A note on the form of current-potential curves

Electroanalytical Chetmstrv and lnto factal Electlochemtstl 3, 49 (1974) 407-413 (~ Else,~ler Sequom S A, Lausanne - Printed m The Netherlands

A N O T E O N THE F O R M O F C U R R E N T - P O T E N T I A L

407

CURVES

I RUZIC * Northwesteln Untverszty, Evanston, Ill 60201 ( U S A ) (Recewed 21st August 1973)

INTRODUCTION

A general equation for the simple cathodic current-potential curve can be given for the controlled-potentml regime of electrolysis m the following form1: l ..~- ld ~ t ~ ) / ( l -~.-ep )

(1)

where p --- nF/R T [ E - E o + ( R T / n F ) I n (Do, J;ed/Ored

£x)]

(la)

and g'(~) =rr ~ c'e~-~ erfc ~, 0(~) = rc~

+

4 = 2t ~ stationary plane model 2

( - ly ~ B

g'(~)= g/(1 + 5,),

~',

~- 62/0 ~

~ = 2(3t/7) ~

expanding plane model 3'4

steady state approach 3- s

(lb)

with = (~zDt) ~

for stationary plane-'

~ (3rcDt//)-~ ~

exp

for expanding plane 3 4 vxdx dz

for convective diffusion 5

0

(lc)

and 2 = (k,/D -~)e -~P ( 1 + ev),

D = O ~ -~ D~cd

Bk = F [ ( 3 k + 7)/14]/F[(3k+ 14)/14]

(ld)

Here ta is the diffusion limiting current. E 0 as the standard potential, Dr~a. fred, and Do,, Jo,'are diffusion and activity coefficients of different forms of depolarizer, respectlvely, k. is the heterogeneous standard rate constant and c~ is the cathodic transfer coefficient. In the case of a very fast electrode reaction (reversible current-potential curve) the ~ term becomes very large and the g,(~) function becomes equal to unity Therefore /re, = /d/( 1 +fro) * On leave fiom "Rudjer Bo~kovxC' Institute, Zagreb. Yugosla~la

(2)

408

I RU2Id

or

log [( ia -- ire, )/ir~, ] = ( n F log e/R T) [E - E o + ( R T/nF) ln Dot Jred/O~d Lx] = (nF log e / R T ) ( E - E ~ )

(2a) ~

where E r is the so-called reversible half-wave potential. In a semi-logarithmic diagram such a current-potential curve gives a straight hne and Tome~ 6 was the first who proposed such a kind of analysis of current-potential curves. However, slow electrode reactions or some electrode reaction coupled with a homogeneous reaction (preceding, parallel, or following one) show a departure from the linearxty described by eqn. (2a) Several authors 7 proposed the use of different variants of semi-logarithmic diagrams so that, for each case, a straight hne is obtained. The disadvantage of such an approach is the use of different, relatively cumbersome, procedures for analysas of results which are not universal and can be used only in some special cases. However, the usual loganthmxc analysis has been shown to be a powerful method, which can be apphed in every case (with special treatments in each different case), even in the cases of relatively complex reacUon mechanisms s. On the other hand several approximate methods have been proposed which give incorrect results at some condmons and have only limited apphcation. A good example is a case of nonreversible current-potential curves which has been treated by several variants of rigorous procedures 3 9 l o and several approximate methods 4, 1 1--13. There are two possible reasons that some approximate procedures based mostly on the steady-state form of the solution yleld inaccurate results' (i) Improper estimation of some parameters from experimental data by which then a detailed analysls ~s performed. (ii) Inconvenient form of approximate solution which at some conditions d~sagrees with the rigorous solution. The most c o m m o n case of the first type of error ~s an inaccurate determination of the reversible half-wave potential, as was discussed m more detail m earlier work 13. In the present work the second type of possible error will be discussed RIGOROUS SOLUTION FOR VERY SLOW ELECTRODE REACTIONS

In the Introduction it has been mentioned that eqn. (2a) is valid for all solutions in the case of very fast electrode reactions (reversible current-potential curve). The reason for this is the fact that for very large ~ parameters all solutions reduce to the same simple equation (eqn. 2). However this is not the case for slow electrode reactions. In the case of a very slow electrode reaction (totally Irreversible current-potential curve) the 0(~) function reaches significant values only at potentials more negative than E 0 (m the case of cathodic process) and therefore the e ° then can be neglected m eqn. (1). Therefore in such a case: I -m- l d ~ ( ~ )

(3)

The steady state approach then yields the following results: , = ia ~/( 1 + ~) or

(4)

409

C U R R E N T - P O T E N T I A L CURVES

1og[(id--,)/Z] = --log ~ = --log 2rS+log (Dr~6)~ = ( ~ n F log e / R T ) ( E - E * ) + log(Dr~d) ~-

(4a)

where E * = E o +(RT/~nF)ln

k(t/D) ~

So, the steady-state result for a very slow electrode reacnon also has a hnear logar!thmlc dmgram, but wlth, a smaller slope than that for eqn (2a) However, for very slow electrode reactions the steady-state approach ~s not a very good approxlmatxon and the rigorous solution (1) follows a more complex shape. In Fig. 1 the corresponding logarlthmm diagrams for the stationary plane, expanding plane model, and steady-state approach are presented. As can be seen, however, at the beginning of the current-potentml curve (more posmve potentials in the case of cathodic process) the steady-state result (4a) can be used as an asymptotic solution w~th 6=(rcDt) ~ in the case of a statmnary plane and 6=(3~cDt/7) ~"m the case of the expanding plane model. 1og[(~d--t)/t]--(c~nF log e/R T)( E - E* ) - log 7r~ stationary plane log[(~a-z)/~] = (~,~F log e/RT)(E-E*)-log(37r/7)~ expanding plane

(5) log(td/I ).

At these potentmls t~ta and ]og[(ld--I)/l ] i n eqn. (5) can be replaced by This has been used m procedures for analysis of totally irreversible waves using the rigorous solution of Koutecky 3 The eqns. (5) have very often been used for

Itog2 D3 o

! O .I

/

/ 3

i

11

i

I

i

~

1OO mV " ~

I E ~* - P o t e n t l o [

Fig I Logarlthmlc analysls for a totalJ3 11~eve1~ible electrode reactlon or an irre,~erslb]e part m more genera] cases ( ] ) S t a t l o n a r y plane model, (2) steady state approach [Wlth 6 = ( r c D t ) ¢ ] , ( 3 ) expanding plane model--lnstantaneous current, (4) expanding plane model--mean current

410

I

RU2I(~

interpretation of the shape of a whole wave However, such an approach ~s not correct, particularly an the case of a stationary plane electrode. Close to the top of the wave (more negauve potentials m the case of catho&c process) the following asymptonc solutions can be used.

1og[(Id --,)/i] = (nF log e/RT)(E- E*)-log 2 stationary plane ]og[(,a -~)/~] = (~,~F log e / R r ) ( E - E * ) - l o g ~

expanding plane

(6)

In the half-wave region there is a transition between the two asymptotic solutions (5) and (6). In the case of the stauonary plane model of diffusion the rigorous solution can be described very well by an antllogarlthmic linear combinaUon of corresponding asymptotic expressions and the expanding plane model by a straight line with a slope higher than cozF log e/RT (see Table 1):

l/(ta-I) ~ 2(3,,'7)~e-P* + 7r~e -~'~ stationary plane

log[(~d-,)/~] ~ 1 l(~nf log e / R r ) ( E - E * ) - l o g 2(3/7)~ expanding plane

(7) where

p* = n F ( E - E*)/R T TABLE 1 SLOPES OF THE IRREVERSIBLE PART OF LOGARITHMIC PLANE MODEL OF DIFFUSION PolentlaI range ( E - E* )/mI ~

Alean curl ettt

1 000 1 012 1 036 1 069- i

1 000 1 000 1 017 1 026

- 200

to 0 to - 4 0 to - 8 0 to - 1 2 0 to - t60 to - 2 0 0 t o - 240

1 100~1 0 9 5 + 0 0 1 ~ 1 126~1 0849 1 026 1 022 1 016 1 000

1 042"] 1 056[ O47 + 0 0 0 5 1 0 4 41" ( 1 045) 0993} 1 0 0 t ) 0 994_+0 004 0 990J

Mean

values

1 041 + 0 035

1 0 1 4 + 0 025

40 0 -40 -80 - 120 -

160

160 120 80 40

FOR EXPANDING

Slope~ in multlples oj ~nF log e/R T Instantaneous current

240 to 160 to 120 to 80 t o

DIAGRAMS

|

I

|

These are expressions valid for instantaneous current, in the case of mean current, the expanding plane model of diffusion results in (see Table 1)" log [(-/a -7)fi] = (co,F log e/R T)(E - E * ) - log 0.8

7 ~ i,,

log [(id --1)/7] ~ 1.0S (c~,,F log e/RT,)(E-E*)-Iog ~r½/2

log[(Td--i)fi] ~ (c~nF log e / R T ) ( E - E * ) - l o g Tr}/2

~ ~id

l

~.

ld/2

(8)

CURRENT-POTENTIAL CURVES

411

Note that the constant r@2=0.886, the value proposed by Weber and Koutecky 14 and the coefflcaent 1.05 is close to that proposed by Matsuda 4 (i 04) However, experimental errors are very often close to the devaanon of the mean current diagram from the slope c~nF log e / R T m the case of the expanding plane model of dlffusmn, and therefore the third eqn. in (8) can be used as a good approxxmatlon m the whole range of potennals with measurable mean current QUASI-REVERSIBLE CURRENT-POTENTIAL CURVES Inside the range of quasl-revers~blllty the steady-state equation is itself a good approxlmanon and therefore eqn (1) yields I = Id ~/( [ q- e l) )( 1 -Jr-~) = ld ( 6 L / D ½ ) / (

1 + ~2/O ~)

(9)

where g = 2/(1 + e ° ) = k~ e-~"/D -~ Therefore

(ld--l)/i=

- I + ( I + ( V ~ / D ~ ) / ( ( ~ L / D ~ ) = ( D i / 6 L ) + e " = ( D / d k , ) e ~p + e p

(9a)

The first term m the final expression (9a) represents the antilogarithm of eqns. ( 5 ) a n d (6)for expanding plane with c~=(rcDt) 4 and is called the arreverslble part. The second term is the antilogarithm of eqn. (2a) and is called the reversible part. The total diagram can be obtained as an antllogarlthm~c sum of these two parts. Therefore for different models of diffusion one can write the following equanons' For the stationary plane model ( / d - ')/l = e°*/[2(3/7) ~ + re} e' 1 -~>'~] + e°

(10)

For the instantaneous current, expanding plane model (ld --l)/1 = (e cq)*/TO½) + e p

for E ~ -,')¢v

(l d - - l ) / l = [ e a p * / ( 3 G / 7 ) -~-] --~-e0

( , 6 - , ) / I ~ e ~ ~"' /2( 3/7) ½ +e °

for E--, + <

for E ~ E*

(10a)

For the mean current, expanding plane model (~d --~)/7 ~ [e"~/(n~/2)] +e'

(10b)

In the case of quasl-revers~ble instantaneous current-potential curves with expanding plane model of diffusion very often up to now the corresponding equanon from (5), with O=(37rDt/7) ~ for the irreversible part, has been used. However, th~s equanon describes the ~rrevers~ble part at the beginning of the wave, where e° IS predominant and close to the llmmng current values where the irreversible part is predominant the corresponding equation from (6) should be used w~th 6=(rcDt) t. Therefore in procedures for analysas of quasi-reversible instantaneous current-potential curves the first and third equations from (10a) can be used and the second equanon mostly used up to now has no practical ~mportance.

412

I RU~Id

CONCLUSIONS

(I) Instantaneous currents in logarithmic dlagrams have significantly different forms for the irreversible part with stationary and expanding electrodes. The beginning of the wave with the stationary plane model and the end of the wave with the expanding plane model have the same equation for the logarithmic diagram (slope omF log e/RT and ordinate at E* potential equal to - l o g rc~). The logarithmic diagram at the end of the wave with a stationary electrode has a slope nF log e/R T, even if the electrode reaction is irreversible. The logarithmic diagram at the beginning of the wave with the expanding electrode has the same slope as at the end, but with the ordinate extrapolated to E* shifted by log(7/3) ~ to lower values (for log [i/(ld --l)]). (li) Close to the half-wave potennal, the stationary plane model can be described by a hnear comblnanon of the equation for the beginning and the end of the wave ( antilogarlthmic linear combination). The expanding plane model (instantaneous current) shows, at the same range of potentials, a slope 10%o h~gher than at the beginning or the end of the wave In the case of mean current, the corresponding deviation from the slope c~nF log e/R T is lower than in the case of instantaneous current and has an order of magnitude close to the error of measurements. (lil) in the case of quasi-reversibility (both reversible and irreversible part have significant influence) steady state is a good approximation. However, in such a case the beginning of the irreversible part cannot be seen because there the reversible part is predominant. Because only at the beginning of the wave the irreversible part follows the expanding plane approximate solution [3=(3~zDt/7) ~] in the case of quasl-reversxble electrode reaction such a solution is no more correct and very often the same solution as that for the beginning of stationary plane model [3=(rcDt) ~] gives more accurate results. The same is valid also in the case of more comphcated mechanism such as two-step electrode reaction 15. At the lower hmlt or quasl-reversablhty an influence of the irreversible part around the E* potential should be considered. In such a case the E* potential can be determined by extrapolation from the end of the wave using the corresponding equation from (6) and then subtracting the irreversible part close to the potential determined by the corresponding equation from (7). The mean current can be treated by earlier methods if special accuracy is not needed. ACKNOWLEDGEMENTS

The author is grateful to Prof. D. E. Smith and Mr. R. Schwall for helpful discussions The completion of this work was assisted by NSF Grant GP-28748X. SUMMARY

The forms of the current-potentml curves obtained for different models of diffusion by rigorous and approximate methods are compared. It is found that for instantaneous current the irreversible part of the logarithmic dmgram (log ( i/id -- i) VS. potential) obtained by rigorous methods at suffi-

CURRENT-POTENTIAL CURVES

413

clently positive potentials agree very well with a p p r o x i m a t e results (with c o r r e s p o n d i n g diffusion layer thickness (rcDt) -~ for stationary a n d (3rcDt/7) } for expanding plane model). However, at more negative potentials the slope of the logarithmic d i a g r a m is no longer equal to c~nF/RT m the case of a stationary plane model of diffusion. In the case of an expanding plane at these potentials the diffusion layer thickness is no longer (3~zDt/7) ~ but equal to that of the stationary plane (~zDt) -~. Therefore the analysis of quasi-reversible p o l a r o g r a p h i c waves a n d related curves should be p e r f o r m e d without the factor (3/7) ~ ( e r r o n e o u s l y ) u s e d earlier, because the irreversible part of the logarithmic d i a g r a m at potentials near the reversible potential has no effect on the observed current (which is controlled by diffusion a n d not by heterogeneous charge transfer) a n d it is only at these low potentials that the exp a n d i n g plane diffusion layer solution should be used The analys~s of m e a n current is also d~scussed. REFERENCES 1 J R Delmastro and D E Smith. d Phy; Chem. 71 (1967) 2138, D E Smith. T G McCord and H L Hung, Anal Chem,39(i967)1149 2 M. Smutek, Chem Ltsty, 45 (1951) 241 3 J Koutecky~ Collect Czech Chem Commun, 18 (1953)597 4 H Matsuda and Y Aiyabe~Bull Chem Soc Jap, 28 (1953) 422 5 V G Levlch, PhystcochemwaI Hydrodynamics, Prentice-Hall, New York, 1962 6 J TomeS, Collect Czech Chem Commun, 9 (1937)12 7 K B Oldharn and E P Parry, Anal Chem, 40 (1968) 65. J M Hale, J Electroanal Chem, 8 (1964) 181. 8 i Ru~16, J. Electroanal Chem, 25 (1970) 144~ 29 (1971) 440, 36 (1972) 447 9 J M Hale and R Parsons, Collect Czech Chem Commun. 27 (1962)2444. J Elecrtoanat Chem, 8 (1964) 247 10 S Sathyanarayana, J Etect~oanal Chem, 7 (1964)403 11 J Koryta, EIectrochun Acta, 6 (1962) 67 12 R Tamamushl and N Tanaka, Z Phvs Chem F~anl@~rt, 39 (1963) 117 13 I Ruble. A Ban6and M Bramca, J Elect~oanat Chem 29(1971) 411 14 J Weber and J Koutecky, Collect Czech Chem Commun. 20 (1955) 980 15 I Ru~l~;, in preparation