Q88
Anal. Chem. 1981, 53, 988-991
of the original double bond positions are extremely weak. In methane CI mass spectra, the numerous EI-like ions are recorded (6). Isobutane CI mass spectrometry should be particularly useful in structural studies of polyunsaturated fatty acids from natural sources, because it gave prominent molecular ion and characteristic fragment ions corresponding to the cleavage of the original double bond positions.
(4) Wolff, R. E.; Woiff, G.; McCloskey, J. A. Tetrahedron 1977, 22, 3093-3101. (5) Hakomori, S-I. J. Biochem. (Tokyo) 1984,55, 205-208. (6) Ariga, T.; Suzuki, M.: Morita, I.; Murota, S.4.; Miyatake, T. Anal. Biochem. 1978,90, 174-182. (7) Schmltz, 6.; Egge, H. Chem. Phys. Lipids 1979,25, 287-298. (8) Nlehause, W. Q.; Ryhage, R. Anal. Chem. 1988, 40, 1840-1847. (9) Minikin, D. E. Chem. Phys. Lipids 1978,21, 313-347. ( I O ) Dommes, V.; Wirtz-Peitz, F.; Kunau, W. H. J . Chromafogr. Sci. 1978, 14. 360-365.
LITERATURE CITED (1) Murata, T.; Ariga, T.; Arakl, E. J. LipidRes. 1978, 19, 172-176. (2) Ariga, T.; Arakl, E.;Murata, T. Anal. Blochem. 1977, 83, 474-483. (3) Araki, E.; Ariga, T.; Murata, T. Biomed. Mass Spectrom. 1978,3 , 182-185.
RECEIVED for review November 14,1980. Accepted February 25,1981. This study W a s supported in part by a Grant-in-Aid for Cancer Research from the Ministry of Health and Welfare.
Theory of Differential Normal Pulse Voltammetry in the Alternating Pulse Mode for Totally Irreversible Electrode Reactions Timothy R. Brumleve’ and Janet Osteryoung” Deparfment of Chemistry, State University of New York at Buffalo, Buffalo, New York 14214
The theory of differential normal pulse voltammetry In the alternatlng pulse mode for totally lrreverslble reactions is presented. The results are compared wlth those of the general kinetic theory for double potentlal step chronamperometry, and the utility of the slmple solutlon for thls special case is Illustrated. For pulse widths of ca. 50 ms the response for a symmetrical double pulse uslng thls waveform should be about 5 tlmes greater than that for the related technique of dlfferentlal pulse voltammetry.
The technique of differential normal pulse (DNP) voltammetry employs a wave form, shown in Figure 1,in which most of the time is spent at an initial potential, Ei,and, as in normal pulse (NP) voltammetry, pulses with duration t zare applied from that potential to successivelylarger values, El. At some time after the pulse is applied, a second pulse of constant amplitude AE = Ez - El and duration t z - tl is applied. The current is measured at tl and at tz, and the differential current, i(tz)- i(tl), resembles that of differential pulse (DP) voltammetry. Thus the name differential normal pulse has been adopted to suggest the resemblance to NP in which faradaic reaction takes place usually only during the fairly brief periods of pulse duration and to DP in which a differential measurement of two faradaic currents gives rise to peakshaped voltammograms. The rationale for suggesting the use of this wave form is that it might provide many of the advantages of both NP and DP while avoiding some of the liabilities of each. For example, it has the advantage of peak-shaped current-potential response but retains the advantage of NP that the extent of reaction is typically 1%or so of that obtained in DP. Aoki et al. have presented a general theory for the current response to this wave form for charge-transfer kinetics ( I ) . (It may be noted parenthetically that the mathematical problem is that of double potential step chronoamperometry. Present address: Anderson Physics Laboratories, 406 N. Busey
Av., Urbana, IL 61801.
The many previous incomplete approaches to this problem in the literature attest to its intractability (2-5). The solution of Bard and Faulkner (6) is incorrect.) From the results of Aoki et al. (see, e.g., Figure 3 of ref 1) it is apparent that the resulting voltammograms can be highly unsymmetrical for reasonable choices of parameters and hence lack the merit of ready characterization through peak height, position, and width which is so convenient in DP voltammetry. A certain sequence of pulse application, which is referred to as the alternating pulse mode, not only avoids this problem but also provides peaks which do not contain the DC distortion present in DP. Figure 1 illustrates this scheme of pulse application, in which the second pulse of constant amplitude is applied alternately in the same and opposite sense as the first pulse from Ei to El. The differential current is taken as the difference Ai = i(tz, El + AE)- i(tz, El - AE), where AE is the value (with sign) of the constant amplitude pulse in the forward direction. Brumleve et al. have discussed the theory and presented experimental results for this wave form for the case of reversible charge transfer (7). While the contribution of Aoki et al. ( I ) provides a general solution to the problem of current response for the DNP wave form, the solution is, because of its generality, cumbersome and difficult to apply. As shown by Aoki et al. (1) and by many others, the range of parameters over which reactions behave quasi-reversibly is in fact rather small. This is illustrated by the voltammograms of Figure 2. Because of this, and because of the apparent utility of the DNP wave form in the alternating pulse mode for both analysis and physical chemical studies, we present here a simplified solution to the problem of current response for this wave form for totally irreversible reactions, Linearization of the exact solution for small values of the differential pulse height, AE, produces simple equations which relate the observables of peak height, position, and half-width to the kinetic parameters. DERIVATION OF CURRENT POTENTIAL CURVES We consider the case of semiinfinite diffusion to a stationary planar electrode. The solution contains a large excess of supporting electrolyte, the oxidized form is soluble, and there
0003-2700/81/0353-0988$01.25/00 1981 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 53, NO. 7, JUNE 1981
0
Q8Q
< t It1: km(t) = kl
tl < t It ~ :km(t) E kz for Ez = El + hE km(t) = k2, for Ez, = El- A&'
(5)
Explicitly km
= ( D O / D R ) " / ~exp[-4(E,,, ~, - &/z)I m = 1, 2, 2'
(6)
where Ell2 is the reversible half-wave potential, D R is the diffusion coefficient of the reduced form, k , is the standard rate constant, CY is the transfer coefficient, and f = F / R T = 38.9 V-' at 25 O C . Similarly we define
t
Figure 1. Potential-time function for DNP in the atternating pulse mode. The second pulse A€ is applied from potential El in the positive and negative senses on successive pulses. The duration of the first pulse is t l , the duration of the entire double pulse is f2, and the delay time between double pulses at the initial potential, E,, is ti. The value of E is incremented on successive pairs of double pulses as shown in the lower portion of the figure.
Qm
( m = 1 , 2 , 2')
= km/&
(7)
and (8) Applying the definition of the Laplace transform to eq 4 and 5 and noting that k2 = ekl we obtain e = exp(-anfAE)
i/nFA = kl(cCo(O,s) - ( e - 1) &t1Co(0,t)e-5tdt) (9) For t I tl we have the well-known result co(o,t) = CobF(Qi&)
(10)
where F ( x ) = exp(x2) erfc ( x ) . Thus
J t 1 C ~ ( 0 , t ) e - 5dtt = Cob[exp(-stl)F(Ql&
-
Transformation and combination of eq 1-3 yields ~ ~ ( 0 , s=) cob/s - i / n ~ ~ f i
(12)
and combination of eq 9, 11, and 12 yields
~/~FAC~~Q= , &E/&(&
[&e-8tiF(Ql&
-
6+
+ Q ~ -) Q1 erf
(E
- 1) x
&I/(&
+
(13) Inversion as described in the Appendix produces the result Qz)(Qi2 - 8)
nf ( E - E,,*
)
Figure 2. Voltammograrris for DNP, alternating pulse mode calculated from an integral equation procedure (8, 9) or from ref 1. Dependence on heterogeneous rate constant, k,, is shown for three values of CY and for -log k, (cm s-'): (A) -2, (E) 2, (C) 2.5, (D) 3, (E) 4, (F) 5, (G) 6. Do = D, = cm2 si, tl = t2 - t , = 0.05 s, nAE= -50 mV.
-
is only one simple, totally irreversible electrode process, 0 + ne R. The diffusion equation is
dCo(r,t)/dt = D,dzc,(x,t)/dxz (1) where CO is the concentration of 0 and DO is its diffusion coefficient. The boundary and initial conditions are CO(S',t) =
Co(X,O) = Cob
(2)
where Cob is the bulk concentration of 0. At the electrode surface
i/nFA4 = DodCo(O,t)/dx
(3)
and the kinetic equation is given by
i/n FA = k,( t )Co(O,t)
(4)
where k ( t ) is the formal heterogeneous rate constant for the reduction of 0 to R. We write k ( t ) to emphasize that while k is a function of potential, the potential is stepped from El to El f AI3 at time tl, and therefore k is also a function of time. We therefore consider three values of k ,
where if, is the forward difference current (Le., the current when the first and second pulses are in the same direction) given by i ( t 2 )- i(tl), i d = nFACob(Do/d,)1/2, and t, is the duration of the second pulse, t, = t2 - tl. The corresponding equation for the reverse difference current, i,, is obtained from eq 14 by replacing Q2 with Q2, and, noting that Q2= e2Q2(, replacing e with The current signal in the alternating pulse mode is
Ai = if,,, - i,, (15) This solution was evaluated numerically as described in the Appendix for comparison over its range of validity with the results of Aoki et al. (1). Figure 3 illustrates the current response, Ai, and the individual forward and reverse currents for a totally irreversible reaction for typical values of the experimental parameters. The solid curves are calculated from eq 14 and 15. The validity of the solution wm verified by comparison with the exact solution for the quasi-reversible case as given in eq 25-29 of
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ANALYTICAL CHEMISTRY, VOL. 53, NO. 7, JUNE 1981
Evaluation of the integral appearing in the expression for H(X) was carried out by a procedure similar to that described in the Appendix for the evaluation of the integral of eq 14. It is apparent from eq 17 and 18 that the linearized current response is a function of only A, a, and AE for all totally irreversible reactions. A binary search was used to find the value of X which gives the maximum value of H(X). The result is
= 0.513 H(A,,,) = 0.556
A,, I -5
Therefore the peak current and peak potential are given by
- 25
-15
E, = El/2 + (l/anf) In (1.95k,@)
nf(E-E,,,)
Figure 3. Forward, reverse, and dlfference currents for DNP in the alternatin pulse mode calculated for a typical totally irreversible case: k , = 10' cm s-', a = 0.5, Do = D, = lo-' om2 s-', f r = f 2 - f l = 0.05 s, n A E = -50 mV; solid lines, calculated from the exact solution of eq 14 and 15; dotted lines, calculated from the approximate solution of eq 36 of ref 1.
ref 1 and by comparison with the results of a numerical integral equation procedure similar to that employed by O'Dea et al. (8,9).The dotted curves are calculated from eq 36 of ref 1, an approximate solution said to be valid over the range of parameters 10-locm s-l/*5 k, tP1I2I1cm d2, 0.1 I a I 0.9, 1.001 I t2/tl I 6.0 and 0.005 In(El - E2) I 0.3 V. (Note that in ref 1 in the discussion following eq 36 it is stated incorrectly that for t 2 / t l I 10 the term al of eq 36 can be neglected. This should read t 2 / t lI 1.1.) The approximate solution matches the exact solution reasonably well for iforas it should, because the conditions for the approximation are met. Deviation from the exact solution is largest midway between the peak potential and the limiting current region well negative of the peak, as one would expect. For i,,,, however, to which the approximate equation does not apply, the errors in this same potential region are far worse and of the opposite sign. Thus the difference current, Ai, calculated from the approximate equation is seriously in error with respect to peak height, width at half-height, and shape. The solution for the totally irreversible case presented here is therefore practically very useful. Referring once again to Figure 2, it is particulary interesting that the peak shape is uniform and nearly symmetrical and that the peak potential appears to depend linearly on log k, in the totally irreversible region. This suggests that simple relations among peak position, height and width, and the kinetic parameters may be derived from eq 14 and 15. We develop these by linearization of eq 14 for small pulse amplitudes such that lanfhE(
