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The extraction of kinetic parameters from chronoamperometric or chronocoulometric data
J. Electroanal. Chem., 145 (1983) 9-20 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
9
THE EXTRACTION OF KINETIC PARAMETERS FROM CHRONOAMPEROMETRIC OR CHRONOCOULOMETRIC DATA
KEITH B. OLDHAM* Deakin University, Victoria 3217 (Australia)
(Received 13th May 1982; in revised form 26th August 1982)
ABSTRACT Data from chronoamperometric or chronocoulometric experiments under joint control by diffusion and kinetics cannot be linearized. However, if the current semiintegralis computed from the experimental data, simple linear graphs can be constructed that embrace all data points and from which the kinetic parameters can be easily found. Suitable algorithms are presented for calculating the semiintegral from current or charge data. The procedures are shown to be superior to methods based on limiting behaviour and to be capable of discriminating well against any noise that may be present.
INTRODUCTION W i t h good reason, scientists like to m a n i p u l a t e their experimental results into a linear relationship. A n electrochemical example is the replotting of polarographic (current i vs. potential E at c o n s t a n t time t) data as the logarithm of (ia - i ) / i vs. E to take advantage of the theoretical H e y r o v s k ~ - I l k o v i c relationship b e t w e e n c u r r e n t a n d potential. Electrochemical m e a s u r e m e n t s for which such linearizations are not directly possible include current vs. time data following the i m p o s i t i o n of a potential step o n a p l a n a r electrode at which one of the following reaction schemes occurs: kf O+ne~R
(1)
kf O+ne~R kb
(2)
kl +n e O ~ P -~ R k - i fast
(3)
u n d e r the usual v o l t a m m e t r i c conditions. W i t h c b d e n o t i n g the b u l k c o n c e n t r a t i o n of species O a n d D O its diffusion coefficient, the c h r o n o a m p e r o m e t r i c relation for
* Permanent address: Trent University, Peterborough, Canada. 0022-0728/83/0000'0000/$03.00
© 1983 Elsevier Sequoia S.A.
10 the irreversible scheme (1) is [1]
i - n r A c b k f exp(fl~t) erfc(fllX/~)
(4)
where fl~ = k f / ~ o o , F is Faraday's constant and A the electrode area. For the quasi reversible scheme (2) the current time relationship obeys the similar equation [2]
i = nrAcbokf exp(flzt) erfc(flzV~-)
(5)
where r2 = ( k f / ~ o ) + ( k b / D ~ R ) , DR being the diffusion coefficient of the initially absent species R. The relationship between current and time for scheme (3), the so-called CE mechanism, has been shown to be [3]
i = nFAcb~oofl3 exp(fl2t) erfc(~3~t)
(6)
where r3 = k l / k ~ l provided that k 1 >> k r The three reaction schemes share a chronoamperometric response of the form i - i o exp(fl2t) erfc(fiv~)
(7)
where i 0 is the initial current (which is not usually accessible by direct measurement). Because of the intractability of the exp(fl2t) erfc(flv/t) function there exists no way in which, without prior knowledge of i o or r , the current vs. time data may be replotted to give a linear graph. Accordingly, the standard procedure [41 is to expand eqn. (7) and use either its short-time approximant
i - i o - 2iofl~/t/~r
(8)
or the asymptotic approximation i0 i= - -
vr~ flvri
i0
2vr~ fl3t3/2
(9)
valid at long times. The replacement of the exact eqn. (7) by either eqn. (8) or (9) is a retrograde step, because the data set on which the analysis is based becomes restricted to the first few or the last few points, with consequent loss of precision. To put it more crudely: one is throwing away good data. Another method adopted by experimentalists is to compare the ratio i/i d of the current to the diffusion current with a standard curve of the ~ exp(x) erfc(x/x) function. One difficulty with this procedure is that it requires a second experiment at an arbitrarily selected more negative potential, with the assumption that the potential change produces no effect other than to drive fl towards infinity. Another problem is that in some cases (with overlapping waves, for example) it may be impossible to find any potential at which pure diffusion control attains. Finally, comparison of experimental and standard curves is a technique of low and uncertain precision. The analysis of chronoamperometric curves has been discussed by Johnson and Barnartt [5]. Of course, there exist non-linear data-fitting techniques that can handle eqn. (7) without eroding the precision of the data [6]. These statistical procedures are, however, of a complexity that does not appeal to the average chemist. Another solution that has been advocated [7] is to Laplace transform the current-time data
and make a linearization in Laplace space, where this is possible. In the present article a third solution is suggested. The semiintegration of faradaic currents [also k n o w n as their convolution with the (qrt) -1/2 function] has been used to advantage in electroanalysis [8] and in potentiodynamic kinetic studies [9]. Using the results of the fractional calculus [10], it can be shown that the semiintegral of a current that obeys eqn. (7) is m
d-'/2i dt-I/2
i o ioexp(fl2t) erfc(fl~/t) /3 /3
(10)
One can now eliminate the troublesome exp (/32t) e r f c ( / 3 ~ ) function between eqns. (7) and (10) to produce the remarkably simple result
i = i o - /3m
(11)
It follows that a graph of the current i vs. its semiintegral rn should be linear, with intercept i 0 and slope - / 3 . F r o m measurents of the intercept a n d / o r slope the kinetic parameters of the reaction scheme are thus calculable. In contrast to data analysis based on eqns. (8) or (9), the linearization (11) is exact and applies at all times 0 ~ t < oo, not just in the short-time or long-time limit. To test the procedure, a set of data was created that fits eqn. (7) with the parameters i 0 = 10.00/~A a n d / 3 = 1.000 s 1/2. The data are reproduced, ~in part, a s the first two columns of Table 1. Notice that the data pairs are evenly spaced in time, with a time increment between adjacent points of A = 0.05 s, and that there is no datum at t = 0. If ij is used to represent t h e j t h current value (e.g. i 4 = 6.438/~A in Table 1), then the algorithm
mj=
alv/J- ~-bj_l + ~ (aj-aj_,)JffJ~-j+(bj-bj_,)[Jarcsin~/~-~/-j~/J-j
(12)
j=2
m a y be used to calculate the semiintegral of the current at the instant t = JA. The a and b parameters in this equation are defined by
a,=
2 1 / j + l ij
21/jij+ l (13)
and ij -- i j+ 1
b'
S,/7
,/7
(14)
N o t e that the algorithm uses the current values i~, i2,---- i j _ ~ and ij to calculate mj but that m 0 and m~ are not calculable. The basis of eqn. (12) is explained in the Appendix.
12 TABLE 1 C u r r e n t - t i m e data that exactly fit eqn. (7) (with i o = 10.000/~A and ,8 = 1.0000 s - 1/2) and the result of semiintegrating these data using algorithm 0 2 )
t/s
i/#A
rn/l~A
0.05 0.10 0.15 0.20 0.25
7.904 7.236 6.784 6.438 6.157
2.757 3.211 3.557 3.839
].10 1.15 1.20
4.146 4.086 4.028
5.852 5.912 5.970
s t/2
The third column of Table 1 shows the result of applying algorithm (12) to the data in the second column (in fact to less rounded values than are tabulated). The values of m, which have /~A s ~/2 units, increase as the experiment proceeds, as predicted by eqn. (10). Table 1 contains only a small selection of the actual i and m
--I0
illaA
-8
o o •
-6
o
O
0
•
Oo o * %
0
© ©
<,,, >
m/I~ A 2
s '12
0
4
6
Fig. 1. Ctu'onoamperometric data plotted in accordance with eqn. (11). (o) Noise-free data from Table 1; ( O ) faradaic data admixed with gaussian noise from Table 2.
13 data because the numbers themselves are of little interest. Each solid point in Fig. 1 represents a data pair from the unabridged counterpart of Table 1, plotted i n accordance with eqn. (11). The linearity is very evident. From a least-squares analysis o f the 23 data pairs, the values i 0 = 9.968/~A
(15)
and fl = 0.9932 s -1/2
(16)
were recovered. Each value is within 1% of its true magnitude. The superiority of the new technique over prior methodology based on eqns. (8) and (9) is evident from a comparison of Fig, 1 with Figs. 2 and 3. The solid points in the latter graphs are respectively the short- and long-time data from Table 1. Figures 2 and 3 also show the straight lines on which the points should lie if the linearizations were valid: clearly they are not. The small errrors reflected in eqns. (15) and (16) arise not from inexact data, nor from an inadequate theory, but from imperfections i n t h e algorithm: they could be diminished, for example, by using a smaller A. Of more experimental interest is the question of how,well the new technique is able to cope with data that contain extraneous contributions, such as spurious "noise". To investigate this, a data set was created by adding to each ij in Table 1 the quantity [11] ij (noise)= ( i ) 7 2
ln( 199017 I cos( 2~rNj÷ ' ] 199017 S
(17)
where Nj are imegers pseudorandomly spaced in the range 0 ~
-lo
N8
t
m6
•
o
Fig. 2. Short-timechronoamperometricdata from Table 1 plotted to take advantageof eqn. (8). The line is theoretical.
14
O°Oo •
\, --6
( t / s ) -~ /
0~4
/
0.6
I
0.8
I
Fig. 3. Points are long-time chronoamperometric data from Table 1 plotted according to eqn. (9). The points would lie on the ftheoreticalJ line if the linearization were valid.
TABLE 2 Current-time data generated by adding gaussian noise of rms amplitude 1.000 A to the faradaic signal of Table 1. The semiintegral data were produced by algorithm (12)
j
t/s
i/l~A
m/l~ A S1/2
] 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20
8.400 7.457 7.102 5.729 5.358 5.247 4.791 5.292 6.863 3.062 5.002 5.880 4.385 4.279 4.759 3.853 3.639 3.640 3.400 4.423 5.386 4.272 2.718 4.735
2.907 3.376 3.567 3.708 3.876 3.987 4.200 4.684 4.456 4.585 4.954 4.994 5.006 5.154 5.152 5.123 5.141 5.140 5.329 5.668 5.749 5.541 5.749
15 generated by the recurrence Nj+ 1
=
(24298Nj + 99991) mod 199017
(18)
Equation (17) generates a set of noise contributions to the current that displays a gaussian distribution about a mean of zero and with a root-mean square of (i). This last parameter was set to 1.000 /~A and the "seed" N o was arbitrarily chosen as 166764 in creating the data set of Table 2. The currents in Table 2 were processed by algorithm (12) and the results constitute the final column of the table. Each i, m data pair provides one of the open points in Fig. 1. As expected, the noise has destroyed the linearity of the points. Nevertheless, when subjected to least-squares analysis, the 23 data pairs in Table 2 give an intercept i 0 = 9.39/xA
(19)
and a slope from which fl = 0.969 s- 1/2
(20)
is recovered. Evidently the new technique is able to extract i 0 and /3 values containing only modest errors, even when the faradaic data are considerably obscured by noise. The chronocoulometric technique [12] has several experimental advantages over chronoamperometry [13]. However, linearization of data that have a kinetic content is again impossible. In chronocoulometry the primary data are values of the faradaic charge q as a function of time. For electrode reactions that follow one of schemes (1)-(3), the charge vs. time relationship, found by integration of eqn. (7), is 2i o
q=-T
--
t
i o exp(/32t) erfc(flv~-)
/32
(21)
where i 0 and /3 retain their previous significance. Not only is it impossible to linearize eqn. (21) itself, without prior knowledge of i 0 or/3, but even the simpler equation
i = i o -/3io~/t/vr + fl2q
(22)
from which the exp(/32t) erfc(/3v/)-) term has been eliminated by substitution from eqn. (7), cannot be linearized. Thus, the simultaneous recording of current and charge, following a potential step at a kinetically mediated electrode, provides no solution to the linearization difficulty. However, the combination of eqns. (21) and (10) can be recast as m = 2i___9_o_fl___qq
(23)
demonstrating that a plot of m / v ~ vs. q/v~ should be linear with a slope of - / 3 and an intercept of 2io/v~. To implement this procedure requires the generation of a set of mj data from the experimental qj data. This may be accomplished by use of
16
the semidifferentiation algorithm
+ ~., (Aj-Aj_,)J~Z-j+(Bj-Bj_,) j=2
(
Jarcsin
- V ~
)]
(24)
whose basis is discussed in the Appendix. Notice that algorithm (24) differs from algorithm (12) only in minor respects, other than in the definitions of the Aj and Bj parameters, which are
2 j~lqj
21/~qj+ 1
j
Aj =
j+l
(25)
and
Bj =
3qj
3qj+l
2j
2(j+ 1)
~ _
~
(26)
To test the efficacy of this procedure for analyzing chronocoulometric results, a data set was created that exactly obeys eqn. (21) with the same i 0 and fl values that were used previously. Part of the data is reproduced as the first two columns of Table 3. These data were then subjected to algorithm (24), a selection of the output being included as the third tabular column. Note the near-perfect agreement between the m values in Table 1 with those in
TABLE 3 Charge-time data that exactly fit eqn. (21) (with i o = 10.000/.tA and fl = 1.0000 s" 1/'2) and the result of semidifferentiating these data using algorithm (24)
t/s
q/#C
m / / # C s - 1/2
0.00 0.05 0.10 0.15 0.20 0.25
0.000 0.427 0.804 1.154 1.484 1.799
2.761 3.213 3.559 3.840
i.10
~1981
~.853
1.15 1.20
6.186 6.389
5.913 5.970
17
--10
m t-v2/pA
m8
Q
°°-,~
QIII --6
q t-1/2/l~A 81/2 2
4
6
I
I
t
Fig. 4. Noise-free chronocoulometric data plotted in the manner suggested by eqn. (23). The leftmost points represent data collected in the early part of the experiment.
Table 3. The former were generated by semiintegrating current data and the latter by semidifferentiating charge data. The concordance between the two outputs is strong evidence that the algorithms for both semioperations are functioning well. Figure 4 is a plot of m/v~vs, q/~/t using the data in Table 3. According to eqn. (23) the graph should be linear, and this is evidently true. A least-squares evaluation of the 23 data pairs gives an intercept of 11.269 /~A and a gradient of -0.9976 S-1/2, from which the values i o = 9.987/tA
(27)
and fl = 0.9976 s- 1/2
(28)
may be calculated. Agreement with the true values 10.000/~A and 1.0000 s-1/2 is superb. Chronoamperometry and chronocoulometry possess two invaluable features not shared by other voltammetric techniques: (1) Because they are constant-potential methods, there is no non-faradaic contribution to the data after the initial charging. (2) It is not necessary to pre-assume a dependence of electron-transfer rate constants on electrode potential.
18 The new data-processing procedures suggested in this article will, it is hoped, allow these advantages to be more fully exploited. ACKNOWLEDGEMENTS Gratitude is due to Trent University for granting sabbatical leave and to Professor Alan M. Bond for his hospitality and enthusiastic encouragement. The financial support of the Natural Sciences and Engineering Research Council of Canada and Deakin University Research Committee is gratefully acknowledged. APPENDIX
The derivation of algorithms (12) and (24) The temporal semiintegral of a continuous current i(r) over the time period 0 ~
1 ft i('r)d'r
(A1)
m ( t ) = ~ J o ~t77
If the period is subdivided into J intervals each of width A, so that t = JA, then eqn, (A1) may be rewritten
1 J-~fa+SA
m ( t ) = ~--~-- E
i(r)dr
vcg j=o'ja
(A2)
~/t-¢
as the sum Of J integrals. Consider the representative interval jA ~
,
bj
jA<~r<~A+jA
(A3)
will be used to interpolate between these end values. The reason for using this particular interpolative formula is explained later. When eqn. (A3) is inserted into eqn. (A2) and the indicated integrations are performed, the result
(
"-'
m(t)= ~
t arcsin ~ / ~ - - V~-tv~Z~--r)A+jA jA
E aj~/t-r j =O
(A4)
"1- - - ~
is obtained. When the integration limits are inserted and the substitution t = JA is
19 made, this result becomes
(~- J-1
m ( JA ) = --
(
(
Y~ a f f J - j - 1 + b; J arcsin g=O
.,_1
(
) j + 1 _ ~/jv+-T ~ J - j J
_
- 1
) (A5)
Next, t h e j = J - 1 terms are withdrawn from the first summation in eqn. (A5) and the j = 0 terms are withdrawn from the second summation in the same equation. This leads to
mj
= ~-[aovQ-- -~bj_ ~r l + ~
j=l ( a j - a j _ l ) ! / - J - j + ( b j - b j _ l )
Jarcsin
-
(16)
after a redefinition of the summation index j in the first, but not the second, summation in eqn. (15). Returning to formula (A3) one may "anchor" the interpolation at each end of the jA ~< T ~ A +jA interval by setting i ( j A ) = ij and i(A + j A ) = i;+~. The solutions of the two simultaneous equations that this generates are given as eqns. (13) and (14) of the main text. However, because i o is not a known datum, a 0 and bo cannot be determined. A way out of this difficulty is to approximate i(~-) for the 0 ~< ~-~< A interval by extrapolation from the A ~ ~-~< 2 A interval. This is tantamount to setting a 0 = a 1 and b0 = b~. To minimize error introduced by the extrapolation, it is especially important that the interpolation formula be a good match to the true i vs. t function at short times. It was for this reason that interpolation formula (A3), which exactly matches the short-time approximant (8), was selected. When a 0 is replaced by a I and b0 is replaced by bl, eqn. (A6) becomes algorithm (12) of the main text. Turning now to the semidifferentiation algorithm needed for chronocoulometry, we start with one of the definitions [14] of the semiderivative of a continuous function q(~) over the interval 0 < r ~< t
/
m ( t ) = q(O)
1
t
dq
One of the advantages of chronocoulometry over chronoamperometry is that, whereas in the latter technique the initial datum is not measurable, the initial chronocoulometric datum q(0) or q0 is known to be zero. Thus, the first right-hand term in eqn. (A7) can be dropped. Just as in devising algorithm (12) we adopted an interpolative formula that matched the short-time approximant of eqn. (7), so we now adopt the interpolation
20
formula q(r)
Aj'r
2Bj'g 3/2
2A
3A3/2
jA
<~r <~ A + j A
(AS)
to match the short-time approximant q = iot - 4 f l i o t 3 / 2 / 3 v ~
(A9)
of eqn. (21) in time dependence. Differentiation of eqn. (AS) gives dq=
2A
A3/2 dr
(A10)
and this may be substituted into eqn. (A7). From this point the development of algorithm (24) parallels that of algorithm (12) so closely that no details of the derivation need be given. Notice that an extrapolation into the 0 ~ r ~< A interval from the A ~
9
THE EXTRACTION OF KINETIC PARAMETERS FROM CHRONOAMPEROMETRIC OR CHRONOCOULOMETRIC DATA
KEITH B. OLDHAM* Deakin University, Victoria 3217 (Australia)
(Received 13th May 1982; in revised form 26th August 1982)
ABSTRACT Data from chronoamperometric or chronocoulometric experiments under joint control by diffusion and kinetics cannot be linearized. However, if the current semiintegralis computed from the experimental data, simple linear graphs can be constructed that embrace all data points and from which the kinetic parameters can be easily found. Suitable algorithms are presented for calculating the semiintegral from current or charge data. The procedures are shown to be superior to methods based on limiting behaviour and to be capable of discriminating well against any noise that may be present.
INTRODUCTION W i t h good reason, scientists like to m a n i p u l a t e their experimental results into a linear relationship. A n electrochemical example is the replotting of polarographic (current i vs. potential E at c o n s t a n t time t) data as the logarithm of (ia - i ) / i vs. E to take advantage of the theoretical H e y r o v s k ~ - I l k o v i c relationship b e t w e e n c u r r e n t a n d potential. Electrochemical m e a s u r e m e n t s for which such linearizations are not directly possible include current vs. time data following the i m p o s i t i o n of a potential step o n a p l a n a r electrode at which one of the following reaction schemes occurs: kf O+ne~R
(1)
kf O+ne~R kb
(2)
kl +n e O ~ P -~ R k - i fast
(3)
u n d e r the usual v o l t a m m e t r i c conditions. W i t h c b d e n o t i n g the b u l k c o n c e n t r a t i o n of species O a n d D O its diffusion coefficient, the c h r o n o a m p e r o m e t r i c relation for
* Permanent address: Trent University, Peterborough, Canada. 0022-0728/83/0000'0000/$03.00
© 1983 Elsevier Sequoia S.A.
10 the irreversible scheme (1) is [1]
i - n r A c b k f exp(fl~t) erfc(fllX/~)
(4)
where fl~ = k f / ~ o o , F is Faraday's constant and A the electrode area. For the quasi reversible scheme (2) the current time relationship obeys the similar equation [2]
i = nrAcbokf exp(flzt) erfc(flzV~-)
(5)
where r2 = ( k f / ~ o ) + ( k b / D ~ R ) , DR being the diffusion coefficient of the initially absent species R. The relationship between current and time for scheme (3), the so-called CE mechanism, has been shown to be [3]
i = nFAcb~oofl3 exp(fl2t) erfc(~3~t)
(6)
where r3 = k l / k ~ l provided that k 1 >> k r The three reaction schemes share a chronoamperometric response of the form i - i o exp(fl2t) erfc(fiv~)
(7)
where i 0 is the initial current (which is not usually accessible by direct measurement). Because of the intractability of the exp(fl2t) erfc(flv/t) function there exists no way in which, without prior knowledge of i o or r , the current vs. time data may be replotted to give a linear graph. Accordingly, the standard procedure [41 is to expand eqn. (7) and use either its short-time approximant
i - i o - 2iofl~/t/~r
(8)
or the asymptotic approximation i0 i= - -
vr~ flvri
i0
2vr~ fl3t3/2
(9)
valid at long times. The replacement of the exact eqn. (7) by either eqn. (8) or (9) is a retrograde step, because the data set on which the analysis is based becomes restricted to the first few or the last few points, with consequent loss of precision. To put it more crudely: one is throwing away good data. Another method adopted by experimentalists is to compare the ratio i/i d of the current to the diffusion current with a standard curve of the ~ exp(x) erfc(x/x) function. One difficulty with this procedure is that it requires a second experiment at an arbitrarily selected more negative potential, with the assumption that the potential change produces no effect other than to drive fl towards infinity. Another problem is that in some cases (with overlapping waves, for example) it may be impossible to find any potential at which pure diffusion control attains. Finally, comparison of experimental and standard curves is a technique of low and uncertain precision. The analysis of chronoamperometric curves has been discussed by Johnson and Barnartt [5]. Of course, there exist non-linear data-fitting techniques that can handle eqn. (7) without eroding the precision of the data [6]. These statistical procedures are, however, of a complexity that does not appeal to the average chemist. Another solution that has been advocated [7] is to Laplace transform the current-time data
and make a linearization in Laplace space, where this is possible. In the present article a third solution is suggested. The semiintegration of faradaic currents [also k n o w n as their convolution with the (qrt) -1/2 function] has been used to advantage in electroanalysis [8] and in potentiodynamic kinetic studies [9]. Using the results of the fractional calculus [10], it can be shown that the semiintegral of a current that obeys eqn. (7) is m
d-'/2i dt-I/2
i o ioexp(fl2t) erfc(fl~/t) /3 /3
(10)
One can now eliminate the troublesome exp (/32t) e r f c ( / 3 ~ ) function between eqns. (7) and (10) to produce the remarkably simple result
i = i o - /3m
(11)
It follows that a graph of the current i vs. its semiintegral rn should be linear, with intercept i 0 and slope - / 3 . F r o m measurents of the intercept a n d / o r slope the kinetic parameters of the reaction scheme are thus calculable. In contrast to data analysis based on eqns. (8) or (9), the linearization (11) is exact and applies at all times 0 ~ t < oo, not just in the short-time or long-time limit. To test the procedure, a set of data was created that fits eqn. (7) with the parameters i 0 = 10.00/~A a n d / 3 = 1.000 s 1/2. The data are reproduced, ~in part, a s the first two columns of Table 1. Notice that the data pairs are evenly spaced in time, with a time increment between adjacent points of A = 0.05 s, and that there is no datum at t = 0. If ij is used to represent t h e j t h current value (e.g. i 4 = 6.438/~A in Table 1), then the algorithm
mj=
alv/J- ~-bj_l + ~ (aj-aj_,)JffJ~-j+(bj-bj_,)[Jarcsin~/~-~/-j~/J-j
(12)
j=2
m a y be used to calculate the semiintegral of the current at the instant t = JA. The a and b parameters in this equation are defined by
a,=
2 1 / j + l ij
21/jij+ l (13)
and ij -- i j+ 1
b'
S,/7
,/7
(14)
N o t e that the algorithm uses the current values i~, i2,---- i j _ ~ and ij to calculate mj but that m 0 and m~ are not calculable. The basis of eqn. (12) is explained in the Appendix.
12 TABLE 1 C u r r e n t - t i m e data that exactly fit eqn. (7) (with i o = 10.000/~A and ,8 = 1.0000 s - 1/2) and the result of semiintegrating these data using algorithm 0 2 )
t/s
i/#A
rn/l~A
0.05 0.10 0.15 0.20 0.25
7.904 7.236 6.784 6.438 6.157
2.757 3.211 3.557 3.839
].10 1.15 1.20
4.146 4.086 4.028
5.852 5.912 5.970
s t/2
The third column of Table 1 shows the result of applying algorithm (12) to the data in the second column (in fact to less rounded values than are tabulated). The values of m, which have /~A s ~/2 units, increase as the experiment proceeds, as predicted by eqn. (10). Table 1 contains only a small selection of the actual i and m
--I0
illaA
-8
o o •
-6
o
O
0
•
Oo o * %
0
© ©
<,,, >
m/I~ A 2
s '12
0
4
6
Fig. 1. Ctu'onoamperometric data plotted in accordance with eqn. (11). (o) Noise-free data from Table 1; ( O ) faradaic data admixed with gaussian noise from Table 2.
13 data because the numbers themselves are of little interest. Each solid point in Fig. 1 represents a data pair from the unabridged counterpart of Table 1, plotted i n accordance with eqn. (11). The linearity is very evident. From a least-squares analysis o f the 23 data pairs, the values i 0 = 9.968/~A
(15)
and fl = 0.9932 s -1/2
(16)
were recovered. Each value is within 1% of its true magnitude. The superiority of the new technique over prior methodology based on eqns. (8) and (9) is evident from a comparison of Fig, 1 with Figs. 2 and 3. The solid points in the latter graphs are respectively the short- and long-time data from Table 1. Figures 2 and 3 also show the straight lines on which the points should lie if the linearizations were valid: clearly they are not. The small errrors reflected in eqns. (15) and (16) arise not from inexact data, nor from an inadequate theory, but from imperfections i n t h e algorithm: they could be diminished, for example, by using a smaller A. Of more experimental interest is the question of how,well the new technique is able to cope with data that contain extraneous contributions, such as spurious "noise". To investigate this, a data set was created by adding to each ij in Table 1 the quantity [11] ij (noise)= ( i ) 7 2
ln( 199017 I cos( 2~rNj÷ ' ] 199017 S
(17)
where Nj are imegers pseudorandomly spaced in the range 0 ~
-lo
N8
t
m6
•
o
Fig. 2. Short-timechronoamperometricdata from Table 1 plotted to take advantageof eqn. (8). The line is theoretical.
14
O°Oo •
\, --6
( t / s ) -~ /
0~4
/
0.6
I
0.8
I
Fig. 3. Points are long-time chronoamperometric data from Table 1 plotted according to eqn. (9). The points would lie on the ftheoreticalJ line if the linearization were valid.
TABLE 2 Current-time data generated by adding gaussian noise of rms amplitude 1.000 A to the faradaic signal of Table 1. The semiintegral data were produced by algorithm (12)
j
t/s
i/l~A
m/l~ A S1/2
] 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20
8.400 7.457 7.102 5.729 5.358 5.247 4.791 5.292 6.863 3.062 5.002 5.880 4.385 4.279 4.759 3.853 3.639 3.640 3.400 4.423 5.386 4.272 2.718 4.735
2.907 3.376 3.567 3.708 3.876 3.987 4.200 4.684 4.456 4.585 4.954 4.994 5.006 5.154 5.152 5.123 5.141 5.140 5.329 5.668 5.749 5.541 5.749
15 generated by the recurrence Nj+ 1
=
(24298Nj + 99991) mod 199017
(18)
Equation (17) generates a set of noise contributions to the current that displays a gaussian distribution about a mean of zero and with a root-mean square of (i). This last parameter was set to 1.000 /~A and the "seed" N o was arbitrarily chosen as 166764 in creating the data set of Table 2. The currents in Table 2 were processed by algorithm (12) and the results constitute the final column of the table. Each i, m data pair provides one of the open points in Fig. 1. As expected, the noise has destroyed the linearity of the points. Nevertheless, when subjected to least-squares analysis, the 23 data pairs in Table 2 give an intercept i 0 = 9.39/xA
(19)
and a slope from which fl = 0.969 s- 1/2
(20)
is recovered. Evidently the new technique is able to extract i 0 and /3 values containing only modest errors, even when the faradaic data are considerably obscured by noise. The chronocoulometric technique [12] has several experimental advantages over chronoamperometry [13]. However, linearization of data that have a kinetic content is again impossible. In chronocoulometry the primary data are values of the faradaic charge q as a function of time. For electrode reactions that follow one of schemes (1)-(3), the charge vs. time relationship, found by integration of eqn. (7), is 2i o
q=-T
--
t
i o exp(/32t) erfc(flv~-)
/32
(21)
where i 0 and /3 retain their previous significance. Not only is it impossible to linearize eqn. (21) itself, without prior knowledge of i 0 or/3, but even the simpler equation
i = i o -/3io~/t/vr + fl2q
(22)
from which the exp(/32t) erfc(/3v/)-) term has been eliminated by substitution from eqn. (7), cannot be linearized. Thus, the simultaneous recording of current and charge, following a potential step at a kinetically mediated electrode, provides no solution to the linearization difficulty. However, the combination of eqns. (21) and (10) can be recast as m = 2i___9_o_fl___qq
(23)
demonstrating that a plot of m / v ~ vs. q/v~ should be linear with a slope of - / 3 and an intercept of 2io/v~. To implement this procedure requires the generation of a set of mj data from the experimental qj data. This may be accomplished by use of
16
the semidifferentiation algorithm
+ ~., (Aj-Aj_,)J~Z-j+(Bj-Bj_,) j=2
(
Jarcsin
- V ~
)]
(24)
whose basis is discussed in the Appendix. Notice that algorithm (24) differs from algorithm (12) only in minor respects, other than in the definitions of the Aj and Bj parameters, which are
2 j~lqj
21/~qj+ 1
j
Aj =
j+l
(25)
and
Bj =
3qj
3qj+l
2j
2(j+ 1)
~ _
~
(26)
To test the efficacy of this procedure for analyzing chronocoulometric results, a data set was created that exactly obeys eqn. (21) with the same i 0 and fl values that were used previously. Part of the data is reproduced as the first two columns of Table 3. These data were then subjected to algorithm (24), a selection of the output being included as the third tabular column. Note the near-perfect agreement between the m values in Table 1 with those in
TABLE 3 Charge-time data that exactly fit eqn. (21) (with i o = 10.000/.tA and fl = 1.0000 s" 1/'2) and the result of semidifferentiating these data using algorithm (24)
t/s
q/#C
m / / # C s - 1/2
0.00 0.05 0.10 0.15 0.20 0.25
0.000 0.427 0.804 1.154 1.484 1.799
2.761 3.213 3.559 3.840
i.10
~1981
~.853
1.15 1.20
6.186 6.389
5.913 5.970
17
--10
m t-v2/pA
m8
Q
°°-,~
QIII --6
q t-1/2/l~A 81/2 2
4
6
I
I
t
Fig. 4. Noise-free chronocoulometric data plotted in the manner suggested by eqn. (23). The leftmost points represent data collected in the early part of the experiment.
Table 3. The former were generated by semiintegrating current data and the latter by semidifferentiating charge data. The concordance between the two outputs is strong evidence that the algorithms for both semioperations are functioning well. Figure 4 is a plot of m/v~vs, q/~/t using the data in Table 3. According to eqn. (23) the graph should be linear, and this is evidently true. A least-squares evaluation of the 23 data pairs gives an intercept of 11.269 /~A and a gradient of -0.9976 S-1/2, from which the values i o = 9.987/tA
(27)
and fl = 0.9976 s- 1/2
(28)
may be calculated. Agreement with the true values 10.000/~A and 1.0000 s-1/2 is superb. Chronoamperometry and chronocoulometry possess two invaluable features not shared by other voltammetric techniques: (1) Because they are constant-potential methods, there is no non-faradaic contribution to the data after the initial charging. (2) It is not necessary to pre-assume a dependence of electron-transfer rate constants on electrode potential.
18 The new data-processing procedures suggested in this article will, it is hoped, allow these advantages to be more fully exploited. ACKNOWLEDGEMENTS Gratitude is due to Trent University for granting sabbatical leave and to Professor Alan M. Bond for his hospitality and enthusiastic encouragement. The financial support of the Natural Sciences and Engineering Research Council of Canada and Deakin University Research Committee is gratefully acknowledged. APPENDIX
The derivation of algorithms (12) and (24) The temporal semiintegral of a continuous current i(r) over the time period 0 ~
1 ft i('r)d'r
(A1)
m ( t ) = ~ J o ~t77
If the period is subdivided into J intervals each of width A, so that t = JA, then eqn, (A1) may be rewritten
1 J-~fa+SA
m ( t ) = ~--~-- E
i(r)dr
vcg j=o'ja
(A2)
~/t-¢
as the sum Of J integrals. Consider the representative interval jA ~
,
bj
jA<~r<~A+jA
(A3)
will be used to interpolate between these end values. The reason for using this particular interpolative formula is explained later. When eqn. (A3) is inserted into eqn. (A2) and the indicated integrations are performed, the result
(
"-'
m(t)= ~
t arcsin ~ / ~ - - V~-tv~Z~--r)A+jA jA
E aj~/t-r j =O
(A4)
"1- - - ~
is obtained. When the integration limits are inserted and the substitution t = JA is
19 made, this result becomes
(~- J-1
m ( JA ) = --
(
(
Y~ a f f J - j - 1 + b; J arcsin g=O
.,_1
(
) j + 1 _ ~/jv+-T ~ J - j J
_
- 1
) (A5)
Next, t h e j = J - 1 terms are withdrawn from the first summation in eqn. (A5) and the j = 0 terms are withdrawn from the second summation in the same equation. This leads to
mj
= ~-[aovQ-- -~bj_ ~r l + ~
j=l ( a j - a j _ l ) ! / - J - j + ( b j - b j _ l )
Jarcsin
-
(16)
after a redefinition of the summation index j in the first, but not the second, summation in eqn. (15). Returning to formula (A3) one may "anchor" the interpolation at each end of the jA ~< T ~ A +jA interval by setting i ( j A ) = ij and i(A + j A ) = i;+~. The solutions of the two simultaneous equations that this generates are given as eqns. (13) and (14) of the main text. However, because i o is not a known datum, a 0 and bo cannot be determined. A way out of this difficulty is to approximate i(~-) for the 0 ~< ~-~< A interval by extrapolation from the A ~ ~-~< 2 A interval. This is tantamount to setting a 0 = a 1 and b0 = b~. To minimize error introduced by the extrapolation, it is especially important that the interpolation formula be a good match to the true i vs. t function at short times. It was for this reason that interpolation formula (A3), which exactly matches the short-time approximant (8), was selected. When a 0 is replaced by a I and b0 is replaced by bl, eqn. (A6) becomes algorithm (12) of the main text. Turning now to the semidifferentiation algorithm needed for chronocoulometry, we start with one of the definitions [14] of the semiderivative of a continuous function q(~) over the interval 0 < r ~< t
/
m ( t ) = q(O)
1
t
dq
One of the advantages of chronocoulometry over chronoamperometry is that, whereas in the latter technique the initial datum is not measurable, the initial chronocoulometric datum q(0) or q0 is known to be zero. Thus, the first right-hand term in eqn. (A7) can be dropped. Just as in devising algorithm (12) we adopted an interpolative formula that matched the short-time approximant of eqn. (7), so we now adopt the interpolation
20
formula q(r)
Aj'r
2Bj'g 3/2
2A
3A3/2
jA
<~r <~ A + j A
(AS)
to match the short-time approximant q = iot - 4 f l i o t 3 / 2 / 3 v ~
(A9)
of eqn. (21) in time dependence. Differentiation of eqn. (AS) gives dq=
2A
A3/2 dr
(A10)
and this may be substituted into eqn. (A7). From this point the development of algorithm (24) parallels that of algorithm (12) so closely that no details of the derivation need be given. Notice that an extrapolation into the 0 ~ r ~< A interval from the A ~