AN A. C. POLARQ‘ GRIAPUIC STUDY OF AQUEOUS Cd (ml
EFFECTS 0?: LARGH AMHUTUDES AND DOUBLE-LAYER
UV "Q‘Rl- \ITEUHS

Thesis; far the Degree 0f £531.31 "
MECH' -AN.. W“ EbNWERSFY
lAMES DE? NES MCLEAN
1957

   

LIBRARY L

Michigan State
University

   

THESJS

 

This is to certify that the
thesis entitled

An A.C. Polarographic Study of Aqueous Cd(II).
Effects of Large Amplitudes and Double-Layer Corrections.

presented by

James D. McLean

has been accepted towards fulfillment
of the requirements for

Ph.D. degree infihemism

Major professor

A c 1 1967
Date “8“ ’

 

ABSTRACT

AN A.C. POLAROGRAPHIC STUDY OF AQUEOUS Cd(II). EFFECTS
OF LARGE AMPLITUDES AND DOUBLE-LAYER CORRECTIONS

by James Dennis McLean

Existing a.c. polarographic equations with which the
kinetic parameters, a, ka,h’ for the electron transfer
process can be readily calculated, were derived with the
assumption that the amplitude of the applied a.c. poten-
tial would not exceed 8/n mv. Experiments were performed
with aqueous solutions of Cd(II) in various supporting
electrolytes at amplitudes from 5 to 50 mv so that the

effect of increasing amplitudes on values of k and a,

a,h
calculated with the aid of the low amplitude equations,

could be tested.

Apparent heterogeneous rate constants, k evaluated

1
from cot ¢ versus w

a,h’
/2 plots for 5.0 x 10-4M Cd(II) in 1.0M
H2804 at 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50 mv were
0.15, 0.14, 0.13, 0.14, 0.14, 0.15, 0.16, 0.15, 0.16, and
0.18 cm/sec, respectively. Charge-transfer coefficients,

a. for the same system at the same reSpective amplitudes
were 0.22, 0.21, 0.20, 0.20, 0.22, 0.20, 0.22, 0.26, 0.26,
and 0.24. Eor an identical Cd(II) concentration in 1.0M
Na2804, the range in ka,h values was 0.06 to 0.08 cm/sec

and the charge-transfer coefficients varied from 0.24 to
0.32 for the stated amplitudes. It is apparent that if ka h

I

is between 0.05 and 0.20 cm/sec, acceptable values for ka h

1

James Dennis McLean
and a can be determined with the aid of the low amplitude
equations up to the limit of amplitudes tested. Deviations
from linearity in the cot 0 versus w1/2 plots become appar-

ent above 30 mv and 800 Hz at E and above 20 mv and 600

R
1/2
Hz at E1/4 and E3/4. The deviations increase with increas—
ing amplitudes in all cases. Intercepts of one on the cot ¢
axis as predicted by the low amplitude equations, were ob-
tained in the cot 0 versus ml 2 plots at all d.c. potentials
and all a.c. amplitudes.

Studies of 5.0 x 10-4M Cd(II) in 1.0M KNo3 demonstrated
that if ka,h is 0.5 cm/sec or faster, the low amplitude

equations do not yield reasonable ka and a values for

,h
amplitudes greater than 5 mv.

Frumkin double-layer corrections were applied to apparent
heterogeneous rate constants for 1.0 mM aqueous Cd(II) in
0.5M Na2804, 1.0M Na2304, 0.5M H2504, 1.0M H2804, 1.0M KN03,
1.0M NaClO4, 1.0M HClO4, 1.0M KCl, 1.0M NaCl, and 1.0M HCl
and the following corrected rate constants were obtained
respectively: 0.77, 0.41, 1.2, 0.81, 0.96, 1.3, 3.0, 1.6,
1.5, and 1.4 cm/sec. These values have considerably less
spread than the uncorrected values, which have a range of
0.063 cm/sec inZLOM.Na2804, to 1.2 cm/sec in 1.0M KCl. The
only cadmium species in chloride media which yields rate

constant values which agree with the remaining corrected

rate constants, is CdCl+.

3 James Dennis McLean

Maximum and minimum probable values of_AQE, the poten-
tial difference necessary for the Frumkin double-layer cor—
rections on the apparent heterogeneous rate constant were
calculated by assuming reasonable error levels of :6 per
cent for the integral double-layer capacitance, i3 per cent
for the bulk concentration of each ionic Species present in
the solution, and i3 mv for the applied d.c. potential and
the same level for the potential at the point of zero
charge. Assuming also an error of i0.03 units in the

charge-transfer coefficient and i20 per cent in k the

a,h’
probable ranges of the corrected rate constants were calcu—
lated. These ranges overlapped for all supporting electro—

lytes with the exception of 1M Na2S04 and 1M HClO4.

AN A.C. POLAROGRAPHIC STUDY OF AQUEOUS Cd(II). EFFECTS

OF LARGE AMPLITUDES AND DOUBLE-LAYER CORRECTIONS

BY

James Dennis McLean

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Chemistry

1967

ACKNOWLEDGMENTS

The author gratefully expresses his appreciation to
Dr. Andrew Timnick for his guidance and counsel extended
throughout the investigation and the preparation of this
thesis.

Recognition is also due Dr. Richard Nicholson for his
many helpful suggestions and discussions and to Mr. Ben
Paulson for his valuable technical assistance.

The author is grateful to the Socony Mobil Oil Com-
pany, the Electrochemical Society, and to the Department

of Chemistry, for financial aid.

********

ii

VITA

Name: James Dennis McLean
Born: November 23, 1940 in Bay City, Michigan

Academic Career: T. L. Handy High School
Bay City, Michigan
(1954-58)

Bay City Junior College
Bay City, Michigan
(1958-60)

University of Michigan
Ann Arbor, Michigan
(1960-62)
Michigan State University
East Lansing, Michigan
(1962-67)
Degrees Held: B.S. University of Michigan (1962)

Scholarships, Fellowships and Assistantships:

Bangor Township Scholarship to Bay City Junior College
(1958—60)

Michigan Public Junior College Scholarship to the
University of Michigan
(1960-62)

Graduate Assistant at Michigan State University
(1962-65, and 1967)

Socony Mobil Fellowship
(1966-67)

Electrochemical Society Fellowship
(Summer 1966)

Memberships: Phi Theta Kappa, Sigma Xi, American Chemical
Society

iii

TABLE OF CONTENTS

INTRODUCTION . . . . . . . .
THEORETICAL . . . . . . . . . . . . . . . . . .
The Reversible A.C. Polarographic Wave
ThetQuasi—Reversible A.C. Polarographic Wave .
The A.C. Polarographic Wave at Large Amplitudes . .
Double—Layer Effects . . . . .
Activity Effects . . . . . . . . . . . . . . . . .
EXPERIMENTAL . . . . . . . . . . . . . . . . . . . .
Instrumentation . . . . . . . . . . . . . . . .
The A.C. Polarograph . . . . . . .
-Calibration of D.C. Currents '. . . . . . . . . .
Calibration of A.C. Currents . . . . . . . . .
Phase-Angle Calibration . . . . . . . . . . . . . .
Other Equipment . . . . .
Reagents . . . . . . . . . . . . . . . . . . . . .
Mercury Purification . . . . . . . . . . . . .

-Preparation of Solutions . . . . . . . . . . . . .

Evaluation of Cd.l.‘ Ki' and Rt . . . . . . . . . .

Evaluation of k and a . .'. . . . . . . . . .

a,h '
DISCUSSION OF RESULTS . . . . . . . . . . . . . .

R
A~Eva1uation of E1/4. El/z, E3/4 and.D . . . . . . .

Evaluation of the Apparent Heterogeneous Rate

Constant, ka , at Large Amplitudes of the Applied
A.C. PotentiéI . .

iv

Page

11
13
18
20
23
25
26
26
27
31
32
33
35
37
38
39
41
43
44

44

TABLE OF CONTENTS - Continued

=Evaluation of the Charge—Transfer Coefficient,

a, at Large Amplitudes of the Applied A.C.
Potential . . . . . . . . . . . . . . . . .

Evaluation of AmE . . . . . . . . . . . . . .

Evaluation of kh, the Corrected Heterogeneous
Rate Constant . . . . . . . . . . . . . . .

Activitvaffects . . . . . . . . . . . . . .
SUGGESTED FURTHER-WORK . . . . . . . . . . . . .
SUMMARY AND coubLUSIONS . . . . . . . . . . .
LITERATURE CITED . . . . . . . . . . . . . . . .

APPENDIX . . . . . . . . . . . . . . . . . . . .

Computer Programs . . . . . . . . . . . . .
Program A.C. . . . . . . . . . . . . . . . .
Program Frumkin . . . . . . . . . . . . . . .

Program Correct . . . . . . . . . . . . . . .

Page

56

70

71
77
79
82
87
91
92
92
97
100

TABLE

II.

III.

IV.

V.

VI.

VII.

VIII.

LIST OF TABLES

Page

R _
E1/2 and D Values for 5.0 x 10 4M Cd(II) in

various supporting electrolytes . . . . . . .

Experimental values for C and Rt’ Constants

d.l.
for evaluation of ka h Th various supporting
electrolytes . . . . . . . . . . . . . . . .

Variation of the apparent heterogeneous rate
constant with changing amplitude of applied a.c.
potential for 5.0 x 10-4M Cd(II) in various
supporting electrolytes evaluated at Efi/z .

Variation of the charge-transfer coefficient
with changing amplitude of applied a.c. poten-
tial for 5.0 x 10-4M Cd(II) in 1.0M H2804 . .

Variation of the charge-transfer coefficient
with changing amplitude of applied a.c. poten-
tial for 5.0 x 10“M Cd(II) in 1.0M Na2804.

-Experimenta1 values for Cd 1 and Rt’ constants
for evaluation of a in various supporting

electrolytes . . . . . . . . . . . . . . .

Values of the applied d.c. potential, E, the
potential at the point of zero charge,th, and
the integral double-layer capacitance, Ki'
used to calculate ADE for 1.0mM Cd(II) in
various supporting electrolytes . . . . . . .-

Comparison of apparent and corrected hetero-
geneous rate constants for 1.0mM’Cd(II) in

various supporting electrolytes . . . . . . .

Probable ranges for values of corrected hetero-
geneous rate constants for 1.0mM Cd(II) in
various supporting electrolytes, calculated

from error estimates . . . . . . . . . . . .

vi

45

52

53

57

58

59

72

73

76

LIST OF FIGURES

FIGURE Page
1. .Essential components for an a.c. - d.c.

polarograph . . . . . . . . . . . . . . . . . 28

2. Schematic of the A.C. MODE . . . . . . . . . 29

3. Schematic of the potentiostat switched to

the calibrating mode . . . . . . . . . . . . 30
4. Polarographic cell . . . . . . . . . . . . . 34
5. Auxiliary and reference electrodes . . . . . 36

6. Variation of the corrected phase-angle with
the square root of angular frequency for
5.0 x 10'4M Cd(II) in 1.0M 142504 at E =

d.c.
R
E1/2 and with an applied a.c. potential of
10 mv O O I O O O O O O I O O O O O O O O O O 46

17. Variation of the corrected phase-angle with
the square root of angular frequency for 5.0 x
10"‘M Cd(II) in 1.0M sto4 at Ed c.
with an applied a.c. potential of 20 mv . . . 47

:'E1/2 and

8. Variation of the corrected phase—angle with
the square root of angular frequency for 5.0 x
10“M Cd(II) in 1.0M H2304 at Ed c.
with an applied a.c. potential of 30 mv . . . 48

= E1/2 and

9. Variation of the corrected phase-angle with the
square root of angular frequency for 5.0 x 10-4M
. _ R .
Cd(II) in 1.0M H2804 at~Ed c. - 31/2 and With
an applied a.c. potential of 40 mv . . . . . 49

10. Variation of the corrected phase-angle with the
square root of angular frequency for 5.0 x 10-5M
Cd(II) in 1.0M H2504 at Ed c. R
-an applied a.c. potential of 50 mv . . . . . 50

=E1/2 and with

vii

LIST OF FIGURES - Continued

FIGURE

11.

12.

13.

14.

15.

16.

17.

Variation of the corrected phase-angle with the
square root of angular frequency for 5.0 x 10-?M
Cd(II) in 1.0M KNO3 at-Ed.c. = Efi/z and with
applied a.c. potentials of 5, 30, and 50 mv .

Variation of the corrected phase—angle with the
square root of angular frequency for 5.0 x IO-IM
Cd(II) in 1.0M sto4 at Ed C

an applied a.c. potential of 10 mv . . . . .

' E1/4 and with

Variation of the corrected phase-angle with the
square root of angular frequency for 5.0 x 10-‘M
Cd(II) in 1.0M H2504 at Ed c

an applied a.c. potential of 20 mv . . . .

= E1/4 and with

Variation of the corrected phase-angle with the
square root of angular frequency for 5.0 x 10-‘M
Cd(II) in 1.0M H2304 at Ed c. - E1/4 and with

an applied a.cs potential of 30 mv . . . . .

Variation of the corrected phase-angle with the

square root of angular frequency for 5.0 x IO-IM
Cd(II) in 1.0M 112304 at Ed 0.
an applied a.c. potential of 40 mv . . . . .

" '31/4 and With

Variation of the corrected phase-angle with the

sqdqre root of angular frequency for 5.0 x IO-IM
Cd(II) in 1.0M H2504 at Ed C.
an applied a.c. potential of 50 mv . . . . .

= E1/4 and with

Variation of the corrected phase-angle with the
square root of angular frequency for 5.0 x 10-4M
Cd(II) in 1.0M H2804 at Ed C

an applied a.c. potential of 10 mv . . . . .

= E3/4 and With

viii

Page

55

60

61

62

63

64

65

LIST OF FIGURES - Continued

FIGURE

18.

19.

20.

21.

.Cd(II) in 1.0M H2804 at E

.Cd(II) in 1.0M sto4 at E

Page

Variation of the corrected phase-angle with the
square root of angular frequency for 5.0 x 10-4M
d.c. = E3/4 and With

an applied a.c. potential of 20 mv . . . . . . 66
Variation of the corrected phase-angle with the
square root of angular frequency for 5.0 x 10-4M
Cd(II) in 1.0M sto4 at Ed c
an applied a.c. potential of 30 mv . . . . . . 67

= E3/4 and with

Variation of the corrected phase-angle with the
square root of angular frequency for 5.0 x 10-4M
d.c. ='E3/4 and With

an applied a.c. potential of 40 mv . . . . . . 68

-Variation of the corrected phase-angle with the

square root of angular frequency for 5.0 x lo-IM
Cd(II) in 1.0M sto4 at Ed c
an applied a.c. potential of 50 mv . . . . . . 69

= "Ea/4 and With

ix

INTRODUCTION

2

Many electrochemical tebhniques currently in use are
based at least in part on d.c. polarography. Heyrovsky (1)
developed this technique in 1922 after studying the behavior
of dropping mercury electrodes. Using the ideas and methods
of Heyrovsky, electrochemists began to examine the total
electrode process by determining the effects of mass trans—
fer to and from the electrode, and electron transfer. Up
to the early 1950's, d.c. polarography was developed for
both qualitative and quantitative analytical applications,
for determining whether an electrode process was reversible,
for evaluating the number of electrons involved in a reduc-
tion or oxidation step, for determining whether step-wise
reduction occurred, for complexation studies, and for evalu-
ating diffusion coefficients.

Since many electrode processes were found to be irre-
versible, electrochemists began to study kinetic effectsto
determine causes of irreversibility. Eyring and Koutecky
were leading pioneers in this field.

Eyring (2) applied the absolute rate theory to elec-
trode processes, and deve10ped the general current-potential
relationship for electrochemical studies in terms of the
kinetic parameters ka,h and a. -The heterogeneous rate con-
stant, ka,h' in cm/sec, is the apparent rate constant for
charge-transfer at the standard electrode potential. The
charge-transfer coefficient, a, represents the fraction of
the potential that favors the forward reaction, and (1 - a)

represents the fraction which favors the backward reaction.

3

Koutecky (3) derived equations for irreversible pro—
cesses which indicated that a and ka,h could be evaluated
by d.c. polarography. In practice, however, applications
were limited and difficult. To extend the evaluation to
faster electron transfer reactions, newer electrochemical
techniques such as the galvanostatic, potentiostatic,
faradaic rectification, and faradaic impedance methods,
cyclic voltammetry, and various forms of a.c. polarography
were deve10ped.

A.C. polarography was first used in 1938 by Mfiller
.EE°.§A- (4) to evaluate d.c. half-wave potentials by noting
the d.c. potential at which the alternating current had
minimum distortion.

‘The first correct interpretation of the electrode
process in the presence of an applied a.c. potential was
reported in 1941 by Grahame (5). He concluded that the
large apparent change in double-layer capacity in the
potential range between the start of the reduction of Cd(II)
and the peak of the wave, was due to the electro-reduction
of Cd(II) and re-dissolution of Cd in phase with the ap-
plied a.c. potential.

The earliest equations reported for the evaluation of
rate constants from faradaic impedance measurements were
derived independently in 1947 by Randles(6) and Ershler (7).

A more complete discussion of the developments in a.c.
polarography up to 1962 is recorded in a monograph by
Breyer and Bauer (8). ~Smith (9), in 1966, presented a

very complete theoretical discussion of the field.

4

Alternating current polarography is an electrochemical
technique in which the a.c. and d.c. potentials applied to
the polarographic cell are controlled and the resulting
alternating current and its phase-angle are measured at a
selected d.c. potential, with the amplitude of the alterr
nating potential held constant. With this technique, only
reversible and quasi—reversible electrode precesses for the
reaction

0 + ne 2:3_ R
can be examined, since irreversible processes yield no
measurable a.c. waves.

From the theoretical expressions, it becomes apparent
that a and ka,h can be evaluated for small amplitudes of
the applied a.c. potential by a.c. polarography from a1-
ternating current measurements. Such currents can be
measured accurately, but the final evaluation of the kinetic
parameters is difficult. The parameters can be more
directly evaluated by measuring the phase-angle between the
applied a.c. potential and the faradaic alternating current.

An experimental instrument was constructed in this
laboratory by Frischmann (10), with which exactly known
constant amplitude a.c. potentials of selected values could
be impressed between the electrodes, and with which phase-
angles could be precisely and accurately measured.

Considerable effort has been expended in the develop-
ment of theoretical equations for the a.c. polarographic

wave which are valid for large amplitudes of the applied

5
a.c. potential. From the complexity of the equations of
Matsuda (11) and Smith (9) it is apparent that it would
be time consuming and difficult to treat large amplitude
a.c. polarographic data exactly.

During the course of this investigation several papers
were published concerning the electrochemical behavior of
Cd(II). Various explanations were offered for the wide
range of values for the apparent heterogeneous rate con-
stants reported in the literature for aqueous Cd(II) in
various supporting electrolytes. :Lingane and Christie (12)
attributed the differences between perchlorate media and
sulfate media to unknown chemical complications or to in-
adequacies of the potential-step method when applied to
fast reactions. Bauer (13) claimed that the potential at
the outer Helmholtz plane governed the values of the rate
constants in various supporting electrolytes, but he gave
no quantitative proof. He discussed chloride, sulfate and
nitrate media. (Hamelin (14) concluded that the differences
were due to complications of the electrochemical reaction
by specific adsorption of the supporting electrolyte. -He
studied chloride, bromide, iodide, sulfate and perchlorate
media.

AFrischmann and Timnick (15) recently evaluated the ap-
parent heterogeneous rate constants for aqueous Cd(II) in
sulfate, nitrate, perchlorate, and chloride media. They found
that the variation in rate constants paralleled the varia-
tion in double-layer capacity, but'they made no double-layer

corrections.

6

Frumkin (16) and his school developed double-layer
corrections in 1933 to account for rate constant differences
observed while studying the reduction of hydrogen ions in
various media. Frumkin's corrections were based on the
Stern modification of the classical Gouy-Chapman theory (17).
In 1947, Grahame (18), presented an extensive critical re-
view of the existing information on the electrical double-
layer and included double-layer capacitance measurements
which he made as well as contemporary measurements made by
others. In 1953, Gerischer (19) pointed out the necessity
of correction for the double-layer in a.c. electrolysis.
Gierst (20), in 1958, investigated the influence of the
double-layer structure in detailed d.c. polarographic studies
of the reduction of chromate ion, and several other related
processes. In 1966,.M0hilner (21) presented a very complete
theoretical discussion of the elements of double-layer
theory.

This work was undertaken to test the extent of ap-
plicability of existing a.c. polarographic equations develop—
'ed for low amplitudes by determining the effect of large
amplitudes on experimentally determined ka,h and d values
for aqueous solutions of Cd(II). Furthermore, it became
apparent that Frumkin double—layer corrections should be
applied to ka,h values obtained at low amplitudes, to de—
termine whether double-layer effects are significant in
explaining the variation of ka,h values for various support-

ing electrolytes. Therefore such corrections were undertaken.

THEORETICAL

Before proceeding with discussion of the theoretical

section which will require the statement of numerous equa-

tions, the definitions of symbolism employed are tabulated

below for ready reference.

A:

Cdiff.

0
II

M
’r?
n

l"‘
(1.
II

electrode area

differential capacity of the electrical
double-layer

measured capacity of the electrical double-
layer

equivalent series capacitive component of
faradaic impedance

concentration of species 1

initial concentration of species i

surface concentration of species i

diffusion coefficient of species i
amplitude of the applied a.c. potential
applied d.c. potential

standard electrode potential

reversible half—wave potential
instantaneous applied potential
potential at the point of zero charge
activity coefficient of Species i
Faraday's constant

total faradaic current (cathode current
postive)

d.c. faradaic current

31
ll

7:
u

9
fundamental harmonic faradaic alternating
current
apparent heterogeneous rate constant for
charge-transfer at E0
heterogeneous rate constant for charge-
transfer at E0, corrected for Frumkin
double-layer effects
heterogeneous rate constant for charge-
transfer at E0, corrected for Frumkin double-
layer effects and for activity effects
integral capacity of the electrical double-
layer
number of electrons transferred in the
heterogeneous charge-transfer step
ideal gas constant
equivalent series resistance of the indi-
cating electrode
equivalent series resistive component of
faradaic impedance
equivalent series resistance of the solution
total uncompensated series resistance
time
absolute temperature
distance from the electrode surface
ionic charge of species i, with its Sign
faradaic impedance (absolute value)

charge-transfer coefficients, 6 - (1 - a)

AD

10
dielectric constant
corrected phase-angle between the applied
a.c. potential and the fundamental harmonic
faradaic alternating current.
potential at the outer boudary of the
Helmholtz layer (plane of closest approach)
potential in the bulk of the solution at
the outer boundary of the diffuse double-
layer
difference of potential between E and m

H S

angular frequency

11

The Reversible A.C. Polarographic Wave

An ideal reversible process is one in which the charge-
transfer rate is so rapid, that the current is purely dif-
fusion controlled and the charge-transfer is described by
the thermodynamic Nernst equation. For the system repre-
sented by the reaction

0 + ne ::3_ R

the partial-differential equations and initial and boundary
conditions for diffusion to a stationary planar electrode
have been developed by Smith (9) as follows:

Fick's law for the oxidized and reduced species:

 

 

ac 52c
0 o
= D (1)
5E” 0 5x2
' 2
$33. = DR 5 CR (2)
t 5x2

Initial conditions:

for t = 0, any X.

c0 = c3 (3)
CR = CR (4)
Boundary conditions:
for t > 0, x -o 00,
CO -—¢ CB (5)
CR -—¢ C; (6)

For t > 0’ X = 0!

ac BCR _ i(t) (7)

D O=—D _.
0 5x R 5x nFA

Equations (1) and (2) imply the assumptions that Fick's law

12

is applicable to.each of the species independently, that
coupled chemical reactions exert no influence, and that
electrode curvature and motion relative to the solution are
negligible. Equation (7) assumes that each reacting species
is soluble either in the solution or electrode phase and
that adsorption effects are negligible.

Other equations necessary for the development of the
final equations which define the reversible a.c. polaro-

graphic wave are

c0 = CHER/DOW? exp (ghee) 41%).]; (a)

and

E(t) - AE sin at (9)

= Ed.c.
Equation (8) is a form of the Nernst equation. Equation
(9) defines the total applied potential for a.c. polaro-
graphy.

Application of the method of the Laplace transforma-
tion to Equations (1) through (7) yields expressions which
together with Equations (8) and (9) finally yield expres—
sions for the faradaic alternating current and the faradaic
impedance. To Obtain these expressions, however, it is
necessary to consider only sufficiently small amplitudes
of the applied alternating potential (AE fi.%-mv), to permit
linearization of the complex exponential forms encountered.
Through the above described mathematical operations the

following equations for the alternating current and impedance

have been derived by Breyer and Hacobian (22), Delahay and

13
Adams (23), Matsuda (11), and Tachi and Senda (24):
2'3 * 1/2
n F.ACO(wDO) AE

I(wt) - 4RT COShz (j/2) sin (at + w/4) (10)

 

 

I<.t> = I... sin (wt + ./4) <11)
2 a
zf 4:T2co:h (34%: (12)
n F ACO(wDO
in which
_ AF. R
' RT (Ed.c. ' El/z) (13)

From Equation (10), it can be seen that the faradaic cur-
rent is proportional to the square root of the angular
frequency (w) and that the phase-angle is equal to 450

(U/4) for all frequencies.

The Quasi-Reversible A.C. Polarographic Wave

A quasi—reversible electrode process is one in which
the current is diffusion and charge-transfer controlled.
Theoretical treatment of the a.c. polarographic wave for
this case, requires only minor modification in the mathe-
matical formulation of the reversible case. Since the
assumption of Nernstian behavior is no longer valid, Equa—
tion (8) is replaced by the absolute rate expression,
which is

i t

nF
= c exp [- L(E(t) - E°)]
nFAka,h OX=o RT

 

(1 - o)nF
_ CRX=0 exp l: RT (E (t) — EOE] (14)

14

By employing Equations (1) through (7) and Equations (9)
and (14), rigorous equations describing the a.c. polar-
ographic wave were derived by Matsuda (25). Smith (26)
slightly modified Matsuda's equations for the controlled
a.c. potential technique. Both-Matsuda's and Smith's solu-
tions are limited to small amplitudes of the alternating
potential.

Smith's expression for the faradaic alternating cur-

rent for a plane electrode is

1/2
2

m 1/2 2 sin(dt + ¢)
(1 + (1 +9—%-—) (,5)

I(wt) = Irev.F(t)

for which Irev is defined by Equations (10) and (11), and

[(ae'j - 5),,1/2 00(t)
1 + .

 

 

E(t) = (16)
ka,h
k f
A = 783%; [e—QJ + em] (17)
1/
¢ = cot-1 ( 1 + ZwK 2 ) (18)
i (t)
00(t) “'1,- (19)
nFACBDO 2
6 a
f — f0 fR (20)
_ B a
D — DO DR (21)

5 = 1~- a (22)

15

Delmastro and Smith (27) considered Spherical dif—
fusion for both the reversible and quasi-reversible cases.
It is necessary to apply correction factors for the faradaic
current expressions for both cases. However, the expres—
sions relating the phase-angles to the apparent heterogene-
ous rate constants are unaltered, regardless of whether
diffusion is to a plane or spherical electrode or whether
the reduced phase is soluble in the electrode.

Evaluation of the apparent heterogeneoUs rate constant
(ka,h) and charge—transfer coefficients (a,5), by either
Equation (15) or Delmastro and Smith's (27) modified equa-
tion, from an a.c. polarogram would be extremely cumber—
some. However, the mere knowledge of ¢ , the phase—angle,
allows one to determine readily the desired parameters as
will be shown from the following considerations.

By substituting Equation (17) into Equation (18), one

obtains

1/‘
cot ¢ = 1 + - (2&9; 2. . (23)

ka,h f(e-Q‘J + e63)

 

Equation (23) predicts that a plot of cot ¢ versus wl/z at
a fixed d.c. potential should yield a straight line with

an intercept at cot ¢ = 1 and a

 

1/2
slope = . (Zggj Bj (24)
ka,h (e + e )
where
k' = k f (25)

a,h a,h

16
It can also be seen that as ka,h approaches an infinite
value, cot 0 will approach unity (e = 45°) as predicted
for the reversible case by Equation (10L

The apparent heterogeneous rate constant, ka un-

,h

corrected for activity effects is the term calculated in
this work.

For the special case where j = 0, that is the ap-
plied d.c. potential is equal to the reversible half-wave
potential, one can calculate from the slope and D, defined
in Equation (21), a value of ka h by the following simple

I

expression
i/2
ka h - (D) 1] (26)
' slope (2) 2

 

The charge-transfer coefficients, (0:5) can be calcu-
lated from the d.c. potential dependence of the cot ¢ versus
1
w /2 plots. The variation of cot 0 with changing frequency
15 Obtained for two applied potentials, Ed.c.,1 and Ed.c.,2'
From Equation (23) it is apparent that the values of the

lepes of cot 0 versus wlé? from the plots at Ed c 1 and

Ed c 2 may be written (if activity effects are neglected)

: 1/2 _ - - I
Slope 1 J£%1—-- (e ajl + e531) (27)
a,h

i/ - a .
BEL—.2; (erajz + e532) (28)

slope 2
ka,h

Thus, one can write

 

 

slo e 1 _ (e-ojg + ij2) _ e-ajz(1 + e32) (29)
810pe 2 (6-031 + e531) e-aj1(1 + e31)

17

 

or
slope 1 = e-d(j2-ji) (1 + e12) (30)
slope 2 (1 + e31)

Taking the natural logs of both sides of Equation (30)

 

 

 

 

gives
1n g%ffifi3€% = ln EEEEfEEE) - o(j2 I ji) 1 (31)
Rearranging yields
: 2(.303 . log 1 tie—3:2 - -—E——Sl° e 1) (32)
32 ‘ 31) 1 + e31 slope 2

 

 

 

The above equations are derived Solely for faradaic
current, that is, current arising from the electro-chemi—
cal reaction. However, current from the polarographic cell
consists of faradaic current and double-layer capacitive
currents. To calculate the faradaic current from experi—
mentally Obtained total a.c. currents, it is necessary to
select an equivalent circuit for the polarographic cell.
Several models are possible, since the double-layer capaci-
tance may be placed either in series or in parallel with
the faradaic capacitance. Delahay (28) has Shown that the
same final results are obtained, regardless of the model

selected. The following series equivalent circuit is

selected, since calculations are simplified:

 

18
The polarographic current must pass through resistance
offered by the solution and the indicating electrode. Rf
and Cf are the faradaic impedances, Cd.l. is the capacity
of the double-layer, RS is the resistance of the solution,
and Re is the resistance of the indicating electrode.

To evaluate kinetic parameters for the electrode
processes, the faradaic phase—angle must be known. Any
measured phase-angle must be corrected for the effects of
double-layer capacitance and total uncompensated series
resistance. Bauer and Elving (29) derived the following

relationship to calculate the corrected phase-angle of the

faradaic current:

 

 

 

Al-(cos 0' - éfi-Rt) (33)
cot ¢ = AB A

M AIR t1 2AIR

ZE' Sin 0 -dcd l. (1 + ( AE ) - AE cos ¢ )

where AI is the amplitude of the alternating current
measured and ¢' is the measured phase-angle between the
applied alternating potential and the resulting alternating

current.
The A.C. Polarographic Wave at Large Amplitudes

Matsuda (11) derived the first expression for an a.c.
polarographic wave which is applicable even when very large
amplitudes are applied. MatsUda's equation is valid only
for reversible a.c. polarographic waves for the amplitude
of the entire complex periodic signal, including higher

harmonics. Bauer derived several formulas for large

19
amplitudes for the special case of very small frequencies
(ka,h >>IJE5), but these are of very limited use and apply
only to the reversible case (30).
Smith recently derived equations for the fundamental-

harmonic component for a quasi-reversible system valid for

 

larger amplitudes (AE.: 35/n mv) and ka h > 10—2 (9).
i/ 2
nzeAC*D 211E 2
- 0 0 EEAE. .

 

A1 + (nEAE/RT)2 A2

 

sin wt + cot- 2
B15+ (nFAE/RT) 32

 

 

where
_ (2w)1/2 (1 + 21/22) ,

A1 - 1/ 2 (35)

4 cosh2 (j/2)[1 + (1 + 2 22) 1
1/

Bi = (2m) 2 1/ 2 (36)
4 cosh2 (j/2)[1 + (1 + 2 2z) ]

A2 = [(2d>)1/2 256 cosh3(j/2)[1 + 3.4142 + 3.82822 +

-1
4.82823 + 2.0024] x [ 1 + (1 + 21/2z)2i] } x

[ 8(d3 + S3)(1 + 4.8282 + 8.65522 + 10.24123 +
8.82724 + 2.82825) x cosh (j/2) - 12(ae-j/2—5ej/2)

(2a —1) x (1 + 4.8282 + 9.51722 + 10.51723 +

20

-j/2 _, j/2 2
6.35624 + 1.88525 + 12(a:osh(j/g§ ) (1 + 4.3572 +

 

6.10422 + 4.16123 + 1.33324) — 12(n2e‘3/2 + SZeJ/z)

(1 + 4.3572 + 7.71322+-6.71323 + 2.00024)} (37)

1
B2 = ((2w) /3 256 cosh3 (j/2)[1 + 3.4142 + 3.82822 +

1/ 2 -1
4.82823 + 2.0024] x [1 + (1 + 2 2z) i] } x
[8(a3 + 53)(1 + 3.4142 + 3.82822 + 4.82823 + 2.0024)
cosh (j/2) - 12(oze-j/2 - Bel/2)(2a - 1) x (1 +

2.7472 + 3.74722 + 2.55223 + 0.86224) +

12(ae-j/2 _ fiej/2)2'

'2 '3 ..
cosh (j72) (1 + 1.8052 + z + 0.2762 )

12(a2e'3/24—52e3/2)(1 + 2.4712 + 2.60922 + 0.27623 -

0.66724)} (38)

1/2
2 = w

/A (39)

Double-Layer Effects

In the absence of strong interactions between the
electrode material and species in solution (Specific ad-
sorption), electrode processes may still be perturbed
significantly by forces operative within the diffuse
double-layer. These influences were not considered in the

preceding theoretical discussion.

21
The following model will Show the effects to be con—

sidered:

   

 
  
   

l
/ | l
/ ' |
/ 8‘ m | Bulk of
h I Solution
.—4 m I
as A |
-H o |
u o N l
5 8 3 |
s ‘6 0 *I '
m o E I
E‘: E I |
m | Diffuse D
/ f—Double—Layer S
I l

 

 

Distance

The Helmholtz layer boundary represents the plane of
closest approach to the electrode for charged species.
These species are prevented from reaching the electrode
surface by the primary water layer at the electrode sur-
face, and by their hydration spheres.

The potential measured experimentally is the po-
tential difference between the electrode surface and the
bulk of the solution. Since the electroactive species
does not reach the electrode surface, and the concentra—
tion of reacting particles in the double—layer is differ-
ent from the bulk of the solution, the electrical energy
available for assisting or retarding the electrode reac-
tion is less than the measured value. The potential dif—

ference between the plane of closest approach (TH) and the

22
bulk of the solution (as) can not be measured experimentally.
Therefore, to correct the rate constants for these effects,

it is necessary to calculate a value for (EH - m~ -The

S)'
following equation based on the Gouy-Chapman theory is

given by Delahay 25, a1. (31):

Ki[(E - Ez) - (EH - 53)} =

 

(40)
. . ) v.
+ RTe 2 CS ex _ ziF(DH fi?§__ _ 1 }
‘ 2w 1 p RT .
Since the equation is non-linear in the variable (DH - ms).

it can not be solved exactly. .Equations of this type can
be solved by numerous graphical and numerical methods of
analysis. -The well-known Newton-Raphson numerical method
(32) is selected, Since it converges rapidly to give good

approximations of the values of (m ,- m Calculations

’ H S)'
are performed for each supporting electrolyte with the aid
of Program Frumkin which is described in the Appendix. -To
calculate (DH - ES) values from Equation (40), it was as-
sumed that the integral double-layer capacitance-Ki was
equal to the differential capacitance Cdiff . This is true
for the limiting case (21), as the d.c. potential ap-
proaches the point of zero charge, Ez,that is

11m.Ki = Cdiff.(Ez) (41)
E *9 E
z
The differential capacitance is equal to the experimental

double-layer capacitance divided by the area of the electrode.

Using the calculated values for (D -~m

H S)' and the folloWing

23
equation developed by Frumkin as presented by Smith (9),
it is possible to calculate corrected values for the ap-
parent heterogeneous rate constants, making the assumption

that sinusoidal variations in ADE may be neglected.

(an - 20)FA6%] (42)

ka,h = kh exp[ e RT
where

ADE = EH - Us (43)

These calculations are performed with the aid of Program

Correct which is described in the Appendix.

Activity Effects

The heterogeneous rate constants for cadmium in vari-
ous supporting electrolytes can be corrected for activity
effects according to the following formula (9):

kh = fi f: k; (44)
While activity coefficients for the reduced form (cadmium
amalgam) are known, it is difficult to estimate reasonable
values for the activity coefficients of the oxidized forms.
Since the supporting electrolytes employed were either
0.5M or 1.0M, Debye-Hfickel type calculations can not be
used. Furthermore, to make calculations, it is necessary
to know the exact form of the oxidized species. This is
not readily apparent, since competing complexes are present

in some cases, and the electroactive Species present in the

double-layer may not be the same as the prevalent cadmium

24
species in the bulk of the solution. In view of these

facts, activity corrections were not attempted.

EXPERIMENTAL

25

26
Instrumentation

The instruments employed and their applications in
this investigation were as follows:

Sargent S-30260 potentiometer to measure all d.c.
potentials applied to the cell:

Tektronix Model 502 oscilloscope to measure the amp-
litude of the a.c. potential applied to the cell and to
monitor the tuning of the frequency sensitive amplifier;

Sargent Model S.R., one second full scale deflection,
recorder to measure the maximum d.c. currents, rectified
a.c. currents, and signals proportional to the phase-angle;

Hewlett—Packard Model 202A signal generator to provide
all a.c. potentials;

Hewlett-Packard Electronic Counter Model 521A to
calibrate the frequency of the Hewlett-Packard signal
generator;

Philbrick Model R-300 power-supply to provide up to

300 ma of r 300 v d.c.

The A.C. Polarograph

The a.c. polarograph employed was constructed and
described in detail by Frischmann(10). The instrument
employed Operational amplifiers and was similar to one con—
structed by Smith, (33) except for modifications in the
phase—angle detecting circuit which permitted direct linear

response recording of the phase-angle to an accuracy of

27

r 0.50, with no high or low level current limitations.
Figures 1, 2, and 3 show block diagrams of essential com-
ponents of the instrument. The main components of the
potentiostat consist of operational amplifiers A, B, and
C shown in Figure 1. A.C. MODE and D.C. MODE are the units
utilized for measuring a.c. or d.c. Signals. Amplifier D
attenuates the signal to the recorder.

The d.c. potential source, at Ed.c.’ supplied 0 to
i 3 volts with a long term instability of at most 0.1 mv.
From this source, any desired potential could be obtained
by adjustment of a 10 turn Helipot. The Sargent potenti-
ometer was used to monitor the desired potential before
each measurement.

The a.c. potential source provided ranges of 0 to
5.00 i 0.05 mv or 0 to 50.0 i 0.5 mv, and any desired
potential could be obtained by adjustment of a 10 turn
Helipot. The a.c. potential was monitored with the Tek-
tronix oscilloscope before each experiment. The frequency
of the signal generator was calibrated with the aid of the
electronic counter and typically corresponded to within
0.6% of the dial reading in the frequency range of 30 to
12009H2.

The d.c. scanning unit, ES ,was not used in this

can

investigation.

Calibration of D.C. Currents

For d.c. mode Operation (Figures 1, 3) a 100K resistor

28

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31
is introduced between amplifiers C and D. This makes
amplifier D a unity gain inverter.

For current calibration, the electrodes are taken out
of the circuit and connection to the precision 100K re-
sistor, R3, is made as shown in Figure 3. By applying
Zero volts across R8, the zero current level on the re-
corder is adjusted. A d.c. potential, dependent on the
current range to be measured, is applied across R3 to at-
tain full scale recorder deflection. After the above cali-
bration step, the electrodes are switched into the circuit,
and the current flowing through the polarographic cell at

a selected potential is recorded.

Calibration of A.C. Currents

To measure fundamental a.c. signals free from noise,
amplifier F (Figure 2) must be tuned sharply to the fre-
quency of the signal to be measured. This condition is
attained by selection of the proper values of R1, R2, and

R3. The role of R1 in determining the frequency is obvious

_!._..
27TR1C

parallel T network, R1 must equal R2 and R3 must equal

from the familiar expression, :1 = . In the twin
R1/2 for sharp tuning. R1,.R2 and R3 are 10 turn.Helipots.
The frequency of the a.c. source is adjusted until the out-
put of amplifier F is at a maximum. 'R4, R5, R3, or R7
(Figure 3) is then selected to provide the desired current
range. A selected a.c. potential is applied across the

chosen resistor and the recorder is adjusted to full scale

32
,deflection. rTo set the low end of the recorder scale, one-
tenth of the selected potential is applied across the re—
sistor and the recorder is adjusted to yield one-tenth of
the full scale deflection. The low end of the recorder
scale is not adjusted to zero reading because of the con-
ductance limitations of the diodes employed in this circuit.
Thus the working range of the recorder scale extended from
one-tenth of full scale to full scale (from one inch to 10
inches on the recorder chart paper). A

After the calibration step, the electrodes are switched
into the circuit, and the rectified a.c. current from the

cell is recorded.

PhaseeAngle Calibration

With amplifier F tuned to the desired frequency, the
phase-angle detector is switched in as shown in Figure 2.
.With an a.c. potential applied across the luf capacitor in
the calibrating circuit (Figure 3) the recorder is adjusted
to indicate the 90° phase shift reading, since the a.c. sig-
nal through the luf capacitor leads the applied a.c. po-
tential by 90°. With an a.c. signal across R4 or R5 in the
calibrating circuit, the phase-shift level of zero degrees
is set on the recorder, since the a.c. signal through a
resistor is in phase with the applied potential. After the
calibration step, the electrodes are switched back into the

circuit and the phase-angle from the cell is recorded.

33

The long term stability of the employed commercial
signal generator was about 0.5°, therefore the above cali-
bration was performed before and after each phase-angle
measurement.

The accuracy of the phase-angle detector was tested
at both large and small amplitudes with the following series
RC networks (components of 1% accuracy): 500 ohm and 1.0
uf; 100 ohm and 1.0 uf; 50 ohm and 1.0 uf. The ampli—
tudes employed were 5.00, 25.0, and 50.0 mv. .The phase-
angles measured agreed within experimental error (~:3 per
cent) to the calculated values throughout the 30 to 1100
H2 frequency range. Data were obtained at fifteen differ-
ent frequencies. These RC circuits and frequencies yielded
a current range from 0.93 us to 300 ua and a phase-angle
range from 89.6° to 17.6°.

To perform the experiments at large amplitudes (10.0
to 50.0 mv), it was necessary to decrease the magnitUde Of
the resiStors in the calibrating mode (Figure 3) so that
the recorder could be calibrated for larger currents from
the polarographic cell. This was the only necessary instru-
mentation change, since the instrument was previously pro—

vided with a variable a.c. potential source.

Other Equipment

The polarographic cell employed in this work is shown
in Figure 4. Through the side-arm with the two-way stop-

cock, nitrogen can either be bubbled through the solution

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34

 

 

 

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35
by way of a coarse frit at the bottom of the cell, or
blown over the top of the solution. The cell is fitted
with a 3-hole rubber stOpper through which the dropping
mercury electrode, saturated calomel electrode, and the
auxiliary electrode (platinum coil) are introduced.

The constructional details of the reference electrode
employed are shown in Figure 5. The reference electrode
was separated from the polarographic solution by an iso-
lation compartment which contained the supporting electro-
lyte. Both the reference electrode and the isolation com—
partments were tightly stoppered to minimize any flow of
solution through the frits. Isolating the reference
electrode minimizes contamination of the polarographic
solution by foreign ions from the Side arm of the reference
electrode.

The auxiliary electrode, a platinum wire coil, is
separated from the polarographic solution by a fine frit.
Its isolation compartment always contained the polaro-
graphic solution. The frit prevented the free diffusion

of oxidation products to the dropping mercury electrode.

Reagents

Reagents were employed without further purification.
The chemicals, and sources are as follows:
Cadmium Nitrate, Tetrahydrate Baker's Analyzed Reagent

Disodium Salt of Ethylenediamine Mallinckrodt Analytical
tetraacetic Acid (EDTA) Dihydrate Reagent

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36

 

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37
Hydrochloric Acid

Mercury

Nitrogen

Perchloric Acid

Potassium Chloride
Potassium Nitrate

Sodium,Chloride

Sodium Perchlorate Monohydrate

Sodium Sulfate (Anhydrous)

Sulfuric Acid

Mercury Purification

Baker's Analeed Reagent

Two sources: Chicago
Apparatus Company, Triple
Distilled, Chemically Pure;
Purified in this Laboratory.

Prepurified, 99.996%
The Matheson Company

Baker's Analyzed Reagent

(Matheson, Coleman and Bell

A.C.S. Reagent

Matheson, Coleman, and Bell

A.C.S. Reagent
Baker's.Analyzed Reagent

G. Frederick Smith
Chemical Company

Baker's Analyzed Reagent

Baker's Analyzed Reagent

Mercury which had previously been used for electro-

chemistry was purified by the following process.

Mer-

cury was filtered into a large filter flask to remove gross
contamination, and one liter of 1M nitric acid was then
added. A glass tube inserted through a rubber stopper
located in the neck of the flask extended below the mercury
surface. When suction was applied to the Side arm of the
flask, the air entering through the glass tube provided
vigorous mixing. ~After one day of aeration and nitric acid
washing, the wash solution was removed, and the mercury was

rinsed several times with distilled water. One liter of 1M

38
sodium hydroxide was added to the filter flask and the
washing process continued for one day. The sodium hydroxide
was used to remove grease and other such organic impurities
from the mercury. The wash solution was then removed and
the mercury was again rinsed several times with distilled
water. This process was continued for one week, alternating
the nitric acid and sodium hydroxide treatment each day.
In the final wash, the mercury was rinsed with 1 liter of
distilled water to which a few drops of concentrated nitric
acid were added to prevent oxide formation. The mercury
was separated from the water and the last traces of water
were removed by blotting the mercury surface with filter
paper. It was then filtered twice through pinholes in fil-
ter paper and then stored in special hard glass bottles.
The cleaned mercury was used for both a.c. and d.c. polar-
ography, cyclic voltammetry and stripping analysis. No
anomalous results were obtained in any of these methods,
indicating that the mercury was equal in quality to com-
mercial triple-distilled mercury used for electrochemical
purposes. It was possible to clean up to 35 pounds of mer-

cury at one time, using this method.

Preparation of Solutions

A 0.0100M Cd(II) stock solution was prepared from
Cd(N03)°4H30 and standardized by EDTA titration (35).
The supporting electrolyte solutions were prepared by

weighing solids, or pipetting electrOIYte solutions, and

39
diluting to volume. The acid supporting electrolytes were
standardized with previously standardized sodium hydroxide.

The sample solutions were prepared by appropriate dilu-
tion of the cadmium nitrate stock aliquots to which the de-
sired amount of electrolyte had been added. None of the
solutions contained maximum suppressor.

Half of the solutions were prepared from the laboratory
distilled water supply and half were prepared from triple-
distilled water. Duplicate experiments run on similar solu-
tions prepared with water from the two sources yielded,‘
within experimental error, no different results.

Dissolved oxygen was removed from all solutions in the
polarographic cell by bubbling with nitrogen for a period
of 20 minutes. .Nitrogen was also swept continuously over
the surface of the solution during the duration of an ex-
periment. Last traces of oxygen were removed from the
nitrogen by bubbling it through an acid vanadous sulfate
solution, dilute sodium hydroxide, and water, before passing
it into the polarographic cell.

~All polarograms were run at 24 1 1°.

Evaluation of C . Ki and R

d.1. t

The determination of the capacity of the double-layer
(Cd 1 ) and total uncompensated resistance (Rt) is ac-
complished by applying a 5 mv a.c. potential of known fre—

quency to the cell containing only supporting electrolyte,

40
and measuring the amplitude of the resulting a.c. current
and its phase-angle.
Since the capacity of the double-layer changes with
the d.c. potential and the electrode area, current and phase~
angles are determined at maximum drop size at each d.c.
potential of interest.

The capacity of the double—layer is calculated by

AI

 

Cd.1. = wAE sin ¢ (45)
and total series resistance by
R = QE-cos ¢ (46)

t AI

The evaluation of Cd.l. and Rt was performed at.
fifteen different frequencies. It was found that the values
obtained were independent of frequency within experimental
error. However, since the value of cos ¢ changes rapidly
at phase-angles greater than 88°, only the values at fre-
quencies greater than 100Hz were used. This resulted in
averaging ten values to obtain Cd.l. and Rt in each support—
ing electrolyte at each d.c. potential.

The above experiments were also performed with an ap-
plied a.c. potential of 50.0 mv for the large amplitude
studies. .The data thus obtained, were used to evaluate

and R for the various supporting electrolytes. Since

‘Cd.1. t
the values obtained at large amplitudes agreed within ex-

perimental error to the values Obtained at small amplitudes,

41
the values of Cd.l. and Rt determined at 5 mv were used for
all calculations at all amplitudes.

To evaluate Ki' the value of the integral double-layer
capacity per unit electrode area, it was necessary to deter-
mine the area of a mercury drop, at maximum drop size, in
each supporting electrolyte, at the desired d.c. potential.
This was accomplished by weighing 20 drops of mercury de-
livered by the capillary in the desired supporting electro-
lyte, and then calculating the drop area assuming that the

drop is a sphere, using a value of 13.54 g/cc for the density

of mercury at 24° (36).

Evaluation of ka h and a

Values for the apparent (uncorrected) heterogeneous

rate constants, (k were calculated with the aid of

a,h)

Equations (21) and (26). The slope necessary for use in
Equation (26) was determined from the cot 0 versus wl/z
plot (Equation (23)) evaluated at a d.c. potential cor-
responding to Efi/z. The data necessary for this plot were
determined by measuring the a.c. current and the phase-angle
at fifteen different frequencies between 30 and 1200 Hz.

All measurements were made at maximum drop size. The ex-
periments were performed with a.c. potentials of 5.00, 10.0
15.0, 20.0, 25.0, 30.0, 35.0, 40.0, 45.0, and 50.0 mv. All

phase-angles were corrected for double—layer capacitance

and total uncompensated series resistance effects by

42
Equation (33), using Program AC which is described in the
(Appendix.' With this program the least squares slopes for
cot 0 versus w1/3 plots of these corrected phase—angles
can also be calculated.

Values for the<dnrge~transfer coefficients (a,B) were
calculated with the aid of Equations (22) and (32). The
slopes necessary for use in Equation (32) included the
slopes evaluated above at Elli/2 , and lepes evaluated in
the same way from similar experiments carried out at E1/4

and E3/4.

DISCUSSION . OF ‘RESULTS

43

44

Evaluation of.E1/‘, Efi/z’ E3/4 and D

 

Values of the experimental half-wave potential, Efi/z,
and the calculated diffusion coefficient, D, for aqueous
Cd(II) in various supporting electrolytes are tabulated in
Table I. The values of E1/4, Efi/z, and E3/4 used in this
work were those determined by Frischmann (10) from d.c.

polarograms. The values were obtained from the E versus

1 -i d.c
log i plots at points where the intercepts were equal

 

to 0.477, 0, and -0.477. E1/4 and 33/4 were displaced 14
mv from EE/s in all media.

The values of Do, the diffusion coefficient of the
oxidized species, (Cd(II)), in various supporting electro—
lytes, necessary for the calculation of D, were determined
by Frischmann (10). The values were calculated from d.c.
polarographic data, with the aid of the Lingane-Loveridge
equation (37). Employing these values and the value of DR’
the diffusion coefficient of the reduced species, (Cd(Hg)),
of 1.61 x 10-5 cmz/sec obtained by Bauer and Elving (38),

values for D were calculated with the aid of Equation (21).

Evaluation of the Apparent Heterogeneous Rate Constant,

3a,]' at Large Amplitudes of the Applied A.C Potential

1 R
Typical plots of cot E versus w /2 at Ed c = E1/2 at

various amplitudes of the applied a.c. potential are shown
in Figures 6-10, for 5 x 10-‘M Cd(II) in 1.0M sulfuric

R
acid. The data were obtained at 31/2: since maximum

45

R _
Table I Ei/z and D Values for 5.0 x 10 4M Cd(II) in

various supporting electrolytes.

 

 

. R
Supporting Electrolyte E1/2 versus SCE,v D* x 106 cmz/sec

 

 

1.0M Nazso4 -0.606 5.51

1.0M H2504 -0.597 7.54

1.0M KN03 -0.585 , 10.20
*D a DB D9

46

 

 

 

 

 

 

 

 

 

 

/
2.00 93 '
O /
/
Q’/
1.80 G/
/
1.60 ‘
cot ¢ ‘/
1.40L
0 AE = 5 mv
1.20 '. ----- AE - 10 mV
1.00 i l I l
O 20 40 60 80
1 -
w /2, radiansl/2 sec 1/2

Figure 6. Variation of the corrected phase-angle with the
square root of angular firequency for 5.0 x 10 4M Cd(II) in

1.0M H2504 at E

of 10 mv. d.c.

- E1/2 and with an applied a.c. potential

47

 

2.00»

1.80‘.

1.60‘

cot ¢

1.40

1.20

 

 

 

 

1.00 l l I I
0 20 ‘40 60’ 80

wl/z, radiansl/z sec-l/2

Figure 7. Variation of the corrected phase-angle with the
square root of angular frequency for 5.0 x 10-4M,Cd(II) in
1.0M H2804 at Ed.c. = E1/2 and with an applied a.c.
potential of 20 mv.

48

 

 

 

 

/
<5/
(s/
2.00&——
/
C).
/
/'
/
1.80~—— /
'/
/
/
1.60-—— /
[D
/
COt O . /
——- /
1.40 ‘ '—————'AE = 5 mv
p; ----- AE - 30 mv
1.20"_ o .
q
1.00 1 l l 1
0 20 40 60 80

wl/z, radiansl/2 sec-l/z

Figure 8. Variation of the corrected phase-angle with the
square root of angular frequency for 5.0 x 10-4M Cd(II) in
1.0M 11,304 at E R
of 30 mv.

d c = E1/2 and With an applied a.c. potential

49

 

2.00 __.
/__9
.. . ,’@ O
1.80 b— . 9
/<6
2
fl/
___ -/'
1.60
./
/'
cot 0 //
0 —"—_AE = 5mv
1.40 L— .
----- AE = 40 mv
.)
/

1.20 -- '

 

 

1.00 i I l J

0 20 4O 60 80

 

wl/z. radiansl/a sec-1/2

Figure 9. Variation of the corrected phase-angle with the

square root of angular grequency for 5.0 x 10-‘M.Cd(II) in

1.0M H3804 at Ed.c. = El/g and with an applied a.c. poten-
tial of 40 mv.

50

 

 

 

 

2.00r——-
1.80 ——. ’-
9’9 g
(9' '
/'
2’
1.60 “— 9,@
O/
cot ¢ //
ZG
1.40 "“ (2’
‘ /
‘ 7/
' /’
. /’ -—__—'AE = 5 mv
___ /'
1.20
' / ----- AB = 50 mv
./
///
1.00 l l l |

wl/z, radiansl/2 sec-'l/2

Figure 10. Variation of the corrected phase—angle with the
square root of angular frequency for 5.0 x 10-4M Cd(II) in
1.0M H2504 at E
tial of 50 mv.

d.c. - 31/2 and with an applied a.c. poten-

51
deviation in quantities measured to evaluate ka,h would be
expected at this d.c. potential. Several supporting elec-
trolytes were employed, to obtain a variation in the value
of ka,h' The values necessary for correcting the phase-
anglasused in these plots are listed in Table II for both
large and small amplitudes.

Table III lists the variation of the apparent hetero—
geneous rate constant with changing amplitude of the applied
a.c. potential for 5.0 x 10-‘M.Cd(II) in various supporting
electrolytes, as well as the values of the lepes from the
cot 0 versus wl/z plots used in the calculations. -From
Table III it is apparent that if ka,h is between 0.05 and
0.20 cm/sec, its value, as determined by the low.amplitude
eqUation does not vary markedly to the upper limit of
amplitudes tested.

This is true even though the slopes demonstrate some
variation, since ka,h values are not strongly sensitive to

changes in the slope. Although the ka values show good

,h
agreement, some deviations in the data become apparent above
30 mv as is shown in Figures 9 and 10. These deviations

occur as a distinct falling off from linearity in the cot ¢
versus wl/z plots at frequencies above 800 Hz, with the
magnitude of the error increasing directly with the frequency.

In all of these plots, the intercept is still one on the

cot 0 axis at zero frequency, as is predicted by Equation (23).

Table II. Experimental values for C

52

d.l.

and.R , Constants

t

for the evaluation of ka h in various Supporting

electrolytes.

I

 

 

Supporting Electrolyte

~Cd.1.' ”3*

 

10 mv 100 mv

*
Rt , ohms

10 mv 100 mv

 

1 .0M N32304
1.0M H2804

1 .0” .m03

0.833 0.834
0.952 0.956
1.14 1.14

77.2 78.7
58.9 60.4
76.4 78.0

 

* R
Determined at El/z-

53

 

 

 

 

 

 

Table III. Variation of the apparent heterogeneous rate
constant with changing amplitude of applied a.c.
potential for 5.0 x 10-2M.Cd(II) in various
supporting electrolytes evaluated at Efi/z.

1.0M N32304: 1.0M sto4 1.0M K1103
cm/sec cm/sec cm/sec
5 2.69 0.06 1.32 0.15 3.66 0.62

10 2.28 0.07 1.35 0.14 2.11 1.07

15 2.40 0.07 1.45 0.13 2.41 0.94

20 2.24 0.07 1.38 0.14 2.39 0.95

25 2.26 0.07 1.37 0.14 2.43 0.93

30 2.21 0.08 1.30 0.15 (2.18 1.04

35 2.20 0.08 1.21 0.16 1.82 1.24

40 2.14 0.08 1.26 0.15 1.46 1.55

45 2.03 0.08 1.21 0.16 1.50 1.51

50 2.13 0.08 1.06 0.18 1.13

2.00

 

54

In a recent monograph (9), Smith presents theoretical
a.c. polarograms and a.c. pOlarographic phase-angles,for a
Specific hypothetical system for AE of 5, 20, 30, and 40 mv.
He demonstrates that cot 0 should decrease with increasing
values of AE at the d.c. potential equal to Efi/z. ~Although
the magnitudes of these changes cannot be compared directly
with the experimental values listed above due to differences
in constant parameters involved, one can see that the
changes follow the same pattern. Smith's theoretical figk
ures, as well as the above experimental data indicate that
the changes are relatively small if the apparent heteroe.
geneous rate conStant is in the range 0.01 to 0.20 cm/sec,
as is the case foer(II) in the various sulfate media.

-Apparent heterogeneous rate constants for Cd(II) in
1.0M KN03 calculated from slopes obtained from Figure 11
show much greater variation than the values obtained in
sulfate media. This is primarily due to the fact that the ‘
heterogeneous rate constant in this medium is considerably
faster, and thus changes in the slope produce a much greater
change in the value of ka,h' Rate constants of the order
of 1.0 cm/sec or faster are difficult to determine accur-
ately by the technique of a.c. polarography. Figure 11
data were included to demonstrate what might be expected
if large amplitudes are employed on these systems. Although
the values of ka,h Show deviation as AE increases, no
deviation from expected behavior, that is linearity in the

cot 0 versus w1/2 plot with a resulting intercept of one on

55

 

A. AB = 5 mv
B, AB = 30 mv A
1.30 _ CI AE = 50 mV
‘ .A
‘ B
1.20 r- A
‘ o.
0‘ C
cot 0 ‘,a‘

 

 

 

I I

0 20 40 ' 60 80

1.00

wl/z, radiansl/2 sec—l/2

Figure 11. Variation of the corrected phase-angle with the
square root of angulaerrequencyifor 5.0 x 10-4M.Cd(II) in

1.0M KN03 at Ed.c. = El/zgand with applied a.c. potentials

of 5, 30, and 50 mv.

56
the cot ¢ axis.at zero frequency, is observed until AE ex-
ceeds 30 mv. At this point, the cot ¢ intercept at zero
frequency becomes greater than one, the value predicted by
theory. -Again, the error becomes increasingly large as AE

increases.

Evaluation of the Charge-Transfer Coefficient, a, at Large

Amplitudes of the Applied A.C. Potential

The values of the charge-transfer coefficient, a. ob-
ACOt ¢)

1
Am 2
d.c. potentials are tabulated in Tables IV and V as well

tained from the ratio of slopes ( at two different
as the values of the slopes used in the calculations. The
Cd 1 and Rt values used to correct the phase—angles to
evaluate these slopes are listed in Table VI. Typical
plots of cot 9 versus wl/z, necessary for the evaluation

of the above slopes, at Ed.c. =‘E1/4 andEd = E3/4 at

various amplitudes of the applied a.c. potential are shown
in Figures 12-21 for 5 x 10-4M Cd(II) in 1.0M sulfuric acid.
From Tables IV and V it can be seen that a values, as
determined by the low.amplitude equation, do not vary
markedly up to an dmplitude of 40 mv. (Although the a values
.show good agreement, some deviations in the data become
apparent above 20 mv, as is shown in Figures 13-16, and
18-21. These deviations occur as a distinct falling off
from linearity in the cot 0 versus wl/z plots at frequencies

above 600 Hz, with the magnitude of the error increasing

directly with the frequency. In all of these plots, the

57
Table IV. Variation of the charge-transfer coefficient with
changing amplitude of applied a.c. potential for

5.0 x 10'4M Cd(II) in 1.0M H2804.

 

 

 

AE SlgpeEj/103, SlgpeE:/:Oz, da .ab.

5 8.71 1.57 0.23 0.22
10 8.29 1.58 0.20 0.21
15 8.00 1.46 0.22 0.20
20 8.21 71.59 0.20 0.20
25 8.26 1.54 0.21 0.22
30 8.27' 1.58 0.20 0.20
35 8.37 1.54 0.22 0.22
40 8.23 1.40 0.25 0.26
45 8.31 1.39 0.26 0.26
50 7.76 1.38 0.23 0.24

 

aObtained from the ratio of slopes at E

Equation (
b

of slopes at E1

tion (32).

32).

3/4 and E1/4, see

Obtained from an average of a's evaluated from the ratio
/2Iand 31/4: and-E3/4 and E1/2' see'EqUa_

58

-Table V. Variation of the charge-transfer coefficient with
changing amplitude of applied a.c. potential for

5.0 x 10"M Cd(II) in 1.0M Na2804.

 

 

 

AE SlgpeEf/ioz, .SlgpeEZ/ioz, aa Aab
5 -1.77 .3.15 0.24 0.24
10 1.50 2.58 0.25 0.25
15 1.48 2.69 0.23 0.25
20 1.45 2.55 0.24 0.24
25 1.59 2.73 0.25 0.25
30 1.55 2.60‘ 0.26 0.26
35 1.49 2.42 0.28 0.28
40 1.52 2.43 0.28» 0.28
~45 1.49 2.32 0.30 0.30
‘50 1.62 2.41 0.32 0.32

 

3Obtained from ratio of slopes at E3/4 and.E1/4, see
Equation (32).

bObtained from an-average of a's evaluated from the ratio

of s10pes at E1/2,and.E1/4 and at E3/4-and.E1/2, see
Equation (32).

59

 

 

 

 

Table VI. Experimental values for Cd 1 and Rt’ cOnstants
for the eValuation of a in various supporting
electrolytes.

* *
S t' E1 t l t E1/4 E8/4
. uppor ing ec ro y e
Cd 1.uf -Rt,ohms Cd.l.’”f Rt,ohms
1.0M st04 0.966 56.0 0.912 56.0
1.0M‘Na2504 0.844 78.7 0.816 74.3

 

*All values listed at E3/4, as well as the values for 1.0M
H2804 at E1

44 were obtained with a different D.M.E. This
D.M.E. was

lso used to obtain the other data necessary
for the evaluation of a in these media.

60

 

 

 

 

/’
/'
/’/’
1.60 —— /®
o/(9
1.40 L— '
cot ¢ //
1.20*—— o "‘ AE - 5 mv
o ----- AB = 10 mv
1.00 | l J L

(01/2,-radiansl/2 sec-l/2

Figure 12. Variation of the corrected phase-angle with
square root of angular frequency for 5.0 x 10-5M.Cd(II)
1.0M H3804 at E
tial of 10 mv.

the

in

d c = E1/4 and with an applied a.c. poten-

61

 

 

 

 

 

1.60 h—-
”®
//,—” (D
@o G
/@
1.40 —- G)
. /
cot ¢ /’
. /
1'20 "' . AB = 5 mv
‘ ----- AB = 20 mv
1.00 J I l J
0 20 40 60 80
wl/z, radiansl/2 secul/2

Figure 13. Variation of the corrected phase—angle with the
square root of angular frequency for 5.0 x 10'4M Cd(II) in
1.0M H2SO4 at E
tial of 20 mv.

d c = E1/4 and with an applied a.c. poten-

62

 

1.60‘

 

 

 

 

 

1.40‘
cot ¢
1.20‘
1.00 -
0 20 40 60 80
wl/z, radiansl/2 sec-.l/2
Figure 14 .

Variation of the corrected phase-angle with the

square root of angular frequency for 5.0 x 10-4M Cd(II) in
1.0M H2504 at E

d.c. : E1/4 and With an applied a.c. poten-
tial of 30 mv.

63

 

1.60

1.40,

cot ¢

1.20

 

 

1.00

 

 

wl/z, radiansl/2 sec-l/2

Figure 15. Variation of the corrected phase-angle with the
square root of angular frequency for 5.0 x 10-4M.Cd(II) in

1.0M H2804 at Ed c = E1/4 and with an applied a.c. poten-
tial of 40 mv.

64

 

 

 

 

 

1.60
”E
//
/ G) @
VG) <9
/
1.40— ,/
cot 0 G '//
C) /'
C) /’
1.20... (D //
'---'AE = 5 mv
G
----- AB = 50 mv
1.00 J l l |
O 20 4O 60 80

wl/Z, radiansl/2 sec—l/2

Figure 16. Variation of the corrected phase-angle with the
square root of angular frequency for 5.0 x 10-4M Cd(II) in

1.0M H3804 at Ed c = E1/4 and with an applied a.c. poten-
tial of 50 mv.

65

 

2.20-—-

1.80—

cot ¢

1.40....

 

 

 

1.00 I J l I

0 20 40 60 80

 

radiansl/2 secul/2

Figure 17. Variation of the corrected phase-angle with the

square root of angular frequency for 5.0 x 10_4M Cd(II) in
1.0M H2504 at E

d c = E3/4 and with an applied a.c. poten-
tial of 10 mv.

66

 

 

 

 

 

 

2.20‘
‘1)
. . C)
1.60 99’
./
cot ¢ . /'
. AB = 5 mv
1.40_—_ a
' ----- AB = 20 mv
1.00 I l l l
0 20 40 60 80
wl/z, radiansl/2 secul/2
Figure 18.

Variation of the corrected phase-angle with the

square root of angular frequency for 5.0 x 10_4M Cd(II) in

1.0M.HZSO4 at Ed c ='E3/4 and with an applied a.c. poten-
tial of 20 mv.

67

 

2.20

 

1.80 _

cot ¢

1.40 -—-

 

 

1.00 | | l l

0 20 40 60

 

80

wl/z, radiansl/2 sec-l/2

Figure 19. Variation of the corrected phase-angle with the

square root of angular frequency for 5.0 x 10-4M Cd(II) in
1.0M 112504 at E

d c = E3/4 and with an applied a.c. poten_
tial of 30 mv.

68

 

2.20

1.80

cot ¢

1.40

 

1.00

 

 

 

Figure 20. Variation of the corrected phase-angle with the

square root of angular frequency for 5.0 x 10-4M.Cd(II) in
1.0M H2804 at E

d c = E3/4 and with an applied a.c. poten—
tial of 40 mv.

69

 

2.20‘L—-

 

 

 

 

 

1.80 ”'-
cot ¢
1.40 —
1.00 it I I
0 20 40 60 80
wl/z, radiansl/z sec-1/2
Figure 21.

Variation of the corrected phase-angle with the

square root of angular frequency for 5.0 x 10-‘M Cd(II) in
1.0M H2804 at Ed c '=

E3/4 and with and applied a.c. poten-
tial of 50 mv.

70‘
intercept is one on the cot ¢ axis at zero frequency, as
is predicted by Equation.(23).

By comparing Figures 12—21 to Figures 5-10, it is ob-
vious that although the deviations from linearity in these
plots occur at lower amplitudes and frequencies at E1/4 and
33/4 than at Efi/z, the changes in the slope
smaller at E1/4 and.E3/4. 'This leads to the conclusion
that acceptable values of a Can be obtained at large ampli-
tudes even though a is more sensitive to changes in the
slope of cot ¢ versus wl/z plots than is ka,h’ ‘Smith's (9)
plots of cot ¢ versusEd c. for a hypothetical system at
amplitudes of 5, 20, 30, and 40 mv predict smaller slope
changes at E1/4 and E3/4, than at Efi/z, in agreement with
the results obtained in this study. These conclusions are
only valid for charge-transfer coefficients of systems
having apparent heterogeneous rate constants in the range
0.01 to 0.20 cm/sec as is the case for Cd(II) in the vari—
ous sulfate media.

Charge-transfer coefficients for Cd(II) in 1.0MMKNO3
were not evaluated, since the ratio of the uncertainties
in the slopes became competitive with the values of the

ratio of the slopes.

~§yaluation of AQE

Values of the applied d.c. potential, the potential

Q

at the point of zero charge and the integral double-layer

capacitance used to calculate Am for 1.0mM Cd(II) in

E

71

various supporting electrolytes, are tabulated in Table VII.
The values of E listed are equal to Efi/z for 1.0 mM Cd(II)
in each supporting electrolyte, respectively, since the ap-
parent heterogeneous rate constants were evaluated at this
d.c. potential. The concentrations employed in the calcu—
lation of AER were the bulk concentrations of each ionic
species expressed in moles/cm3 and the charges (zi) were

the ionic charges with their respective signs.
Evaluation of kt' the Corrected Heterogeneous Rate Constant

Table VIII lists the uncorrected and corrected values of
the heterogeneous rate constants for 1.0mM Cd(II) in various
supporting electrolytes, as well as the AEE and charge-
transfer coefficients used for the corrections. The values

of ka and a listed in Table VIII were obtained in a pre—

,h
vious study in this laboratory by Frischmann (10). Low
amplitude a.c. potentials (5 mv) were employed and the ex-
periments and calculations were performed exactly as de-
scribed in this work. The value of 20 necessary for the
calculation of the corrected heterogeneous rate constant is
the ionic charge (with the appropriate sign) of the oxidized
form. The oxidized form was assumed to be Cd++ in all sup—
porting electrolytes except the chloride media,.where the
various chloride complexes of cadmium were considered.

From Table VIII, it is apparent that the only cadmium

species which gives a value for the corrected heterogeneous

rate constant which correlates with corrected values in the

72

 

 

 

 

Data from Conway (39).

cData from Kolthoff and Lingane (40).

Table VII. Values of the.applied d.c. potential, E, the
potential at the point of zero charge, Ez, and
the integral double-layer capacitance, Ki’ used
to calculate AmE for 1.0mM Cd(II) in various
supporting electrolytes.

giggiigigge Ea versus SCE,v ~Ez versus SCE,v 'Ki,uf/cm2

0.5M Na2804 -0 .604 -0 .427b 22 .0

1.0M Na2SO4 40.606 —0.427b 22.6

0.5M 92304 -0.594 -0.427b “23.2

1.0M sto4 -0.597 -0.427b 25.7

1.0M KN03 —0.585 -0.573C 30.9

1.0M H0104 -0.607 -0.544“C 29.5

1.0M Nac104 -0.573 —0.544C 31.5

1.0M KCl -0.642 -0.517b 29.9

1.0M NaCl -0.635 -0.518b 29.1

1.0M HCl —0.644 -0.517b 30.2

aE = Efi/z.

b

73

 

 

 

Table VIII. Comparison of apparent and corrected hetero-k
geneous rate constants for 1.0mM Cd(II) in
various supporting electrolytes.

Supporting Am a,v a ka,h k.k:cm/sec

Electrolyte E cm/sec ~h

0.5M Na2S04 0.0375 0.21 0.076 0.77

1.0M Na2804 0.0301 0.20 0.063 0.41

0.5M 32804 0.0377 0.27 0.14 1.2

1.0M H2504 0.0321 0.24 0.12 0.81

.1.0M.KNO3 0.0108 0.50 0.63 0.96

1.0M Hc104 0.0431 0.26 0.25 3.0

1.0M NaClO4 0.0254 0.31 0.34 1.3

1.0M Kcl 0.0676 0.44 1.2 1.640dc1f)

23. (Cd++)
0.12(CdC12)
0.0084(Cdc1g)
0.0006(CdCl:)

1.0M NaCl 0.0636 0.46 1.2 1.5(CdCl+)
1.0M HCl 0.0687 0.42 0.94 1.4(CdCl+)

 

‘aCalculated with the aid of Equation (40) using data from
Table-VII. ‘

b

Calculated with the aid of Equation (42).

74
other supporting electrolytes, is CdCl+. This is not un-
reasonable, since various cadmium-chloride complexes are
undoubtedly present in the bulk of the solution, and the
species most prevalent in the bulk of the solution, are
not necessarily the Species most prevalent in the elec—
trical double-layer, where the charge-transfer rate is
determined.

It should be emphasized that the corrections which
were made did not take specific adsorption into account.
Therefore if adsorption is an important complication in the
chloride media, the conclusions which were drawn about the
species which undergoes charge-transfer are not valid.
However, no evidence of adsorption comppications was observed
by Frischmann while recording a.c. polarograms with the
chloride media.

From Table VIII, it is also apparent that values of
corrected heterogeneous rate constants for Cd(II) in vari-
ous supporting electrolytes cover a considerably smaller
range than the apparent (uncorrected) rate constants. The
range of uncorrected values is from 0.063 cm/sec for 1.0mM
Cd(II) in 1.0M Na2S04 to 1.2 cm/sec for 130mm Cd(II) in
1.0M KCl, while the range of corrected values in the same
respective supporting electrolytes is 0.41 cm/sec to 1.6
cm/sec.

-Even though the overall range of the corrected values
is definitely smaller than the overall uncorrected range,

several anomalies are obvious in Table VIII. The corrected

75
rate constants in the various sulfate media are not in as
good agreement as would be-eXpected, and the corrected
value of kh in 1.0M HClO4 appears to be too large. These
facts, however, may be explained by the following considera-
tions. -Values for corrected rate constants depend directly
on experimentally determined values of the apparent rate
constants which are at best known within :20 per cent.
Therefore, the fact that uncorrected rate constants for
Cd(II) in 0.5M and 1.0M Na2804 have very similar values
may be fortuitous.

‘Since errors in the values of quantities used to calcu-
late the corrected rate constants would also contribute to
the above deviations, it was decided to calculate a prob—
able range of values for the corrected rate constants in
each supporting electrolyte. -To accomplish this, maximum
and minimum values for AbEwere calculated for each sup-
porting electrolyte, by assuming the following reasonable
error levels in experimentally determined values necessary
for the calculation of AGE: -Ci r 3 per cent, E and E2
i 3 mv,-Ki i 6 per cent.

The stated error levels were employed with the aid of
Equation (40) to calculate the two extreme values for AGE
for various supporting electrolytes listed in Table IX.

n

These values for AmEmax and AEEml , values of Ra h‘i

per cent, and a iCHJ3 units, were employed with the aid

20

of Equation (42) to calculate the probable ranges for

values of the corrected heterogeneous rate constants for

76

 

 

 

Table IX. Probable ranges for values of corrected hetero-
geneous rate constants for 1.0mM Cd(II) in
various supporting electrolytes, calculated from
error ektimates.

Supporting A“Emax'v Ammin,v kfiax,cm/sec kgin’

Electrolyte .E 9E cm/sec

0.5M Na2804 0.0397 0.0353 1.2 0.50

1.0M Na2S04 0.0321 0.0277 0.61 0.27

0.5M H2804 0.0399 0.0354 1.8 0.76

1.0M H2804 0.0341 0.0297 1.1 0.54

1.0M.KN03 0.0178 0.0047 1.6 0.59

1.0M HC104 0.0493 0.0362 6.0 1.5

1.0M NaClO4 0.0327 0.0182 2.6 0.68

1.0M xc1 0.0728 0.0614 2.6(CdCl+) 1.1

45. (Cd++) 12.
0.22(CdC12) 0.061
0.020(Cd01;) 0.0036
0.0018(CdCl:) 0.0002
1.0M NaCl 0.0688 0.0574 2.3(CdCl+) 0.96
1.0M HCl 0.0740 0.0626 2.4(CdCl+) 0.87

 

77
1.0mM Cd(II) in various supporting electrolytes listed in
Table IX.

The ranges of values for the corrected heterogeneous
rate constants for Cd(II) overlap in all cases, except in
1.0M Na2804 and 1.0M.HC104. Even in these media, the
limits of the probable ranges are close to overlapping the
limits of rate constants in the other supporting electro-
lytes. This would indicate that the Frumkin double—layer
effect is an important factor in explaining the reported
ranges of values for aqueous Cd(II) in the supporting
electrolytes considered in this work. Exact agreement
among the rate constants corrected in this manner would
not be expected, since activity effects and migration of
the electroactive species within the diffuse double—layer

also have an effect on the rate of the electron-transfer.
-Activity Effects

Although activity corrections were not attempted be-
cause of the high concentration of supporting electrolytes,
some conclusions can be drawn concerning the results of
such corrections. From Equation (44) it can be shown that
corrections for activity coefficients different from unity
would result in an increase in the corrected heterogeneous
rate constants. Since the activity corrections should be
larger in the various sulfate media than in the perchlorate

or chloride media, the probable overlap in the rate constant

78
values corrected for both Frumkin double-layer effects, and
for activity effects would be more complete than the over-

lap apparent in the values tabulated in Table IX.

SUGGESTED FURTHER WORK

79

80
Since it has been demonstrated that it is possible to
apply considerably larger a.c. potentials than was formerly

accepted, to evaluate k and a with the aid of cot ¢

a,h
versus w1/2 plots for Cd(II), it would also be interesting
to test experimentally, phase-angle dependence on the d.c.
potential, for this system atIa selected low and high
frequency, employing the larger a.c. amplitudes. Plots of
cot 0 versus d.c. potential would be obtained, from which
conclusions regarding the mechanism of the electrode pro-
cess could be drawn. I

If a preceding or succeeding chemical reaction pro-
ceeds at a rate comparable to the charge-transfer rate,
departure from linearity is observed in the cot ¢ versus
w1/3 plots as well as changes in the normal bell-shaped
wave obtained when cot 0 is plotted against applied d.c.
It would be interesting to test at large amplitudes, sys—
tems which demonstrate the kinetic complications mentioned
above at low amplitudes, to establish whether the expected
deviations are enhanced.

The small amplitude a.c. polarographic experiments
for Cd(II) Should be repeated with the supporting eleCtrolyte
concentration reduced to 0.05 to 0.1M. This would allow
an estimation of the activity coefficients for Cd(II) in
-the various supporting electrolytes with the aid of the
extended Debye-Huckel equation. The apparent heterogeneous

rate constants could then be corrected for Frumkin double-

layer effects and activity effects to determine if the

81
rate constant values converge to a single value. *This
work would also demonstrate the effect of supporting elec—
trolyte concentration on the charge-transfer rate.

The small amplitude experiments should also be re-
peated for Cd(II) in supporting electrolytes containing
ions such as I_ or.SCN_ which are readily adsorbed. Ad-
sorption corrections could then be attempted for the
various known adsorbing supporting electrolytes at various
concentrations, to test the validity of accepted adsorp-
tion corrections.

Double—layer corrections including adsorption effects
as well as the Frumkin correction (no adsorption) could
then be made and the corrected rate constants could be
compared to those obtained for media containing ions which

are not strongly adsorbed.

SUMMARY AND CONCLUSIONS

82

83

Experiments were performed with aqueous solutions of
Cd(II) in various supporting electrolytes at amplitudes of
the applied a.c. potential from 5 to 50 mv, to determine
the effect of increasing amplitudes on values of ka,h and
a,;calculated with the aid of the low amplitude equations.

Apparent heterogeneous rate constants evaluated from
cot 9 versus w1/2 plots for 5.0 x 10-‘M.Cd(II) in 1.0M
H2804 at 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50 mv
were 0.15, 0.14, 0.13, 0.14, 0.14, 0.15, 0.16, 0.15, 0.16.
and 0.18 cm/sec, respectively. ~Charge-transfer coeffici-
ents for the same system at the same respective amplitudes
were 0.22, 0.21, 0.20, 0.20, 0.22, 0.20, 0.22, 0.26, 0.26,
and 0.24. For an identical Cd(II) concentration in 1.0M

Na2SO4, the range in ka values was 0.06 to 0.08 cm/sec

,h
and the charge-transfer coefficients varied from 0.24 to
0.32 for the stated amplitudes.
Although acceptable values of a and ka,h were obtained
for the above systems, deviations from linearity in the
cot 0 versus w1/2 plots above 30 mv and 800 Hz at Efi/z and
above 20 mv and 600 Hz at E1/4 and.E3/4 were observed. The
deviations increased with increasing amplitude in all cases.
Intercepts of one on the cot 0 axis as predicted by the
low amplitude equations, were obtained in.the cot ¢ versus
w1/2 plots at all d.c. potentials and all a.c. amplitudes.
Studies of 5.0 x 10_4M Cd(II) in 1.0MMKN03 gave ka,h

values from 0.62 cm/sec at 5 mv, to 2.0 cm/sec at 50 mv.

Intercepts of greater than one on the cot 0 axis were

84

obtained in the cot ¢ versus w1/2 plots, a departure from
accepted low amplitude theory, at applied amplitudes greater
than 30 mv. Charge—transfer coefficients could not be ob-
tained for this system, since the ratio of the uncertainties
in the slopes necessary for the calculation, became com-
petitive with the values of the ratio of the slopes.

‘The results of this study show that if the apparent
heterogeneous rate constant is in the range 0.01 to 0.50
cm/sec, acceptable values of a and ka,h can be obtained
with the aid of low amplitude equations, with no deviation
from theory up to amplitudes of at least 30 mv. This is
considerably larger than the 8/n mv limit of low.amplitude
theory, which would be 4 mv for-Cd(II).

'Frumkin double-layer corrections were applied to ap-
arent heterogeneous rate constants for 1.0mM aqueous Cd(II)
in 0.5M Na2804, 1.0M Na2304, 0.5M H2S04, 1.0M H2804, 1.0M
K1103, 1.0M NaClO4, 1.0M 8010,, 1.0M;KCl, 1.0MNac1, and
1.0M HCl, and the following corrected rate constants were
obtained respectively: 0.77, 0.41, 1.2, 0.81, 0.96, 1.3,
3.0, 1.6, 1.5, and 1.4 cm/sec. These values have consider-
ably less spread than the uncorrected values, which have a
range of 0.063 cm/sec in 1.0M.Na3504, to 1.2 cm/sec in
1.0M KCl. The only cadmium species in chloride media which

yields rate constant values which agree with.the remaining

‘.
O

corrected rate constants, is CdCl+.

_Maximum and minimum probable values of Am the poten—

El

tial difference necessary for the Frumkin double-layer

85

corrections on the apparent heterogeneous rate constant were
calculated by assuming reasonable error levels of :6 per
cent for the integral double-layer capacitance, :3 per cent
for the bulk concentration of each ionic Species present

in the solution, and 13 mv for the applied d.c. potential
and the same level for the potential at the point of zero
charge.

Assuming also an error of i 0.03 units in the charge-
transfer coefficient and i 20 per cent error in ka,h' the
probable ranges of the corrected rate constants were calcu-
lated. These ranges overlapped for all supporting electro-
lytes with the exception of 1MNa2504 and 1M HC104.

Activity corrections were not made, since it is dif-
ficult to Obtain values of the activity coefficient of the
oxidized Species in the various 1.0M supporting electrolytes,
but it is predicted that the corrections would make the
overlap of the rate constants in the various media even
more complete.

The results of this study Show that Frumkin double-
layer corrections cause the range of the apparent hetero-
geneous rate constants for aqueous Cd(II) in the media
investigated, to converge. Thus, other complications such
as preceding reactions or specific adsorption do not sig-
nificantly influence the rate of the charge transfer.

Even though the Frumkin double—layer corrections help
to explain the observed range of rate constants for Cd(II)

in the various media it would appear that Cd(II) is not

86
the ideal choice as an electrochemical test ion, since it
is known that Cd(II) forms various complexes with commonly

used supporting electrolyte anions.

LITERATURE CITED

87

1.
2.

dampen

q

10.

11.
12.

13.
14.

15.

16.
17.
18.
19.
20.

21.

88

J. Heyrovsky, Chem. Listy, léu 256 (1922).

S. Glasstone, K. J. Laidler, and H.-Eyring, "The

Theory of Rate Processes", McGraw—Hill,-New.York,

J. Koutecky, Chem. Listy, 41, 323 (1953).

R. H. Mflller, R. L. Garma, M. E. Droz, and J. Petras,
Ind. Eng. Chem., 19” 339 (1938).

D. c. Grahame, J. Am. Chem. Soc., ESL 1207 (1941).

J. E. B. Randles, Faraday Soc. Discussion, 1, 11 (1947).
B. Ershler, Faraday Soc. Discussion, 1, 269 (1947).

B. Breyer and H. H. Bauer, "Alternating Current Polaro-
graphy and Tensammetry", Interscience Publishers, New

York 1963.

D. E. Smith, "Electroanalytical Chemistry", Vol. I,
Chapu I, A. J. Bard, Editor, Marcel Dekker, Inc.,

*New York 1966.:

J. K. Frischmann, Ph.D. Thesis, Michigan-State University,
East Lansing, Michigan (1966).

H. Matsuda, Z. Elektrochem., 61, 489 (1957).

P. Lingane and J. Christie, J. Electroanal. Chem., 19”
284 (1965).

H. H. Bauer, J. Electroanal. Chem., 12, 64 (1966).
A. Hamelin, C. R. Acad. Sc. Paris, 262, 520 (1966).

J. K. Frischmann and A. Timnick, Anal. Chem., 39” 507
(1967).

A. N. Frumkin, Z. physik. Chem., Alfig, 121 (1933).

O. Stern, Z. Elektrochem.,.§QJ 508 (1924).

D. C. Grahame, Chem. Rev., 41, 441 (1947).

H. Gerischer, z. physik. Chem., 292, 293 (1953).

L. Gierst, Ph.D. Thesis, University of Brussels (1958).

D. M. Mohilner, "Electroanalytical Chemistry", Vol. I,

-Chap. 4, A. J. Bard, Editor, Marcel Dekker, Inc.,
»New York 1966.,

22.

23.

24.

25.
26.
27.

28.

29.

30.
31.

32.

33.
34.

35.

36.

37.

38.

89

B. Breyer and S. Hacobian, Australian J. Chem., 8,
322 (1955).

P. Delahay and T. J. Adams, J. Am. Chem. Soc., 14,
5740 (1952).

I. Tachi and M. Senda, Bull. Chem. Soc. Japan, 2g“
632 (1955).

H. Matsuda, Z. Elektrochem., QgJ 977 (1958).
D. E. Smith, Anal. Chem., 36” 962 (1964).

J. R. Delmastro and D. E. Smith, Anal. Chem., 38”
169 (1966).

P. Delahay, "New Instrumental Methods in.Electro-
chemistry", Interscience, Inc., New York, 1954, pp.
146-8.

H. H. Bauer and P. J. Elving, J. Am. Chem. Soc.,
§Z, 2091 (1960).

H. H. Bauer, Australian J. Chem., lén 13 (1962).

4M. Breiter, M.-Kleinerman, and P. Delahay, J. Am. Chem.

Soc., fig, 5111 (1958).

R. W. Southworth and S. L. Deleeuw, "Digital Computa-
tion and Numerical Methods", McGraw-Hill, New York,
1965, pp. 183-6.

D. E. Smith, Anal. Chem., _§, 610 (1963).

D. D. Deford, "Stabilized Follower Amplifier", Appli-
cations Bulletin, G. A. Philbrick Researchers, Inc.,
Boston, Mass. -

G. Schwarzenbach, "Complexometric Titration", Inter-
science, New York, 1957, pp. 55, 83-86.

"Handbook of Chemistry and Physics", 42nd edition,-C.
D. Hodgman,-Editor, Chemical Rubber Publishing Co.,
Cleveland, Ohio, 1960-1, p. 2144.

J. J. Lingane, B. A. Loveridge, J. Am. Chem. Soc.,
22, 439 (1950).

H. H. Bauer and P. J. Elving, Anal. Chem., 39, 341
(1958).

39.

40.

90

B. E. Conway, "Electrochemical Data", Elsevier Pub-
lishing Co., London, England, 1952, pp. 223-228.

I. M. Kolthoff and J. J.

Interscience Publishers,

Lingane, "Polarography",
Inc., New York, 1941, p. 106.

APPENDIX

91

92

Computer Programs

Many of the calculations reported in this work were
performed on a Control Data type 3600 digital computer.
The programs were written in FORTRAN language, since this
system is widely used and compatible with most modern
computers. The programs and instructions for their use
are listed below. Derivations and references for the

formulas employed in the programs are included in the text.

Program A.C.

This program is used to correct the measured phase—
angle for uncompensated solution resistance and double-
layer capacitance effects. It is also used to calculate
the least squares slope of the resulting data assuming
an intercept of one at zero frequency. The following data
are read in:

Card 1 - IM, ID, IY, which is the date in numbers,

columns 1 — 2,: month

3 - 4 day
5 - 6 = year
Card 2 - N RUN, which is the total.number of sets
of data
columns 1 - 2
Card 3 - SENT, which is a sentence describing the set
of data

columns 1 - 80 - will print out any numbers,

letters or symbols

93
Card 4 - N DATA, which is the total number of data
points in a given set of data

columns 1 - 2
Card 6 - constants for a given set of data

columns 1 — 10, VOLT which is AB in volts in the

text

columns 11 - 20, R which is Rt in ohms in the text.
Card 7 - data

columns 1 - 10,PHIP, which is ¢'in degrees in

the text
columns 11 - 20, OMEG, which is the frequency in
Hz in the text

columns 21 — 30, CURR, which is the current in

. amperes in the text

(must be in E field and right hand justi-
fied e.g. 6.925 x 106 = 6.925E - 006)
columns 31 - 40, D.C., which iS‘the d.c. potential
with its Sign in the text
columns 41 - 50, CAPDL, which is the measured
double-layer capacity,-Cd.l., in
farads in the text
(must be in.E field and right hand
justified)
Cards 8 — etc. - new data, same format as card 7
Repeat cards 3 - 7 if desired

The output involves printing.the date, the sentence, PHIP,

94

and SLOPE, where PHIP is now the angle in radians, OMEG is
the angular frequency, VOLT, R, CAPDL, CURR, and D.C. are
the same as on the data cards, and COTAN is the cotangent
of the corrected phase-angle which is cot ¢ ,in the text,
ANGLE is the corrected phase—angle in degrees, ROMEGA is
the square root of the angular frequency, which is w}/%
in the text, and SLOPE is the least squares lepe with an
intercept of 1.0 for the plot of cot ¢ versus wl/z. —All
of the above numbers are printed in scientific notation
with 6 figures.

The performance of this program was checked by hand

calculation.

95

mz<\. H u Boas

, . -. mmzms\m2meu mz<.
AAmHmmvmmoo*m*.m I .N**m + .Hv*Homao*omzo I AmHmmvmsz4<n mmzma.

Am.I AHHmmvmmoov44 n mama
”7.4" m
aHo>\mmso u a

Amazovmsmom.n.omzom.

omzo*mmHvH. 6*. m I muse
. .owH\mHmm*mmmH¢H. m u aHma

A6. oHH 6. on 6. 6Hm 6. onnv Search
2 HHo> 6m cams

A\\\aomzommm onm
05mm xHH mquzamm _xw zasoomm xcH masons xx HamaummH

om

.MHH .mmH xHH .aHo>mw x6 .aomzomm .xm mHmmmw xvv aasmomomm

mm aszm
A\\\\\NH \mH,.NH \mH .NH maco.ms .xooHV aczmom
sH..oH..sH.Hoo Sszm
OHH
AM\\\\\\\\\m¢n Hovv aazmom

uH .AHvazmmv .Hm sszm

A\\mmHoz< mmamm omaommmoommm xovv Basses
om aszm

A\\\\\\\\HHHV aazroa

asaczuoz

mH Hszm

ANHV Hazmom

«Baez 6H 9am

Amach amazon

AoH.HuH .AHVHZMmV .m cams
zpmz .oH cams

AmHmV eazmom

SH oH esH .oom cams
Aomvs.Aomvx.Aomvazmm onmzmsHo
o¢:zamoomm

How

HN
ON

\I 0

«H

OH

m
N

com

96

UHZOM.UQ

92m,

m.ov.ovAZpuzva

Am.mHm u mmonmsvsazmom
m.moHaszm

A\\\veasmom

HNH

Nxm\ mewxmvu m
Hvx+xmuxm
CKLWW§§W§m

N**A

BZHmm

W+Nxmuuwm

ng H OOH

.oImxm
.ouxm
uwxm

H I_zomz.u zsmz
mazHazoo
H.mv.vams¢ozva
H,I means I
Am.mHmoHv sazmom

mmmHVH.m\mez«*.omH
AmHozavmzaam

mZ¢W H N
GmSGMH

.MHUZ¢..MZ¢ .MMDU .Ham¢U..m .EAO> .UMZO .mHmm .ov

¢H¢Qz

BZHmm
MAGZd
flflwz<

H x.
H+HHH.

mv
”OH

HNH

OOH

,mv

ow

97

.1

Program Frumkin

This program calculates a value for ROOT, which is

(0H - ms)

in:

or AOE in the text. The following data are read

Card 1 - ANAM, which is a sentence describing the set
of data
columns 2 - 80 - will print out any numbers,
letters, or symbols
Card 2 — data
columns 1 - 10, IN, which is the number of guesses
columns 11 — 20, AI, which isKi in uf in the text
columns 21 - 30, POT, which is (E - E2) in volts,
in the text
columns 31 — 40, CP, which is C+s in moles/ems,
in the text
columns 41 — 50, CMH which is Ci_in moles/cm3,
in the text
columns 51 - 60, ZP, which is 2+ in the text
columns 61 - 70, ZM, which is Z__without its Sign,
in the text
Card 3 - etc. XG, the gueases, 1 per card
columns 1 - 10
Repeat dards 1 - 3, if desired
The output involves printing the sentence, the ROOT.
which is the answer for AGE in volts to 9 figures with its

Sign, and GUESS, the value printed as XG on card 3. The

98
output also includes the number of trials necessary to find
the ROOT, or SYSTEM DIVERGES, with the value when.
the number of trials necessary to calculate the ROOT exceeds
99.

The performance of this program was checked by calcu-
lating values reported in the literature by Delahay (32),
using Delahay's data. IAnother check was made-by putting
computer values for AGE back into Equation (40) to be cer-

tain that this made both Sides of the equation equal.

99

as;

os oo
A Boom osz oa mmH vHaononm 6. man mmmpomsxm m. HHmI soommmvaazmom HH
H ox.Hx HH aszm ooH
A mmumm>Ho zmsmsmmmHvaazmom oH
. 6H eszm moH
moH.moH.noHAmmIva
H+HIH
mxIHx HoH
HoH.ooH.ooHA.moooo.IAmx\A AHxImxvvammava
xmeoEoEI qumx _
A HHS: mamomI HxIson: *HHIIxzom
Amm¢>+AHxImNImo. mmIVHHHHIHNIHoIHooI muse. Hvrmma>IHarxmmo
Aerzs.6o. mmvamxmerISUIHooImmso. Hunma>
AAHmm>vmamom\. HVIm. Immas
AsoIAHHISNIoo mmvmmxu420+moIAHx*mN*mo. mmIVHHHHIHUVImoOImmus. NIHmH> moH
HI H.
oxIHx
Av.on vaazmom m
ox.mo¢mm
zH . HHSOOHOQ...
Av.ono.oHHvH¢zmom NH
mooImo.H*HaIH<
2N.mu.zo.mo som H4..zH. «Hoamm
A\m<oH\\\vaazmom v
AoH.HInA ASVSHZHV vsszm moH
moH «Aoo. momva
.. AmmoHWeazmom n
AOH.HIn.Aszaz« .moamm H
AoHvzaza onmzmzHo
szzsmm sarcoma

100

Program Correct

~
0

This program calculates a value for HET, which is kh,
or kg, the corrected heterogeneous rate constant in the
text. This rate constant may be corrected for activity ef-

fectS,-Frumkin doubleelayer effects (kh),

or both (kg).
The following data are read in: .
Card 1 - a sentence describing the set of data
columns 1 — 49 - will print out any numbers, letters,
or symbols
Card 2 - data

columns 1 - 10, AHET, which is k the apparent

a,h'
heterogeneous rate constant
in the text

columns 11 - 20, ON, which is n in the text
oblumns 21 - 30, 20, which is 20, with its Sign,
in the text
columns 31 - 40, DELPHI, which is ADE in volts,
in the text
columns 41 - 50, F0, which is fO in the text
columns 51 - 60, FR, which is fR in the text
columns 61 - 70, A, which is a in the text
Repeat cards 1 - 2 if desired.
The output involves printing the sentence, and the cor-
rected heterogeneous rate constant to Six figures.

The performance of this program was checked by hand

calculation.

101

PROGRAM CORRECT
READ 2 .

FORMAT(49H

IF(EOF,60)4,5

PRINT 2

READ 1 [AHET vQN’I ZOIDELPHII F0: FR IA

FORMAT(7F10.0)

B=1.0-A

HET=AHET/(F0**B*FR**A*EXPF(39.06*DELPHI*(A*QN-ZO)))

PRINT 3,HET

FORMAT( /*CORRECTED HETEROGENOUS RATE CONSTANT =*.F11.5///)
GO TO 1

END