Electroanalytical Chemistry and lnterfacial Electrochemistry, 61 (1975) 251-263 251 © Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

CONVOLUTION POTENTIAL SWEEP VOLTAMMETRY

PART IV. HOMOGENEOUS FOLLOW-UP CHEMICAL REACTIONS

J. M. SAVt~ANT and D. TESSIER

Laboratoire d'Electrochimie de r UniversitO de Paris VII, 2, Place Jussieu, 75221 Paris Cedex 05 (France)

(Received 7th February 1975)

ABSTRACT

The application of Convolution Potential Sweep Voltammetry to mechanism analysis and rate determination in electrochemical processes involving homogeneous chemical reaction is discussed. The formal analysis of the transition between pure diffusion control and pure kinetic condition is treated in the case of a first order follow-up reaction. The practical applicability of the method is then tested on the reductive pinacolization of acetophenone in acetonitrile using as operational param- eters the sweep rate, the initial concentration and the water content of the medium.

INTRODUCTION

It has been shown previously1 that the advantage of convolution potential sweep voltammetry (CPSV) over conventional linear sweep voltammetry (LSV) in mechanism analysis of electrochemical processes involving homogeneous chemical reactions coupled with the charge transfer steps, essentially concerns the "pure kinetic conditions", i.e. the situation where a stationary state is established by mutual balance of chemical reaction and diffusion. In such conditions, indeed, the convoluted current:

, = 4 f ' i(v) ~z~ jo (t--o)~ dv

can be combined with the current i itself into a simple relationship involving the electrode potential, which characterizes the mechanism under study.

For a first order EC mechanism, for instance:

A+ne ~ B (standard potential: E °)

B ~ C (rate constant: k)

the logarithmic linear relationship is:

E = E°+ (Rr/2nr) In k+ (RT/nF) In [(I~-l)/i]

where It is the limiting value (for E = - oo) of the convoluted current I. Such a possibility is not restricted to first order kinetics and applies as well

for dimerisation, disproportionations, etc. The forms of the logarithmic linear rela-

252 J.M. SAVI~ANT, D. TESSIER

tionships have already been listed for the basic reactions schemes involving ante- cedent and consecutive reactions as well as the ECE-disproportionation sequences (see Table 1 in ref. 1).

The main advantage that can be expected from the use of convolution procedures in treating the LSV data is that the accuracy in the mechanism deter- mination is better, since the information available along the whole i-E curve is entirely used instead of only the peak values.

The main goal of the present work was to test experimentally the applicability of the method to a system involving a simple and known electrochemical/chemical reaction sequence. We selected for this purpose the electrohydrodimerization of acetophenone in acetonitrile with various amounts of water added. Although this reaction was mainly studied in protic media there is little doubt that it proceeds by way of a radical-radical coupling leading to the pinacol (see e.g. ref. 2 and references therein), the effect of water addition being to increase the solvation of the anion radical which results in an acceleration of the dimerization reaction 3. The reaction scheme is therefore of the type:

A + l e ~ B

2B ~ product

Three parameters are thus available in order to vary the kinetic factor 2 = ( R T / F ) k d / v which features the effect of the chemical reaction on the diffusion process4: the sweep rate v, the initial concentration of the substrate c ° and the concentration of water from which the rate constant k is an increasing function.

In pure kinetic conditions the logarithmic analysis corresponding to this reaction has already been established1:

E = E ° + (R T/3F) In (~k/FSD ~) + (R T/F) In [(I, - I)/i ~]

where D is the diffusion coefficient of the substrate A. The effect of the sweep rate and initial concentration which are formally absent in the above equation occurs through i for the sweep rate (i is indeed proportional to v~), and through (11 - I ) / i ~ for the concentration (i, I, and I1 are proportional to c°). The effect of v and c ° appears more clearly in the equivalent formulation:

E = E ° + (R T/3F) In (2kc°/v) + (R T/F)In {(1 - (1/11))/(i/l~ v½) ~ }

in which the last term does not depend upon these parameters. The combined analysis of the i-E and I -E curves does not therefore raise

any particular difficulty when the kinetic parameter 2 is large enough or conversely when it is small, i.e., when pure diffusion control prevails. In this last case the logarithmic linear analysis to be performed is I :

E = E ° + (RT/F) In [(/j - I)/I]

In the intermediate situation, which may well correspond to sweep rates in the experimentally accessible range if the chemical reaction is moderately fast, convolution analysis offers no significant simplification over the conventional analysis of the LSV data. The application of CPSV to electrochemical/chemical reaction

cPsv. FOLLOW-UP CHEMICAL REACTIONS IV 253

sequences therefore requires the chemical reaction to be fast enough for the pure kinetic situation to be reached in not too narrow a range of sweep rates. It follows that it is useful, in order to discuss the practical applicability of the method, to evaluate the order of magnitude of the intermediary kinetic zone in terms of 2 as regards the convoluted current, and the shapes and slopes of the logarithmic analysis.

Such an evaluation will be presented here in the case of a first order EC reaction scheme although the test example we have selected corresponds to second order kinetics. The reason for this is that the calculations are much simpler in the first order case and are adequate in so far as only trends and orders of magnitude are looked for.

EXPERIMENTAL

The experiments were performed in acetonitrile (ACN) with 0.1 M tetra- ethylammonium perchlorate as supporting electrolyte. The water content of the solution was 0.3~o as determined by the Karl Fischer method. The acetophenone was a Prolabo product and was distilled before use.

The working electrode was a mercury droplet hanging on a gold disc; its surface area was about 4 mm 2. The reference electrode was an Ag/AgCIO4 (10-2 M) electrode. The temperature of experiment was 22°C.

The cell, LSV apparatus, digitalization and convolution procedures were the same as already descr ibedl5 .The largest usable sweep rate (~300 V s -1) was limited by the maximal speed of the digital acquisition system (30/~s for 8 bits accuracy), and the minimal number of points (50) on each curve.

The correction for sphericity and uncompensated resistance were the same as previously described LS. The sphericity correction factor D~/ro was typically 0.1 s -~, and the residual resistance remaining between the working and the reference electrode after positive feedback compensation was 12 ~).

RESULTS AND DISCUSSION

Transition between pure kinetic and pure diffusion control For a first order EC reaction scheme the current is given in dimensionless

form by the following integral equation6:

-~ f ~ ~t {~(u ) / ( z -0 ) ~} { l + e x p ( - ~ ' ) exp [ -~ . ( z -u ) ]}du = 1

0

where the various dimensionless variables have the following meaning: current: ~ = i/nFSc°D~(nFv/R T) ~ reaction rate: ;~ = (R T/nF) k/v time: z = (nF/R T) vt potential: ~ = - (nF/RT)(E- E °) During the forward scan, in cyclic voltammetry, the electrode potential E

varies with time according to:

E = E~- vt (E~ initial potential)

and during the reverse scan according to:

254 J.M. SAVI~ANT, D. TESSIER

E = E f + v ( t - O )

where Er and 0 are the potential and time at the point of inversion (El= Ei-v0) . According to these definitions, the dimensionless current-potential curves can

be computed as a function ~P of the dimensionless potential during the forward (increasing values of 4) and the reverse (decreasing values of ~) scans, for each value of the kinetic parameter 2. The backward current trace depends moreover on the inversion potential i.e., in dimensionless form, on: I f = - ( n F / R T ) ( E f - E ° ) .

Once, for a given value of 2 and of ~r, the ~(~) function has been computed, the convolution integral in dimensionless form:

1~ T-/ = ~--½ [~/(0)/("C-- o)½]du (l~-'l=I/Ii) ~0

can be calculated as a function of the dimensionless potential ~ for the forward and backward scans,

This was done numerically, evaluating the integrals by the half-interval rectangle method as classically done in LSV analysis7; the interval width was taken as 0.01.

The 1~(~) values in the forward scan were then treated by two different logarithmic analyses:

(a) as if the process were purely diffusion controlled (DO),

- ~ = I n [ ( 1 - I~)/I~P] i.e., E = E°+(RT/nF) In [(11-1)/1]

(b) as if it were under a pure kinetic control (stationary state situation) (KP),

- ~ = ½ In 2 + l n [ ( 1 - 1 ~ ) / ~ ]

i.e., E = E° + ( R T/2nF) In k + ( R T/nF) In [(11-1)/i]

Some of the results.are represented on Figs. 1-4 for 2= 10 -3, 10 -1, 1 and 10 ~. Each diagram shows the forward and reverse [fiE) and I(E)] curves for a few values of the inversion potential, and the curves representing the logarithmic analysis for the DO and KP assumptions.

For 2 = 10- 3 (Fig. 1) practically pure diffusion control is reached: the forward and reverse I(E) curves are superposable whatever the value of the inversion potential. The DO analysis gives a straight line with the correct slope, i.e., - 1 as concerns and, e.g., with n = 1 and 22°C, 58.5 mV per decade as concerns the electrode potential. Conversely the line obtained with the KP analysis is somewhat curved and its slope at the half-wave potential of the I(/E) curve is -- 1.23, i.e., 71.8 mV per decade for n = 1 and 22°C which is far from .tt4e theoretical value of 58.5 mV.

Conversely, for 2= 104 (Fig. 4) practically the pure kinetic conditions are achieved:the i( E) pattern exhibits an irreversible behaviour and the correct analysis is the KP one which gives a slope of 58.5 mV (n = 1.22°C) whereas the DO analysis results in a curved line with a slope at E½ of 49.5 mV. It is seen that the reverse I(E) trace is no longer superposable on the forward trace and that it depends upon the value of the inversion potential. It tends toward zero more slowly when the inversion is made at potentials more and more cathodic to the peak potential.

In the intermediate range (Figs. 2 and 3) these last features are also observed

CPSV. FOLLOW-UP CHEMICAL REACTIONS IV 255

, f

-1 0.5 0.2-

-1 i -05

I I

-127 - 3 8 2

E/mV Fig. 1. First order EC reaction scheme Current, convoluted current and logarithmic analysis. Kinetic parameter: 2=10 -3. ( 0 ) DO, ( x ) KP. (n=l, 22°C).

_c

(DO) (KF')

.,., o..

o _ ~ - - ~ / o - ~ - o

-1 DO* ~

~KP

E, /mV •

%

-02" -0.5

Fig. 2. First order EC reaction scheme. Current, convoluted current and logarithmic analysis. Kinetic parameter: 2 = I0- L ( 0 ) DO. ( × ) KP. (n = 1, 22°C).

256 J.M. SAV]~ANT, D. TESSIER

(DO) (KP) ~ ~

/ 0.5

"' -02-

-1 --1 ~ -Q5

DOe ",', K P

I I • 0 -25,4

E/mV

Fig. 3. First order EC reaction scheme. Current, convoluted current and logarithmic analysis. Kinetic parameter: 2= I. (O) DO, ( x ) KP. (n= l, 22°C).

_(DO)

~ o.2-

. ~ ~ -o o - - ~ c - - o ,-.~-.~

B ~

• \ -O2- ,' ~ -1 [- --1 DO. \ -0.5

"~ KP ./ - 1 0

[

~ \ o

e/my

Fig. 4. First order EC reaction scheme. Current, convoluted current and logarithmic analysis. Kinetic parameter: 2= 104. (O) DO, ( x ) KP. (n= 1, 22°C).

CPSV. FOLLOW-UP CHEMICAL REACTIONS IV 257

together with a partial reversibility of the reverse i(E) trace. Neither of the two logarithmic analyses give a perfectly straight line with the right slope. The difference from the correct slope for each analysis can therefore be taken as a quantitative criterion of the departure from complete reversibility or irreversibility.

This is shown on Fig. 5 where the slopes at E½ of the I(E) forward curve are represented for both analyses as a function of the kinetic factor )~.

70

E

to

50 f

-123

s~ -1

' ' 6 ' - 4 '

log (kRT/vnF)

Fig. 5. First order EC reaction scheme. Slopes at E, of the pure diffusion control (DO) and pure kinetic control (KP) analyses as a function of the kinetic parameter, s¢=slope in dimensionless form. sE = slope in terms of potential for n = 1 and at 22°C.

The experimental uncertainty on the slope measurement is, at best 5, 1 mV. Consequently, the range of intermediary kinetic behavior extends from log )~ = -0 .85 to log 2 = 1.05, i.e. over about two orders of magnitude.

If the chemical reaction is moderately fast, one can achieve diffusion control by raising the sweep rate, and the pure kinetic condition by decreasing it. At least two orders of magnitude of sweep rate are necessary to pass from one limiting situation to the other. If the experimental system is such that this is possible, the standard potential E ° can be determined in the pure diffusion situation, i.e., at high sweep rates, from the intersection of the DO straight line with the E-axis, and [E ° +(RT/nF)lnk] from the intersection of the KP straight line with the E-axis at slow sweep rates when the pure kinetic conditions are achieved. The rate constant k is then obtained from these two measurements.

258 J .M. SAVI~ANT, D. TESSIER

*-..., " \

z~- o , - _ z-I< _£ % . , . -7

-,,,,,,, -

"~•~, ~mV

-2.~o -&5 -z~o -

E/V Fig. 6. Acetophenone 5 mmol 1-1 in ACN+0.1 M Et4NCIO4+ 1.3% H20. v=337 V s-L Logarithmic analysis. The number on each curve is the value of the slope at E~ in mV/decade.

\

I (DE)) *~ ~~~A

; r - o ,

\

ElY

Fi~. 7. Acetophenone 5 mmol 1 -~ in ACN+0 .1 M Et4NCIO4+ 1.3% HzO. v=3.37 V s - t . Logarithmic analysis. The number on each curve is the value of the slope at E l in mY/decade.

(KP)

-5

-6

- 7

- 2.~10

CPSV. FOLLOW-UP CHEMICAL REACTIONS IV 259

Electrochemical pinacolization of acetophenone The i(E) curves were recorded as a function of the sweep rate for three

different media; without addition of water (the remaining concentration of water in the medium was 0.3~), with addition of 1~ and then 5~o water. An acetophenone concentration of 5 mmol 1-1 (and in some cases 1 mmol 1-1) was used. Eight sweep rates were used ranging from about 0.1 to 300 V s- 1 every half-order of magnitude. For each set of values of the experimental parameters the forward i(E) trace was convoluted with the function (nt) -½, and the resulting I(E) curve was then analysed according to the pure diffusion (DO) and to the pure kinetic (KP) logarithmic relationships and the slopes at E½ of the I(E) curve were determined.

The results obtained with 1% water added are the most illustrative of the transition from pure kinetic conditions to pure diffusion control on raising the sweep rate. It is seen in Fig. 8 which corresponds to 0.11 V s-1 that the KP analysis gives a slope close to the theoretical value of 58.5 mV/decade, whereas the DO analysis leads to 42.2 mV/decade. The system is thus under pure kinetic conditions.

c

(Do) AX~XA (KP)"

-1

\

I

EIv

- 4

=

- 5

-6

Figl 8. Acetophenone 5 mmol 1 ±1 in ACN+0.1 M Et4NCIO4+ 1.3~o H20. v=0.11 V s -~. Logarithmic analysis. The number on each curve is the value of the slope at E~ in mV/decade.

Conversely at 337 V s-1 it is under practically pure diffusion control. It is seen, in Fig. 6 that the DO analysis leads to the correct slope whereas the KP analysis gives 83 mV/decade.

At an intermediate sweep rate, for instance at 3.37 V s-1 (Fig. 7) neither the DO nor the KP analysis leads to the right slope: the first one gives 45.9 mV/decade and the second one 64.5.

260 J. M. SAVI~ANT, D. TESSIER

"7 •

> E

O

8 0

7 0

÷

.i)/

5C

4ol

J

f f° ÷ / ÷

Jog (v/V s -1)

Fig. 9. Acetophenone 5 mmol 1-1 in ACN+0.1 M Et4NCIO4+l.3 ~ H20. Slopes of the logarithmic analyses as a function of the sweep rate.

In Fig. 9 the complete variation of the slopes characterizing the DO and KP analyses over the 3.5 decades of sweep rate are shown, and illustrate the transition from one limiting situation to the other.

The behavior of acetophenone in the present medium provides good con- ditions for determining the rate constant k. The standard potential E ° is first measured at 337 V s- 1 which gives E ° = - 2.263 V. Then at the slowest sweep rate, 0.11 V s -1, the potential:

E k = E ° + (R T /3F) In (2k /3FSD ½)

is provided by the intersection of the KP analysis straight line with the potential axis: E k = - - 2 . 1 1 3 V. FSD ~ = l~/c is obtained, knowing C °, through the measurement of/1 which was anyway necessary to construct the logarithmic plots. This leads to: k = 4 x 105 mo1-1 1 s -1.

The variation of the slopes with the sweep rate for 5% water added are shown in Fig. l0 for c°= 5 mmol 1-1. It is seen that the system is shifted toward pure kinetic conditions due to the acceleration of the dimerization reaction: the D O

CPSV. FOLLOW-UP CHEMICAL REACTIONS IV 261

behavior is no longer attainable in the available sweep range and the KP conditions are achieved for 1.5 decades of sweep rate starting from the lowest one.

80

70

%

> 60

50

40

KP

+

log (v/v s - I ) Fig. 10. Acetophenone 5 mmol 1-1 in ACN+0.1 M Et4NClO4+5.3% H20. Slopes of the logarithmic analyses as a function of the sweep rate.

The same kind of diagram is shown in Figs. 11 and 12 in the case of no water added (residual concentration: 0.3%) for c ° = 5 and 1 mmol l --1, respectively. Now the system is shifted toward pure diffusion control: the KP situation can no longer be reached and the DO behavior is observed for 0.5 decade of sweep rate for 5 mmol 1-1 starting from the highest one and 1.5 decades at 1 mmol 1-1 showing the second order character of the overall kinetics. The standard potential, as determined from the intersection of the DO straight line with the potential axis, is now - 2.322 V. The difference between this value and that obtained with 1% water added, i.e., 59 mV reflects the increasing solvation of the acetophenone anion-radical as the water content of the medium is raised.

The increase in solvation results in a smaller coulombic repulsion between the anion radicals and thus in an acceleration of the dimerization. This effect can be evaluated using the slope curves of Figs. 9-12 which allow the coupling rate constant to be estimated at 0.3 and 5.3% H20 according to the following procedure. As seen above 'with 1.3% H20 one can pass by raising the sweep rate from pure

262 J .M. SAVI~ANT, D. TESSIER

80[ 70

"7

8

> 60

o_ o

+

+

50 /0

40

-1 0 1 2

log(v/V s- 1 )

80

70

"7 aa ~d

6c

f f l

50

40

+ / KP

+

log (v/v s-')

Fig. 11. Acetophenone 5 mmol 1-t in ACN + 0.1 M EtaNC104 + 0.3% H20. Slopes of the logarithmic analyses as a function of the sweep rate.

Fig. 12. Acetophenone 1 mmol 1 - l in ACN+0.1 M Et4NCIO4+0.3% H20. Slopes of the logarithmic analyses as a function of the sweep rate.

kinetic control to pure diffusion control. It follows that the experimental points cover practically all the useful part of the slope curves as can be seen in Fig. 9. Then it is possible by horizontal shifting to make the slope curves of Fig. 10 (5.3% H20) to coincide with the curves of Fig. 9. The shift in log v which is necessary for this thus provides the logarithm of the ratio of the dimerization rate constants passing from one concentration of water to the other. With 1.3% water the rate constant was determined through an E k measurement as being 4 x 105 mol-1 l s - 1 It follows that for 5.3% HaO k = 1.3 x 106 mol - 1 1 s- 1. The same procedure can be applied to the curves of Figs. 11 and 12 and the respective values found for k are then 6 × 10 4 and 3 × 10 4 mol - 1 1 s- 1 corresponding to 1 mmol 1-1 and 5 mmol 1-

CPSV. FOLLOW-UP CHEMICAL REACTIONS IV 263

initial concentration. The difference in these last results gives an idea of the error in such a determination of the dimerization rate constant.

These results show that the acceleration of°the dimerization reaction is less pronounced as the water concentration increases as can be expected chemically. Indeed as the water concentration increases, owing to preferential solvation, a water shell tends to build up around the anion radical which is therefore less and less sensitive to further additions.

ACKNOWLEDGEMENTS

The work was supported in part by the C.N.R.S. (Equipe de Recherche Associ6e No 309: Electrochimie Organique). Prof. M. Hulin, Universit6 de Paris VI, is thanked for the permission to use the 1130 IBM computer of the Groupe de Physique des Solides de rEcole Normale Sup6rieure, Paris.

REFERENCES

1 J. C. Imbeaux and J. M. Sav6ant, J. Electroanal. Chem., 44 (1973) 169. 2 L. Nadjo and J .M. Sav6ant, J. Electroanal: Chem., 33 (1971) 419. 3 E. Lamy and J. M. Sav6ant, J. Electroanal. Chem., 50 (1974) 141. 4 J. M. Sav6ant and E. Vianello, C.R. Acad. Sci. Paris, 256 (1963) 256. 5 L. Nadjo, J. M. Sav6ant and D. Tessier, J. Electroanal. Chem., 52 (1974) 403. 6 L. Nadjo and J. M. Sav6ant, J. Electroanal. Chem., 48 (1973) ll3. 7 H. Matsuda and Y. Ayabe, Z. Elektrochem., 59 (1955) 494.