Mechanistic Studies Using Double Potential Step Chronoamplometry: The EC, ECE, and Second-Order Dimerization Mechanisms R. Joe Lawson’ and J. T. Maloy D e p a r t m e n t of C h e m i s t r y , W e s t Virginia University, M o r g a n t o w n , W. V. 26506
Chronoamplometry, the study of the current semiintegral m ( f ) in the potential step experiment, is suggested as a method for investigating the rates and mechanisms of chemical reactions following electrode reactions. Working curves obtained through exact methods and through digital simulation are presented to assist in the determination of reaction mechanisms and in the evaluation of rate constants for the EC, ECE, and second-order dimerization mechanisms. Because of the constancy (or nearconstancy of m ( t ) at all points in the double potential step experiment, the uncertainty in the determination of m ( f ) is shown to be small with respect to that associated with the determination of either the current or the charge; thus, chronoamplometry is proposed to be superior to either chronoamperometry or chronocoulometry in the study of subtle variations necessary for the elucidation of reaction mechanisms.
Semiintegration of current-time d a t a in potential step and potential sweep experiments has been proposed as an analytical method because the current semiintegral may be proportional to the concentration of the electroactive species present ( I ) . More importantly. the current semiintegral has been shown to achieve a constant value independent of t h e way the potential is changed from its initial value, so long as the concentration of electroactive species is virtually zero a t the electrode surface (2). If a reactant R is oxidized or reduced to a product P in the electrode reaction
It f n e
-
P
THEORY The current semiintegral has been defined (3,4 ) as
(3) and may be regarded as that operation which, when performed twice on the function i ( t ) ,results in Q ( t ) ;hence, performing the operation once, as in Equation 3, results in semiintegration. In the double potential step experiment a t a planar electrode when the potential is stepped from a n initial potential Eo, where no faradaic process takes place, to a potential El, where the electrolysis of R takes place at a diffusion-limited rate for the time interval t f , and then stepped back to the potential E2, where the electrolysis of P occurs under diffusion-limited conditions, the current in the absence of kinetic complications is given ( 5 )by
In this equation. S t , ( t ) is the unit step function which is defined
(1)
a t a potential a i which the concentration of R a t the electrode surface is zero. the current semiintegral m ( t ) a t any time t is given by
nz(t) = nFACD”?
ies. Contained herein are working curves obtained from the semiintegration of digitally simulated double potential step current-time curves; these working curves illustrate t h e utility of the current semiintegral in kinetic studies and serve as a theoretical basis for this novel use of the semiintegral in mechanistic studies.
S . ( t )= 0
Fort
ti
Substitution of Equation 4 into Equation 3 yields
(2)
where n is the number of moles of electrons transferred per mole of R, F is Faraday’s constant. A is the electrode area, C is the bulk concentration of R, and D is the diffusion coefficient of species R (2, 3 ) . The invariability of m(t)renders it particularly inviting to those engaged in t h e study of the kinetics of homogeneous reactions following a n electrode reaction: variations in current, i ( t ) . or charge, Q ( t ) ,during a short time experiment limit the effectiveness of these parameters in the study of the rates and mechanisms of following reactions. Thus, double potential step chronoamplometry, the study of m ( t ) in a double potential step experiment, may offer important advantages over conventional chronoamperometric and chronocoulometric techniques in these mechanistic studPresent address. School of Chemical Sciences. University of 11linois at Urbana-Champaign, Urbana. Ill. 61801, ( 1 ) K B. Oldham, A n a / . Chem.. 4 4 . 196 (1972) (21 M Grennessand K €3. Oldham. Anal. Chem . 44. 1121 11972) ( 3 ) P E Whiston H W VandenBorn and D H Evans A n a / Chem 45, 1298 (1973)
a result identical to Equation 2, and
Fort
>
t , (6)
Thus, in kinetically uncomplicated double potential step chronoamplometry, the disappearance of the semiintegral is observed at all times after the second potential step. If the product of the initial electrode reaction is depleted so rapidly by the following chemical reaction t h a t there is no current after the second potential step, the semiintegral may be written
m(t) riFAC‘DIi-
’G i v 1
=
dh
-2 - X
sin-j,& F o r t >/ t
(7)
( 4 ) K . E. Oldham and J. Spanier, J. Elecfroanal. Chem. lntertaciai Electrochem., 26, 331 (1970). ( 5 ) T. Kambara. Bull. Chem. SOC.Jap., 27. 523 (1954).
so t h a t in the limit of very fast following reaction, m ( 2 t f ) is equal to one half n(t f ) . Equations 6 and 7, then, represent t h e extreme conditions in the semiintegration of i ( t ) in t h e double potential step experiment. To investigate the effects of kinetic complications a t intermediate times, digital simulation was performed to obtain kinetically perturbed z( t ) curves for semiintegral transformation. In each of the mechanisms considered. the reaction given by Equation 1 was assumed a t El while its reverse
P r n e -R
(8)
was assumed at Ez, both occurring under diffusion-limited conditions. The effects of three different kinetic complications were simulated; these mechanisms are similar to three of those considered by Imbeaux and Saveant (6) in their recent development of convolutive potential sweep voltammetry. In each case the kinetic perturbation leads to the electroinactive entity X: Mechanism EC1 (first-order ec)-in this case, the kinetic complication was
P L X
the kinetic
k
P A P'
(10)
--
with P' subsequently electroactive a t El
X
Mechanism ECZ (second-order dimerization)-with kinetic complication
P+Ph?'X
(11)
At
=
t,/L
~ ( t=) ( n F A C D ' / ? / m X 2 -
(17)
The diffusion coefficients of all species were taken to be equal. During each iteration, the dimensionless current was calculated as the quantity Z(t ) where
(18) The value of ht,CJ-' was defined as a n input parameter at the outset of an individual simulation, and Z ( t ) was obtained a t A t intervals for all t in the range 0 < t 5 2t,. The resulting current-time curve was numerically semiintegrated and integrated to obtain dimensionless representations of both m ( t ) and Q ( t ) .A new value of h,t[CJ-l was then selected, and the entire process was repeated. The numerical semiintegration of Z ( t ) was obtained through the series equivalent of Equation 3
the
(12)
where k z is a second-order rate constant. After the effect of one of these kinetic perturbations upon I ( t ) had been determined for a particular value of the dimensionless rate constant h,t,C/-l (where J is the order of reaction in the rate determining step), the simulated current-time curve was semiintegrated a t selected points along the curve. For the E C l E mechanism the current on the forward step has been shown (7) to be
(13)
fbr t 5 t,. This may be substituted into Equation 3 in an attempt to obtain a n analytical solution for the kinetically perturbed semiintegral
Although this integral resists evaluation with elementary functions. it may he expressed
where IO ( h l t / 2 ) is the zero order modified Bessel function of the first kind. This modified Bessel function may be computed through the expansion ( 8 )
( 6 ) J . C lmbeaux and J. M. Saveant, J . Electroanal. Chem. lnterfaciai Electrochem., 4 4 . 169 (1973). ( 7 ) G . S Alberts and i . Shain, A n a / . Chem.. 35, 1859 (1963). ( 8 ) M Abrarnowitz and I . A . Stegun. "Handbook of Mathematicai Functions. Applied Mathematics Series No. 5 5 , " National Bureau of Standards. U . S Government Printing Office, Washington, D.C., 1965. p 375.
560
DIGITAL SIMULATION The digital simulation procedures for obtaining modified current-time curves have been described previously (9) and copies of the Fortran program used are available upon request. In these simulations, t f was represented by L iterations so t h a t S t , the duration of a single iteration, was given by
(9)
where h l is a first-order rate constant. Mechanism E C l E (first-order ece)-where complication was
P' f ne
to obtain a numerical solution for m ( t ) a t any value of h1t 5 k l t f . The numerical evaluation of m ( t ) may be compared with that obtained through digital simulation to test the validity of the simulated curve.
A N A L Y T I C A L C H E M I S T R Y , V O L . 46, NO. 4 , A P R I L 1974
where t = k A t and h = j A t . The term l/At was included in the denominator of the sum because Z ( t ) , and hence i ( t ) , was calculated a t the mid-point of each At interval; thus, each current term in the sum was divided by the square root of its distance from the upper limit of integration. Combination of Equations 17, 18, and 19 results in the form used to semiintegrate Z( t )
RESULTS AND DISCUSSION The semiintegrated results of several simulations are shown in Figure 1. T h a t there is good agreement between the simulated results and Equations 5 , 6, and 7 is shown by curves a and e. The rounded portion of curve a does not agree with Equation 6, but this is to be expected within a few iterations of the switching point. There are two possible sources for this deviation between Equation 6 and curve a in the vicinity of the switching point: the digital simulation of the current-time behavior or the numerical semiintegration of the simulated i ( t ) curve. Since numerical semiintegration of the exact i ( t ) behavior (Equation 4 ) results in a nearly identical m ( t ) curve, one may conclude that this discrepancy is due primarily to the method selected to semiintegrate the simulated i ( t ) curve rather than a n error in the simulation itself. The method employed herein has been found to give slightly better results than one employed previously (2) in those cases where exact solutions exist for comparison; thus, it seems reasonable t o assume that the goodW. V . Childs, J. T. Maloy, C.P. Keszthelyi, and A . J Bard. J. Electrochem. SOC. . 118, 874 (1971).
2
.
0
4
I
n F A C d’*
-2
2
0
log kf
t-
0.oy , 0
,
,
,
1
I
1
,
I
Figure 2. T h e c h r o n o a m p l o m e t r i c w o r k i n g c u r v e tor tne ‘jingle p o t e n t i a l step s t u d y of t h e ECIE m e c h a n i s m The c u r v e I S t h e r e s u l t of t h e digital Simulation a n d t h e p o i n t s r e p r e s e n t the e x a c t s o l u t i o n ( E q u a t i o n 15)
,a 2
‘4, Figure 1. S i m u l a t e d r e s u l t s of double p o t e n t i a l step c h r o n o a m p l o m e t r y u n d e r t h e i n f l u e n c e of k i n e t i c c o m p l i c a t i o n s All curves are the resuiis of simulations with L = 1000. Curve a: Mechanism EC,, ECIE. and ECr wilh k,tfCJ-’ = 0,001. Curve b: Mechanism EC2 with kZfiC = 1.0. Curve c: Mechanism EC, with k i t , = 1.0. Curve d: Mechanism EC,E wUh k i t i = 1.0. Curve e: Mechanisms EC1 and ECz in the limit of large k,rfCJ-’ Curve f : Mechanism ECIE in the limit of large k l I ,
ness-of-fit in the vicinity of the switching point may be improved only by increasing the number of po’ints sampled in this region. This was not deemed necessary, however; as t approaches 2t,+ and the number of iterations becomes large (-2000), both curves a and e approach the values predicted by Equations 6 and 7 . This indicates that the numerical semiintegration is most reliable when the numoer of iterations is large. For this reason, points were selected for the working curves presented below so as to avoid time intervals represented by a small number of iterations. Curves b, c, and d show the effect of intermediate kinetic perturbations upon m(t ) . Each curve was simulated with k,t,C-’-’ set equal to unity. It may be noted t h a t m ( t ) is neither zero nor constant after the switching point if there are kinetic complications; however, the slope of m ( t ) after the switching point is small enough (maximum relative slope at t = 2 t f is -0.3) that its value can be determined a t any point with a high degree of certainty. Curves u, d, and f of Figure 1 show typical results expected with a system governed by the EClE mechanism. Here, m i i ) is riot constant after the first potential step because the second electrode reaction contributes to the current. Since n is the same in both reactions, m ( t ) doubles in tne limit of large h1t. The simulated variation of m i r ) with k l t is shown as the working curve in Figure 2. Superiniposed on this working curve are several points evaluated through Equations 15 and 16; the agreement between the simuiated working curve and t h a t obtained tnrough the modified Bessel function is within the uncertainty of electrochemical measurement. This working curve is similar to another previously published for this niechariism ( 1 0 ) ; however, the working curve in Figure 2 shows the variation of m ( t ) rather than the variation of i( t).tl 2 . Sirice analog instrumentation is presently available to obtain m ( t ) directly (IZ), this working curve en(101 M . U. hawiey and s. W Feldberg. J. Phys. Chem., 70, 3459 (1966). ( 1 1 1 K B Oloharn. Aoai. Chern.. 45. 39 (1973).
Table I. Numerical Working Curves [m( 2 t ~/lm ) ftj) us. k,t,Cj-l) for Chronoampllometric Studies of Reaction Mechanismsa Mechanism
kjtfCj-l
0.001 0.01 0.10 0.40 1.o
3 .O 10. 100. cob
aIn each simulation, L EC2, k d f C = 500.
ECI
0.005 0.013 0.055 0.167
Mechanism EClE
Mechanism EC2
0,009 0.015
0.008 0.013 0.055 0.148 0.245 0.354 0.431 0.487 0.495
0.071 0.200
0.301 0.440
0.326
0.489
0.478 0.500 0.502
0.502 0.503 =
0.434
1000. ’For E C I and E C S ,
klrj
= 1000; for
ables one to study the E C l E mechanism through comparison with directly obtainable experimental data, thereby eliminating the need for tedious data treatnient. While single potential step methods are sufficient for the E C l E mechanism, double potential step techniques must be employed for the other mechanisms treated because m ( t ) is constant before the second potential step in each of these reaction schemes. One acceptable working curve for these mechanisms is the quantity m ( 2 t , ) / m ( t j ) as a function of k , t , + C I - l , and this is given for each mechanism in Table I. The working curves presented in this table and Figure 2 enable one to distinguish between these mechanisms and to dei:ermirie the rate constants for the homogeneous reactions through double potential step chronoamplometry . These working curves are shown graphically in the upper three panels of Figure 3. For the sake of comparison, the quantities i ( 2 t , ) / i ( t f )and Q(Zt,)/C&tr) tor each of the mechanisms are also shown. While times other than t f and 2 t f could have been selected for these working curves, measurements a t these particular times are obtained easily experimentally, and both -the sirnularion and numerical semiintegration are reliable a t these points. lf these times are used, it may be seen that the m ( t ) ratio undergoes a greater change with variatlc’ns in h,t,C’-l than the i ( t ) ratio; in fact, the ratio for m ( t ) almost shows as much variation as the ratio for Q ( t ) . This observation is important in mechanistic studies because it is often only through slight variations in curvature that one mechanism can be distinguished from another. and the greater the relative change in the measured parameter. the more reliably the curvature can be determined. The chief advantage of double potential step chronoamA N A L Y T I C A L CHEMISTRY, VOL. 46, NO. 4 , A P R I L 1974
561
fo Kt,)
1
t
U
Iog k,t,Ci-’ Figure 3. Working curves and uncertainties for the double potential step study of reaction mechanism
plometry in mechanistic studies, however, is the constancy (or near constancy) of m ( t ) .Even though m ( t ) is not a constant after the second potential step for any mechanism, the variation of m ( t ) is less t h a n the variation of i ( t ) or Q ( t )for all times during which the homogeneous chemical reaction influences these parameters. T o illustrate this point, a quantity proportional to the relative uncertainty in the measurement of the value of the time-variant parameter a t t f and Zt, was calculated for each f ( t ) [either i ( t ) , Q ( t ) , or m ( t ) ]at several values of h,tfC’-l. This was determined by numerically evaluating the derivative of each f ( t ) to obtain the relative slope I f ‘ ( t ) / f ( t ) ] a t t f and 2 t f . The absolute values of these quantities were then added to obtain U, a quantity proportional to the relative uncertainty in the determination of the ratiof(2tf)/f(tf):
tion with variations in tf. (Obviously, since i(2tf)/i(tf) under the influence of kinetic perturbation approaches zero in the limit of long t f while m ( 2 t f ) / m ( t , )approaches zero as t f approaches zero, the absolute uncertainties in each of these ratios approach zero in the same limits.) In addition, this development completely ignores the problem of charging currents which also contribute t o the overall experimental uncertainty. Indeed, these may even be a more serious cause for experimental uncertainty in the determination of f ( Z t r ) / f ( t f ) than those discussed above. However. any discussion of the utility of the current semiintegral in the determination of charging effects is beyond the scope and intent of this communication. The alteration of chronoamplometric d a t a by these effects is presently being investigated and it is anticipated that the results of these investigations will be the subject of a future publication.
CONCLUSIONS The variation of U with k , t f C J - l for each of the mechanisms is shown in the lower three panels of Figure 3. From these graphs, it is clear that if U is a valid measure of ) be deteitotal relative uncertainty, m ( 2 t f ) / m ( t f may mined with a greater degree of relative certainty than either i ( 2 t , ) / i ( t j )or Q ( Z t f ) / Q ( t t ) for any of these mechanisms regardless of t f . This direct result of the invariance of m ( t )suggests t h a t double potential step chronoamplometry offers a more reliable way to compare experimental kinetic d a t a with theory than either chronoamperometry or chronocoulometry. The actual experimental uncertainty in the f ( Z t f ) / f ( t f ) ratio, of course, is not represented by U. Multiplication of Lr by dt, the experimental uncertainty in the recording of time, would yield relative uncertainty in the ratio determination. Further multiplication by the ratio itself would yield the absolute uncertainty ip the measurement. Even if IYis multiplied by f ( 2 t f ) / f ( t fto) obtain a quantity proportional to the absolute uncertainty in the ratio determination, this quantity is significantly less for m ( t ) over most of the range in which kinetic complications cause a ratio varia-
562
A N A L Y T I C A L C H E M I S T R Y , V O L . 46, N O . 4 , A P R I L 1974
Chronoamplometry has been shown to offer some important advantages over conventional chronoamperometry and chronocoulometry in the study of reaction mechanisms. In the study of the E C l E mechanism, variation of m ( t ) may be used directly to elucidate the mechanism and determine the rate constant. In double potential step studies of any of the mechanisms treated herein, chronoamplometry has been shown to lead to more reliable results because the semiintegral is either constant or varies slowly with time. The working curves presented above serve as a theoretical basis for mechanistic studies presently under way.
ACKNOWLEDGMENT The authors wish to thank Keith B. Oldham and John Gruninger for their helpful discussions. Received for review August 6, 1973. Accepted Xovember 13, 1973. Research supported by the donors of the Petroleu m Research Fund administered by the American Chemical Society.
