TEuE-EQREFICAL EVALUATION OF EFFECTS

QF AMALGAM FQRMATEON 0N

STATEON‘ARY ELECTROEDE POLARQGRAPHY

‘e‘x’i'TH APFUCATEGN YO REDUCTION OF

ALKAU METALS {N ACETQMFRELE

The“: 5011' ”we Degree of M. S.

MECHEGAN STATE UHWEIEETY

Floyd Hilbert Beyerlein

1967

Ihnbm

LIBRARY ' .

Michigan State:

University

ABSTRACT

THEORETICAL EVALUATION OF EFFECTS OF AMALGAM FORMATION

ON STATIONARY ELECTRODE POLAROGRAPHY WITH

APPLICATION TO REDUCTION OF ALKALI

METALS IN ACETONITRILE

by Floyd Hilbert Beyerlein

The theory of stationary electrode polarography has been

extended to include influence of amalgam formation. The

mechanism treated is O+ne a: R(Hg) where the charge transfer

is Nernstian and the electrode is spherical. Effects of

finite electrode volume are shown to be negligible for reason-

able experimental conditions, and therefore have been ignored.

For the single scan experiment results are presented in terms

of a semiempirical spherical correction term. For the cyclic

experiment results are summarized in tabular form. Important

results of the theory include the prediction that the ratio

of anodic to cathodic peak currents is greater than unity.

In addition, enhanced peak potential separations also are

predicted under some conditions.

The theoretical calculations have been tested experi-

mentally for the reduction of cadmium at a hanging mercury

drop electrode, and the agreement between theory and experi-

ment is excellent. In addition, reduction of several of the

Floyd Hilbert Beyerlein

alkali metals in acetonitrile has been studied with stationary

electrode polarography, and the theory of amalgam formation

has been used to explain some apparent anomalies.

THEORETICAL EVALUATION OF EFFECTS OF AMALGAM FORMATION

ON STATIONARY ELECTRODE POLAROGRAPHY WITH

APPLICATION TO REDUCTION OF ALKALI

METALS IN ACETONITRILE

BY

Floyd Hilbert Beyerlein

A THESIS

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

MASTER OF SCIENCE

Department of Chemistry

1967

VITA

Name: Floyd Hilbert Beyerlein

Born: April 15, 1942, in Frankenmuth, Michigan

Academic Career: Frankenmuth High School

Frankenmuth, Michigan--1956-1960

Michigan State University

East Lansing, Michigan-~1960-1964

Michigan State University

East Lansing, Michigan--1964-1967

Degree Held: B. S. Michigan State University (1964)

ii

ACKNOWLEDGEMENT .

The author wishes to express his appreciation to

Professor Richard S. Nicholson for his guidance and en-

couragement throughout this study.

Thanks are also given to Sandra M. Beyerlein, the

author's wife, for her encouragement and understanding.

iii

TABLE OF CONTENTS

INTRODUCTION . . . . . . . . . . . . . . . . . . . .

THEORY O O O O O O O O O O O O O O O O O O O 0 O C 0

Boundary Value Problem. . . . . . . . . . . . .

Integral Equation Form of Boundary Value

Problem. . . . . . . . . . . . . . . . . .

Numerical Solution of Integral Equations. . . .

Results of Theoretical Calculations . . . . . .

Single Scan Method . . . . . . . . . . . .

Cyclic Triangular Wave Method. . . . . . .

“PERIMENTALI O O O O O O O O O O O C O C O O . O O O

Instrumentation . . . . . . . . . . . . . . . .

Potentiostat . . . . . . . . . . . . . . .

Function Generator . . . . . . . . . . . .

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

Chemicals. . . . . . . . . . . . . . . . .

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

Evaluation of the Instrument. . . . . . . . . .

Conventional Polarography. . . . . . . . .

Potentiostatic Electrolysis. . . . . . . .

Stationary Electrode Polarography. . . . .

Comparison with Experiment of the Theoretical

Calculations for Amalgam Formation . . . .

Electrochemistry of Alkali Metals in Aceto-

nitrile. . . . . . . . . . . . . . . . . .

Conventional Polarography. . . . . . . . .

Stationary Electrode Polarography. . . . .

Comparison of Stationary Electrode Polar-

ography of Alkali Metals with Amalgam

Formation Theory. . . . . . . . . . .

CONCLUSION . . . . . . . . . . . . . . . . . . . . .

LITERATURE CITED 0 O O O C O O O O O O O O O O O O O

APPENDICES . . . . . . . . . . . . . . . . . . . . .

iv

Page

13

15

22

29

29

29

55

58

58

59

59

59

4O

4O

41

47

47

48

48

53

54

56

TABLE

II.

III.

IV.

LIST OF TABLES

Page

Empirical Spherical Correction Parameters as a

Function of Potential. . . . . . . . . . . . . . 25

Peak Potentials and Ratio of Anodic to Cathodic

Peak Currents as a Function of $0. . . . . . . . 25

Peak Potentials and Ratio of Anodic to Cathodic

Peak Currents as a Function of 7 . . . . . . . . 26

Peak Potentials and Ratio of Anodic to Cathodic

Peak Currents as a Function of EA' . . . . . . . 27

Variation of Triangular Wave Frequency with Time

Constant of Integrator . . . . . . . . . . . . . 57

FIGURE

LIST OF FIGURES

Theoretical cyclic polarograms showing effects

of $0 with 7 = 1. . . . . . . . . . . . . . . .

Spherical correction as a function of potential

Variation of spherical correction with $0 for

v = 1 O O O O O O O O O O O O O O O O O O O O 0

Circuit diagram of potentiostat . . . . . . . .

Circuit diagram of function generator . . . . .

Stationary electrode polarogram for reduction

of ferric oxalate . . . . . . . . . . . . . . .

Stationary electrode polarogram for reduction

Of cadnlium. O O O O O O O O O O O O O O O O O O

Stationary electrode polarogram for reduction

of sodium in acetonitrile . . . . . .’. . . . .

vi

Page

15

18

21

51

55

45

46

51

LIST OF APPENDICES

APPENDIX 4 Page

A. Reduction of Boundary Value Problem to

Integral Form. . . . . . . . . . . . . . . 57

B. Reduction of Boundary Value Problem to Two

Simultaneous Integral Equations. . . . . . 65

C. Relation of the Functions (é! _ and5U or r-ro

(BE-r=ro to Current. . . . . . . . . . . . 65

D. Computer Program . . . . . . . . . . . . . 66

vii

INTRODUCTION

The original objective of this research was to extend

the technique of stripping analysis (27) to determination of

trace concentrations of alkali metals in nonaqueous solvents

such as acetonitrile. Stripping analysis consists first of

a constant potential concentration of the metal ion in a

mercury microelectrode, such as the hanging mercury drop

electrode (27). After the deposition step, the concentration

of metal in the electrode is determined by stationary electrode

polarography.

In contrast with conventional polarography the influence

of ohmic potential losses on stationary electrode polarog-

raphy cannot be eliminated by simple application of Ohm's law

(9,15,17). Therefore, with stationary electrode polarography

it is essential that ohmic potential losses be negligible. In

principle this could be a serious problem when working in

nonaqueous solvents where low conductivities are encountered.

However, by using a three electrode configuration, it is

possible to compensate electronically for ohmic potential

losses (2). Therefore, before studies of stripping analysis

were begun an electronic instrument for recording stationary

electrode polarograms was constructed. A description of this

instrument together with its evaluation based on electronic

tests and chemical experiments is described in a later sec-

tion.

Before attempting to analyze trace concentrations of

alkali metals, conventional cyclic stationary electrode

polarographic experiments with a hanging mercury drop electrode

were performed on millimolar solutions of the metals in aceto-

nitrile. Although in general the alkali metals were well-

behaved, peak potential separations (difference of cathodic

and anodic peak potentials) were considerably greater than

the 57/n_mv. usually assumed for reversible electron transfer

(19). This increased peak potential separation could not be

accounted for by ohmic potential losses, because of the

electronic compensation of ig_drop mentioned above. The ob—

served peak potential separations also could not be explained

in terms of kinetic effects of the electron transfer, because

the peak potential separations were independent of scan rate

(18). A third possibility was the fact that amalgam formation

was taking place. All previous theoretical treatments of

stationary electrode polarography--including the calculation

of 57/g_mv. peak potential separations for reversible electron

transfer—-have been based on plane electrode geometry, or in

a few cases on spherical electrode geometry. In every case,

however, effects of amalgam formation have been ignored.

Ignoring amalgam formation for a plane electrode is justified

because no mathematical distinction exists between the cases

of Species soluble in the solution phase or electrode phase

(6). Most applications of stationary electrode polarography,

however, involve the use of spherical electrodes (hanging

mercury drop electrode), and in this case a mathematical dif-

ference does exist between the two caSes (28). In spite of

this fact previous theoretical treatments of spherical

electrodes have not considered amalgam formation (19), on the

assumption that sphericity would be the only important effect.

Recent work in other areas, however, indicates that this

assumption may be in serious error. For example, Stevens

and Shain (50) for the case of potentiostatic electrolysis

and Delmastro and Smith (8) for the case of a.c. polarography

have shown that in some cases consideration of amalgam forma-

tion is essential to correct interpretation of experimental

results. Therefore, effects of amalgam formation appeared

to be a possible explanation for the observed behavior of

the alkali metals.

In addition to the effects described above, it recently

has been shown that the kinetics of exchange reactions of

metal ions with ligands such as EDTA can be studied by oxi—

dizing the metal amalgams from a hanging mercury drop elec-

trode with stationary electrode polarography (15,29). In

each of these cases, however, theory was used which ignored

effects of amalgam formation, again on the assumption that

these effects would be unimportant.

For these reasons it seemed important to investigate

quantitatively the effects of amalgam formation for stationary

electrode polarography with spherical electrodes before de-

veloping stripping analysis methods. Therefore, the major

portion of this thesis reports on the theory of stationary

electrode polarography for amalgam formation, and as will be

shown these effects can be very important to correct interpre-

tation of polarographic curves.

THEORY

To treat rigorously the case of amalgam formation

0 + ne ‘——. R(Hg) I

for stationary electrode polarography with a spherical elec-

trode appears to be very difficult (8,25,50). However,

following the discussion of Reinmuth (25) the problem can be

simplified greatly by ignoring the influence of finite elec-

trode volume and considering only the effects of sphericity

and the convergent nature of the diffusion process. This

restriction is perfectly justified for the case of stationary

electrode polarography with a hanging mercury drop electrode,

because Reinmuth has shown that for typical electrodes,

effects of finite volume become important only for electroly-

sis times of the order of 40 seconds. For stationary electrode

polarography this corresponds roughly to scan rates of 8

mv./sec. or slower. Scan rates of this magnitude are at least

a factor of three smaller than the slowest scan rates normal-

ly employed. Therefore, the mathematical treatment which

follows neglects finite electrode volume according to Reinmuth's

suggestions.

The only mechanism to be considered is I where the charge

transfer is assumed to be Nernstian. Although the more

general case could have been treated, the most pronounced

effects of amalgam formation would be expected for the

reversible case. Therefore, the treatment of the reversible

case should serve to define qualitatively all of the trends

to be expected for more complicated cases.

Boundary Value Problem

The boundary value problem based on Fick's diffusion

equations for a spherical electrode, and considering the

restrictions cited above, is

§99 = BZCQ 2. 8C

8t D0[ or + r EEG] (1)

5CR _ 520R 2 5C (2)gr "DR[§FZ’+F 5:31

t = O; r 2_ro Co = 00* (5)

t=07r20CR-O

(4:)

t>o;r—>oo co—->co* (5)

t>O;r—h0 cR—»o (5)

. _ 5 ._ 5C (7)t>0,r-ro DOB-go-DR-a-ER

99.: BE. _CR eXPI-(RTHE E0)] (8)

In the above equations §_represents concentration as a

function of time, t, and radial distance, r, from the center

of a sphere of radius, £9. 90* is the initial analytical

concentration of the oxidized species, 0, and it is assumed

that the reduced substance, R, is generated in_situ (see

Equation 4). The remaining terms and equations have their

usual significance and are discussed by Reinmuth (24).

Equation 8 is the Nernst equation consistent with our

assumption of reversible electron transfer. For the case of

stationary electrode polarography the potential in Equation 8

is a triangular wave function of time, that is

E=Ei-vt o<tgx (9)

E=Ei-2v)\+vt tZA (10)

where E1 is the initial potential, y_is the scan rate (dE/dg),

and A_is the time at which the direction of potential scan is

reversed.

By substitution of Equations 9 and 10 in Equation 8,

boundary Equation 8 can be written in the abridged form

t > o; r = r0 a? = e Sx(t) (11)

where e = exp [(%%)(Ei - E°)] (12)

exp(-at) at g.aA

Sk(t) = (13)

exp(at-Zak) at 2_aA

and a = fi¥l . (14)

The above boundary value problem cannot be solved ana-

lytically because of the nonlinear and discontinuous nature

of Equation 11. Nevertheless useful numerical solutions can

be obtained as described in the following sections.

Integral Equation Form of Boundary Value

Problem

Although the preceding boundary value problem can be

solved only by numerical methods of analysis, the numerical

treatment is greatly simplified by first reducing the boundary

value problem to integral equation form. Reduction to a

single integral equation is developed in Appendix A. The

result is (Equation A29):

1 t

CO*7JB:=fO fO(T){J?Tt—?T—-— -g-exp[%2(t-T)]erfC(J§E¢t-T ])dT

”SA—98%(t) t fo (T {WE-2:)— “ID—R eXP[LR2 (t-rHerfcfw—RJt-TJ

+OZJDR

r exp [- 232 (t-T)]] dT . (15)

o 0

Because of the complex form of the kernels in integrals of

Equation 15, direct numerical solution of Equation 15 would

be very difficult, and therefore an alternate approach was

sought. This alternate approach, which is described in

Appendix B, resulted in a set of two simultaneous integral

equations, each of which was considerably less complex than

Equation 15. This system of equations is (Equations B4 and B5):

av68 (tN. DR ( —) ”ad'r

A ft or r= = roC0* + 2Co*r56' Vt

JV 0 Jt-Tr J}?

in (16)

-*JD‘ t ( r )r=rodT

J-T: o Vt-T

.1 _ _ _ (_D_Q _ DR ).es)\(t)~/DR

r0 EE-r=ro r0 5r r=ro r02 rozesx(t) ~f__

v

av

ft (31—; r=erT

0 Jt—T (17)

All of the kernels in Equations 16 and 17 are identical and

considerably less complex than those of Equation 15.

Therefore, Equations 16 and 17 are more amenable to numerical

solution than Equation 15.

Numerical Solution of Integral Equations

For numerical solution of Equations 16 and 17, it is

important to have these equations in dimensionless form so

that results are not dependent on particular values of experi-

mental parameters (A, Q, 2, etc.). Reduction to dimensionless

form can be accomplished with the following change of variable

T = z/a (18)

and the following definitions

y = at (19)

_~/‘n$

250 ' :07; (20)

Do

7 =' i; (21)

U_ 0 5m

x (y) — comf? (5r,r=ro (22)

w (y) = 919—— §Y—) (25)

wow; 5r r=ro

10

Equations 16 and 17 now become

(z)dz _ 2Jy U y X (zzdzGS (y) y 2_______ 1 + _____Q.- 24

ax (om T (om H

and

= [7¢0953x(y) 'go/PY] fy ‘i’(Z) dz

Jfir- 0 y-z

(25) 7x(y)-Y(y)

respectively. The functions x(y) and Y(y), once calculated,

are related to current by the following expressions which_are

derived in Appendix C (Equation C2):

i = nFACBJaDO F(y) (26)

where

_ _ _ _ 1(1) 95 1(y)F(y) —-J77'[X(y) (x(y) v ) (982%(y)-%2)] (27)

Values of F(y) are directly related to potential by recalling

the definition of Sak(y) (Equation 15)

(E - E°)n = %1 ln esax(y). (28)

Interestingly the functional dependence of current on

experimental parameters can be deduced directly from Equation

26 without actually solving the integral equations. Thus,

current is directly proportional to bulk concentration,

Also, dependence of current on electrode geometry and

hence amalgam formation is embodied in the dimensionless

1 (Equation 20). The magnitude of U in turnparameter, g; -0

11

depends on the experimental parameters £9 and 3% (see

Equations 14 and 20). Thus, as either £9 or y_increases,

.go approaches zero. With g0 zero, Equation 25 reduces to

7x(y) = Y(y). , (29)

Equation 29 substituted into Equation 27 gives

Fo(y) =IJET'x(y) (50)

where the subscript on the function, Fo(y), is taken to mean

the general current function, F(y), evaluated for‘gO equal

zero. Combination of Equations 29 and 50 with Equation 24

results in the following single integral equation

fY EQLELQE = 1

O “(y-Z 1+795a)\(y)(51)

Thus, for the case of go

calculated directly from Equation 51. Equation 51 is an Abel

sufficiently small, currents can be

integral equation, the closed form solution of which has been

given previously (19). In addition, Equation 51 is exactly

the equation which describes the case of a planar electrode

of semi-infinite volume (19). Reduction of Equations 24 and

25 to this case is entirely reasonable, because go approaches

zero as £9 approaches infinity in which case the electrode

would become a plane of semi-infinite volume. For this case

no distinction can be made between R soluble in the electrode

or solution phase (6).

12

Results of these observations are twofold. First a

check of the numerical solutions of Equation 24 and 25 is

provided, because these results must reduce to previously

published solutions of Equation 51 as g0

Also, the results indicate that the parameter

approaches zero.

g0 can be

regarded as a spherical correction term which simultaneously

includes effects of amalgam formation. Interestingly this

parameter is of exactly the same form as the spherical term

derived by Reinmuth (25) for the case of diffusion to a

spherical electrode, but with both 0 and R soluble in the

solution phase. The differences between these two cases is

discussed in the following section.

Aside from these limiting cases the exact form of the

current-potential curves and their dependence on go can only

be obtained through solution of Equations 24 and 25 explicitly

for x(y) and F(y).

Equations 24 and 25 were solved by two different numeri-

cal techniques, the step functional method (19) and the

method due to Huber (11). Solutions obtained with both methods

converged to the same values, but because Huber's method is

inherently more accurate, most of the results reported here

were obtained by that method. All calculations were performed

on the Michigan State University Control Data 5600 digital

computer, and the FORTRAN program for Huber's method is

listed in Appendix D.

15

Results of Theoretical Calculations

Results of the numerical solution of Equations 24 and 25

in terms of the current function, F(y), for two values of

g6 and 1_equal one are shown in Figure 1. The curve for

Q6 = 0.001 is identical (within 1%) to the previously pub-

lished solutions of Equation 51 discussed in the preceding

section. In addition, for values of $6 3 0.001, the solution

of Equations 24 and 25 was found to be independent of values

of 1_and 9, provided 79 was larger than EEE (6.5). This ob-

servation also is consistent with previous solutions of

Equation 51 (25) and simply corresponds to the fact that

polarograms are independent of initial potential provided the

initial potential is sufficiently anodic.

Further discussion of results of the numerical calcula-

tions is most conveniently divided between the single scan

and cyclic experiments.

Single Scan Method. Effects of amalgam formation can be

treated quantitatively only for the single scan method be-

cause of the complicating influence of switching potential

for the cyclic experiment. To determine the influence of

amalgam formation on the single scan experiment, calculations

were performed in which both E and 1_were varied independently._O

From variations of 1_with fixed gb effects of 7 were

found to be relatively minor causing primarily small variations

of the cathodic peak potential. For this reason and because

Figure

1.

Theoretical

cyclic

polarograms

showing

effects

of

gwith

y=

1.

Dashed

lines

are

extensions

of

cthodic

scans.

14

(KM

0.2--

0.01—-

15

l

lJ

|

40

o-40

-80

-120

(E-EJ!)n,

mv.

Figure

1

16

20 and ER usually are not markedly different, the remaining

discussion is limited to the value of 1_equal one. In addi-

tion only values of go in the range of 0<g0<0.1 are con-

sidered, because this includes all experimentally reasonable

values of the parameter (19).

To evaluate effects of amalgam formation it would be

useful to compare results of the solution of Equations 24 and

25 with results of Reinmuth's work where amalgam formation

was ignored (both 0 and R soluble in the solution phase).

Reinmuth's results are presented in terms of a "spherical

correction" which must be added to values of the current for

the case of a plane electrode (Equation 26 with F(y)=Fo(y)).

Thus, solutions of Equations 24 and 25 can be treated in this

manner simply by subtracting at identical potentials values

of Fo(y) (see Equation 50) from values of F(y). This dif-

ference, referred to as "spherical correction" is plotted

versus potential in Figure 2 for two values of go. Also in—

cluded in Figure 2 are the analogous spherical correction

terms due to Reinmuth.

From the data of Figure 2 amalgam formation apparently

has a marked influence on the spherical correction term.

For example, in the absence of amalgam formation the spherical

correction is always a smooth sigmoid-shaped curve which is

everywhere positive. In other words in the absence of

amalgam formation currents at a spherical electrode always

are larger than they would be for a plane electrode. This

Figure

2.

Spherical

correction

as

afunction

of

potential.

Solid

line-amalgam

formation

Dashed

line-no

amalgam

formation

17

uorqoelloo Teorlequ

0.10

0.08

0.06

0.04

0.02

0.00

-30

—80

(E5E%)n,

mv.

Figure

2.

-150

-180

18

19

result is in sharp contrast to the case of amalgam formation

where the spherical correction is negative for potentials

anodic of E9, and is positive for potentials cathodic of E9.

Moreover, the potential region over which the correction

ranges from negative to positive is relatively small (ca. 50

mv.), which accounts for the distorted appearance of some of

the polarograms (see Figure 1). Also, the magnitude of the

spherical correction is actually larger near the peak than

the maximum value of Reinmuth's correction, but approaches

Reinmuth's value at negative potentials. Thus, the assump-

tion often made that sphericity should have little effect on

peak currents (20) is not valid when amalgam formation is

present.

Although the data of Figure 2 indicate the importance of

amalgam formation, unfortunately such data are not very use-

ful in practical terms because every value of go encountered

experimentally would require a new computer solution of

Equations 24 and 25. In an effort to circumvent this problem

the dependence of the Spherical correction term on g6 at

fixed potential was investigated, and typical results are

shown in Figure 5. From Figure 5 it can be seen that the

spherical correction is nearly a linear function of D ,

except for some potentials where deviation from linearity is

observed for large values of go. Because these deviations

from linearity are never large, however, it is possible to fit

(within 1%) all of the data of Figure 5 to a parabola, and,

thereby define empirically a spherical correction as

Figure

5.

Variation

of

spherical

correctionwith

$0

for

7

(a)(E-Eé)n,

mv.

1.

20

0008‘—

0.06_

uorqosxxoo Teorxaqu

0.00

88.65(a)

-9.82

1.28

129.75

29.54

1L

Ll

0.00

0.01

0.02

0.03

Figure

5.

0.04

0.05

0.06

0.07

0.08

0.09

21

22

spherical correction = a0: + 600. (52)

There g_and §_are coefficients of the parabola and are func-

tions of potential only. With the aid of Equation 52, there-

fore, it is possible to define the current function, F(y),

of Equation 26 as follows

F(y) = Fo(y) + agg + 000 . (53)

Since values of Fo(y) can be found in the literature, with

the aid of Equation 55 it is a simple matter to calculate

currents (see Equation 26) for amalgam formation provided the

constants g_and Q are known. These values of g_and Q,

accurate within 1%, are listed in Table I. For convenience

the values of Fo(y) are also included. It should be emphasized

that the data of Table I are strictly applicable only forjl

equal one and g0.g 0.1, but as already mentioned this includes

most cases of interest.

Cyclic Triangular Wave Method. Data of Figure 1 show

that amalgam formation is especially important for a cyclic

experiment--that is, the anodic portion of the curve is even

more strongly influenced than the cathodic portion. The

reason for this is the product of electrolysis, R, is confined

to the finite volume of the electrode. Thus, even though the

diffusion process during oxidation is divergent, the actual

anodic current is larger than it would be in the absence of

amalgam formation, and the ratio of anodic to cathodic peak

Table I.

1

__

25

Function of Potentiala

Empirical Spherical Correction Parameters as a

(E-Eé)n, mv. Fo(y) a B

120 0.009 -0.004' -0.018

100 0.020 -0.005 -0.024

80 0.042 -0.011 -0.056

60 0.084 -0.055 -0.065

50 0.117 -0.068 -0.085

45 0.158 -0.095 -0.087

40 0.160 -0.122 -0.090

55 0.185 -0.162 -0.086

50 0.211 -0.211 -0.082

25 0.240 -0.270 -0.066

20 0.269 -0.540 -0.041

15 0.298 -0.424 -0.005

10 0.528 -0.507 0.042

5 0.555 -0.594 0.100

0 0.580 -0.676 0.174

-5 0.400 -0.749 0.261

-10 0.418 -0.805 0.548

~15 0.452 -0.842 0.459

-20 0.441 -0.861 ‘0.520

-25 0.445 -0.862 0.615

-28.50 0.4465 -0.848 0.670

-50 0.446 -0.841 0.692

-55 0.445 -0.800 0.768

-40 0.458 -0.740 0.845

-50 0.421 -0.607 0.951

-60 0.599 -0.476 1.019

-80 0.555 -0.258 1.098

-100 0.512 -0.095 1.107

-120 0.280 -0.021 1.097

-150 0.245 -0.002 1.080

aCurrent for a spherical electrode is given by

i .

= nFACSJaDO [Fo(y) + aD0

r0 a

7155+ B

r07?

]

24

currents (anodic peak currents are measured to the extension

of the cathodic curve (19)) can be greater than unity.

Tables II, III, and IV summarize the effect of the three para—

meters 0 and §_ (switching potential) on the ratio of—0’ 1’ A

peak currents. The conclusion drawn from these data is that

whenever Sphericity is important, the ratio of peak currents

will be larger than unity. This fact is especially important

because previous results indicated that only kinetic effects

could cause the ratio to differ from unity (19). In fact,

it was suggested that this ratio be used as a diagnostic test

to demonstrate the presence or absence of coupled chemical

reactions. In light of the present results, however, these

diagnostic tests must be revised. The ratio of anodic to

cathodic peak currents also has been used to measure homo-

geneous rate constants (15,29). It now is clear that such

measurements can be in error if amalgam formation is involved.

Tables II, III, and IV also summarize effects of the three

parameters 0 , y, and E_ on peak potential separations. These

A

effects on peak potentials all are consistent with the

Nernstian model assumed for the electrode process. For ex-

ample, for constant g0 and §_ (Table III) increases of theA

parameter y_(VQO.WJQR) cause both peaks to shift anodically.

This result is reasonable because an increase of 1_corres-

ponds to a decrease of in which caSe the surface concentra—-D-R

tion of R relative to 0 would increase (R diffuses into the

electrode more slowly). This effect causes a Nernstian shift

Table

II.

Peak

Potentials

and

Ratio

ofAnodic

to

Cathodic

Peak

Currents

as

a

Function

of

00

a

mv.

i/i

go

(Epc‘

E%)n,

mv.

pl

pa

pC

(E

-E

)n,

mv.

nAE

Pa

'2

0.001

0.01

0.02

0.04

0.06

0.08

0.10

-29.54

-29.54

-52.11

-54.68

-54.68

-57.25

-59.82

29.54

29.54

29.54

26.97

26.97

24.40

21.84

59.08

59.08

61.65

61.65

61.65

61.65

61.66

1.00

1.05

1.10

1.21

1.52

1.45

1.56

25

(EA

-Eé)n

=-129.75

mv.

and

v=

1.

Table

III.

Peak

Potentials

and

Ratio

of

Anodic

to

Cathodic

Peak

Currents

as

a

Function

of

7a

(E

-E%)n,

mv.

Pc

Pa

(E

-Eyn,

mv.

nAEp,

mv.

-r'l

Pa

Pc

-55.47

-42.21

-40.10

-57.25

-56.51

-57.56

-57.42

-57.54

-55.51

8.75

19.44

21.56

24.40

25.54

26.66

26.80

26.68

28.91

64.22

61.65

61.66

61.65

61.65

64.22

64.22

64.22

64.22

10.17

1.96

1.75

1.45

1.55

1.28

1.26

1.25

1.24

a

(El

-Eyn

-129.75

mv.

and

00

=0.08.

26

Table

IV.

Peak

Potentials

and

Ratio

ofAnodic

to

Cathodic

Peak

Currents

as

a

Function

of

EA

a

(EA

-E§)n,

mv.

(Epa

-E%)n,

mv.

nAE

,mv.

-70.64

26.97

-91.19

26.97

-129.75

24.40

-150.28

24.40

-170.85

24.40

64.52

64.52

61.95

61.95

61.95

1.29

1.55

1.45

1.49

1.54

a00

=0.08

and

y=

1.

b

27

28

of the waves along the potential axis to more positive values.

Since both cathodic and anodic peaks shift simultaneously,

the difference of the peak potentials remains relatively

constant. Similar interpretations can be given for the in-

fluence of 0-0 and EA on peak potentials.

EXPERIMENTAL

Instrumentation

As described in the Introduction electrochemical experi-

ments performed in nonaqueous media require electronic compen—

sation of ohmic potential losses. Because no suitable .

instrument was available commercially, one was constructed

from solid state Operational amplifiers (Nexus Model SA-1

except for amplifier §_of Figure 4 which was Philbrick Model

P-2). Power for all of the operational amplifiers (i.15 volt,

400 ma.) was supplied by two Elcor zener-regulated power

supplies (Elcor Electronics, Type A215-400). These power

supplies were selected because of their excellent isolation

characteristics from ground.

The instrument which was constructed consisted of essen-

tially two different sections, the three electrode potentio—

stat and the function generator. For convenience each of these

sections is discussed separately.

Potentiostat. A block diagram of the potentiostat is

shown in Figure 4. The circuit is of conventional design and

its operation is described elsewhere (5). The load resistor,

3L, used to control current sensitivity was a Heath Model

EUW-50 decade resistance box.

29

Figure

4.

Circuit

diagram

of

potentiostat.

RE:

CE:

WE:

F.G.:

Ei:

SIG.:

C.A.:

F:

C.F.:

RL:

35b:

Reference

electrode

Counter

electrode

Working

electrode

Function

generator

input

Initial

potential

Extra

input

available

for

additional

signal

sources

Control

amplifier

Voltage

follower

amplifier

Current

follower

amplifier

Load

resistor

(decade

resistance

box)

Connects

amplifier

Fto

the

recorder

50

100K,1%

F.G.-W-

100KI1%

0.01

f.

E1cyAmww—

”

100K,1%

II

.A

.5b

IF

100K.

1%

CE

FW—

RE

WE

100

Pf,

1)

,__

RECORDER

[4h

Figure

4.

51

52

The d.c. operating characteristics of the potentiostat

were evaluated first with the aid of a resistive dummy cell.

An accurately known d.c. potential was applied at the input

labeled E5 of the control amplifier (C.A.) with a portable

precision potentiometer (Biddle Gray Model CAT. 605014).

The potential then was measured with a potentiometric re-

corder (Sargent Model SR) at the output of the current

follower (amplifier C.F.) for various values of 3L. These

potentials always were consistent with the potential at Ei

within the 1% tolerance of the resistors used.

To ensure that frequency response of the potentiostat

was adequate for the experiments to be performed (frequencies

used never exceeded 25 Hz.) the rise time of the potentiostat

also was measured.

The high frequency characteristics of a potentiostat are

determined by the bandwidth of the operational amplifiers and

their output current capabilities (2). Because amplifier E

had a relatively small bandwidth (75 KHz compared with 1.5

MHz for the others) and maximum current outputs were.i 2 ma.,

frequency response of the instrument was expected to be

fairly limited.

The rise time of the potentiostat was measured with a

real cell which contained millimolar ferric ion in 0.2 M

potassium oxalate-—0.2 M oxalic acid supporting electrolyte.

The working electrode (WE) was a hanging mercury drop and the

reference electrode (RE) was a saturated calomel electrode

55

with a Luggin capillary probe. All inputs in Figure 4 were

grounded except the one labeled "SIG." at the input of

amplifier C.A.: this input was connected to a pulse generator

(Tektronix Type 161). The pulse generator provided a gate

(rise time 0.5 microseconds) of 0.5 volt amplitude, which was

sufficient to drive the potential of the working electrode

well into the limiting current region for reduction of ferric

oxalate. Rise time was then determined as the time required

for the potential between the working and reference electrodes

to reach 0.45 volts (90% of the potential applied at the in—

put of the potentiostat). The rise time determined in this

manner was 5 milliseconds, which corresponds to a useful

bandwidth of about 55 Hz (2). Thus, the bandwidth of the

instrument was suitable for all of the experiments described

in this thesis.

Function Generator. The circuit diagram for the function

generator is shown in Figure 5. This function generator is

similar to one described by Underkofler and Shain (51), and

provides triangular and square waves in either a triggered or

free—running mode. The circuit of Figure 5 actually differs

in two major respects from the one of Underkofler and Shain.

These differences are: (1) a ten turn potentiometer was in—

serted in the integrator circuit (amplifier I) so that

frequencies between various settings of the fixed input

resistance and feedback capacitance could be obtained; and

(2) the amplitude of the square and triangular waves was

Figure

5.

Circuit

diagram

of

function

generator.

D.A.:

I:

A:

131,132:

.033

Sla‘

81b:

32:

35a:

55:

S6:

Difference

amplifier

(square

waves)

Integrator

amplifier

(triangular

wave)

Attenuator

amplifier

Zener

diodes

Solid

state

diode

Controls

polarity

of

trigger

Puts

diode,

D3,

in

and

out

of

the

circuit

Applies

trigger

to

amplifier

D.A.

for

single

shot

mode

(D3

in

circuit)

Selects

square

or

triangular

wave

Changes

input

resistance

of

amplifier

I

Selects

value

of

feedback

capacitor

for

amplifier

I

54

4.7K

:1K

(TEN

TURN)

50K

100K

1II

D1

1

_\W

TIN1515A) D2_

5a

10K

TRIGGER

0A

TO

POTENTIOSTAT

10K

(TEN

TURN)

1K

Figure

5.

55

56

controlled by an operational amplifier attenuator, rather

than a potentiometer. The reason for the latter modification

was to ensure that the output impedance of the function

generator always would be negligibly small compared with the

input resistors of the control amplifier of the potentiostat.

To evaluate the function generator the frequency of the

triangular waves was determined as a function of changes in

resistance and capacitance in the integrator circuit. The

frequencies were measured with a memory oscilloscope (Tektronix

Type 564), and the results are shown in Table V. Except for

very high frequencies the frequency change is tenfold for

each tenfold change in R_or 9, At fixed values of R_and Q

the ten turn potentiometer at the integrator input could be

used to vary the total input voltage to the integrator by a

factor of ten, so that in principle the frequency also could

be varied continuously by a factor of ten. In practice this

did not work for input resistances which were comparable with

the resistance of the potentiometer, because in this case not

only the input voltage was changed but also the total input

resistance to the integrator was changed. The effect of this

was to cause frequency to be a nonlinear function of the

position of the center tap on the potentiometer for small inte-

grator resistors. This effect presumably could be eliminated

by placing a voltage follower between the potentiometer and

switch S5 to provide impedance matching with the integrator

input. The follower was not actually installed, however,

57

Table V. Variation of Triangular Wave Frequency with Time

Constant of Integrator

R, ohms C, uf. RC f, Hz.

10.0M 1.0 10.0 0.00526

10.0M 0.1 1.0 0.0515

1.0M 1.0 1.0 0.0515

1.0M 0.1 0.1 0.515

100.0K 1.0 0.1 0.466

100;0K . 0.1 0.01 4.66

1020K 1.0 0.01 2.65

10.0K 0.1 0.001 24.40

58

because the instrument functioned satisfactorily for our

purposes--i.e., frequency was essentially infinitely variable.

Cell and Electrodes. The cell and electrodes were of

conventional design and are described elsewhere (1).

Chemicals. All chemicals were reagent grade and used

without further purification with the following exceptions.

Acetonitrile (Matheson, Stock No. 2726, B.P. 80.5-82.50C) was

purified by distillation according to the procedure of Mann

(21). The tetraethylammonium perchlorate, which was used as

supporting electrolyte for work in acetonitrile, was prepared

by metathesis of tetraethylammonium bromide with sodium

perchlorate according to the procedure of Kolthoff (12).

The product was recrystallized three times from water and

dried at 800C.

RESULTS AND DISCUSSION

Evaluation of the Instrument

To ensure that the instrument which had been constructed

would give reliable results, it was evaluated experimentally

with several different techniques and chemical systems

which already had been well characterized. Results of these

experiments, which are described below in terms of the dif-

ferent techniques employed, indicated that the instrument

gave reliable results within 1% in every case.

Qpnventional Polarography. The polarographic reduction

of cadmium was studied. This system has been characterized

in terms of half wave potential and diffusion current con-

stant under a number of different conditions, and results

can be found in the tabulation by Meites (16). We chose the

system 1.0 x 10‘3M cadmium in 0.10M_potassium nitrate,

buffered with 0.10M_acetic acid and 0.1QM_§odium acetate.

Polarograms were recorded with a potentiometric recorder

(Sargent Model SR) without damping (1 second pen response),

and a scan rate of 4.0 mv./ sec. The dropping mercury elec-

trode was calibrated in the usual way (14) and diffusion

current constants were calculated from maximum currents.

The half wave potential obtained was -0.577 volt Kg. S.C.E.

59,

40

and Id was 5.55 as compared with the values -0.58 volt Kg.

S.C.E. and 5.55 given by Meites.

Potentiostatic Electrolysis. This experiment was per-

formed with the instrument used as a.potentiostat, the

applied potential being derived from the square—wave section

of the function generator. The working electrode was a

hanging mercury drop of radius 0.0599 cm. The system investi-

gated was 1.0 x 10'3M ferric oxalate with 0.2 M potassium

oxalate and 0.2 M oxalic acid; the measured pH was 2.50.

For potentiostatic electrolysis under limiting current

conditions with a spherical electrode of radius, £9, current

is given by the expression (28)

i = nFADcZ§[ «hr—5t-

é‘

.1r

] . (54)

0

Thus, a plot of i_versus 3f should be linear with slope and

intercept proportional to the diffusion coefficient, 2,

For the ferric oxalate system all such plots were found to

be linear. The diffusion coefficient calculated from the

slope was 7.05 x 10’6 cm.2/sec. compared with the literature

value (22) of 6.51 x 101‘6 cm.2/sec.

Stationary Electrode Polargggaphy. The instrument also

was evaluated with stationary electrode polarography, because

this technique was used for most of the experimental work

reported in this thesis. The ferric oxalate system described

in the preceding section was studied, since it is known to

41

behave reversibly (22). A typical stationary electrode

polarogram for a single cathodic scan is shown in Figure 6

(points). Also included in Figure 6 is a theoretical polaro-

gram (solid line) calculated from literature data with the

aid of the diffusion coefficient determined in the preceding

section. The theoretical data were placed arbitrarily on the

potential axis to coincide with the experimental peak poten-

tial. This fit corresponds to an apparent half wave potential

of —0.209 volt lg. S.C.E. compared with the literature value

of —0.198 volt Kg. S.C.E. (22).

Comparison with Experiment of the Theoretical

Calculations for Amalgam Formation

To check experimentally the validity of the theoretical

calculations for the case of amalgam formation requires a

system for which both go and 2R are accurately known so that

.go and 1_could be calculated. Moreover, the electrode re-

action should be Nernstian to conform with the model on which

the theoretical calculations were based. Although there are

very few systems for which all these data are known,

fortunately the system cadmium-cadmium amalgam, which is

Nernstian, was recently studied carefully by Stevens and

Shain (50). For millimolar cadmium in a supporting electro-

lyte of 1.0 M_potassium chloride buffered with 0.1 M acetic

acid and 0.1 M sodium acetate, Stevens and Shain found

Figure

6.

Stationary

electrode

polarogram

for

reduction

of

ferric

oxalate.

Lines,

theoretical.

Points,

experimental.

Scan

rate

25.0

mv./sec.

42

(E)O

*DVJu/T

0.4....

0.5(__

0.2

__.

0.0

__

0

'I

ll

0

J

—--(o

.20

-0.25

-0.50

.S.C.E.,

volts

-0.10

-0.15

-

E

“’l.'>

Figure

6.

-0.55

45

44

0.755 x 10'5 cm.2/sec.D

ll

0

DR = 1.6 x 10’5 cm.2/sec.

y = /.%0.= 0.678.

. R

Because of the availability of these data reduction of cadmium

under experimental conditions identical with those of‘Stevens

and Shain was selected for quantitative comparison with theory.

A number of current-voltage curves were recorded on a

X-Y recorder (Honeywell Model 520), and a typical one is

shown in Figure 7 (points). The experimental curve of Figure

7 was obtained at a scan rate of 21.5 mv./sec. with a hanging

mercury drop electrode of radius 0.0676 cm. From these

values of experimental parameters the value ofgO is calcu-

lated to be 0.051.

To compare the experimental curves of Figure 7 with

theory, the current function, F(y), (see Equation 27) was

calculated for 1_= 0.678 and = 0.051. With these valuesgo

of F(y), current was calculated from Equation 26 using the

above values of electrode area, diffusion coefficient, and

scan rate with the number of electrons, g, equal 2. The

theoretical current calculated in this fashion has been in-

cluded in Figure 7 (solid line). For Figure 7 the theoreti-

cal polarogram was shifted arbitrarily along the potential

axis to obtain the best agreement between theoretical and

experimental peak potentials. This best fit corresponded to

Figure

7.

Stationary

electrode

polarogram

for

reduction

of

cadmium.

Lines,

theoretical.

Points,

experimental.

45

1.5-O—

10.0

5.0-__

2T1";

u

0.0-—-

c

-10.0

—_. -0957

“0.59

0

O

0

0

I;

‘I

lb

I

31

‘O°51

-0.65

O

0

0

G

0

Figure

7,

-0.65

Volts

~—

EVS.

S.C.E.,

-0.67

-0.69

46

47

an apparent half wave potential of -0.645 volt s. S.C.E.

compared with the literature value -0.642 volt s. S.C.E.

(10).

The excellent agreement between theory and experiment

shown in Figure 7 establishes the validity of the theoretical

calculations. This agreement between theory and experiment

should be indicative in general of the extent to which theo-

retical calculations for stationary electrode polarography

can be expected to apply to real systems, provided the cor-

rect theoretical model has been chosen.

Electrochemistry of Alkali Metals in

Acetonitrile

Acetonitrile was used as solvent for study of three of

the alkali metals (sodium, potassium, and lithium). Aceto-

nitrile was selected because it has been shown to be a useful

solvent for electrochemical studies, and moSt of the alkali

metals already have been studied polarographically in acetoe

nitrile (4).

The alkali metal solutions were prepared from the anhy-

drous perchlorate salts. A discussion of the electrochemical

investigations of the alkali metals presented according to

the experiments performed follows.

Conventional Polarography. Polarographic investigation

of the alkali metals was used to confirm previous work which

indicated that alkali metals give well-defined polarographic

48

waves in acetonitrile (4). Polarograms were obtained for

millimolar solutions, and results were in good agreement

with the literature.

Stationary Electrode Polarography. The analytical ap—

plications of stationary electrode polarography are similar

to those of conventional polarography where peak current is

the parameter analogous to limiting current. For example,

from Equation 26 peak current should be a linear function of

bulk concentration in the case of reversible amalgam forma-

tion. Thus, if reduction of the alkali metals follows this

model, peak currents for the alkali metals should provide a

means of quantitative analysis of the metals.

Actually, from the analytical viewpoint stationary elec-

trode polarography has at least two advantages over conven-

tional polarography. First, the analysis time is significantly

less; and second, under optimum conditions the technique is

actually more sensitive than conventional polarography (7).

For these reasons the dependence of peak current for the

alkali metals was investigated over a range of concentrations

(1.0 x 10'3 M.to 5.0 x 10'5 M). Within this range peak cur-

rents varied linearly with bulk concentration for each of the

metals. Thus, stationary electrode polarography can be re-

garded as a useful analytical technique for alkali metals in

nonaqueous media.

Comparison of Stationary Electrode Polarography of Alkali

Metals with Amalgam Formation Theory. The alkali metals are

49

known to behave reversibly in acetonitrile (12), and since

they form amalgams readily we also compared theory with ex-

periment for these metals.

Values of 20 were calculated from the cathodic peak cur-

rent of a stationary electrode polarOgram and Equations 26

and 51. Values of 2R were obtained from material transport

measurements in liquid amalgams (26).

As an example, typical of the other alkali metals, the

results for sodium will be used for purposes of illustration.

For sodium the following values of 20 and D were found-R

D0 = 0.58 x 10"5 cm.2/sec.

DR = 0.86 x 10“5 cm.2/sec.

giving

7 = E? = 0.668.

A comparison of theory with experiment is shown in

Figure 8 for a millimolar solution of sodium with 0.1 M

tetraethylammonium perchlorate as supporting electrolyte.

The solid line is theoretical and points are the experimental

polarogram. The experimental curve of Figure 8 was obtained

at a scan rate of 161.4 mv./sec. with a hanging mercury drOp

electrode of radius 0.0552 cm. From these values of experi-

mental parameters 0 was calculated to be 0.007._0

Calculation of the theoretical curve was the same as

already described for the case of cadmium, except the above

Figure

8.

Stationary

electrode

polarogram

for

reduction

of

sodium

in

acetonitrile.

Lines,

theoretical.

Points,

experimental.

50

15.0(._.

10.04—-

5.0(_.

9T1";

'l

(I

0

LG

I

O,

-10.0,__.

O

'1

'1

'1

Oo

00

J1

lI

-1075

Figure

8.

-1080

-1085

-1090

-1095

EXi»

S.C.E.,

-2.00

volts

51

52

values of go and y_were used. The theoretical curve was

shifted along the potential axis to obtain the best agreement

between theoretical and experimental peak potentials. This

best fit corresponded to an apparent half wave potential of

-1.878 volt yg, aqueous S.C.E. compared to the literature

value of -1.855 volt yg, aqueous S.C.E. (4).

Although the agreement between theory and experiment

shown in Figure 8 is satisfactory, it clearly is not as quanti-

tative as in the case of cadmium reduction (Figure 7).

In particular, the peak potential separations are slightly

larger than theory would predict. However, in view of the

way in which diffusion coefficients were obtained for this

system, these minor differences are not unreasonable.

CONCLUSION

Based on theoretical and experimental results presented

in this thesis the importance of amalgam formation for

stationary electrode polarography with spherical electrodes

has been established. The most important effects are on

ratios of anodic to cathodic peak currents and peak potentials.

In both cases consideration of amalgam formation has been

shown to be essential to correct interpretation of experi—

mental results. When these effects are considered the agree-

ment between the theory presented and experiment is excellent.

The theory developed also is capable of explaining some appar-

ent anomalies associated with reduction of alkali metals in

acetonitrile.

55

1.

2.

10.

11.

12.

15.

14.

15.

16.

LITERATURE CITED

Alberts, G. S., and Shain, I., Anal. Chem. 55, 1859 (1965).

Booman, G. L., and Holbrook, W. B., Anal. Chem. 55, 1795

(1965).

Churchill, R. V., "Operational Mathematics," p. 55,

McGraw-Hill Book Co., New York, 1958.

Coetzee, J. F., McGuire, D. K., and Hedrick, J. L.,

J. Phys. Chem. §1J 1814 (1965).

DeFord, D. D., Division of Analytical Chemistry, 155rd

Meeting, ACS, San Francisco, Calif., April.1958.

Delahay, P., "New Instrumental Methods in Electrochemistry,"

p. 52, Interscience, New York, 1954.

Ibid., p. 140.

Delmastro, J. R., and Smith, D. E., Anal. Chem. 58, 169

(1966).

DeVries, W., and Van Dalen, B., J. Electroanal. Chem. 19,

185 (1965).

Frischmann, J., Ph. D. thesis, Michigan State University,

East Lansing, Mich., 1966.

Huber, A., Monatsh. Mathematik und Physik. 41, 240 (1959).

Kolthoff, I. M., and Coetzee, J. F., J. Am. Chem. Soc.

12. 870 (1957).

Kuempel, J. R., and Schapp, W. B., 155rd Meeting, ACS,

Miami, Fla., April 1967.

Meites, L., "Polarographic Techniques," p. 56,

Interscience, New York, 1955.

Ibid., p. 71.

Ibid., pp. 250-295.

54

17.

18.

19.

20.

21.

22.

25.

24.

25.

26.

27.

28.

29.

50.

51.

55

Nicholson, R. 8., Anal. Chem. 51, 667 (1965).

.;5;5,, p. 1551.

Nicholson, R. S., and Shain, I., Anal. Chem. 55, 706 (1964).

Nicholson, R. S., and Shain, I., 5555,, 51, 190 (1965).

O'Donnell, J. F., Ayres, J. T., and Mann, c. K., Anal.

Chem. fl. 1161 (1965).

Olmstead, M. L., and Nicholson, R. 8., Anal. Chem. 55,

150 (1966).

Reinmuth, W. H., Anal. Chem. 55, 185 (1961).

Reinmuth, W. H., $555,, 54, 1446 (1962).

Reinmuth, W. H., J. Am. Chem. Soc. 15, 6558 (1957).

Schwarz, W., Z. Elektrochem. 55, 555 (1955).

Shain, I., in "Treatise on Analytical Chemistry,“

Kolthoff and Elving, eds., Part I, Sec. D—2, Chap. 50,

Interscience, New York, 1965.

Shain, I., and Martin, K. J., J. Phys. Chem..§§, 254

(1961).

Shuman, M. S., Shain, I., Great Lakes Regional Meeting,

ACS, Chicago, 111., June 1966.

Stevens, W., and Shain, I., Anal. Chem. 55, 865 (1966).

Underkofler, W. L., and Shain, I., Anal. Chem. 55, 1778

(1965).

APPENDICES

56

Reduction of Boundary Value Problem

APPENDIX A

to Integral Form

It is convenient first to reduce Equations 1 and 2 of

the text to parabo

accomplished with

U(r,t)

V(r,t)

lic form. This transformation can be

the following functions

rCO(r,t)

rCR(r,t).

(A1)

(A2)

In terms of the functions g_and y, the boundary value problem

‘ given by Equations 1 through 8 of the text becomes

an _ 020

(St — D0(6r2)

av = a v(5;) DR(5;29

t = 0; r > r0 U = rCE

t = 0; r 2_0 v = 0

t > 0; r -$-oo U -’-rCS

t > O; r “9'0 V —”0

D0 8U D0t > 0; r — r0 r0 (5r)r=ro - r02 Ur_r0

. DR RY. _ DR

;;'(5r)r=ro -_é Vr=ro

57

(A5)

(A4)

(A5)

(A6)

(A7)

(A8)

(A9)

t > 0; r = r0 —£5£Q = esk(t) .(A10)

Equations A5 and A4 can be integrated easily with the aid of

the Laplace transformation, for which we adOpt the following

definition and notation

_. co

;E[U(r,t)} = U(r,S) = U = f [exp(-St)][U(r,t)] dt

o

-(A11)

Thus, taking the Laplace transformation of Equation A5, and

applying Equation A5 one obtains

626 SUI_ £_ *

32' - DO - DO CO .(A12)

A general solution for Equation A12 is

*

C r

U'= -g—-+ A exp(r dS/DO) + B exp(-r-JS/DO) (A15)

where A_and g are integration constants. From Equation A7,

however, A_is evidently zero, so that Equation A15 becomes

__ Car

U =-§- + B exp(-r 757D; ) .(A14)

The value of §_can be determined by evaluating Equation A14

at _F_Q

*

B = (U. - roCo ) exp (r0 JS/DO ) .(A15)

S

Equations A14 and A15 combine to give

-)(-

Cor

S

*

E = + (UFrO - Inga) exp[ 7"“‘5/13O (rO-r)] .(A16)

Because current is calculated in terms of flux at the

59

electrode surface, Equation A16 is differentiated with

respect to g-and evaluated at gfigo.‘ The result is

*-

BU' = C0 - rQCE(EF'r=ro 5—-— 73/00 (Ur=ro - s ) .(A17)

With the aid of Equation A1, Equation A17 can be written in

terms of concentration and flux of substance 0

EEO =cgvb'; [ 1 * 1

We "—7, SEEM?) +Co F(J?+JD-O)r0 3

_ 1 f0 (5) (A18)

735 Js_+~/—700'ro

where the function fo(£) is the surface flux of 0:

= 0CQ(r,t)fo(t) D0 [ 5r ]r=ro .(A19)

The inversion of Equation A18 to the real time domain can be

accomplished with the aid of tables of Laplace transform

pairs and the convolution theorem (5)

t

_ * _ 1

COr=r _ CO D f0 fO(T)

J‘—' D0 .r—'

[:-——3¥———-- —QQ- exp(-—2 (t-T»erfc( EQJJt-T )] dT

m??? r0 1‘0 ro

Treatment of the equation in V(£,§) is similar to that

for U(£,§). Application of Laplace transformation to Equa-

tion A4, together with Equation A6, leads to the equation

analogous with Equation A15

60

Vl= C exp(-r JS/DR )+'D exp(rnJS/DR) .(A21)

Application of boundary condition A8 gives the following

relationship between integration constants

The remaining integration constant can be determined as

before, and the final result is

exp(-r'JS/DR ) - exp(r'JS/DR )

exp(-ro «ls/DR ) - exp(ro 'JS/DR )

<l n <|

. (A25)

Proceeding as before an expression proportional to surface

flux of R is obtained by differentiation of Equation A25 and-

evaluation at E? o

exp(-rd~/ S/DR ) + exp(rod S/DR )

exp(—rd~/§_/-D;) - exp(rdJ S/DR )

. (A24)

By recognizing that the exponential terms of Equation A24

are of the form of the hyperbolic cotangent, one can write

Equation A24 as

(éy- = V; 'JS/D coth r0 JS/D .(A25)6r r=ro =ro R R

With the aid of Equation A2, Equation A25 can be written in

terms of concentration and flux of R

61

cR = fR(S) 7 1 .(A26)

r=ro '73— 7-8 coth'r'o's/S/DR - 'JDR7ro

R

Inversion of Equation A26 cannot be obtained in closed form.

However, as discussed in the text the finite volume contri—

butions are to be neglected, and Reinmuth has shown that

this is equivalent to setting the 5555 term equal to unity.

Under this restriction Equation A26 then becomes

fR( s) ‘

R = .(A27)

r=ro N/D_R (J‘s-— " VDR;r0 )

Inversion of Equation A27 is relatively straightforward, and

leads to the following

JD— D D

- i exp(E-Eg-(t-Tflerfd Eli/F?)

t

R -—1- 1w[———,__ .r=ro VIZ; O F(t-T) 0

2"(DR

ro

+

D .

exp(- r—Eg (t-T) )] dT, . (A28)

At this point Equation A20 and A28 can be combined with

Equation 11 of the text to give the following single integral

equation

D

C* --;- ft f (T) -—;————-- 29.exp(-92(t-T))erchJ§975:T) GT0 J13; o 0 M r0 r0 r0

_ as (t) JD— f0—

- A ft f (T) '——;‘“—' ’ '53 exP(B'lz"2(t-T))erfc(“"5'"(timf )

75;. o O 'JT(t—T) r0 r0 r0

+ 275; D

r0 exp(- :32 (t-T)) dT .(A29)

62

The solution of Equation A29, fO(§), is related to current

by Fick's first law

i = nFAfO(t). (A50)

APPENDIX B

Reduction of Boundary Value Problem to Two

Simultaneous Integral Equations

Expressions developed in Appendix A can be used to derive

a set of two simultaneous integral equations. From Equation

A17 one has

1* *"

—' = roCQ + C0‘90 Do (8UU __) .(B1)r=ro S 3372 ‘ "' 6r r=ro

JS

Inversion of B1 leads to

-x- -x- t (gy- dT

Ur=ro= roCO + 2CdJDO §_ -JDQ f0 6r r=rQ .(B2)

7 J v Jt—T

Likewise, inversion of Equation A25 with the finite volume

restriction leads to

_ D A!

vr=ro — gg' 'fl: (0r)rf;rodT .(B5)

0 Jt-T

With the aid of Equation A10, Equation B5 can be combined

with Equation B2 to yield

8v

esk(t) DR t 5?r=rodT =

J? o m

,—— 8U

roe; + 2anJDO -% - .29 ft (5E'r=rQ9T .(B4)

7? ° 'Jt-T

65

64

Equation B4 is a single integral equation involving two

unknown functions, (5%)r—r and (gg'r-r .

- 0 - 0

Thus, to solve for

these two functions a second independent integral equation

is required. This second integral equation can be derived by

combining Equations A9 and A10 with Equation B5 above. The

result is

Pg. (gy— ._ _D_R. (Q31) =

r0 6r r=ro r0 6r r=ro ”

§y_(39 _ DR \ (05km) JDR ‘ft 5r)r=rod1 (135)

ro‘2 r0298)\(t) I J? I 7 o «(t—:17

Further treatment of this system of integral equations is

described in the text.

APPENDIX C

5t;6r r=ro to Current

Relation of the Functions (%¥)r=ro and (

Current is calculated from Fick's first law

. _ aco _

1 — nFADO(-5;—)r=ro— nFAfO(t) (C1)

Thus, the problem reduces to relating the functions (3%)r=ro

8U . . .and (0r r=ro to f0(£). By combining Equation A9 agd A10 and

. . . V

factoring, one can obtain gf=ro as a function of (5r)r=ro and

3E) _ . This value of U _ can be substituted in ther r-ro —r—ro

lefthand side of Equation A9 to give fo(§) as a function of

8v 8U . . .(8E7r=ro and (5:9r=ro° The resulting equation can be rewritten

in terms of go, 1, x(y), and Y(y) (see Equations 20 through

25 in the text) and combined with Equation C1 to give

. as (y)

1 = nFAcgdaDOJ? [x(y) - (X(Y) - ELM” a)\ )1- (02)7 esax(y)-1_

y2

65

APPENDIX D

Computer Program

The numerical calculations reported in this work were

performed on a Control Data 5600 digital computer, and the

programs were written in Fortran language. Since this sys—

tem is widely used and compatible with most modern computers,

a Fortran program is listed below for the numerical method

of Huber. The following data are read in: NTOT, which is

the total number Of times the program will be executed;

NRUN, which is the total number Of sets of go and 1_used;

THETA, which is e in the text; DELTA, which is the length Of

an integration interval; SSCAN, which is the number Of single

scans; LIMIT, which is the total number of integration inter-

vals; GAMMA, which is y_in the text; SQUIG, which is A.in the

text; and PHIO, which isgO in the text. The output involves

printing the above data and the values of Fo(y), F(y),

(Riga, SPHERICAL CORRECTION, (Q - §°)n, F(y) for extension

of cathodic scans, (§,- 5%)5 for the extended cathodic scan,

CHI, which is x(y) in the text, and PSI, which is F(y) in the

text.

66

67

PROGRAM AMALGAM

ODIMENSION CHI(1000), PSI(1000), SQR(1000), SQRD(1000).

isw(1000), swp(1000), CUR(1000), CHIIT(1000), PSIIT(1000).

28WIT(1000), SWPIT(1000), CURIT(1000), CHIP(1000).

5AL(1000), BET(1000), ALIT(1000), BETIT(1000)

600

100

105

10

15

READ 600, IM, ID, IY

FORMAT (512)

READ 100, NTOT

READ 100, NRUN

FORMAT (12)

READ 105, THETA, DELTA, SQUIG, SSCAN, LIMIT

FORMAT (4F10.0, 110)

s = 0.

z = s + 1.

Q=1.

I = 1

SW(I) = EXPF(THETA + S*DELTA*SQUIG - DELTA*Q)

SWIT(I) = SW(I)

Q=Q+1.

I = I + 1

IF(Q-Z*SQUIG)5,5,10

s = s + 1.

z = s + 1.

IF(Z-SSCAN)15,15,50

SW(I) = EXPF(THETA - z*DELTA*SQUIG + DELTA*Q)

SWIT(I) = EXPF(THETA — DELTA*Q)

25

50

55

110

68

Q Q + 1.

I = I + 1

IF(Q-Z*SQUIG)15,15,25

s = s + 1.

z = s + 1.

IF(Z-SSCAN)5,5,50

A=(DELTA)**(5./2.)

A=(4./5.)*A

SQR(1) = 1.

SQRD(1) = 1.

Q = 1.

D0 55 I = 2,LIMIT

Q = Q + 1.

SQR(I)= Q**(5./2.)

SQRD(I) = SQR(I) - SQR(I-1)

READ 110, PHIO, GAM

FORMAT (2F10.0)

PHIR = PHIo/GAM

R0 = (GAM*DELTA)/A

DO 91 N = 1,LIMIT

POT = 1.77245/(A*(1. + GAM*SW(N)))

L = N - 1

K = N

CHIP(N) = POT

D0 92 J = 1,L

CHIP(N) = CHIP(N) — CHIP(J)*SQRD(K)

69

92 K = K - 1

91 CONTINUE

CHIP(1)= DELTA*CHIP(1)

D0 15 I=2,LIMIT

15 CHIP(1)= CHIP(I-l) + DELTA*CHIP(I)

Q=1.

D0 50 N = 1,LIMIT

SWP(N) = (GAM*PHIO*SW(N) - PHIR)/1.77245

L = N — 1

K = N

RHSi = 1. + (2.*SQRTF(Q*DELTA)*PHIO)/1.77245

RHS2 = 0.

Q= Q + 1.

DO 55 J = 1,L

RHs1 = RHSl - A*AL(J)*SQRD(K) - A*SW(N)*BET(J)*SQRD(K)

RHS2 = RHS2 - GAM*DELTA*AL(J) + DELTA*BET(J) + A*SWP(N)*BET(J)

2*SQRD(K)

55 K = K - 1

ROERHS1 - RHSZBET(N)

BET(N) BET(N)/(DELTA+ A*SWP(N) + GAM*DELTA*SW(N))

AL(N) = RHs2 + BET(N)*(DELTA + A*SWP(N))

AL(N)/(GAM*DELTA)AL(N)

50 CONTINUE

JP = SQUIG

D0 44 K = 1,JP

AL(K)ALIT(K)

7O

44 BETIT(K) = BET(K)

JPP = JP + 1

Q= JPP

D0 45 N = JPP,LIMIT

SWPIT(N) = (GAM*PHIO*SWIT(N) - PHIR)/1.77245

L = N - 1

K = N

RH81 1. + (2.*SQRTF(Q*DELTA)*PHIO)/1.77245

RHSZ = 0.

Q = Q + 1.

D0 47 J = 1,L

RHS1 - A*ALIT(J)*SQRD(K) - A*SWIT(N)*BETIT(J)*SQRD(K)RHS1

RHS2 = RHSZ + A*SWPIT(N)*BETIT(J)*SQRD(K) - GAM*DELTA*ALIT(J)

2+ DELTA*BETIT(J)

47 K = K - 1

BETIT(N) = R0*RH31 - RHSZ

BETIT(N) = BETIT(N)/(DELTA + A*SWPIT(N) + GAM*SWIT(N)*DELTA)

ALIT(N) = RHS2 + BETIT(N)*(DELTA + A*SWPIT(N))

ALIT(N) = ALIT(N)/(GAM*DELTA)

45 CONTINUE

PRINT 115

115 FORMAT (1H1////////)

PRINT 120

120 FORMAT (45x, 17HAMALGAM F0RMATION/)

PRINT 121

121 FORMAT (51x, 5HHUBER//////////)

PRINT 601, IM,ID,IY

.71

601 FORMAT (100x, 7H DATE 12, 1H/ 12, 1H/ 12////)

PRINT 150, THETA, DELTA

150 FORMAT (11H LN(THETA)= F6.5, 8H DELTA= F8.4/)

PRINT 155, SQUIG, LIMIT

155 FORMAT (10H LAMBDA= F6.1, 8H LIMIT= 15/)

PRINT 145, SSCAN

145 FORMAT (24H NUMBER OF SINGLE SCANS= F6.0////)

PRINT 140, PHIO

14o FORMAT (21H SQRT(D0)/R0*SQRT(A)= F8.4/)

PRINT 141, PHIR

141 FORMAT (21H SORT(DR)/R0*SQRT(A)= F8.4/)

PRINT 142, GAM

142 FORMAT (19H SQRT(D0)/SQRT(DR)= F8.4/////)

PRINT 150

1500FORMAT (7x, 7HCURRENT, 7x, 7HCURRENT, 44x, 8HBASELINE, 5x,

18HBASELINE)

PRINT 151

1510FORMAT (7x, 6HPLANAR, 7x, 9HSPHERICAL, 4x, 8H(E-E1/2), 4x.

110HCORRECTION, 5x, 6H(E-EO), 6x, 9HSPHERICAL, 4x, 8H_E-E1/2).

27x, 5HCHI, 11x, 5HPSI///)

CHI(1) = DELTA*AL(1)

CHIIT(1) = DELTA*ALIT(1)

PSI(1) = DELTA*BET(1)

PSIIT(1) = DELTA*BETIT(1)

D0 22 I=2,LIMIT

CHI(I) = CHI(I-1) + DELTA*AL(I)

22

61

62

65

60

81

85

84

85

82

72

PSI(I) = PSI(I-1) + DELTA*BET(I)

CHIIT(I) = CHIIT(I-1) + DELTA*ALIT(1)

PSIIT(I) = PSIIT(I-l) + DELTA*BETIT(I)

CONTINUE

D0 60 I = 1,LIMIT

COE = SW(I) — 1./(GAM*GAM)

IF(COE)61,62,61

COE = SW(I)/COE

GO TO 65

COE = 0.

CUR(I) = CHI(I) — (CHI(I) - (PSI(I))/GAM)*COE

CUR(I) = 1.77245*CUR(I)

CONTINUE

D0 81 I = 1,JP

CURIT(I) = CUR(I)

D0 82 I = JPP,LIMIT

C0E = SWIT(I) - 1./(GAM*GAM)

IF(COE)85,84,85

COE = SWIT(I)/COB

GO TO 85

C0E = 0.

CURIT(I) = CHIIT(I) - (CHIIT(1) - (PSIIT(1))/GAM)*COE

CURIT(I) = 1.77245*CURIT(I)

DO 87 I = 1,LIMIT

POTO = 25.68857*LOGF(SW(I))

POT12 POT0 + 25.68857*LOGF(GAM)

75

POTIT = 25.68857*LOGF(GAM*SWIT(I))

DI = CUR(I) - CHIP(I)

OPRINT 155, CHIP(I), CUR(I), POT12, DI, POT0, CURIT(I), POTIT,

1CHI(I), PSI(I)

155 FORMAT (1x, 2F14.8, F12.4, F14.8, F12.4, F14.8, F12.4, 2F14.8)

87 CONTINUE

NRUN = NRUN - 1

IF(NRUN)75,75,1

75 NTOT = NTOT - 1

IF(NTOT)74,74,2

74 CONTINUE

END

“‘4... «.4.- - *' 7M ‘W ‘0

1

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