Analytic Solutions for Double Potential Step Chronoamplometry for First-Order Mechanisms at Planar Electrodes Robert C. Bess, Stacy E. Cranston, and Thomas H. Ridgway" Department of Chemistry, Texas A& M University, College Station, Texas 77843
Analytical solutions for the double potential step chronoamplometric responses of the irreversible pseudo-first-order EC, ECE, ECEC, One-Half Order Regeneration, and Catalytic mechanisms have been developed. The results for the EC and ECEC mechanism with infinitely fast second reaction are in reasonable accord with previous results. Working curves allow the determinationof reaction mechanisms in the group covered and permit evaluation of the rate constants.
The semiintegral (1-4) of current-time data obtained in double potential step experiments has been proposed as a viable alternative to the more common current and charge methods for evaluation of the mechanistic path of the decay of electrode products ( 4 ) . A recent paper by Lawson and Maloy ( 4 )presented working curves for the EC, Second-Order Dimerization, and a special case of the ECEC (second chemical reaction infinitely fast) mechanism. Their results, obtained by finite difference simulation of the currents followed by a numerical semiintegration, clearly demonstrated some of the advantages of the semiintegral presentation. This report presents analytic solutions for the semiintegral for the EC, ECE, ECEC, Catalytic, and One-Half Order Regeneration mechanisms under the influence of double potential step excitations. The results show reasonably good correspondence with the simulation results of Lawson and Maloy ( 4 ) for the EC and ECEC mechanism with a n infinitely fast second chemical reaction.
In the ECEC mechanism a (pseudo-first-order irreversible ecec case), the electrode product D is allowed to undergo an irreversible reaction with rate constant k2 in the presence of large excesses of X to yield the inert product F .
X+D-F
kz
The Catalytic mechanism (CAT) assumes that the product
B undergoes an irreversible pseudo-first-order reaction with Y to reform A . B+Y-A+W
(7)
The one-half order regeneration mechanism (OHOR) assumes that the product B reacts irreversibly with species Y , present in great excess, to yield one-half of a parent species.
B
kl + Y----+'/zA +W
(8)
The form of this mechanism which is most commonly encountered in the case of organic species actually involves two steps, the formation of C via Equation 4 followed by k :i
C+B-A+Z
(9)
with Equation 4 being the rate limiting step. The procedure used here will be to utilize Equation 2 to develop R ( s ) followed by a reversion to the time plane.
THEORY The current semiintegral has been defined in convolution form by (5, 6)
where i(t) is the current. An alternate form in Laplace space ( 3 ) is given by rn(s) = I(s)/s1/2
(2)
where s is the Laplace variable while l ( s ) and A ( s ) are the transforms of current and the semiintegral, respectively. For all of the mechanisms treated here, the parent material A undergoes an nl electron transfer step to yield the electrochemical product B .
Afnl*B
(3)
In the EC mechanism (a pseudo-first-order ec case), the product B decays irreversibly, with a rate constant k l , to an inert species C in the presence of a large excess of reactant 2.
Z+Bkl-C
(4)
In the ECE mechanism (a pseudo-first-order irreversible ece case), the chemical product C is electrochemically active and can undergo an np electron transfer step a t the electrode to yield a chemically stable species D .
Cfn2=D
(5)
DEVELOPMENT OF THE RESPONSES For all of the mechanisms considered, the assumption of semiinfinite linear diffusion is made and the diffusion coefficients for all electroactive species are assumed equal. On the forward step (0 < t < 7 ) , the surface concentrations of species A (and C when present) are zero. On the reverse step ( 7 < t < 27), the surface concentrations of species B (and D when present) are zero. For the EC and Catalytic cases, the chronoamplometric responses are derived from published current or charge expressions by multiplication of the current or charge expression in the Laplace plane by sP1l2 or s112, respectively, and return to the time plane. EC Mechanism. The transform of the current for this mechanism has previously been shown (7) to be i(S)
nFACfi
1
=I&
where T is the potential switching time. The symbols ( a ) , and
M ( a , b, z ) are Pochhammer's symbol (8) and the confluent hypergeometric function ( 8 ) ,respectively, and are defined in Equations 11and 12. (a), = a(a
+ l ) ( a + 2 ) . . . ( a + n - 1); (11, = n; (a)()= 1 (11)
ANALYTICAL CHEMISTRY, VOL. 48, NO. 11, SEPTEMBER 1976
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05 04
03
rn(2T) -
m(T)
02
X [(kz - k l ) ( t 01
00 -2
0
-1
2
1
Log (kT) Figure 1. Double potential step chronoamplometric working curves for the EC and ECEC mechanisms A comparison of the analytic solutions and finite difference predictions (4) of m(27)lm(r)vs. log ( k , ~for ) the irreversible first-order EC, ECE, and ECEC (k2
= m ) mechanisms. (a) EC mechanism. Open circles represent data from reference 4. (b) ECE mechanism. (c) ECEC mechanism. Solid circles represent data from reference 4
M ( a , b, 2 ) =
- 7)InM[n + I/., n + 1, (k2 - k1)7] (Ma)
There are a number of important sub-cases of this mechanism, such as the forms when k 2 = 0, k z = k l , and k l T . The form of $Mz(k 1, t , T ) which will be of most interest is when t = 27 (Le., at twice the switching time). It can be shown that for this special case (12), Equation 16a can be greatly simplified. @Mp(k,27,
7)
= e-2k7M(1/2,1, 2 h r )
(16b)
ECEC Mechanism. The chronoamplometric response is again obtained by multiplication of the transform of the current response by s - l I 2 or the transform of the charge response (10, 1 1 ) by s+lI2 and retransformation to give 1620
+
+
T)]
+ n2[1 - u ( ~-)e-/?lfM('/2, 1, kit) + aMZ(k1, t,
dMz(k, t , T ) = e-kt
[k(t - 7)lnM[Yz,n + 1 (l)n(n + 1)/(2n + 1 ) ' k ( t - T ) ] M ( ~ 3/2, IZ 2, k7) (20) I t can be shown by application of recursion relationships (8) and multiplication theorems (12),or by direct computation that the function d M 4 ( k ,27, T ) has the value of unity for all values of k T . CAT Mechanism. The derivation for the charge response has been published (14) and, from this information, it is simple to show that the chronoamplometric response is given by X n
n=O
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DISCUSSION OF THE RESPONSES EC Mechanism. The behavior of the r n ( 2 ~ ) / r n ( 7 )vs. log ( k ~ plot ) of the analytic solution, Equation 14, is shown in Figure 1. In this figure, the solid line represents the analytic solution and the individual points are from the paper by Lawson and Maloy ( 4 ) .The simulated data are consistently somewhat greater than the analytic solution, but the differences are essentially minor. The analytic solution is based upon charge and current expressions whose validity has been extensively confirmed by finite difference simulations of those responses (9, 10). ECE Mechanism. The mechanism treated by Lawson and Maloy ( 4 ) is not the ECE mechanism in this notation, but rather a special case of our ECEC mechanism. Prior to the second step, the results for the mechanisms should be the same. Equation 15 for t less than T looks somewhat different than their Equation 15 but since (8) e-ktM(1/2, 1,k t ) = e-ktf/210(ht/2)
(22)
this yields the form expressed in their terminology. The plot
SEPTEMBER 1976
2.0
05
P
I
0 4
rn(2T) _-
0 3
m(T)
0 2
01
00
-2
-1
0
1
2
Log(kY) Figure 2. Double potential step chronoamplometric working curves for the EC, ECE, ECEC, OHOR, and CAT mechanisms
Figure 3. Single potential step chronoamplometricworking curves for the ECE, ECEC, OHOR, and CAT mechanisms
m ( 2 ~ ) l m (vs. ~ ) log (klr) working curves for all mechanisms considered. ( a )EC mechanism. ( b )ECEC mechanism, k2 = a.( c )ECEC mechanism, k2 = kl. (d) OHOR mechanism. (e) CAT mechanism. (0 ECEC mechanism, k2 = 0.1 kl. (9) ECEC mechanism, k2 = 0.01 k , . ( h )ECE mechanism and ECEC mechanism,
(a)CAT mechanism. (6)ECE and ECEC mechanisms. ( c )OHOR mechanism
k2 = 0
of m ( 2 r ) / m ( ~ vs.) log (h7) shown in Figure 1 reveals two striking features, a limiting value of 0.25 instead of 0.5 which one might expect and a local maximum in the plot with values higher than 0.25. It is possible to predict just this limiting value by considering what is happening in the case of an infinitely fast rate constant. Under these circumstances, the first charge transfer step yields nl electrons on the forward step and no current on the reverse step. The species B produced a t the electrode immediately decays to C, undergoes a further n2 electron exchange to form D, which diffuses out into the solution until the reverse step. Since species D is assumed stable, the overall result is the same as if one had a reaction which exchanged nl n2 electrons on the forward step and only 122 electrons on the reverse step. This means that the limiting behavior must be the same as for the sum of a reversible n2 transfer plus a totally irreversible EC nl transfer case.
+
Equation 23 clearly predicts m ( 2 r ) / m ( r= ) 0.250 if nl = n2 as was observed in Figure 1. The maximum observed does not arise from the same kinds of phenomena which produces the Napcurrent and charge maximum in the nuance ECE mechanism (15)and does not represent any kind of inflection in the individual chronoamplograms themselves, nor is it the result of buildup of computational errors, but rather is a property of the system itself. If the maximum were a result of computational problems, the error would increase with increasing h r and a maximum would not be observed. What is in fact occurring can be seen by combining Equations 15a with 15b to give
Every amplomb (amp sec1/2and Oldham ( 1 6 )have also been suggested) produced in Equation 5 in the forward step by formation of C is precisely cancelled a t 27 into the back step by the consumption of D . In a mathematical sense, the maximum is produced by QMl(h1, 27, 7) decreasing somewhat more rapidly than the confluent hypergeometric function in the denominator. E C E C Mechanism. The m ( 2 7 ) / m ( rvs. ) log (hl.r) plot for
a number of sub cases of this mechanism are shown in Figure 2. The results for hz = 0 are identical to those of the ECE mechanism as expected. For non-zero values of the k z rate constant, the limiting value now approaches one half, although in the case of k2 = 0.01h1, this limit is not reached in the range covered in Figure 2. The final limit of one half can be understood by considering the case where both k l and k2 are very large. On the forward step, the current observed will look like a (nl n2) electron process, while on the back step no current will flow, so that the limiting form of the chronoamplogram will be
+
m ( t )=
(nl
+ n2)FAC fi
(25) In the intermediate cases, where h 1 > k z , the two-step nature of the mechanism is clearly revealed in the double wave seen in the response ratio. The response is very sensitive to the follow-up step in Equation 6 for values of hp < h but as k:! becomes larger than hl the change in response becomes much smaller, and in fact, the case where k z = 10 k l is virtually indistinguishable from the response for h2 = a,the case treated by Lawson and Maloy ( 4 ) . Their results are compared to Equation 15 in Figure 1. As in the EC case, their results are consistently somewhat higher than ours. The shapes of the curves are in fact slightly different and are not simply shifted along the time axis. OHOR Mechanism. The m(2r)/m(r) vs. log ( h 1 7 )plot for this mechanism is also given in Figure 2 and shows a limiting value of one half, which is consistent with the limiting behavior one would expect. For infinitely fast rates of reaction, the forward step should look like a 2nl electron transfer process while on the back st&p,the absence of B in solution should lead to zero current flow. The chronoamplometric response should therefore resemble the limiting behavior of the EC mechanism with twice as many electrons exchanged. CAT Mechanism. The m(27)/m(r) vs. log ( h l r ) plot for this mechanism as shown in Figure 2 is characterized by a limiting value of 4 - 1 rather than one half or one quarter. This value can be obtained by manipulation of the large argument expansion for confluent hypergeometric functions (8) of the type found in Equation 21. Physically this limit arises from the fact that for large values of h l , the current response will look essentially like a step function until the potential switching time tau, when the current abruptly goes to zero, there being no B remaining in solution. The function to ] the chronoamplobe semiintegrated is thus [l - U ( T ) and gram increases as t1I2while on the back step the response is tu2
- ( t - 7)1/2*
ANALYTICAL CHEMISTRY, VOL. 48, NO. 11, SEPTEMBER 1976
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Table I. Numerical Working Curves m ( 2 r ) / m ( vs. ~ ) log ( k 1 ~for ) Double Potential Step Chronoamplometric Studies of Reaction Mechanisms Log (
k l ~ )
-2.00 -1.90 -1.80 -1.70 -1.60 -1.50 -1.40 -1.30 -1.20 -1.10 -1.00 -0.90 -0.80 -0.70 -0.60 -0.50
-0.40 -0.30 -0.20 -0.10 0.00
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90
EC 0.005 0.006 0.008 0.010 0.012 0.016 0.019 0.024 0.030 0.038 0.047 0.058 0.072 0.088 0.108 0.131 0.158 0.187 0.220 0.256 0.292 0.328 0.361 0.391 0.416 0.437 0.452 0.464 0.472 0.478 0.483 0.487 0.490 0.492 0.494 0.495 0.496 0.497 0.497 0.498
0.005 0.006 0.008 0.010 0.012 0.015 0.019 0.024 0.029 0.036 0.045 0.055 0.067 0.081 0.097 0.115 0.134 0.155 0.176 0.197 0.215 0.232 0.245 0.255 0.262 0.266 0.267 0.268 0.268 0.267 0.266 0.265 0.264 0.263 0.262 0.261 0.260 0.259 0.258 0.257
0.005
0.005
0.006 0.008 0.010 0.012 0.015 0.019 0.024 0.029 0.036 0.045 0.055 0.067 0.081 0.097 0.115 0.135
0.006
0.155
0.176 0.197 0.216 0.233 0.247 0.257 0.265 0.270 0.273 0.275 0.278 0.280 0.283 0.287 0.292 0.298 0.307 0.316 0.328 0.342 0.357 0.374
Double Potential Step Chronoamplometry as a Diagnostic Tool. Inspection of Figure 2 indicates that, in principle, the technique may well be useful for diagnostic and mechanistic studies. For example, the ECE and CAT mechanisms have very characteristic behaviors. Further, it will be simple to at least determine the presence of the second chemical step in the ECEC mechanism, even for quite small values of hplk I , by carrying out experiments of sufficient duration. Evaluation of the true ratio of hzlh will be difficult for values in excess of unity by this technique, as may be seen from the similarity of the h 1 = h2 and h2 = m plots in Figure 2. The value of h can be obtained in all ECEC cases from N,, vs. log ( h ~ tplots. ) This form of data presentation will also serve to discriminate between the OHOR and ECEC mechanisms as may be seen from Figure 3. The major problems associated with this method are the present uncertainties about the treatment of charging currents ( 4 )and the mechanics of semiintegration itself. There are two present methods for semiintegration, analog by means of a pseudo-transmission line ( I 7), and the various digital schemes ( I , 2 , 4 , 5 , 1 8 )for direct numerical semiintegration, although a convolution by means of Fast Fourier transforms may prove to be more convenient. The analog method seems to work well for time scales in the range of 0.050-15.0 s but on either extreme appears to produce somewhat erroneous results (17). 1622
0.008
0.010 0.012 0.015
0.019 0.024 0.029 0.036 0.045 0.055 0.067 0.081 0.098 0.116 0.136 0.157 0.179 0.201 0.222 0.242 0.259 0.274 0.288 0.300 0.313 0.327 0.342 0.358 0.376 0.394 0.411 0.427 0.442 0.454 0.463 0.471 0.477 0.481
0.005 0.006 0.008 0.010 0.012 0.015 0.019 0.024 0.030
0.037 0.046 0.057 0.070 0.085 0.103 0.124 0.147 0.173 0.202 0.232 0.262 0.293 0.323 0.350 0.375 0.397 0.416 0.431 0.444 0.454 0.462 0.468 0.473 0.477 0.481 0.484 0.486 0.488 0.490 0.491
0.005 0.006 0.008 0.010 0.012 0.016 0.019 0.024 0.030 0.038 0.047 0.058 0.072 0.088 0.108 0.131 0.157 0.187 0.220 0.255 0.291 0.325 0.358 0.386 0.409 0.427 0.441 0.451 0.459 0.466 0.471 0.475 0.479 0.482 0.484 0.486 0.488 0.490 0.491 0.492
0.005 0.006 0.008 0.010 0.012 0.015 0.019 0.024 0.030 0.037 0.046 0.057 0.0.71 0.086 0.105 0.127 0.151 0.179 0.209 0.240 0.272 0.303 0.333 0.359 0.381 0.400 0.416 0.428 0.439 0.447 0.454 0.460 0.466 0.470 0.474 0.477 0.480 0.482 0.484 0.486
0.005
0.006 0.008 0.010 0.012 0.015 0.019 0.024 0.030 0.037 0.046 0.057 0.069 0.085 0.102 0.123 0.146 0.171 0.198 0.226 0.253 0.280 0.304 0.325 0.343 0.358 0.370 0.379 0.386 0.392 0.397 0.400 0.403 0.405 0.407 0.409 0.410 0.411 0.411 0.412
How this method will fare in double potential step experiments is presently unknown. The long time limit, per se, is not a problem since it is well into the range where one expects convective or electrode sphericity effects to become important. The short time limit may be overcome by altering the time constants utilized in the pseudo transmission line. The digital methods appear to be satisfactory for analytical studies (2) and seem to produce reasonable results in charge transfer kinetic studies by potential step methods ( 1 8 ) and cyclic voltammetry ( 5 ) .Saveant and co-workers (19-22) have made extensive use of an essentially identical numerical deconvolution method in their studies of cyclic voltammetric responses. The small but real discrepancies between the EC and ECEC ( k z = a)cases presented here and those obtained by Lawson and Maloy ( 4 ) cause some concern. It is not presently known if these differences result from some interaction between the finite difference simulation method with the numerical semiintegration algorithm employed or if the algorithm itself is solely responsible. It is well known (23)that finite difference methods are most inaccurate a t short times in the simulation of potential step responses. The semiintegral technique weights short time contributions more than long time contributions and this interaction may be the source of the discrepancy.
ANALYTICAL CHEMISTRY, VOL. 48, NO. 11, SEPTEMBER 1976
The charging current problem will have to be solved before the method has any practical application. This is a non-trivial problem since the double layer response is given by
where R is the uncompensated solution resistance, cd]the double layer capacitance in farads and A E the step size in volts. This form is equivalent to Oldham's Dawson integral form (2). The properties of this function can be best evaluated by making the substitution X = t/ncdl, after which it is simple to show that the peak value of 0.61 ECdl/n amplombs occurs a t t = 0.854 RCdl s by using t h e properties of this class of function (8).At times longer than 100 ncdl the function decays smoothly and will have a value of 0.564AECdlld-t amplombs. One can easily compute the fractional contribution of the double layer to the total signal in this time region. A more productive measure is to compute the time required for the double layer to have decayed to some fraction X of the faradaic signal. This is given by
t=
0.564AEC*dl 96.5nC*Xv%
(27)
where Cdl* is the double layer capacitance in pf per cm2 and C* is the concentration of electroactive species in moles per liter. Substituting common values of 50 pf for the double layer, 1 mM for the concentration, one electron and 5 X lov6 cm2/s into Equation 27 yields t = 0.050/X2 s. It is important t o recognize that altering the solution resistance has no effect upon the long term response, only the time required to reach the simple decay is affected. A recent publication by Maloy et al. (24) suggests a method based upon clipping the current during the short time portion of the cell response when the double layer charging contribution is dominant. The procedure seems to be satisfactory for the single step method and may prove useful in the double step technique. The results obtained in this work are summarized in Table I and may be used to construct working curves of useful size or as input to curve fitting or interpolation routines for computerized data reduction. Direct evaluation of the analytic solutions is possible although the convergence of some of the functions is moderately slow. The confluent hypergeometric functions can be evaluated by direct use of their defining equations (8)but negative arguments should be avoided by means of the Kummer transformation (8) to prevent erroneous convergences.
CONCLUSIONS Analytic solutions for the double potential step chronoamplometric responses of the EC, ECE, ECEC, one-half order regeneration, and irreversible catalytic mechanisms have been presented. The analytic solutions for the EC and ECEC(h2 = a) cases differ somewhat from the previously published finite difference simulation results of Lawson and Maloy (4). The differences are small but detectable, and the source of the difference is under study. A combination of single and double potential step working curves allows evaluation of kinetic rate constants and mechanism within the family of reactions considered.
LITERATURE CITED (1)K. B. Oldham. Anal. Chem., 44, 196(1972). (2)M. Grenness and K. B. Oldham, Anal. Chem., 44, 1121 (1972). (3)R. L. Birke, Anal. Chem., 45, 2292 (1973). (4)R. J. Lawson and J. T. Maloy, Anal. Chem., 46, 559 (1974). (5) P. E. Whiston, H. W. VandenBorn, and D. H. Evans, Anal. Chem., 45, 1298 (1973). (6)K. B. Oldham and J. Spanier. J. Electroanal. Chem., 26,331 (1970). (7)W. M. Schwarz and I. Shain, J. Phys. Chem., 69,30 (1965). (8)M. Abramowitz and I. A. Stegun. "Handbook of Mathematical Functions, Applied Mathematics Series No. 55". National Bureau of Standards, U.S. Government Printing Office, Washington, D.C., 1965. (9)T. H. Ridgway. R. P. Van Duyne, and C. N. Reilley, J. Electroanal. Chem., 34,267 (1972). (10)T. H. Ridgway, Ph.D. Dissertation,University of North Carolina, 1971. (11)T. H. Ridgway, R. P. van Duyne, and C. N. Reilley. J. Nectroanal. Chem., in press.
(12)"Higher Transcendental Functions", Vol. I, "Bateman Manuscript Project", A. Erdelyi et al., Ed., McGraw-Hill, New York, 1953,p 283. (13)T. H. Ridgway, R . P. Van Duyne, and C. N. Reilley, J. Electroanal. Chem., 67, l(1976). (14)J. H. Christie, J. Electroanal. Chem., 13, 79 (1967). (15)M. D.Hawkey and S. W. Feldberg, J. Phys. Chem., 70,3459 (1966). (16)D. K. Roe, Anal. Chem., 46,8R (1974). (17)K. B. Oldham, Anal. Chem., 45,39 (1973). (18)J. H. Carney and H. C. Miller, Anal. Chem., 45, 2175 (1973). (19)C. P. Andrieux, L. Nadjo, and J. M. Saveant, J. Electroanal. Chem., L6, 147 (1970). (20)J. C. Umbeaux and J. M. Saveant, J. Nectroanal. Chem., 44, 169 (1973). (21)F. Ammar and J. M. Saveant, J. Electroanal. Chem., 47, 115 (1973). (22)L. Nadjo, J. M. Saveant, and D. Tessier, J. Nectroanal. Chem., 52, 403 (1974). (23)S.W. Feldberg and A. J. Bard, Ed., "ElectroanalyticalChemistry", Vol. 3, Marcel Dekker, New York, 1969,pp 199-295. (24)S. C. Lamey. R. D. Grypa, and J. T. Maloy. Anal. Chem., 47,610 (1975).
RECEIVEDfor review August 22,1974. Resubmitted May 10, 1976. Accepted May 10, 1976. Research sponsored by the Organized Research Fund of Texas A&M University.
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