The effect of uncompensated resistance on the potential-step method of investigating electrochemical kinetics

JOURNAL

THE

OF

EFFECT

STEP

I<.

ELECTROAXALSTICXL

OF

METHOD

CHEMISTRY

171

UlSCOMPEi\;‘SATED

OF

ISVESTIGATIXG

RESIST_ANCE

ON

THE

ELECTROCHEMICAL

POTENTL~LKINETICS

13. OLDHAM-

h_oztA rlmencalr .J vintion Scic2zce Cczter. Tlrozrsand Oaks. Califovnra Califorrria I?zsiiiufc of TechnoZogy, Pmadena. CaZrforrLin (bTi.S.d ) (Received

May

=Sth,

91360

c U.S._4

_)

1965)

The potential-step method of studyin, 1, developed by GERISCHER AND VIELSTICH variously Agreement betxveen parameters, might be hoped”, and a recent publication3 standard rate constant upon concentration.

= the kinetics of electrode reactions, first has been employed by several authors_ determined, is seldom as satisfactory as reports an apparent dependence of a The present paper seeks to elaborate the

role of uncompensated resistance in the potential-step method, to show how corrections for it may be made, and to demonstrate that disregard of this factor may lead to anomalous results. In the potential-step method, an electronic potentiostat is used to control the potential, E, between a working electrode, WE, and a reference electrode, RE (see Fig. r)_ Prior to electrolysis, this potential is maintained at a value E, such that negligible current flows, but a sudden change to Em causes a current i to pass between the working electrode and a counter electrode CE. Cell design and choice of supporting electrolyte enable the ohmic resistance between CE and WE to be minimizedNevertheless, this resistance cannot be entirely eliminated and a fraction of it will inevitably be developed between WE and the tip of the reference electrode. Thrs -leads to an ohmic drop, ie, latter resistance-the “uncompensated resistance”, for which even an ideal potentiostat is powerless to correct. It should be noted that uncompensated resistance may also reside in the extemdl circuit (e-g_, in the capillary of a dropping mercury electrode). Although with many cell designs the major source of uncompensated resistance may be esternaI, this is not considered here since redesign of the cell (e.g., as by I~XDLES _%SD sOMERTOS4 for the dropping electrode) will lead to its virtual elimination. Figure I shows an equivalent circuit of a cell with uncompensated resistance. Cdl denotes the double-layer capacity of the working electrode and 2 its faradaic impedance_ The effect of e is threefold: Fi7st effect. The potential, E, across WE is changed from E, to (E, -tie), leading to a diminution of the faradaic current, +, and a destruction of the potentiostatic condition. Second effect. Whereas in the absence of e the double-layer capacity could be charged instantaneously by an ideal potentiostat, enabling the full applied potential * Permanent

address : University

of Newcastle-upon-Tyne,

England.

J_ Etectroamal.

Chem..

II

(1966)

171-187

to be developed immediately introduces a charging delay. ohmically

limited

across 2, the presence of an uncompensated The delay arises because the maximum total

resistance current is

and causes E to rise only slowly from E, to (E,+z&)As far as a current-measuring device in the external circuit is current going to charge the double-layer is indistinguishable from

Third effect. concerned,

Fig.

that

the

I. Equivalent

circuit

part of the solution

equipotential

faradaic is a p&e

which

current.

of

traverses

In the

of negligible

ccl1

and

resistance the

tip

absence

duration

associated circuiti. The uncompcnsatcd between the surface of the working of the reference electrode.

resistance, ,o, is electrode and the

lying

of uncompensated which

may

be easily

resistance, igndred.

the charging transient will contribute to the measured length of time. Before developing the theory of the potential-step ukmpensated resistancei relationships applicable to cited.

this

charging

For a finite

current method the ideal

for

current

e, however,

an appreciable

in the presence case e =o will

of be

wzcom$ensated resistance Consider the working electrode WE to be of constant area A Snd immersed in a_&olution co&akin& a &dk concentration c of the oxidized member 0-z of the redox cotipi&

No

Ox+ne-

z? ,Sd

(I)

its reduction product Rd.b&ng soJcble but initially absent- At time t=o. the potential ------.; E apphe& tb -IwE-_is changed discontinuckly ffom-Er- to ‘Em as a result of which-a ca_tioz&c _fFtidtic c&rent ir coku-nences to flow. Ass&ng transport to be solely .by -iine_ardiffusitin.- the &rre@t after. an klectiolysi5 time t is5

EFFECT

OF

UNCO_MPEX-SLaTED

;f = if0 exp{Q*t)

RESISTANCE

I73

erfc{Ql/t}

where

(2)

D and D’ being the diffusion coefficients of Ox and Rd while kl and k-1 are the heterogeneous rate constants for the forward and reverse directions of the electron transfer reaction (I). In derivirig eqns. 12). activity coefficients have been ignored and neither Ox nor Rd is considered to be adsorbed at the electrode or to be involved in any taken as unimolecular. The initial reaction other than (I), which is, moreover, faradaic current, z&, is given by ;ro = nAFklC The

(3)

potential-dependence

of the rate

unF(&-EJ

kr exp 1

RT

I

=

ks =

constants

k-1 exp

is represented

--(I (

--(x)nF(&-

by the equation Es)

RT

(4)

1

in whicha is the cathodic transfer coefficient and k, the value acquired by either rate constant at the standard potential, Es, of the redox couple_ The potential E across the working electrode is, in the absence of uncompensated resistance, equal to the applied potential E. Equation (2) may be replaced by a series expansion in I/t of which the first three terms are if zio

--r-Q

(5)

Truncation of the series after these three terms leads to a maximum error of less than r”/oforI I zf -/ ZfO I 0-S; retention of only the first two terms introduces up to 5% error over the same range. Kinetic information may be obtained from potential-step data by analyzing the potential dependence of either if0 or Q_ In the absence of uncompensated resistance, the electrolysis current is wholly faradaic except at the instant t= o. Therefore if0 is determined by a short extrapolation to zero time of an i VS. I/t graph. Since In if0 = In nAFk.& kigandu

are then found Because

-

g

(E-Es)

by plotting

(6)

the logarithm

of ZfOvs. potential_

(7) Q may

be found

from

the initial

considerable range, a log and (I --do) to be madea.

Q vs.

gradient

Em

graph

of the i vs. p enables

plot.

independent

If EaD is varied evaluations

over

a

of ke, oc

J_ EZeclroanaZ. Chem.. II (1966) 171-187

K. B. OLDH_%M

174

We

shall use the word

“quasipotentiostatic”

w-hen referring

to a potential-step

electrolysis in which the uncompensated resistance is small but not negligible_ In a final section we discuss the effect of completely ignoring a small uncompensated we mean one which multiplied resistance. By a “small” uncompensated resistance, by the initial faradaic current yields a potential not exceeding AT/nF. During a truly potentiostatic electrolysis, the quantities E, RI, km1 and Q are constant

for all t?

In discussing (E=E,)+(E=E,)

where static

o, but

under

the puasipotentiostatic it is convenient

quasipotentiostatic

this is no longer

true.

electrolysis which ensues from the potential to define a potential EF such that

conditions

step

Z&Z is the initial faradaic current that would be observed in a truly potentioelectrolysis with an applied potential jurnp ET+Ez. This definition will be

clarified

by reference

to Fig.

2.

Fig-z The worse of two potentiostatic and one_quasipotentiostatic electrolyses plotted to show thevariationoffaradaiccurrcntwithpotential. Underbulypotentiostatic conditionsthefakadaic durrentrisesinstantaneo~~yalong~erIsing curvethenfallsvertically to zero. In aquasipoten-tiostatic electrolysis, the faradtic current follows a similar riskg curwe, although more slowly, reach= a ~imumandthendeclin~toapproachthediagonalstraightline [ofslope r/g) asymptotically.

Later in this paper, measur ements made on a quasipotentiostatic electrolysis -fol@ving the_appliFd pbtential $np Er+Ea with aq uncompensated resistance, e, will be-related to-results that would be obtained without uncompensated resistance foU&ving th& potkntial jumps E;+;E,-or E,GEz. SubsQ5pts_ I’, _s, _2 and ‘a tie used throughout this paper to denote that the _ti$kcripf+‘quantit~~ r&tes to-the de&ignated-constant potential. Thus if02 means. the initial-f&daic c&rent-fohoting_a potenti&at&‘jump to Ez.

EFFECT

OF

UNCOMPENSATED

of

Magnitude

RESISTANCE

zwtcom$ensated

175

resisfance

Fast electrode reactions have been studied almost exclmively at spherical mercury electrodes and this shape of working electrode is assumed in this section. For a spherical working electrode paired with a large and remote counter electrode, the geometry of the latter is largely irrelevant to the cell resistance, which equals I/~-T, z being the specific conductivity of the solution and r the radius of the working electrode. The uncompensated resistance between WE and the naxrow tip of a probe reference electrode is a fraction c/(r+ c), of the total resistance where 5 is the distance from the WE surface to the tip of the RE, the latter being taken as small enough not to shield the WE. Hence the product of electrode area and uncompensated resistance is given by Ae

”

=

z(r+

5)

It happens that it is the Ae product which is important in a quasipotentiostatic experiment. We see from eqn. (9) that this product is diminished by decreasing either 6. Although closer electrode r or f ; a similar conclusion was reached by B..~RN_.R~ spacings have been used’, it is unlikely that an effective 5 much less than the 0-033 cm employed by B...RXARTT can be achieved 6_ More t_ypical values would be 0.05 cm for r and 0.10 cm for an effective 5 _ A supporting electrolyte chosen for its low resistivity would have, for solvent water, a z in the range O-I---1.0 X2-km-l. Hence 0.03-0.3 0 cm4 would be a typical range of values for dig, in a well-designed quasipotentiostatic experiment_ If no attention is given to these consideratrons, AQ can be as large as IO Q cm%_ Qrcasipotentiostatic

electroljsis

Current-time behavior during a potential-step experiment with uncompensated resistance of any magnitude can, in principle, be determined by solution of the following system of equations (see appendix for complete list of symbols). E

E-

~_

i,=i-if=;,=

e

-

nAF

k&(0,

=

t) exp

= II) a_?: dC (o,t) =

__c&_

RT -omF

d&

d_t

(s-ES))--kC'(0.t) exp{(r--31)g(8--Es)]

-D
(OJ)

(X, t) = 0 = C(x:

0,

t

10)

E(t
=c; E(t>

C’(_z1

0,

t

10)

=o

o) =Em

These equations cannot be solved directly. For small e, however, one can be certain - that the qualitative result is as shown by solid lines in Fig. 3_ After a sufficient length of time z$, the charging transient has decayed to a negligible residue and the total current then equals if. Beyond this time, if decays J_ Electround.

CXmz..

II

(1966)

q-r-x87

K_ B. OLDHAM

176 quasipotentiostatic%lly condenser, value E, identical

Lz.,

there

but remains

free

of

only

complications the

“first

effect”.

arising Since

from

the

E finally

double-layer achieves

the

applied tributed

when if-has decayed to zero, the shape of the if VS. t curve for t > tQ must be with that for a condenser-free quasipotentiostatic electrolysis following the the presence of the condenser has conpotential step E,+E,. However, a second effect-the time delay tC. The magnitude of tC results frolm extra-

polating

back

to if=z&,a

as in Fig. 3.

._.__._

\

MC

\

Fig. 3. In a quasipotentiostatic electrolysis the non-faxadaic current i, decays monotonically to become negligible after time T+ During this time interval the faradaic current achieves a maximum value which falls short of &z- There are two reasons for this: concentration polarization has developed leadine to a loss, di/c~, in faradaic current; condenser charging has prevented the buildup of potential, leading to a loss Aiflrc.

The fall in current from & ZYafter long times is due to concentration polarization and for t >t+ we, can set +=&,s - 5& where 5ircp denotes this loss due to concentration polarization.‘ At very short times, on the other hand, if is less than i& not because of concentration polarization but because the condenser charging is rpbbing the farad& path of current that the latter would otherwise draw. Denoting the faradaic current-lost on account of the condenser by 5& we can set +=&,zAi& at-short times: At intermediate times, both Ai fcp and AZ& are important and +=&-5daj, 5&. Now we shall ~sume.di& and d&to be independent of each other, so that 5&. is -calcirlated.on the basis of no condenser and A& is calculated on the basis of no.c_onceqtiation polar&&ion_ This assumptiori is only reasongble if the crossover - _ p@+!(X on Fig. -i &t which 5t;;p =5&k) does riot- lie much below &z_ Since other

EFFECT OF USCOMPEKSArED RESISTANCE

=77

considerations (see Fig_ 6 below) require + at tO to exceed 0.93 &a, the requirement easily met To evaluate tc we assume, folIowing LAITISEX, TISCHER _SXD ROE*, that rc = 7

I

ZfO2

s

CD A+, 0

dt

is

(10)

The justification for this assumption is that the integral in this equation represents the faradaic charge lost on account of the double-layer condenser. Had this charge not been lost, it could have generated the constant current if02 throughout the charging period o < t tzc_ The assumption will introduce little error provided that little concentration polarization has ensued by time r+_ The “other considerations” we have mentioned limit concentration polarization at -rrsto less than 7%; therefore eqn. (IO) may be confidently adopted. The “third effect” means that only after time tO can the measured currents be used to provide kinetic data. We shall define tQ as the time required for the transient current to fall to a smaI1 fraction # of zios, i.e., Q is characterized by the identity Qz-+ov=i+Az-f~-ifo~_ We now proceed to discuss each effect in turu (although in reverse order), indicating how each may be evaluated quantitatively. We assume that the interest in the experiment is the determination of the kinetic parameters and that efforts have therefore been made to minimize e as much as possible_ Attempts to minimize uncompensated resistance will tend to make the exact value of e unknown and we assume, therefore, that the experimenter is not initially aware of the value of e_

Ignoring concentration behaves as a time-independent characteristic if = TEAFt?k, exp Considering if =

-mtF ( RT

polarization, impedance

2

the faradaic path through the cell with the following current-voltage

(e--E,))

(rr)

the circuit in Fig. I, ir and E are also inter-related, E- &X3

e

+

for t > o, by

C.Zg

so that by eliminating

if between

W4 these equations

we derive

a relation

between

E

and t. It is useful to define three dimensionless

CWF

3

=RT(

zr =

parameters

E-&J

~(Er-Em) unF_

a = yj+_fae

by means of which, and eqn_ (S), we can combine eqns. (12) and (II) into J_

EZectroamzL

Chem.,

XI

(1966)

171-187

;I$8

f+tion,

K.

From its definition, z_+ is the time for the current q5, of its non-transient vaIue; for no concentration + _

iTif

1

B. _-

OLDHAX

to rise to within a small polarization this means

,;a

(r4)

ZfC?l so- thBt ri+9)

ti -= C&p

._ P_+

_

& (IS)

a esp&-2)-z

Values of this integrai have been evaluated for likely magnitudes of a and zr_ Table I +hotis ‘the results of such calculations for q5= 3%. The corresponding zroo% or ~1% values mdy‘be determined by subtracting 1-15 from, or adding 1.05 to, the tabulated *~%-&&S.. -. TAl3LE=-l

TABLE VXLUES 2,/a

2

3. ? 6' 7 9 IO

I $5

7-9 8.2 8.5 827

.8.8 -9.0

9.1 g-2

_

-L2-’

.9,4

i4. k6 IS-

g-5 9:7 :gs

2b

_Q‘-9

:

6.0 6-4 R-7 7-O 7.2 7-3 7-5 716 7-7 7-Q 8.1 812 8.3 8-5

4-3 48 5-I 5-4 5-6 s-8 5-9 6.X .62 6.4: 6.6 -6-7 6.9 7-o

OF

r&ate

0.0

0.1

o-3

2

I-3’

I.19

3

I.69 I-97 2.19 2-37 2-53 2.66

1-56 1.84 2-06 2.24 2-53

O-99 I-35 1-63 1.85 z-03 2.18 z-31

2-78 2.88 3-07 3-22 3-35

2-65 2.75 2-93 3.09 3-22

2-43 e-51 2-72 2.87 -3.01

3-47 3-55

3-34 3-42

3-=3 3.20

: 6 s7 9 TO 12 I4 16 I8 20

2

2.40

Th_etabIe-showi that for the-representative values zC=5. a=o.I, it takes seven t@@the (ciduble&yei c&pacity-_X uncompensated resistance) time constant for the 6-an@tint So: c&y to. 3% _6f -tie -faradaic current. As we shaJ.I see_ below, r+ is the t+~:~_$$ &ch -an‘ i-vsZ-@ plot de-$&t&Cfrom linearity by the-factor q5_ With the usual expy~e~tal &cisiop: deviations ivould-need to be at least 3% to be detectable. -geri‘ce:ge_adopt .zi (q5 &PjO) ‘~fthe&intof_noticeabIe de&ture from linearity_ --_

_

EFFECT

OF

UNCOMPENSATED

RESISTANCE

=79

Values of this integral which is a function solely of a and zr are-listed in Table z_ For the representative values zr = 5, a = 0.1, te is about twice the (double-layer capacity X uncompensated resistance) product_ First

effect

electrode subjected to the applied A condenser-free quasipotentiostatic potential step E,+E, experiences initially the potential E2 and so draws current z&s where these terms are defined by eqn. (S). This is, therefore, the current that will be found by extrapolating to t = t =, currents recorded beyond tG in a real quasipotentiostatic experiment _ Without knowledge of e, eqn. (S) shows that there is no direct method of determining &, or a from measurements of zjo~ at a series of values of E,. One method of getting round this difficulty is to introduce deliberately extra uncompensated resistance P (see Fig. I) in series with the cell and study the effect on ifO:! of changes in P at constant E,. It follows from eqn. (8) that

so that plotting a graph as in Fig. 4 will enable if -, a and e to be determined. From of eqn. (6). However, the variation of ifwith E,, k, can be found by application exploitation of this method requires more accurate data than experiment is likely to provide. A better method is to vary concentration_ Since

-if03e =

2tA Fk,

exp

-anF (Em + &a RT

p

-_E.))

(r9)

it follows that a plct as in Fig. 5 will yield the standard rate constant. Experiments with added uncompensated resistance will enable oc and e to be determined also. So far in this section, we have assumed that an extrapolation to give ifOz is possible; we now.justify this. It has been shown9 that the initial phases of a condenserfree quasipotentiometric electrolysis obey the equation

to the same accuracy and over the same range that a truly potentiostatic electrolysis obeys eqn. (5)_ As explained below, the term 4Cf--a)/ 7~ is usually much smaller than unity, - so that the major effect ‘of uncompensated resistance over the range ~_>i&~z

>oS

is- to diminish

the

slope

of -the

ijr VS. p_ graph

from

~&%&&&,)

to

~~((z&&)/(r + a)) but to retain its linearity. -_ Tliough exact for sm&ll values-ofa, eqn. (20) is based upon approximations9 wgch become seriously invalid a& approaches unity_ We can therefore set 0.5 as the upper limitof-permissible values of ti. With v&es-of. Ae between 0.03 and 0.3 Q cmz, &is implies an_up$er limit to cu&ent-density of the-order of-o-3 A cm-z_ ThefermJis defined by J_ EZectromraZ:

Chem..

II

(1966)

-171-187

K.

: I?+.---‘/

P /P

OLDH_431

______ ____-__-_ r 1

i

Lug ~i,02~p.0

I

/

c 92

B.

Q

Pp

Fig, 4. .A diagram illustrating ‘tiostatic electroIyses carried

how the out with

unknowns e, cx and ifadded uncompensated

can be determined resistance_

from

quasipotcn-

EFFECT

OF

USCOMPENS4TED

RESISTANCE

IS1

(21) a). are & &-g&#T. ~Q.‘sJp_vJer, (f-a! I& 1J%&y so that f!P2 estre_meYal~EE- ef . . (fhave a much smaller magnitude than this, since LYwill usually be in the vicinity of 0.5 and a typical value for Ez would be close to the polarographic half-wave potential and Ea equal to the half-wave potential, (E,+ (RT!nF) In l/o’/D) ; for oc=0_5 (f-n) = o. If e is small enough that the linearization

if02 =if-exp{-cza)zs

(22)

zf-(1-a)

is valid, then the approximatizm

A&,_ Qm

I-L-2

1-f

is equally valid. Under these conditions, $ ‘us. \/t graph is

the initial gradient of the quasipotentiostatic

(24) In the limit of small e, therefore, the intercept of the if vs. vf plot is diminished by the factor (1-a) and its initial slope by the factor (I -3a+f). if02 may be found by extrapolating the linear section of the i us. I/t plot to fi=& But since u and e, on which ta depends, are initially un’known, how is the extrapolation to be performed? An iterative method may be adopted in which tc is guessed, e and orare then measured approximately, tc refined, and so forth. A less tiresome method, which is probably of sufficient accuracy, hinges on the observation that Tables I and z show 0.1s I tc/ &/i7+ lie5 always in the range z+
of method

We have seen that the determination of kinetic parameters by a potential-step method in the presence of an uncompensated resistance requires analysis of the intercept (at i/8 = v’r=) andjor slope of the linear section of the i 3s. 5/‘2plot (see Fig_ 6). The section which is linear to within about 4% extends from r/l = VT+ to I/t= C/TL where 7~ is the time at which i = o . 8 &,s. Clearly, for the method to be at all applicable, XL must be sufficiently greater than zti. say four times as great or more, at at least one potential. Since Qz is arnk-knum when Ez is close to ES5, TV is then a maximum and conditions are optimal for kinetic determination_ It is the magnitude of k, which primarily determines TV, whereas td is relatively insensitive to this parameter. Figure 7 skws how the characteristic times values of other parameters: tc, t+ and 7~ depend on k, for the following t=ical 1. Electroanal. Chem., II (1966)

171-187

K. B. OLDHAM

Fig. 6. An illustration showing hbw an cstrapolation TL 2 4 t+ is seen to imply io 2 o-93 rSofe_

n=2,

a=

Ez=Ea,

ro-km%ec-1,

o-5, D=D'=

can

RT/F=2.6

bc made

x 10-2

to determine

V,

Cdl/A =2

i/oa- The

condition

- 10-5

F cm-z,

or 10-e mole cm-3 and ,+4 =0.03 or 0.3 Q cmz. 4,ro-5 ll cm? offers the best conditions for cm-a, +I =0.03

E,=E,+o.qV,C=ro-

It is k&dent that C Aro-5 mole extrapolation and that, with the parameters selected, represents the upper limit for measurable rate constants_ a valid

k,=o.rg

cm

set-1

Uncmq+sated resistance z@w7ed In this final section, we discuss the effect of ignoring a small uncompensated resistance and analyse the conditions under which the apparent kinetic parameters, lisnptid (~a~, so determined, are valid approximations to the true values. Equation (8) shows that uncompensated resistance will destroy the linearity of a log if0 vs. E, graph and thus invalidate the most widely practiced method of determining K, and OTfrom potential-step results. However, curvature of this plot may be concealed -beneath experimental scatter and it is therefore of interest to determine what -is the best straight line that can be drawn through points obeying eqn. (8)_Fojr a set of evenly spaced Em-values -centered about E,, the straight line -giving least square deviation has the equation log

L

if0

=

log (mAFk<]

~aF(Ern--Es) - zj303-R7.

-

-

2_3o:kd

sinhad

EFFECT

OF

\

UKCOMPENS_4TED

RESISTAKCE

183.

\

Fig_ 7_ Compar+ive values of the square roots values of the parameters R,. iZ and PA. Dashed violated.

of the characteristic lines indicate that

times tc, t+ the condition

J, Ekctvoavral-

Chem.,

and TL for various cr < o-5 has been

II (1966)

171-187

K. B, OLDHAM

184 Since

the

errors

of only

rapidly

approximation 17%

~-1

sinh x=

and

and IoO/~, respectively,

with decreasing X, eqn.

log z-lo = log

I

3X-‘[cash

for x values

(25) can be simplified

(a4Fk.C) - b

x -_x-1 of unity,

sinh ;c3 = I

introduce

the error diminishing

to

+ (I - b~;oyT-Em)

(27) z-303 while retaining most of its precision, provided ocd does not exceed unity_ Figure 8 compares points determined directly from eqn. (S) with the straight line generated by eqn. (27). It is e\-ident that the experimental precision would need

I 003

I 0.02

I

I -0.01

I 0

09 (Em

-E,)(V)

I -0.02

I -003

-

Fig. S_ Showing that curvative of the log if us. E relationship may not be pronounced, even whcr, the effect of uncompensated resistance is considerable. The data correspond to a value of o 3 for b_

to be

considerable

to

detect

significant

curvature

in the

experimental

points.

The

0-026 V, and k,t?=r_3 x ro-7 mole parameters PA =0.3 SL ems, n=2, cx=o_z. RT/F= cm-2 see-1, were used in constructing this figure. If +, data collected in the presence of a small uncompensated resistance are analysed on the assumption that e =o, eqn_ (27) shows that the apparent kinetic parameters are related to the true ks anda by the relations log ksap = log k, and &aP

=cK(I

--b)

Sitice b is-a +itive

-

&

2-303

(28)

(29)

quantity proportional to concentration, it follows that both the apparent standard rate constant and the apparent transfer are too small and that both decrease with increasing concentration; these are precisely the results observed experimentally3. The true-a-value could be obtained by extrapolating an map ‘us. c g=$ to zero concentration and the true log lie by sinGlarly extrapolating a log kaap vs. c graph. For values of b 5 d.05~ Keap and asp will differ by less than 5% from the true values, i.e., they,wiIl be well &ithin- the usual experimental error associated with electrode kinetic stud&s. For a: [email protected], rt =.z and RT/F =0.026 V, this implies k,CeA -2 GF: xmO-Sitiole Cl %ecyl and if C= ~6-5 -mole cm-a and eA = 0.03 fi ems. k, must be

EFFECT

OF

UXCOhIPENSATED

RESISTANCE

185

less than 2 - 10-2 cm set-1. Referring to Fig. 7, we see that for c= IO-J mole cm-a, eA=0.03 R cm’, KS=2 - 10-2 cm set-1, an i WY.I/t plot is sensibly linear from l/t=2 IO-~-]&= 14 - ro-3 secf, while l/&=x - x0-3 se&. Under these conditions, ignoring t= will lead to an overestimate in &, of only z%, so that completely disregarding uncompensated resistance will, for KS-values of less than 2 - 10-2 cm set-1, lead to no significant error in either k, or u. This conclusion applies, of course, only when the various parameters have the optimum values selected for this example, only if log ZrOdata are collected in an applied potential range not exceeding (E8f_ RT/anF) and only to kinetic evaluation from if0 (greater dependence on e attends the determination of k, andcr from Q) _ ACKNOWLEDGEMEST

I thank many colleagues for their constructive criticism and valuable discussions. This article is based on studies which were supported in part by the U.S. Army Research Office (Durham) and is contribution No. 3239 from the Gates and Crellin Laboratories of Chemistry of the California Institute of Technology. SUMMARY

If uncompensated resistance is ignored, the standard rate constant, k,, and the transfer coefficient oc, as determined by the usual potential-step method, will both be too small. The error may not be large, however, provided that k, (2 . IO-” cm see-1 and that the experiment is designed to minimize the uncompensated resistance effects. A method is described by which& and k, may be determined in the presence of uncompensated resistance; the method may be used for k, magnitudes as large as 0-15 cm see-l.

A C(x, t) C’(x. t) c CdL

CE D D’ E E, EOO EZ 2% F OX

Q QQ

electrode area (cm’) ; concentration of 0-z at distance x and time t (mole cm-z) ; concentration of Rd at distance x and time t (mole cm-s) ; bulk concentration of Ox (mole cm-a) ; double layer capacity (F) ; counter electrode ; diffusion coefficient of Ox (cm” set-1) ; diffusion coefficient of Rd (cm2 set-1) ; potential applied to WE vs. RE (V) ; valueofEatt o(V); the potential (E, +&e) (V) ; standard potential of Oz/Rd couple (V) ; Faraday’s constant (g-65 x 104 C equiv.-1) ; an electroreducible species ; parameter, KID-k + k-1D’-s (set-a) ; kl&-* + k-l&‘-s ; value of Q when .s= E,(sec-a) ; J.

EZeclroanaZ_

CLern_,

II

(1966) 171-187

K.

k--leDI-+ ; value of Q when E= Ea (set-*) gas constant (8-32 J deg-lmole-1) ; reduc-tion product of 0~ ; reference electrode ; absolute temperature (“K) ; working electrode ; faradaic impedance; K12D-*

+

dimensionless

parameter

cmF if0 p;‘RT ;

dimensionless an electron ;

parameter

CHGA FZk,Ce/RT;

complementary

;

error function,

total cell current, (le +if) (A) ; valueofiatt=r+ (A); nonfaradaic current (A) ;

faradaic current (A) ; initial faradaic current (A) ; initial faradaic current following potential step to E = EaD (A) initial faradaic current following potential step to E = Ee (A) rate constant for reduction of Ox at potential E (cm set-1) ; value of kl when e = Ea (cm set-1) ; value of kl when E = EZ (cm set-1) ; rate constant for oxidation of Rd at potential E: (cm see-1) ; va+e of k- I when E = Em (cm set-1) ; value of R-I when .s=E2 (cm set-1) ; cornmon value of k1 and k-1 when E = Es (cm set-1) ; apparent value of ks (cm set-1) ; number of faradays to reduce one mole of OX (equiv. mole-l) radius of spherical WE (cm) ; time since onset of electroly&s (set) ; an arbitrary variable ; distance from electrode surface (cm) ; the. dimensionless ptiameter (&-Em) unF/RT; .the dimensionle_ss p ammeter (E, -Em,) mF/RT; rritium and mi+m& values of (E, - E,)+zF/RT; decrease in + due to concentration polarization (A) ; faradaic current lost on account of double-layer charging (A) micompensated resistance added-in series with cell (SL) ; t+ cat$iodicStiansfer coefficient ~dirnensionless) ; -ap+rent value df L*;. _distarice&om..W& totip of a narrow RE probe (cm) ; poten++ of %V&s: adjkcent layer of solution_(V) ; spck& conductance of s&t&n (n-1 cm-r) ; _-npe~~~~~d_~~~~~~~~ ($2)-; e_ffective~.duration of char&&_-delay (sec) ; _

; ;

;

;

B.

OLDHAM

EFFECT

OF

UNCOJIPEXSATED

RESISTAETCE

187

upper time limit of linear if VS. VCrelation (set) ; time for transient to decay to a fraction q5 of & (set) ; a small fraction, e.g. 3% ; an*arbitrary variable ; REFERENCES

I H. GERISCHER AXD W. VIELSTICH, Z. PJlysck. CJlem. (2%ankfzwt), 3 (1955) IG; 4 (1955) IO. 1 N. TANAKA AND R. TAMAMUSHI, Electrochim. Acta, g (1964) gC+_ Y:‘. OKINAKA, S_ TOSHINA XXD H. OKIPCIWA, TaZanCa, II (1964) 203. 3 J_ H. CHRISTIE, G. LXUERXND R. A. OSTERYOUSG,]. EZecfromanZ. Chem., 7 (1964) 60. 4 J-E. l3. R~XDLES AND K. W. SOJIERTON,TY~IIS. Faraday Sot_, 48 (x953)937_ Interscience Publishers Inc., 5 P. DEL_%HAY, New I~aslrzrmentaZ Methods in EZectrocJ~emistry. NewYork, Igs+ G S. BARNARD. /_ EZecirocJzem. SOL, 108 (IgGI) 102. 7 W. B. SCHAAP AKD P. S. MCKINSEY, _-l?aaZChem., 3G (1964) 1251. 8 H. A. LMTINEN, R. P. TISCHER AND D. K_ ROE, J. EZectrocJzem.Soc., 107 (1960) 546. g I<. B. OLDHAM, EZectvocJbian. Acta, in press. /_

EZectroanyZ.

Chem.,

II (rgGG) 171-187