure 6. Before evaluating the effect of large
overpotentials, it seemed advisable to compare the experimental
data of Figure 6 with predictions of the low overpotential
model. To do this it is necessary to know To and TR' because
then Equation 8 can be plotted as W/Yi (equivalant to AE/AEi of
Figure 6) 23, time. Thus,~values of To and TR' were calculated

from Equations 6 and 7 using experimentally determined values

32

of CO, CR, and double-layer capacitance. The literature
value13 for DO was used for both DO and DR' In view of the
fact that three oxalate anions are coordinated to both
oxidized and reduced forms of the depolarizer,1“ the assumpt-
ion of the equality of diffusion coefficients seems to be a
reasonable one.

Using these calculated values of To and TR' it was
then possible to compare the experimental data of Figure 6
with theory. The corresponding theoretical plot is labeled
as Curve B in Figure 6. Clearly, from Figure 6 there is an
alarming lack of agreement between theory and experiment. A
number of different experiments were therefore performed in
an attempt to identify the source of this discrepancy; results
of these experiments are presented next.

Effect of Electron Transfer Kinetics. It is clear from
Figure 6 that the observed relaxation is much slower than pre-
dicted by diffusion control. One possibility would therefore
appear to be that the relaxation is kinetically controlled
rather than diffusion controlled. It is reported in the
literature that ks for the ferric oxalate system is greater than

1 cm/sec. Using this value of ks and Reinmuth's more general

Figure 6.

33

Comparison of theoretical and experimental
relaxation curves.

Curve A: experimental, ferrous-ferric oxalate
Curve B: theoretical assuming diffusion control

Curve C: theoretical with simultaneous kinetic
and diffusion control

Parameter: D0 = DR = 6.3 x 10"6 cmz/sec

c0 .19 x 10"3 a

CR = .17 x 10-3 13
Ks = l cm/sec

c = 30.6 x 10"6 f/cm

1

:3
II

a = .5

AE

AE-

0.3

 

 

 

1—

t, m sec.

Figure 6

35

equation6 for a simultaneous kinetic and diffusion controlled
relaxatior, Curve C in Figure 6 is obtained. This curve on the
time scale of Figure 6 differs from the diffusion controlled
case (Curve B) by only 2%. Thus, either the value of ks used
to calculate Curve C.is incorrect, or else the discrepancy
cannot be explained by a kinetically controlled relaxation. In
fact, to approach the relaxation rate observed experimentally
ks would have to be some three orders of magnitude smaller than
the literature value. Independent measurements in this

‘ laboratory15 have shown that this is not the case.

Errors in To and TR- Another possible source of the

 

discrepancy between theory and experiment would be large errors
in the values of To and TR used to calculate the theoretical
curve. Since errors in To and TR would have to be very large
(22.90%) to account for the discrepancy, the most likely sources
are the bulk concentrations and double layer capacitance, be-
cause these quantities are squared in the expressions for To
and TR' Since the iron oxalate used to prepare solutions was
weighed on an analytical balance, and concentrations were check-
ed polarographically, errors in concentration cannot account

for the necessary errors in r and 1R. Nevertheless, double-

0

36

layer capacitance was determined by performing coulostatic
experiments on blank solutions —— i.e., solutions containing
only supporting electrolyte. The double layer capacitance

was then determined by dividing the number of coulombs by the
potential reached in the relaxation. The number of coulombs

was determined by recording the potential across the 100 ohm
resistor in series with the cell (see Figure 5) and then
integrating the resulting current-time curve. An average value
of 29.5 ufd/cm2 was obtained by this method. Second, to ensure
that the presence of depolarizer had no appreciable effect on
the double-layer capacitance, the method of measuring the slope
of the initial charging curve described earlier was employed.
These measurements must necessarily be performed at very short
times, where considerable ringing was observed, and therefore
these measurements are less precise than those above. Neverthe-
less, with this method an average value of 28.7 ufd/cm2 was
obtained. Finally, to determine the potential dependence of

the double-layer capacitance, cyclic voltammetry on a blank
solution was performed. Typical results are shown in Figure 7.
When it is recalled that for this curve double-layer capacitance

is simply i/(dE/dt), where dE/dt is constant, it apparent that

Figure 7.

37

Stationary electrode polarogram for supporting
electrolyte without depolarizer.

Species: potassium binoxalate
Concentration: . 15 M
Scan rate = 165 v/sec

area of electrode = .051 cm2

38

 

 

 

Figure 7

. '-- .
q

T ,__._-._
W
+—-
l V l 1 1 J
-.125 ‘-.15 -.2o -.25 -.30
L_ E 223 SCE, volts

 

39

over the small potential range covered in the relaxation
experiments, double layer capacitance is constant. The
average value of the double layer capacitance calculated from
Figure 7 is 30.6 ufd/cmz. Thus, the results of all these
experiments make it very unlikely that errors in doublerlayer
capacitance can account for large errors in To and 1R.

Effect of finite current injection time. Theoretically
it is assumed that current injection occurs in zero time.
Experimentally, this is impossible and therefore the effect of
finite injection time was investigated as a possible source of
error.

This source of error would apparently describe at least
the direction of discrepancy observed between theory and
experiment. Thus, if appreciable Faradaic reaction were to
take place during the charge injection process, then after
cessation of the impulse there would already be some concen-
tration polarization at the electrode surface. This fact would
require that depolarizer diffuse to the electrode from further
out in the bulk of the solution. This in turn would make the
relaxation slower than predicted by the theory, which implicitly

assumes the absence of Faradaic reaction during charge injection.

40

To determine the effect of finite charge injection time,
experiments were performed in which the injection time was pro-
gressively decreased from 1.0 to 0.1 u sec. As the injection
time was decreased, the amplitude of the current impulse was
progressively increased so as to keep the total coulombic
content of the pulse constant. The potential time curves were
virtually identical for these experiments so that no trend was
observed on decreasing the pulse duration. Thus, the effect
of finite injection time apparently is not a serious source of
error.

Errors in AEi. The only remaining logical source of

 

error is in the parameter AEi, the electrode potential
immediately after cessation of the current impulse. Thus, AEi
was used in normalizing the experimental data to compare them
with theory, so that large errors in AEi could cause the
apparent discrepancy between theory and experiment. Unfortun-
ately, AEi could not be measured directly because of ringing
observed in the relaxation curves immediately after cessation
of the current impulse. Thus, AEi had to be determined by
calculating from the equation

AEi = a (13)
C

41

where q us the change in total coulombic charge in the double-
layer following the impulse, and C is double-layer capacitance.
Errors in AEi, therefore, could arise either from errors in q
and/or C. The total charge injected is accurately known, and
therefore the only possible source of error in q is if
appreciable charge were consumed by Faradaic reaction during
the charge injection. This possibility, however, has already
been eliminated (see discussion above concerning finite in-
jection time).

Likewise, experiments described above make it unlikely
that errors in C are very large. Finally, although ringing
obscured the initial relaxation, values of AEi estimated from
photographs of relaxation curves agreed with values of AEi
calculated from Equation 13.

Other sources of error. It is believed that the experi—
ments described above cover all logical sources of error. Thus,
the only explanation for the large discrepancy between theory
and experiment appears to be serious inadequancies in the
theoretical model, or previously unknown complications in the
electrochemistry of the iron oxalate system. The latter could

include coupled chemical reactions or adsorption, either of

42

which could account for the observed behavior. To invoke
specific explanations of this type, however, would be entirely
speculative, and serve little purpose at this time.
Effect of Large Overpotentials

The original objectives of this research were to evaluate
the effect of large overpotentials on coulostatic relaxation.
Obviously, in view of the results reported above, conclusions
from large overpotential experiments can not be very meaningful
in terms of the original objectives. Nevertheless, some
experiments were performed using overpotentials as large as
43 mV. Results of these experiments are summarized in Figure 8.
These data show that the rate of relaxation for the iron oxalate
system increases with increasing overpotential. This result is
contrary both to theory and intuition for electron transfer in
an uncomplicated redox couple, and is further evidence that the
electrochemistry of iron oxalate is more complex than previously

assumed.

Figure 8.

43

Relaxation curves for larger perturbations
from equilibrium on the ferrous-ferric oxalate
system.

Curve A: relaxation for 7 mV perturbation
Curve B: relaxation for 26 mV perturbation
Curve C: relaxation for 43 mV perturbation

perameters: CO = .19 x 10"3 M

CR

.17 x 10-3 13.

c = 30.6 x 10-6 f/cm2

0.6

0.3

 

44

Figure 8

11.

12.

13.

14.

15.

LITERATURE CITED

Delahay, P., Anal. Chim. Acta 31, 90 (1962).
Reinmuth, W. H., Anal. Chem., 36, 211 R (1964).
Delahay, P., Ibid., 24, 1267 (1962).

Delahay, P., "New Instrumental Methods in Electro-
chemistry," Interscience Publishers Inc., New York,
N.Y.' 1954' P. 35.

Nicholson, R. 5., Anal. Chem., 21 667 (1965).
Reinmuth, W. H., Ibid., 24 1272 (1962).

Randles, J. E. B., and Somerton, K. W., Trans. Faraday
Soc. 48, 937 (1952).

Lauer, G., Anal. Chem., 28, 1277 (1966).
Daum, P. H., and Enke, C. G., Ibid., 653 (1969).

Weir, W. D., and Enke, C. G., J. Phys. Chem. ll,
275 (1967).

Daum, P. H., Ph.D. Dissertation, Michigan State
University, 1969.

Alberts, G. S., and Shain, 1., Anal. Chem., _2, 1859
(1963).

Olmstead, M. L., and Nicholson, R. 5., Ibid. 38, 150
(1966). """" '—

Kolthoff, I. M., and Lingane, J. J., "Polarography,"
Interscience Publishers, Inc., New York, N. Y., 1941,
P. 168.

Nicholson, R. 8., personal communication, 1970.

45