Chronopotentiometry (1)

7/30/2019 Chronopotentiometry (1)

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CHAPTER 7

CHRONOPOTENTIOMETRY

In this technique the current flowing in the cell is instantaneously stepped from

zero to some finite value. The solution is not stirred and a large excess of supporting

electrolyte is present in the solution; diffusion is the only mass transfer process to be

considered. Electrolysis at constant current is conducted with the apparatus schematically

presented in Figure (1). where P is a power supply whose output current remains constant

regardless of the processes occurring in the cell. The potential of the working electrode

E1 against the reference electrode E2 is recorded by means of instrument V.

For a simple reaction as described by Equation (1), a chronopotentiogram will

typically look like the plot in Figure (2).

RneO =+ (1)

As the electrolysis proceeds, there is a progressive depletion of the electrolyzed species at

the surface of the working electrode. As the current pulse is applied there is an initial

sharp decrease in the potential as the double layer capacitance is charged, until a potential

at which O is reduced to R is reached. There is then a slow decrease in the potential

determined by the Nernst Equation, until the surface concentration of O reaches

essentially zero. The flux of O to the surface is then no longer sufficient to maintained the

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applied current, and the electrode potential again falls more sharply, until a further

electrode process occurs.

7.1 Initial and Boundary Conditions

Unless otherwise stated, the following conditions are assumed to be achieved: (1)

the solution is not stirred; (2) a large excess of supporting electrolyte is present in

solution, and the effect of migration can be neglected; (3) conditions of semi-infinite

linear diffusion are achieved.

Substance O is reduced at a plane electrode and the product of electrolysis R is

soluble in solution. Since the current density is maintained constant during electrolysis,

the following equation

( )0

,

=

=

x

o

oox

txCnFDi (2)

can be written from the definition of the flux. Equation (2), which is the first boundary

condition, can be written in the following form

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( )=

=0

,

x

o

x

txC (3)

with

o

o

nFD

i= (4)

The second boundary condition is obtained by expressing that the sum of the

fluxes for substance O and R at the electrode surface is equal to zero. Thus

( ) ( )0

,,

00

=

+

== x

RR

x

oo

x

txCD

x

txCD (5)

The initial conditions can be selected a priory, and generally one can assume that the

concentration of substance R is equal to zero before electrolysis and that the

concentration of substance O is constant. Thus:

(1) CR(x, 0) = 0, and Co(x, 0) = Co.

(2) The functions Co(x, t) and CR(x, t) are bounded for large values of x. Thus,

Co(x, t) Co

and CR(x, t) 0 for x

Variation of The Concentrations Co(x, t) and CR(x, t).

The solution of the above boundary value problem was reported by Karaoglanof,

who calculated the concentrations of both species. The concentrations are

( )

+

=

2121

2

21

2121

24

exp2

,

tD

xxerfc

tD

xtDCtxC

oo

oo

o

(6)

( )

=

2121

2

2121

21

24exp

2,

tD

xerfc

D

xD

tD

x

D

tDtxC

RR

o

RR

oR

(7)

where the notations "erf" and "erfc" represent the error integral defined by the formula:

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( )dzzerf =

0

2

21exp

2 (8)

The function erf{} is defined under the form of a finite integral, having zero as lower

limit and as upper limit of integration. Therefore, values of erf{} are determined only

by the variable , "z" being simply an auxiliary variable. The variations of erf {} with

are shown in Figure (3) for values of comprised between 0 and 2. The error

function is zero when its argument is equal to zero, and the function approaches unity

when becomes sufficiently large. "erfc" is defined by the equation:

( ) ( ) erferfc =1 (9)

An example values of Co(x, t) are plotted against x in Figure (4) for various times

of electrolysis and for the following data: io =10-2

A cm-2

, n=1, D = 10-5

cm2

s-1

, Co =

5x10-5

M /cm3. All the curves of Figure (4) have the same slope at x=0, because the flux

and consequently the derivative ( ) ttxCo , is constant at x=0.

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7.2 Potential - Time Curves

The potential is calculated from the Nernst equation, the concentrations Co(0, t)

and CR(0, t) being written from (6) and (7). Thus

21

21

21

21

lnlnPt

PtC

nF

RT

Df

foD

nF

RTEE

o

oR

Ro ++= (10)

2121

2

o

o

nFD

iP

= (11)

The sum of the first two terms on the right-hand of equation (10) is precisely the potential

E1/2 defined by equation:

i

ii

nF

RTEE d

+= ln21 (12)

21

21 ln

+=

R

o

o

Ro

D

D

f

f

nF

RTEE (13)

When a mercury electrode is used, the potential E1/2 is the polarographic half-

wave potential. Hence, equation (10) can be written as follows

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21

21

21 lnPt

PtC

nF

RTEE

o += (14)

The potential calculated from equation (14) is infinite when the numerator in the

logarithmic term is equal to zero, i.e. when the time t has the value defined by the

following relationship

PCo=21 (15)

Actually the potential at time increases toward more cathodic values until a new

reaction occurs at the electrode: such a process can be the reduction of water or the

supporting electrolyte.

By introducing in Equation (14) the value of Co expressed in terms of the time

defined by equation (15), one obtains the following potential-time relationship

21

2121

21 lnt

t

nF

RTEE

+=

(16)

The above equation has the same form as the equation of a reversible polarographic

wave, the diffusion current and the current being replaced by21

and by t1/2

,

respectively. Thus, the properties of potential-time curves can be deduced by simple

transposition of the theory of reversible polarographic waves. The potential E1/2

corresponds to a value of t equal to 4 as can be seen from Figure (5). Equation (16)

also shows that a plot of the decimal logarithm of the quantity ( ) 212121 tt versus

potential should yield a straight line whose reciprocal slope is 2.3

RT/nF. Logarithmic plots are linear as predicted by equation (16), and the potentials E1/2

as shown in Figure (6) are in good agreement with the polarographic half-wave

potentials.

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Butler and Armstrong coined the term "transition time" to designate the time

defined by equation (15). According to equations (11) and (15) the transition time is

o

o

i

DnFC

2

212121 = (17)

The square root of the transition time is proportional to the bulk concentration of

substance reacting at the electrode and inversely proportional to the current density io.

Thus, transition times can be greatly changed by variation of the current density. The

limits between which the transition time can be adjusted are determined by the

experimental conditions:

(1) convection should not interfere with diffusion.

(2) the fraction of current corresponding to the charging of the double layer

should remain negligible in comparison with the total current through the cell. In practice

the transition time should not exceed a few minutes. Because of the charging of the

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double layer, transition times shorter than one millisecond cannot be measured with

reasonable precision.

According to equation (17), [called Sand equation (1-3)], the product 21oi

should be independent of the current density.

Potential-Time Curves for Totally Irreversible Processes:

The rate of totally irreversible process is correlated to the current density by the equation:

( )

= RT

FEntCknF

io

o

hfo

exp,0, (18)

The combination of (16) and (18) yields

( )

=

RT

nFEPtCk

nF

i oohf

o exp21, (19)

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The transition time is determined by the condition Co(0, ) = 0 as for reversible

processes. Hence, Co

= P 21 , and the corresponding value of can be introduced in

equation (19). The equation for the potential-time curve is thus

o

hf

o

k

D

nF

RTt

nF

RTE

,

21212121

2ln)ln(

= (20)

or in view of equation ()

=

21

2121 1ln)ln(

t

nF

RTt

nF

RTE (21)

An example of potential time curve is shown in Figure (7) for the reduction of iodite.

The potential should rise at time t=0 according to equation (21). It is seen from

equation (21) that the shape of the potential-time curve depends on the product n, and

that the transition time is independent of the kinetic of the electrochemical reaction.

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The potential at time zero depends on the parameters n, and on the current

density. Potential-time curves for the totally irreversible processes may thus be shifted to

a certain extent in the potential scale by variation of the current density.

o

hfk ,

A plot of decimal logarithm of {1-(t/ )1/2

} versus potential is a straight line

[Figure (8)] whose reciprocal value is 2.3RT/nF. Thus, n is readily calculated. The

rate constant k is calculated from the potential at time zero by application of equation

(21).

o

hf,

7.3 Two Consecutive Electrochemical Reactions Involving Different Substances

When two substances O1 and O2 are reduced at different potentials, the potential-

time curve exhibits two distinct steps. The first transition time 1 corresponding to the

reduction of substance O1 can be calculated from the treatment explained above. The

second step cannot be determined from this simple treatment. As the electrolysis

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proceeds after the transition time 1 , the potential of the polarizable electrode adjusts

itself to a value at which substance O2 is reduced. Substance O1 continues to diffuse

toward the electrode at which it is immediately reduced. As a result the constant current

through the cell is the sum of two contributions corresponding to the reduction of

substances O1 and O2, respectively. The transition time 2 for the second step in the

potential-time curve is reached when the concentration of substance O2 becomes equal to

zero at the electrode surface.

( )oC221

2

(

Initial and boundary conditions have to be described to derive the transition time

2 . In the writing of these conditions it is convenient to take as origin of the transition

time 1 . The time in the new scale will be represented by the symbol t', and the

relationship between t and

t' is

1' = tt (22)

The transition time for the second step of the potential-time curve 2 is determined by the

condition ( ) 0,0 22 =OC by the equation

o

O

i

FDn

2

2

21

21

112

=+ (23)

The quantity on the right-hand is proportional to

) 21121

21 + (24)

As a result, the transition time 2 depends on the concentration of substance O1 which is

reduced at less cathodic potentials. The order of magnitude of the increase in 2 which

results from the contribution of O2 can be judged from the particular case in which

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oo CC 21 = , 21 nn = , 21 OO DD = ; (25)

equation (23) yields for such conditions 12 3 = .

Stepwise Electrode Processes-The Boundary Value Problem:

A substance O is reduced in two steps involving n1 and n2 electrons, the reduction

product being R1 and R2. Potential-time curves for such processes exhibit two steps,

when R1 is reduced at markedly more cathodic potentials than substance O. After the

transition time 1 for the first step substance O1 continues to diffuse toward the electrode

at which it is directly reduced to substance R2 in a process involving n1 + n2 electrons.

Furthermore, substance R1 produced during the first step diffuses toward the electrode at

which is reduced in a process involving n2 electrons.

The distribution of substance R1 at the transition time 1 is given by equation (7)

in which t is made equal to 1 . The resulting expression is the initial condition for the

present problem. The boundary condition is obtained by expressing that the current is the

sum of two contributions corresponding to the reduction of substances R1 and O,

respectively. Thus

( )( )

( )F

i

x

txCDnn

x

txCDn o

x

o

o

x

R

R =

++

== 0

21

0

12

',',1 (26)

Functions and are bounded for large values of x. The following

equation for

( txCo , ) )( txCR ,1

the concentration of substance R1 at the electrode surface is reported in the literature (4)

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( ) ( )

+

+=

21

1

21

1

1

21

21

2

211'

2',0

1

tn

nn

FDn

itC

R

o

R

(27)

Transition Time for the Second Step of the Potential-Time Curve:

The transition time 2 is obtained by equating the right-hand member in equation (7) to

zero. The resulting expression can be written in the form:

( ) ( )

+

+=

21

1

21

1

1

21

21

2

211'

2',0

1

tn

nn

FDn

itC

R

oR

(28)

which shows that the relationship between the transition times 1 and 2 is remarkably

simple. When n1=n2, the transition time 2 is equal to 3 1 .

Experimental data for the reduction of oxygen confirm the correctness of the

foredoing analysis. Potential-time curves for these substances are given in Figure (9).

Oxygen is reduced in two steps involving two electrons each, and consequently 2 = 3 1 .

Cathodic Process Followed by Anodic Oxidation:

A substance O is reduced to R and the direction of the current through the electrolytic cell

is reversed at the transition time corresponding to the reduction of O. Substance R is

now oxidized and a potential-time curve is observed for this process.

The concentration of substance R at the transition time is expressed by

equation (7) in which the time t is made equal to . The resulting expression is the initial

condition for the present boundary value problem, since it is now the concentration of

substance R which is to be calculated. Initial and boundary conditions are the same as for

single electrochemical reaction (see equation 4).

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( )'

',

0

=

=x

R

x

txC

R

o

nFD

i ''= (29)

the intensity i in the reoxidation process may not necessarily be adjusted at the same

value as in the reduction of O and R.

'o

Hence the current density ' is introduced in equation (29). The function

for

oi

( ) 0', 6txCR 6

The concentration of substance R during the re-oxidation is:

( )( )

( ) ( )

+

+

+=

'2'4exp2',

21

221

tD

xxerfc

tD

xtDRtxC

RR

R

( ) ( )

++

+

2121

2

'2'

'4exp

''2

tD

xxerfc

tD

xtD

RR

R

(30)

with

R

o

nFD

i= (31)

Variations of the concentration CR(x, t) with distance from the electrode are

shown in Figure (10) for the same data as those used in the construction of Figure (4) and

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for the following numerical values: 2210' cmAii oo== , DR=Do=10

-5cm

2s

-1. The CR(x,

t') versus x curves of electrolysis larger than t' = 0 exhibit a maximum. The concentration

of R at a sufficient distance from the electrode becomes slightly larger than the

corresponding initial concentration at time t'=0; this results from diffusion of substance R

toward a region of the solution in which the concentration of R is lower than at the

maximum of the CR(x, t') vs x curve.

The concentration of substance O during reoxidation when the diffusion

coefficients Do and DRare equal is

( ) ( )',', txCCtxC Ro

o = (32)

7.4 Transition Time for the Re-oxidation Process

The transition time is determined by the condition CR(0, ')=0. By writing (32)

for x=0 and solving for the transition time ' for the re-oxidation process one obtains

( ) 222

''

+= (33)

When = ', when the current densities i and ' are equal, equation () takes very

simple form

o oi

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31'= (34)

which shows that the transition time for the re-oxidation process is equal one third of the

transition time for the initial cathodic process, the current density being the same in both

processes. An example of potential-time curve is given in Figure (11).

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EXPERIMENTAL

Required equipment and supplies

Pt working electrode, A=0.5 cm2.

1x10-3

M

( ) 36

CNFe = 1x10-3

M.

[KCl] = 1M

PAR Model 352

Three compartment electrochemical cell.

Objectives:

The objectives of this experiment are by using chronopotentiometry and

chronopotentiometry with current reversal to determine: (1) the dependence of the

transition time on the bulk concentration of electroactive species reacting at the electrode

and (2) the dependence

of the transition time on applied current density.

The electrochemical cell employed for these studies should be conventional three-

compartment design with contact between the working electrode compartment and

the reference electrode via a Luggin probe. The chronopotentiometric experiments should

be carried out using standard calomel electrode (SCE) in (1) 1x10-3

M ( ) 36

CNFe and

1M KCl.

Data Analysis:

Values of (1) the transition times as a function of the applied current densities for

constant concentrations of ( ) 36

CNFe and (2) for transition times as a function of

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different concentrations of ( ) 36

CNFe at constant current densities and (3) potential vs.

21212/1log tt were obtained by Popov and Laitinen (unpublished results) and are given

in Table (1), Table (2) and Table (3).

Table 1. Values of Transition Time for Different Cathodic Currents Obtained in

1x10-3

M Fe(CN)6-3

+ 1M KCl

Concentration

10-3

M

Current Density

A/cm3

Transition Time

Measured, (s)

Relative Standard

Deviation

1.0 60 5.2 1.2%

1.0 55 6.2 0.6%

1.0 50 7.5 1.4%

1.0 45 10.7 0.8%

1.0 40 12.2 1.2%

1.0 35 15.5 0.3%

Table 2. Values of Transition Time Obtained for Different Concentrations of

Fe(CN)6-3

at Constant Cathodic Current Density of 50 A/cm2.

[Fe(CN)6-3

]

10-3

M

Applied Current

A/cm3

Transition Time

Measured, (s)

Relative Standard

Deviation

1.5 50 15.5 0.5%

2.0 50 31.0 1.16%

2.5 50 51.8 1.1%

3.0 50 72.5 1.8%

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Table 3. Potential vs21

2121

log

t

Potential (mv) [SCE]

21

2121

log

t, (s)

180 -0.78

190 -0.6

210 -0.3

227 0.0

255 0.4

265 0.6

275 0.8

Plot 21i vs i; i vs 1/i; 21i /C vs C; 21 vs C

Compute (a) the diffusion coefficient of the electroactive species in 1M KCl. (b)

the number of electrons involved in the process (c) Discuss the logarithmic plot for

reversible electrode processes (d) compare the "n" value obtained from the slope of the

logarithmic plot for reversible electrode processes and "n" value obtained from Sand

Equation.

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REFERENCES

1. B. N. Popov and H. A. Laitinen,J. Electrochem. Soc., 117, 4, 482, (1970).

2. B. N. Popov and H. A. Laitinen,J. Electrochem. Soc., 120, 10, 1346, (1973).

3. R. Cvetkovic, B. N. Popov and H. A. Laitinen, J. Electrochem. Soc., 122, 12, 1616,

(1975).

4. Boris B. Damaskin, "The Principles of Current Methods for the Study of

Electrochemical Reactions, Editor, Gleb Maamntov, McGraw-Hill Book Co., New

York, (1968)

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Chronopotentiometry (1)