Amalgam concentration at the surface of a spherical mercury electrode following a potential step

41

J Electroanal Chem., 218 (1987) 41-52 Elsewer Sequoia S A, Lausanne - Prmted m The Netherlands

AMALGAM MERCURY

CONCENTRATION AT THE SURFACE OF A SPHERICAL ELECTRODE FOLLOWING A POTENTIAL STEP

JEFF E ANDERSON Murray State Vmversrty, JANICE

C MYLAND

Murray, KY (V S A ) and KEITH B OLDHAM

Trent Vmversrty, Peterborough (Canada) (Recewed

1st July 1986, m rewed

form 15th September 1986)

ABSTRACT This article describes a procedure for deternumng the concentration of a metal m mercury at the surface of a spherical electrode by convolvmg the current wth the appropriate convolution function The procedure IS apphcable to a vanety of voltammetnc techmques, mcludmg chronoamperometry, and to an electron-transfer reaction of any degree of reverslbtity To test the procedure, currents that obey an exact equation for a reversible reduction followmg a potential step are convolved the resultmg surface concentrations agree wth theoretical values A test usmg expenmentaldata 1s smularly successful m generatmg concentration values that match theoretical values and that fit the Nemst equation An algonthm to carry out the procedure 1s presented

INTRODUCTION

A compamon article [l] developed a procedure for the spherical convolution of current data from a spherical electrode, such as a hangmg mercury drop, to determme the concentrations of electroactive species at the electrode surface That treatment was restrrcted to species that are present 111the electrolyte solution Many potent&step expenments, however, generate metals that dissolve m the mercury electrode. The mathematics of the amalgam problem IS &stmctly mfferent from that of the former case, and IS the subject of tlvs artde. The dfference between the solution-soluble and mercury-soluble cases mses from two sources Fust, the diffusion mto solution from the electrode surface IS dlvergent, whereas It IS convergent from the electrode surface mto the mercury sphere Second, whereas the &ffuslon of product species mto the electrolyte IS effectively semnnfuute, the fnute size of the mercury drop may restnct the domam of dlffuslon m the amalgam at experrmentally sqmflcant times. 0022-0728/87/$03

50

8 1987 Elsewer Sequoia S A

42

These two effects were taken mto account m the denvation [2] of the general equation cs - Cb =

&+c

fa( Dr FF7)d7

that perrmts the calculation of cS - c b, the surface concentration excursion of metal m the amalgam at tune t, from measured faradsuc currents I recorded durmg the 0 G 7 G t mterval In tis equation V 1s the volume, 4ar3/3, of the mercury sphere, F 1s Faraday’s constant and n 1s the number of electrons reqmred to generate each metal atom. D IS the diffusion coefficient of the metal m mercury and cb 1s the ongmal amalgam concentration, which we henceforth treat as zero. The function fa(z) 1s defined m ref. 2 and will be &scussed m the followmg section A procedure to Implement spherical copvolutlon via eqn (1) has been pubhshed previously [2] but It IS unsmtable for potent&step expenments because of the large lmtlal currents In the present article, we present an alternatlve procedure that 1s apphcable even to infinite lrntlal currents We ~rlll vahdate’ the procedure by applymg It to synthetic data and to expenmental data for the Cd’+ + 2 e- + Cd reactlon. THE CONVOLUTION

FUNCTION

The function fa ~rlth which the current IS convolved m eqn (1) was defined m ref 2 as the inverse Laplace transform of E;(a)

=

1

=--- 1 &2 - 1 u”’ coth( u~‘~) - 1 Xexp{ -2(k

+ 1)~“~)

(2)

where u 1s the “dummy” Laplace vanable correspondmg to z Usmg the ldentlty

and bmonual expansions, eqn. (2) may be rewntten as

-2E (-)kk!,go/,(kZ’l), k-0 x [@(k, I, u) + 2#(k, I+ 1, u) + #(k, I+ 2, u)]

(4

where 6 1s a transformed function defined by 8(k

I ) = exp{ -2(k+l)u”‘} 9 ,Q u1/2( J/2 - 1)’

(5)

43

The expansion m eqn (2) converges rapidly for large u (1.e short times) but an alternative formulation, namely

-2--

G(a) = ; + f

k-l 0

(6)

+uL’

1s more useful for small u (long times) Here YL 1s the kth poslhve root [3] of the equation Y = tan(Y) Equations (4) and (6) define identical functions and both are exact In fact, the function fa does not appear m our convolution procedure Instead, as ~11 be shown m the next section, we reqmre its semuntegral d-‘/*fa(r)/dz-‘I*, mtegral ifa( z) and double mtegral ufa( z) Expressions for the Laplace transforms of the semuntegral and mtegral may be found by &vldmg -4~) by &* and u respectively For short times, usmg eqn. (4), this leads to

=l9(-l,l,

z)-2E(-)kkli k-0

x[fI(k,

2’ ,_o l’(k - 1)’

1+1, z)+@(k,

1+2,

z)]

(7)

after inversion, and lfa( z) = i’fa(S) =8(-l,l,

d{ z)-l-2&-)kkli

2’ 8(k, i+2, ,so I’(k - 1)’

k-0

z)

(8)

where 8(k, 1, z) 1s the Laplace mvert of &k, Z, a). For I = 0 or 1, tbs function 1s

B(k, 0, z) =

&ev( -‘kz+ “‘)

(9)

or B(k, 1, z) = exp(z - 2k - 2) erfc((k + l)z-‘/*

- zl/*)

00)

and other cases are accessible via the recursion formula 6’(k, I+ 1, z) = (2/1)[(z

- k - l)B(k, I, z) + ze(k, I- 1, z)]

For long times, from eqn (a), the correspondmg

s,,(,)

01)

results are

= (j(5)*‘* + & 2 daw(;:z1’2J k-l l/2

+2x-+1+3+.

m

7rl/*z1/* k-l

1 vl’

2Y:z

4y,6z*

02)

44

where daw denotes Dawson’s mtegral [4], and O3 1 - exp( -yEz)

lfa(z)=3z+2z

03) Yk’ Expressions for the double mtegral at short times can be obtamed by direct integration of eqn (8) Thus k-l

nfa(z)=/‘lfa(j) 0

d{=8*(-1,

1, z)-z k

01

2’

8*(k,

-2k~o(-)kk',~o~l(k-~)l

1+ 2, z)

(14)

where 8* denotes the integral of B, examples bemg 8*(k, 0, z) = 2( i)l’*

exp( -(“:

“‘)

- 2(k + 1) erfc( s)

05)

and 8*(k,

1, z) = exp(z - 2k - 2) erfc{(k + l)z-‘I* -2(z/7r)l’*

exp( -(k+

1)*/z)

- zl/*} + (2k + 1) erfc((k + l)z-‘I*)

(16)

For long times, direct mtegratlon of eqn (13) produces O” 1 - yiz - exp( -yiz)

ufa(z)=$-2x

07)

Yk’

k-l

Though they appear very different, eqns. (14) and (17) actually define identical functions, as do eqns. (7) and (8), as well as eqns. (12) and (13). It 1s these six equations that fmd use m the followmg section for numerical evaluation of the convolution. Of course, the mfuWe summations m those equations wdl be smtably truncated FORMULA

TO EVALUATE

THE CONVOLUTION

INTEGRAL

As m the compamon article [l], we assume that current data are avalable as a set of values I~, 12, , I,, . , I,, equally spaced m time Here I, denotes the current at the instant 7 = 0 - 1 + +)A where A 1s the tune interval between pomts and + 1s a fraction or whole number Proceedmg as m ref 1, we amve at

t

Dt-Dr

J ( I fa

d7=g(rA)“*~

r*

0

+$

fa($)+$ifa($)

[I,-a-g(J-l++)-“*I

nfa(b)

i +J~1[rJ_,-a-g(J-~-1++)-1’2] J-1 x

[nfa(

bJ

-

b)

-

2 11fa(bJ)

+

llfa(

bJ

+

b)]

08)

45

where a=[(~+l)1’2~2-$V*,J/[(~+l)1’2-~~*]

,p”“($B+l)l’*(r, - I*)/[(++

g=

(19)

(20)

l)l’* +*]

and b= DA/r*

(21)

In practice, to a&eve accuracy of 1 part m 10’ or better, the semuntegral term IS calculated from

ew( $) erfc( -D1~r1’2) - 2::-lkkfio f,ctt f),

y

sfa(

7)

=

x [B(k, I + 1, Df/r*) 2Dl/*t’/*

3+e’/*r

+ B(k, I + 2, Dt/r*)]

r* r4 1ODt + 700D2t2

+

t < 3r2/2D

r6 10500D3t3

1

t)‘z

3r2

(22)

The short-time version of thrs formula arises directly from eqn (7) by emplncal truncation usmg K = Int{(3Dt/r*) - 0 2) Equations (9)-(11) are used to evaluate the 19terms. The long-term version uses the asymptotic expansion m expression (12) and the known sums CY; * = l/10, CYi4 = l/350 and CY; 6 = l/7875 Snndarly, the da and nfa terms are calculable from ‘exp( $)

erfc( -D1rt1’2)

Dt

ifa =C -2;(-)‘k$ ( r* i k-0

- 1

2’ 8 ( k,1+2,$ ,cO I’(k - I)’

t < 3r2/2D

(23)

t 2 3r2/2D

and exp($)erfc(

-“1~t1’2)-1-3($)1’2-$

t
exp( -yiDt/r*)

t > r2/20D

YZ (24) These formulae are the basis of the algonthm that 1s presented m Appendix A

TEST ON SIMULATED

DATA

The formulae we have derived are not restncted to reversible reactions, or even to potential steps However, we shall Impose these snnphfymg restnctlons 111this section, so that we may apply the Nemst relation&p CSR -=exp(g(E,-E))=K 4 to the electrode reactlon O(soln) + n e- + R(amal)

(26) where Es 1s the standard (more properly the formal or condltional) potential. To venfy the algonthm m Append= A, and consequently the convolution procedure from the previous section, we convolved a set of theoretical current data and compared the result v&h theoretlcal surface concentration data. Both data sets correspond to a potential-step expemnent on a mercury drop electrode at which reaction (26) occurs reversibly The literature [5] contams an expresslon for the current flowmg under these condltlons That expresaon 1s vahd only for sphenclties rmlder than are requved for a strmgent test of the convolution procedure Accordmgly we have used an exact expression [6] vahd for any degree of sphenclty In terms of a farmly of functions H(k, 1, z) of which the first two members are H(k, 0, z) = (TZ-~”

exp{ -(k + l)‘/z}

(27)

and H(k, 1, z) = exp{ y2z - 2(k + 1)~ } erfc{ (k + l)~-‘/~

- YZ’/~}

(28)

the subsequent members bemg calculable from H(k, 1+Lz)=$(vz-k-l)H(k,

I, z)+zH(k,

1-1,

z)]

(29)

this expression is I=

nAFDoc;K

1

r(K+u)

u-

[

+;H(-l,O,y)+&

(Y + 4’ Ky

H( -1, l,?) &,(

Kyu)“i

(KI=0

+y+u)2H(k,

/+2,?)+2(y+u)H(k,

I.,,?

u)~-‘(~KY)’ (k - 1)W ) +H(k,

&?)}I (30)

where u = ( Do/D,)1/2 and y=(K-u’)/(K+u) The second column of Table 1 hsts a selection of the 100 current values generated via eqn (30). Using the algonthm of Append= A all the current data were

47 TABLE

1

Simulated data used to test the convolution algonthm The followmg constants were employed n = 2, V==9195~10-~* m3, A=-02s,N=lOO, +=05, DR=166X10-10 m* SC*, Do=73X10-‘o mz s-l, c& = 0 931 mol rnm3, K - 1024

t/s 05 15 25 35 45 55 65 75 85 95 10 5 115 12 5 13 5 14 5 15 5 16 5 175 18 5 19 5

r/uA

2 286412 1281774 0 961872 0 785987 0 668887 0 582723 0 515387 0 460694 0 415114 0 376446 0 343212 0 314363 0 289120 0 266883 0 247179 0 229629 0 213923 0 199809 0 187073 0 175539

ck /mol mV3 vta algonthm

vlaeqn

0 42029 0 45530 0 48043 0 50148 0 52014 0 53719 0 55303 0 56788 0 58188 0 59511 0 60762 0 61947 0 63070 0 64133 0 65141 0 66096 0 67002 0 67862 0 68679 0 69457

0 42006 0 45514 0 48029 0 50135 0 52002 0 53708 0 55292 0 56777 0 58177 0 59500 0 60751 0 61936 0 63059 0 64122 0 65130 0 66085 0 66992 0 67852 0 68669 069446

(31)

sphencally convolved to produce values of CA, some of which are tabulated m the third column. Data m the fourth column were calculated via the eqn [6]

(31) usmg the same parameter values as were used Hrlth eqn (30) The nearly perfect agreement between columns 3 and 4 vahdates both the algonthm and the theory upon which it 1s based The parameters hsted m the heading of Table 1 were chosen to match the values appropnate to one of the expenments reported m the next se&on

48 TEST ON EXPERIMENTAL DATA

The mstrumentatlon used connsted of an Apple II Plus rmcrocomputer mterfaced to an IBM Model EC/225 Voltammetnc Analyzer The mterface conslsted of a 12-bit analog to &@tal converter (ADC) wth three 16-bit tuners (TecMar Inc. Model TM-AD213) and a 12-bit >al to analog converter (DAC) wth 16 I/O bits (TecMar Inc Model TM-DAlOl) The DAC was mo&fled such that 1 bit corresponded to 1 mV. The DAC was used to control the potential of the workmg electrode m the potential-step expenments It was routmely cahbrated using a separate cahbration program One of the control bits was used to control the drop knocker of the Static Mercury Drop Electrode (SMDE) obtamed from EG & G Prmceton Apphed Research Corporation The ADC was used to measure the current and was cahbrated dady usmg the DAC to apply vanous potentials to the dummy cell of the IBM potentlostat at vanous sensltlvltles The cahbratlon routme generated a cahbratlon file used to convert ADC code to rmcroamps The potentmlstep expenments were performed by a machme language routme called from an Applesoft program All tunmg operations were performed usmg the processor clock and the timers associated wth the ADC Interface board DIstilled delomzed water, Baker Analyzed Reagent grade KCl, and mossy cadmmm were used to prepare a 9 31 x lop4 M Cd(I1) solution m 10 M KC1 Solutions were degassed for 8 mm A senes of five potential-step expenments was performed uang the “small” hangmg mercury drop generated by the SMDE These expenments, performed at 25 “C, conslsted of steppmg the potential from - 0 500 V vs an Ag/AgCl reference electrode (10 M KCl) to - 0 630 V, - 0 650 V, - 0 670 V, -0 690 V, and -0 710 V The current was measured 0 100 s after the step and then once every 0.200 s Each of the potential-step expenments was performed e&t times on four &fferent sensltlvlty ranges. Therefore, a total of 32 runs were averaged for each of the five potential-step expenments In ad&tlon, the same procedure was used to evaluate the background current on a blank (10 M KCl) After eight runs were averaged on a gven sensltlvlty, the data obtamed on the blanks were subtracted and the resulting current/time data was saved on disk A separate Basic (Applesoft) program was used to retneve the data from disk and average the results obtamed on different sensltlvltles In some cases an mdlvldual datum was rejected if the current measured was off scale on the particular sensltlvlty range used Therefore, not all data are averages of 32 values The area of the “small” mercury drop was evaluated by collectmg and weqhmg 100 drops generated at the uutial potent& This procedure was camed out three times and the average mass of a smgle drop was used to evaluate the drop volume of 9 915 X lo-” m3 Literature values of the diffusion coefficients of Cd(amal) and Cd*‘(aq) namely D, = 16 6 X lo-lo mz s-l [7,8] and Do = 7 3 X lo-” m* s-l [9] were usually used However, to evaluate the effect of uncertamty m the diffusion coefflclent of cd m mercury, some data were reconvolved usmg D, = 18 3 x lo-lo m2 s-l A selectlon of the current data from the -0 500 V to -0 670 V step expenment

49 TABLE 2 Expenmental chronoamperometnc data from the -0 500 V to - 0 670 V step expenment and the denved concentration values The final two columns correspond to a DR value Inflated by 10%

f/S

l/WA

cb/mol

05 15 25 35 45 55 65 75 85 95 10 5 115 125 13 5 14 5 15 5 16 5 17 5 185 19 5

2 2606 12395 0 9086 0 7342 06194 0 5600 0 5062 044% 0 4086 0 3913 0 3627 0 3275 0 3097 0 3020 0 2887 0 2723 0 2430 0 2352 0 2215 0 1893

0 435 0458 0 476 0 493 0 508 0 524 0 542 0 557 0 571 0 586 0 602 0 615 0 628 0 642 0 655 0668 0 679 0 689 0700 0 707

m-3

c&/m01 me3

Web

CR/m01 mm3

CA/G

0 393 0444 0 477 0 501 0 523 0 538 0 550 0 563 0 576 0 586 0 594 0604 0 614 0 620 0 627 0 633 0642 0 650 0 655 0 665

111 103 100 0 98 097 0 97 0 99 0 99 099 100 101 102 102 103 105 106 106 106 107 106

0417 0441 0460 0 478 0 494 0 512 0 530 0 547 0 562 0 578 0 594 0608 0 621 0 636 0 650 0 663 0 674 0 684 0 695 0 703

106 099 096 0 95 095 0 95 096 097 097 099 100 101 101 102 104 105 105 105 106 106

1s gven m the second column of Table 2. Values m the third column were calculated with the ad of the algonthm m Appenti A. These ck values are drrectly comparable with those theoretlcal values hsted m Table 1 because ldentlcal parameters have been used. The agreement 1s good Further conflrmauon of the procedure may be obtamed by calculatmg the ci/c& ratio, which should be tune-mdependent for a reversible electrode reaction Since c& 1s the concentration of the solution-soluble Cd2+ Ion, we used the algorithm from ref 1 and the bulk concentration cb, to generate the values of cb hsted m column 4 Column 5 1s the ratio CL/C& wluch should equal the Nernstmn constant K In fact the experrmental ratlo has a mean and standard devlauon equal to 1024 f 0.039 The final two columns m Table 2 reflect the effect of mcreasmg D, by 10% Evidently, the ck values are rather msensltive to thts change, as 1s the ck/c& ratio The procedure summaruzd m Table 2 was repeated for each of the five step expenments, producmg a set of ci/cS, ratios The loganthms of these ratios are plotted versus potentml m Fig 1 The error bars represent the standard devlatlons m each ratio and the stra&t hne 1s an unwe&ted hnear regresslon through the first four pomts The hne has a slope of 76 0 f 0.4 V-’ compared to the Nernstian value 2F/RT = 77 8 V-’ The large error bar on the last point anses from ratios averagmg about 23 at the shortest tunes, passing through a nummum of about 13,

E/V(vs Fig 1 Plot of In(ci/c&)

Ag/AgCl>

vs potential See text

and nsmg to about 30 after 20 s We do not understand fully this behavlour winch is msenstttve to adjustments in such parameters as D, and ck ACKNOWLEDGEMENT

We wish to thank the Natural Sctences and Engmeermg Research Council of Canada for their generous fmanctal support. APPENDIX

A

Here we present the algortthm to perform spherical convolution and generate values of the surface concentration cs (or cs - cb d the electrode is mtially an amalgam) It is assumed that the quanuttes given m Table 3 have been preloaded m the specified umts Set g=(z,

- z*)/[ +-lj2 - (++ 1)-“2]

Set (I = [ z2#-l12 - z,(+ + P2]

g/(z1-

Set b = DA (4~/3V)~‘~ ForldJGNreplacez, SetPandQ=O

byz,-n-g(J-1++)-“2

12)

51 I

Set z=Jb+b If z > 0.05 go to (1) Set R = 4/(4

+ z)

For 4 2 k 3 1 replace R by 1 + (0.5 - k)Rz/k(k For O
+ 0 5)

ReplaceRby1+2(z/r)“2+z-[1+2R(z/a)”Z]exp(z) Go to (2) (1) Set R = - T

- $ + I& l+

- exp( - 20.191~) 203.83

I(2 ) Set h,=2Q-P-R

exp(-39 5z) 8 74

1)

and P=QandQ=R

Replace g by g(ab)‘j2 Set G = A/(289455nVb2) Setz=b(J-l++) [fz>15goto(6) Sets=-ubandK=Int(O8+3z) St

x

=

kz-

l/2 _ z’/2

If I x I > 1 5 go to (3) SetL=l+Int(9JxI)and

w=L/(L+x’)

ForL>/>lreplacewby1+[(05-1)wx*]/[I(I+O5)] Replacewbybexp(z-2k)[l-(2w~/n’/~)] Go to (4) 3) Set L = 2 + Int(32/x)

and w = 1/(2l/*x)

For L z I> 1 replace w by l/( wl + 2l/*x) For l
For Odk
Replace w by b(2/~)“~w ‘4) SetQ=w/bandw=(o+g)w Ifk=Ogoto(5) Set M=(-)‘2band

For O
P=(lrr)

-1’2exp(-k2/z)and

If I # 0 replace M by 2 M(k - I)// Set R=2[(z-k)Q+zP]/(I+l) Replacewbyw+M[gQ+(n+g)R] i SetP=QandQ=R

:5) ReplaceSbyS+w Go to (7)

6) sets=2*~[~]1’2(3+&(l+$-(l+~)))+ab(3z+f) 7) ForOgJdJ-lreplaceSbyS+r,_,hl Output(J+l++)A,GS

exp( - k*/z)

w=O

52 TABLE Data

3

to be preloaded number of electrons volume of electrode sphere (m3) hme interval between data pomts (s) total number of data pomts fraction of A between step and first pomt dlffunon coefflclent (m* s-l) current values

n

V A N + D 11rl2r

TABLE

T’N

4

Algonthm test values Preload n = 2, V=lO-lo, r,-4x10-6, $=3x10-6,

D=10-9, N -4, Q-2X10-6

A=4,

+=30,

~=5X10-~,

J

r/s

CR/m01 mm3

1 2 3 4

120 124 128 132

415 415 416 416

1251 7615 1963 4263

The algonthm ut&zes eqns (18) through (24) m a striughtforward fashion, wth some ad&tlonal roundmg where this does not degrade the accuracy below one part m lo5 The error function complement 1s evaluated m the algorithm via a pubhshed routme [lo] By an mstructlon such as “For 0 d J < N - 1” we mean that the bracketed routme 1s repeated with J successively set equal to O,l, , N - 1 In many computer languages this may be implemented by a FOR-NEXT or DO loop In ad&bon to provldmg storage for the N + 6 preloaded constants specified m Table 3, space must be allocated for the vanables g, a, b, J, P, Q, z, J, R, k, G, S, K, X, L, W, 1, A4 and an array h,, h,, ., h,_,. The quantltles output are the samphng times (m s) and the correspondmg surface concentration excursion (m mol me3 or mM). The test values hsted m Table 4 are designed to exercise the algor&m maximally they do not represent reahstic values Duphcatlon of the output values hsted m Table 4 confirms that the algonthm 1s functlomng correctly REFERENCES 1 J C Myland, K B Oldham and C G Zoslu, J Electroanal Chem, 218 (1987) 29 2 J C Myland, K B Oldham and C G Zoslu, J Electroanal Chem, 193 (1985) 3 3 M Abramov&z and IA Stegun, Handbook of Mathematical Functions, NatIonal Standards, Washmgton, 1964, Table 4 19 4 Ref 3, Sect 7 1 5 A Bond and K B Oldham, J Electroanal Chem , 158 (1983) 193 6 J C Myland and K B Oldham, unpubhshed denvatlon, 1986 7 V M M Lobo and R Mlls, Electroc~m Acta, 27 (1982) 969 8 Z Galus, Pure Appl Chem , 56 (1984) 635 9 J K Fnchman and A Tlmmck, Anal Chem , 39 (1967) 507 10 K B Oldham, J Electroanal Chem , 136 (1982) 175

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