'1'“ t
" ‘ .-. _'

_. , .
..._q t -\ .'?

..
WI
. .7 -‘.~'.~—.\-.
,3! : .~ .. I.” /‘I f‘" ~, ‘ '_
ms 4:2; x’"
— I. .' ' .
i d r
" i- . ,
‘1; n- _v '71" '
o “‘4' l“. 4.1.“; " E'l
‘CAJ ‘9. ‘ ‘ ~‘
is: ' *1 .
9'23"" .‘~‘ ( “F ‘7 ’7 ‘:
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means

This is to certify that the

thesis entitled
TRANSFER COEFFICIENTS AND HETEROGENEOUS RATE
CONSTANTS FOR Cd(II) IN VARIOUS SUPPORTING

ELECTROLYTES EVALUATED BY PHASE-ANGLE
MEASUREMENTS WITH A MODIFIED
A. C. POLAROGRAPH

presented bg

Joseph Karl Frischmann

has been accepted towards fulfillment
of the requirements for

Ph .D . degree in Chemistry

. 7 ‘ ‘
///:f (If? ".1": ‘3’”2’ .7— ? ..’.— 21' {c%

Major professor

Date March 7, 1955

0-169

 

ABSTRACT

TRANSFER COEFFICIENTS AND HETEROGENEOUS RATE CONSTANTS
FOR Cd(II) IN VARIOUS SUPPORTING ELECTROLYTES
EVALUATED BY PHASE-ANGLE MEASUREMENTS WITH
A MODIFIED A. C. POLAROGRAPH

by Joseph Karl Frischmann

An improved a.c. polarograph was constructed. Its
general features are those of an instrument constructed by
Smith (37). The following modifications were introduced.

A low—gain operational amplifier was interposed be—
tween the cathode bias of the potentiostat and the tuned
amplifier. This assured maintaining a virtual ground at
the input of the cathode bias so that a 5mv a.c. potential
could be accurately placed between the reference and at the
indicating electrodes at all frequencies employed.

Electrochemical kinetic parameters are difficult to
evaluate from alternating current measurements. These
parameters are more simply and directly evaluated from
phase—angle measurements. An improved phase—angle measur—
ing circuit was designed and incorporated.

With a test RC series circuit constructed from com-
ponents with 1% tolerance, measured phase—angles agreed
with calculated values to within 1% through the frequency

ranges of 30 to ZSOcps and 500 to iZOOcps.

Joseph Karl Frischmann

Transfer coefficients evaluated for the reduction of
1.0mM Cd(II) at the dropping mercury electrode at 24 .i 10,
for the following supporting electrolytes, 0.5M Na2804,
1.0M Na2804, 0.5M H2804, 1.0M H2804, 1.0M NaClO4, 1.0M HC104,
1.0M KN03, 1.0M HCl, 1.0M NaCl, 1.0M KCl were: 0.21, 0.20,
0.27, 0.24, 0.51, 0.26, 0.50, 0.42, 0.46, 0.44 respectively
h’

for the same order of the above supporting electrolytes were:

and the values for the heterogeneous rate constants, k

0.076, 0.065, 0.14, 0.12, 0.54, 0.25, 0.65, 0.94, 1.2, 1.2,
cm/sec respectively.

Straight line cot¢ — mi plots, with intercepts cot¢ =
1, extrapolated to zero frequency, were obtained through the
frequency range 50-1100cps for all supporting electrolytes
containing Cd(II).

Linearity in the above plot, and agreement between

experimental and calculated cot ¢ values, employing an

appropriate equation derived for the electrode process
0 +-ne ——+= R

is evidence that the electrode process is only diffusion
and charge—transfer controlled.

The order of double—layer capacitance for anions of
the various supporting electrolyte solutions at a d.c.
potential of 0.6v.verCE was as follows: C1_ > N03- >
C104_ > 804:. This order parallels the order of adsorba-

bilities of the above anions on mercury.

Joseph Karl Frischmann

The decrease in heterogeneous rate constants for the
reduction of Cd(II) in the above supporting electrolytes

parallels the decrease in double-layer capacitance.

TRANSFER COEFFICIENTS AND HETEROGENEOUS RATE CONSTANTS
FOR Cd(II) IN VARIOUS SUPPORTING ELECTROLYTES
EVALUATED BY PHASE-ANGLE MEASUREMENTS WITH

A MODIFIED A. C. POLAROGRAPH

BY

Joseph Karl Frischmann

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Chemistry

1966

O \
a} “Q
I0 \‘5

7
W

ACKNOWLEDGMENTS

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

Recognition is also due Dr. Richard Nicholson for
his many helpful suggestions and discussions, James McLean
and John Holland for their valuable technical assistance.

The author is grateful to the Socony Mobil Oil
Company, the National Science Foundation, and to the
Department of Chemistry for financial aid.

Special thanks go to my wife, Mary, for her encourage-
ment and understanding throughout graduate school.

**********%

ii

VI TA

Name: Joseph Karl Frischmann

Born: September 20, 1957 in Milwaukee, Wisconsin

Academic Career: Rufus King High School
Milwaukee, Wisconsin
(1951-1955)

Marquette University
Milwaukee, Wisconsin
(1955-1961)

Michigan State University

East Lansing, Michigan
(1961-1966)

Degrees Held: B. S. Marquette University
(1959)

M. S. Marquette University
(1961)

iii

INTRODUCTION . .

THEORETICAL . .

EXPERIMENTAL . .

TABLE OF CONTENTS

Instrumentation.
The A. C. Polarograph. .
Calibration of D. C. Currents.
Calibration of A. C. Currents.
Phase-Angle Calibration.
Other Equipment.

Reagents .

Preparation of Solutions
of E
of D

Evaluation
Evaluation
Evaluation
Evaluation

0

i,

O

O

E , E from D
and D g'.

C O

of Cd,1, and Rt . .
of d and kI .

DISCUSSION OF RESULTS.

Evaluation
Typical A.
Evaluation
Evaluation

kh .

h

0

R
C. P larSgram . .

of Transfer Coefficient,

Polarograms

a. O O O O

of the Heterogeneous Rate Constant

SUGGESTED FURTHER WORK

SUMMARY AND CONCLUSIONS.

LITERATURE CITED

0

iv

Page

20

21
21
51
55
54
55
58
59
59
59
40
41

45
44
46
50
54
69
71

74

LIST OF TABLES

TABLE Page

I. E5, i and Do Values for 1.0mM Cd(II) in
Vérious Supporting Electrolytes . . . . . . . 45

II. Variation of E:_with Concentration and Type
of Electrolyte in Isolation Compartment.
Sample Solution is 1.0mM Cd(II) in 1M HCIO4 . 47

III. Variation of Cd.l. and Phase—Angle Correction
with Changing Ed.c. for 1.0mM Cd(II) in 1M
Na2804 at 822cps. . . . . . . . . . . . . . . 51

IV. Values of the Transfer Coefficient, d, 0b—
tained from Slope Measurements for 1.0mM
Cd(II) in Various Supporting Electrolytes . . 52

V. Values of the Transfer Coefficient, d, for
1.0mM Cd(II) in Various Supporting Electro-
lytes . . . . . . . . . . . . . . . . . . . . 55

VI. Values for the Heterogeneous Rate Constant,
k , for 1.0mM Cd(II) in Various Supporting
EIectrolytes. Values for Cd,1, and Rt, for
Electrode Used are Included . . . . . . . . . 56

VII. Effect of Cd(II) Concentration on the Hetero—
geneous Rate Constant, kg . . . . . . . . . . 57
VIII. Tabulation of Heterogeneous Rate Constants
and Transfer Coefficient. Values Determined
by Various Investigators. . . . . . . . . . . 64

LIST OF FIGURES

FIGURE

1.

10.

11.

12.

15.

14.

15.

Variation of cot ¢ with.changingrixi.potential
for 1) preceding, 2) no, 5) following first
order chemical reaction. . . . . . . . . . . .

Essential components for an a.c.-d.c. polaro-
graph. . . . . . . . . . . . . . . . . . . . .

Schematic of the.i d.c. potential source . . .
Schematic of the a.c. potential source . . . .
A.c. Signal mEasuring component. . . . . . . .
Schematic for full—wave rectifier. . . . . . .
Schematic of circuit for detecting phase dif-
ference between polarographic and reference

signal . . . . . . . . . . . . . . . . . . . .

Schematic of potentiostat switched to the
calibrating mode . . . . . . . . . . . . . . .

Electrolysis cell. . . . . . . . . . . . . . .
Auxiliary and reference electrode. . . . . . .

Typical a.c. polarogram for 1.0mM Cd(II) in
1M Na2804 at 55.8cps . . . . . . . . . . . . .

Typical a.c. polarogram for 1.0mM Cd(II) in
1M Na2SO4 at 822cps. . o . . o . . . . o o o 0

Variation of cot ¢ with changing a.c. fre-
quency for 1mM Cd(II) in various supporting
electrolytes . . . . . . . . . . . . . . . . .

Variation of cot m with changing Ed,c, at 54.8
cps for 1mM Cd(II) in 0.5M Na2804. . . . . . .

Variation of cot T with changing Ed.c. at 825
cps for 1.0mM Cd(II) in 0.5M Na2804. . . . . .

vi

Page

17

24

25

26

28

29

50

52
56

57

48

49

55

59

'60

LIST OF FIGURES — Continued

FIGURE page

16. Variation of cot ¢ with changing Ed.c. at
54.4cps for 1.0mM Cd(II) in 0.5M H2804. . . . 61

17. Variation of cot ¢ with changing Ed.c. at 828
cps for 1mM Cd(II) in 0.5M H2804. . . . . . . 62

18. Variation of double—layer capacitance with
changing applied d.c. potential for various
supporting electrolytes . . . . . . . . . . . 65

19. Variation of double—layer capacitance with

changing applied d.c. potential for various
supporting electrolytes . . . . . . . . . . . 66

vii

I NTROD UC TI ON

 

For a long time after the discovery of electrochemi-
cal phenomena, investigators were concerned with gross
effects such as the relationship between the quantity of
electricity consumed and the amount of electrochemical re—
action which occurred at electrodes, or the relationship
between the potential of electrodes and the concentration
and nature of species influencing the potential, or the
relationship between the number and Species of ions in solu—
tion and amount of current conducted by the solutions. Later
efforts were directed to the examination of many aspects of
these phenomena and were put on sound theoretical basis
through classical thermodynamics. Even though good agree—
ment between the magnitude of some electrical quantity and
the kinds and number of species was obtained when expected
and measured quantities were compared, still no exact ex-
planations of processes at electrodes were revealed through
the above efforts.

During the 1920's, the attention of some electrochemists
was directed to behavior of microelectrodes resulting in
the development of polarography by Heyrovsky in 1924. Now
the electrochemists began to consider the individual contribu—
tions of electron transfer, mass transfer to the electrode,
and removal of reaction product to the total electrode
process. Up to the early 1950's, d.c. polarography was ex-
ploited for analytical applications, for determining whether

an electrode process was reversible, for the evaluation of

 

the number of electrons involved in a reduction or oxi-
dation step, for the determination of whether or not step—
wise reduction occurred for complexation studies, or for the
evaluation of diffusion coefficients.

Since many electrode processes were irreversible,
causes for irreversibility had to be identified and attention
was now drawn to kinetic effects. Pioneers in this field
among others were Eyring and Koutecky.

In the application of the absolute rate theory to
electrode processes, Eyring (21) developed the general current-

potential relationship for the reaction
0 + ne-——¥ It.

The equation is

i(t) = n FAké [CO exp (- %%'G(E(t)-EO))-

 

CR exp ((1RT d)nF (E(t)- E0)}

in which i(t) is the current flowing at time t, at an applied
potential E(t), E0 is the standard electrode potential, CO
and CR are the surface concentrations of oxidized and reduced
species, n, F, R, T, have their usual significance and d,

the transfer coefficient, and kg, the heterogeneous rate
constant, are the kinetic parameters.

l
The heterogeneous rate constant, kh in cm/sec, is the

rate constant for charge-transfer at the standard electrode

 

potential. The following is a simple explanation of the
significance of d as well as its indirect definition.

For an electrochemical reaction to occur in a potential
field an energy barrier must be overcome. This existing
potential will favor the reaction in one direction and hinder
the reaction in the opposite direction. If the potential
that favors the forward reaction is OE, then (1—d)E is the
potential that favors the back reaction, so a simply is the
fraction of the existing potential which drives the reaction
in one direction.

Koutecky (26) derived equations for irreversible pro-
cesses which indicated that d and kg could be evaluated by
d.c. polarography but applications were limited. To extend
the evaluation to faster electron transfer reactions, newer
electrochemical methods such as cyclic voltammetry, galvano—
static, potentiostatic, faradaic rectification, faradaic
impedance, and various forms of a.c. polarography were de-
veloped.

A. c. polarography was first used in 1958 by Mfiller
and co-workers (51) 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 (22). 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 up to the limiting current, was due to the electro-

reduction of Cd(II) and re-dissolution of Cd in phase with

 

the applied a.c. potential.

The earliest equations reported for the evaluation of
rate constants from faradaic impedance measurements were
derived independently in 1947 by Randles (54) and Ershler (19).

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

Until very recently, all the equations for the a.c.
methods were derived for diffusion to a plane electrode.
Delmastro and Smith (17, 18), have shown that all the a.c.
wave equations previously derived do not adequately describe
the relationship between current and kinetic parameters for
electrode processes occurring at a spherical electrode. The
equations for diffusion to a plane and spherical electrode
are given in detail in the theoretical section.

From the theoretical expressions it becomes apparent

 

that d and kg

alternating current measurements. Such currents can be

can be evaluated by a.c. polarography from

measured accurately but the final evaluation of the para—
meters is difficult. The parameters can be more directly
evaluated by measuring the phase-angle between the faradaic

alternating current and the applied a.c. potential.

 

Experimental instruments with which exactly known,
steady small amplitude a.c. potentials could be impressed
between the electrodes and with which phase-angles could be
accurately measured, did not exist when this study was
undertaken. Furthermore, there were wide discrepancies in
values of d and kg for Cd(II) reduction determined by various
methods and for various supporting electrolytes or in values
determined by different investigators employing similar
methods (see Table VIII).

This study was undertaken to construct an instrument
with which a 5 mv a.c. potential could be impressed accurately
between the electrodes and with which the phase angles could
be precisely measured. The capabilities of the instrument

O

were to be tested with the evaluation of d and kg for Cd(II)
in various supporting electrolytes and to establish whether

or not the double-layer capacitance affected directly the

charge-transfer process.

 

 

 

THEORETICAL

 

Alternating current polarography is a technique in
which the d.c. and a.c. potentials applied to the polaro-
graphic cell are controlled and the resulting alternating
current is measured as function of the d.c. potential with
the amplitude of the alternating potential held constant.
With this technique only reversible and quasi—reversible

electrode processes for the reaction

0 + ne R

——>
(_.
can be examined since irreversible processes yield no a.c.
waves.

Before proceeding with the discussion of the theoretical
section which will require the statement of numerous equations,

the definitions of symbolism employed are tabulated below for

ready reference.

A = electrode area
d,8 = mass transfer coefficients, 8 = (1 - a)
Cf = faradaic capacitance

c: = initial concentration of species i

D1 = diffusion coefficient of species i

Ed c = applied d.c. potential
E0 = standard redox potential
R

E%_= reversible half—wave potential

E(t) = instantaneous value of potential
AB = amplitude of the applied a.c. potential
F = Faraday's constant

erfc = error function complement

I(wt)

AI

id.c.

k1,k2

(D

X

Zf

activity coefficient of species

first harmonic faradaic alternating current at
time t

amplitude of measured first harmonic alternating
current

d.c. faradaic current component
equilibrium constant

first order chemical reaction rate constants

hetegogeneous rate constant for charge-transfer
at E

number of electrons transferred

phase-angle between applied a.c. potential and
fundamental harmonic faradaic alternating current

gas constant

resistance of indicating electrode
faradaic resistance

spherical electrode radius

solution resistance

total series resistance

absolute temperature

time

angular frequency

distance from the electrode surface

faradaic impedance

An ideal reversible process is one in which the charge—

transfer rate is so rapid that the current is purely diffu—

sion controlled.

10

The expressions for the faradaic alternating current
and the faradaic impedance associated with diffusion only
have been derived by Breyer and Hacobian (11), Delahay and
Adams (16), Matsuda (29), Tachi and Senda (40) for diffusion
to a plane electrode. The expression for faradaic alternat—
ing current when AEIg 8mv, and whether or not the reduced

form is soluble in the electrode, is

n2F2ACO*(wDO)éAE

 

_ ' .1
_ ' E
I(wt) — IreV.Sln (wt + 4 ) (2)
in which
. R
j:%(Edc “El; (5)
for an applied potential
E(t) = E — AE sin (wt) (4)

d.c.

For the faradaic impedance, the expression is

z = 4RT cosh2(j/2) (5)
f n2F2AC3—IwDofg

The faradaic impedance can be considered equivalent
to a series RC circuit. The phase relationship for a series

network is as follows

Zf ”Cf

 

 

11

Rf and Cf are the resistive and capacitative components
of the faradaic impedance and from the above diagram, it

is obvious that

 

Cot ¢ = Rf w Cf (6)
and
(R)2+(1)2=(Z)2 (7)
f “Cf f

For an ideal reversible process the alternating current
leads the applied alternating potential by 450 (cot ¢ = 1),

then
R _ 1 _ 4RT cosh2 (j/2)
f _ ——— _ ,
n2F2ACo*(2wDo)

 

wa (8)

In a recent study by Delmastro and Smith (18), for the
influence of spherical diffusion on the alternating current
polarographic wave, the following more rigorous expression
for I(wt) was developed for the reduced form being soluble

in the electrode:

 

.1. A.
2 l.
1(wt) = Irev 1 + D? I‘Dfi L [1 - exp(b2t) erfc(bt2)]
° 8 D02 _ DR2
sin (wt +-E ) (9)
or simply
. U
= +-—
I(wt) Irev. HS Sin (wt 4 ) (9a)

and
j A _ 11:
b = e D0 D3 (10)
r0 (1 + 8])

 

12

Equation (9) differs from equation (1) by the quantities
enclosed in the large brackets and these quantities consti-
tute a correction term, (HS), to account for the difference
between diffusion to a spherical electrode and a plane
electrode.

A quasi-reversible electrode process is one in which
the current is diffusion and charge-transfer controlled.
The equations which follow were derived with the assumption
that the d.c. process was reversible.

Matsuda (50) has derived rigorous equations for the
quasi—reversible system. Smith (59) slightly modified
Matsuda's equations for the controlled a.c. potential
technique.

Smith's expression for the faradaic alternating current

for a plane electrode is
2 -§
(2.9%) 2
A

 

I(wt) = I F(t) sin (wt + @)

rev‘ 1 + (1 +

(11)
for which Irev is defined by equations (1) and (2), and

(ae-j -(3) D% Qo(t)

 

 

F(t) = 1 + (12)
kh
k f _ ' '
A = —§7- e aj+_ eBZ} (15)
D2
é -
e = cot-1(1 + 12%l-—) (14)
i (t)
Qo(t) = d'c‘ %_ (15)
nFA (30* DO
f = £05 fRa (16)

D '= no8 DRQ (17)

15

Delmastro and Smith (18) also considered spherical
diffusion for the quasi-reversible case and arrived at an
expression which has the same form as equation (11) and
equations(15)—(17) pertain. In the new expression F(t) is
identical with the Spherical correction term, HS, defined
in equations (9) and (9a).

Evaluation of the heterogeneous rate constant, kh'
and mass transfer coefficients, (a, 8), by either equation
(10) or Delmastro and Smith's (18) modified equation from
an a.c. polarogram would be extremely cumbersome. However,
the mere knowledge of w, the phase—angle, regardless of
whether diffusion is to a plane or spherical electrode,
allows one to readily determine the desired parameters as
will be shown from the following considerations.

By algebraic manipulation of equations (15), (5) and
(14), one obtains

%
cot ¢ = 1 + i2 w D). . . (18)

khf(e--OLj + e83)

 

.I
Equation (18) predicts that a plot of cot ¢ versus w2 at

a fixed d.c. potential should yield a straight line with an

 

intercept at cot ® = 1 and a
( J2
slope = , g p) . (19)
k (e 00 + e53)
h
where
kh = khf . (20)

14

The apparent heterogeneous rate constant, kh, uncorrected

for activity coefficients is the term calculated in this work.
For the Special case where j = o, that is, the applied

d.c. potential is equal to the reversible half—wave potential,

one can calculate from the slope and D, defined in equation

(17), a value of k by the following simple expression

 

h
.L
. (0)2 l
kh = slope (2)2 (21)

There are two methods for obtaining the mass transfer

coefficients. In the first method, as equation (18) predicts,

cot ¢ will exhibit a maximum when the term (e—aj-+ e53) is at
a minimum. This will occur at
R
RT a
2 + — ._
Ed.C. Efi HF In B (22)

The second method consists of determining the slopes result—

%-

ing from the plot of cot ® versus w at two different d.c.

potentials. The value of d can be determined by

 

: 1_ _ 2.505 slope]
d 2 2(j2 _ ji) log slopeg (25)

One can obtain the reversible faradaic alternating
current equation (1) from equation (11) by letting kh ap-
proach 00.

The quantities in the latter equation defined by equa—
tions (12), (13), (14) will then go to 1, oo, and 21

respectively. A form of equation (11) for spherical diffusion

reduces to equation (9) when kh approaches a).

 

 

15

Smith (57) has also derived equations for frequency
and potential effects on phase-angles for systems with
preceding or following first order chemical reactions
accompanying the charge-transfer.

For the system with a preceding chemical reaction,

k
Y —;l-+~ 0 + ne-?——* It
2

the cot ¢ was Shown to be

 

 

 

 

p§£y%+.;L_. {._i_ [(11‘<32)%+ 9]%I+.LE_ + e3 }
cot e = A 1+e3 1+K 1 +32 1+K
1 1 (1+g?)%i-gg'§ K j
Egi 1+K [ 1 + g2 1 + 1+K + e } (24’)
where
K = E: (25)
g = k; + kg (26)

It can be seen that either large K value or high frequency

potential will reduce equation (24) to

 

 

 

l.
2
lim cot ¢ = lim cot ® = 1 + (2?) (27)
K —+-a3 g —é~o
and at low frequencies, equation (24) reduces to the
following:
(2 )5. 1 + j
lim cot ® = 1 + i ( K e j ) (28)
9"“) TIEJre‘
w # 0

By performing an experiment at high frequencies, A

can be evaluated (equation (27)). Once A is known, a low

16

frequency experiment performed under conditions so that
j = 0, K may be evaluated.
For the system with a following chemical reaction
k
0 + ne ———+» R -j;a—> 3?
l

the cot T was derived to be

 

 

 

 

—-——( 2(1)) % 1 '. fl ( 1+gi) é‘+g % Kelj

cot T = 7‘ + (1+ej) £1+K [ II+ 92 J + Iii + 1} (29)
1 , ej (1+g2)% — % K 3
fiaflfizl 11.923] +-1—_f—K+1}

where K and g are defined by equations (25) and (26)
respectively. When the limits K —+-oo, g —9'0 are imposed

equation (29) reduces to equation (27) and when g —+:oo,

w # 0,
it j
lim cot T = 1 + £%QL ('£—:—§—') (50)
5131+:
1+K

As for the previous system, K can be evaluated by high and
low frequency experiments.

Equations(24) and (29) do not predict a linear relation-
ship between cot T and 6% and non-linearity in a cot T — w2
plot indicates that processes in addition to diffusion and
charge-transfer are occurring. Since both systems yield
similar non-linear plots, it is not possible to distinguish
between the systems on this basis. The two systems can be
distinguished by the different variation in cot T with d.c.
potential when a constant, low—frequency potential is applied.

This difference is illustrated by Figure 1.

 

 

 

 

 

.H u s .oofloflu tax .m.o u o “cOAooeou
HOOHETQO Hmpuo pmuflm mQHBOHHOm Am_.oc AN .mcflpmomum AH
How Hmflucouom+.o.p,mflfimflmzo SDHB e #00 mo coaumfium> .fi ousmflm

Owl OMI ONI Ofil O Ofi ON Om ow

L _ _ a

 

17

m a 9 OOO

 

 

 

18

The assumptions which were made for the derivations
of equations pertaining to the faradaic current and phase—

angle dependence are:

 

(1) Fick's law (15) can be applied to each Species.

(2) The absolute rate equation (21) can be applied

to the electrode reaction kinetics.

(5) Charge-transfer involves one rate-determining

step.

(4) Diffusion and chemical reaction are the sole

means of mass transfer.

(5) Adsorption effects are negligible.

The above equations are also derived solely for the
faradaic current, that is, the current arising from the
electrochemical reaction. However, the current from the
polarographic cell consists of the faradaic current and
double-layer capacitative currents. The polarographic
current must pass through resistance offered by the solu-

tion and the indicating electrode.

Rf Cf

'—’\/er 'L
H Rs Re

 

 

 

 

 

/
\

l
l
Cd.1.

The above is an equivalent circuit for the polarographic

cell. Rf and Cf are the faradaic impedances, Cd 1 is the

 

19

capacity of the double-layer, R is the resistance of

5
solution, and Re is the resistance of the indicating
electrode.

To evaluate kinetic parameters for the electrode pro-
cesses the faradaic phase-angle must be known. Any measured
phase-angle must be corrected for the effects of double—
layer capacitance and total series resistance. Bauer and

Elving (2) derived the following relationship to calculate

the phase-angle of the faradaic current:

AI AI

 

_ AE (COS ()3 - ZE- R't)
COt ¢ 7 AI AIR 2AIRt (51)
_ ' I _ t 2 _ I
AE Sin T w Cd.l.(1 +( AE ) AE cos T )
where
AI is the amplitude of the alternating current

measured.

AB is the amplitude of the alternating potential

applied.

T' is the phase-angle measured between the alternat-
ing current measured and the alternating potential
applied.

w is the angular frequency.

Cd 1 is the capacity of the double—layer.

Rt = RS + Re (total series resistance).

 

 

EXPERIMENTAL

20

 

21

Instrumentation

 

The instruments employed in this investigation are
as follows:

A Sargent S—50260 potentiometer was used to measure
all d.c. potentials applied to the cell.

A Tektronix Model 502 oscilloscope was employed to
measure the amplitude of the a.c. potential applied to the
cell and to monitor the tuning of the frequency sensitive
amplifier.

The Sargent Model M.R. and S.R. one second full scale
deflection recorders were employed to measure the maximum
d.c. currents, rectified a.c. currents, and signal propor-
tional to the phase—angle.

A Hewlett—Packard Model 202A signal generator provided
all a.c. potentials.

A Hewlett-Packard Electronic Counter Model 521A was
employed to calibrate the frequency of the Hewlett—Packard
signal generator.

A Philbrick Model R-500 power—supply provided up to
500ma of.i 500 v.d.c.

A Chicago—Stancor transformer was used as a 20 ampere,

6.5V filament supply.

The A. C. Polarograph

 

Previous to 1965, a.c. polarographs were constructed

using conventional circuitry but during that year there was

 

22

considerable activity in exploiting operational amplifiers
as elements in electrochemical instrumentation (55). With
experience gained at the Oak Ridge National Laboratory (20)
during the summer of 1965, the construction of the instru-
ment used in this study was undertaken.

The potentiostat of the instrument was essentially
that designed by Kelley, Jones, Fisher (25). It employed a
Deford follower (12) which places a high impedance path at
the reference electrode so that current flows between the
indicating electrode and auxiliary electrode instead of
between the indicating electrode and the reference electrode.
This type of potentiostat was similar to one constructed by
Smith (57), but instead of proceeding on to the tuned ampli-
fier as Smith had done, another low—gain operational ampli-
fier was interposed between the cathode bias (amplifier C,
Figure 2) and the tuned amplifier (amplifier F, Figure 5).

Since Smith's (57) phase-angle detecting circuit had
high and low level current limitation, this part of the
instrument was modified by eliminating the diodes, which
broke down at high currents, from the feedback loop. The
operational amplifier, now at open loop performed simply as
a voltage detector which responded to input Signals from
50mv to 100v. Thus the high level current limitation of
Smith circuit was overcome without sacrifice of low current
sensitivity. Details of the modified phase—angle detector

will be discussed subsequently in more detail.

 

25

Conventional circuitry for full-wave rectification and
Signal recording was employed. Pertinent details of the
components comprising the instrument will now be discussed.

Figure 2 shows a block diagram of the instrument. The
potentiostat consists of operational amplifiers A, B, and C.
A.c. mode and d.c. mode are the units utilized for measuring
a.c. or d.c. signals. Amplifier D attenuates the signal for
recording.

The constructional details for the stable d.c. potential

source to supply the E are shown in Figure 5. From 0 to

d.c.
i.5 volts could be obtained from this source with the aid of
the 1K Helipot. The long term instability of the supply was
at most 0.1mv.

The constructional details for the Ea.c. supply are
shown in Figure 4. A 1000 fold or 100 fold attenuation of
the 5 volt signal from the signal generator produces 5 or
50mv across the 10 turn Helipot from which ranges of 0 to
5.00 i.0.05mv or 0 to 50.0 i_0.5mv were available. The
frequency of the signal generator was calibrated and
typically corresponded to within 0.6% of the dial reading
in the frequency range of 50 to 1200cps.

A d.c. scanning unit, Ed.c. scan, not used in this
investigation, was also incorporated in the instrument.

To record an a.c. polarogram, a 5mv a.c. potential

together with the desired d.c. potential were applied it

Ea c and Ed c . The output of amplifier C, related to

 

 

24

 

 

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27

current passing through the polarographic cell, is amplified
by amplifier E (Figure 5). The use of amplifier E makes
possible the gain adjustment of amplifier C so that C can
maintain its input at a virtual ground at all frequencies
employed. If a resistance of 5K or greater were put in the
feedback loop (RL) of amplifier C, a small a.c. potential of
the frequency (> 500cps) introduced would appear at the input.
This resulted in current measurement errors of about 10% and
phase-angle errors of about 50 at 1000 cps. When RL was
lowered to 470 ohms, this problem was eliminated in the
frequency range of 50 to 1200 cps.

To obtain fundamental—harmonic a.c. polarograms which
are undistorted by effects of extraneous a.c. signals
(60cps noise or higher harmonic faradaic components) and to
make phase-angle measurements, a highly frequency—selective
amplifier (Figure 5, amplifier F) was introduced as an
integral part of this instrument. The amplifier has two
frequency ranges which enable it to amplify selectively any
frequency in the range of 50 to 250cps or 500 to 2500cps.

The Signal from amplifier F is rectified by the circuit
shown in Figure 6 and recorded.

In Figure 7 amplifiers H, I, and J are high gain limit-
ing voltage detectors which give square wave outputs of the
same frequency and phase as their a.c. input signals. The
amplitude of these square waves after clipping with back—to-
back Zener diodes, is independent of the amplitude of the

original a.c. signal.

28

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51

The signal from the tuned amplifier is converted by

amplifiers H and I to a constant amplitude square wave.

A reference a.c. signal with a known phase relationship to

the potential being applied to the cell is also converted

by amplifier J to a constant amplitude square wave. The

two square waves are then added and rectified. The signal
resulting from these operations has a d.c. component dependent
on phase relationships between the square waves, this signal
being a maximum when the square waves are in phase and zero
when they are 1800 out of phase.

Amplifiers H and I are in series to insure that even
small signal levels from the tuned amplifier are converted
to square waves. The rise time of amplifiers H, I, and J
is improved by positive feedback via capacitative coupling
of the output and the positive input of the operational
amplifier. This causes pulsing of the amplifier during the
rising portion of the square wave. It is very essential to
minimize rounding of the edges of the square wave by small
Signals at high frequencies otherwise the phase—angle sig—

nals vary with amplitude of input signal.

Calibration of D.C. Currents

 

The d.c. mode (Figure 2) consists of a 100K resistor
which makes amplifier D a unity gain inverter.

For current calibration, the electrodes are taken out
of the circuit and connection to the precision 100K resistor is

made as shown in Figure 8. Since the input of operational

52

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55

amplifier C is a virtual ground, the potential appearing
at amplifier B is placed across the 100K resistor and the
current flowing through the resistor is EB/100K. Since D
is a unity gain amplifier, its output potential will be
equal to the output of C but of opposite sign.

For d.c. operation the resistance of RL is 100K. To
calibrate the recorder response, 0 volt, introduced by the
d.c. potential source (Figure 5) is measured to within a
millivolt and placed across the 100K resistor. This will

define the zero current on the recorder. Then EB measured

 

to within a millivolt is placed across the 100K resistor.
This corresponds to a current of EB/100K. The corresponding
signal from amplifier D is then attenuated to give a con-
venient recorder deflection. The ultimate accuracy of the
current measured will depend upon the accuracy of the 100K
resistor. At this point the electrodes may be switched back

into the circuit and a potential—current curve recorded.

Calibration of A.c. Currents

 

To measure fundamental a.c. signals free from noise,
amplifier F (Figure 5) must be tuned sharply to the fre-

quency of the signal to be measured. The proper R1 value

1),

is selected to provide the desired frequency (fl = 2TR_C
1

R2 is set equal to R1, and R3 is adjusted until oscillations
in amplifier F cease. The frequency of the a.c. source is
adjusted until the output of amplifier F is at a maximum.

R1, R2, or R3 (Figure 8) is selected to provide the desired

54

current range (I = EB/R). EB is measured across R1, R2 or
R3 with the oscilloscope. Amplifier D is adjusted until a
convenient deflection on the recorder is attained.

The lower calibration limit should never be zero cur-
rent. The reason is that diodes are not perfect conductors
in the forward direction at low current levels. Therefore
the current response of the recorder would not be linear from
zero to the upper current limit. The lower current calibra-
tion should be 0.5ua or greater. This is no limitation
since capacitative currents in this work were typically 1 to
50ua depending upon the frequency of the applied a.c.

potential.

Phase—Angle Calibration

 

With amplifier F (Figure 5) tuned to the desired fre—
quency, the phase-angle detector is switched in as shown in
Figure 7. With the calibrating circuit switched to the 1uf
capacitor, the phase-Shifter (Figure 7) is adjusted until
the recorder gives nearly a zero reading. The bucking
potential (Figure 6) is adjusted until the recorder gives a
zero reading. The calibrating circuit is then switched to
a resistor, R1, R2, or R3, and amplifier D is attenuated to
a convenient recorder setting. Since the a.c. signal through
the 1uf capacitor leads the applied a.c. potential by 900,
the lower limit setting represents a Signal 900 out of phase.
Since the a.c. signal through a resistor is in phase with

the applied potential, the upper limit represents a 00

 

55

phase-Shift. The long term stability of the employed com-
mercial oscillator was about 0.50, therefore the above
calibration was performed before and after each phase-
angle measurement.

The accuracy of the phase—angle detector was tested
with the following series RC networks (components of 1%
accuracy): 500 ohm and 1.0uf; 100 ohm and 1.0uf; 50 ohm and
1.0uf. The phase-angles measured agreed to within 1% of
the calculated values throughout the 50 to 1100cps frequency
range. These RC circuits and frequencies yielded the cur-
rent range from 0.95ua to 55 ua and the phase-angle range

from 89.60 to 17.60.

Other Equipment

 

The electrolysis cell employed in this work is shown
in Figure 9. Through the side—arm with the two-way stop
cock, nitrogen can either be bubbled through the solution 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 5—hole
rubber stopper through which the electrodes (dropping mercury
electrode, saturated calomel electrode, and the auxiliary
electrode) are introduced.

The constructional details of the reference electrode
employed are shown in Figure 10. The reference electrode
was separated from the polarographic solution by an isola—
tion compartment which usually contained the supporting

electrolyte. Both the reference electrode and the isolation

 

56

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58

compartment 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, is separated

from the polarographic solution by a fine frit. Its isolation
compartment always contained the polarographic 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

Matheson, Coleman and
Bell. A.C.S. Reagent

Potassium Chloride

Sodium Chloride

Perchloric Acid

Disodium Salt of Ethylenedi-
aminetetraacetic Acid (EDTA)
Sodium Sulfate

Hydrochloric Acid

Sulfuric Acid

Sodium Perchlorate Monohydrate

Potassium Nitrate

Nitrogen

Mercury

Baker's Analyzed Reagent
Baker's Analyzed Reagent
Mallenckrodt Analytical
Reagent

Baker's Analyzed Reagent
Baker's Analyzed Reagent
Baker's Analyzed Reagent
G. Frederick Smith
Chemical Company

Matheson, Coleman,
A.C.S. Reagent Grade

Prepurified, 99.996%
The Matheson Company

Purified in this laboratory

and Bell

59

Preparation of Solutions

 

A 0.0100M Cd(II) stock solution was prepared from
Cd(N03)2-4H20 and standardized by EDTA titration (56).

The supporting electrolyte solutions were prepared by
weighing solids or pipeting liquid electrolytes and diluting
to volume.

The sample solution was prepared by appropriate dilu-
tion of cadmium stock aliquots to which the desired amount
of electrolyte had been added.

None of the solutions contained maximum suppressor.
All solutions were prepared from the laboratory distilled
water supply. These solutions yielded no different results
than when triply distilled water was used.

Dissolved oxygen was removed from all solutions in
the electrolysis cell by bubbling with nitrogen for a period
of 20 minutes. Last traces of oxygen were removed from the
nitrogen by bubbling it through an acid vanadous sulfate
solution, dilute NaOH, and H20 before passing it into the
electrolysis cell.

All polarograms were run at 24.1 10.

Evaluation of El} E1, E3 from D.C. Polarograms
4 2 4

 

The current—potential values for the sample solution
were determined manually every 10mv from the foot of the
wave to well into the limiting current region. In the same
potential region the current—potential relationship was

determined for the supporting electrolyte. The current of

40

the sample solution was corrected for the residual current.
The values of Bi, Efi’ E3, were determined by the log equal
4

to 0.477, 0, -0.491 intercepts of a E versus log id-i/i

d.c.
plot.

Evaluation of DO and D

 

DO, the diffusion coefficient of the oxidized species
was calculated with the aid of the Lingane—Loveridge equa—
tion (27). The parameters necessary for this evaluation
are id(average diffusion current), C (millimole concentration

of electroactive species), m (mg/sec of mercury delivered by

 

the electrode), t (drOp time).

To obtain average diffusion currents, the maximum
diffusion current was multiplied by 6/7. The capillary
characteristics (m and t) were evaluated at the potential at
which the diffusion current was measured. The drop time,
(t), was evaluated by timing the fall of 20 drops and m
from the weight of the 20 drops and known t.

The evaluation of D was accomplished with the aid of
equation 17 and using the value of DR equal to 1.61 x 10‘5

cma/sec obtained by Bauer and Elving (5).

Evaluation of Cd.l. and Rt

The determination of the capacity of the double—layer

 

( ) and total resistance (R ) is accomplished by apply—

Cd.l. t
ing a 5mv a.c. potential of a known frequency to the cell
containing supporting electrolyte, and measuring the ampli—

tude of the resulting a.c. current and its phase—angle.

41

Since the capacity of the double—layer changes with
d.c. potential and 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.l. _ wAE sin T (32)

 

and total series resistance by

_ AF:
Rt — AI cos T (55)
The evaluation of Cd 1 and Rt was determined at least

at two different frequencies. It was found that the values
obtained were independent of frequency within experimental
error. However, since the value of cos T changes rapidly
at phase-angles greater than 880, these experiments were

performed at 500 and 550 cps to obtain smaller phase-angles.

Evaluation of d and kh

 

By performing three experiments, the mass transfer
coefficient (a), the heterogeneous rate constant uncorrected
for activities (kg), and mechanism of the electrode process
can be determined by measuring the following:

(1) The phase—angle dependence on potential at low

frequency (4250cps).

(2) The phase—angle dependence on potential at high

frequency (4x820cps).

 

42

(5) The phase-angle dependence on frequency at d.c.
potentials corresponding to Bi, Bi” E&”

The experiments were performed with an a.c. potential
of 5mv. All phase—angles were corrected for double—layer
and total series resistance effects with a Control Data 5600
computer.

The mass transfer coefficients were calculated by
equation (22) and equation (25).

Equation (22) depends upon the location of the d.c.
potential where cot T is a maximum. This is best accomplished
by employing the high frequency data of experiment 2, since
in these experiments cot T has larger values, and the effect
of chemical reaction on cot T are negligible.

Equation (25) depends upon determining the slope
(equation (19)) from the phase—angle dependence on frequency
(equation (18)) at two different d.c. potentials.

The heterogeneous rate constant, kg, can be readily
calculated with the aid of equation (21) and (17), once
the slope from cot T Versus 6F plot (equation (18)) has
been evaluated at a d.c. potential corresponding to ES,

Once the parameters kg, d, 8, D are evaluated, they
are introduced into equation 18, and a phase—angle dependence
on d.c. potential is calculated at the low frequency em-
ployed. The calculated plot is then compared to the experi—
mental one in order to determine if the mechanism involves
just diffusion and charge—transfer or some other rate control—

ling steps.

DI SCUSSI ON OF RESULTS

 

45

44

Evaluation of E1, BE, E3 and DO
2

4 4

 

Values of the experimental half—wave potential, E:_,

and the calculated diffusion coefficient, D0, of the oxi—
dized species for the various supporting electrolytes, as
well as quantities needed to calculate D0, are tabulated in
Table I.

Shifts of E:_to more negative potentials usually
indicate complexation of the reducible species, the more
stable the complex, the more negative E:_.

On the basis of this expected behavior, the E:_values
indicate that the relative strengths of the complexes formed
between Cd(II) and the anions of the supporting electrolyte

follow the order:

C1- > SO4- > N03_ > c104”.

This is not the exact order which would be expected from
published values (7) of the log of the formation constants
which for Cd(II) and the anions in the order listed above

are respectively:

2.00, 2.42, 0.4, 1.42(?).

This is an indication that other factors affecting
E:_Shifts must also be considered. One of these could be
junction potentials which exist at boundaries between dis-
similar electrolytes.

This becomes obvious when the E:_value for 1M NaClO4

is compared to the one for 1M HC104. In both experiments the

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45

 

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46

reference electrode was separated from the sample solution
by an isolation compartment containing 1M NaN03 instead of
1M HClO4 to prevent the precipitation of KClO. at the SCE
arm. That the E: values are dependent on the concentration
and type of electrolyte in the isolation compartment is shown
in Table II. This dependence must be due to a junction
potential arising from the difference of diffusion rates of
ions through both junctions, that is, saturated KCl/isolation
electrolyte, isolation electrolyte/1M HClO4. 0n changing the
isolation electrolyte from 1M NaNOs to 0.1M NaNOs, E: shifted
to a more negative potential, and when changed to 1M HN03
shifted it to a more positive potential.

A point of interest is that the value of the slope,

from the plot of log id-i/i versus E plot (Tables I and II),

d.c
is independent of the concentration and type of electrolyte
in the isolation compartment. The current—potential curves
retain their expected shape and only their relative position
compared to the reference electrode is shifted.

Ei and E& values were determined from the intercepts
at log equal to 0.477 and —0.491 respectively on a log id—i/i

versus plot. Bi and Eg were displaced 14mv from

Ed.c.
R . .
E% in all media.

Typical A.c. Polarograms

 

Figures 11 and 12 illustrate typical total fundamental—

harmonic a.c. polarograms of 1.0mM Cd(II) in 1M Na2804 at

47

Table II. Variation of E:_with Concentration and Type of
Electrolyte in Isolation Compartment. Sample
Solution is 1.0mM Cd(II) in 1M HClO4

 

 

 

 

a
Isolation ER b
Compartment v5; SCE, v Slope
1M NaN03 —0.607 0.0296
0.1M NaN03 —0.646 0.0294
1M HN03 -0.586 0.0294

 

aDetermined at the log = 0 intercept from the plot log
id-i/i VS. Ed C .

0.059
2

 

bAE/(A log id-i/i) =

48

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49

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50

55.8 and 822cps respectively. It is evident from Figure 12
that from the base-line level, the total alternating cur-
rent is made up largely of capacitative current at high fre-
quencies.

Table III compares the measured phase-angle with the
corrected (faradaic) phase—angle. It becomes apparent that
the correction becomes greater as the faradaic current be—

comes small compared to the capacitative current.

Evaluation of Transfer Coefficient, a

 

The values of the transfer coefficient, a, obtained

A cot T)
A w

potentials are tabulated in Table IV. Transfer coefficients

from the ratio of slopes ( at two different d.c.
of Cd(II) in the chloride and the nitrate media were not
calculated in this manner since the ratio of the uncertainties
in the slopes became competitive with the values of the ratio
of the slopes.

Transfer coefficients were also calculated from the
d.c. potential (equation (22)) where cot T is a maximum.
Table V compares the values of the transfer coefficients
calculated by equation (22) and (25).

Both methods for calculating the mass transfer co—
efficients give comparable values as long as the charge-
g > 10-4

cm/sec). As the charge—transfer rate becomes larger, the

transfer rate is sufficiently slow (0.5cm/sec > k

evaluation of d becomes more uncertain by either method.

However, even for large charge-transfer rates (kh gglcm/sec)

Table III.

51

Variation of Cd,1, and Phase-Angle Correction
with Changing Ed.c. for 1.0mM Cd(II) in 1M
Na2804 at 822cps

 

 

 

(E vs. SCE),v AI ua Cd 1 ,uf T'Measured T correcteda’b
(degrees) (degrees)
0.55 25.50 0.905 59.9 56.7
0.56 25.45 0.891 52.5 59.8
0.57 27.90 0.880 41.5 55.0
0.58 51.15 0.865 51.0 50.7
0.59 55.45 0.855 25.5 25.2
0.60 55.80 0.840 20.0 20.5
0.61 51.60 0.826 21.5 17.7
0.62 27.65 0.815 27.5 16.4
0.65 25.70 0.806 58.1 17.4
0.64 21.10 0.795 49.6 18.5
0.65 19.85 0.782 59.5 19.2

 

aCorrected for C

b

R

t

d.l.

= 77.2 ohms.

and Rt berquation (51).

 

 

52

 

 

 

 

Table IV. Values of the Transfer Coefficient, a, Obtained
from Slope Measurements for 1.0mM Cd(II) in
Various Supporting Electrolytes
. a a a
Supporting Slope at Slope at Slope at b
Electrol te E E E a
Y i- 2 *2
0.5M NaESO. 1.42 x 10‘2 2.26 x 10“2 2.72 10‘2 0.20 i 0.05
1M NagsO4 1.67 x 10'2 2.65 x 10‘2 5.05 10‘2 0.22.: 0.05
0.5M H2804 0.911x 10‘2 1.42 x 10‘2 1.58 10‘2 0.24.: 0.05
1M H2804 1.01 x 10‘2 1.57 x 10-2 1.77 10"2 0.25 1.0.05
1M HClO4 5.78 x 10‘3 8.55 x 10"3 9.11 10‘3 0.50 1.0.04
1M NaClO4 4.11 x 10-3 6.11 x 10-:3 6.55 10-53 0.28 :1: 0.04
a . A cot T .
Determined from A w at a constant d.c. potential
. 2

bObtained from ratio of slopes
(23).

Ei/El' and 31/33: see equation
2 2 4

55

Table V. Values of the Transfer Coefficient, d, for 1.0mM
Cd(II) in Various Supporting Electrolytes

 

 

Supporting a b

 

Electrolyte (Ed.c.—E:)’mv d d C

0.5M Na2804 16.5 0.21 i 0.05 0.20 i 0.05
1M NagsO4 17.5 0.20 i 0.05 0.20 i 0.05
0.5M H2804 12.5 0.27 i 0.05 0.24 i 0.05
1M H2504 14.5 0.24 i 0.03 0.25 i 0.05
1M KNO3 0.0 0.501 0.05 ---

1M HC1O4 15.5 0.26 i 0.04 0.50 i 0.04
1M NaClO4 10.0 0.51 i 0.04 0.28 i 0.04
1M KCl 5.0 0.44 i 0.05 ---

1M HCl 4.0 0.42 i 0.05 ---

1M NaCl 2.0 0.46 i 0.05 ---

 

aPotential at which cot T is maximum

bCalculated when cot T was maximum by equation (22).

CAverage value determined from 2 COt ¢ at two different d.c.

potentials, see equation (23). w 2

54

the location of the d.c. potential where cot T is a maximum

can be determined to within i 5mv.

Evaluation of the Heterogeneous Rate Constant, kh

i'at

Typical plots of cot T versus w

 

_ R
Ed.c. — 13% are
illustrated in Figure 15 for 1.0 mM Cd(II) in various sup—
porting electrolytes. All of the plots yield straight lines
up to frequencies of about 1100cps with intercepts at

cot T = 1, extrapolated to zero frequency, which is expected

for the following reversible electrode process

0 + ne ———%: R .

Such linearity was not attained by previous investigators

(5, 4, 5).
Table VI lists the slope (%j&g§—$ at Ed c = E:_), D,

and the values of the heterogeneous rate constants uncor-
rected for activity coefficients for 1.0mM Cd(II) in various
supporting media. The values of the heterogeneous rate
constant, kg, ranged from 0.065cm/sec in 1M Na2804 to 1.2cm/
sec in both 1M NaCl and 1M KCl.

The effect of Cd(II) concentration on the kg is Shown
in Table VII to be constant within experimental error. The
larger deviation of kg in 1M Na2804 is caused by the larger
error in the corrected phase—angle. The problem arises any
time the faradaic current becomes small compared to the

total current. For 0.2mM Cd(II) the faradaic current is

equal to the capacitive current at 55cps and is about 10%

55

.mmumaoupomao mCaDHommSm msoaum> Ca
AHHVUO 25a How mocmsqmnm .0.m mcamcmno LuaB e #00 mo coapmaum> .ma Tnzmam

 

 

m 0mm mmcmapmu . m8
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a.U ms nu mo rt n. .L O .0 t c o
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56

 

 

 

 

Table VI. Values for the Heterogeneous Rate Constant, k g
for 1.0mM Cd(II) in Various Supporting Electro-
lytes. Values for Cd,1, and Rt, for Electrode
Used are Included.

Slopea C c '
Supporting AcotT 2 Dx106, d.l., Rt, k ,
Electrolyte Aw; X10 cma/secb uf ohms cg/sec
0.5M Na2804 2.26 5.89 0.816 87.5 0.076
1M Na2804 2.65 5.51 0.855 77.2 0.065
0.5M H2804 1.42 7.54 0.857 64.4 0.14
1M H2804 1.57 7.05 0.952 58.9 0.12
1M KN03 0.558 10.20 1.14 76.4 0.65
1M HClO. 0.855 8.58 1.09 60.6 0.25
1M NaClO. 0.611 8.57 1.11 81.5 0.54
1M KCl 0.189 10.54 1.11 72.6 1.2
1M KCl 0.255 9.68 1.12 60.2 0.94
1M NaCl 0.189 10.56 1.08 78.0 1.2
aAt El:
2

bD = DOB DRO‘

CAt E = ER vs SCE
d.C % o .

57

Table VII. Effect of Cd(II) Concentration on the Hetero-
geneous Rate Constant, kh

 

 

 

Supporting Conc. Slope,a kg .
Electrolyte Cd(II),mM ficotT x 102 cm/sec
w
1M Na2504 1.0 2.65 0.065
1M Na2804 0.2 2.28 0.075
1M H2804 1.0 1.57 0.12
1M H2804 2.0 1.64 0.11
1M KN03 1.0 0.558 0.65
1M KNOB 2.0 0.576 0.60

 

58

of the total alternating current at 822cps. There is better

h
both H2S04 and KN03 pointing out the desirability of working

agreement of k with varying concentrations of Cd(II) in
in a concentration range which produces a faradaic current
at least one-half the magnitude of the total alternating
current measured.

Figures 14, 15, 16, 17 illustrate the comparison be—
tween calculated and experimental variation of cot T with
changing d.c. potential at low and high frequency for 1.0mM
Cd(II) in 0.5M Na2SO4 and 0.5M H2S04.

The data (kg, 0. D) in Table VI were used to calculate
the theoretical plot for 1mM Cd(II) in all the supporting
media with the aid of equation (18). In each case, the
experimental cot T, at high or low frequency at various d.c.
potentials, agreed well with the calculated cot T. If a
maximum uncertainty of.i 0.50 in the corrected experimental
phase—angle values was assumed, each cot T value would vary
by the amounts indicated by the vertical line drawn at each
experimental point.

The low frequency plots further substantiate that the
reduction of Cd(II) in the sulfate and perchlorate media

is a simple quasi-reversible electrode process

+
O ne 7——%* R

The low frequency data for the nitrate and chloride

media neither support nor dISprove the above electrode

59

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65

process, since the charge—transfer rates are so rapid in
these media, that the electrode process becomes dependent on
diffusion only and the detection of other kinetic processes
becomes impossible. For kg values between 0.6 and 1.2 cm/sec
the value of the phase—angle at 55cps for these media should
lie between 44 and 450 (450 indicates that the electrode
process is only diffusion controlled), and the experimental
values agreed typically to within 0.20 of the calculated
value.

Table VIII lists the values of kg and d.obtained by
other investigators employing various techniques. The reason
for the disagreement With some of the investigators is not
clear. In general, one can find values of the relative rates
of charge-transfer for the reduction of Cd(II) determined by
various investigators which are in agreement with our observa—
tions, that is, rates decrease in the order, expressed in

terms of the anions of the supporting electrolyte,

c1" > N03 > 0104‘ > 804: .

Because of the heterogeneous nature of electrode pro—
cesses, it is not too surprising that changes in the double—
layer structure or electrode surface would affect the
charge—transfer rate. An interesting correlation between the
capacity of the double—layer and the charge—transfer rate is
illustrated in Figures 18 and 19, that is, the higher the
capacity of the double-layer, the more rapid the charge—

transfer.

64

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67

The capacity of the double-layer at a constant d.c.
potential can be affected by adsorption of reactants, products,

and supporting electrolyte. Since kh is relatively constant
with changing Cd(II) concentration, it is usually assumed (9)
that adsorption of the electroactive Species is negligible.
In a recent monograph Delahay (14) shows that the para—
meter, kg should decrease with increasing supporting electro—
lyte concentration, and that Specific adsorption of an anion
on the electrode should increase the charge—transfer rate for
reduction of a non-adsorbed cation and be decreased by the
presence of specifically adsorbed cation. The above gener—
alities however are often complicated by formation of com-
plexes between the reducible cations and anions in solution.
Delahay (15) lists the amounts of specifically adsorbed

anions on mercury at a constant d.c. potential. The degree

of adsorption is

C1- > N03_ > 804: ,

while Grahame (25) in earlier work determined the order to be
c1“ > N03—> 010.7 .

No comparison of the relative adsorbabilities of C104- to
so.: could be found in the literature. In the d.c. potential
region investigated, there is little probability that any

significant amount of cation is being adsorbed on the

electrode.

 

68

 

From Figures 18 and 19, it can be seen that the order
of capacitance of the double-layer for the anions of the

supporting electrolyte solutions is
c1“ > N03— > c104" > SO4— .

The values of kh for these media decrease in this same order.
These results provide experimental evidence to substantiate
the statements made by Delahay (14) that the adsorption of

anions increases the rate of reduction of non—adsorbed

cations.

 

Limited evidence is also presented to support the state—
ment that increasing the concentration of supporting electro—
lyte Should decrease the rate constant. In Table VI this
trend becomes apparent. The value of kg in 0.5M Na2804 is
0.076 cm/sec while in 1M Na2804 it is 0.065 cm/sec. Also in
0.5M H2804 it is 0.14 cm/sec while in 1M H2804 it is 0.12
cm/sec. Admittedly the significance of these differences in

kh may be open to question until work at higher and lower

supporting electrolyte concentrations is carried out.

SUGGE S TED F UR THER WORK

 

69

On examining the experimental results obtained with
the instrument certain experiments should be undertaken.

Since the charge-transfer rate in chloride media
appears to be so rapid and so slow in sulfate, various
small concentrations of chloride could be added to the
sulfate medium to determine whether the rate could be en-
hanced. The effect of the added chloride in the double-
layer capacity could also be determined.

The effect of supporting electrolyte concentration
on the charge—transfer rate should be examined.

It would be of interest to duplicate the experiments
of the completed study with increased a.c. potential ampli—
tudes to test at which amplitude level the rate constant

becomes amplitude dependent.

 

 

SUMMARY AND CONCLUSI ONS

 

71

72

An improved a.c. polarograph was constructed. Its
general features are those of an instrument constructed by
Smith (57). The following modifications were introduced.

A low-gain operational amplifier was interposed between
the cathode bias of the potentiostat and the tuned amplifier.
This assured maintaining a virtual ground at the input of
the cathode bias so that a 5mv a.c. potential could be
accurately placed between the reference and at the indicating
electrodes at all frequencies employed.

Electrochemical kinetic parameters are difficult to
evaluate from alternating current measurements. These para-
meters are more simply and directly evaluated from phase—
angle measurements. An improved phase-angle measuring
circuit was designed and incorporated.

With a test RC series circuit constructed from com—
ponents with 1% tolerance, measured phase—angles agreed with
calculated values to within 1% through the frequency ranges
of 50 to 250cps and 500 to 1200cps.

Transfer coefficients evaluated for the reduction of
1.0mM Cd(II) at the dropping mercury electrode at 24 1.10,
for the following supporting electrolytes, 0.5M Na2804,
1.0M Na2504, 0.5M H2804, 1.0M H2804, 1.0M NaClO4, 1.0M HC104,
1.0M KN03, 1.0M HCl, 1.0M NaCl, 1.0M KCl were: 0.21, 0.20.
0.27, 0.24, 0.51, 0.26, 0.50, 0.42, 0.46, 0.44 respectively

and the values for the heterogeneous rate constants, kh, for

the same order of the above supporting electrolytes were:

 

75

0.076, 0.065, 0.14, 0.12, 0.54, 0.25, 0.65, 0.94, 1.2, 1.2.
cm/sec respectively.

Straight line cot T - wé plots, with intercepts cot T =
1, extrapolated to zero frequency, were obtained through the
frequency range 50-1100cps for all supporting electrolytes
containing Cd(II).

Linearity in the above plot, and agreement between
experimental and calculated cot T values, employing an

appropriate equation derived for the electrode process

-———+
0 + ne E, R

 

is evidence that the electrode process is only diffusion
and charge-transfer controlled.

The order of double-layer capacitance for the anions
of the various supporting electrolyte solutions at a d.c.

potential of 0.6V VS.SCE was as follows:
Cl' > N03_ > c104- > S04: .

This order parallels the order of adsorbabilities of the
above anions on mercury.

The decrease in heterogeneous rate constants for the
reduction of Cd(II) in the above supporting electrolytes

parallels the decrease in double-layer capacitance.

 

LI TERATURE CI TED

 

74

10.
11.

12.

15.

14.

15.

16.

17.

18.
19.

20.

75
Bauer, H. H., Elving, P. J., J. Am. Chem. Soc. 50, 541
(1958).
Ibid., 82, 2091 (1960).
Ibid., 82, 2094 (1960).
Bauer, H. H., Elving, P. J., Anal. Chem. 50, 554 (1958).
Ibid., 50, 541 (1958).
Biegler, T., Laitinen, H. A., Anal. Chem. 51, 572 (1965).
Bjerrum, J., Schwarzenbach, G., Sillen, L., Stability

Constants, Part II Inorganic Ligands, pp. 55, 84, 105,
111, The Chemical Society London, 1958.

 

 

 

Breyer, B., Bauer, H. H., Alternating Current Polarography
and Tensammetry, in Chemical Analysis, P. J. Elving and

I. M. Kolthoff, eds., Vol. 15, pp. 1-25, Interscience,

New York, 1965.

 

 

 

 

Ibid., p. 150.
Breyer, B., Bauer, H. H., Australian J. Chem. 9, 425 (1956).
Breyer, B., Hacobian, S., ibid., 8, 522 (1955).

Deford, D. D., Stabilized Follower Amplifier, Applications
Bulletin, G. A. Philbrick Researches, Inc., Boston, Mass.

 

Delahay, P., New Instrumental Methods in Electrochemistry,
p. 48, Interscience, New York, 1954.

 

Delahay, P., Double Layer and Electrode Kinetics, p. 206,
Interscience, New York, (1965).

 

Ibid., p. 61.

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

Delmastro, J. R., Smith, D. E., J. Electroanalyt. Chem.
2, 192 (1965).

Ibid., Anal. Chem. 52. 169 (1966).
Ershler, B., Faraday, Soc. Discussion 1, 269 (1947).

Frischmann, J. K., unpublished work.

21.

22.
25.
24.

25.

26.

27.

28.

29.
50.

51.

52.

55.

54.

55.

56.

57.
58.
59.

40.

41.

76

Glasstone, S., Laidler, K. J., Eyring, H., The Theory
of Rate Processes, pp. 575—77, McGraw-Hill, New York,
1941.

 

 

Grahame, D. C., J. Am. Chem. Soc. 55, 1207 (1941).
Grahame, D. C., Chem. Rev. 4;, 441 (1947).
Head, w. E., Anal. Chim. Acta fig, 297 (1960).

Kelley, M. T., Jones, H. C., Fisher, D. J., Anal. Chem.
5;. 488 (1959).

Koutecky, J., Chem. Listy $1, 525 (1955).

Lingane, J. J., Loveridge, B. A., J. Am. Chem. Soc. 15,
459 (1950).

Lingane, P. J., Christie, J. H., J. Electroanalyt. Chem.
19. 1284 (1965).

Matsuda, H., Z. Elektrochem. 5;, 489 (1957).
Ibid., 62, 977 (1958).

Mfiller, R. H., Garman, R. L., Droz, M. E., Petras, J.,
Ind. Eng. Chem. 19, 559 (1958).

Nicholson, R. 8., Anal. Chem. 51, 1551 (1965).

Operational Amplifier Symposium, Anal. Chem. 55, 1770—1845
(1965).

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

Randles, J. E. B., Somerton, K. W., Trans. Faraday Soc.
éé. 951 (1952).

Schwarzenbach, G., Complexometric Titration, pp. 55,
85-86, Interscience, New York, 1957.

 

Smith, D. E., Anal. Chem. 55, 602 (1965).
Ibid., 55, 1811 (1965).
Ibid., §§, 962 (1964).

Tachi, I., Senda, M., Bull. Chem. Soc. Japan 55, 652
(1955).

Tanaka, N., Tamamushi, R., Electrochim. Acta 5, 965
(1964).

 
 
 
 
 
 
 
 
 

.b.

 

 

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