Anal. Chem. 1994,66, 1866-1872
Quasireversible Linear-Potential-Sweep Voltammetry: An Analytic Solution for Rational a Jan C. Myland and Keith B. Oldham' Trent University, Peterborough, Canada K9J 7B8
An analytic solution is presented to the problem of predicting the shape of a linear-potential-sweep voltammogram for a quasireversible electrode process under the conditions usually encountered in voltammetric experiments. It is necessary to assume that the transfer coefficient is a rational number, but in practice, this is not an important restriction. The solution takes the form of a series of exponential functions, the coefficients of which are calculable by a simple recursion formula. The series diverges except in the potential range corresponding to the early stages of the experiment, but numerical methods nevertheless permit all interesting regions of the voltammogram to be investigated. The exact location of the peak, as a function of a and of the kinetic parameter k(RT/nFvD)l/*, has been delineated. Comparison of these results with data generatedby the DigiSim software is reported. If one includes its "cyclic" variant, linear-potential-sweep voltammetry is probably the most frequently practiced form of voltammetry. Yet, analytic expressions for the shape of a linear-potential-sweep voltammogram are known only for the special case of reversibility.' Sometimes integral equations are solved,2but the usual procedure for predicting the shapes of linear-potential-sweep voltammograms under nonreversible circumstances is by way of digital ~ i m u l a t i o n .In ~ this paper it is demonstrated that an exact analytic solution can be found that applies to linear-potential-sweep voltammetry in the more general, or quasireversible, case.
THE MODEL Consideration will be given to the n-electron oxidation
occurring at an electrode under the following conditions, which are those usually assumed in voltammetric studies: (a) Initially the species R is present, at a uniform concentration ck, but its oxidation product 0 is absent when the experiment commences. (b) Because the solution is quiescent and contains excess supporting electrolyte, diffusion is the only operative transport mechanism. The diffusivities of the electroactive species are DR and DO, which are not necessarily equal. (c) The working electrode is planar, or can be treated as if it were, so that diffusion occurs along flux lines that are effectively parallel. (d) The transport field is semiinfinite, i.e. in front (1) Oldham, K. B. J. Electroanal. Chem. Interfacial Electrochem. 1979, 105,
373-375. (2) Nicholson, R. S.;Shain, I. Anal. Chem. 1964, 36, 706. (3) Britz, D.Digita/SimulationinElectrochemistry(LectureNota in Chemistry, Volume 23); Springer-Verlag: Berlin, 1981.
1166
Analytical Chemistry, Vol. 66,No. 11, June 1, 1994
of the working electrode there is an unimpeded zone of width far exceeding (DRAt)lI2, where At is the duration of the experiment. (e) Reaction 1 is not complicated by any concurrent reactions. (f) Adequate corrections have been made for ohmic polarization and nonfaradaic current. (g) Frumkin effects can be ignored. If these seven conditions are fulfilled, then it is known4 that the semiintegral m of the faradaic current density satisfies the twin relationships
where each?I denotes a concentration at the electrode surface and F is Faraday's constant. Let it further be assumed that the oxidative and reductive members of process 1 obey first-order kinetics with respect to species R and 0,respectively; then Faraday's law permits the anodic current density i to be expressed as
- I= k
nF
c s - k cs ox
R
(3)
rd 0
where ko, and krd are the potential-dependent heterogeneous rate constants. The ratio of the latter must be related by (4)
to the electrode potential E in order that eq 3 should reduce to the Nernst equation where i = 0. Here EO'is the conditional (or formal) potential of the R/O couple, while R and T have their usual significances. Equations 2-4 may be combined into
the universal equation of voltammetry. For the present purpose, it is more convenient to rewrite this as
-- - mL-m-mexp-(E{it iDR1I2 kOX
-E)
1
(6)
where mL is an abbreviation for nFc@,'12 and Eh is a "benchmark potential", equal to Eo' + (RT/2nF) h(DR/Do). For a reversible reaction, the half-wave potential coincides (4) Oldham, K. B.; Myland, J. C. Fundamentals of Electrochemical Science; Academic Press: New York, 1993; Section 7:12. 0003-2700/94/0366-1866$04.50/0
0 1994 American Chemical Society
with Eh and therefore the benchmark potential is often described as “the reversible half-wave potential”. It is usual to assume that the potential dependence of the oxidative rate constant k,, can be described by the ButlerVolmer equation nF k,, = ko’ exp (1 - cu)-(E RT
1
eqs 8 and 9 may be combined into
i mL expjbt} = m exp(bt) + m + -expjabt) Ab’/’
(13)
- Eo’)\ =
1
k o r ( ~ ) ( 1 - a )exp((1/2 a)RT\E nF -Eh) (7)
where ko’ is the common (to the forward and reverse processes) heterogeneous rate constant for the electron transfer at the conditional potential E O ’ . With this assumption, eq 6 may be recast as
; :1
mL - m - m exp -(E
1
- E ) (8)
This equation is valid for any voltammetric experiment that meets conditions a-g, provided that the electrode reaction obeys the kinetic laws that have been assumed.
THE LINEAR-POTENTIAL-SWEEP MODEL In the linear-potential-sweep voltammetric study of an oxidation, the electrode potential is made to increase linearly with time. It is convenient to adopt a time scale that has, as its origin, the instant at which the potential reaches the benchmark potential Eh. With that choice,
Equations 10 and 13 each independently link the threevariables i, t , and m. In principle, therefore, it should be possible to eliminate m between these two equations and so derive an explicit relationship for i in terms of t ; that is, it should be possible to predict the shape of the linear-sweep voltammogram. Before proceeding to a general solution, three limiting cases will be discussed. Each of these corresponds to one of the three right-hand terms of eq 13, all of which are positive, being negligible in comparison with the other two.
REVERSIBLE SOLUTION Reversibility corresponds to an overwhelmingly large value of the conditional rate constant ko‘, so that A- 0 3 . Equation 13 reduces to m=
mL e x p w 1 + expjbt)
(14)
in this limit. To find the current density i, eq 14 must be semidifferentiated. To accomplish this operation, one first expands the [l + exp(bt)]-’ term
m = -m,E(-)’ explibt) E = Eh+ ut
J=1
(9)
where u is the sweep rate, a positive constant. As is usual in linear-potential-sweep and cyclic voltammetries, it will be assumed here that the sweep commenced at such a negative potential (for an oxidation) that the initial faradaic current was immeasurably small. The experiment is therefore equivalent to one that started at t = - 0 3 , and accordingly, the faradaic current density and its semiintegral are related by the definition
with minus infinity as the lower limit of the convolution integral. With the convenient definitions of a frequency parameter b = -nFu RT
and a dimensionless reversibility index5 ( 5 ) Bard, A. J.; Faulkncr, L. R. Electrochemical Methods: Fundamentals and Applications: John Wiley & Sons: New York, 1980; p 224.
and then semidifferentiates term-by-term. This produces (see the Appendix)
(16)
Though his derivation was considerably different from that just given, result 16 was first obtained by Reinmuth6 in 1962. It provides an exact expression for the early portion of a reversible linear-potential-sweep voltammogram. Unfortunately, its straightforward applicability is limited to negative values of t (i.e. to potentials up to the half-wave potential), which constitutes the less interesting half of thevoltammogram. An alternative series representation1 describes the entire reversible linear-potential-sweep voltammogram (and, of course, the forward branch of a cyclic voltammogram) exactly. A related expression’ has been shown to govern the reverse branch of a reversible cyclic voltammogram.
IRREVERSIBLE SOLUTION Total irreversibility prevails for oxidation 1 when the backward reduction proceeds at a negligible rate at all (6) Reinmuth, W. H. Anal. Chem. 1962, 34, 1446. (7) Myland, J. C.; Oldham, K. B. J . Electroanal. Chem. Interfacial Eleclrochem. 1983, 153, 43-54,
Analytical Chemistry, Vol. 66,No. 11, June 1, 1994
1867
electrode potentials that elicit a measurable anodic current. This means that the subtractive term in eq 3 may be ignored. If the consequences of neglecting kr,cb are pursued, one finds that the m term in eq 13 disappears, so that the simpler relationship
--i - ( m , - m ) exp((1- a ) b t )
(17)
formula
define all other values of Aj. The general formula for the Aj coefficients is A j = ( - l / A ) j [ ( l+ a ) ( l + 2a)
holds. Again we have two independent equations, (10) and (17), interrelating the variables i, t, and m. It is straightforward to demonstrate that the explicit formula -A (1 -
1-
1
j
a)1/2
0'p
j = 1, 2 , 3, ... (25)
which may be written more succinctly as
exp{(l - a ) j b t ) (18) A, =
and its semiderivative (see the Appendix)
... ( 1 + j a -
i
( - T [)(:)]'I2 a1/2
j = 1 , 2 , 3 , ...
(26)
where ( ) j denotes a Pochhammer polynomial.9 Hence the equation describing the foot of the linearpotential-sweep voltammogram is found to be
satisfy eq 17. Therefore eq 19 describes the time dependence of the current density in irreversible linear-potential-sweep voltammetry. This result was derived some time ago.8
after eqs 22 and 26 are combined and simplified.
FOOT OF WAVE At the foot of the linear-potential-sweepvoltammetric wave, m is small in comparison with mL. Accordingly, the first right-hand term is negligible in eq 13. The remaining terms are
QUASI REVERSIBLE SOLUTION We now seek a solution of the unsimplified eq 13. The transfer coefficient a is a number in the 0 Ia I1 range. Let us assume that it is a rational number
m , exp{bt)= m
+
1
exp{abt)
(20)
and this equation may be solved by assuming that m
m = m L E A je x p { ( l + j a ) b t )
(21)
where q is a positive integer larger than unity and p is another positive integer smaller than q. With this substitution, eq 13 may be reorganized into
]=o
m = - -Ab'/2 e x p e } - m exp{btj
+ mL exp{bt]
(29)
where each Ai is a constant. Then, using the results of the Appendix, semidifferentiation gives m
i = m L b l i 2 E A j ( l+ ja)'/2exp((1 + j a ) b t ) ]=O
Now, let it be assumed that there exists a solution to eq 29 which leads to an expression for the semiintegral of the current density that is of the form
When eqs 21 and 22 are substituted into eq 20, the result (30) 1
1 = E A j expliabt} + - C A j ( l + ~ ' a ) 'exp{(j /~ ]=O
+ 1)abtj
A1=0
emerges after division by mL explbt). This equation is satisfied by assigning the value of unity to A0 and letting the recursion (8) Goto, M.; Oldham, K . B. And. Chem. 1976.48,
1868
1671-1676.
Analytical Chemistry, Vol. 66,No. 11, June 1, 1994
where the B, coefficients (which depend on A and a but not on t) are presently undetermined. The corresponding expression for the current density itself is found to be (9) Spanier, J.; Oldham, K. B. An Arlas of Functions; Hemisphere Publishing Corp. and Springer-Verlag: Washington, DC, and Berlin, 1987; Chapter 18.
m
i = m , [ b / q ) ' / 2 ~ j ' i 2 Bexp j -
(31)
I=,
Of course, the final term is zero, involving as it does a B coefficient of negative index. The penultimate term may also be zero, but if it is not, it may be replaced by another application of formula 37, generating
by semidifferentiation (see the Appendix). Substitution of eqs 30 and 31 into (29) gives
1
"
exp(bt1- - x j ' / ' B j
The final term will likely contain a B coefficient of zero or negative index and hence be zero. If not, the procedure may be repeated as often as necessary until
exp
Aq'/21=1
B, = 1
This complicated equation takes the more compact form m
.
m
m
is established. Similar arguments, employing formula 37 as many times as are needed to obtain a right-hand side involving only B coefficients of nonpositive index, may be used to establish that Bj = 0 for j
when e is used as an abbreviation for exp(bt/q). Replacement of j by j - p in the second summation and by j - q in the third summation now gives 1 C B j d = eq - ]=l
Aq'I2
f:0' -p)1/2Bj-pd f:Bj+d
j-p+l
-
(40)
0 are needed in eq 30, and we are therefore free to define Bo and all B coefficients with negative indices as equal to zero. This permits the unification of the lower limits of the three summations
It is now possible to equate the multipliers of d for various valid values of j . In the special case when j = q, (36)
because BO = 0. For all other values of j , one finds
This last is a recursion formula by which each coefficient may be calculated from two other coefficients of lower index. That B, invariably equals unity may be demonstrated by the following reasoning. Because q - p is a number necessarily unequal to (and less than) q, one may use formula 37 to replace the B , term in formula 36. This produces
Evidently, when augmented by rules 40-42, eq 3 1 is a valid solution to the problem and exactly describes the time dependence of the faradaic current density in linear-potentialsweep voltammetry when the electrode reaction behaves quasireversibly. When the mL and b abbreviations are abandoned, this equation may be converted to
in which form it expresses the current density as a function of electrode potential. Equation 43 suffers from the same serious limitation as series 16, the Reinmuth equation. To ensure that the series converges, the argument of the exponential function must be negative. Moreover, even when convergent, the numerical evaluation of eq 43 presents a series challenge because values of B, tend to grow in magnitude as j increases, which makes the retention of adequate precision difficult with ordinary (or even double-precision) arithmetic. AN EXAMPLE Here we shall first evaluate the leading values of the Bj coefficients for the typical case a = 0.4. Then q = 5 and p = 2. Rules 40 and 41 show that B , = B, = B, = B , = 0
(38)
(44)
and Analytical Chemistry, Vol. 66,No. 11, June 1, 1994
1869
B5 = 1
(45)
Table 1. Data Used in Calcuiatlng the Shape of an a = 0.4 Linear-PotentlaCSweepVoltamoaram
i
Rule 42, which takes the form
formula for calculating Bj
B , = - - B1
A 5
- B = - -1 A
(47)
the second equality being a consequence of rules 40 and 41. In fact, this last equation generalizes to B, = -l/A for all a values. Expressions for the first few coefficients in the p = 2 , q = 5 case are included in Table 1. It is evident that, as j increases, the algebraic expressions for Bj become steadily more complicated, but their numerical values remain calculable from the recursion formula. Some values for the A = 0.5 case are also listed in Table 1, along with the cumulative sum Zj1/2B,expunF(E- Eh)/qRT) when E = Eh. These numbers amply demonstrate the difficulty in summing the series in eq 43 unless A is large and/or the exponential term is small. The first few terms in the expression for the current density are found to be
&1/2Bj
14 5
0 1
0 2.236 068
-(1/A)(5/5)'lzB5 Bz = -l/h 8 -(l/h)(6/5)1/2BgB3 = 0 9 -(1/A)(7/5)1/zB, B4 = (7/5)1/2(1/Az) 10 -(l/~i)(8/5)'/~B* Bs = -1 11 -(l/A)(9/5)1/2Bs Bg = -(63/25)'/2(1/A3) -
-2
-3.055 435
0
-3.055 435
4.732 864
11.143 157
-1
7.980 879
-12.699 606
-34.138 950
12 -(1/h)(10/5)1/2B10B, = (1+ 21/2)/A
4.828 427
-17.412 788
50
-9.444 883 X 1013 -3.119 252 X 10"
500
1.765 556 X 1P70 ' 3.392 186 X lP1
7
in this instance, then permits the remaining coefficients to be evaluated. As an example of the application of this formula, the j = 7 instance provides an interesting and simple case. One finds
Bj for A = 0.5
approach a discernable limit, as illustrated in the final column of Table 1. We applied the Pad6 techniqueI0J1 to overcome this problem. As many as one million Pad6 operations were needed to provide a single point on the voltammogram. The second difficulty arises because, especially at the more positive t values, the magnitude of successive summands generally increases, as the penultimate column in Table 1 demonstrates. Some terms are positive, others negative, so that precision is lost on forming the sum. The only solution to this problem is to retain more digits in the calculation. We used the facilities of the MapleVI2 software and utilized as many as 500 digits in our calculations.
RESULTS Via the numerical procedures just described, we have used eqs 40-43 to explore the dependence of quasireversible linearwhere, as before but now specialized to a = 2/5, t = exp(nF(E potential-sweep voltammograms, and especially their peaks, - Eh)/5RT) and A = ~ " ( R T / ( ~ F D R ~ / ~ D O Notice ~ / ~ U ) )on~the / ~parameters . a and A. Data were generated at intervals that as A 0 3 , most of the summands vanish and the remaining of 0.2 in nF(E - E h ) / R T and used to construct curves via a terms duplicate eq 16, the reversible instance. In fact, eq 16 cubic spline. In Figure 1, a is held constant at '12 and A takes is a special case of expansion 43 that is a-independent. values of 0.2, 0.5, 1, 2, 5, and 0 3 , the final value being the Similarly, eq 27 is another special case of expansion 43. reversible limit. This figure demonstrates that, as the Because total irreversibility is a special case of quasirereversibility index decreases, the peak of the voltammogram versibility, one might also expect eq 19 to be a special case becomes much broader, its position shifts to more positive of eq 43, but this is not so. These two series are fundamentally potentials (for an oxidation), and its height decreases. Figure different in that expansion 43 is a summation of negative 2 shows a family of curves which possess a common value of powers of A, whereas series 19 involves increasing positive the reversibility index: A = 0.5. Here a takes values of l/5, powers of the reversibility index. One might surmise that 2/5, 3/5, and 4 / ~ . As this figure shows, the voltammogram series 19 is a special case of a yet-to-be-discovered expansion changes with increasing a to become broader, its peak shifting that is an alternative representation of quasireversible linearto more positive potentials and diminishing in height. These potential-sweep voltammetry. trends agree with the well-known results of other w o r k e r ~ . l ~ , ~ ~ Notice the intriguing feature of Figure 2, that there is a NUMERICAL PROCEDURES common, a-independent point. We have confirmed that this There are two concurrent difficulties in attempting to use series 43 to compute numerical i-versus-E (or i-versus-f) data (IO) Spanier, J.; Oldham, K. B. An Arlas ojFunctions; Hemisphere Publishing Corp. and Springer-Verlag: Washington, DC, and Berlin, 1987; Section 17: for quasireversible linear-potential-sweep voltammetry. These 13. two problems are distinct and require different strategies for ( I 1) MacDonald, J. R. J . Appl. Phys. 1961, 35, 3034. (12) Char, B. W.; Geddes, K. 0.; Gonnet, G. H.; Leong. B. L.; Monagan, M. B.; their resolution. Both strategies must be applied simultanWatt, S.M. MapleVLunguage Reference Manual; Springer-Verlag: Berlin, 1991. eously. (13) Nicholson, R. S.Anal. Chem. 1965, 37, 1351-1355. The first difficulty is that, for wide ranges of potential, (14) Bontempelli, G.; Mango, F.; Mazzocchin, G.-A.; Seeber, R. Ann. Chim. (Rome) 1989, 79, 131. series 43 is divergent, i.e. successive partial sums do not
-
1870
Analytical Chemistry, Vol. 66, No. 11, June 1, 1994
0.5
I
I
s
0.4
3
- 0.44
- 0.42
Q)
P,
0.3
- 0.40
2.0
4
f
/
0.2
0.1
1.9
\
-Id
- 0.38
- 0.36
0.34
0.0 I
I
-4
-2
I
I
I
0
2
4
1.8
I
0.2
0.4
0.3
0.5
0.6
0.7
0.8
a
nF(E-Eh)IRT Flgure 1. Linear-potential-sweep voltammograms calculated by eqs 40-43 for the values of A indicated. The transfer coefficient a is throughout. The current density i has been normalized by division by the constant $(n3?Qv/RT)”2.
Flgure 3. Dependence of peak height and position on the transfer coefficient. “Peak position” denotes ne€* - P)/RTand “peak height” denotes ( ~ ~ $ ) ( R T / ~ P EPk ~and V ) ~being ~ ~the , coordinates of the voltammetric peak. The line connects points determined in this study; points shown as open symbols are from the literature.”
b
0.5
Table 2. Comparlson of Peak Coordlnates Determlned by the Analytlc Formula 43 wlth Those Found from the DlglSlm SOttware.
0.4
E,& A
5 2 1 0.5 0.5 0.5 0.5 0.5 0.2
0.2
0.1
0.0 I
I
-4
-2
I
I
I
0
2
4
E+-E”
CY
eq 43
(mV), simulation
i d , simulation
0.5 0.5 0.5 0.2 0.4 0.5 0.6 0.8 0.5
28.6 33.4 40.6 51.3 56.9 64.9 69.7 74.9 78.6 107.2
0.4462 0.4361 0.4232 0.4069 0.4465 0.4096 0.3868 0.3602 0.2931 0.3646
28.5 33.4 40.5 51.3 56.8 64.9 69.7 74.9 78.5 107
0.446 0.436 0.423 0.407 0.447 0.410 0.386 0.360 0.293 0.365
infinity
0.3
- E”
bV), eq 43
i+,
The peak current density i~ has been normalized by division by the constant ~ ~ ( n ~ ~ D ~ u / R ! t ‘ ) ’ / ~ .
nF(E-Eh)lRT Flgure 2. Linear-potential-sweep voltammograms calculated by eqs 40-43 for the values of a Indicated. The reverdbility Index A is 0.5 throughout. The current density i has been normalized by division by the constant $(n3?DRvlRl)’/2.
behavior is not unique to A = 0.5, but we have failed to identify any theoretical explanation of this phenomenon. The coordinates of each peak were determined by fitting a parabola to three points in the peak‘s vicinity, using augmented data. Figure 3 shows the effect of a on the peak height and position when A = ?r1I2/2. This value of A was chosen as it corresponds to J/ = 0.5 used in the theoretical voltammograms reported by N i c h o l ~ o n and ’ ~ Bontempelli et al.I4 Points obtained from the graphs published by the latter workers15 have been inserted into Figure 3; the agreement is as good as could be expected from reading small-scale published plots. (1 5) Note that our a = 0.3 correspondsto their 0.7, and vice versa, because we treat an oxidation whereas the other workers analyzed a reduction.
Another way of checking our results is via the recently marketed DigiSim software.16 Using this simulation program17 and its built-in peak-finding routine, we determined the coordinates of the first peak of simulated cyclic voltammograms for assortedvalues of A and a. Some of these results are displayed in Table 2, where they are compared with values calculated by the analytic procedure described in the present article. Essentially, the tabulated values coincide. This not only confirms our derivation but is a tribute to the efficacy of the DigiSim software.
CONCLUSIONS An exact analytical formulation has been found that describes the shape of a linear-potential-sweep voltammogram (16) Rudolph, M.; Reddy, D. P.; Feldberg, S. W. DfgiSim 1.0 Software; BioAnalytical Systems Inc.: West Lafayette, IN, 1993. (17) We employed the parameters n = 1, T = 298.15 K,& = DO = 1 X lo-’ cm2 s-I, A = 1 cm2,c i = 0.010 364 3 M, ck = 0, u 2569.27 V s-l, stepsize = 0.001 V, E,,,,,/E,~i,h = E Di 0.4 V, and ¶, = 0.5, the latter parameter governing the rate of exponential increase of the simulation’s distance scale. These valucs were selected so that ko‘/cm s-l and A are numerically equal and to give the the value of 1.000 A. normalizing facor c~(n3F’DRu/Rr)’‘*
Analytical ChemMry, Vot. 66, No. 11, June 1, 1994
1871
under quasireversible conditions. To reiterate, eq 43 expresses the current density as a function of electrode potential, whereas the formula that relates the current density to time is
of Canada. Access to the DigiSim software was kindly provided by Professor Alan Bond and Dr. Stephen Feldberg.
APPENDIX The notation1* d-Il2flt)/ [d(t m ) ] - * / z indicates the operation of semiintegration of the functionflt) with respect to t with a lower limit of minus infinity. This operation may be defined by
+
where t is the time scale that has its origin at the instant that the potential sweep reaches the benchmark potential ah.The Bj coefficients are to be found be applying the rules for j < q
+
Here, A represents the reversibility index k o ' ( R T / n F W Dol-uu)llzand the quotientplq equals the transfer coefficient
Similarly the dV2f(t)/ [d(t m)]l/z notation represents the operation of semidifferentiation of the functionflt) with respect to t with a lower limit of minus infinity. This operation is defined by
"(
d t [d(td-1'2f(t) +
a.
The restriction to rational a values is not a problem in practive because any number between zero and unity may be expressed as a ratio of two integers to any desired precision. The advantages of the analytical approach over simulations or solutions of integral equations include the following: (a) Answers provided by an analytical solution are exact, whereas questions of accuracy may arise with the other methods. One of the most useful aspects of analytical results is the check that they provide of simulations. (b) Whereas simulations and integral equations lead to progressive solutions, Le. the calculation of one voltammetric point relies on the validity of the results for all previous points, this is not the case with an analytical solution. This aspect of the analytical approach enables one to "zoom in" on a voltammetric feature of interest. (c) Further processing (e.g. differentiation) of an analytical solution is possible without loss of precision. On the other hand, the analytical approach is undoubtedly less flexible than simulation.
ACKNOWLEDGMENT The authors gratefully acknowledge the financial support of the Natural Sciences and Engineering Research Council (18) Oldham, K. B.; Spanier, J. The Fracfionol Culculus; Academic Press: New York. 1974.
1872 Analytical Chemistty, Vol. 66, No. 11, June 1, 1994
I
That is, a semiderivative is the derivative of a semiintegral. These two operators perform very simple transformations, namely d-ll2 exp(bt + c) = b-"' exp(bt + c } [d(t + m)]-l/'
(A3)
d'l2 exptbt + c) = bl/' expibt + c ) [d(t + m)]1/2
(A4)
and
when applied to exponentials of a linear function o f t . Recehred for review November 22, 1993. Accepted March 4, 1994.
*
*Abstract published in Aduance ACS Abstracts. April 1, 1994.
