Study of a simple redox system with adsorption of both reactant and product at the DME when a time dependent potential is applied. Pulse polarography

J. Efecrroana/ Chem, 183 (l?85) Elsevier Sequoia S A.. Lausanne

73

73-89

- Pnnted

m The

Netherlanas

STUDY OF A SIMPLE REDOX SYSTEM WITH ADSORPTION OF BOTH REACTANT AND PRODUn AT THE DME WHEN A TIME DEPENDENT POTENTIAL IS APPLIED. PULSE POLAROGRH’HY PART IV. LANGMIJIRIAN POLAROGRAPHY

ADSORPTION

FRANCESC

FAUST0

MAS, JAUME

PUY,

SAN2

Deparramenr de Quimrca Fiwca, Fact&at de Quimlca, 08028 - Barcelona (Spat) (Recewed

15th August 1984;

IN NORMAL

PULSE

and JOSE

VIRGIL1

Unwersrrat

de Barcelona, Au Diagonal, 647,

in revised form 21st September

1981)

ABSTRACT The cakulauons camed out III Part II (F. Mas. J. Puy. F Sara ad J. Vlr@, J Elecuoanal. Chem , 183 (1985) 41) to obtwn the current-tune curves in pulse polarography when both reactant and product follow a Langrnuir ~sotherrn are apphed to the case of normal pulse polarography (NPP) The morphology of Ihe exact polarographic curves IS analyzed with respect to the adsorpuon parameters and the srccuracy of the approximate curves is dlscussed A special comment IS made about the lirmtmg current and its dependence on the adsorption parameters Fmally. a method of obtamng the adsorptron parameters lrom the approximate e-quatlons is outhned.

INTRODUCTION

In Part I [I ] a rigorous treatment for stuclying reversible diffusion-controlled reactions with adsorption of the reactant and/or the product of the reaction has been developed for the case when a square wave voltage is applied to the Dh4E. In Part II [2] the exact I = i(r) curves for pulse polarography have been numerically calculated assuming that the reaction components follow a Langmuir isotherm. The only attempt to deduce an approximate analytical equation for NPP currents corresponds to that of Van Leeuwen et al. [3,4] but it is restricted to the limiting current assuming other restrictive conditions. They assume either that the reaction components follow a linear isotherm thus considering the low concentration limit or that full coverage is attained before pulse application under limiting mass-transport conditions thus considering the higher concentration limit. The same treatment was later extended to the stationary drop mercury electrode (SDME) [5] which seems to the authors to be more suitabie when comparing with the experimental results. This was done in a subsequent paper [6] for methylene-blue and Pb2+ in acid medium showing that some features predicted by the theoretical equations are accomplished_ On the other hand, numerical results have been presented by Flanagan et al. [7] for NPP using the digital simulation technique [8] with several adsorption isotherms. 0022-0728/85/903.30

0 1985 Elsevier Sequoia S A.

74

The present paper has been devoted to establishing the theoretical poiarograms obtained as a response in the NPP techruque following the treatment developed in Parts I and 11 m either a numerical or an approximate analytical way. THE PROBLEM General condrtrons

In normal pulse polarography, almost at the end of the drop hfe (at time fO) the potential is pulsed from E, (the base potential wkch is constant durmg the run) to E, whrch is drfferent for each drop. At a delay trme after pulse application which coincides with the drop time (r,), the current lFIpp (E,I,,~,) is sampled obtaming frnaily a current-potential curve (NP poiarogram). The system parameters E,. t, and I, are experimentally controlled. The eiectrochemrcal reaction consrders a simple redox process R + n e- + P on I the mercury electrode wtth the following assumptions: (1) The charge transfer rate is much faster than the ‘diffuston rate of the components towards the DME. Then, the Nemst reverstbihty reiatton holds. (2) Both components may be adsorbed on the electrode surface. Such a process is mstantaneousiy governed by an adsorption Isotherm. In particular, and foliowmg the calcuiatrons in ref 2, the Langmuu Isotherm IS chosen for both reactant and product of the reaction. (3) The diffuston towards the electrode will be the hmrting rate step of the whole process. (4) The base potential _!I, IS chosen so that no faradais reaction occurs. So respecE, - 00 and E, - - m (S, -+ 0) describe a reduction and an oxidation, tively. The extenston to any other potential E, so that faradal,: reaction can occur is trtviai withtn the treatment that follows. The mathematrcal formufa~~oil

We will cdli rlNpp(6) the dunenstoniess resultant current in pulse poiarography at

tlNPPW

= %+Od)

2

response functton in NPP, gtven ttme t = I~. Thus,

which is the

(1)

where the base potentrat is 8 = 0 and the pu!se amplrtude IS included as a parameter. Under the assumptions considered above and choosing the Iikovic planar eiectrode model. the general relattonship between the current and the time IS given by 111:

(2) or

75

where T~~(X~;~)

is the so-called

direct current

term, given by

(4) The followmg

notation

S(i)

= (VP(~))~GCz)

p(r)

= exp((nF/RT)[E(r)

f,(t)

= C,(O,t)/C*

QJ(I) = [I+

has been used (see ref. 9) x e 7’13 G ( t/t,, )“3 - &I}

W)lf.(~)

t,, = (77r/12)(

r,‘,/c*‘D,)

d,(r)

= r,(t)/rn,

V(X)

= 0)/%i(C,,)

I~( I,,) = 4m2nF,1(7/3a)c*D;“t;(6 Here, the functions 0, (A> = ~,c,(W

~7114

=

_

J/,

d,(X)

+x-V7

and q(X)

are defmed

by the system of equations (5)

/?dWd~b

(BR(I~Jh);EK

>

(6)

c~(O.~)/C~(O,~)=P(C)=~~~{~(~~~/RT)[E(~)--~]}

(7)

where eqn. (6) represents a general type of adsorption isotherm. As we have pointed out previously. this system of equations does not have an explicit analytical solution for the isotherms which are most frequently used, but a numerrcal solution ,an always be worked out. A numerical calculation IS given Lselow for the Langmuir isotherm because in this Isotherm only a small number of parameters have to be specified and the qualitative conclusions so obtained can be extended to other experimental isotherms [lo]. Nevertheless, it IS also desirable to find approvlmate analytical expressions In order to evaluate its dependence on the system parameters_ Tbereforc. the expressrons based on the method of successive approximatrons obtarned for the current-time curves [l] will be used in the case of NPP, then a comparison will be made with the numerical results. Their accuracy has already been drscussed for a Langmurr isotherm m Part II [2]. From the approximate equation for the current obtained in section A.2 of ref. 1 and accordkg to eqn. (1). the response in NPP can be written:

l

The notation 0,(X:6 = 0) ISequivalent to 8,(X) in ref. 1 stressmg

the case wthout

reaction

76

where q,,(Xd; S) is given by eqn. (4). Equation (8) has been deduced by extending the potentiostatic solution, variable in the interval (0, A;), to the complete interval (0. A,) and then making the approximation: ~~((~‘(~)/~~)dp=(P.(~)~~(l/~~)dp which is independent of the isotherm considered. establishes a relationship between the dimensionless and the dimensionless volume concentration around approximated according to section B.2 of ref. 1. e,(h)=A3”4[1

The integral, eqn. (5), which superficial concentration B,(X) the electrode q(X) can also be

-q(A)]

e,(X)=fI,(h;S=o)

X(X,

q(h)=QI(h;S=O)

(9)

and IfX>X, B,(X)-A”“4[1 e,(x;S=o)=A”‘ql

-‘p(X;)]

-A-‘q-X&(X)-qJ(h,)] -rp(h;S=O)]

whch represents an extension of the Levlch approximation [ll] to pulse polarography. By means of this approximation, the most accurate results have been obtained for the I = I( r j curves [2] when a Langmuir isotherm is consIdered. It constitutes a good approximation for a large range of the adsorption parameters, and it is especially suitable in the case of weak adsorptlon. Alternatively, the response m NPP can be approximated according to section B.3 m ref. 1 by

which IS based on the assumption of maximum flux introduced by Koryta [12]. It is an extension of the Koryta-Lavlron approximation [9] to pulse polarography and becomes suitable in the case of strong adsorption. The dimensionless concentration functions 6?(A) and q(X) are then defined by

(11) (12) both Independent of the isotherm considered. increases as the adsorption coefficients increase.

The

accuracy

of these

functions

77 NUMERICAL

RESULTS

ONES LANGMUIRIAN

FOR

qNpp(h)-AND

ITS COhlPARISON

As we mentioned above our calculattons isotherm because this case has already been was also used with the simulation technique The Langmuir adsorption isotherm takes

can be associated

6(~)/&(~)

THE

APPROXIMATE

have been carried out for a Langmmr studied with other techniques [lo] and [7].

the form

l=R,P

Kc,(O,r)=Q0)/[1-~,(0] whxh

WITH

ADSORPTION

to the reversibility

= K ,CP(OJ)/&CR(OJ)

From

eqn. (13) we can also obtain

UX)

= +G?G)/[1+

relatron,

this leading

= (K,jK,)/p(t)

= ,I:f)

to

f?,(t) explicitly

++f0)1

where W(x)

= PR [l + K(A)I/[l

PR = KRc*

P = ( K,/K,

+ a(x)]

= P, 11 + Ps(x)l/[l

+ s(x)]

>im

Then, eqn. (15) together wtth the integral eqn. (5) allows us to calculate the resultant current in NPP by means of etther eqn. (2) or cqn. (3). For this kind of isotherm the direct current term becomes

The numerical solution of the system of eqns. (5) and (15) has been worked out by Huber’s method [13], fohowmg the procedure indicated in Appendrx A of ref. 10. Frgures l-8 show the results so obtained for several values of the different parameters. Some general considerations can be pointed out. In the processes with adsorption, the limitin, a value obtamed for the current when the electrode potential takes values more negatives than E,(S + co) (redl_rction process) is smaller than the limiting current when there is no adsorptron. Accordmg to both the ideas of Barker and Bolzan [14] and the results of Flanagan et al. [7] this depression is proportional to the increase of the adsorption parameter bR (Fig. 1). Moreover, we can observe that the decrease depends only on the parameter flR being independent of the ratio of adsorption coefficients j3 (Ags. 2 and 3). The adsorption modifies the shape of the wave (Fig. 1). Thus, when K, > K, (Ftgs. 1 and 2) the wave shifts towards positive potentials as rBR increases, a& for strong adso_rPtion a prewave develops, analogous to what happens tn chtsstcal polarography. When K, c K, (Figs. 2 and 3) the wave shifts towards negative

-v Fig 1 Curves of 7 NPP = r,,(r,)/r,!r,,) vs 1’ = elemenlal rrducrlon process has been considered ~,,,=10~6molm~~and~~r=IO~6~~s~‘:~*=00957molm~~. A Pulses towards negative potenwds have been there 1s no reaction (In Ihe calculations 6 = 0.) pariicuiar case p = ( Xi,,/h’,)dm = 100 and Approxlmnre results oblalned from eqns (8) and (10) and (12) (strong adsorption) and (- - -) case

Fig

2

As

Fig

1 *lth

PR = lOand~=100(0).

(nF/‘/RT) (E - E,) (NP polarograms) In all Clgures an with n = 2. T= 298 16 K. D,= D, =lO-” m’ s-‘. r,=2001 sandr,(r,)=1.51613xlO-’ for which applied at ~~ = ~~/r,, = 1 3 from a poIenlla1 CurrenI \ampllng Itme Q = fd/f,, = 1 446337 For thls 10 (X). l(*). 0.1 (+) (pR = c*KR = IO3 (0). ) -) approximate results obtnlned from eqns (9). ( wIthout adsorptIon

IO(x).01

(*)OOl

(+)

79

potentials and a current peak devebps which is similar to the ones ohtalned In the DPP response function (Part 111 [I 51). This behavtour has already been reported

[3,6.7.14] both in simulated polaro$rams and in some expertmental systems. The peak appears only when the reactant IS adsorbed more strongly than the product (p G 0.1) and IS enhanced when the pulse duration I, decreases (compare Fogs. 2 and 3). As can be seen In Fig. 4. when the product is adsorbed more strongly than the reactant, even for small samphng tunes (1% 0 f the ratio I,_,/z~), thts behaviour does not appear and only a wave shift towards positrve potentials takes place. The shape of the polarograms in Fig. 4 is due to the small value of both the sampling ttme and the ratio tp/fd used. As r,, mcreases. matntaining the ratto cr/fJ. two-wave polarograms develop (Frg. 5) and the same effect is observed when the ratro t,/t, is increased (Ftg. 6). The Increase of etther the samphng time or the pulse duratron can then constitute a qualitatrve tesr for recognizing adsorptron in the Bbvholeprocess. In the case of strong adsorptton. when I, < I,,,. the ltmtting current comcides In ali polarograms (Fig. 7) and equals the value for PR inftnite. Identical to the dc !tmtting current. Thts can be considered as the minimum value for an adsorption-rnfluenced NPP limtting current. A one-wave shape is observed for every value of the parameter p, the waves only dtffertng in the shift towards either posttive or negative potentrals (dependtng on p > 1 or fl -C 1, respectively). No peaks appear when /3 c 1. contrary to the case when adsorption IS weak (Ftg. 8). Nevertheless, even in the case of weak adsorptton if the reactant concentration is low enough. the maxtmum flux hypothesis holds and the current peak does not appear. From an experrmcntal point of view, it is also interesting to study the evolutton of

NP polarograms when- the bulk concentration of the reactant is changed, because this variable is easy to control. In our treatment, an increase of the concentration implies both an increase of the adsorption parameter )BR ( PR = KRc*) and a decrease then A,, Ad). This situation can be correlated with of t,,, the time scale (increasing

% 4.0 -

,’

-V 4 As Fig

= 1000

1 wrth y, = 1.187. Q = 1 2995. fiR = 0 and BP = c*K,

( *).

100 (x).

10 (0)

04

0.0

-8

-6

FIN 5 As FIN 1 with 70=18311d~d=199692(~)

-4

-2

-v

0

2

/3 = 100. fin = 10: -rO= 1.3 and TV= 1.54572 (0):

4

6

q, = 1.6 and TV= 1.77712

(x);

81

that observed in ftgs. 1, 2 and 5. As both flR and ho (h, > 1) increase, the one-wave polarogram can evolve into a two-wave polarogram. while the limiting current is affected in an opposite sense: while an increase in jlR leads to a depression of the limiting current, an increase in A, leads to an enhancement of it, this enhancement

--v Fig 6.As FIN.1 wth B =lOO, BR =lO. q,= 1.3and rd=1.36447 (0):

FIN 7. AS Rg 1 wth T~=O.D. ~,=0542663; p=lOO /?=0.1and~R=106(+),/3=0.01and/3,=107(*)

and fiR=

1.446337(X). 1.54672(*).

lo3 (0); /3=10 and pR=lO1

(x).

82

usually prevaihng. If only the product is absorbed. only the increase of A, has to be considered and the hmiting current follows the same evolution as seen in the absence of adsorptIon (see Fig. 7 m ref. 7). The approximate results for qNpp (6) have also been plotted (full !ines In the figures) together with the exact ones. The approximate expression, cqn. (8). is especially sultable for weak adsorption and reproduces the exact results well whatever the value of the adsorption parameters. Even In the case of strong adsorptlon the agreement IS satisfactory and the discrepancies with the results from the strong adsorption approximation (eqn. 1C) are very small. The accuracy decreases wlthln a polarogram when going towards negative potenteals (6 + co) since d pulse of increasing amphtude 1s applied. On the other hand, the accuracy decreases when the samphn, 0 time becomes shcrter. Both effects have already been commented on in Part II [2] for the I = l(t) curves in pulse polarography. The best agreement between the exact results and the approximate ones IS found for the case of strong adsorptlon when t, < t,, (FIN. 7) where both approximations colnclde In this case. eqn (10) enables us to deduce a simple expression for the half-wave potentral which Justlfles the potential shift of the wave with respect to p (FIN. 7). So, maklng qNPP(8,,?) = +qNp,,(8 + 00) we obtain: E ,,? = E, + (RT/nF)

In (K,/K.)

(17)

Summarizing. the approximate expression. eqn. (8). can be used for every value of the adsorption parameters. Although eqn. (10) IS simpler, its applicablllty IS restrlcted to the case of strong adsorption.

FIN 8 AsFlg

1 w~th~-,=0495.s,j=05006.~R=10and~=10(0);01

(.),!,a1

(+),0(x)

83 ANALYTICAL

EXPRESSIONS

FOR THE LIMITING

CURRENT

The limiting current in normal pulse polarography is measured at the sampling time fd when at time to(to -Z td) a potential pulse has been apphed brmging the system from the base potential (8 = 0) to a Fotentral 6 = cc. If imtially only the reactant R is present in the solution, for X c X, there is no faradaic reactron and then, f.(A)=&(X)=0

q( h;6 = 0) = fX( h)

For h > A,, due to the conditron relation_ eqn. (7) we obtain lim f, = f,6-‘{( S-CC

LIP/D,)

6 = 30 and

taking

into

account

the reversibility

= 0

(18)

whtch means that the application of a pulse with infmlte amplitude leads to the instantaneous transformation of all thz reactant around the electrode into product. The superfictal concentration 8,, direct!y related to f, by the Isotherm, will also be null after the apphcation of the pulse. Then the function q(A) can be expressed as f,(h;B

l

cp(h)=

h
= 0)

J-f&6=

co)

(19)

h>h,

The expression for the limitmg current can nous be found by taking the hnut 6 ---, cc either in eqn. (2) or in eqn. (3) and considering eqn. (19). From eqn. (3)

where fR(X) and consequent!y fX(h) are defined by means of both the integral (5) and the isotherm, eqn. (IS), which in this particular case read

eqn.

O,(X)

(21)

=

A33/14

-

$x-V

jX(f,hd/‘~~)d~

X

e

A,

0 PRfRw=bd~v[~

-

&20)1

X-CA,

(22)

Thus, the limiting current depends only on the adsorptron coefficient K,, so justifymg what was observed in the figures. Approximate expressions can also be obtamed for the limiting current using either eqn. (8) or eqn. (10) and taking into account the conditions of eqn. (18) or eqn. (19). Thus, from eqn. (8) ~FIpp( 6 = Kl) = P/l4 + x”dj(f&

.d = o)/(J_)

+fX-5/7[7XBk(Xd;~~=O)f2BR(Xd;~=0)] where f n( X; 6 = 0) and B&h;

6 = 0) are defined

4,“’ +2Jk(X,;S=0),/mo-3 by eqn. (9) as

(23)

84

e,(x;s=o)=P4[1

-fa(X;8=0)]

(24)

together wrth the isotherm, eqn. (15). Although the system of eqns. (23), (24) and (15) does not have a simple analytical solution, it does not incorporate integral terms (memory effects are eliminated) and rt reproduces within a good degree of accuracy the value of the limiting current for a large range of the adsorption parameters. As we previously mentioned, the accuracy decreases as the adsorption of the reactant becomes stronger, in agreement with this kind of approximatton, wluch is suitable for weak adsorptron. It is not easy to analyze the dependence of the limiting current on either the adsorption coefficient K, or the bulk concentration c*, except for limiting cases. However, a numerical optimrzatron can be made, as we will describe below. iising the approximation indicated for the case of strong adsorption (eqn. lo), the expression for the limiting current hecomes the simplest one.

where q(X,) and q)(h,,) are defined by eqn. (12). As we commented m Frg. 7, this approxrmation correctly reproduces the lirmting current when X, < 1, taking a value which IS independent of the adsorption parameters There is no dependence on PR in eqn. (25), because this approximatron considers a priori p8, = cc. So, it is not posstble to calculate It. Nevertheless, the non-dependence on pR becomes a crrterion in establishing that there is a strong adsorption of the reactant. In this case (lo c r,), the dependence on the bulk concentration c* ~111 be linear, as in classical polarography, and the half-wave potential is given by eqn. (17). Particular cases (I)

Case without adsorprlon

In this case q( h j = 1 and 0,(X) the NP polarogram

= 0 and a sample exact

equation

is obtained

for

(26) whose hmrting value as 6 --, cc will be (27)

The behaviour is similar to that observed in classical polwography, current being directly proportional to the bulk concentration.

the hmiting

(2) Only the reactant IS a&orbed

Equations

(2) and (3) can be used to consider

the particular

case when there is

85

only adsorption

of the reactant. So, takmg 6’,(X)

= 0

(28) (29)

where fR(h)

f?,(h)

and q(h)

are defmed

by the integral

eqn. (5) which is now

( 30)

The limiting current for tlzs case takes the same expression as eqn. (20) but now f R( X) and BR( h) are given by the integral eqn. (30). Approximate expressions can also be obtained for this case from eqns. (8) and (10). Thus, from eqn. (8)

this equation being like eqn. (13) in ref. 4, reported by Van Leeuwen et al. for a linear isotherm. In that case f, and OR are known analytically, but for a general isotherm they must be calculated from eqn. (30) taI.ing into account the specific type of isotherm. From eqn. (10) the approximate limiting current will be:

where y(A,) and v(X,) are given by eqn. (12). Equation (32) is also a more general equation than eqn. (20) in ref. 4, deduced high electrode coverages before the pulse of potential is applied. (3)

Oniy the product Using the same

IS a&orbed procedure

as in

the

previous

case

but

taking

into

for

account

86

B,(A)

= 0, the ltmiting

current

will be

whtch is the same limtt as in the case without adsorption (Fig. 4). This is reasonable because the lrmiting current depe,lds only on the adsorption of the reactant. This result IS also obtained m the calculattons via simulation of Flanagan et al. [7]. NUMERICAL

TREAThlENT

TO OBTAIN

THE PARAhfE-I-ERS

The analytical expressrons etthl:r for the total NPP polarogram or even for the hmiting current rnvolve a complex dependence on the parameters. So, the dtrect calculatton of the parameters IS difficult. Thus, we thmk it convement to develop a numerrcal method in order to yield the parameters that grve the best reproduction of the expenmental results. Although the reproduction of e cperimental results could be made from the exact theorettcal ones, the calculation time becomes too large when an optinuzation process to obtain the parameters 1s requtred. Therefore the approximate expression eqn. (8) has been used to optinuze the parameters in order to reproduce the expenmental results. As we stated above, the agreement with the exact results IS remarkable whatever the value of i he adsorption parameters. The optrmizatton process IS car-red out by minnnizmg the sum of squares of the differences between the approxtmcte and the experimental results by means of Powell’s algortthm [16] TABLE

1

Optlmzauon q, = I .44634

of

the adsorptIon

Value of the parameter m the exacl calculation P

BR

100

10

100

1

100

0.1

10

10

10

1

10

01

parameters

p and PR

Potcnual range whch contams the points selected m the oprlmlzaalIon

Number of points

(6. - 6) (6. - 2) (6. - 6) (6. - 2) (6. - 6) (6. - 2) (69-6) (6.-l) (6. - 6) (6.-l) (6.-6) (6.- 1)

50 33 50 33 50 33 50 29 so 29 50 29

The

parameters

consldered

Value of the parametcr obtamed from the optlmlzatlon process P

BR

76 16 90 11 11903 11915 154 18 230 07 9240 9 893 11.46 11.47 12 11 13 18

8 183 10 49 0 7165 0.7070 0 06724 0 04066 5 232 6704 0.7145 0 6891 0 07687 0 06527

are

70 = 1 3 and

Vanance 106s’

3310 673 24.3 33 7 211 177 138 17.7 2 08 0 290 1.78 0.152

87

In our case, we consider as experimental results those obtained by means of the exact calculations with a predetermmed value for the adsorption parameters. From the optlmizatlon process with the approximate expressions we find the optimum values for the parameters and then these can be compared to the former ones. Tables 1 and 2 summarize some results obta!ned for the case of Intermediate values of the adsorption parameters when the NP polarograms are reproduced. The results pre analogous to the DPP technique [15] and the optimized parameters are of the same order of magnitude as the exact ones. The results are not Improved either when the optimization process is made at larger values of I, or when the points of the optimization are selected In a restricted potentlal range. On the other hand, the reproduction of the limltlng current enables the calculaTABLE

2

Optimizatmn

of

the ndsorptmn

parameters

/3 and

JgR The

parameters

consldersd

are

T,, = 1.8

and

Q = 1 99692 Value of the parameter

Po~entlal range

Number

Value of the pnrnme-

VZWlXKc

in the exact calculnt~on

which contains

of po1nls

ter obtamed

I06S’

B

B;(

the points selec-

the optrmrzarlon

ted in the

cess

opts-

mlwllon

10

100

1

100

01

100

10

10

1

10

01

10

TABLE

from

B

pro-

BR

(6. - 6)

SO

69 79

6 566

(6. - 2)

33

79 49

8 473

(6. - 6)

50

11402

0 7093

(6. - 2)

33

114.00

0 7017

(6. - 6)

50

154 56

0 06537

(6. - 2)

33

256 46

0 03505

(E.-b) (6.-l) (b.-6) (6.- 1) (6. - 6)

50

8402

4 920

29

9544

6 667

50

1185

(6.-l)

29

12 52

50

11.17

29

11 17

0.7138 0 7140 0 07736 0.06930

3110 1580 3 17 2 46 191 131 261 37 9 0 137 0 210 0.770 0.0637

3

Optlrmzatlon of the ndsorprlon parameter pR for the NPP hmlung current The optlrmzatlon has been camed out wrth 25 points calculated a1 several iD and q values. Other parameters consIdered are t’=-lOand/3=10 Value of the PR

Value of the BR

Vanance

parameter m the exact cnlculnrIon

parameter obwmed from the optlm=-

lo9 sz

10 1 01 0.01

5.3358 0 7510 0 08035 0 008408

uon pr-*

107 174 1.42 1 16

88

tion of the adsorption parameter p, in NPP. Thus, the result obtained (Table 3) using &he approximate eqn. (23) for the limiting current when different values of I, and zJ are considered gives an accuracy similar to that obtained when the total polarogram is reproduced. CONCLUDING

REMARKS

The exact expresstons for normal pulse polarography have been deduced under the ass_tmptions given above. Numerical calculations on these expressions allow us to descnbe the different shapes of the polarograms and then evolution with every parameter. Analysrng the polarograms we ca-r conclude: (1) When an adsorption process takes place, the hmitmg current is always depressed wr:h respect to that without adsorptton as has already been mentioned rn the literature. This depression increases as the adsorptton parameter PR increases and tt is independent of the adsorptton of the product. This fact has also been demonstrated mathemattcally from the exact eqn. (2). (2) The one-wave polarogram obtained without adsorption is distorted (lo > f,,,) and shifts towards either positive or negative potentials dependmg on whether K,>K, or K,> K,, respectively (case of a reduction process), leading finally to two-wave polarograms when both adsorption parameters are large enough. The degree of drstortion depends strongly on both the trme of pulse application t,, and the sampling time I, showing whether the adsorption exrsts. Thus, when K, > K, a current peak may appear as was prevrously reported, dtsappearing when either both components are adsorbed strongly or X, becomes small enough. When the adsorpnon decreases the peak current is enhanced and this feature becomes clearer. When the behavrour IS simtlar but only one-wave lo < I,,. and for strong adsorption, polarograms can be obtained. (3) Within the treatment developed here, a change in the reactant concentratron Implies a change of both the adsorption parameter and the time scale through c,. Thus, an increase of the bulk concentration leads to two-wave polarograms while the limrting current is usually enhanced. (4) The approximate expression eqn. (8) accurately reproduces the shape of the polarograms for every value of the adsorptron parameters. So, it can be used in the reproduction of polarograms whatever the expertmental cozcirtions, even for small values of the ratto t,/f,. The results can also be easily reproduced with the aid of a personal microcomputer. In the case of strong adsorption a simpler expression (eqn. 10) can be used. An equation for the half-wave potential is given in this case which reproduces the potential shift very well. Thus, the direct determination of the ratro of the adsorption parameters, 8, is only possible in this case. Moreover, analytical expressions for the limiting current can only be obtained in a simple way from eqn. (101. (5) Finally, in order to complete the comparison between the experimental and the theoretical results a numerical method is proposed from which optimized adsorption parameters can be found: from the limiting current BP can be obtained

89

and from the total palarogram both fi and fiR can also be obtained. The optimized parameters are of the same order of magnitude as the exact ones and we thmk that a combined use of both NPP and DPP [15] techmques can help to defme the system parameters. The analysis carried out in this paper is based on the use of a Langmutr isotherm which convementiy describes the situation at low concentrations when the effect of thrs analysis can be the interaction between mo!ecules is mirumiz ed. Nevertheles, extended to any other kind of isotherm, thus making possible the study of systems where the concentrations of the reactants are higher. The exact calculations were carried out on an IBM-4341 computer (in FORTRAN language) and the approximate ones were carried out using a BASIC-operating Hewlett-Packard 85A microcomputer connected to a Hetwett-Packard 7225A plotter. REFERENCES 1 2 3 4 5 6 7 8

J. Puy. F Mas. F. Sanz and J. Vu& J Electroanal. Chem , 183 (1985) F. Mas. J Puy. F. Sanz and J Vugh. J. Electroanal Chem . 183 (1985) H.P. van Leeuuen, J Eleetroanal Chem , 133 (1982) 201 H P. van Leeuwen. M Sluyters-Rechbach and K Holub. J. Electroanal. K Holub and H P van Leeuwen. J Eleclroanal. Chem , 162 (1984) 55. H.P van Leeuwen. J. Electroanal Chem . 162 (1984) 67 J B Flanagan, K. Takahashl and F C Anson. J Electroanal. Chem , 85 S N Feldberg m #._I. Bard (Ed.), Elecu oanalyucal Cherrusrry. Vol. 3, 1969. pP 199-296

9 F. Mas, J Puy, F. Sanz and J Vqih. J Electroanal Chcm . 158 (1983) 10 J Puy, F Mas. F. Sanz and J Vu& J Electroad Chem . 158 (1983) 11 V G. Leach. B I. Khatkin and E D. Belokolos. Eleklrokhmuya. 1 (1965) Sov. Electrochem . 1 (1965) 1137. 12 J Koryta, Collect. Czech. Chem. Commun , 18 (1953) 206. 13 A. Huber. Monatsh Math. Phys ,47 (1939) 240. 14 G C. Barker and J.A Bolzan, 2. Anal. Chem , 216 (1966) 21.5 15 F. Sanz. J. Puy, F Mas and J. Vugm. J. Electroanal. Chem , 183 (1985) 16 M J D Powell, J. Computer, 7 (1964) 155; W.I. Zangwrll. J Computer,

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