Application of double potential-step chronocoulometry to the study of reactant adsorption. theory

ELECTROA~~YTICA

236

APPLICATION TO

THE

JOSEPHI

N.

Gates

and

(U.S_d

_)

_Xodh

OF

STUDY

_Artre&xn

(Received

DOUBLE

OF

ROBERT

Labordories

~5th~

A-

CHRONOCOULOBIETRY

in revised

form

THEORY

.tzox

OSTERYOUXG *, Cdiformia

of Cherrrislry

1966:

IX\‘TEFU?_4CI_4Im ELECTROCHEMISTRY

ADSORPTION.

Science Cenfeu. Thousand

ArGabiom

February

AND

POTENTIAL-STEP

REACTANT

CHRISTIE,

CreZZin

CHENISTRY

Iwsditute

FRED

Oaks, California

May

20th.

C.

_XNSONt

of Technology.

(USA

Pasadetza.

Californza

)

1966)

IXTRODUCTIOX

Previous applications of the chronocoulometric method to the study of adsorption of electroact;ve s~bstaces at electrodsl-a have suffered somewhat from the lack of a rigorous double-layer charging correction in the presence of adsorption. A double potential-step chronocoulometric method has been suggested4 which offers significant promise as a method for the study of adsorption, since an internal measure of the true double-layer charging correction can be obtained with this techniqueThis paper presents the theoretical treatment of double potential-step chronocoulometry as applied to the study of reactant adsorption. THEORY

AND

DISCUSS103

-4 double potential-step Consider the redos couple O++zewith

standard

potential

chronocoulometric

experiment

is performed

as follows

-_R

(=)

Potential,

is adjusted

:

EO_ In

to a value

a solution

EI sufficiently

containing anodic

0,

but

no

R,

the electrode

of EO that no current

fLows- At

t = o, the po&ntial is stepped to a value Ef sufficiently cathodic of Eo that the concentr&ion of 0 at the electrode surface is immediately reduced to zero andreaction (I) proceeds to the right at a rate limited by the diffusion of 0 to the electrode. After a time, t, the potential is steppedback to Et and the concentration of R at the electrode surface is immediately reduced to zero_ The resulting current is limited by the rate pf diffusion of R- back to the electrode. The measured parametkr is the total-charge passed as a fr&tion of time. Figure I shows the qualitative Q-t behavior observed with and tcithout adsorption of O_ ~Im the foIlowing discussion, we assume that any adsorbed sub&ance reacts immediately lwhen the potential is stepped to_ any value & which the corresponding unadsorbed~~ubstance

reacts-

%ud

that

double-layer

charging

also

takes

place

in-

~SORPTIOW

33Y DOUBLE

POTENTIAL-STEP

CHRONOCOULOMETRY

237

stantaneously_ We need, therefore, to calculate only the charge passed from the reactionofmaterialarrivingattheelectrodebydiffusion,and thento addinterms corresponding to reaction of adsorbed material and double-layer charging,tdobtainthe total charge-timeresponsetothe double potential-step function-

OS-

0.4

Y

‘;

c

m

-

03-

-a= L

cc?-

0.1 -

0

a1

02.

0.3

0.4 e

-o.l-

Fig.

I_ Qualitative

Q-f

behavior

without

(a), and with

0.5

0.6

0

s

(b). adsorption

(see eqns.

8, g. 13, 15)_

of I - (Z/-Z) &n-l m ~5. @lvr- The points are calculated:thelineistheleast-squares Line. ul(O/l/T) + a~, to which the pomts have been fitted_

Fig. z- Plot

straight

Inlineardiffusion probfemswithNeumann potentiometry), the integral equations I

Co(o,t)=*Co

-

____ d+cDo

t

r(e)

I j/mo

de

boundary

conditions (e-g.,chrono-

(24

havebeenverywidely applied(see REINMUTR~ foradiscussion of their application). In chronocoulometry, the electrode reaction is diffusion controlled and the boundary conditions zre writlxeninterms of the surface concentrations,Tather than the surface fluxes. The problem therefore has Dirichlet boundary conditions. The analogs of eqns. (za) and (zb) forthistypeofboundaryconditionsmaybeobtained asfollows:Consideringfirsteqn. (za),we changevariables and ttoqand8,respectively, divide through by I/t--eandinte~ate from o to t,

Since

&qn. (2a)-becomes

J_ H. CHRISTIE,

23s

Ii.

A.

OSTERYOITNG,

F.

C. ANSOK

or

Operatingon

Since

(zb)in asimilarmanner$ves

eqn_

we

will consider

only

cases in which

*CR,=O

and

C~.(o,t)=o,

t>t,

eqn. (3E) becomes

(&) (A derivation eIsewheree.)

of these equations

In this C,(o,

case,

t> =

it is well

Ifg

by means

known7

*CO

The charge passed by reaction

that

of the Laplace

the surface concentration

of R

t
is given

transformation

is

(5) material

is then

given

by

(6) Q(t’t) Equation

=

(7)

(6) is the integral of the fannli ar Cottrell equation for the diffusion-limited in the charge, Qdl, consumed in double-layer chargiug, gives

cm-rent. Add-ng

(9) Thereisno double-layerchargingte~~iT1 eqn. (g) becausethe electrodepotentialis back at its ini& potential for t > t and no nei -double-layer charging has occurred. The net charge, Q==Q(z)-Q(t>t), passed for t>t is given by

(IO)

ADSORPTION

BY

DOUBLE

POTENTLXL-STEP

As has been pointed arid wiIl pass through Also

plots

identical

out by AXSOP,

the origin

a plot of Q(t P-G) ZJS.l/t-if-

if there is no adsorption

of Q(t < t) ZJ.S_ dt and Qr vs. 0 = ($z slopes

(27zF1/Do

Equalslopes

"c,/+)

of either

t will be linear

reactant

+ 1/z - jt) wih be linear

andidenticalintercepts

andintercepts

239

CHROKOCOGLOMETRY

or product. and will have

(Qdl)_

are also obtainedintheratherspecialcasethatthe

amounts of adsorbed 0 and R are exactly the same. Let us assume that 0 is initially adsorbed to the extent of r moles/cm’ and that the adsorbed 0 reacts at t = o with the consumption of nFT C/cm’ to form r moles/cm’ of adsorbed R. Let us also assume that no further adsorption or desorption of R occurs in the time t I t and that the adsorbed R is reconverted to adsorbed 0, with the consumption of nFT C/cm’, when the potential of the electrode is returned to its original value. In this case, the intercepts of the Q(t< t) ZJUS. $6 and Qt. XJLIS. 0 plots will be identical, but will be equal to

Qa+qzFr_

If the adsorption is extensive, these Values for the intercepts will be larger than would be expected for double-layer charging alone, and in any case will be different from the values for the amount of charge needed for double-layer charging as given by the difference m the electronic charge on the electrode at Ei and Em (see ref _ 4 for a convenient method for measurement of electrorric charge on the electrode). Under these conditions, a plot of Q(t >t) ZX_ I/t-j= will still pass through the origin and would give no hint that adsorption was present in the system_ For the case in which

R is less extensively

Adsorrption

of 0, but If the reactant step chronocoulometric

Consider

adsorbed

than

is 0,

see below.

not of R (0) is adsorbed. but the product (R) is not, the double potentialprocedure can be used to determine the amount of adsorption

the case in which

0 is initially

adsorbed

to the extent

of TO moles/cm2

and the adsorbed 0 reacts at t = o with the consumption of nFTo C/cm2. The reaction of the adsorbed material can be considered as an instantaneous source of R located at x = o, t = o, which gives rise to a concentration profile of R in the solution (o< t c t) given bys

Bythesuperpositiontheorem,thetotalconcentrationof Ratanypomt inthesolution resultingfromtheinstantaneoussource, is given by the sum of the concentration, CR=, and the concentration7, Cnr, resulting from the continuous source of 0 diffusing to the electrode

CR@,

t) = CR=@,

t) +cR=(Z,

*CO erfc

tions

t)

=

($&=)

+

exp

&&

( --Z?/4Dnt)

tt-~

(12)

The charge passed during the forward process (t 5 t) will be the sum of contribufrom diffusing material, adsorbed material, and double-layer charging (13) For

t >t;

eqn.

(4b)

is applicable

to CR’ and Cnn J;

independently

Electvoanal.

Chem..

13

(1967)

a36244

240

J_

IL

CHRISTIE,

R_

iLOSTERYOUNG,

F_

C.

AXSON

(I@) Combining

(re)

and

(Iqb) gives

Thisequationdiffersfromeqn.(g)onlybytheterm, 4zFT0[(2jz) sin-im], arisingfrom there-oxidation of Rformedfrominitiallyadsorbed 0. Asin eqn. (g) t-hereisnoterm corresponding to double-layer charging since it is assumedthatthe electrodehas been restoredtoit-sinitial conditions fort>t and thatnonet double-layer chargehas been sassmnption implies thatboththeamount required_ It shouldbe recognized &at&i ofadsorbed 0, To, andthesurface concentration,Co (0,t>t),retumimmediatelyto their initial values when the potential is returned to Ei_ In practice, no error is incurredsolongasthis conditionisachievedbeforethefirstdatapointaftert_ In order to provide the material to be r-adsorbed onthesurface,enoughtimemustelapsfor the sum of the anodic faradaic charge passed andtheamountofOthatdiffusesback to the electrode to reach cuFro_ Combining (13) and (15)gives

Thisequationdiffersfromeqn.(~o),thecasein~vhichthereisnoadsorption,o~y by the term, n_FTo [I- (2/n) sin-1 jr/f], arising fromthe re-oxidationofmaterial formed frominitially adsorbedO.Ifwetemporarilyignorethisterm,weseethatthelntercept of theQr US. 0 wouldbe simply Qdl, the true double-char,@ngterminthe presence of adsorption_ We will now show that the term nFTo [I-(z(z)sir-1)/t/t] has littleeffecton the linearity of the Q=---@ plot and a calculable effect on the slope and intercepts. A quantitative measure of the amount of adsorption is therefore possible_ Wemdkeeqn.(I6)dimensionlessbydividingbytheCottrell(diffusing)charge, Qc, passed during the forward process 07) Wealsoomi-tthedouble-layerchargingtermsinceitisassumedtobeequaltothatfor the forward process and its inclusion is nnnecessar$ for our present discussion- We thenhave

the asterisk(*)indicatesthatQaihasbeensubtracted fromthe exper-hent~y measurable quantity. The effect ofthesecond term on the slope, S; of a v&c ZJS-i(t/t)-r +I-_ ~&=Q/fiplotmaybe obtained by drfxerentiationr

where

J:EZedroanaZ.Chm,

13 (1967) 236-244

ADSORF’TION

BY

DOUBLE

POTENTIAL-SIIEP

GARONOCOWLOMETIIY

24’

Fort== x.x 7;, S = I -i- 0.236 nli;l;r/Qc; for t = 2 t, S = I A- x_oS -rzF.Fo/Qt~_Since FLFFCJQC is rarely as la~~geas unity, the plot is very nearly linear (fcx t=zrtozt,thetypica2 v experimental range). Figure 2 showsaplotofx(2/z) sin-1mus_Q/lfz_ Thepointsarethecalculated val.ues of I- (2fn> sin-l [rii; the straight line has been fitted to thepointsbyleastsquares adjustment (all points weighted equally)..The values c?fO/jk chosen here co~~pondtothoseusedinoure~perimentalstudyof~eadsorption afCd(II) in the presence of thiocyanatee. The values of the slope and intercept of the least-squares tchosen. ForFig.z,thevalues a~just~d~n~d~pe~d only s~~ht~youthev~~es~f~~~ aftheinterceptandtheslope are a~= -0 0688 anda~=o.g~o,respectively.F$ure3 shows plots of "(?& c 3s. O/{-E calculated for several assumed values of nF..fTo/Qc. III all cases, deviations of the points tiomthe lea&-squares line are quite sma.K

1 01

0

I

03

I cc3

i

0.4

1 a5

8

3_ Plots

Fig

4.

straight

square5

of *Qr/Qc Zincs are those

0

a.1

‘

a.2

r a.3

r 0.4

I

1

t

05

OS

a7

VS. O/flz. calculated from eqn. (rG) for assumed values of ?zFra/Qcgiven by eqn. (20). The value of ~FPo]$& is givea on each line.

Plot of (-r/z)sin-1 m sfzraight line, 6% (j/G;

Q

Qc

=

-ZZF.Fo

z -I-

The

vs. p/G - ~!*-w~i I_ The pointi ax-e calculated ; the line is the least-- r/f/z - I} +- 21 ,,. to which the points have been fitted.

Accepting the fact that {I- (z(z) sin-1 m) can be well approximated linear function of the form a~(O/1/2)-+-ao,we can rewrite eqn. (x8),

3

0

dpX$=i

-vE

Fig.

t

,

06

Qc

\v&re al ad ~0 are theslope andtbeixx*ercept _ Similarly, eqn. (~6) can betitten

of the 1&e shown

in Fig_ 2.

by a

J.

“42

The straight

line fitted

[r t arnEro/Qc] straight line even and intercept “Q From the calculated

-4 check

S,

-

Sr

The

value

data

CHRISTIE,

will then have

R.

A

OSTERYOUNG,

a slope Sr =

[znFm*Co/fi]

of nFro

may

be obtained

from

the relationship

TZFTO

I+-1

(23)

____

Qc

of Qc needed

here is obtained

from the relationship

[see eqns.

Qc=Q(t=t)-0Q The ratio of the slopes calculated observed ratio of the slopes The double-layer charging

Qt.11= The

F_ C_ ANSOY

and an intercept oQr = Qd1-b a0 ~FI’o. The Q(d < r) vs_ jf plot is a &me in the presence of adsorptron of 0 and has a slopeSr = z~zE;‘~Do*c~/~~ = Qdl+nFrO_ intercepts of these two plots, the amount of adsorbed 0 can be

on this value

=

to the Q=-o

H.

CQr-aoOQ I--a0

(13) and (r7)], (241

in this term

way

is given

should

agree

with

the

experimentally

by

(35)

value

of Q dl determined from eqn. (25) should be equal to the difference in eleccharge on the electrode at El and Er_ rt can also be shown that (z/ 7~) sin-r jq may be well approLulmated by a linear function of the form, br ifi-jm] + bo (see Fig. 4). and eqn. (rg) can therefore be written tronic

(26) The values of 60 and br calculated for Fig. 4 [using the same values of t/r as in Figs. 2 and 3) are o-0937 and o-970, respectively. The value of 7cFJ10 could be calculated directly from the intercept of the Q(d >z) wx it - G plot, but since bo is smalls it is felt that the procedure described above is more accurate, as well as more revealing, since it allows several checks for internal consistency_ ~. If both cc)and R are adsorbable, but R to a lesser extent than 0, we can conceive of the following situation_ The reactant 0, initially adsorbed to the extent of TO moles,‘cm2, reacts at t = o to form F~L moles/cm” of adsorbed R and (~CI -TR) molesjcm3 of I;: free to diffuse away from the electrode. No further adsqrption or desorption of R occurs (t s r} and at t the adsorbed R is reconverted to adsorbed 0 --ivitb the corismnption~of nFrR C/ems.Under these conditions, the effective source strength becomes (X’o -X’&) and a -term must be added to eqn.m (16) to account for the reaction of a&orb&d--R at tWe fhen h&ve

ADSORPTION

BY

DOUBLE

POTENTIAL-STEP

243

CEiRONOCOULOMETRY

(13) is unaffected. thesameapproximationfor_Tr-(~JJz)sin-~ Jt/t]asabove,thedifferencein the intercepts of the Q(tc~) ZJ~.if and the Qr US_ 0 will give Equation

With

“Q - "Qr

=

(2s)

7ZF(rO--R)

I-&

and

the ratio of the slopes

S, - = I+al SK We

g

will be

(ro-rFL)

-

(29)

see that in this case only the difference TO-TR will be measured. Equation (25) will now give an apparent double-layer charging term

OQ=-aoOQ I--a0 which nFr&.

= Qai-TLFTR

will belargerthan

(30)

the true

double-layer charging term bythefaradaicterm,

Ifthetruedouble-layercharging contribution canbe obtained(e.g.,fromdropextrusion e_uperimentsa), nFT= may be obtained from eqn_ (30) and this value of nFrR maybeusedto calculatethevalueof nFT’o fromtheobserveddifferenceinthe intercepts of the Q(~KT) ZIS_)ltand the Q= nus_0 plots [eqn_ (a@]_ If Ris adsorbable, but Ois not, orif Ris more e_tiensivelyadsorbed than 0. amodelfortherateatwhich Risadsorbedmustbeintroduced.Thetreatmentofsuch casesismorecomplicatedth~thetreatmentpresentedhereandwiUbe discussedin a subsequent communication. CONCLUSIONS

The double potential-step chronocoulometric method shows great pro-mise for the study of reactant adsorption, particularly when used in conjunction with drope--ion e-uperiments. In this treatment we have made only the very reasonable assumptions that the-adsorbed material reacts immediately when the potential is changedtoavalueatwhichreactioncan occurandthatthereisnocouplingbetween thereactionofthe adsorbedandthe diffusiugmaterial.Theresults obtainedwiththis technique shouldbeless ambiguousthanthose obttiedwith othermethodsusedfor the study of adsorption, e-g:., chronopotentiometry, faradziciIupedance,andsingle potential-step chronocoulometzy as previously applied.

We

wishtothank

JANET

G. JONES

for sev&alhel@ful

cGcussions

during the

course of this work. Th.istiorkwassuppo~edinpa@b~thg~.

S.A.rmyRese-archOffice(Durham).

244

J.

H.

CXRISTIE,

R.

3.

OSTERYOUKG,

F.

C.

ANSOK

The~~eoryofane~vtechniqueforthestudycfreactantadsorptionisdeveloped. It is shown

that

a rigorous

double-layer

correction

can

be made

and

that

a quantita-

ofreactantadsorptionispossible. By combinationof determin 'ation ofthe amount thistechniquewith othermethods,such as the drop-exkusionmethod,forthedeterminationofdouble-layercharge,itispossible,insomecases,to determinetheamounts tive

of both reactant and product adsorbedREFERENCES I

z 3 4 5 6 7 3

F_ C. -4NSOx. ANal. CJzem., 36 (x96+) g3z_ J_ H_ CEIRISTIE, G. LATTER _ND R_ -4. OSTERYOCNG, J. EZecfvoanaZ. C7iem.. 7 (1964) 60_ R. A_ OSTERYOUNG AND F. C_ AKSOS. Anal. C7zem.. 36 (1964) 975. F_ C. AKSOK, Axal. Chem_. 38 (1966) +p_ TV_ H. REIS-MUTH, AmaL. Ch.zw_, 34 (1962) 1446. J_ H_ CHRISTIE. J_ EZe?ctroanaZ. Chem., 13 (1967) 79. P_ DE--4~. _Vew 1+zshurnenlaZ _Methoh+ in Elecfvochemisfvy. Interscience. New York. 1954. p- 53. Cunduc~unr Heat zqzSolfds, Clarendon Press, Oxford, London, H. s. CARSIAW AND J. c. J AEGER.

2nd ed.,

1959.

9 F_ C_ Axsox. J.

EEedroanaL

of

p_ 259.

J- H_

Chem

CHRISTIE

, 13 (1967)

AXD

R.

236234

_4_

OSTERYOUNG,

J-

Elecfroanal

Chem.,

13 (1967)

343.