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Influence of the adsorption of the depolarizer or of a product of the electrochemical reaction on polarographic currents
ELECTROANALYTICALCHEMISTRYAND INTERFACIALELECTROCHEMISTRY Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
333
I N F L U E N C E OF T H E A D S O R P T I O N OF T H E D E P O L A R I Z E R OR OF A P R O D U C T OF T H E E L E C T R O C H E M I C A L R E A C T I O N O N P O L A R O GRAPHIC CURRENTS XVII. THEORETICAL STUDY OF A REVERSIBLE SURFACE REACTION FOLLOWED BY A FIRST ORDER CHEMICAL REACTION IN LINEAR POTENTIAL SWEEP VOLTAMMETRY E. LAVIRON Laboratoire de Polarographie organique associ~ au C.N.R.S., Facult~ des Sciences, 21 Dijon (France)
(Received 22nd July 1971 ; in revised form 7th September 1971)
In a series of previous papers ~-3, we have presented a theoretical study of adsorption peaks appearing in linear potential sweep voltammetry, when the electrochemical reaction is reversible and when the different species involved are strongly adsorbed. In the first article a the case of a simple red-ox reaction was considered; the two following papers 2'3 dealt with the case in which the product of the electrochemical reaction undergoes a second order chemical reaction. In this paper we present a theOretical study of adsorption peaks with the same hypotheses (reversible reaction, strong adsorption of Ox and Red), when the electrochemical reaction is followed by a first order chemical reaction, according to the scheme : O+ne ~
R
R -k p We shall consider the case of a reduction for the forward peak and of an oxidation for the backward peak; the reverse case can be readily deduced. FORWARD PEAK Equation o f the peak
As in the previous cases 1 3 we have, by assuming that the adsorption obeys a Langmuir isotherm : -
Fo/FR= 1/z = ~/
(1)
z = exp [ - a(E - Ep,n)]
(2)
with
a = n F / R T, F o and F R are the surface concentrations (mol c m - 2 ) and Ep, n is the peak potential of the normal peak obtained in the absence of chemical reaction. E is given by the relationship E = E i - v t ; Ei is the initial potential and v the rate of potential change in V s-1. As previously 1- 3, Fo(tl) will represent the value of the surface concentration at time t = 0 (t 1 delay time). We shall assume as previously ~- 3 that: J. Electroanal. Chem., 35 (1972)
334
E. LAVIRON
(a) O, R and P are so strongly adsorbed that the number of molecules which can diffuse away from the electrode is negligible when compared to the number of those which remain adsorbed. Although it is difficult to determine theoretically the limiting values of the adsorptivity which are necessary for this condition to hold, an experimental verification can easily be made: the amount of molecules adsorbed (obtained e.g. by graphical integration of the current) must correspond to that given by the equation of Koryta 7. (b) the rate of potential scan v is large enough, so that the amount of molecules which reach the surface during the scan is negligible when compared to the amount already adsorbed at time tl, and that the area can be considered as constant (it can be shown that under the usual conditions the rate must exceed about 1 Vs- 1). (c) Fo(q) is smaller than its maximum value tim" Then, at time t: the value of Fo is equal to Fo (t 1), minus the amount which has disappeared by reduction : -
-
l lo idt
Fo =Fo(tl) -
-
nFA
(3)
the sum of the surface concentrations is equal to fro(t1):
ro + r. + r . = ro(tl)
(4)
The rate of the chemical reaction is given by: dFp/dt = kF R
(5)
By combining (1)-(5), the following differential equation is obtained: d F o / d t + ( k + a v ) Z(I +Z) -1 Fo = 0
(6)
For t = 0 and when E i is sufficiently large when compared to Ep,n, (mathematically for Ei-Ep,n--~oo), X'--~O and Fo must tend towards Fo(tl). By taking this condition into account, from (6): Fo=Fo(tl)
(I+z) -(l+t)
(7)
with I --
k av
RTk nF v
whence : FR = Fo(tl) Z(1 +;0 -(1 +')
(8)
The current is given by i = - n F A d F o / d t ; thus: i= [ ( n F ) 2 / R T ] A v F o ( t l ) ( i + 1)Z/(1 _{._•)2 + /
(9)
or
i = (F2/R T)n 2 A v F o (tl) f(x, l)
with F 2 / R T = 3.76 x 106 at 25°C. The units are as follows: i (A); A (cm2); v (V s - l ) ; Fo(tl)(mol cm-2). J. Electroanal. Chem., 35 (1972)
(10)
335
ADSORPTION OF DEPOLARISER OR PRODUCT. XVII
Shape of the peak When the ratio v/k tends towards infinity (rate of potential sweep much larger than the rate of the chemical reaction),/---,0; eqn. (9) reduces to the equation obtained in the simple case without chemical reaction 1. The duration of the electrolysis is too short for the chemical reaction to occur. Variations of the function f()~, l) for different values of 1 are shown in Fig. 1. As i is proportional to f(x, l), these curves give the shape of the peak. When 1increases (v decreases) the peak shifts towards positive potentials (see next paragraph) and becomes narrower and asymmetrical. The width t5 at mid-height varies from 90.6/n mV for l= 0 to 66/n mV for l--,oe. f CX,L) 4 3
n (E-Ep,n)/mV
Fig. 1. Variations off(;(,/) with n(E-Ep,n), l: (1) 0, (2) 1, (3) 10, (4) 100.
Variation of the peak potential Ep I f ~(p
is the value of ~( corresponding to Ep, one obtains readily :
Zp= 1/(1 + l)
(11)
Hence, at 25°C, in view of eqn. (2):
n(Ep- Ep,n) -- 0.0592 log(l + l) The variation of n(Ep-Ep,n) as a function of log l is given in Fig. 2. When l - - ~ , Ep tends asymptotically towards the value: Ep = Ep,n+ (2.3RT/nF) log l or
2.3RT
Ep=Ep,n-t- ~
RT
2.3RT
log~ff + ~
log k
2.3RT log v nF
(12)
When the ratio v/k is small enough, the peak potential thus varies linearly with log v, with a slope of 59.2/n mV per logarithmic unit. In the case of a volume reaction 4'5, the value of the slope is 29.6/n inV. J. Electroanal. Chem., 35 (1972)
336
E. LAVIRON
Log I,
!,
26
Io
_!,
-2 i
CEp.Ep,n)/,,,V Fig. 2. Variations of n(Ep-Ep,n) (peak potential) with log I. )2+1 ~,4
k._
Log L
2,
II
-ll
-21
Fig. 3. Variations of fp(X, l) = [(1 + 1)/(2+ 1)]2 +' with log I. Variation of the peak current T h e m a x i m u m fv(;(, l) of f(;(, l) is given b y :
fp(Z, l ) = [(1 +l)/(2+l)] 2+~ Figure 3 shows the variation of the m a x i m u m as a function of log 1. T h e value J. Electroanal. Chem., 35 (1972)
ADSORPTION OF DEPOLARISER OR PRODUCT. XVII
337
of the maximum passes from 0.25 for l = 0 to e -1 =0.368 for l---}oo. By introducing the value of fp(z, l) into eqn. (10), one obtains for the peak current iv: ip= (F2/RT) n 2 AvFo(tl)[(1 +/)/(2 +/)]2 +,
(13)
ip can also be written in the form: ip = n F A k F o ( q ) I - 1fp(g, l)
For a given value of Fo(tl), nFAkFo(tl) = C .~. To study the variation of ip as a function of v, it is sufficient to study the variation of l- 1 fp(Z, l). The variation of log l- 1fp (X, l) is shown in Fig. 4 as a function of log l [log ip = f ( - l o g v)]. As can be seen from the Figure, the peak current varies proportionally to v when v is either very large or very small in comparison to k. For intermediate values, the variation coefficient is a little smaller than 1, but its value is always about 1 (ip~C-~. × v°'9 when - 0 . 5 < log l < 0.5). It can be concluded from this result that a study of ip as a function ofv cannot give valuable information on the chemical reaction. V++t)
--
~1
,0
,-1
togt
Fig. 4. Variations of log {1 1fp(~, 1)} with log l (log ip with log v). (1) Asymptote log 1+ log 0.25, (2) asymptote l o g / - l o g e. BACKWARD PEAK
Equation of the peak The reaction scheme is still the same, but E is now given by E = Ei + vt, so that :
(14)
d z / d t = - avz
The surface concentration Fo is proportional to the amount of eiectricity which has been consumed: Fo =
(nFA)-I
f'o
i dt J. Electroanal. Chem., 35 (1972)
338
E. LAVIRON
(the minus sign shows that anodic currents are negative). F~ is the initial concentration of O. The sum of the surface concentrations are equal to Fo(q) so that eqn. (4) still holds, and (5) is still valid. By combining (1), (4), (5) and (14), the following differential equation is obtained : dFo/dt + (k - av) Z(1 + ~)-1Fo = 0
whose resolution gives for Fo: Fo = Ct.(l +Z) z-1
(15)
with 2 = (RT/nF)(k/v) Hence for FR: FR = c:x(1 + z)~- '
Taking into account the condition FR= F~ (F~ initial concentration of FR) when E = Ei, the value of the constant can readily be calculated :
C: = r~z~- I (1+ zi)1-~ with Zi = exp [ - a (E i - - Ep,n)] The current is then given by: i = - n F A dFo/dt = - (nF)Z/R T A v q Z ; 1 (1 + or
i= - (F2/R T)n2 AvF~ ¢(Z,
Z i )1 - 2
(1 -- 2) Z (1 +Z)z-
2, Zi)
2
(16) (17)
The value of F~ is given by eqn. (8) in which Z is replaced by Xi. The current can then be written: i= - (F2/R T)
n 2 a v F o (t 1)(1 +
Zi)- (l + a)(1 _ 2) Z(1 + Z)~- 2
(18)
or
i= - (F2/R T)n 2 A v r o (tl) 4' (Z, Zi, l, 2)
(19)
The backward peak depends on the same parameters (rate of potential sweep, rate constant of the chemical reaction) as the forward peak, and on a new parameter, the initial potential E i of the reverse sweep. Shape of the peak The variations of • as a function of n ( E - E p , n) are shown in Fig. 5 for different values of 2 and for n(Ei-Ep,n) = - 0 . 2 V. The function ~ ( E - E p , , ) passes through a maximum for Z---Xp=I/(1-2), when 2 < 1, which gives for the peak potential Ep:
Ep = Ep, n --~(2.3R T/nF) log (1-2)
(20)
The peak potential Ep theoretically varies from Ep, n to -o(3 when 2 varies from 0 to 1 ; as can be seen from Fig. 5 and eqn. (20), the peak potential remains virtually equal to Ep, n when 2 is sufficiently different from 1. By replacing Z by Zp in eqn. (18), one obtains for the peak c u r r e n t i v ( 2 < 1): ip = - (F2/R T)n 2 A v r o (tl)(1 + zi)- (' + z~[(1 - 2)/(2 - 2)] 2- a J. Electroanal. Chem., 35 (1972)
(21)
339
ADSORPTION OF DEPOLARISER OR PRODUCT. XVII n iE- Ep,n)/mV
100,
Q
_100.
.
.
.
5
Fig. 5. Variations of function ~bwith , ( E - Ep,,) for ,(E i -Ep,.)= - 0 . 2 V. 2:(1) 0, (2) 0.02, (3) 0.04, (4) 0.08, (5) 0.12, (6) 0.20.
As shown by eqn. (16) or (18), i is always equal to zero when 2 = 1, i.e. when v = R T k / n F = 0.0256 kin at 25°C. The significance of this result is as follows: at the initial potential Ei, a certain amount of O and R are present. When the potential is scanned in the anodic direction, F~ always decreases and Fo varies in such a way that the ratio Fo/FR is at all times equal to )~.The variations o f f o and F~ can occur in two ways : through the chemical reaction, which makes R decrease, and through the electrochemical reaction, which transforms R into O or O into R. If the rate of the chemical reaction is small enough when compared to the rate of the sweep (2< 1), the decrease of R and the increase of O occur mainly through the electrochemical reaction, which causes an anodic current to appear. When 2 = 1, the rate of the chemical reaction is just sufficient to cause a decrease in R such that the ratio FR/Fo is equal at all times to )~; as shown by eqn. (15), Fo is then constant. There is no electrochemical reaction, and i is equal to zero. If 2 > 1, the sign of the current changes (eqn. (16) or (18)), and a reduction current is obtained. This can be explained as follows : the chemical reaction causes too large a decrease in R, which is compensated by an electrochemical transformation of O into R: hence the appearance of the reduction current. This reduction current will however remain virtually equal to zero if Ei is negative enough, for the amount of 0 present at the initial potential Ei will then be very small. If the initial potential of the backward sweep is not too negative, i.e. if the forward sweep is stopped before the current has decreased to zero, the cathodic current can on the contrary have an important value, as shown by Fig. 6. The curves in this Figure have been obtained in the following way: the forward curve is given by the function f(L l) (eqns. (9) and (10)), and the backward curve by the function 4)' (Z, 2, )~i, l) (eqns. (18) and (19)). The rates of potential sweep are assumed to be equal for the forward and for the backward sweep. The potential at which the potential scan is reversed J. Electroanal. Chem., 35 (1972)
340
E. LAVIRON
has been taken equal to Ep,n(~i = 1). Analogous curves are obtained if the scan is reversed at any point along the peak. These results can be used advantageously for the determination of the value of k (see next paragraph).
t (X,L)
I,:I, (x,x,,t) Fig. 6. Variation of current for different values of I. The forward and the b a c k w a r d rates of potential sweep are the s a m e (l= 2). The potential scan has been reversed for E = Ep,.. ( , ) l = 2 = 0.2, ( ,, -)l= 2 = 1,
(,,,)l=~=2.
DETERMINATION OF THE VALUE OF THE RATE C O N S T A N T
k
Comparison of the forward and backward peaks As in the case of a second order reaction 2'3, a comparison of the height of the backward peak with that of the forward peak allows the rate constant k to be calculated. A different rate of potential sweep could be chosen for the forward and backward scans ; experimentally it is however simpler to use the same rate (I = 2) : the height of the peaks will then be of the same order of magnitude, which leads to a better precision ; on the other hand capacity currents will have the same importance. In the general case, the absolute value y of the ratio of the backward peak current iv,r to the forward peak c u r r e n t ip,a is obtained by dividing (21) by (13):
•
fl
=
ip,atP"--(l+zi)-~'+~)
~ 1-2~2-~ (2+1~ 2+/ \2~--21
\~1
This equation is valid for any value of Zl; for l = 2, it can be written as : J. Electroanal. Chem., 35 (1972)
(22)
ADSORPTION OF DEPOLARISER OR PRODUCT. XVII
, 2111--I~2-1(2"}-1~
Y=(I+zi)- ~2~l-l)
341
2+1
\~l/
(23)
The variations ofy as a function of log I are shown in Fig. 7 for different values and for l = 2.
o f n (E i - - Ep,n)
Lo9 L
Fig. 7. Variations of the ratio y ip,r/ip, a with log 1 for the same sweep rate for the forward and for the backward sweep n(Ei-Ep,,): (1)0, (2) -0.2, (3) -0.4. =
Equation (23) and Fig. 7 allow the calculation of the value of k from the experimental value of y. Let Ye be this value, obtained for a value ve of v. From Fig. 7, the corresponding value le of I can be obtained. Equation (7) gives :
k=(nF/RT)/eV~=38.9 l~nv~at
25°C
(24)
k is expressed in s- 1, v in V s- 1. If for example v~ is chosen so that the height of the backward peak is half that of the forward peak, y~ = Yo.5 =0.5. Table 1 gives the corresponding values of lo.5 and k for different values of n(Ei-Ep,n) TABLE 1
n(EI-Ep,,)/V
lo.5 =2o.s
k/s -1
0 -0,2 -0.4
0.340 0.0425 0.0218
13.23 nvo.5 1.65 nvo.5 0.85 nvo.5
As in a previous case 2, the initial potential E i of the backward scan can be easily chosen if n(Ei-Ep,.) is negative enough, by taking the peak potential of the backward peak as a reference. For example, eqn. (20) shows that the difference between nEp and nEp,n is smaller than 1 mV f o r / = 2 = 0 . 0 4 2 5 or 0.0218. If n(Ei-Ep,n) is not very negative, the precision is not so good. For example, for Ei = Ep,n and for l = 2 = 0.34, the difference is about 10 mV, and varies rapidly with 2. It is therefore more advantageous to use negative values of n(Ei-Ep,n) to determine k.
Study of the backward current As was shown, the backward current is constantly equal to zero for 2 = 1, J. Electroanal. Chem., 35 (1972)
342
E. LAVIRON
whatever the initial potential E i of the backward scan; Vo being the corresponding value of v, k = 38.92 nvo
(25)
Experimentally (see e.g. Fig. 6), the potential scan will be reversed when the forward current still has an important value (e.9. in the vicinity of the peak potential), and v will be adjusted to such a value that the backward current is constantly equal to zero. Although the precision in the determination of vo is likely to be smaller than in the determination of ve, the method can be advantageously used in certain cases, for the value of vo is smaller than the corresponding value of re. As the upper value of k which can be determined is limited by the values of v which can be used experimentally without deformation in the curves (e.g. because of the i-R drop), higher values of k can be determined by using eqn. (25) rather than (24). For n ( E i - E p , n ) = - 0 . 2 V, for example, v0.5 =23.6 Vo. APPLICATIONS
The application of the present theory to the case of the reduction of azobenzene to hydrazobenzene, followed in an acid medium by the benzidine rearrangement, will be presented in a following paper 6. As in the preceding cases 2'3, the present theory can be applied to the same reaction scheme in thin layer voltammetry 6. SUMMARY
A theoretical study of the peaks appearing in linear potential sweep voltammetry is presented under the following conditions : (a) the electrochemical reaction is reversible; (b) the depolarizer and the products of the electrochemical reaction are strongly adsorbed; (c) the product of the electrochemical reaction undergoes a firstorder reaction according to the scheme:
O+ne ~- R
Rk-~z
The equations for the reduction and oxidation peaks have been calculated, and their characteristics are described. The rate constant of the chemical reaction can be determined on the basis of this theory. The results of this theoretical study can be directly applied to the same reaction scheme in thin layer voltammetry. REFERENCES 1 2 3 4 5 6 7
E. LAVIRON, Bull. Soc. Chim. France, (1967) 3717. E. LAVIRON, Electrochim. Acta, 16 (197l) 409. E. LAVIRON,Jr. Electroanal. Chem., 34 (1972) 463. J. M. SAVEANTAND E. VIANELLO, C. R., 256 (1963) 2597. R. S. NICHOLSON AND I. SHAIN, Anal. Chem., 36 (1964) 718. E. LAVIRON, to be published. J. KORYTA, Collect. Czech. Chem. Commun., 18 (1953) 206.
J. Electroanal. Chem., 35 (1972)
333
I N F L U E N C E OF T H E A D S O R P T I O N OF T H E D E P O L A R I Z E R OR OF A P R O D U C T OF T H E E L E C T R O C H E M I C A L R E A C T I O N O N P O L A R O GRAPHIC CURRENTS XVII. THEORETICAL STUDY OF A REVERSIBLE SURFACE REACTION FOLLOWED BY A FIRST ORDER CHEMICAL REACTION IN LINEAR POTENTIAL SWEEP VOLTAMMETRY E. LAVIRON Laboratoire de Polarographie organique associ~ au C.N.R.S., Facult~ des Sciences, 21 Dijon (France)
(Received 22nd July 1971 ; in revised form 7th September 1971)
In a series of previous papers ~-3, we have presented a theoretical study of adsorption peaks appearing in linear potential sweep voltammetry, when the electrochemical reaction is reversible and when the different species involved are strongly adsorbed. In the first article a the case of a simple red-ox reaction was considered; the two following papers 2'3 dealt with the case in which the product of the electrochemical reaction undergoes a second order chemical reaction. In this paper we present a theOretical study of adsorption peaks with the same hypotheses (reversible reaction, strong adsorption of Ox and Red), when the electrochemical reaction is followed by a first order chemical reaction, according to the scheme : O+ne ~
R
R -k p We shall consider the case of a reduction for the forward peak and of an oxidation for the backward peak; the reverse case can be readily deduced. FORWARD PEAK Equation o f the peak
As in the previous cases 1 3 we have, by assuming that the adsorption obeys a Langmuir isotherm : -
Fo/FR= 1/z = ~/
(1)
z = exp [ - a(E - Ep,n)]
(2)
with
a = n F / R T, F o and F R are the surface concentrations (mol c m - 2 ) and Ep, n is the peak potential of the normal peak obtained in the absence of chemical reaction. E is given by the relationship E = E i - v t ; Ei is the initial potential and v the rate of potential change in V s-1. As previously 1- 3, Fo(tl) will represent the value of the surface concentration at time t = 0 (t 1 delay time). We shall assume as previously ~- 3 that: J. Electroanal. Chem., 35 (1972)
334
E. LAVIRON
(a) O, R and P are so strongly adsorbed that the number of molecules which can diffuse away from the electrode is negligible when compared to the number of those which remain adsorbed. Although it is difficult to determine theoretically the limiting values of the adsorptivity which are necessary for this condition to hold, an experimental verification can easily be made: the amount of molecules adsorbed (obtained e.g. by graphical integration of the current) must correspond to that given by the equation of Koryta 7. (b) the rate of potential scan v is large enough, so that the amount of molecules which reach the surface during the scan is negligible when compared to the amount already adsorbed at time tl, and that the area can be considered as constant (it can be shown that under the usual conditions the rate must exceed about 1 Vs- 1). (c) Fo(q) is smaller than its maximum value tim" Then, at time t: the value of Fo is equal to Fo (t 1), minus the amount which has disappeared by reduction : -
-
l lo idt
Fo =Fo(tl) -
-
nFA
(3)
the sum of the surface concentrations is equal to fro(t1):
ro + r. + r . = ro(tl)
(4)
The rate of the chemical reaction is given by: dFp/dt = kF R
(5)
By combining (1)-(5), the following differential equation is obtained: d F o / d t + ( k + a v ) Z(I +Z) -1 Fo = 0
(6)
For t = 0 and when E i is sufficiently large when compared to Ep,n, (mathematically for Ei-Ep,n--~oo), X'--~O and Fo must tend towards Fo(tl). By taking this condition into account, from (6): Fo=Fo(tl)
(I+z) -(l+t)
(7)
with I --
k av
RTk nF v
whence : FR = Fo(tl) Z(1 +;0 -(1 +')
(8)
The current is given by i = - n F A d F o / d t ; thus: i= [ ( n F ) 2 / R T ] A v F o ( t l ) ( i + 1)Z/(1 _{._•)2 + /
(9)
or
i = (F2/R T)n 2 A v F o (tl) f(x, l)
with F 2 / R T = 3.76 x 106 at 25°C. The units are as follows: i (A); A (cm2); v (V s - l ) ; Fo(tl)(mol cm-2). J. Electroanal. Chem., 35 (1972)
(10)
335
ADSORPTION OF DEPOLARISER OR PRODUCT. XVII
Shape of the peak When the ratio v/k tends towards infinity (rate of potential sweep much larger than the rate of the chemical reaction),/---,0; eqn. (9) reduces to the equation obtained in the simple case without chemical reaction 1. The duration of the electrolysis is too short for the chemical reaction to occur. Variations of the function f()~, l) for different values of 1 are shown in Fig. 1. As i is proportional to f(x, l), these curves give the shape of the peak. When 1increases (v decreases) the peak shifts towards positive potentials (see next paragraph) and becomes narrower and asymmetrical. The width t5 at mid-height varies from 90.6/n mV for l= 0 to 66/n mV for l--,oe. f CX,L) 4 3
n (E-Ep,n)/mV
Fig. 1. Variations off(;(,/) with n(E-Ep,n), l: (1) 0, (2) 1, (3) 10, (4) 100.
Variation of the peak potential Ep I f ~(p
is the value of ~( corresponding to Ep, one obtains readily :
Zp= 1/(1 + l)
(11)
Hence, at 25°C, in view of eqn. (2):
n(Ep- Ep,n) -- 0.0592 log(l + l) The variation of n(Ep-Ep,n) as a function of log l is given in Fig. 2. When l - - ~ , Ep tends asymptotically towards the value: Ep = Ep,n+ (2.3RT/nF) log l or
2.3RT
Ep=Ep,n-t- ~
RT
2.3RT
log~ff + ~
log k
2.3RT log v nF
(12)
When the ratio v/k is small enough, the peak potential thus varies linearly with log v, with a slope of 59.2/n mV per logarithmic unit. In the case of a volume reaction 4'5, the value of the slope is 29.6/n inV. J. Electroanal. Chem., 35 (1972)
336
E. LAVIRON
Log I,
!,
26
Io
_!,
-2 i
CEp.Ep,n)/,,,V Fig. 2. Variations of n(Ep-Ep,n) (peak potential) with log I. )2+1 ~,4
k._
Log L
2,
II
-ll
-21
Fig. 3. Variations of fp(X, l) = [(1 + 1)/(2+ 1)]2 +' with log I. Variation of the peak current T h e m a x i m u m fv(;(, l) of f(;(, l) is given b y :
fp(Z, l ) = [(1 +l)/(2+l)] 2+~ Figure 3 shows the variation of the m a x i m u m as a function of log 1. T h e value J. Electroanal. Chem., 35 (1972)
ADSORPTION OF DEPOLARISER OR PRODUCT. XVII
337
of the maximum passes from 0.25 for l = 0 to e -1 =0.368 for l---}oo. By introducing the value of fp(z, l) into eqn. (10), one obtains for the peak current iv: ip= (F2/RT) n 2 AvFo(tl)[(1 +/)/(2 +/)]2 +,
(13)
ip can also be written in the form: ip = n F A k F o ( q ) I - 1fp(g, l)
For a given value of Fo(tl), nFAkFo(tl) = C .~. To study the variation of ip as a function of v, it is sufficient to study the variation of l- 1 fp(Z, l). The variation of log l- 1fp (X, l) is shown in Fig. 4 as a function of log l [log ip = f ( - l o g v)]. As can be seen from the Figure, the peak current varies proportionally to v when v is either very large or very small in comparison to k. For intermediate values, the variation coefficient is a little smaller than 1, but its value is always about 1 (ip~C-~. × v°'9 when - 0 . 5 < log l < 0.5). It can be concluded from this result that a study of ip as a function ofv cannot give valuable information on the chemical reaction. V++t)
--
~1
,0
,-1
togt
Fig. 4. Variations of log {1 1fp(~, 1)} with log l (log ip with log v). (1) Asymptote log 1+ log 0.25, (2) asymptote l o g / - l o g e. BACKWARD PEAK
Equation of the peak The reaction scheme is still the same, but E is now given by E = Ei + vt, so that :
(14)
d z / d t = - avz
The surface concentration Fo is proportional to the amount of eiectricity which has been consumed: Fo =
(nFA)-I
f'o
i dt J. Electroanal. Chem., 35 (1972)
338
E. LAVIRON
(the minus sign shows that anodic currents are negative). F~ is the initial concentration of O. The sum of the surface concentrations are equal to Fo(q) so that eqn. (4) still holds, and (5) is still valid. By combining (1), (4), (5) and (14), the following differential equation is obtained : dFo/dt + (k - av) Z(1 + ~)-1Fo = 0
whose resolution gives for Fo: Fo = Ct.(l +Z) z-1
(15)
with 2 = (RT/nF)(k/v) Hence for FR: FR = c:x(1 + z)~- '
Taking into account the condition FR= F~ (F~ initial concentration of FR) when E = Ei, the value of the constant can readily be calculated :
C: = r~z~- I (1+ zi)1-~ with Zi = exp [ - a (E i - - Ep,n)] The current is then given by: i = - n F A dFo/dt = - (nF)Z/R T A v q Z ; 1 (1 + or
i= - (F2/R T)n2 AvF~ ¢(Z,
Z i )1 - 2
(1 -- 2) Z (1 +Z)z-
2, Zi)
2
(16) (17)
The value of F~ is given by eqn. (8) in which Z is replaced by Xi. The current can then be written: i= - (F2/R T)
n 2 a v F o (t 1)(1 +
Zi)- (l + a)(1 _ 2) Z(1 + Z)~- 2
(18)
or
i= - (F2/R T)n 2 A v r o (tl) 4' (Z, Zi, l, 2)
(19)
The backward peak depends on the same parameters (rate of potential sweep, rate constant of the chemical reaction) as the forward peak, and on a new parameter, the initial potential E i of the reverse sweep. Shape of the peak The variations of • as a function of n ( E - E p , n) are shown in Fig. 5 for different values of 2 and for n(Ei-Ep,n) = - 0 . 2 V. The function ~ ( E - E p , , ) passes through a maximum for Z---Xp=I/(1-2), when 2 < 1, which gives for the peak potential Ep:
Ep = Ep, n --~(2.3R T/nF) log (1-2)
(20)
The peak potential Ep theoretically varies from Ep, n to -o(3 when 2 varies from 0 to 1 ; as can be seen from Fig. 5 and eqn. (20), the peak potential remains virtually equal to Ep, n when 2 is sufficiently different from 1. By replacing Z by Zp in eqn. (18), one obtains for the peak c u r r e n t i v ( 2 < 1): ip = - (F2/R T)n 2 A v r o (tl)(1 + zi)- (' + z~[(1 - 2)/(2 - 2)] 2- a J. Electroanal. Chem., 35 (1972)
(21)
339
ADSORPTION OF DEPOLARISER OR PRODUCT. XVII n iE- Ep,n)/mV
100,
Q
_100.
.
.
.
5
Fig. 5. Variations of function ~bwith , ( E - Ep,,) for ,(E i -Ep,.)= - 0 . 2 V. 2:(1) 0, (2) 0.02, (3) 0.04, (4) 0.08, (5) 0.12, (6) 0.20.
As shown by eqn. (16) or (18), i is always equal to zero when 2 = 1, i.e. when v = R T k / n F = 0.0256 kin at 25°C. The significance of this result is as follows: at the initial potential Ei, a certain amount of O and R are present. When the potential is scanned in the anodic direction, F~ always decreases and Fo varies in such a way that the ratio Fo/FR is at all times equal to )~.The variations o f f o and F~ can occur in two ways : through the chemical reaction, which makes R decrease, and through the electrochemical reaction, which transforms R into O or O into R. If the rate of the chemical reaction is small enough when compared to the rate of the sweep (2< 1), the decrease of R and the increase of O occur mainly through the electrochemical reaction, which causes an anodic current to appear. When 2 = 1, the rate of the chemical reaction is just sufficient to cause a decrease in R such that the ratio FR/Fo is equal at all times to )~; as shown by eqn. (15), Fo is then constant. There is no electrochemical reaction, and i is equal to zero. If 2 > 1, the sign of the current changes (eqn. (16) or (18)), and a reduction current is obtained. This can be explained as follows : the chemical reaction causes too large a decrease in R, which is compensated by an electrochemical transformation of O into R: hence the appearance of the reduction current. This reduction current will however remain virtually equal to zero if Ei is negative enough, for the amount of 0 present at the initial potential Ei will then be very small. If the initial potential of the backward sweep is not too negative, i.e. if the forward sweep is stopped before the current has decreased to zero, the cathodic current can on the contrary have an important value, as shown by Fig. 6. The curves in this Figure have been obtained in the following way: the forward curve is given by the function f(L l) (eqns. (9) and (10)), and the backward curve by the function 4)' (Z, 2, )~i, l) (eqns. (18) and (19)). The rates of potential sweep are assumed to be equal for the forward and for the backward sweep. The potential at which the potential scan is reversed J. Electroanal. Chem., 35 (1972)
340
E. LAVIRON
has been taken equal to Ep,n(~i = 1). Analogous curves are obtained if the scan is reversed at any point along the peak. These results can be used advantageously for the determination of the value of k (see next paragraph).
t (X,L)
I,:I, (x,x,,t) Fig. 6. Variation of current for different values of I. The forward and the b a c k w a r d rates of potential sweep are the s a m e (l= 2). The potential scan has been reversed for E = Ep,.. ( , ) l = 2 = 0.2, ( ,, -)l= 2 = 1,
(,,,)l=~=2.
DETERMINATION OF THE VALUE OF THE RATE C O N S T A N T
k
Comparison of the forward and backward peaks As in the case of a second order reaction 2'3, a comparison of the height of the backward peak with that of the forward peak allows the rate constant k to be calculated. A different rate of potential sweep could be chosen for the forward and backward scans ; experimentally it is however simpler to use the same rate (I = 2) : the height of the peaks will then be of the same order of magnitude, which leads to a better precision ; on the other hand capacity currents will have the same importance. In the general case, the absolute value y of the ratio of the backward peak current iv,r to the forward peak c u r r e n t ip,a is obtained by dividing (21) by (13):
•
fl
=
ip,atP"--(l+zi)-~'+~)
~ 1-2~2-~ (2+1~ 2+/ \2~--21
\~1
This equation is valid for any value of Zl; for l = 2, it can be written as : J. Electroanal. Chem., 35 (1972)
(22)
ADSORPTION OF DEPOLARISER OR PRODUCT. XVII
, 2111--I~2-1(2"}-1~
Y=(I+zi)- ~2~l-l)
341
2+1
\~l/
(23)
The variations ofy as a function of log I are shown in Fig. 7 for different values and for l = 2.
o f n (E i - - Ep,n)
Lo9 L
Fig. 7. Variations of the ratio y ip,r/ip, a with log 1 for the same sweep rate for the forward and for the backward sweep n(Ei-Ep,,): (1)0, (2) -0.2, (3) -0.4. =
Equation (23) and Fig. 7 allow the calculation of the value of k from the experimental value of y. Let Ye be this value, obtained for a value ve of v. From Fig. 7, the corresponding value le of I can be obtained. Equation (7) gives :
k=(nF/RT)/eV~=38.9 l~nv~at
25°C
(24)
k is expressed in s- 1, v in V s- 1. If for example v~ is chosen so that the height of the backward peak is half that of the forward peak, y~ = Yo.5 =0.5. Table 1 gives the corresponding values of lo.5 and k for different values of n(Ei-Ep,n) TABLE 1
n(EI-Ep,,)/V
lo.5 =2o.s
k/s -1
0 -0,2 -0.4
0.340 0.0425 0.0218
13.23 nvo.5 1.65 nvo.5 0.85 nvo.5
As in a previous case 2, the initial potential E i of the backward scan can be easily chosen if n(Ei-Ep,.) is negative enough, by taking the peak potential of the backward peak as a reference. For example, eqn. (20) shows that the difference between nEp and nEp,n is smaller than 1 mV f o r / = 2 = 0 . 0 4 2 5 or 0.0218. If n(Ei-Ep,n) is not very negative, the precision is not so good. For example, for Ei = Ep,n and for l = 2 = 0.34, the difference is about 10 mV, and varies rapidly with 2. It is therefore more advantageous to use negative values of n(Ei-Ep,n) to determine k.
Study of the backward current As was shown, the backward current is constantly equal to zero for 2 = 1, J. Electroanal. Chem., 35 (1972)
342
E. LAVIRON
whatever the initial potential E i of the backward scan; Vo being the corresponding value of v, k = 38.92 nvo
(25)
Experimentally (see e.g. Fig. 6), the potential scan will be reversed when the forward current still has an important value (e.9. in the vicinity of the peak potential), and v will be adjusted to such a value that the backward current is constantly equal to zero. Although the precision in the determination of vo is likely to be smaller than in the determination of ve, the method can be advantageously used in certain cases, for the value of vo is smaller than the corresponding value of re. As the upper value of k which can be determined is limited by the values of v which can be used experimentally without deformation in the curves (e.g. because of the i-R drop), higher values of k can be determined by using eqn. (25) rather than (24). For n ( E i - E p , n ) = - 0 . 2 V, for example, v0.5 =23.6 Vo. APPLICATIONS
The application of the present theory to the case of the reduction of azobenzene to hydrazobenzene, followed in an acid medium by the benzidine rearrangement, will be presented in a following paper 6. As in the preceding cases 2'3, the present theory can be applied to the same reaction scheme in thin layer voltammetry 6. SUMMARY
A theoretical study of the peaks appearing in linear potential sweep voltammetry is presented under the following conditions : (a) the electrochemical reaction is reversible; (b) the depolarizer and the products of the electrochemical reaction are strongly adsorbed; (c) the product of the electrochemical reaction undergoes a firstorder reaction according to the scheme:
O+ne ~- R
Rk-~z
The equations for the reduction and oxidation peaks have been calculated, and their characteristics are described. The rate constant of the chemical reaction can be determined on the basis of this theory. The results of this theoretical study can be directly applied to the same reaction scheme in thin layer voltammetry. REFERENCES 1 2 3 4 5 6 7
E. LAVIRON, Bull. Soc. Chim. France, (1967) 3717. E. LAVIRON, Electrochim. Acta, 16 (197l) 409. E. LAVIRON,Jr. Electroanal. Chem., 34 (1972) 463. J. M. SAVEANTAND E. VIANELLO, C. R., 256 (1963) 2597. R. S. NICHOLSON AND I. SHAIN, Anal. Chem., 36 (1964) 718. E. LAVIRON, to be published. J. KORYTA, Collect. Czech. Chem. Commun., 18 (1953) 206.
J. Electroanal. Chem., 35 (1972)