A.C. polarography and faradaic impedance of strongly adsorbed electroactive species

J. Electroanal. Chem., 105 (1979) 25--34

25

© Elsevier Sequoia S.A., Lausanne - - Printed in The Netherlands

A.C. POLAROGRAPHY AND FARADAIC ADSORBED ELECTROACTIVE SPECIES

IMPEDANCE

OF STRONGLY

PART II. THEORETICAL STUDY OF A QUASI-REVERSIBLE IN THE CASE OF A FRUMKIN ISOTHERM

REACTION

E. LAVIRON Laboratoire de Polarographie Organique associ~ au C.N.R,S. (L.A33), Facultd des Sciences Gabriel, 6 Boulevard Gabriel, 2 1 0 0 0 Dijon (France)

(Received 5th February 1979; in revised form 15th May 1979)

ABSTRACT The expression of the faradaic impedance is calculated in the case of a quasi-reversible system O + ne ~ R under the following conditions: (a) both the oxidized and the reduced forms are strongly adsorbed; (b) the adsorption rate is large, and does not control the kinetics of the system; (c) the adsorption of both O and R obeys a Frumkin type isotherm (interactions between the adsorbed molecules are CO.nsic]~red). The results are compared to those obtained previously in the case of a Langmuir isotherm. The cotangent of the phase angle is still proportional to (A/k s (k s = rate constant of the electrochemical reaction), so that still tends towards 90 ° when co/k s -+ 0, but the slope of the straight line obtained is a complicated function of the superficial concentration and of the interaction coefficients. The height of the a.c. wave is no longer proportional to the bulk concentration of the reactant, to T7/6 (~ = drop time) and to h -1/2 (h = height of the mercury reservoir). When co/k s -+ O, the shape of the a.c. wave is identical to that of the linear sweep voltammogram.

In the first paper of this series [1], we gave the expression of the faradaic i m p e d a n c e a n d o f t h e a.c. p o l a r o g r a m s in t h e c a s e o f a q u a s i - r e v e r s i b l e r e a c t i o n , when both the oxidized and reduced forms O and R are strongly adsorbed, and when the adsorption follows a Langmuir isotherm. We treat here the same problem, when there exist interactions due to forces between the adsorbed m o l e c u l e s , i.e. w h e n t h e a d s o r p t i o n o b e y s a F r u m k i n t y p e i s o t h e r m . THEORY (1) General expression

of the faradaic resistance and capacity

T h e c o n d i t i o n s a r e t h e s a m e as in r e f . 1: (a) O a n d R a r e s t r o n g l y a d s o r b e d ; ( b ) t h e t o t a l c o v e r a g e o f t h e e l e c t r o d e is s m a l l e r t h a n 1; (c) t h e r a t e o f t h e a d s o r p t i o n p r o c e s s is s o l a r g e t h a t i t d o e s n o t l i m i t t h e k i n e t i c s o f t h e s y s t e m . W e w i l l u s e in t h i s p a p e r t h e s a m e s y m b o l s a n d t h e s a m e sign c o n v e n t i o n s as in r e f . 1 ( a n o d i c c u r r e n t s a r e r e g a r d e d as p o s i t i v e ) . I n t h e c a s e o f a d r o p p i n g m e r c u r y e l e c t r o d e , w h e n o n l y O is p r e s e n t in s o l u t i o n , t h e s u m o f t h e s u p e r f i c i a l

26

concentrations of O and R is given by: F O + Fn = F w=

0.74 c o D h n t i n

(1)

Co and Do being the bulk concentration and the diffusion coefficient of O. (For a detailed discussion of the mathematical description of the problem in the case of a strong adsorption, see refs. 1--3.) The derivation of the faradaic impedance carried out in ref. 1 is very general, so t h a t the results can be used directly here. We will still assume t h a t no change of the capacity of the electrode will result from the application of the alternative potential, since the total coverage of the electrode remains practically constant. If R a and ca are the faradaic resistance and capacity, we have R a = 0 =

(~E/~i)ro, r R

l / C a = a = ( n F A ) - l ( f i o - - fiR)

(2) (3)

with flo = (~E/aFo)~, r~

(4)

~R = (~E/~FR)~, ro

(5)

A being the area of the electrode. (2) T h e c u r r e n t p o t e n t i a l c h a r a c t e r i s t i c

We will suppose that the area occupied by one molecule of 0 is the same as that occupied by one molecule of R. We will designate by Fm the m a x i m u m value of either Fo or FR, and by 0o and OR the coverage of O and R, respectively (0o = FO/Fm and OR = FR/Fm). Let us consider the reaction O ~ activated complex ~ R; we will make the following hypotheses concerning the influence of interactions between the • molecules, by taking the system w i t h o u t interactions as a reference; (a) the energy of the molecules of O varies linearly with the coverage 0o, owing to interactions between molecules of O. (b) the energy of the molecules of O also varies linearly with the coverage OR, owing to interactions between molecules of O and molecules of R. If AGo represents the energy of the molecules of O, we can write (the factor 2 has been introduced to facilitate a comparison with the coefficients of the Frumkin isotherm; see below): AGo/RT

= - - 2 p O o - - 2rOR

(6)

(C) likewise we can write t h a t the energy of the molecules of R varies linearly with OR (interactions between molecules of R) and with 0o (interactions between the molecules of O and R). A G R / R T = --2qOR - - 2rOo

(7)

(d) the energy AGa¢t of the activated complex also varies linearly with 0o and OR, owing to interactions with O and R AGac t/RT

= --2VOo - - 2wOR

(8)

27

(We will assume t h a t p , q, r, v and w are i n d e p e n d e n t o f the p o t e n t i a l , which is justified b y the fact t h a t t h e range o f p o t e n t i a l studied is n o t v e r y large ( 2 0 . 1 V).) We can thus write, f o r the c a t h o d i c activation e n e r g y AGe: AGc/RT

= (AGo

+ AGact)/RT

= --2(19 + v) Oo - - 2(r + w) 0Z

(9)

and f o r the anodic activation e n e r g y AG a: AGa/RT

= (AGR + A G a c t ) / R T

= --2(q + w) 0 R - - 2(r + V) 0o

(10)

L e t p + v = fi, r + w = 3', q + w = h and r + v = p; if we assume t h a t the rate constant k s o f t h e e l e c t r o c h e m i c a l r e a c t i o n is i n d e p e n d e n t o f the coverage, the curr e n t p o t e n t i a l relationship can be written: i = nFAksFm

{B-a~? 1 -~exp(--2}~B-R -- 2PB-O) -- B-ol?-~exp(--2flB-o - - 2~'B-R)}

(11)

B-O and B-a are the coverages in t h e absence o f alternating c u r r e n t , and ~ is still d e f i n e d by: ~? = exp ( ( n F / R T )

( E - - E°')}

(12)

in which the surface s t a n d a r d p o t e n t i a l E °' is given by: E °' = E ° - - ( R T / n F )

ln(bo/ba)

(13)

E ° being the v o l u m e s t a n d a r d p o t e n t i a l , and bo and bR the a d s o r p t i o n coefficients o f O and R. A t the equilibrium, we obtain, b y e q u a t i n g eqn. (11) t o zero (we will still designate b y E the equilibrium p o t e n t i a l ) : 77 = (0O/0R) e x p ( 2 ( p --fl) 0 o + 2(~ --~,) ~a}

(14)

or

~? = (B-O/~R) e x p ( 2 ( r - - p ) B-o + 2(q -- r) B-R}

(14 bis)

E q u a t i o n (14 bis) shows, as was t o be e x p e c t e d , t h a t , at equilibrium, o n l y the c o n s t a n t s p , q and r relative t o O and R have t o be t a k e n into a c c o u n t and t h a t the c o n s t a n t s relative to t h e activated c o m p l e x have disappeared. In a previous p a p e r [3], we have established, in the case o f a F r u m k i n isot h e r m , t h e following relationship b e t w e e n the equilibrium p o t e n t i a l and the coverages, valid for the c o n d i t i o n s w h i c h we c o n s i d e r here (eqn. 8, ref. 3, with Fo,m = FR.m = Fro, i.e. n o = nR = V): = (B-O/B-R) e x p {2VB-O(aOR -- ao) + 2VB-R(aR -- aOR)}

(15)

V is the n u m b e r o f m o l e c u l e s o f solvent (or clusters o f solvent molecules [ 4 ] ) displaced b y o n e m o l e c u l e o f O or R; ao, a a and aOR are the c o n s t a n t s o f intera c t i o n b e t w e e n m o l e c u l e s o f O, molecules o f R, and m o l e c u l e s o f O and R, respectively (ai is positive for an a t t r a c t i o n and negative f o r a repulsion). B y i d e n t i f y i n g eqns. (14) and (15) we obtain: 2P(aOR -- ao) = 2(p -- fl) = 2(r - - p ) 2V(aZ -- a o a ) = 2(k - - 3') = 2(q -- r) which shows that: p = V a o , q = v a a , r = VaOR.

(16) (16 bis)

28

9o o r

i

c ~~' "°~ .(E EO% -0.1

o.1

0

Fig. 1. OO/OT (or anodic d.c. reversible polarogram) as a function of the potential. (AB and CD) Stable branches. (BC) Unstable branch. The value of vG0 w is indicated on each curve.

Substraction of (16) from (16 bis) gives: V(ao + aR --

2aoR)

= v G = ?~ - - 7

--P

+ [3 = p

(17)

+ q -- 2r

Addition of (16) and (16 bis) yields: (18)

V ( a R - - a o ) = ~. - - ~[ + P - - fl = q - - p

As we showed in ref. 3, the quantity defined by eqn. (17) determines the shape of the reversible d.c. wave or the shape of the variations of either 0o or OR with the potential (Fig. 1). Let us designate b y 0T the total coverage of the electrode: (19)

~O + ~-R ----0T

If p G 0 w = 0, the wave has the shape of a normal reversible polarogram. If v G O T < 0, the wave is more drawn o u t than the normal polarogram. If ~G0 w > 0, the wave is steeper than the normal wave. For pV0 w = 2, the slope of the tangent at mid-height becomes infinite. For pG0 w > 2, the theoretical curve becomes S-shaped. For certain values of the potential, three different compositions of the film of O and R are theoretically possible. In fact, the part of the curve with a negative slope represents a state of instability and cannot be obtained experimentally. In the case of a dropping mercury electrode the coverage is initially equal to zero and then increases, so that the situation of instability can be reached at some time during the drop life (see ref. 3 for a detailed discussion). In practice, the appearance of unstable films will possibly result in non-reproducible results. On the other hand, the quantity ( R T / n F ) (aR -- ao)/JOT (cf. eqn. 18) determines the shift of the half-wave potential [3], which is given by: E 1 / 2 , a = E O' + ( R T / n F )

(aR - - a o ) /20T ----E°'+

(RT/nF)

(q - - p ) 0 T

(20)

We will designate b y s the quantity V ( a R - - a o ) . If s > 0, the wave is shifted towards positive potentials; if s < 0, towards negative potentials. The meaning of eqn. (20) is the following: as shown by the preceding discussion, 2RTaovOw

29

represents the energy which has to be spent, or which is released during the total reduction or oxidation because of the gradual disappearance of the interaction forces between molecules of O. At the half-wave potential one half of the molecules of O have appeared (oxidation) or disappeared (reduction), hence the term in RTaovOw . If for example a o > 0 (attraction forces between molecules of O), the peak potential becomes more negative; in the case of a reduction, for example, more energy has to be spent in order to overcome the attraction forces. The term in RTvaaO w can be interpreted similarly. (3) Derivation o f R a and Ca

By taking into account eqn. (19), eqn. (11) can be written: i = n F A k s F m (m-Oa~?1-~exp(h~a) -- lOo~-~exp(gOo)}

(21)

with: g = 2(7 --/3) h = 2(p -- ~) l = exp(--23'0T) m = exp(--2p0T) By using eqn. (2), we derive from (21), taking eqn. (12) into account: R a = 0 = ( R T / n 2 F 2 A k s F m ) X-~

(22)

with X = (1 -- a) m~-al?i-aexp(h~TR) + alOoT?-aexp(gOo)

(23)

By using the relationship (see e.g. ref. 5) (F~ = Fo or FR; Fk = FR or Fo): (~E/OFj)i, rk = --(~E/Oi)rj, rk (~ i/~ Fj)E, rk

(24)

we obtain from (4), (5), (12) and (21) 1/C~

= a =

ksO[

(25)

with = m(1 + hOR) ~'/'-"exp(hYR) + l(1 + g~o) ~7-%xp(g~-o)

(26)

(4) Phase angle and a m p l i t u d e o f the current

The phase angle is given by: cotg ~ = ¢ORaC a = ¢oO/a

(27)

or

cotg ~ = ( w / k s ) ~-1

(28)

The general expression of the amplitude, established in ref. 1, is still valid: I = e0-1 cos ~

(29)

30 By introducing in eqn. (22) the value of k s derived f r o m ( 2 8 ) , we obtain: 0 = ( R T / n 2 F ~ A C O F m ) ( ~ / X ) cotg ~

(30)

Substitution of this value in (29) gives, after normalization of the amplitude: = I / ( n ~ F 2 A e C O F T / R T ) = 0T1(×/~) sin ~

(31)

which gives in particular the general expression of the a.c. wave, when the potential is varied. DISCUSSION IN THE CASE OF A D.C. REVERSIBLE REACTION Let us consider the case of a "d.c. reversible" reaction: the reaction rate is large enough for the reaction to appear reversible during the growth of the drop, i.e. the equilibrium corresponding to eqn. (14) or (15) between the mean~ concentrations Po and r R is achieved. Explicit equations for the phase angle and the amplitude cannot be obtained, and t h e y can be only defined implicitly. ( 1 ) P h a s e angle

It is defined by eqns. (12), (14), (19), (26) and (28). As shown by eqn. (28), still tends towards 90 ° when co/k s + O, as in the case of a Langmuir isotherm. As ~ is independent of the frequency, cotg ~ still varies linearly with co, but the slope 1 / k s ~ is a complicated function of the superficial concentrations and of the interaction coefficients, so that k s cannot be deduced from the experimental results. As shown in the following paper, the linear variation of cotg ~ with co is a general property of strongly adsorbed redox systems, valid for any isotherm. Let us designate by cotg0 ~ the value of cotg ~ in the case of a Langmuir isotherm; we have: cotg0 ~ = (oMks) 07 -~ + ~,~1-o~)-1

(32)

so that the ratio of cotg ~ for a Frumkin isotherm to cotg0 ~ is equal to: Q = cotg ~/cotg0 ~ = (~-a + r/1-") ~-1

(33)

This ratio is independent of co and k s. We have studied it numerically at constant potential when 0T varies. A few curves are given in Figs. 2 and 3. The following comments can be made: (a) no predictable trend can be deduced from the individual values of the interaction coefficients. Q can be larger or smaller than 1; as shown in Fig. 2, it can even first decrease, then increase; (b) even if G = 0, the ratio Q depends on the individual values of the interaction constants (Fig. 3) (when G = 0, the reversible d.c. wave [3], or the reversible a.c. wave (see below), do not depend on the individual values of the interaction constants); (c) the error on k s can be very large, if eqn. (32) is used instead of (28); (d) cotg ~ tends to become infinite in the region of the discontinuity (Fig. 2); (e) as shown in Fig. 2, negative values of cotg ~ are theoretically possible,

31

C}

0

i

'

' o.~ I

',

'

~

¢

1(

0

-2 i

I

-6

BT

o

ols

-]

Fig. 2. V a r i a t i o n s o f Q. F r u m k i n i s o t h e r m . ~ = 0, ~' = - - 3 , ~ = - - 1 , p = - - 2 ; PG = 4, s = 0. E -E ° ' , n = 1, 0~ = 0.6, t = 2 5 ° C . ( 1 ) C o r r e s p o n d s t o b r a n c h A B , ( 2 ) t o b r a n c h BC, ( 3 ) t o b r a n c h C D o f Fig. 1. Fig. 3. V a r i a t i o n s o f Q. F r u m k i n i s o t h e r m . ~ = ~/= )k =/~; t h e i r c o m m o n e a c h c u r v e ; P G = O, s = O. S a m e v a l u e s o f E , n, (~ a n d t as in Fig. 2.

v a l u e is s h o w n o n

i.e. the system could eventually present an inductive behaviour. In practice however, it will not be possible to observe such a behaviour, because the negative values of cotg ~ correspond to the unstable branch of the i--E curves (Fig. 1) (the theoretical inductive behaviour corresponds to the situation where Fo decreases and I~R increases when the potential becomes more positive). (2) A.c. polarogram

It is defined by eqns. {12), (14), (19), (23), (26) and (31). We will distinguish the "reversible" (co/k s ~ 0, ~ + 90 ° ) and "less reversible" (~ ¢ 90 °) cases. When the reaction is "reversible", (co < < ks) a numerical resolution of the above mentioned equations shows that the a.c. wave has exactly the same shape as the corresponding linear sweep voltammograms [3,6], which, as we showed earlier, are proportional to the derivative of the d.c. polarogram with respect to the potential (compare Fig. 4, this paper and Fig. 1, ref. 6). In particular, the peak potential Ep is equal to the half-wave potential defined by eqn. (20). This result can also be confirmed directly: although eqn. (31) does n o t take the form of the equation of the voltammograms given in refs. 3 and 6, it is possible to calculate directly for the "reversible" a.c. wave an expression which is identical to that of the voltammograms, by starting with the equations given in ref. 3, and by assuming a sinusoidal variation of the potential [7]. A few examples of a.c. waves are given in Fig. 4. The peaks have the same shape, whatever the

32

I

0

"

--

L ~/V

Fig. 4. T h e o r e t i c a l " r e v e r s i b l e " a.c. w a v e s . F r u m k i n i s o t h e r m . (~ = 0.6, k s = 104 s -z , f = 5 0 Hz,'t=25°C.(1)fi=0,'y=3, k=l,p=2,0 T=0.25;vG=-4;s=0;vG0 T=-1.(2)/3='),= = p = 0 ( L a n g m u i r i s o t h e r m ) . ( 3 ) , ( 4 ) / 3 = 0, 7 = - - 3 , ~ = - - 1 , p = - - 2 ; v G = 4 , s = O. ( 3 ) 0 T = 0 . 2 5 ; p G0 w = 1. ( 4 ) 0 w = 0 . 3 7 5 ; / ) G O T = 1.5.

values o f fi, 7, X, p and ~, so long as co < < k s. When/)GOT = 0 , the peak has the shape of a normal wave, and its width at mid-height is equal to 90.6/n mV at 25°C. As shown in ref. 3, this can happen when 0W -~ 0, or when G = 0 (Langmuir isotherm, or compensation between the interaction forces). When VGOT< O, the peak becomes broader, and when

0.3'~

oo

~'~1000 ~ ° ' ) / V

-0.1

0.1

-0.,

0

0.1 . (~_Eo,)/V

Fig . 5. T h e o r e t i c a l " i r r e v e r s i b l e " a.c. w a v e s . F r u m k i n i s o t h e r m . / 3 = 0, ~' = - - 3 , k = - - 1 , p = - - 2 , 0 w = 0 . 3 7 5 ; ~G = 4, s = 0 ; v G 0 T = 1.5. (~ = 0.6. k s = 5 0 0 s -1. t = 2 5 ° C . T h e f r e q u e n c y ( i n H z ) is i n d i c a t e d o n e a c h c u r v e . 1

Fig . 6. I n f l u e n c e o f i n d i v i d u a l v a l u e s o f t h e i n t e r a c t i o n p a r a m e t e r s a t c o n s t a n t V G a n d s o n t h e " i r r e v e r s i b l e " a.c. w a v e ( t h e o r e t i c a l ) . F r u m k i n i s o t h e r m . / 3 = 7 = k = P ( t h e i r c o m m o n v a l u e is i n d i c a t e d o n e a c h c u r v e ) ; 0 T = 0 . 3 7 5 ; vG = 0, s = 0 ; v G 0 w = 0.0~ = 0 . 6 , k s = 5 0 0 s -1 , f= 200 Hz, t = 25°C.

33

2000.

-I).1

0

0.1

Fig. 7. T h e o r e t i c a l a.c. w a v e s . F r u m k i n i s o t h e r m . ~ = 0, ~ / = - - 3 , X = - - 1 , p = - - 2 , 0 w = 1; ~G = 4, s = 0; ~'GO w = 4. o~ = 0 . 6 , k s = 5 0 0 s -1 , t = 2 5 ° C . T h e f r e q u e n c y ( i n H z ) is i n d i c a t e d o n eaeh eurve.

vGOv > 0, it becomes narrower. For p G O T ~ 2, three mathematical branches are obtained (Fig. 7, curve 1); as shown in ref. 3 in the case of the voltammograms, the shape o f the peak can no longer be defined mathematically. Figures 5 to 7 give a few examples of a.c. peaks when the reaction is no longer reversible. It is in particular interesting to not e (cf. variations of Q), t hat the wave depends on the individual values of the interaction constants, even if G = 0 (Fig. 6). When p G O w > 2 (Fig. 7) the peak theoretical curve is modified when co mp are d to the reversible curve, but the peak cannot be defined mathematically. (3) Dependence of the wave height on the diverse parameters Variations with the frequency. In eqn. (31), × and ~ are i n d e p e n d e n t of the frequency; as shown above, the variation of ~ with the frequency is the same as in the case of a Langmuir isotherm, with an apparent rate constant ks~ instead of ks. The variation o f I with the f r e q u e n c y is thus the same as in the case of a Langmuir isotherm, with an apparent rate constant different from k s. Variations with the concentration. In the case of a dropping m ercury electrode, 0 w is still a linear function of e.g. the bulk concent rat i on of O in the case of a reduction (cf. eqn. l i , but ×, ~ and ~ are complicated functions of 0W. The peak height should n o t be proportional to the bulk concent rat i on of O, cont rary to the case o f Langmuir isotherm. Variations with the drop time T and the height h o f the mercury reservoir. Ow varies p r o p o r t i ona l l y to t 1/2. X, ~ and ~ being however complicated functions of 0W, I should n o t be proportional to T7/6, as was the case for a Langmuir isotherm. F o r the same reason, the variations with h -in should n o t be linear. We have verified experimentally the above predictions by taking benzo(c)cinnoline at pH 11.5 as an example. As we have shown in a previous paper, a

34 Langmuir isotherm is obeyed when 0 T ~ 0.1 (C ~'~ 10 -9 mol cm 3) [1], and the above relationships are followed. When the c o n c e n t r a t i o n is increased (experiments were c o n d u c t e d for c = 2 × 10 -s and 5 X 10 -8 mol cm -3, i.e. 0W = 0.13 and 0.32), the a.c. waves become distorted and the above relationships are no longer followed. CONCLUSION

We have assumed in the derivation of the mathematical results t hat ks is i n d e p e n d e n t of the coverage. This hypothesis is n o t a priori the most representative o f the real situations. It is well known, for example, t hat when the coverage increases, the orientation of the molecule can change [ 8,9], which could conceivably cause a change in ks. Such effects would be interesting to study; u n f o r t u n a t e l y , our results show t hat this does n o t seem possible, if interactions are present, which is the case for m os t organic molecules. In order to obtain the value of ks, it is necessary to be able to calculate the value o f ~ (eqns. 26 and 28), which requires the knowledge of the interaction constants p, q, r, v and w. p(ao), q(aR) and r(aoR) can at least in t h e o r y be obtained f r o m interfacial tension or differential capacity measurements [ 9], plus the analysis o f the polarograms or linear potential sweep voltammograms [ 3]. If no assumption a b o u t the influence of the interactions on the activated complex is made, it will n o t be possible to determine v and w; changes in the coverage will cause u n k n o w n changes in ~ and in ks, so that neither v, w nor k s can be d e t e r m i n e d . This should be true for any ot her electrochemical m et hod. Our results show t hat the only significative value of ks will be the value obtaihed by extrapolation when 0W -~ 0, i.e. when the interactions between the molecules become negligible (Langmuir isotherm). In particular a normal shape for the "reversible" a.c. wave is n o t a p r o o f that a Langmuir isotherm is obeyed, since this can result from a compensation between the interaction constants; as shown above, such a situation can be det ect ed by studying e.g. cotg ~0 when 0W increases. We will present in following papers of this series the complex plane analysis o f the impedance, as well as the t r e a t m e n t of the case when a surface and a volume reaction occur simultaneously. REFERENCES 1 2 3 4 5 6 7 8 9

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