Linear sweep voltammetry: Kinetic control by charge transfer and/or secondary chemical reactions: I. Formal kinetics

Electroanalytical Chemistry and Interfacial Electrochemistry, 48 (1973) 113 145 113 ~ Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

LINEAR SWEEP VOLTAMMETRY: KINETIC CONTROL BY CHARGE TRANSFER AND/OR SECONDARY CHEMICAL REACTIONS

I. FORMAL KINETICS

L. NADJO and J. M. SAVI~ANT

Laboratoire d'J~lectrochimie de l'Universit~ de Paris VIL 2 place Jussieu, 75221 Paris C~dex 05 (France)

(Received 29th January 1973)

The variation of the peak potential in linear sweep voltammetry (1.s.v.) with experimental parameters such as the sweep rate and the initial concentration has been shown to be a valuable source of diagnostic criteria for the mechanistic analysis of systems involving follow-up chemical reactions. The experimental studies in this field have concerned mainly the discussion of dimerization mechanisms a-5 and of the e.c.e, vs . disproportionation problem 5'6 in the reduction of .organic compounds. In every case, the electron transfer at the electrode was assumed to be so rapid that the rate determining factor is the chemical reaction and/or the diffusion process. The corresponding formal kinetics relationships, ignoring the eventual role of the charge transfer kinetics a'7's, were therefore applied. The satisfactory fit of the experimental results with the predicted behaviour must tSe related to the observation that charge transfer to large organic molecules is fast owing to the small change in solvation 9.

However, even with fast charge transfers it may well be that an increase in the rate of the deactivation chemical reaction would hinder the reoxidation step at the electrode, so that an irreversible behaviour kinetically controlled by the reduction step would be observed.

It is the purpose of the present paper to discuss this problem and to state quantitatively the conditions to be fulfilled by the rate constants, the sweop rate and, for second order chemical reactions, the initial concentration, in order that the kinetic control be either by the chemical reaction, or the charge transfer or the diffusion process, or possess a mixed character. This problem is of particular relevance in the mechanistic analysis of organic electrochemical processes in organic solvents, where adsorption does not play in general an important role. Indeed, in this condition, the kinetic characterization of the chemical steps involved in the mechanism requires that the initial charge transfer is not the rate determining step.

The following reaction schemes will be successively considered: first order deactivation of the reduction product, consecutive dimerization, e.c.e, and disproportionation mechanisms. The discussion is developed for the case of reduction. Transposition to oxidation is immediate.

The conclusions drawn in this paper will be illustrated in the second paper of this series by the reduction of some carbonyl compounds in alcoholic media.

114 L. NADJO, J. M. SAVI~ANT

Recently, a discussion of the same problem in the case of a first order deactivation and of a dimerization has appeared 1°. A simulation method was used that consists 11 of an explicit finite difference resolution with direct discretization of time and space. The main advantages of such a method are: (i) con- vergence seems generally good, (ii) the relationship between the physical situation and the computation is simple and direct--the computation simulates the physical problem. So, cumbersome mathematical formulations and analysis are avoided. In fact, the second feature is rather a subjective one. The counterpart of avoiding mathematical analysis is that particular solutions corresponding to limiting values of the parameters which feature particular physical situations are not readily deducible from the results of the calculations. These particular solutions need indeed the mathematical analysis to be performed as far as possible in order to be reckoned up, rigorously stated and 6haracterized. The lack of such a mathematical analysis led, in the present case to conclusions that are not quite correct although the numerical calculation of each individual l.s.v, curve seems perfectly accurate. In particular, the ranges of parameter values where the system behaves, within experimental errors, like a fast reaction following a Nernstian charge transfer were underestimated.

On the other hand, the polarization problem can be stated, as shown further, in the form of a boundary value problem, in the general case for first order kinetics and in a wide range of parameter variations for second order kinetics. The numerical computation of the resulting'integral equations is much simpler and less time-consuming than any finite difference computation of the starting partial derivative equations since only one variable discretization (time variable) instead of two (time and space) is required. The boundary value formulation is thus of interest for further numerical computation of the polarization curves.

The method used here involves the use of dimensionless variables, functions and parameters (see the list of symbols, refs. 7 and 12 and references therein).

FIRST ORDER CONSECUTIVE REACTION

Then

A + ne ~ B (standard potential E °, rate constant k~, transfer coefficient ~) B ~ products (first order rate constant k) The polarization problem is formulated as: c3a/c3z = c32a/c3y 2

~b/t~z = t~2b/t~y 2 - ~b

z=O, y~O and y=oo, z~>O: a = l , b=O

y=O, z~>O: ~=Oa/~y= -Ob/Sy, ~ = A e x p ( ~ ) [ a - b exp( -~) ]

tg a o = I - P P , b o = n - ~ (~-q) -~7 t e x p [ - 2 ( ~ - q ) ] d q

-u

It follows that the current function ~ is the solution of the following integral equation:

~A -~ e x p - ( - c ~ ) = 1 - ~ -~ { l+exp ( -~ ) exp[ -2(~- t / ) ]} (~- t / ) -~ ~dr/ (1) -u

LSV CHARGE TRANSFER AND CHEMICAL KINETIC CONTROL 115

The polarization curves therefore depend upon the parameters c~, A, and 2. If ~ is considered as a fixed quantity, limiting behaviours are obtained for extreme values of A and 2.

1. A ~ ( K O )

Equation (1) becomes:

(* rc -~ [1 + e x p ( - ~) e x p - ) ~ ( ~ - tl)] ( ~ - tl)- ~ ~ d ~ = 1 (2)

i.e. the integral equation featuring a first order irreversible reaction consecutive to a Nernstian charge transfer 13" 14. If, moreover

(a) 2--,0 (DO): a pure diffusion control situation is observed which is characterized by the peak values ~ 2

~up=0.446, ~ v = l . l l , ~p-~p/2---2.20 (3)

(b) 2 - - . o (KP): a stationary state is established by mutual compensation of the chemical reaction and of the diffusion process. In other words pure kinetic conditions associated with a Nernstian charge transfer are reached.

Defining a new potential dimensionless variable:

# ' = 4 + ½ 1 n 2 - R T E - E ° + ~ l n vnF]_l (4)

( u ' = u - ½ In 2)

( u ' = u - ½ In 2) eqn. (2) becomes1 :

e x p ( - ~') = 1 - I ' ~

for which the peak values are ~' ~ 2 (5)

7Jp = 0.496, ~'p = 0.78, # ~ - ~/2 = 1.85 (6)

The peak current is proportional to v ~ as in the pure diffusion Nernstian case. The best diagnostic criterion pertains to the peak potential which varies linearly with log v by 29.6 n mV at 25°C in the cathodic direction for a tenfold increase. The wave in cyclic voltammetry (c.v.) is completely irreversible.

L B L E 1

)NSECUTIVE FIRST ORDER REACTION

pne DO ~ K O KP ( 7- 0.952)

l (0.276)

I ~g2

- - ~ p / 2

- ~ -1.0 -0.8 -0.6 -0.4 1.110 1.038 1.000 0.945 0.866 - ~ -0.113 0.079 0.254 0.406 0.446 0.450 0.452 0.456 0.460 2.204 2.178 2.164 2.145 2.119

- 0.2 0.0 0.2 0.4 0.6 0.8 0.759 0,622 0.456 0.265 0.058 -0.159 - 0.529 0.622 0.686 0.726 0.749 0.762 0.783 0.465 0.471 0.476 0.482 0.486 0.489 0.496 2.086 2.048 2.009 1.971 1.940 1.916 1.859

116 L. NADJO, J. M. SAVEANq"

t'N

.<

II

r/3

Z <

o .<

L~

,d

, . . ,e

O'

O

O ' , t " q O ¢ ' q ~


These two particular cases can be characterized without the help of any dimensionless parameters--all the experimental factors are included in the definition of the dimensionless variables and functions. In the KO case, featured by eqn. (2), the dimensionless polarization curves depend on a single parameter, 2. The peak current does not depart significantly from the proportionality to v ~ when varying 2. The peak potential varies more significantly 13"~4 and its determination is rather used as a diagnostic criterion. The variation of the peak values with log 2 are figured in Table 1. 2 mV is a reasonable estimation of the error on the peak potential determinations. At 25°C and for n = l this corresponds to z~4=0.08. From the values shown in Table 1 it follows that if log 2 ~< -0.95 the observed kinetics are the same, as concerns peak potentials, as for pure Nernstian diffusion control, and if log 2 ~> 0.28 the observed kinetics are the same as for pure kinetic control by the irreversible chemical reaction within experimental error. So three kinetic zones are defined: DO or pure Nernstian diffusion control, KP or pure kinetic control by the chemical reaction, and KO or mixed chemical-diffusion control.

2. 2--~0 ( Q R ) Equation (1) becomes:

~gA -1 e x p ( - a 4 ) = 1 - [1 + e x p ( - 4)] I~g (7)

i.e. the integral equation featuring a quasi-reversible behaviour controlled by the diffusion and the charge transfer processes 12.

(a) A~oo (DO): a Nernstian behaviour is again obtained (eqn. 3). (b) A-~0 (IR): a completely irreversible behaviour is obtained.

Performing the following transformations:

4" = ~4 + In As- ~ (8)

u* = a u - l n Aa -~ (9)

~*= ~,~-~ (10) eqn. (7) becomes:

7 j* e x p ( - 4") = 1 - I * ~* (11)

i.e. the same as eqn. (5). Thus the peak values are the same as given in eqn. (6). The peak height is again proportional to v ~. The peak potential varies linearly with log v, by 29.6/an mV (at 25°C), i.e., 59.1/n mV for ~--0.5, in the cathodic direction for a tenfold increase.

For a given value of a the polarization curves in the QR case depend on the parameter A. The corresponding peak values are listed in Table 2 in the case where a=0.5. The kinetic zones DO, IR (kinetic control by the charge transfer process) and QR (mixed kinetics) have been determined along the same procedure as above.

3. 2~oo ( K I ) A stationary state is reached as concerns the concentration of B. It follows

that

118 L. NADJO, J. M. SAVF, ANT

u~

<

Z o

< u~

©

Z © r..)

e,I

q*o.


ao = ~'~-~ (12)

and thus the polarization curves are given by the integral equation

T A - 1 e x p ( - ~ ) + 7/2 -½ exp( -~) = 1-I~g (13)

The corresponding c.v. wave exhibits a completely irreversible behaviour. Introducing the quantities ~*, u*, T* (eqns. 8-10) and the new kinetic

parameter:

p -~ 0~ (a - 1 ) / 2~A l /%~-~ = ( a n F v / R T ) (~ - 1 ) /2~k~/~k-~ D - i/2a (14)

i.e., for a--0.5:

p = 2~A22 -~ = 2 ~ ( n F v / R T ) - ~ k ~ k - ~ D - 1 (15)

eqn. (13) becomes:

T* exp(-~*)+pT'* exp(-~*/a) = 1 - I * T* (16)

i.e. for a = 0.5:

T* exp( -~*)+pT* exp( -2¢*)= 1 - I * T* (17)

(a) p ~ 0 (IR): eqn. (16) tends toward eqn. (11). Kinetic control is now by the charge transfer. As can be seen in eqns. (14--15) this can occur, for a given value of the charge transfer rate kG, through an increase of the chemical reaction rate constant and/or an increase in the sweep rate.

(b) p ~ (KP): introducing T, ~' and u' (eqn. 4) instead of T*, ¢* and u*:

~* = a~'+~ In p / a ~ (for ~=0.5; ~.__~1~,.17~ In p+¼ In 2) (18)

eqn. (16) can be rewritten as:

c~ '~ - ' ) / 2p -~ T exp( -c~ ' )+ T e x p ( - ¢ ' ) = 1 - I ' T (19)

when p~oo this equation tends toward eqn. (5), i.e. the rate determining step is now the chemical reaction in pure kinetic conditions associated with a Nernstian charge transfer.

Equation (17) has been computed for various values of the parameter p according to a numerical procedure of the same type as already used v's'~2. (The Fortran IV program is available on request.) The results as concerns the peak are shown in Table 3. Again, the peak current does not depart significantly from the proportionality to v ~ passing from one zone to the other. The determination of the kinetic zones has therefore been performed according to the variations of the peak potential as above.

The various kinetic zones so far determined are reported in Fig. 1 on a log A - l o g 2 diagram where each point represents a state of the polarization problem.

It remains now to estimate the zone KG corresponding to the general case, i.e. to eqn. (I) without any simplification. The numerical computation of eqn. (1) (for e=0.5) has been performed along the same procedure as above. The results concerning the peak values are shown in Table 4.

From the comparison of the values of the peak potential variables (~p,


4

3

' ~2

kG

DO KO

( (~Ep -- ~t-----~)25oc =0

| . . . . .

t ~Ep ~ 29.6 mV -~l-'T~gv/25°C = D

/

ref 10 / / / /

/

OR ~KG//" KI

log

Fig. 1. First order reaction following the charge transfer. Diagram of the kinetic zones.

DO KO

IR

KP

KI

2.0

1.0

0.5

0.0

-Q5

-1.5 -1.0 -0.5 (3.0 0.5 1.0 log 2

Fig. 2. First order reaction following the charge transfer. Details of the central zone of the kinetic zone diagrams.

3" or ~p) of Table 4 successively with those in Table 1, 2 and 3, the boundaries between the KG zone and the other zones have been determined using the same condit ion on the error A~p =0.08 (note that, accordingly, ACp = 0.08 but A~* = 0.04 since ~ = 0.5). The resulting kinetic zone diagram is represented in Fig. 2. The two hatched parts represent c o m m o n parts of zones D O - K P and zones QR-KI . In these parts the system behaves as in any one of the two zones of the couple, within experimental error.

TABLE4

CONSECUTIVE FIRST ORDER REACTION, GENERAL CASE (ct=0.5)

The numbers for a given couple of log A and log 2 values represent successively ~p, 4*, ~ , Up, Cp- ~p/2.

log A ~ ~ g 2 -- 1.2 -- 1.0 --0.8 --0.6 --0.4 --0.2 0 0.2 0.4 0.6 0.8

1.3

1.1

0.9

0.7

0.5

0.3

0.1

-0 . I

-0.3

-0 .5

-0 .7

1.113 1 .088 1 .050 0 .996 0 .918 0 .813 0 ,677 0 .514 0 .328 0.127 -0.084 3.896 3 .884 3 .865 3 .838 3 .799 3 .746 ,3.679 3 .597 3 .504 3 .403 3.298

-0.269 -0.064 0 .129 0 .305 0 .457 0 .582 0 .667 0 .744 0 .789 0 .818 0.837 0.446 0 .447 0 .449 0 .452 0 .456 0 .461 0 .466 0 .471 0 .475 0 .478 0.480 2.214 2 .205 2 ,192 2 .173 2 .148 2 .117 2.081 2 .045 2 .012 1.985 1.967 1.141 1 .117 1.079 1.025 0 .948 0 .844 0 .710 0 .548 0 .365 0.167 -0.040 3.450 3 .438 3 .419 3 .392 3 ,353 3.301 3 .234 3 .154 3 .062 2 .963 2.859

-0.240 -0.035 0 .158 0 .334 0 .488 0 .613 0 .710 0 .779 0 .826 0 .858 0.881 0.444 0 .446 0 .448 0 ,450 0 .454 0 .458 0 .463 0 ,468 0 .471 0 .474 0.475 2.229 2 .220 2 .207 2 .189 2 .165 2 .135 2 .101 2 .066 2 .035 2 .011 1.996 1.186 1 .162 1.125 1,071 0 .995 0.892 0.761 0 ,602 0 .423 0 .230 0.030 3.012 3 .000 2 .982 2 ,955 2 .917 2 .865 2 .799 2 ,720 2 .630 2 .534 2.434

-0.195 0 .010 0 .204 0 .381 0 .535 0.662 0.761 0 ,833 0 .884 0 .921 0.951 0.442 0 .443 0 .445 0 .447 0.451 0 .455 0 .459 0 .463 0 .466 07467 0.468 2.253 2 .244 2 .232 2 .215 2 .192 2 .163 2 .131 2 .099 2 .072 2 .052 2.042 1.257 1 .233 1 .197 1.144 1.070 0 .969 0 .841 0 .687 0 .515 0.330 0.140 2.587 2 .575 2 .557 2 .530 2 ,493 2 .443 2 .379 2 .302 2 .216 2.123 " 2.028

- 0.125 0.081 0 .276 0 .453 0 ,609 0 .739 0.841 0 .918 0 .975 1.021 1.061 0.438 0 .439 0 .441 0 .443 0,446 0 .449 0 .453 0 .456 0 .458 0 .458 0.457 2.290 2 .282 2.271 2 .255 2 ,233 2 .207 2 .178 2 .151 2 .129 2 .117 2.114 1.366 1.343 1.308 1 .257 1.185 1.089 0 .966 0.821 0 .658 0 .487 0.314 2,181 2 .169 2 .152 2 .126 2 .091 2 .042 1.981 1 .908 1 .827 1 .742 1.655

-0.016 0 .192 0 .387 0 .566 0 .725 0 .858 0 .966 1.051 1 .119 1.178 1.235 0.432 0 .433 0 .435 0 .436 0 .437 0,441 0 .444 0 .445 0 .446 0 .445 0.443 2,348 2 .341 2 .331 2 .316 2 .298 2 ,275 2 .252 2.231 2 .218 2 .216 2.225 1.534 1.513 1 .480 1 .432 1 .364 1.274 1 .160 1.028 0 .883 0 .733 0.585 1.805 1 .794 1.777 1 .753 1 .719 1 .674 1.618 1.551 1 .479 1.404 1.330 0.153 0 .361 0 .559 0 .741 0 .903 1 .043 1 .160 1.258 1 .343 1 .424 1.506 0.424 0 .425 0 .426 0 .427 0 .429 0 .430 0 ,432 0 .432 0.431 0 .429 0.426 2.435 2 .430 2.421 2 .410 2 .395 2 .379 2 ,363 2 .353 2 .353 2 .365 2.392 1.790 1.769 1.739 1 .696 1.635 1 .555 1,457 1 .344 1.225 1.108 0.997 1.471 1.461 1 .446 1.425 1 .394 1 .354 1.305 1.249 1 .190 1.131 1.075 0.407 0 .618 0 .818 1 .005 1 .174 1.325 1.457 1 .575 1 .686 1 .799 1.918 0.413 0 .414 0 .414 0 .415 0 .416 0 .416 0 .416 0 .415 0 .413 0 .410 0.406 2.564 2 .560 2 .555 2 .547 2 .538 2.531 2 .527 2 ,532 2 .549 2 .582 2.629 2.162 2 .145 2.•20 2 .084 2 .034 1 .970 1.893 1.811 1 .728 1.651 1.583 1.197 1.189 1 .176 1.158 1.133 1.101 1 .063 1,022 0 .980 0 .942 0.908 0.781 0 . 9 9 4 1.199 1.393 1 .574 1 .740 1.893 2 ,041 2 .188 2 .342 2.504 0.400 0 .400 0 .400 0 .401 0 .401 0 .400 0 .399 0 .397 0 .394 0.391 0.387 2.744 2 .742 2 .741 2 .739 2 .739 2 .742 2 .753 2 .776 2 .814 2 .865 2.929 2.684 2.671 2 .653 2 .626 2.591 2 .548 2 .499 2 .449 2 ,404 2 .365 2.334 0.998 0 .991 0 .982 0 .969 0.951 0 .930 0 .905 0 .880 0 ,858 0 .838 0.823 1.302 1 .520 1 .732 1 .935 2 .130 2 .317 2 .499 2 .679 2 ,864 3 .056 3.255 0.386 0 .386 0 .386 0 .386 0 .385 0 .384 0 .383 0 .380 0 ,378 0 .375 0.372 2.973 2 .974 2 .977 2.981 2.990 3 .005 3 .030 3 .067 3 .116 3 .174 3.237 3.360 3 .352 3 .341 3 .325 3 .305 3 .282 3 .257 3 .235 3 ,215 3 .200 3.190 0.875 0 .871 0 .866 0 .858 0 .848 0 .836 0 .824 0 .813 0 ,803 0 .796 0.790 1.979 2 .201 2 .420 2 .635 2 .845 3 .052 3 .257 3 .465 3 .676 3 ,891 4.111 0.373 0 .373 0 .373 0 .372 0 .372 0.371 0 .369 0 .368 0 ,366 0 ,364 0.362 3.222 3 .226 3.231 3 .240 3 .254 3 .275 3 .305 3 .342 3 ,385 3 ,431 3.476 4.160 4 .156 4 .150 4 .143 4 .134 4 .124 4 .114 4 .105 4 .099 4 .094 4.091 0.815 0 .813 0 .810 0 .806 0 .802 0 .797 0 .792 0 .787 0 .784 0 .782 0.780 2.778 3 .005 3 .229 3 .452 3 .673 3 .894 4 .114 4 .336 4 .559 4 ,785 5.012 0.364 0 .363 0 .363 0 .363 0 .362 0 .361 0 .360 0 .359 0 .358 0 .357 0.356 3.440 3 .445 3 .450 3 .459 3 .472 3 .490 3 .512 3 .538 3 .565 3 .592 3.616


On the left corner of Fig. 1 is represented in direction and magnitude (logarithmic) the effect of variations in k, kc and v on the system. It is seen that an increase of the sweep rate shifts the representative point along a descending straight line with a slope 0.5 from the right to the left. The trend is thus to pass from a situation kinetically controlled by the chemical reaction to one controlled by the charge transfer rate. Each of these straight lines corresponds to a fixed value of the ratio, A2/2 = l~/Dk.

The main practical interest of these diagrams is to evaluate the conditions to be fulfilled by the experimental parameters (rate constants and sweep rate) in order that the system belongs to the zones KP, IR or DO where the peak potential varies linearly with log v (KP, IR) or does not vary at all with v (DO) and where therefore simple diagnostic criteria are available as concerns the mechanism analysis. Comparison of Figs. 1 and 2 shows that a rough estimation of the K G zone as represented by dashed lines in Fig. 1 suffices in most cases for this purpose.

It is seen in Fig. 1 that the conditions for the system to conform to the formal kinetics of a Nernstian charge transfer followed by an irreversible chemical reaction within a 2 mV error on the peak potential (for n= 1, at 25°C) are with a good approximation:

log p~>2.17 for l og2>0 .28 (KP) (20) and

log A>~l.06 for log2<0 .28 ( K O + D O ) (21)

The conditions stated in ref. 10 for obtaining the same situation are shown in Fig. 1. It is seen that they are too restrictive; in particular the KP zone is minimized.

Suppose, e.g., t h a t / % = 1 0 0 n½D ½ cm s -1 (---0.57 cm s -1 for D = 1 0 -5 cm z s- l ) . Conditions (20) become: log k+log(nF/RT)v<4.96 and log k-log(hE~ RT)v >0.28. This is represented in Fig. 3. The hatched part corresponds to systems in the KP zone. It is considered that the available sweep rate range extends from 0.1 to 1000 V s -1. The maximum sweep rate for the system to remain in the KP zone and thus to conform to Nernstian-chemical reaction kinetics is 5.5 V s -1 (for n = l at 25°C). The maximum value of k is then 417 s -1. If the sweep rate is less, higher values of k are allowed. For example, for v = 0.1 V s-1, k can be as high as 23000 s-1.

Still higher values of v and k are found if kc is larger. If, for example, kc is tenfold larger (~---5.7 cm s - l ) * k may reach 2.3 x 108 s -1 at v=0.1 V s -1 and 4.17x104 s -1 at v=550 V s -1 (Fig. 3). For this last value of k most of the currently available range of sweep rates corresponds to the 2916/n mV variation (at 25°C) of the peak potential with log v.

When the determination of the chemical reaction rate constant is to be per- ' formed by the peak potential method a part of the peak potential vs. log v diagram should belong to the KP zone (slow scan) while the other part (fast scan) should be horizontal (diffusion controlled Nernstian behaviour). The determination of the

* This is in the upper range of rate constant values, uncorrected for the double layer effect, that are to be found with organic aromatic molecules (see refs. 9 and 15).


I

, • . - . - - /

-L Ei iii -//

2

~ . ~ 1; 2.6 3.6 4.610g(ffV/~ 1

-10 0~0 110 2'.0 .~0 lOg V(n=l 25°C) Fig. 3. First order reaction following the charge transfer. Conditions for obtaining a pure kinetic Nernstian behaviour (t3Ep/c~ log v = - 2 9 . 6 / n at 25°C)for two particular values of the charge transfer rate constant k~.

rate constant then results from the measurement of the sweep rate at the inter- section of the oblique and horizontal parts of the diagram 7. This also may be useful in mechanism analysis when changes in the medium are involved *. Half an order of magnitude in sweep rate appears as a minimum for each linear part of the E p - l o g v diagram. As can be deduced from Figs. 1 and 2 a first condition for this is (see, e.g., the oblique line A in Fig. 2):

log A >~ 1.4 + 0.5 log 2 (22)

i.e. log(g~/O ~) >1 1.4+0.5 log k

As concerns the horizontal portion, the condition is that v =(R T/nF)[k/2b- 0.5)] ~< 103 V s-1 and thus k ~< 103 (nF/RT)(2b- 0.5), where 2b is the boundary value between zones DO and KO. For the oblique linear portion, v =(RT/nF)[k/(2R + 0.5)] >7 10- V s- 1 and thus k/> 10- ~ (nF/RT)(2B + 0.5), where 2 B is the K O / K P boundary value.

This is represented in Fig. 4 for n = 1 and 25°C. As k increases the oblique linear part tends to be larger than the horizontal portion of the Ep - log v diagram). It

- - I 1 is seen, for example, that, for k~=0.57 cm s -~ (with D=10 -5 cm z s '.), k cannot be larger than 50 s -~ and that the oblique part covers 'about ohe order of magnitude in v. For kc = 5.7 cm s-~ k can reach 2300 s-2, the horizontal portion covering then half an order of magnitude.

If now with increasing k and/or decreasing/%, condition (22) is no longer fulfilled, the oblique linear part of 29.6/n slope, if any, is no longer followed by a horizontal portion. This tends to bend. As long as zones K G and QR are concerned the corresponding slope is not very large. On the contrary, when the


~ 1 . 5 ~

~ 0.5

- 0 . 5

1.0

29.6/n mY

1.5 2.0 2.5 3,0 Iog (k /s -1)

Fig. 4. First order reaction following the charge transfer. Conditions for the Ep-log v variations to be as represented in the insert, allowing the determination of the chemical reaction rate constant.

150

100

> 50

o

I..u I

3" -5o

g v = 59.1 mV

v = 38.7 mV

dEp/blog v= 29.6 mV

- 1 .0 Q O 1.0 2.0 log p

Fig. 5. First order reaction following the charge transfer. Variation of the peak potential with the sweep rate in the KI zone (see Fig. 1).

-100

log v 1


system enters the KI zone, the Ep diagram does not present two parts anymore but a slow curvature corresponding to slopes between 29.6/n and 59.1/n mV at 25°C (for ~--0.5). This is seen in Fig. 5 where:

2(~* - In p) = - (nF/R T)(Ep - E °) - In (l~/Dk) (23)

is represented as a function of log p (eqn. 15). A variation of the ratio l~ /Dk merely results in a translation of the vertical scale. On the other hand, for specified values of k~, D and k the scale for log p is double that for log v. It follows that the slopes on the diagram of Fig. 5 directly reflect those on an E o - l o g v diagram. On the left part of the diagram, the slope tends toward 59.1/ n mV (at 25°C for ~ = 0.5) while on the left part it tends toward 29.6/n mV. 6.3 orders of magnitude in v separate the first region from the second one. It follows that an experimental system may well remain in the transition region for all the available sweep rates. Under these conditions, it is seen in Fig. 5 that the Ep - l o g v variations may appear as approximately linear with slopes between 59.1/n mV and 29.6/n mV if the experimental excursion in the sweep rate range does not exceed 2 or 3 orders of magnitude.

The slope increases when the ratio k~/Ok ~ decreases, and conversely.

CONSECUTIVE DIMERIZATION

Then

A + n e ~ B (standard potential E °, rate constant k~, transfer coefficient e) 2B ~ products (second order rate constant kd) The polarization problem is formulated as:

8a/& = 82a/SyZ

Ob/ ~z = O2b/ ~y2 - - i~ d b 2

z =0 , y~>0 and y=oo, r>~0: a = l b = 0

y=0 , z~>0: 7J - -Sa /Sy=-3b /Sy

~U=A e x p ( a ~ ) [ a - b e x p ( - ~)]

(24) (25) (26)

(27) (28)

a0 = 1-I~P

The calculation of b 0 in the general case requires the numerical resolution, by a finite difference procedure, of eqn. (25). The problem is of the same degree of complexity as for a Nernstian charge transfer 7, the only change being in the boundary condition (eqn. 28). However this numerical calculation can be avoided by reference to the results obtained above in the case of a first order reaction. Indeed, it can similarly be considered that the delimitation of the KG zone can be performed by extrapolation of the boundary lines between the other zones. Among these last ones the boundary values of log A for DO/QR/IR are the same as above. The boundary values of log 2 for D O / K O / K P can be derived from previous results7:

DO/KO: log 2 = -0.43, KO/KP: log 2=0.13

In the KP zone, the pure kinetic polarization curves are characterized by the following integral equations:


c5

Z - - " - -

o

.<

o

. ~ ~

I . . . . . . .

o o ~ v ~ e ,4 ~ l l OO

c5~Hc5 o , ~ ~i

l c 5 E c 5 o ~ ~H

c5 ~ c 5 c5 .~ ~

~ i ~ o o o ~


with 7"~ e x p ( - ~ ' ) = 1 -1 '7" (29)

~ ' = ~ +½ In 22d - nF _ o + In (30) 3 R T ~ 3v ~ -

(u'= u-~ In 2d)

7"p=0.527, ¢~=0.502, Cv-¢v/2=l.512 (31)

The peak potential varies therefore linearly with log v, by 19.7/n mV at 25°C, in the cathodic direction for a tenfold increase. It also varies linearly with log c o by the same amount but in the opposite direction.

It remains only to formulate the expression of the current function in the zone KI, i.e. when pure kinetic conditions prevail (2d~oo). From eqns. (24-27) it ensues that:

a o = 1 - I 7 " , b o = T~(Z2d/3) -+ (32)

Then, taking eqn. (28) into account and introducing ~u,, ¢, and u* as defined by eqns. 8-10, it follows that:

7'* exp( - ~*) +pT'*~ exp( - ~*/e) = 1 - I* ~u* (33) with

p=(~)~ ~( z~- 3)/6~ al/~2~ ~ (34)

for ~=0.5:

p = 6~(A2/2~) = (6) ~ (nFv/R T)-~k~ kd- ~ c °- +D-1 (35)

When p~0 , eqn. (33) tends toward eqn. (11) which features a complete kinetic control by the charge transfer forward reaction (zone IR). When p - ~ , eqn. (33) tends toward eqn. (29) which features the kinetic control by the dimerization reaction, the charge transfer remaining Nernstian (zone KP).

Equation (33) was numerically computed for a =0.5 according to a procedure similar to those previously described (see ref. 7 and refs. therein). The results are shown in Table 5. The determination of the boundary values of p for the transition between zones KI, KP and IR was as above.

The whole kinetic diagram js represented in Fig. 6. The effects of variations in kG, kd, v and c o are represented in direction and magnitude (logarithmic) in the left corner. The main differences from the first order case are: (i) the kinetic factor 2 d now depends on the initial concentration c °, (ii) the boundary lines between the zones KP/KI / IR have now a slope of 1/6 instead of 1/4.

A variation in the sweep rate still shifts the system along a straight line with a slope 0.5 and this corresponds now to a constant value of the ratio k~/D kd c °.

As can be seen in Fig. 6, the conditions for the system to conform to the formal kinetics ofa Nernstian charge transfer followed by an irreversible dimerization within 2 mV (for n = 1, at 25°C) are:

log p>2.35 for log 2>0.13 (KP) (36) and

logA~>l.06 for log2<0.13 ( K O + D O ) (37)

5£)1 4.C

x o

2.C

1£

o.c

-1.c

kG

l co ~ kd

v

DO

--(a~, Ep )25oc =0 o log v

• ~Ep ~ 25oC = 0

QR

i i

-4,0 -3.0 -2.0

a Ep 19.7 - ( ~ l ~ - v ) 2 5 0 c = n

( ~Ep 19,7 ~" ~ ) 2 5 " c n I ~ /

ref lO K p 1 /

I / KI I / ~ ' - " ~

t l ~ ~ t d Ep ~ 59.1 l/ ...--- ~dlogv '25°C= n

IR . ~Ep ) (~I~ 25"C =0

i

-1.0 0.0 1.0 2.0 3.0 4.0 5£) log

Fig. 6. Dimerization following the charge transfer. Diagram of the kinetic zones.

12c\

lO£

9 0

8 o

70

~, RG= 5 7 c m s - /

5.o

4.0 ~

30 %=0.57 x

2D : ~

10 /

o o.6 ~16 L6 3 ; ' o~-,, 46 Iog(~Tv/s ,

1.5

b

& o.e

~ o.c "6 o -0.5

4.0

3.5

c,l

3.0

2

--~-2.5

2.0

~ v

I -1.0 O0 1.0 2 0 3'.0 1.0 1.5 2.0 2.5 3.0

log v(n=l, 25°C ) log( kdC%~ 1 )

Fig. 7. Dimerization following the charge transfer. Conditions for obtaining a pure kinetic Nernstian behaviour (dEp/Olog v=-19.7 /n mV at 25°C) for two particular values of the charge transfer rate constant kG.

Fig. 8. Dimerization following the charge transfer. Conditions for the Ep-log v diagram to be as represented in the insert, allowing the determination of the chemical reaction rate constant.

LSV C H A R G E TRANSFER AND CHEMICAL KINETIC C ONTR OL 129

Again, the conditions stated in ref. 10 in this connection are shown in Fig. 6. Taking the same numerical example as for the first order case ( k c =

100 n~D~cm s-1-~0.57 cm s -1 for D = 1 0 -5 cm 2 s - l ) , conditions (eqn. (34) become:

log kd c o + 2 log (nF/R T) v < 7.22

log kdC°-log(nF/RT)v>O.13

This is represented in Fig. 7 where the hatched part features the systems be- longing to zone KP. The maximum v a l u e o f v is then 5.8 V s -1 ( n = l , 25°C). The corresponding maximum value of kac ° being 312 s -1. For v=0.1 V s - i , kdc ° can reach 10 6 S - 1 . If kG is tenfold larger (5.7 cm s -1) the maximum value of v is 580 V s -1 and of kdC °, 3.12x104 s -~. For this last value of kac ° most of the available range of sweep rates corresponds to the 19.7/n mV variation of the peak potential with log v.

The conditions for obtaining an E p - l o g v diagram with a linear oblique portion of 19.7/n mV (at 25°C) slope and a horizontal portion, corresponding each to at least half one order of magnitude on v are represented (for n = 1, 25°C) in Fig. 8 which is the strict analogue of Fig. 4, for second order kinetics. The conditions concerning kd c o are close to those for k with first order kinetics.

For systems in the KI zone the slope of the Ep - log v pattern passes gradually from 19.7/n mV to 59.1/n mV at 25°C for ~=0.5. This is shown in Fig. 9 which represents:

15C v = 59.1 mV

100

o

~ 50

I ,g.

.o y o

- 5 0

#Epl 0 l o g v = 3 9 . 4 m V

log v 1 t

r,

13) o

-1~ QO 1# 2 0 log p

Fig. 9. Dimerization following the charge transfer. Variations of the peak potential with the sweep rate in the KI zone (see Fig. 6).


5 0

7 / # Epl OIog c°= 9 . 9 r I("

log c °

2 0 0

> E

"----

c

~* IO0

I

"T

-1.0 O.O 1.0 2.0 log P

Fig. 10. Dimerization following the charge transfer. Variations of the peak potential with the initial concentration, in the K! zone (see Fig. 6).

(2RT/nF)(¢* _ 3 In p) = - (Ep ± E °) - (RT/nF) In 3(k~/Dkd c °)

as a function of log p. When log p varies by 1, log v varies by 1.5. Figure 9 thus shows the slopes of the Ep-log v patterns when crossing the zone KI. As in the first order case it is seen that such patterns may well give the impression of being linear. Here the apparent slope will be between 59.1 and 19.7 mV (n = 1, 25°C).

Similarly the slopes of the Ep-log c o diagrams are shown in Fig. 10 which represents:

(2R T/nF) ~* = - (E, - E °) + (2R T/nF) In 2 ~ {kG/[D¢(nFv/R T) ½] }

as a function of log p. The currently available variation range of c o does not exceed one order of

magnitude and such a variation results only in a change of 0.33 in log p. It follows that the Ep-log v diagram will appear linear in practice with a slope between 0 and 19.7 mV per decade (n = 1, 25°C).

E.C.E. A N D D I S P R O P O R T I O N A T I O N

(1) Let us consider now the following reaction scheme:

A + ne ~ B (E °, kc, ct)

LSV CHARGE TRANSFER AND CHEMICAL KINETIC C ONT R OL

k~, (2) B ~ C (k = kf + k b , K - - kf/kb)

kf

(3) C + ne ~ D

131

(this second reduction is assumed to be much easier than the first one so that (Cc)o=0 along the entire wave)

kd

(4) B + C ---* A + D

As shown earlier (see ref. 5 and refs. therein) three important limiting cases can be considered:

(i) Reaction (4) does not occur and kb~>k f. This is the e.c.e, reaction scheme with an irreversible chemical reaction interposed.

(ii) Reaction (3) does not occur significantly and reaction (2) is the rate determining step. This is the disp. 1 scheme.

(iii) Reaction (3) does not occur significantly and reaction (4) is the rate determining step, (2) being at equilibrium. This is the disp. 2 scheme.

E.c.e. The polarization problem is formulated as follows:

t~a/&c = 0 2 a/~y 2

•b/•z = 02b/~y2-2b (2=RTkb /nFv )

~c/~r = ~2 c/~y2 + )~b

z=0, y>~0 and y=oo , z > ~ 0 : a = l , b = 0

y=0 , z>~0: tPl=~a/dy= -Ob/gy, c = 0

IP 1 = A e x p ( ~ ) [ a - b e x p ( - ~ ) ]

tP2 = (c3c/c3y)o and ~ = 7~t + ~z

It is easily shown that the polarization problem can be formulated under a boundary value form, as in the case of the Nernstian e.c.e, case TM, which leads to the following system of two integral equations:

tl~l A- 1 e x p ( - a ~ ) = 1 - n -~ [1 + e x p ( - ~ ) e x p - 2 ( ~ - q ) ] ( ¢ - q ) - ~ ~ l d q (38) --U

n -~ f ¢ [ e x p - d ~ ( ~ - t l ) ] ~ P l d r l = I ~ l - I ~ 2 (39)

1. A ~ o o (KO) . Equations (38) and (39) reduce tO the integral equations featuring the Nernstian e.c.e, case 8"a6. The polarization ~ problem depends upon a single parameter ;L For small values of 2(DO) a reversible diffusion controlled n-electron wave is obtained. As 2 increases the peak height increases from ~u o = 0.446 to 7~p=0.992. The variations of ~p, ~p and ~p-~p/2 with log 2 have been already reported s'16. For 2 = oo, i.e. for pure kinetic conditions (KP), the wave is exactly double that corresponding to a first order consecutive reaction. The peak potential then varies linearly with log v by 29.6/n mV, at 25°C, in the cathodic direction for a tenfold increase in v.

132 L. N A D J O . J. M . S A V I ~ A N T

t'q

~ o o t r~ o ~

eq

t'q

~5 ,--; ~ ,..:

~ . . - :

~Sr- i

~5¢q

~ ,::5 t-,i

, - - o

,:5 ¢'q

c5¢- i

e,i

. I

LSV CHARGE TRANSFER AND CHEMICAL KINETIC CONTROL i33

The boundary values of 2 separating the kinetic zones can be derived from previous results 8 on the basis of an error on the peak potential determination corresponding again to A~p--0.08:

DO/KO: log 2=0.03, KO/KP: log 2= 1.31

2. 2--*oo (KI) . Then, bo--- 7"2 -~ and eqns. (38), (39) become:

½7"A -1 exp(~)+½7"2 -~ e x p ( - 4 ) = 1-½ 17" (40)

It follows that the wave is exactly twice the wave obtained for a consecutive first order reaction (see eqn. 13). The variations of the peak potential are thus exactly the same. In particular the boundary values of the parameter p (eqns. 14 and 15) are also log p = -0.815 and 2.17.

3. 2---,0 (QR). The characteristics of the wave are exactly the same as with the other reaction schemes.

4. A~O (IR). Introducing ~*, u*, 7"* (eqns. 8-10) and 7"]'= 7"1 ~t--~, 7"~= 7"z e - l , eqns. (38), (39) become:

rc [ e x p - 2 * ( ~ * - q ) ] ( ~ * - q ) ~7"]`dq = 1"7"*-1"7"* (41) U*

7"] ̀exp( - ~*) = 1 - 1 7"* (42)

2*= 2~-1 is now the only parameter upon which the problem depends. When 2*--*0 (IR1) 7",~0, thus 7"*= 7"~'. An n-electron irreversible wave

kinetically controlled by the forward charge transfer reaction is obtained as in the previous reaction schemes.

When 2*-~ov (IR2), 7"2"--*7"~' and thus:

½7"* exp( - 4*) = 1-½I* 7'* (43)

It follows that the wave is exactly twice the height of the above, the peak potential and peak width remaining the same.

The variations of the peak value in the transition range of 2* values is shown in Table 6 (numerical calculation were along the same procedures as above). The boundary values of 2* for the transitions IRI/IR and IR/IR2 are figured in the Table.

As in the case of a Nernstian e.c.e, reaction scheme the peak height varies significantly (by the ratio of 2) with the sweep rate and the rate constant of the chemical reaction. From these variations the value of the rate constant can be determined once e is known using the working curve shown in Fig. 11, provided that either the IR1 or the IR2 zone is attainable experimentally.

The diagram of the kinetic zones according to a 2 mV error on the peak potential determination is represented in Fig. 12.

The evaluation of k~, k and v for which a Nernstian pure kinetic behaviour is met is very similar to that discussed in the case of a first order reaction consecutive to the charge transfer. Similarly also in the KI zone the Eo-log v plot exhibits a quasi-linear character with a slope between 59.1/n and 29.6/n mV at 25°C for e=0.5.


0.9

08

:go_

07

O.e

0.5

.V .?.?-V'V:~ ~7

.V~ ~ 00.0 ~Cr

/ s ° /V p disp 1

Z o ° 7d /

o

J

-10 QO 10 RT kb

Iog(~n F- --if--) Fig. 11. E.c.e. and disp. 1 reaction schemes. Slow charge transfer. Variations of the peak current with the sweep rate and the chemical reaction rate constant.

5.0

4.0

3.0

~ 2.0

0.(3

1.0

~G - k

J v

DO

~Ep -- (a i--]-~-g v )25oc= 0

~ _ . dEp 29.6 'a--G~-g~ )za°c- T

KO _ ~

OR / / KG ~ / i /-

. . . . . . . . ( OEp ) _ 59.1 - ~ 25oc- T

IR1] _( aEo ) 59.1 / IR IR 2

dlogv 25"c,=--~, / ,

-zE) -3.0 -2.0 -1.0 0.0 1£) 3.0 4.0 5.0 log j1

Fig. 12. E.c.e. reaction scheme. Diagram of the kinetic zones.

Disp. 1 T h e p o l a r i z a t i o n p r o b l e m is n o w as followsS :

~?(2a + b)/& = c~ 2 (2a + b)lcgy 2

tVb/t~z = t ~ 2 b / t ~ y 2 - 22b

z = O , y~>O and y - - o o , z>~O: a = l , b = O


y = O, z >1 0: ~P = ~3a/c3y = - 8b/dy

~ = A exp(~4) [ a - b e x p ( - 4 ) ]

This is again a linear problem that can thus be formulated under a boundary value form:

171 1+2 e x p ( - ~) f* ~ A - 1 e x p ( - ~ ) = 1 2 2re ~ .1 ~ ( ~ - q)-~exp [ - 2 2 ( ~ - t/)] dq

- - U

(44) 1. A-ooo ( K O ) . Equation (44) reduces to the integral equation featuring

the Nernstian disp. 1 case 5. For small values of 2 (DO) the reversible diffusion controlled n-electron wave is obtained. As )~ increases the peak height increases from 0.446 to 0.992. For large values of 2 (KP) the wave is exactly twice that corresponding to the first order reaction, and thus the same as for the above e.c.e, case making allowance for the introduction of 22 instead of 2:

~' being defined as:

4 ' = 4 + ½ In (2/2)

4'0 -o0.783 and thus 4p -o 0 .783-3 ln(2/2)

• However the peak values in the intermediate zone (KO) are not strictly the same as for the e.c.e, case. This has already been noted as concerns the peak height 5. The peak values corresponding to the DO zone are listed in Table 7. From these, the boundary values of 2 can be calculated on the basis of a 2 mV error on the peak potential (for n = 1, 25°C):

D O / K O : log 2 = - 1.20, K O / K P : log 2 = - 0 . 0 8

It is seen in Table 7 that a definition of the kinetic zones according to a 5Vo error on the peak current would have lead to a significantly larger KO zone.

2. 2--*oo ( K I ) . Equation (44) becomes:

½~(½A) -1 exp( - ~ ) + ½ke(½,~.)- ½ exp( - 4) = 1 - ½ I ~ (45)

i.e. the same as eqn. (40) (e.c.e.) dividing A and 2 by 2. What has been said about the KI zone in the e.c.e, case and in the case of a first order reaction is thus readily transposable to the present case. The kinetic parameter to be in- troduced is now:

p, = 2 (a - 2)/2~ 5 ( ~ - 1 )/2~ A ~/~ 2 - ~

(for c~=0.5:p'=(½)AZ2 -~) instead ofp. It follows that the equations of the boundary lines are:

IR2/KI :log p' -- - 0.815, i.e.: log A = ¼ log 2 - 0.257 KI /KP: log p' = 2.17, i.e.: log A = ¼ log 2 + 1.24

3. A-oO ( I R ). Equation (44) becomes:

A- 1 exp ( - 0~) = 1 171 1 r |~ ~ ( ~ - q ) - l e x p [ - 2 2 ( ~ - q ) ] d q (46) 2 2re I 3_,

136 L. N A D J O , J. M. S A V I ~ A N T

.<

Z <

r .~

z

t,,1

i t ) ¢r~ ¢-q

I c5 ,-.: ~ t-,i

t t ' ~ t - . q . - . ~ t . ~

I

I

I

i

I

e,i t-i

LSV CHARGE TRANSFER AND CHEMICAL KINETIC C O N T R O L 137

i.e. : I* ~* 1 W

J ~* (~* - q)- ~ exp [ - 2* (~* - q)] dr/ 7 j* exp ( - 4*) = 1 2 2re -~ _,,

(47)

2* being: 2* = 22~- a When 2*--,0 (IR1) the standard charge-transfer controlled n-electron wave

is obtained ( ~* = 0.496, 4" = 0.783) When 2*--,oo (IR2), eqn. (47)becomes:

½~* e x p [ - ( ~ * - l n 2)] = 1-½I* ~* (48)

The wave is now double the height of the preceding one, the wave width is the same and the peak potential corresponds to:

4" = 0.783 + In 2 = 1.476

Table 8 shows the variations of the peak values as functions of 2* and the values of 2* corresponding to the boundaries IRI/ IR and IR/IR2.

Again the peak current exhibits a larger variation than the peak potential in the intermediary zone. These variations are shown in Fig. 11 together with those featuring the e.c.e, case. These curves can be used as working curves in order to determine the magnitude of the rate constant of the rate determining step (B~C) .

The diagram of kinetic zones can be derived from the above results• It is shown in Fig. 13. The conclusions as concerns the peak potential are quite similar to those of the e.c.e, case and the first order consecutive reaction.

Disp. 2 The dimensionless formulation of the problem is now as follows (2~ =

2R T ka c°/nFKv).

5.0

4.0

3.0

2.O

0.0

-1.0

kG

DO I) Ep

- - (~ i---'T'o~g v )25. c =0

!

KO

i OR / KG

i I / - -

i

alogv 25°C- o

_ _ , , IR1 [

• (}Ep 29.6 -- ( ~ ) 2 5 . C = n

KP

"

" ~ " _ , dEo ,, _ 59.1 I - ' ~ l-T°~g v '25"c = -'n"-" LIR 2 , , ,

i

5.0 -4.0 -3.O -1.0 0.0 1.0 2.0 3.0 4.0 Io9 2

Fig. 13. Disp. 1 reaction scheme. Diagram of the kinetic zones.

138

oo

~q

c5c5~

c5 c5 -~

c5c~

oo I

~ 0 ~

~o

~ 0 ~


~?a/ Oz = d e a/ ~y 2 - )~i b2/2 (49)

~b/~ = ~ b/~y 2 - ,~ b 2 (50)

~=0, y>~0 and Y - - - ~ ' l z ~ > 0 : a = l , b = 0 y = 0, • ~> 0: 7* = ?al ly = ~- ~b/~y

7* = A exp ( ~ ) [ a - b exp( - 4)]

Through linear combination of eqns. (49) and (50) and integration, the boundary integral condition:

2 7*A -1 e x p ( - ~ 0 + b 0 [ l + 2 e x p ( - 0 ] = 2 - 1 7 . (51)

is obtained. The resolution in the general case amounts to solving numerically eqn.

(50), by a finite difference method, with eqn. (51) as a boundary condition. The degree of complexity of such a computation is exactly the same as for the Nernstian case s.

1. A--+oo ( K O ) . This corresponds to the Nernstian disp. 2 scheme already considered s. Increasing )td, the peak current varies from 7.p=0.446 (DO) to 7*p = 1.054 (KP) on the other hand, the peak potential, first independent of v and e ° (DO), becomes a linear function of log v and log c o (KP), the rates of variation being -(19.7/n) mV and (19.7/n) mV at 25°C.

The variations of the peak values with the kinetic parameter 2~ are shown in Table 9. The values of 2~ separating the three zones DO, KO, KP can be derived from a specified error, say 5~o, on the peak current measurements as already done 8. They can be also derived from the variations of the peak potential assuming a 2 mV error (for n = 1, 25°C). The 2~ values are then:

DO/KO: log 2~ = - 1.0, KO/KP: log 2~ = 2.2

For continuity in the argument these values will be used here for constructing the diagram of the kinetic zones.

2. 2'd---,OO ( K I ) . A pure kinetic situation is met which involves that ~b/Oz=O so eqn. (50) can be converted in a boundary integral equation:

bo = (22~)-~ 7.~ (52)

Taking eqn. (51) and the fact that b0 is small into account:

2 7.A-1 exp( - ~4) + 2(22~/3) -~ 7.~ exp( - 4) = 2 - 1 7 . (53)

This equation is very similar to the corresponding one obtained in the case of a consecutive dimerization. Indeed, introducing the following variables and parameters: 4 + =c~4+ln(A/2~ ~*) instead of 4" (eqn. 8), 7*+ = 7./2a ~= 7**/2, and p = 6 ~ 2 - a/%~ z~-3)/6C~ A1/~ 2'd- ~ , for ~=0.5:

p=3*~Z-1AZ2'a-+=[3~-/(2x2*')](nFv/RT)-~kZ K~k~+c° -+D -1 (54)

instead of the parameter p defined by eqns. (34) and (35). Equation (53) becomes:

7*+ e x p ( - ~ + ) + p T * +,= exp( -4+/c0 = 1 _ I + 7* +

i.e. exactly the same as eqn. (33).

1 4 0 L. N A D J O , J. M. SAVI~ANT

~le,i +

H

Z <

Z

m ~

< b~

I

I

I

I

i o ,.-:

I

LSV CHARGE T R A N S F E R AND CHEMICAL KINETIC C O N T R O L 141

The numerical results are thus readily deducible from those given in Table 5. The current is now double that corresponding to dimerization. The variation

of the peak potential with the sweep rate and the initial concentration are of the same type. Pseudo-linear behaviour will thus be also encountered in the KI zone.

When A ~ the Nernstian pure kinetic 2n-electron curve is met (zone KP). When A ~ 0 (zone IR2), ~p+ =0.496 and ~p+ =0.783, i.e. ~p=0.992 ~½ and ~o= (1/a)[0.783-1n(A/27{)].

The current is thus twice as high as in the IR1 zone and the peak potential is slightly different.

The boundary values of p defining the three zones KP/KI/IR2 are the same as for the consecutive dimerization (see Table 5).

3. A ~ O ( I R ) . Raising 2~ one passes from zone IR1 to zone IR2. Accordingly the wave doubles in height. The dimensionless wave is now obtained as follows: eqn. (50) is rewritten, introducing ~* and 2*:

~ b / ~ * = t32b/~y*2-2*b 2 (55)

(2* = 2~/ct, y*=yc~ ~, (Ob/#y*)o= - ~*)

eqn. (55) is numerically solved 7' 8 taking

2~u* exp ( -~* ) +b o = 2 - l*~U* (56)

as a boundary condition.

1.0 ,.v.-v"v~

V/V" / 0.9 /V/Vo/oO0 °°0°~'-

( a )/g DO c~(b )

. o .Q8

~/(::~90/'0 . 0

~ 7 .d ~ 0 /

d' o.e ~,

/ ,o4

i . • i

O0 1.0 2.0 (a): log 2RTkd/DFKv (b): log 2RTkd/enFKv

Fig. 14. Disp. I reaction scheme. Variations of the peak current with the sweep rate and chemical reaction rate constant (a) for a Nernstian charge transfer, (b) for a slow charge transfer.

142

.< [-.,

. 1

,.,,,

,,,,,

t"q

- , 5

t'q

L. N A D J O , J. M . S A V I ~ A N T


The results are shown in Table 10. From them the values of 2~ featuring the limits between the zones IR1, IR, IR2 are readily derived using, e.9. the peak potential as the measured quantity, the error being again 2 mV (for n = 1, 25°C).

The variations of the peak current (~*) are shown in Fig. 14 together with those obtained in the case of a Nernstian behaviour 8. The peak current is, in this case, much more sensitive to change in 2~', as determined e,g. by variation of the sweep rate, than the peak potential.

8 ~

5.c ~G

4.(3 K kd

v DO 3.C 0 Ep

- - (~l-T~-Tg v )25oc=0

2.0 ~Ep (dl---~gcO)25°c =0

KO

--( ~Ep )25 _ 19.7 dlogv *c - "--'~"

( ~Ep ) 19.7 2s.c =--a-

K P

J

!

QR / o.o .

~)Ep , _ 59.1 I IR -1"0 - - ( ~ ) 2 5 ° C - n

( ~ ) 2 5 o , c = 0 IR,2 / . , ,o ~ o z o -~o ao lO Lo 3:0 41o

log 2

Fig. 15. Disp. 2 reaction scheme. Diagram of the kinetic zones.

/ /

/ (G / / KI

I -(~l-~ -~')2~°c=-~- IR2 ( ~ ) 2 5 ° C = 0

i 5.0

From these results, the diagram of the kinetic zones featuring the disp. 2 mechanism can be derived. It is shown in Fig. 15 for c<=0.5.

The orders of magnitude of kG, k and v for which a Nernstian pure kinetic behaviour is met are the same as those discussed in t l~ case of a consecutive dimerization.

Some qualitative conclusions may be derived from the preceding results as concerns the analysis of organic electrochemical mechanisms:

(i) For a specified value of the chemical rate constant the system tends to deviate from the Nernstian behaviour as the sweep rate is increased, and conversely.

(ii) Diagnostical use of the Nernstian characteristic slopes of the l.s.v. Ep-log v (log c °) plots is nevertheless legitimate in a wide range of chemical rate constants and sweep rates for apparent charge transfer rates of the order of 1-10 cm s-~, i.e. for large molecules.

(iii) For smaller molecules, more caution must be exercised in the handling of data. In particular, changes in slope on the same Ep-log v plot must be carefully looked at. In any case, if a Nernstian, reversible, diffusion controlled pattern is obtained at high sweep rate, this provides evidence that the Nernstian characteristic slopes can be safely used at smaller sweep rates.


ACKNOWLEDGEMENT

The work was supported in part by the C.N.R.S. (Equipe de Recherche Associ6e no. 309: Electrochimie Organique). The numerical calculations were performed on the IBM 370 digital computer of the CIRCE (C.N.R.S., Orsay, France). Prof. M. Hulin is gratefully thanked for permission to use the IBM 1130 computer terminal of the Laboratoire de Physique des solides de l'Ecole Normale Sup6rieure, Paris, France.

SUMMARY

The 1.s.v. formal kinetics of systems involving kinetics control by the initial charge transfer step and/or secondary chemical reactions is derived for the following reaction schemes: first order deactivation, consecutive dimerization, e.c.e. and disproportionation. Emphasis is laid on the conditions in which the diagnostic criteria obtained for a Nernstian behaviour of the charge transfer remain applicable. This leads to a discussion of the practical limitations in the diagnostical use of the peak potential variations for the analysis of organic electrochemical mechanisms.

LIST OF SYMBOLS

Definit ions

c~ concentrations of the reacting species c o initial depolarizer concentration D diffusion coefficient (assumed to be the same for the depolarizer and the

initial reduction product) E electrode potential, E = Ei - vt Ei initial potential v sweep rate E ° standard potential ko electron transfer rate constant a transfer coefficient k first order reaction rate constant k a second order reaction rate constant i current S electrode surface area

Dimensionless quantities Time Space Potential Current Concentrations Rate constants

Convolution integrals I s ~ = re- ~ ~ ( ~ - t/) - ~ dr/ - - u s

The subscript 0 indicates y = O.

z = ( n F / R T ) vt y = x (nFv /R T D) ~

= -- ( n F / R T) (E - E °) = ( n F / R T ) v t - u, u = (nF/U T)(Ei ~ E °) = i/nFS c o D~ ( n F / R T ) ½ v ½

a = CA/C °, b = CB/C °, C = Cc/C ° A = kG (D nFv /R T ) - ½, 2 = R Tk/nFv, 2 d = R T kd c° /nFv

s = none , * or ')


REFERENCES

1 J. M. Sav6ant and E. Vianello, C. R., 256 (1963) 2597. 2 A. M. Khopin and Z. I. Zhdanov, Elektrokhimiya, 4 (1968) 228; English transl.: Soy. Electrochem.,

4 (1968) 200. 3 C. P. Andrieux and J. M. Sav6ant, d. Electroanal. Chem., 26 (1970) 223. 4 E. Lamy, L. Nadjo and J. M. Sav6ant, J. Electroanal. Chem., 42 (1973) 189. 5 L. Nadjo and J. M. Sav6ant, J. Electroanal. Chem., 33 (1971) 419. 6 C. P. Andrieux and J. M. Sav6ant, J. Electroanal. Chem., 33 (1971) 453. 7 C. P. Andrieux, L. Nadjo and J. M. Sav6ant, J. Electroanal. Chem., 26 (1970) 147 8 M. Mastragostino, L. Nadjo and J. M. Sav~ant, Electrochim. Acta, 13 (1968) 721. 9 M. E. Peover and J. S. Powell, J. Electroanal. Chem., 20 (1969) 427.

10 D. H. Evans, J. Phys. Chem., 76 (1972) 1160. 11 S. W. Feldberg, Digital Simulation in Electroanalytical Chemistry, Vol. 3, Marcel Dekker, New York,

1969, pp. 199-296. 12 H. Matsuda and Y. Ayabe, Z. Elektrochem., 59 (1955) 494. 13 R. S. Nicholson and I. Shain, Anal. Chem., 36 (1964) 706. 14 J. M. Sav6ant and E. Vianello, Electrochim. Acta, 12 (1967) 629. 15 R. Dietz and M. E. Peover, Discuss. Faraday Soc., 45 (1968) 154. 16 R. S. Nicholson and I. Shain, Anal. Chem., 37 (1965) 190.

Documents

Linear sweep voltammetry: Kinetic control by charge transfer and/or secondary chemical reactions: I. Formal kinetics