Effect of sphericity on the voltammetric (polarographic) log plot for reversible amalgam formation

J Electroanal Chem, Elsewer Sequoia S A

218 (1987) 1-13

, Lausanne - Pnnted m The Netherlands

EFFECT OF SPHERICITY ON THE VOLTAMMETRIC (POLAROGRAPHIC) LOG PLOT FOR REVERSIBLE AMALGAM FORMATION

JEFFREY

E ANDERSON

Murray State Unrversrty, Murray, KY (U S A ) ALAN M BOND Deakm Unrversrty, Waurn Pod,

Vwtona (Austraha)

KEITH B OLDHAM Trent Unrversrty Peterborough (Cad) (Recewzd 30th October 1985, m revwd form 28th August 1986)

ABSTRACT

An Important source of thermodynanuc data 1s measurement of the halfwave potentials of reversible polarograpluc or voltammetnc waves Usually such potentials are determmed by performmg a “log plot” analysts on expenmental current-voltage data The theory of such analyses 1s based on planar electrodes but m practice mercury spheres are the favounte electrodes The prdction that sphenclty affects the mtercept (and, to a lesser extent, the slope) of the log plot w-&out destroymg its hneanty has been venfled expenmentally for the reduction of Cd(I1) m 10 M KCI at a static mercury drop electrode Three methods of correchng for sphenclty are exanuned and are shown to yield tnne-independent halfwave potentials of thermodynannc vah&ty Apphcations to chronoamperometry, normal pulse voltammetry, classical polarogaphy and normal pulse polarography are discussed

INTRODUCTION

One of the most fnutful sources of thermodynanuc data has been the study of reversible electron-transfer reactions by transient electrochenncal techmques For example, perusal of the hterature [1,2] on stabhty constants of metal complexes m solution reveals that the polarographlc method [3] 1s one of the most popular techmques. In many of these stu&es eqmhbnum constants are calculated from the standard electrode potential E O, which 1s itself calculated from (or 1s assumed equal to) the reversible halfwave potential Alternatively, the dependence of the halfwave potential on hgand concentration 1s analyzed [3]. 0022-0728/87/$03

50

6 1987 Elsewer Sequoia S A

2

Accurate halfwave potentials have usually been determmed by the “log plot” method m which the loganthm of (z,, - z)/z 1s graphed versus electrode potential E, or by an equivalent hnear regresszon Here I IS the current at a particular value of E and zd 1s the “&ffuaon current”, that IS, the current measured at extreme polanzatlon (sufflclently negative potent& m the case of a reduction) but under otherwzse Identical comhtlons Makmg the log plot IS valuable m that its hneanty and slope provide confnmatlon that the expenmental data do, indeed, frt the theoretlcal relationstip ln[(z,-z)/z]

=(nF/RT)(E-E”‘)-4

ln(&/Do)

(1)

on whch the method 1s based However, as will be demonstrated below, the plot may be hnear and have the correct slope even when eqn (1) 1s mvahd In tlus equation Do and D, are the dlffuslon coefficzents of the electroactzve species, E”’ 1s the formal (or condltzonal) potential of the O/R couple under the prevahng expenmental condtions, and other symbols have then usual electrochenucal agmflcances Imphclt m eqn (1) 1s the condition that I and zd are measured at the same instant, an interval t after electrode polanzatlon Equation (1) 1s denved from the thermodynanuc Nemst equation on the basis of five condltlons that E IS constant, that the electrode 1s planar; that there 1s no motion of the metal/solution Interface, that transport of 0 and R to or from the electrode 1s solely by dlffuslon, wzth a dzffuslon field that 1s effectively serm-mfmte, and that the electron-transfer reactzon 1s reversible In expenmental practice the condztlon of electrode plananty 1s almost always violated, especzally with the favounte electrode, a spherical mercury drop A recent pubhcatzon [4] addressed the shapes of log plots for spherical electrodes It was prticted theoretically that the neglect of spherzclty could cause a marked perturbation of the log plot, even when the other four condltlons are met Dlscrepancles from the simple theory were predicted to be most pronounced for cases of the reversible reduction of metal ions to amalgam-formmg metals Iromcally, these are precisely the systems that have been most prohflc m yzeldmg thermodynarmc data The expenmental techmque adopted m the present study 1s potentml-step chronoamperometry at a static mercury drop electrode at which a reversible electrode reactlon occurs this method meets all five of the condztlons except that of plananty In these circumstances, it 1s predicted [4] that the (zd - z)/z ratio will depend on time m such a way that, for a gzven value of t, the loganthm of (zd - z)/z wdl remam wtually hnear wzth potenttal whereas the slope and intercept of this hnear relation&p wdl depend on t, mvahdatmg eqn (1) The present article wdl address the question of how serious,, m expenmental practice, these sphenclty-induced departures from eqn (1) are, for the sample electrode reaction Cd*+(aq)

+ 2 e- @ Cd(amal)

(2)

We shall evaluate the recently-proposed [4] method of correctmg for the sphenclty effect by modzfymg the loganthnuc functzon that 1s plotted versus potential as well as mtroducmg two new correction procedures Our exclusive concern wzll be wzth

3

expernnents camed out under condttions of stnctly constant potential at a mercury sphere electrode and urlth reversible amalgam-formmg reactlons Normal pulse voltammetry consists of numerous potent&step chronoamperometnc experunents camed out at progressively mcreasmg polamatlon, the conclusions of this article are therefore apphcable Though It uses a non-stationary electrode, normal pulse polarography [5] exploits the later stages m the hfe of a growmg mercury drop, when the expansion of the sphere 1s slow, so that our results should apply, at least approxnnately These two popular pulse techmques employ bnef mtervals of polanzation and the effect of electrode spherrclty 1s expected to be small As we shall see, however, this effect may not always be neghgble. Several of the five mandatory condltlons are violated m classical polarography. Nevertheless, eqn (1) has been frequently employed to analyze polarographlc results and the hterature 1s replete wth halfwave potentmls so determmed. With a contmuously polamed dropping mercury electrode the ( I~ - I)/I ratio depends not only on t (the drop time) and potential, but also on the charactemtlcs of the capillary Attempts to model the behavior of droppmg electrodes mathematlcally, so as to predict polarographc I versus t relation&ups [6], have had only hrmted success, doubtless due to the comphcated and lrreproduclble hydrodynanuc regme that prev;uls early m drop hfe. Accordmgly, absolute halfwave potentials determmed by classical polarography should be viewed wth some sceptlclsm, though differences m polarographlc E,,, values may be quantitatively vahd when two expenments are camed out under sun&r conditions Fortunately, the successful commerclahzatlon of the stationary (but renewable) mercury drop electrode means that electrochenusts no longer need attempt to extract precise thermodynarmc mformatlon from polarographic results, except where the droppmg electrode’s advantages outwelgh its disadvantages. There 1s some amblgulty m the use made of the term “halfwave potentml” and of Its usual symbol, El,* Here we shall use E,,, in a purely expenmental context it is the potentml at which z = 1~/2 It 1s also the pomt at wluch a plot of ln[(zd - 1)/z] versus E crosses the potential axis In contrast, we shall use the symbol E, to denote the potential at which the hnear graph of ln[(r, - I)/I] versus E would cross the potential axis rf the expenment did obey eqn (1) That 1s to say, we define E, = E”’ + (RT/2nF)

ln( D,/D,)

Ths potential 1s somemes called the “reversible halfwave potent&’ or the “thermodynarmc halfwave potential” and notations such as Ei,* or (E:,,), are encountered. The letters “r” and “p” refer to the reqmrement of reverslbtity and planmty but, m fact, all five of the mandatory condltlons must be met before E, and El,2 are equal EXPERIMENTAL

A solution of 9 31 x 10V4 M Cd(I1) m 1.00 A4 KC1 was prepared from Qstdled daomzed water, Baker Analyzed Reagent grade potassmm chlonde and mossy

4

cadmmm of sun&r grade Solutions were deoxygenated by passage of argon for 8 mm pnor to each experunent The temperature was 25°C The workmg electrode was a “small” hangmg mercury drop generated by the EGCG/PARC Model 303 Static Mercury Drop Electrode umt The vohune of the drop was found by collectmg and we&mg three samples, each of 100 drops, generated at the mttlal potential The counter and reference electrodes were respectively platmum and sdver/sdver chlonde All potentials are quoted wrth respect to the Ag/AgCl/KCl (1.00 M) reference Each expernnent consisted of applymg a potential step from an lmtial value of - 500 mV to one of the followmg final potentials -630 mV, -650 mV, -670 mV, - 690 mV, - 710 mV or - 800 mV The current was measured 0 10 s after the step and then once every 0 20 s Each of the potent&step expenments was performed 8 tnnes on 4 different senslhvlty ranges, so that a total of 32 runs were camed out for each of the 6 steps In ad&tlon, the same procedure was used to evaluate the background on a blank (1.00 M KCl) solution The mstrumentation used consisted of an Apple II Plus nucrocomputer mterfaced to an IBM EC/225 Voltammemc Analyzer The interface consisted of a 12-bit analog to dlgtal converter (ADC) ~th three 16-bit tuners (TecMar Model TM-AD213) and a 12-bit dlptal to analog converter (DAC) mth 16 I/O bits (TecMar Model TM-DAlOl). The DAC was modified such that one bit corresponded to 1.00 mV The DAC was used to control the potential of the workmg electrode 111 the potential-step experunents. It was routmely cahbrated using a separate cahbratlon program One of the control bits was used to control the drop knocker of the Static Mercury Drop Electrode. The ADC was used to measure the current and was cahbrated daly usmg the DAC to apply various potentials to the dummy cell of the IBM potentlostat at vanous sensltlvlhes The cahbratlon routme generated a cahbration file that was used to convert ADC code to nucroamperes The step expenments were performed by a machme language routme called from an Applesoft program. All tnnmg operations were performed using the processor clock and the tuners associated wth the ADC Interface board

DATA TREATMENT

After the 8 sets of current data for each sensltlvlty were averaged, the background currents were subtracted and the resultmg current/tune data were saved on &Sk A separate Applesoft Basic program was used to retrieve the data from disk and to average the results obtamed on different sensltlvlties In some cases an mdlmdual datum was rejected because It was off the scale of the particular sensltlvlty. Therefore not all data are averages of 32 values. The currents for the step to -800 mV were treated as zd values and used to compute In[(z, - z)/z] for z values recorded from steps to the other five potentials In this computation, of course, the zd and z values were selected to relate to the same time mstant t and the tunes t = 0 3 s, 0 5 s, 0 7 s, 1 1 s, 17 s, 2.7 s, 3.9 s, 5 9 s,

5 TABLE 1 Regression results for uncorrected data t/s

slope/v-l

El,, /mv

r

03 05 07 11 17 27 39 59 89 13 3 19 9

75 19 78 11 7840 78 76 76 21 7933 80 66 83 49 83 91 89 12 95 56

-6676 -6680 -6686 -6698 -6720 -6729 -6749 -6769 -679 1 -6825 -6870

0 99938 0 99984 0 99984 0 99982 0 99988 0 99997 0 99983 0 99995 0 99972 0 99996 0 99952

8 9 s, 13 3 s and 19 9 s were chosen Thus particular set of times was selected because each tune represents about 150% of its predecessor The slopes and mtercepts were deterrnmed by a standard hnear regresslon procedure The results of thus analysis are presented m Table 1 and Fig 1 The columns headed r m the tables hst the correlation coefflclents of the hnear fit The

-0 63

-0 65

-0.67

-0.69

-0. 7

Fig 1 Regresslon hnes for log plots based on uncorrected data

closer these values are to unity, the better the hneanty of the data It is only for the longest and shortest times that the data display anythmg but a trivial departure from hneanty, as evtdenced by the correlatton coefficient. Thrs hneanty ensures that the mtercepts reflect the experimental E1,2 values accurately Though the hneanty of the uncorrected log plots is satisfactory, their slopes and mtercepts are certainly not. At the shortest ttmes the slopes may be considered to be w&m experimental error of the 2F/RT = 77 84 V-’ value predicted by eqn. (1) but there 1s a progresstve steepenmg trend as t increases Still worse are the intercept results. Whereas the halfwave potentral should be a constant accordmg to eqn (l), it is seen to drift negatively as time progresses, becommg-as much as 20 mV m error at the longest times These discrepancies m slope and intercept, with mamtenance of hneanty, are exactly those predicted to occur as a consequence of electrode sphencrty [4]. The need for sphenclty correction IS clear m our results Of course, we have dehberately chosen a rather small sphere and rather long ttmes so as to accentuate the sphencity effect and test the correction procedure under the most stringent circumstances It could be argued that we have contrived our experiment to demonstrate the effect of sphernxty, the latter bemg neghgible under more typical condittons Tins is not so As will be demonstrated below, the sphencuy-mduced error m a halfwave potential, when measured by the usual log plot method, 1s approxrmately

z

In

1 + ( *Dot)1’2/a i 1 - (TrD,t)“*/a

(4) 1

Here a is the electrode radius, Do and D, are the diffusion coefficrents of the metal ion m solution and metal atom m mercury, &d t IS the measurement time To make\ this quantity thermodynamtcally neghgible (say less than 1 mV) under typical expenmental conditions (n = 2, a ~4x10~~ m, Do=D,=10d9 m* s-l) would require expenments to be completed m less than 0 1 s, a rare occurrence m chronoamperometry Smaller radu, larger diffusion coefficients, or fewer electrons would make the error worse CORRECTION

PROCEDURE

To correct for sphenclty m the case of a reversible reduction to an amalgam-formmg metal, ref 4 advocates that mstead of plottmg ln[(ld - J)/J] versus potential, one plots the left-hand side of eqn. (5) versus potential dJ,-J)-(Dd”*(md-m) ar + (

Do)“2m

1=$E-E,)

(5)

The slope should be nF/RT and the mtercept on the potential axis should be the “thermodynarmc” halfwave potential E, New symbols m eqn (5) are the semnntegrals m and md, whtch are defined m the Appendtx.

To test this correction procedure we used the current data that were used previously for the uncorrected log plot analysu. The 1 data were semnntegrated off-hne, usmg the algorithm described m the Appendix, to generate values of m at the eleven chosen tunes Lkewse J* values were semnntegrated, agam usmg a spacmg of 0 2 s, to provide the correspondmg md data The measured 1.244 mg mass of an mdlvldual mercury drop translates to a drop radius of a = 2 80 x lop4 m

(6)

We used the hterature values [7,8]

for the diffusion coefficient of Cd2+, and D, = 1 60 X 10m9 m* s-l

(8)

[9,10] for the dlffuslon coefficient of Cd Thus the charactenstlc with the dlffuslon of cadmmm metal v&urn the mercury drop, a*/rD,

time associated

= 15 6 s

1s comparable urlth the duration of the step expenment CORRECTEDRESULTS Regression analyses of ln[a(z, - I) - ( D,)‘i2 (md - m)]/[ar + ( Do)1/2m] versus E were camed out m exactly the same way that was used for the uncorrected data, usmg the constants gven 111eqns. (6)-(8) regression lines are plotted as Fig 2

The results are hsted m Table 2 and the

TABLE2 Regression results for corrected data

t/s 03 05 07 11 17 21 39 59 89 13 3 19 9

slope/v - l

&JmV

r

74 19 77 86 78 01 7900 7131 77 22 77 22 78 91 77 34 76 62 7103

-6645 -6641 -6640 -6636 -6642 -663 8 -6641 -6637 -6627 -6624 -6600

0 99932 0 99984 0 99984 0 99986 0 99914 0 99997 0 99985 0 99960 0 99888 0 99914 0 99740

8

-0.63

-0 65

-0 67

-0 69

-0 ’

E/V Rg

2 Regrewon

hnes for log plots corrected via eqn

(5)

It 1s very evldent that the slopes and intercepts of the corrected log plots are markedly more constant than the uncorrected ones. For data obtamed at times less than about 15 s, the average slope and the correspondmg standard devlatlon are (77 73 f 0.80) V-l

(10)

while the correspondmg halfwave potential 1s E, = ( - 663.6 f 0.6) mV

01)

Such a l~gh degree of constancy 1s very gratifymg and amply venfles that the correction procedure 1s vahd. Moreover, result (10) 1s remarkably close (m fact, fortmtously close as Judged by the standard deviation) to the theoretical value 2 F/RT = 77.84V-l. At times greater than 15 s agreement 1s less satisfactory We believe that there are two reasons for tis First, the corrections at long -es represent a very major component of the loganthn~c argument and the prase values of the constants (6)-(8) become crucial Second, the denvation [4] of eqn. (5) 1s based on the “quas1semunfuute” assumption. ‘Ills 1s tantamount to assummg that, durmg the duration of the expenment, the centre of the mercury drop remams devoid of cadnuum. Because the tnne that It takes for a agmflcant concentration of cadmmm amalgam to bmld up at the drop centre 1s of the order of the tnne reported m eqn. (9), one would expect to see the fadure of the quaslsemunfmte approximation at

9

about thts tune. Recall that one of the five mandatory condtttons requrred to vahdate eqn (1) 1s that the Qffusron fields for 0 and R be effectrvely semunfmte the same 1s true for eqn. (5) Very recent work [ll] shows how rt 1s p’osstble to avord makmg the quansemnnfmte approxnnatton, but such extreme correctrons ~111 probably seldom be needed m routme voltammetrrc work We advocate that the procedure provrded by eqn (5) be routmely adopted m studtes whose goal 1s the generatron of precrsron thermodynamtc values, even m cucumstances m whrch drffusron coeffrcrents must be unperfectly estimated In work of less ngour, the simpler approxrmate correctton techmques, now to be described,, wrll sausfactonly remove spherrctty errors under condltrons of modest spherrcrty SIMPLER CORRECTION PROCEDURES

Even under reversrble potenhostatrc condrtrons, the Cottrell equatron [12] 1s not obeyed at a spherrcal electrode Nevertheless, under condttrons of mrld sphencrty, the dommant term 111the equatton descrrbmg the time dependence of current 1s mversely proportronal to the square root of t k = constant

, 3 kt-‘12

The semuntegral of tell2

02) IS srmply 7r112 and therefore

m = ka’i2 3 I( at)1’2

03)

Srmrlarly md = I~( rt)‘12 and rf these approxrmatrons are mserted mto eqn (5) one finds that the loganthmrc argument may be factored mto two terms, permtttmg rearrangement to In(

Id - 1 1

1

E _ g In h nF

=$?-g [

1 + (7rDot)1’2/a i 1 - ( &,t)“2/a

II

(14)

lh upproxwnute treatment unphes that the standard log plot, I e. ln[(r, - I)/I] versus E, v&l gwe a strarght hne The slope of thts strarght hne IS predicted to be a constant, nF/RT, but the E,,, mtercept on the potential axrs 1s predcted to be time dependent and equal to E, only m the t + 0 hmtt. We can use the data 111 Table 1 to analyze how senously m error these predrctrons are The predrcted constancy of the slope 1s evrdently erroneous. However, for times less than about 5 s its vanatton IS small and the average value (78 09 f 1 9) V-’

(15)

compares well wrth the theorettcal77 84 V-’ value Table 3 has been compiled to test the predrctron concemmg the mtercept on the potentral axrs The quantrty (4) 1s the correctron term that must be added to the mtercept to produce the halfwave potentral. Tins correctton has been mcorporated mto the final column of the tabIe, whrch shows that the procedure does approxr-

10

TABLE 3 Data relevant to the approximate correction procedure l+(sDoly/(l

EI/Z /mV

Correctlon/mV

Et, /mV

-6676 -6680 -6686 -669 8 -6120 -672 9 -6749 -6769 -6791

30 39 47 60 76 100 12 5 16 6 23 2

-6646 -6641 -6639 -663 8 -6644 -6629 -6624 -6603 -6559

1 - ( rDRfy’*/a 126 136 144 159 181 2 17 2 64 3 63 608

03 05 01 11 17 27 39 59 89

mately remove the vanablhty m halfwave potential for times up to about 5 s Beyond this tune, the ad&bon of term (4) overcorrects for sphenclty The average value of the fast seven tabular entnes 1s E,=(-6637fO8)mV

(16)

whch 1s m full agreement with the exact result (11) We conclude that the simpler correction procedure 1s adequate for times, m our expenments, of up to 5 s The second column m Table 3 then suggests that, generally, values of (1 + (Iroot)‘/2/u)/(1 - (~rDat)~/~/u) of up to about 3 may be tolerated For typical diffusion coefficients this condition 1s satisfied m most voltammetnc expenments. Therefore we beheve that the simpler correction procedure should fmd general apphcabtity A vahd obJectlon may be levelled agamst this procedure that it reqmres values of diffusion coefficients, which are not always known unth preclslon If the ( ~Dt)‘/2/a terms m eqn (14) are sufflclently small m comparison mth umty, this equation can be approximated as E l,2=Eh-s

ln

1 + ( 1ZDot)1’2/a 1 - ( 17DRr)1’2/a

_E

_

RT(Dy2+Dy2)(ary2

h

nFa

(17)

wbch shows that a graph of El,, versus t’i2 should be hnear m the short-tune bt and extrapolate to E,. Figure 3 depicts such a graph constructed from the data m Table 1 Evidently Iineanty persists to surpnsmgly long times. The mtercept of the regression hne gves E,=(-6649fO7)mV which 1s consistent unth results (11) and (16) The slope of the hne, (-4

(18) 87 f 0 35)

11

-0.

Fig 3 Halfivave potentmls from Table 1 (excludmg the t = 19 9 datum) plotted versus the square root of hme The lme 1s the regresslou of the ten pomts and has a correlation coefflaent of - 0 99745

mV s-l/* IS, however, somewhat at vanance with the theoretical slope of - 5.35 mV s-‘I* pr&cted by eqn (17) on the basis of the data m eqns (6), (7) and (8) Tlus may reflect erroneous &ffuslon coefflclents and/or an mappropnate extrapolation As a means of fmdmg E,, this graphlcal method IS nevertheless satisfactory CONCLUSIONS

To be thermodynanncally meanmgful, halfwave potentials must be mdependent of the voltammetnc method used for then measurement and of the time parameter employed. Chronoamperometry and normal pulse voltammetry are capable of yleldmg “thermodynamic” halfwave potentials E, only if attention IS gven to the mfluence of sphenclty on the currents observed ‘with amalgam-formmg processes at mercury drop electrodes. Electrode sphenclty may cause the In[(z, - z)/z] versus E graph to @ve an erroneous halfwave potential even when thts plot IS hnear and even d Its slope has the theoretical nF/RT value Studymg z and zd as functions of tune perrmts a sphenclty correction to be apphed, three correction methods are avdable The dependence of the E, - El,* difference on the square root of t means that the hscrepancy between the expenmental and “thermodynamic” values of the halfwave potential may be slgmflcant even at rather short times Thus for t = 50 ms,

12

a value typical of pulse polarograpmc practice, sphenctty can cause a shift of halfwave potenttal exceeding one nnlhvolt It should be stressed that none of the relatlonshtps derived m this article can be expected to apply to classrcal polarography. However, some of the newer polarograpmes (such as normal pulse polarography) that exploit the short-tune potenuostatic behavlour of a dropping electrode grown to mature size under current-free conditions should obey these relattonshtps approximately ACKNOWLEDGEMENTS

The fmanctal support of the Deakrn Umversity Research Comnuttee and the Natural Sciences and Engmeermg Research Council of Canada is gratefully acknowledged Fmanctal support for the purchase of the interface used m tins work was obtamed from the Comrmttee on Institutronal Studtes and Research at Murray State Umversity APPENDIX

The semnntegratton algorithm is based upon segmenting the Rtemann-Liouvllle integral defimtton [13] of the semuntegral m(r)=s-l/2/r

z(dd7

0 (t-$‘2

_-l/2/=

Wd7

+a-l/2

0 (t-7)1’2

J-l z J-2

T I+' /

q

+)dT (t-+'2

into J - 1 components, where TJ = t and where T,, T,, T,, , T, are the times at whtch current data are recorded. The ~(7) term wnhm each mtegral 1s replaced by A, + B,T-~/* where AJ and BJ are deternnned by fittmg the experimental currents zJ measured at q and T,,, The integrals are then evaluated analytically and For the comhtions used m the present experunent m which T/ = (3 - $)A where A = 0.2 s, we are led to the final result ‘/+I

~(~)1’2=Al[(J-f)1’2-(J-l)1’2]

+sarmm($--)‘” 2

+‘~1A,[(J-~)1/z-(J-J-1)1’2] J-1

B arcsln (/ + 9”’ + / L\‘/* (J-g””

A,[(~+f)l’~-

(J - f y]

= (J + +y2r,,,

_ arcSln (J - tY2 (J-+)“’

- (J - +)1’21,

1

13

and B,[(j+#‘2-(~-+)1’2]

=&‘2(/2-+)1’2[t,--,+1]

Thus IS the formula used to calculate m(r) at each of the eleven standard times. For the 0.3 s time mstant, only two current data are avadable to calculate the semnntegral, so that m IS hkely to be unrehable m thus instance For this reason the slopes and halfwave potential values for 0 3 s were excluded from the calculation of the averages m (10) and (11) REFERENCES 1 Chemical Society (London), Stabdlty Constants of Metal-Ion Complexes, Special Pubhcatlon No 17, 1964 2 A M Bond and G Hefter, Cnt~cal Survey of Stablhty Constants and Related Thermodynanuc Data of Fluonde Complexes m Aqueous Me&a, IUPAC Chenucal Data Senes No 25, Pergamon Press, Oxford, 1980 3 A J Bard and L R Faulkner, Electrochermcal Methods Fundamentals and Appbcations, Wdey, New York, 1980, Sectlon 5 4 5 4 AM Bond and K B Oldham, J Electroanal Chem , 158 (1983) 193 5 A J Bard and L R Faulkner, ref 3, SectIon 5 8 2 6 J R Delmastro and D E Snuth, J Phys Chem , 71 (1%7) 2138 and references therem 7 J K Fnchmann and A Tlmmck, Anal Chem ,39 (1967) 507 8 J Heyrovslj and J Khta, Prmaples of Polarography, Acadenuc Press, New York, 1966 9 V M M Lobo and R Mdls, Electrochm Acta, 27 (1982) %9 10 2 Galus, Pure Appl Chem ,56 (1984) 635 11 J C Myland, K B Oldham and C G Zosb, J Electroanal Chem, 193 (1985) 3 12 F G Cottrell, 2 Phys Chem ,42 (1902) 385 13 K B Oldham and J Spamer, The Fractional Calculus, Acadenuc Press, New York, 1974