1-s2.0-002207289403684U-main

ELSEVIER Journal of Electroanalytical Chemistry 382 (1995) 111-127

JOURNAL OF

The use of polarography and cyclic voltammetry for the study of redox systems with adsorption of the reactants. Heterogeneous vs. surface path

E. Laviron Laboratoire de Synth~se et d'ElectrosynthOse Organomdtalliques (Unitd de Recherche associ~e au CNRS 1685), Facult~ des Sciences, 6 Bd. Gabriel

21000 Dijon, France

Received 16 June 1994; in revised form 21 July 1994

Abst rac t

The use of polarography and linear-sweep voltammetry (LSV) for the study of a redox reaction O + ne ~ R when both O and R can be adsorbed (Langmuir isotherm) is examined, on the basis of a rigorous theoretical treatment presented earlier for a rotating disk electrode (r.d.e.) (E. Laviron, J. Electroanal. Chem., 124 (1981) 19 and J. Electroanal. Chem., 140 (1982) 247).

In aqueous medium on a mercury electrode, the reaction practically always occurs via the adsorbed species (surface redox reaction). However, two cases can be distinguished, according to whether the rate of desorption of the product of the reaction (in polarography) or of the adsorbed reactant (in LSV) is large or small when compared with the duration of the measurement (r in polarography, RT/nFv in voltammetry). In the first case, the reaction appears as heterogeneous, with an apparent rate constant khm, which is much larger than the normal constant kh, and which can be determined by using the classical theories for a heterogeneous reaction. In the second case, the reaction has a "surface" character, and the electrochemical surface rate constant k s can be determined by using the appropriate theories. The domain for each reaction can be represented by using adsorption diagrams log~- or logv vs. log(bobR) 1/2 (b o, bR; adsorption coefficients). The advantages of using polarography and cyclic voltammetry rather than r.d.e, voitammetry for the study of the above systems are discussed; they are theoretical (non-steady-state nature of the methods) as well as experimental (use of the dropping mercury electrode).

Keywords: Polarography; Cyclic voltammetry; Adsorption

I . In t roduct ion

Adsorpt ion of the species part ic ipat ing in an electrochemical react ion is extremely frequent, especial ly in the case of organic compounds [1,2]. This is part icu- larly true in aqueous solutions, but adsorpt ion in non- aqueous solvents, although weaker than in water, can still be appreciable [3-7]. The general scheme for a simple react ion is given in Fig. 1, in which Oad s and / Rad s are the adsorbed forms, Oso I and Rso I the unad- /

/ sorbed forms near the surface, 0 and R the forms in / the bulk solution away from the electrode. The surface ~// concentrat ions are F o and FR, the concentrat ions in O/ solution Co(X,t) and CR(X,t) , C and c being their ~/ b , / value for x ~ o0. The surface and heterogeneous elec- ~ / trochemical rate constants are respectively k s (in s - l ) ~ / and k h (in cm s - l ) . In theory, a possibil ity of direct

/ electron exchange between the adsorbed and non-ad- / sorbed systems is possible; however, we shall not con- / / sider it here, because it can be neglected in most / practical cases [8-10].

0022-0728/95/$09.50 1995 Elsevier Science S.A. All rights reserved SSDI 0022-0728(94)03684-5

Considered in its general ity, the mathemat ica l treatment of the problem (solution of the diffusion equations with adequate initial and boundary condit ions) is very complex, as far as transient electrochemical methods are concerned. A full discussion of the diverse

F 0 c o(O,t) c o (x,t) o Co(X --~)

Oad s ~ ~ Oso I ~ "~ 0

ks(s4) kh(Cm s l ) transport

Rads~ ~ Rso I ~ ~ R

F R CR(O,t ) C R(x,t) C~, (x .-),at)

Fig. 1. The reaction scheme.

112 E. Laviron / Journal of Electroanalytical Chemistry 382 (1995) 111-127

works concerning these methods can be found in Ref. [11]. The mathematical difficulties are illustrated, for example, by the theories for faradaic impedance (in particular by the works of Delahay et al., Barker, Sluyters-Redbach, Sluyters et al.; cf. the 30 references quoted in Ref. [11]) or by Wopschall and Shain's classical treatment in cyclic voltammetry [12], which gives complex results, although it is restricted to the case of a reversible reaction. Analytical solutions can be obtained only in a few limiting cases, such as that of a linear isotherm with an immobile electrode, or for Frumkin or Langmuir isotherms when both O and R are adsorbed strongly (surface system) [11], or when only one of them is adsorbed, the other being present in the solution in high concentration [13,14].

However, we showed in two previous papers [15,16] that the problem can be solved simply in the case of voltammetry on a rotating disk electrode (r.d.e.), which is a steady-state method. Easy-to-interpret analytical solutions are obtained, when the rate of the adsorption reaction itself is assumed to be so fast that it is not rate-controlling, and when a Langmuir isotherm is obeyed. Both conditions are usually fulfilled in the case of mercury electrodes, on which the adsorption rates are usually so high that they cannot be measured by existing methods (see Ref. [17] and more especially the discussion by Delahay [18a]), while langmuirian conditions can be attained at low surface concentrations [11]. On solid electrodes, by contrast, adsorption can be slow and often involves the formation of chemical bonds between the molecules and the surface [2,18b,19,20], and difficulties arise owing to the heterogeneous nature of the surface [18c,19].

Transient methods, such as polarography and linear-potential-sweep voltammetry (LSV) are much more advantageous to use, firstly because the non- steady nature of the measurements allows the reactions to be studied in a much larger range of adsorption (see below), and also because of practical reasons (reproducibility and ease of use of dropping or pseudo-stationary mercury electrodes, compared with the diffi- culty of obtaining good mercury films on the r.d.e., larger span of "measurement" times, etc.). We examine in this paper, on the basis of our previous results for the r.d.e., how polarography and LSV can be used to study the problem.

We shall consider the case of a reduction; the results can easily be transposed to that of an oxidation when needed.

In general, the reduction or oxidation current for an organic compound is given by an equation of the form (reduction currents are taken as positive)

i = nFAk~ [~:o e(-anF/RTXE-Ee) _ _ sCRe((l-a)nF/RTXE Ee) ]

(1)

which holds either for a heterogeneous reaction (with k e = kh, ~o = CO, ~R = CR, and E e is the equilibrium potential) or for a surface reaction when a Lang- muir isotherm is obeyed [11,21-26] (with k e = ks, ~o = Fo, ~:R = FR, and Ee is the surface equilibrium potential). Theoretically a can have any value; in practice, however, three cases can occur, i.e. n = 1, a = 0.5, or n = 2 with a = 0.25 or 0.75 [23-26].

2. Rotating-disk electrode: summary of the main results

2.1. Generalization of the theory for any value of a

In our previous study [15,16], we defined the problem for any value of a, but the equations were solved only in the particular case when ot = 0.5. In view of Eq. (1), we need to generalize the results for any value of OL

For a simple heterogeneous reaction without adsorption, we have the equation (Eq. (7) of Ref. [15])

X = 3'[(1 -X)O -'~ --XO 1-a ] (2)

in which X is the dimensionless concentration of the reduced form near the electrode

C = CR(O,t)//C T = cS//CT (3)

i.e. the concentration near the surface, which is constant here, divided by the total concentration c T of O and R in the solution (here, in the case of a reduction, the concentration of O). The parameter 0 is defined by

Co( O,t ) c s 0 - CR(o,t------- ~ cS exp[ (nF /RT) (E -Ee) ] (4)

and 3' is the dimensionless electrochemical rate constant

y = kh~$D -1 (5)

where 8 is the thickness of the diffusion layer, and D is the diffusion coefficient.

The dimensionless current I is given by

i I= nFAcTDS- 1 X (6)

When the reactants can be adsorbed as shown in Fig. 1, the total current, which is the sum of the surface current i a and the heterogeneous current i h, is given by Eq. (21) of Ref. [15], when a Langmuir isotherm is obeyed. The transfer coefficients are assumed to be the same for both reactions; although they could be different in theory, they have been found to be equal in the

E. Laviron /Journal of Electroanalytical Chemistry 382 (1995) 111-127 113

diverse experimental cases which we have examined [23-26].

i = i a + i h = nFAD3-1c s

= nFAks(FO~7 -~ - FRr/ ' - - )

+ nFAG(cSO -" - cSO 1 -~) (7)

with [11,15,27]

= (bo /bR)O (8)

in which b o and b R are the adsorption coefficients of O and R.

Assuming that the adsorption equilibrium is always established and that a Langmuir isotherm is obeyed, and proceeding as in Ref. [15], we easily obtain:

I=X = yM[(1 -X)0 - " +xO 1-~] (9)

with or

m=l+ b_ l r~_ l /Z+rZ, , ( l _x )+r_2( l _ ,O X (10)

The dimensionless parameters or, b and r are defined by (F m maximum surface concentration, assumed to be equal for O and R):

ksrm or = - - (11)

khC T

b =CT(bobR) 1/2 (12)

r = (bo /bR) '/2 (13)

Comparison of Eq. (9) with Eqs. (2) and (6) shows that the global reaction is equivalent to a heterogeneous process with an apparent dimensionless rate constant My, i.e. in view of Eq. (5), an apparent rate constant khM defined by:

k hM = Mk h (14)

Since M is always larger than one, the apparent reversibility of the reaction is always increased. Gener- ally speaking, M varies along the wave, since it depends on the current I=X (Eq. (10)), and also depends on c T through or and b; this point will be discussed below in Section 2.4.

2.2. The three situations

Our previous study [15,16] led to the conclusion that three types of situations can be distinguished (Fig. 2).

In the first situation (Fig. 2, case I), there is no adsorption, and only the heterogeneous reaction (rate constant k h) takes place. As shown in Ref. [15] (see also below), this case should be very seldom encoun- tered, if ever, in aqueous medium, but can exist in non-aqueous solutions.

In the second situation, part (Fig. 2, case IIa) or all (Fig. 2, case lib) of the reaction takes place on the

I

k h

7 ! " froz film a b I

v

11 III

khM = Mk h

Fig. 2. Schematic representation of the reaction scheme for the three situations on the r.d.e. (---,) mass transport; (~) electrochemical reaction.

surface. However, because the desorption rate of R is large enough [15,16], the molecules of O are transported towards the electrode, they are reduced, and then are transported away from it, which is similar to what occurs for a normal heterogeneous reaction. As shown above, the global reaction is indeed equivalent to a heterogeneous process, with the apparent heterogeneous rate constant khM.

The third situation arises when the desorption rate of the molecules of R becomes so slow that we have an immobile ("frozen") film on the electrode [16] (Fig. 2, case III). On a rotating-disk electrode, the electrochemical reaction is again heterogeneous in the sense that the molecules of O are reduced without being adsorbed, in the presence of the film; however, the film can have an influence on the rate (autoinhibition effects [16,27]), so that the rate constant can be different from k h, and its value cannot be predicted.

2.3. The value of ks /k h

Quantitative calculations, using the equations which we have derived, require the knowledge of the ratio kh/k s. Brown and Anson [28] proposed the formula

k Jk h = 6 108 cm - l (15)

calculated on the basis of the theory of Marcus, assuming that the reorganization energies were equal for the surface and heterogeneous processes. More rigorous calculations made by Mohilner [29] yield a value of about 2 x 109 cm -~ for the ratio ks /k h [15]. Another approach, based on a concentration effect (a volume concentration corresponding to the layer of adsorbed molecules is calculated, and it is assumed that the rate constant is k h) yields for the ratio about 2 x 10 7 cm-1 [15]. Experimental values of k S, obtained recently by indirect methods [21,24-26], do indeed have the order of magnitude predicted by these theories. The ratio ks /k h should in any case be very large, and the differences mentioned above do not change the theory and the equations derived below; in particular, the limits between cases II and III do not depend

114

75

5

ff 2s

A

E. Laviron /Journal of Electroanalytical Chemistry 382 (1995) 111-127

H (case I1 )

j T

~ S (case II1 )

/D C

-25 p 0.01 0.5 0.99

-15 -10 -s 0 5 ~ 10 15

0.5 log (bo b ~ ,'cm ~ tool -I ) \0 .5 log (bob ~ )i

'0.5 log (b~ b R )/

Fig. 3. Variations of logM with log(bobR) 1/2 (Eq. 3; see text for the complete discussion) for ks/kh=6lO 8 cm 1, D=410 6 cm 2 S- 1, Fm= 5 10 -I mol cm -2 ' r = 1 and X = 0.5: ( ) CT = 10-7 mol cm -3, ( - - - - - ) c T = 10 -6 mol cm 3. The limit between zones II and III was calculated for LSV (Eq. (76)) with c = 0.05 V s- ~.

on it. We shall use Eq. (8) in what follows when the ratio k Jk h is needed.

2.4. The variations of M with (bobR)1/2

An example of curve logM vs. log(bobR) 1/2 is shown in Fig. 3 for two values of c v and for r = 1. All the curves are similar; only the height of the plateau on the right differs from one curve to the other when ca-, r or a changes (see below). They are made up of three branches A, B and C. Branch A is obtained when b (or bob R) becomes small enough (no adsorption, case I of Fig. 2).

When b (or bob R) increases, two situations can arise. The term r2~(] -- X) + r -Z (1 -~)X in Eq. (10) varies monotonically from r 2a to r -20 -a ) when X varies from 0 to 1, i.e. when the potential varies from +~ to -~ . We can thus find conditions (branch B) such that, whatever the potential,

b lr'~ 1/2 >> r2~(1- X) +r 2(1 a)X (16)

Eq. (10) then reduces to

M = m = 1 + o'br 1/2-~

= 1 + (ks /kh)Fm(bobR) l /2 r 1/2-~ (17)

The reversibility coefficient m then depends neither on X (which means that it is independent of the potential E), nor on the analytical concentration c T. The constant khM, now written

khm = mk h (18)

is thus a true apparent constant, since its value does not vary with E or c m.

In a large range of log(bobR) 1/2 values, logm varies linearly (Fig. 3). The equation of the straight line D is

found easily by noticing that 1 in Eq. (17) then becomes negligible, so that:

logm = log(ks/kh) + logF m + log(bobR) L/2

+ (1 /2 - a ) logr (19)

When, in contrast, we always have

b l r " - ' /2

E. Laviron / Journal of Electroanalytical Chemistry 382 (1995) 111-127 115

.---, % E

2L.

.g / log C T increases "-

by 1 unit

I I I I I -2 -I 0 I 2

log b---9-- b R

Fig. 4. Limiting value log(bobR)~/2 for which Eqs. (17)-(19) are valid, as a function of log(b o /bR) . D = 4 X 10-6 cm 2 s-1, /'in = 5 X 10 10 mol cm-2: ( ) a=0.5 ; ( - - - - - - ) a=0.75; ( . . . . . ) a = 0.25. The limits are shifted downward by one unit when logc T increases by one unit.

aromatic compounds this ratio is usually not very different from unity, which is probably due to the fact that the structure of the molecule, and in particular its aromaticity, does not change very much when it is reduced or oxidized. A large difference between b o and b R should cause an adsorption wave or peak to appear separately from the diffusion wave or peak [27,30] when the reaction is reversible, since in the case of a Langmuir isotherm, the standard surface potential E ' is related to the "normal" standard potential E by [11,31,32] (cf. Eq. 8)

E '= E - (2 .3RT/nF) log(bo /bR) (26)

Pre-waves have indeed often been observed [11], but they are due to the influence of the interactions between the adsorbed molecules [11,33] rather than to a large value of bo/b R. Ratios found experimentally for redox pairs on mercury are: methylene b lue / leucomethylene blue, 0.1 [33]; riboflavine and rosindu- line GG/ reduced form, 1 [33,34]; nitrosobenzene/ phenylhydroxylamine, 0.14 [35]; nitro group/dihy- droxylamino group for 4-nitropyridine [24], 4-nitropyridine-N-oxide [25] and 4-nitroacetophenone [26], = 1. Exceptions can be found, however; for example, when the reduction occurs in the vicinity of the limits of the adsorption domain [36].

2.5. The reaction path

The reaction path is defined as the percentage of the reaction going via the surface path, i.e. as the ratio of the surface current i a to the total current i = i a q- i h. It is given by [15]

or = (27) P '+b- lrc~-l /2+r2~( 1 -X ) +r-2(1-a)X

From Eqs. (10) and (27) we obtain

p = 1 -M- ' (28)

The reaction path becomes independent of E and c T in the region, defined above, where M = rn (Eq. 17). The values of log(bobR )1/2 for which p is equal to 0.01, 0.5 and 0.99 when, for example, r -- 1 are respectively - 2.52, - 0.52 and + 1.47; they are shown in Fig. 3.

3. Heterogeneous vs. surface process in polarography

All the equations given above can be applied directly in polarography in case II of Fig. 2, since the general process (diffusion-reduction-diffusion) is the same.

However, polarography can also be used in case III (strong adsorption, slow desorption of R), because of its non-steady-state nature. The molecules of O diffuse towards the electrode, they are adsorbed and reduced; the molecules of R remain attached to the surface (Fig. 5); this is true whatever the isotherm, since the only condition is that the concentrations co(O,t) and CR(O,t) , which are in equilibrium with the surface concentrations, remain much smaller than the analytical concen-

Oads

III

k S

R f ads Fig. 5. Reaction scheme for strong adsorption in polarography (cf. case III of Fig. 2): (~) diffusion; ( -~) electrochemical reaction (reduction).

116 E. Laviron /Journal of Electroanalytical Chemistry 382 (1995) 111-127

tration ca-, so that the transport of the molecules of O is diffusion-controlled. This process can go on as long as the electrode is not completely covered by a film of O and R, or R. The time t m needed to reach full coverage is given by Koryta's equation [27,37]

t m = 1.82F2/Dc 2 (29)

The theory for this type of situation has been published previously for a reversible reaction [27,32] and in the general case of any degree of reversibility [38].

It is thus of interest to define the conditions for which the reaction practically appears as a purely heterogeneous or as a purely surface process.

3.1. Reversible reaction

The problem, which can be studied by solving the diffusion equations

aC 0 a2Co O - - (30)

at ax 2

aC R 02CR - - - D - - (31) at ax 2

for a planar, immobile electrode, is similar to the problem of the adsorption of an electroinactive sub- stance, which has been treated by Delahay and Tracht- enberg [39], who showed that the results can practically be applied directly to the dropping mercury electrode.

We shall consider the case of a reduction; transposi- tion to the oxidation is obvious.

The initial conditions

Co(X,0) = c v (32)

CR(X,0) = 0 (33)

express that there is no adsorption at time t = 0; the adsorption process will take place as soon as the elec- trolysis begins, as is the case in polarography, since the drop does not exist at t = 0.

The boundary conditions are

--~-x )x=odt+Dfo(~x )x=odt (34) aC o

(no adsorption at time t = 0)

CR(0, t )C (0 ' t ) [ nF _ Eo)] 0 = exp ~--~(E = (35)

F o = boFmco(O,t ) = Koco(O,t ) (36) F R = bRFmCR(O,t ) = KRCR(O,t) (37)

We use here a linear isotherm, which will not limit the validity of our conclusions; we have indeed shown previously [11,33,40,41] that the experiments must be conducted at low coverages, when the Langmuir or the

Frumkin isotherms become linear, in order to avoid the influence of the interactions between adsorbed molecules.

We have also:

Co(O~,t ) -~ c~ (38)

CR(O%t ) ~ 0 (39) The current i is given by

i dF +D[ ac(x't) ] nFA d t ax x = o

dFR [ acR(x't) ] (40) - d t D ax x=0

The problem can easily be solved by using the Laplace transformation [39,42]. We obtain for the concentrations near the electrode

Co(0,t ) = CT0(1 + 0)-111 - - exp(A2)erfc(A)] (41)

cR(0,t ) = CT(1 + 0)-111 -- exp(A2)erfc(h)] (42)

and we get for the dimensionless fluxes (flux divided by CTD1/2~ "- 1/2t- 1/2) at the surface

q~o = Y- 1D = 1 (43) x=0 1+0

(ac R ] F(A) (44) q~R=y-ID~-~X ]x= o 1+0

in which

F(A) = 7rl/2h2exp(A2)erfc(A) (45)

O1/2tl /2(1 + O) O1/2t1/2(1 + O)

A = KR(1 + KoO/KR ) = bRFm(1 + r20) (46)

cTD1/2 y = ~1/2tl /2 (47)

We can also easily derive from Eqs. (36),(37) and (41),(42) the equivalent fluxes corresponding to the variations of F o and F R

dro r20[1 - F(A)] = y-1 - - -- (48)

~0Oeq dt 1 + r20

q~R~q =y- i dF a _ 1 -F (A) dt 1 + r20 (49)

In view of Eqs. (40), (43), (44), (48) and (49), the dimensionless current I is given by

7rl/2tl/2i 1 -F (A) F(A)

I = nFAD1/2c T 1 + r20 + 1 + 0 (50)

This is the general equation of the wave for any degree of adsorption.


For strong adsorption, if b o is not too different from bR, )t ----) 0, F(M ~ 0, and

1 1 I ~ - - (51)

1+r20 1+~7

Eq. (51) has been derived earlier [27,32]. For weak adsorption, h ~ 0% F(A) ~ 1, and

1 I - , - - (52)

1+0

which is indeed the equation of a normal "heterogeneous" polarogram [43].

We can use the equations derived above to express that, as soon as they are formed, the molecules of R leave the electrode (weak adsorption, cases I or II, Fig. 2), or remain adsorbed (strong adsorption, case III, Fig. 5). Two criteria can be used.

(1) We can directly compare the flux of R to that of O, i.e. form the quantity

= 1 + 0 - OF()t) (53)

(2) We can compare the amount of R which remains adsorbed to the total amount which is formed

Req 1 x = (54)

~Req -~- ~0R (1 + r20)F(h) 1+

(1 + 0)[1 -F ( ) t ) ]

For weak adsorption, F()t) --) 1; then x' ~ 1, so that the flux of R equals the flux of O (no molecule of R remains adsorbed), and x --) 0 (~Req


v

I -2 8

0.01 q

I 5 6 7

/

/ S

0.5 log [( b o b a )I/2 / cm 3 mol-I

I I I i I

5 6 7 8 9

log(b R / m 3 mol-l)for b o / b R = 0. I

Fig. 7. Polarographic adsorption d iagram for a reversible reduction (x = 0.5) for diverse values of bo/b R (shown on each curve). The abscissae can be chosen as log(bobR) 1/2, which is valid for all the curves, or Iogba, which must be def ined for each curve (an example is given for bo/b R = 0.1).

We can thus calculate F(A) for given values of r 2, I and x, and get the corresponding value of A from tables. Then we calculate 0

F(A) 0 1 (61)

I(1 -x )

and deduce successively b R from Eq. (46) and b o from the ratio r 2 ----- bo/bg. The calculations were carried out for D = 4 10 -6 cm 2 S -1, /'m ---~ 5 X 10 -1 mo1 cm -2 and 0.01 < r 2 < 100.

The results show that, for a given value of x, the limit does not change much along the wave (0.2 log(bobR) 1/z units when one passes from Ej/4 to E3/4), which means that it will practically still be a straight line. We have shown in Fig. 7 the limits obtained for x = 0.5 and I = 0.5 (E1/2). They are shifted to the right when r > 1, to the left when r < 1, by about 0.5 log units for a tenfold increase or decrease of r. These graphs are valid for reduction; for oxidation, the values of b o and b R must be exchanged.

3.2. Totally irreversible reaction

Resolution of the diffusion equations is very complex in this case. Therefore we have used another method, based on the comparison of the half-wave potentials, for large values of z, when the reaction will tend to be heterogeneous, since there is more time for the reduced molecules to desorb, with those for smaller values, when the reaction will be of a surface nature.

For a totally irreversible reduction, the half-wave potential for a heterogeneous reaction is given by [45]

2.3RT E1 /2 -E - - - l ogO.886(khmz l /2D-1 /2) (62)

anF

in which khm is the experimental constant which is actually measured (cf. Eq. 17).

For surface process we have, if the adsorption obeys a Langmuir isotherm [38],

2.3RT El~ 2 - E ' log( 1.197ksz ) (63)

anF

or, in view of Eq. (26)

2.3RT r _~] El~z-E- ~nF lg[l'197ksr(b/bR) (64)

The transition (x = 0.5) between the heterogeneous (for large values of z) and the surface reactions (for small values of r) takes place when Eqs. (62) and (64) are equal, which gives

logz = - 2log(ks~kin) - 1og(1.830)

+ 2 a log( bo/b R ) (65)

or, in view of Eqs. (18) and (19)

logr =A + (0.5 + a) log(bo/bR)

+ 21og(bo/b R)1/2 (66)

with

A = 21og(0.74FmD-1/2) (67)

For oxidation, we have

log r = A - ( 1.5 - a) log(b o/b R) + 2log( b o /b R ) 1/2 (68)

Eqs. (66) and (67) can also be written

log~" =A + (a - 0.5) log(bo/bR) + 21ogb~ (69)

with b i = b o for a reduction, and b i =b R for an oxidation.

4. Heterogeneous vs. surface process in cyclic voltammetry

Cyclic voltammetry can be applied directly to the study of case II of Fig. 2, using the classical theories [43,46] for a heterogeneous process whose rate constant is khm (cf. Section 2).

Moreover, because of its non-steady nature, it can, like polarography, be used to study the surface reaction (cf. case III, Fig. 2), which is not possible on the r.d.e. On a pseudo-stationary Hg electrode, the potential scan is started at a time t I (delay time) after the formation of the drop; during this time, a certain amount of O has adsorbed at the surface. When the sweep rate is large enough, the current due to the


[

R~ ads Fig. 8. Reaction scheme for a strong adsorption in linear potential sweep voltammetry (cf. case III of Fig. 2). (~) diffusion; (-I~) electrochemical reaction (reduction).

adsorbed molecules will be much larger than that due to the molecules brought by diffusion to the electrode. In other words, the number of molecules of O diffusing towards the electrode, or of R diffusing away from the electrode, will be much smaller than that of the adsorbed molecules (Fig. 8). The reaction will be of a purely surface nature; the theory for this case has been published previously [11,27,32,47]. Solutions having for- mally the same mathematical expressions have also been obtained for the case when only one of the two forms O and R is adsorbed, the other being present in the solution in high concentration [13,14].

It is thus of interest to determine the conditions in which the reaction appears as a heterogeneous (weak adsorption) or as a surface process (strong adsorption).

4.1. Reversible reaction

A rigorous analytical solution cannot be obtained. We shall assume that the "heterogeneous" and the surface currents are independent; at slow sweep rates only the heterogeneous current will indeed be observed, whereas at high sweep rates only the surface current will be seen.

For a heterogeneous process, the peak current is given by [43,46]

iph = 0.45( hE)3 /2(RT) -I/2ACTD1/2 u 1/2 (70)

In this equation, c T is in principle the analytical concentration (its meaning will be discussed later), and v is the sweep rate.

For a surface reaction when the adsorption obeys a Langmuir or a Henry isotherm [11,27]

ips = 0.25(nF)2(RT) -1AFT v (71)

where F T is the surface concentration at the start of the sweep.

Let

ips - - = z (72)

iph

We shall designate by u z the corresponding sweep rate.

We get

L' z = 3 .24(RT/nF) DcZF~r2z 2 (73)

In Eq. (71), F T is the surface concentration Fo(t t) of O at the start of the potential scan, i.e. at time t] after the beginning of the drop formation (or more generally the beginning of the adsorption process).

Since the adsorption is at equilibrium, the concentration of O near the surface in the case of a Henry isotherm is

Fo( t 1) = boFmco( O,t 1) (74)

At first sight, we can thus equate c T and F T in Eqs. (70) and (71) to Co(0,t 1) and F(t]) respectively. Strictly, however, Eq. (70) should not be used, since its deriva- tion [43,46] implies that the solution is homogeneous at the start of the sweep, whereas it is not, because of the adsorption [39].

However, (a) for a weak adsorption, co(0,t l) will have returned to a value not too different from c T at time t t [39], and Eq. (70) will be approximately valid at slow sweep rates, whereas at high sweep rates, Eq. (71) will be applicable, since, as explained above, only the adsorbed molecules are reduced, and (b) for a strong adsorption, the current due to the adsorbed molecules will predominate, even at slow sweep rates.

If we substitute, in Eq. (73), c T by co(O,t 0 and F T = Fo(t 1) by their values taken from Eq. (74), we obtain

logcz = B + 21ogz - 21ogb o (75)

or, since b o = (bo/bR)l /Z(bobu) 1/2

logv z = B + 21ogz - 21ogr - 21og(bobR) 1/2 (76)

For oxidation, we get

logv z = B + 21ogz - 21ogb R (77)

or

logv z = B + 21ogz + 21ogr - 21og(bobR) ]/2 (78)


I I 2 3 4 5 7

log[( b o b R )1/2 / cm 3 tool-']= log[b o / cm 3 mol-'] = Iog[b R / cm 3 mol"]

Fig. 9. LSV adsorption diagram for a 2e reversible reduction or oxidation for b o = b R (Eqs. (76)-(79)); the value of z is shown on each curve.

In these equations

B = 0.51 + l og(DRT/nF) - 21ogF m (79)

Transit ion from the weak to the strong adsorpt ion region will be obtained for z = 1. The intermediate region corresponds to couples of values of z such as 0.1 and 10, 0.05 and 20, etc. Adsorpt ion diagrams are shown in Figs. 9 and 10 for b o = b R and b o = 0.1b R.

The transit ion for z= l and v=0.050 V s -1 is shown in Fig. 3.

4.2. Totally irreversible reaction

In this case, two methods can be used to establish the adsorpt ion diagram; comparison of the peak heights, or intersection of the asymptotes. The limits

6 i _~~_2 ..........

2 3 4 5 6 7 log[( b o b R )1/2 / cm 3 mol-l]( reduction or oxidation

i i i i i i i i 2 3 4 5 6 7 8

log bR ( oxidation I I I I I I I I

6 7 I 2 3 4 5 log bo (reduction

F ig. 10. LSV adsorpt ion d iagram for a 2e revers ib le react ion (Eqs.

(76) - (79) ) for b o = 0.1 and z = 1.

obtained are somewhat different. This type of result is not unexpected; it is found, for example, in the transition between reversible and kinetic domains of kinetic diagrams [48,49], where the limit depends on whether peak heights are compared, or whether intersect ion of asymptotes is considered. In the present case, compari - son of the peak heights is useful to evaluate the intermediate region, but it is preferable to use the intersection of the asymptotes to establish the adsorpt ion diagrams, since the measurements of the constants and the diagnosis of the mechanisms are based on the study of the asymptotes [23,24-26,47].

4.2.1. Comparison of the peak currents For a totally irreversible heterogeneous process, the

peak current for a reduct ion is given by [43,46]

ioh = 0.496(nF)3/Z( RT) - l /eal/ZAcvD1/Zvl/2 (80)

For a surface reduct ion when the adsorpt ion obeys a Langmuir isotherm [27,47]

ips = e- I (nF)Z(RT) - laAvFT (81)

Proceeding as in the reversible case, we obtain

logv z = c - l ogan - log(bo /bR)

- 2 log(bobR) t/2

= c - l ogan - 21ogb o (82)

for reduction, and

logv z = c - log(1 - a)n + log(bo/bR)

- 2log( bob R) 1/2

= c - log(1 - a)n - 21ogb R (83)

for oxidation, with

c = 0.26 + 21ogz + l og (DRT/F ) - 21ogF m (84)

4.2.2. Intersection of the asymptotes For a total ly irreversible heterogeneous reduction,

the cathodic peak potent ia l can be expressed under the form [46]

2.3RT 1.23(rrDanFv/RT) 1/2 Epc - E - - - log (85)

anF khm

For a surface reduct ion when the adsorpt ion obeys a Langmuir isotherm [47]

2.3RT anFv Epc - E ' - - l og - - (86)

anF RTk s

In view of Eq. (26), we can write

2.3RT anFv(bo) ~ Epc - E - (87) anF lg R-----~s


> E

L~A

k~d v

200 H I / ~16"" / ;8

100 ~ i G /

E o, i 2 log ~.s

E ~ log ~.

k~2 -100

Fig. 11. Examples of theoretical variations of gpa and Ep c as a function of logv for a heterogeneous reaction () [37] and for a surface reaction (o) [38] at 20C, for n = 2, a = 0.75, bo/b R = 0.4, when the uptake of the second electron is rate-controlling [23]. The slopes indicated on the asymptotes are expressed in mV per logv unit; 3. = FDV/RTk2m, A s = nFv/RTk s.

Intersection of the two asymptotes defined by Eqs. (85) and (87) (Fig. 11) gives

logv z = 0.68 + log(DRT/F) - l ogan

+ 2log( ks/khm )

- 2a log(bo /bR) (88)

or, in view of Eqs. (18) and (19)

logv z = G - logan - (0.5 + a)iog(bo/bR)

- 2log( b o b R) I /2 (89)

which can also be written

logv z = G - logan + (0.5 -a ) log(bo /bR)

- 21ogb o (90)

For oxidation we have

logv z = G - log( l - a)n + (1.5 - a)log(bo/bR)

- 21og(bobR) 1/2 (91)

or

logv z = G - log( l - a)n + (0.5 - a)log(bo/bR)

- 2logb R (92)

In these equations

G = 0.68 + log(DgT/F ) - 21ogF m (93)

We have shown in Figs. 12 and 13 examples of adsorption diagrams (by definition, z = 1 in the present case).

4.3. Reversibility / irreversibility (R-IR) diagrams

Eqs. (85) and (86) allow us to define the domains of reversibi l i ty/ irreversibi l i ty in the plane logv/lOgkhm or logv / logk S. We shall consider that the transition

5

4 l 2 H

--- : - -2 - - 0

"~ *lR I \ \ o-" X V/ *R }tR

a I K -2 a: ~ L ---~

-3 -4 I I I I I i i I

3 4 5 6 7 8 9 I {I

log (b o bR)l/2 / crn 3 mol i

Fig. 12. LSV adsorption diagram for a 2e totally irreversible reaction for b o = bR, c~ = 0.75 and z = 1 (Eqs. (89)-(192)). For the explanation of the R / IR zones, see Section 4.4. P: equivalent limit for polarography.

from reversible to irreversible behavior occurs when the horizontal asymptote (E = Epc for a reversible reduction) intersects the oblique asymptote defined by Eqs. (85) or (86).

For a heterogeneous reduction when the reaction is reversible [43], Epc=E - 1.1(RT/nF)(cf. Fig. 11). The intersection with the oblique asymptote (Eq. 85) is thus given by

logv = 0.963 - 21og(1.237r 1/2)

DF - log-~-~ - logan + 21Ogkhm (94)

The R- IR diagram is shown in Fig. 14 (in order to facilitate comparison with polarography, lognv is studied instead of logv).

For a surface reduction, the intersection occurs when Eoc = E ' [11,27,32,47], whence, from Eq. (86)

RT logv = log--if- - logan + logk S (95)

The R- IR diagram is shown in Fig. 15. Since the transition from the horizontal to the oblique asymptote

5

3

S 2

_ ~lR IN N. . . . . ~C _ _ 'k l l CR

' I I L -4 I t 4 5 6 8 9 10

log [ (b o bR) t'2 / cm 3 mol - i ]

Fig. 13. LSV adsorption diagram for a 2e totally irreversible reaction for b o = 0.1bR, o~ = 0.75 and z = 1 (Eqs. (89)-(92).


5

-~" 4

3

= 2

"7

; 0

-| e.,-

--~ -2

-3

-4 -4

LSV /

R

P

I I I I I

-3 -2 ~ 1 0 1

log(kh /cms q)

Fig. 14. The R / IR diagram log(m,)= f(logk h) for heterogeneous reduction at 20C. The value of a is indicated on the curves. The limits are the same for oxidation For LSV, the difference between the curve for a = 0.5 and that for a = 0.25 or 0.75 is only 0.06 lognV units P: equivalent limit for polarography.

oxidation, a must be changed to 1 - a in Eqs. (94) and (95), which yields the same limit (logv = -0.75) for the heterogeneous reaction and logv = -0.20 for the surface reaction. We have also represented on the right of the graph the cathodic and anodic asymptotes for Ep c and EPa [23] (cf. Fig. 11). For the reduction, for example, when the figurative point is below C in Fig. 12, the reaction is heterogeneous. At slow sweep rates (below B) it is reversible; when v increases, it becomes irreversible at B. At C, a direct transition to the irreversible surface reaction occurs.

Another example is given in Fig. 13 for the same conditions, except that bo/bR=0.1. We have now logm =5.60; for khm =-2 .2 , the R - IR limit in the heterogeneous region is still obtained for logv = -0.75, but the limits in the surface region are now respectively logv =-0 .80 and -0 .32 for the reduction and the oxidation.

5. Discussion

An LSV sweep rate Veq equivalent to the polarographic drop time can be defined [51] by the equation

is gradual, an intermediate zone which is not shown in Figs. 14 and 15 can be defined, based on the difference (e.g. 2 mV) between the curve and its asymptotes (see, for example, Ref. [50] for a heterogeneous process, and Ref. [47] for a surface reaction).

In the case of oxidation, a should be changed to 1 -a in Eqs. (94) and (95).

RT Ueq nFr (96)

We shall therefore discuss the results for polarography and cyclic voltammetry together.

4.4. Reversibility /irrecersibility in the adsorption diagrams

Domains of reversibility and irreversibility can be defined in the adsorption diagrams (Figs. 12 and 13). However, the R - IR limits in the weak and strong adsorption regions are not independent; their relation- ship depends on the value of log(bobR) 1/2.

Let us consider, for example, a reduction for which log(bobR )1/2 = 6, bo/b R = 1, a = 0.75 and n = 2. From Eq. (19) (cf. Fig. 3), assuming that F m = 5 10- t0 mol cm -2, we find logm = 5.48.

Let us now choose for 1Ogkhm the value -2.2. We can calculate, from Eq. (94), with D = 4 10 -6 cm 2 S-1 (c f . also Fig. 11) that the R - IR limit is obtained for logv = -0.75 (Fig. 12). Using Eq. (18), with logm = 5.48 and IOgkhm = --2.2, we obtain logk h = -7.68. Eq. (19) then gives logk s = 1.1; for this value of logk s, Eq. (86) (cf. also Fig. 15) gives for the R - IR transition logv = 0.68. The results are shown in Fig. 12. For the

4 -7

; 3

2

t

; 0

-t

_o -2

-3

-4 I -4 -3

0.5

IR / R

I I I l I I I l l I

-2 -1 0 1 2 3 4 5 6 7

log(ks / s q)

Fig. 15. R / IR diagram log(nv)= f(logk h) for a surface reduction at 20C. The value of a is indicated on the curves. For an oxidation, the limits for a = 0.5 and 0.75 must be exchanged. P: equivalent limit for polarography.


5.1. Range of experimentally measurable values of khm and k~

A priori, apart from solid electrodes, either a pseudo-stationary dropping mercury electrode or a stationary (hanging drop or coated mercury electrode) can be used in cyclic voltammetry.

The dropping electrode presents several advantages: reproducibility, clean surface, possibility of carrying out rapidly a large number of measurements. It also has a distinct advantage over the stationary electrode, in that the time at which the adsorption begins (i.e. the birth of the drop) is well defined, which ensures that the conditions at the surface are well known and repro- ducible. This allows, in particular, experiments to be carried out using any concentration in solution, since one does not have to wait for the equilibrium between the surface concentration of the adsorbed species and the analytical concentration to be established.

By contrast, stationary electrodes suffer from several drawbacks. They are less easy to reproduce, they do not permit many measurements to be made in a short time, and, more particularly, the time at which the adsorption begins is not well defined. In these conditions, one should wait until the adsorption equilibrium between the surface concentration and the analytical concentration is established; meanwhile, adsorption of impurities can take place. Moreover, platinum, gold or silver mercury-plated electrodes are unsatisfactory, in particular because of the dissolution of the substrate in the mercury [52-54 and references therein], whereas mercury deposits on carbon lack homogeneity because of their poor adherence to the surface [53]; iridium- based electrodes seem more promising [52-54].

Although mercury electrodes are much easier to use because of the reasons enumerated above, work on well-defined solid electrodes in clean solutions is a priori possible. However, if chemisorption occurs (cf. Introduction), the theories developed in the present paper for weak adsorption (case II) may not be applicable. It may nevertheless be interesting to use such electrodes to study case III.

In polarography, drop times of 0.1 to 20 s can be envisaged. In cyclic voltammetry, the area of the electrode must be as small as possible in order to minimize the ohmic drop effects, i.e. a capillary with a small rate flow must be used, and the delay time t~ must be as short as possible. The diameter of the capillary cannot be decreased too much, because the high back-pres- sure which then appears must be surmounted, e.g. by heating the mercury [55], which introduces experimental difficulties. In practice, a mercury flow of about 0.15 mg s -~ and a delay time of about 10 ms can be used [23-26,35]. Under these conditions, the voltam- mograms for small concentrations are free from ohmic drop effects for l~ < 2000 V s-1 [23-26,35].

In view of the above considerations, we can define the range of experimentally measurable values of khm and k~ (Figs. 14 and 15), taking into account the fact that part of the oblique asymptote (for fast sweep rates) or of the horizontal asymptote (for slow sweep rates) must be obtained; we have allowed for that about one logv unit. We find that measurable values are 10 -4 to 0.45 cm s -~ for khm (a comparable upper limit has already been defined for heterogeneous constants determined by cyclic voltammetry with the same upper limit for the scan rate [50]), and 2.5 s ~ to 5 )< 103 s - I for k~. If E or E ' values can be known, for example by extrapolation [24-26], much smaller rate constants can be determined (e.g. as small as about 10 -6 cm s -1 for a heterogeneous process; see Refs. [24,25]).

According to Eq. (15), surface rate constants should be of the order of 10~-109 s -1 since k h values are usually in the range 0.1 to 10 cm s- 1 [24-26], which has been confirmed experimentally by using indirect methods [21,24-26,35].

On the other hand, even for weak adsorption, khm values are in the 104-106 cm s- l range (cf. Eq. 17). As can be deduced from Figs. 14 and 15, such values are quite out of reach of LSV; it is even doubtful whether they can ever be measured by direct electrochemical methods. However, in aqueous medium, the reduction of organic compounds is often made up of a succession of electron and proton uptakes, which can be described by using square schemes or their derivatives (fence, ladder, cubic schemes, etc.) [57-60]. When the proto- nations are fast, i.e. can be considered to be at equilibrium, which has indeed been shown to hold when nitrogen or oxygen atoms are involved [21,24-26,61], the theory shows that the global process is equivalent to a simple reaction, with an apparent electrochemical rate constant which becomes much smaller than the constants k~ or khm for the monoelectronic elementary reactions, so that its value falls in the measurable range. We have investigated diverse reactions using these properties [21,24-26,35].

When b o 4: bR, it must be noticed that the results for a reversible reduction in polarography (Fig. 7) are the reverse of those for an irreversible reaction (Eq. (66)), or for LSV (Eqs. (66), (68), (75), (77), (82), (83), (89), (91)); for bo/b R = 0.1, for example, in the case of a reduction, the limit between the zones of weak and strong adsorption is shifted towards larger log~" (or smaller log(bobR) 1/2) values (Fig. 7), whereas the reverse is true in the other cases. This is a result of the different nature of the process. For a reversible reduction in polarography, a decrease of b o, i.e. an increase of b R for a given value of bob R, results in an increase of the adsorption of R; less molecules leave the electrode. The adsorption zone expands. For a totally irreversible reduction in polarography, in contrast, a


4

3

oo > l H S~ 0 ez0

-I

-2 p

-3 6 9 7

-4 I I I I 4 5 6 7

log[( b 0 b R )12/cm 3 mol.i]

Fig. 16. Limits of the adsorption diagram (x = 0.5 or z = 1) obtained from the diverse methods for bo/bn=l, D=410 -6 cm 2 s - l , / 'm=510 - l tool cm 2: curve (1), Eq. (82), Red, irreversible, n = 2, a = 0.75, and Eq. (83), Ox, irreversible, n = 2, a = 0.25; curve (2), Eqs. (76) and (78), Red, Ox, reversible, n = 2; curve (3), Eq. (89), Red, irreversible, n = 2, a = 0.75, and Eq. (91), Ox, irreversible, n = 2, a = 0.25, and Eqs. (76) and (78), Red, Ox, n = 1; curve (4), Eq. (82), Red, irreversible, n = 2, a = 0.25, and Eq. (83), Ox, irreversible, n = 2, a = 0.75, and Eqs. (82) and (83), Red, Ox, n = 1, a = 0.5; curve (5), Eq. (89), Red, irreversible, n = 2, a = 0.25, and Eq. (91), Ox, irreversible, n = 2, a = 0.75, and Eqs. (89) and (91), Red, Ox, irreversible, n = 1, a = 0.5; curve (6), Eq. (59), Red, Ox, reversible, n = 1; curve (7), Eq. (59), Red, Ox, reversible, n = 2; curve (8), Eqs. (66) and (68), Red, Ox, irreversible, n = 1, a = any value; curve (9), Eqs. (66) and (68), Red, Ox, irreversible, n = 2, a = any value.

decrease of b o d imin ishes the surface nature of the process, since R no longer plays a role in the react ion. In LSV (reversible or i rreversible case) a decrease of b o also d imin ishes the surface nature of the reduct ion, since the quant i ty of molecules adsorbed before the start of the sweep decreases.

5.2. Coherence of the results

At first sight, the diverse methods used for deter- min ing the heterogeneous and surface react ion zones (solut ion of the dif fusion equat ions, intersect ion of the asymptotes, compar ison of the peak heights) seem to yield rather di f ferent results. However, for b o = b R, the l imits are not very di f ferent (Fig. 16). Even when b o 4:b R, the di f ferences remain small. We can, for example, calculate for a reduct ion the ratio ips/ioh when the asymptotes intersect by subst i tut ing the value of v der ived from Eq. (89) into Eqs. (80) and (81), taking in account Eq. (74). We obta in

ips/iph = 0.61( bo/bR ) '~/2-25 (97)

When bo/b R = 0.1, this rat io is thus 0.81, 0.61 and 0.46 for a = 0.25, 0.5 and 0.75 respectively; for bo/b R

= 0.01, we obta in 1.08, 0.61 and 0.34. For oxidation, the exponent is 0 .25- a/2 and the results are re- versed. The ratio of the peaks is not very di f ferent from unity in every case.

5.3. The width of the transition region

Accord ing to the theoret ical results, the in termedi - ate region between the purely heterogeneous and purely surface react ions extends over a large range of logz or logv (cf. Figs. 6 and 9). In practice, however, the t rans i t ion region can be much narrower. Let us, for example, cons ider Fig. 11, in which the di f ference between k s and khm is large enough that the curves intersect in the region of the obl ique asymptotes. A t the intersect ion, by def init ion, the peak potent ia ls for the surface and heterogeneous redox processes are equal. When v is e i ther increased or decreased, the peak potent ia l is determined by one of the two reactions, and the var iat ions closely follow the asymptotes, as is conf i rmed exper imental ly [25,26,35]. There is no diff iculty in character iz ing the asymptotes, whose inter- sections with E or E ' give khm and k s respectively (21,24-26) ( in the present case, the constants are rela- tive to the second e lectron uptake; cf. Fig. 11 and Ref. [23]). If the di f ference between k s and khm is too small, the s i tuat ion becomes more complex (Fig. 17), because the intersect ion of the curves can occur out of the asymptotic region. In Fig. 17, for example, the anodic asymptote for the heterogeneous process (slope, 58 mV) cannot be determined exper imental ly , since the curves for the heterogeneous and surface process coin- cide in this region; even the cathodic asymptote (slope, -38 mV) will be diff icult to character ize. A more careful study, using the whole curve, must be carr ied out [35,61,62].

> E

Z v

200

100

E o, 0 >

E

k~d

K ~ -I00

/6 II i# .4 ##, C/ ##

HiS

~ ~ 2 / I

o

1 / #/

11158 /

3 log k s I I I 2 3 log k

I

I -39~"- ,~ -19 ~

Fig. 17. Example of theoretical variations of EPa and Ep c as a function of logv for a heterogeneous reaction () [37] and for a surface reaction (o) [38] at 20C, for n = 2, a = 0.75, bo/b n = 0.2, when the uptake of the second electron is rate-controlling [23]. The slopes indicated on the asymptotes are expressed in mV per logv unit; h = FDV/RTk~m, A s = nFv/RTk~.


5.4. Definition of an average time of stay of the molecules on the surface

Let us cons ider Eq. (55); when x = 0.5, i.e. when half of the molecu les of R leave the electrode, and half remain adsorbed, a o = 0.43. Let us des ignate by t~ the t ime def ined by

tsl/2 - - - - 0.43bR Fm D- 1/2 (98)

or

t~ = O.lSb2RF2D -t (99)

We can cons ider t~ as an average t ime of stay of the molecules on the electrode. If t >> t~, there is enough t ime for the molecu les to desorb, so that the react ion appears as homogeneous (Fig. 2, case II; Figs. 6 and 7). When t ~l I

L iI 101/ |l I

II I II I

~ - - ~- - ' ,,

I I I J I 2 3 4 5

D

t 1 I

I l l

I I

i i I I I I | iI i 1

7 6 8 13 I 9 14 I

[ J LL I 7 8 9 I 0

Iog(bo ' cm ~ mol 1)( reduction ) or Iog(b R 'cm ~ mol l )( oxidation )

Fig. 18. log(bo/b R) as a function of logb o (for reduction) or logb R (for oxidation): lines (1)-(6), LSV, irreversible reaction, t, = 2000 V s 1, z = 1 (line (1), Red, a = 0.75, n = 2; line (2) Ox, a = 0.25, n = 2; line (3), Ox, a = 0.75, n = 2; line (4), Red, a = 0.25, n = 2; line (5), Ox, a=0.5, n=l and Red, a=0.5, n=l ) ; lines (6)-(9), LSV, irreversible reaction, c = 0.050 V s- l, z = 1 (line (6), Red, a = 0.75, n = 2; line (7), Ox, a = 0.25, n = 2; line (8), Red and Ox, a = 0.75, n = 2; line (9), Red, a = 0.25, n = 2); lines (10)-(12) LSV, reversible reaction, n = 2, e = 2000 V s- 1, Red or Ox (line (10), z = 0.05; line (11), z = 1; line (12), z = 20); lines (13)-(15) polarography, irreversible reaction, n = 1 or 2, Red or Ox, x = 1, z = 2.0 s (Eq, (61)) (line (13), a = 0.75; line (14), a = 0.25; line (15), a = 0.5). In region A, the reaction appears as heterogeneous whatever the sweep rate. In region B, the reaction always has a surface character, whatever c or r. In regions C and E, the reaction has a surface or heterogeneous character according to the sweep rate and the nature of reaction (reduction, oxidation, reversibility). In D, it can appear as a heterogeneous or surface reaction, whatever the type of reaction.

V s-1, for example, the react ion will always appear as heterogeneous in the cond i t ions of Fig. 12 if log(bobR) 1/2 < 4.5, etc. The prob lem can be studied by cons ider ing Eq. (69), in which z is equal to its maximum value, 20 s, or Eqs. (75), (77), (90) and (92), in which v takes ei ther its upper or its lower values, i.e.. 2000 V s -1 and 0.050 V s -1. Graphs giving log(bo/b R) as a funct ion of e i ther logb o (for reduct ion) or logb R (for oxidat ion) can thus be obta ined (Fig. 18). The solid l ines cor respond to x = 0.5 (polarography) or z = 1 (LSV), the dashed l ines on the left to z = 0.05 (5% of surface react ion for l ines 1 and 2), those on the right to x = 0.95 (95% of surface react ion) for l ines 13 and 14. As can be seen, the react ion will always appear as heterogeneous when logb o (for reduct ion) or logb R (for oxidat ion) is smal ler than 2.75, whereas it will always be of a surface nature for logb o or logb R larger than 9. Between these values, both types of react ions can appear , when v (or z) varies. The l imits indicated are valid with in the range of values of c and bo/b R def ined. Data for other condi t ions (e.g. if larger values of u are used on an i r id ium-based mercury electrode; cf. Sect ion 5.1) can easily be calculated.


6. Conclusion

The rigorous mathematical theory established for the rotating disk electrode [15,16] constitutes the nec- essary basis for understanding quantitatively the role of adsorption of the reactants during the redox processes. However, non-steady-state methods are much more advantageous to use for quantitative experimental studies, for several reasons. The first are of a theoretical nature; non-steady-state methods allow both the heterogeneous and the surface reactions to be studied, whereas only the heterogeneous one can be examined using steady-state methods such as r.d.e, voltammetry; also, if a mercury electrode is used, adsorption is generally of a physical nature, so that the theories developed in this paper are directly applicable. An- other reason, which is practical, is linked to the advantages of using a dropping mercury electrode: good definition of the time at which the experiments start, reproducibility, cleanliness, possibility of performing many experiments in a short time.

Application of the methods described in the present paper has allowed us to establish the detailed mechanism of the reduction of diverse compounds [21,24- 26,35,62] in aqueous medium, and in particular to confirm the order of magnitude of the rate constants k~ and khm predicted by theory. Generally speaking, our studies also show that adsorption phenomena, which had hitherto been considered as an obstacle to the study of mechanism in aqueous media, can on the contrary be quantitatively incorporated into the global analysis of the redox processes. We are currently study- ing other types of systems on the basis of the results developed in the present paper, and we are planning to examine non-aqueous media.

Lastly we think that our results can contribute to a better understanding of the influence of the adsorption of the reactants on solid electrodes in certain cases, if, for example, slow chemisorption does not complicate the reaction.

Acknowledgements

We would like to express our thanks to Mrs. Raveau-Fouquet and Mrs. Tilleul for their help in preparing the manuscript.

References

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