Anal. Chem. 1988, 60, 2269-2273
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Diagnosis of Irreversible Electron Transfers and Reversible CEC, CE, and EC Processes from Differential Pulse Polarographic Criteria Lucia M. Rodriguez-Monge, Eulogia Mufioz, Jose L. Avila, and Luis Camacho* Department of Physical Chemistry and Applied Thermodynamics, Faculty of Sciences, University of Cbrdoba, Cbrdoba, Spain
By use of the reactlon layer approxlmatlon we have obtalned analytlcal current-time-potential dlfferentlal pulse polarographic relations for CEC, EC, and CE processes. Such processes are analyzed, along wlth the correspondlng reversible and lrreverslble electron transfers, In order to establlsh sultabk dlagnostlc crlterla, based both on peak current and on peak half-wldth measurements. The proposed crlterla are checked wlth well-known experlmental examples.
There are a host of theoretical differential pulse polarographic (DPP) studies on reversible (&) (1-9) and irreversible (Ei) (10-15) electron transfers but relatively few on the application of the pulse technique to electron processes coupled to chemical stages. Thus, Kim and Birke studied first-order catalytic processes (16) and CE schemes (which involve a chemical stage prior to the reversible electron transfer) (17).. Recently, the reaction layer approximation (1419)was used to study dimerization reactions subsequent to reversible electron stages by DPP (20). The approximation has been found to be accurate for sufficiently large rate constants (20). In this work we have used the reaction layer approximation to derive current-time-potential D P P relations for CE, EC (involving a chemical stage subsequent to a reversible electron transfer), and CEC (with chemical stages prior to and subsequent to a reversible electron transfer) processes corresponding to an expanding plane electrode. The expressions derived and those corresponding to E, and Ei processes have in turn been used to establish appropriate diagnostic criteria. Birke et al. (17, 21) established a diagnostic criterion for CE, E,, and Ei processes based on the use of two DPP recordings, one with cathodic (AE = El - Ez, positive) and another with anodic pulses and the same absolute value. The relation between the peak heights obtained allowed these authors to discern between CE, E,, and Ei processes, as well as to deal with processes involving quasi-reversible transfers (17,21). As shown in this work, this criterion is inadequate on its own to discern between the analyzed processes. Thus, CEC and E, processes are indistinguishable, as are Ei and EC processes, on the sole basis of this criterion. The aim of this work was thus to perform a systematic study of all the potential diagnostic DPP criteria useful for this purpose and to select the ones best suited to the processes dealt with, which are experimentally checked.
EXPERIMENTAL SECTION Experiments were conducted with 5.7 X M Cd(N03)2,4 X lo4 M p-nitrophenol (PNP) in 0.1 M HzSO4, 5 X lo4 M M phenylglyoxal (PG), monoethyl fumarate (FME), 3.75 X M glyoxal (G)in H3PO4, CH,COOH, and &BO,, and 9.9 X 0.1 M each. The pH of all solutions was adjusted with solid NaOH. All products used were Merck p. a. Recordings were made on a Metrohm 626 polarograph. The dropping Hg electrode used had a flow rate of 2.99 mg/s. Pulse 0003-2700/88/0360-2269$01.50/0
Table I.
p,
8, and Aip/Aih Values for Different Schemes Reaction Scheme Values
scheme
P
P
WiDC
Definitionsa
= t'/(tl + t ' ) = t'/t + Ki)/7)'I2for i = 1, 2 a = CA/Co IL= XiKi/(1 + XiKi) 0 = exp(nF(El - EO)/RT} b = CB/Co 22 a c = cC/co v = - a- D - - 'a2 at ax2 3t ax d = CD/C' 7
X i = (3?rtk-, (1
a Key: n, number of electrons; m, H g flow rate (mg/s); Co,concentration (mmol/L); D, diffusion coefficient (cm2/s);Z, transfer
coefficient.
potentials were applied for a time (t') of 0.05 s. AE and drop lifetimes (t) were adjusted individually in each case. The potential scan rate was kept at 2 mV/s throughout. All measurementa were made at 25 "C.
THEORETICAL BACKGROUND Completely Irreversible Electron Transfer (Ei). As usual in DPP, we shall assume potential El a t the dropping mercury electrode to remain constant between t = 0 and t = tl. At this time is applied a potential pulse AE = El - Ez over an interval t'. The DPP current generated is given by Ai = i(t'+ tl, E,) - i(tl, El). We shall also assume that r = t'/(t' + tl) < 0.1. Under these conditions, and after slight modification of Matsuda's nomenclature intended to adapt it to the treatment made below, the current-potential-time relation for a process governed by a completely irreversible electron transfer can be written as (13) Ai/iDc = (1- a) W P ) % / P ) (1) where iD, (Cottrel's diffusion current), u, p , and j3 are defined for this process in Table I. The functions * ( p ) and 9 ( p / / 3 ) can be expressed with errors less than 0.4% by means of (22, 23)
-[
q ( P=) 1293 1 P
+ f-
NP/P) =
([f1' +
1
+ 0 . i 5 ~ ) ' ~ ] (2)
1 0.658
+ [0.342' + (P2/7rp2)11/2
0 1988 American Chemical Society
(3)
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 20, OCTOBER 15, 1988
These functions were used to carry out a comprehensive numerical study of the product \ k ( p ) @(PI@) in terms of the variable p over the p range 103-10-3. From this study was gathered-either numerically or graphically-all the desired information, which is presented in the next section of the paper. CEC Process. Consider the scheme where the electron
Scheme I A
k
~
-
'1
0 5
B
Icg k,
k-1
+ ne- 5 c k-2
Variation of Aip/i,,, vs log k , for a CE mechanism, with 7 = (0.04/0.992), E = +50 mV, n = 1, and K, = lo-'. Soiid line corresponds to the predictions of eq 16 for K , = 0. Circles denote the results given in Figure 2 of ref 17.
transfer is assumed to be reversible. If all the species involved are assumed to have the same diffusion coefficient, then the corresponding mass transfer equations can be expressed as V(a + b ) = 0 (4)
proximation will be obtained later, upon comparing the results of our treatment with the exact solution corresponding to the CE mechanism, solved elsewhere (17). The current can be defined as
B
Figure 1.
k
C S D
+ k - l ) ( K l U - b) + d) = 0 - d ) = - ( i t 2 + k_,)(K,c - d )
V(K,a - b) = -(k1
(5)
V(c
V(K,c
(6) (7)
where all the symbols used are defined in Table I, and co denotes the sum of the concentrations of species A and B in the bulk solution. Under the conditions prevailing prior to the application of the pulse (t < tl), the solutions to eq 4 and 6 are given by the Nernst diffusion layer approximation. Once the pulse is applied (t > t l ) and provided t'> 1. The p, p, and Aip/& expressions derived by applying these conditions are presented in Table I. In reference to CEC processes, application of these conditions will be henceforward implicit. CE Processes. Scheme I and its solutions can be applied to a CE process provided K 2 = 0. The expression obtained can be compared with the exact solution reported by Birke et al. (17). This comparison is made on the basis of the dependence of Aip/iDl on log kl (Figure 1)at constant Kl. The relationship between the Ilkovic diffusion intensity (iDI)and that of Cottrel is given in Table I. The conditions used were the same as those employed in ref 17. The solid line in Figure 1 corresponds to the predictions of eq 16, assuming K2 = 0, while the circles correspond to the exact solution (17). The difference is never greater than 8% under the above-mentioned conditions. If the first chemical stage is assumed to be of a highly kinetic character, then K1 1and K2/Xz>> l), the current will be given by eq 13, where p, 0, and Ai,/iDc are defined in Table I. Reversible Electron Transfer (E,). Again, Scheme I can be applied to this process as long as K1 m, IL = 1,and K2 = 0. The relation Ai/Ai, is given by eq 13, and p, 0, and Ai,/& are defined in Table I. It should be noted that the expression thereby obtained coincides with that reported in the literature for this process (22).
-
RESULTS AND DISCUSSION Analytical Criteria. We have systematically analyzed the influence of the variables AE and t on the experimental parameters Ai,, E, (peak potential), and W (peak half-width). The result was six diagnostic criteria commented on below. Figure 2 shows the variation of Aip/ILiDc vs ItnAE for all five reaction schemes studied. The parameter IL is different from 1 only for CE and CEC processes, while It # 1 for Ei processes only. The different behavior of each reaction scheme allows for clear-cut distinctions to be made between each. In order to fully understand the practical application of Figure 1, let us consider a DPP recording made at AE = +50 mV and another run at AE = -50 mV, as Birke et al. do (21). If the process concerned conforms to a CEC or E, scheme, then the absolute value of Aip will be the same in both cases. If the process conforms to a CE scheme, the anodic peak (negative AE)will be greater-in absolute terms-than the cathodic peak (positive AE),as predicted in the literature from sufficiently large rate constants (17). Opposing predictions (i.e. the cathodic peak will be greater in absolute terms than the anodic peak) can be made for EC and El schemes. The greater the pulses applied, the more this behavior is apparent. A second analytical criterion is based on the t dependence of Ai?, which is markedly influenced by the AE value in question. This behavior can be better understood by analyzing the Ai, values obtained at two rather different times, t = 2 s and t = 0.5 s, and plotting the ratio Ai,(t = 2 s)/Ai,(t = 0.5 s) as a function of %nAEfor all five schemes studied (Figure 3). As can be seen from the figure, all the processes studied tend to a common behavior for AE a. In this case, the ratio between the peak currents tends to 2.52, thus indicating a linear dependence of Ai, on t2f3. Decreasing AE values, however, may result in changes in the t dependence of Ai,, depending on the particular scheme considered. Thus, the Ai,(t = 2)/Aip(t = 0.5 s) ratio increases with a value of over 2.52 for CE processes and decreases for EC and Ei processes, for which it tends to 1.3 when ItnAE m. Finally, the peak
-
-
-80
-40
0
40
80
120
160
in A E (mv 1 Ai& = 2 s)/Aip(t = 0.5 s) as a function of
Figure 3. Variation of %inA€(t= 0.05 s)for E, (---), CEC (---), E, (-), processes.
EC (.-.), and CE (..-)
LAE Figure 4. Variation of ZnW (mV) as a function of Itn A€ (mV) for E, (- -), CEC (- -), E, (-), EC (. .), and CE (. .) processes, with T = 0.1 and t = 0.5 s.
-
-
-
-
current ratio is independent of AE and remains constant at 2.52 for CEC and E, processes. As can be seen, the information obtained by application of this analytical criterion is basically the same as that arrived at by using the previous criterion. In fact, EC and & processes cannot be distinguished on the sole basis of these criteria, and neither can CEC and E, processes. A third criterion would involve studying the AE dependence of E,. As shown, such a dependence is not significantly influenced by the particular reaction scheme, so it cannot be used as an analytical criterion. The variation of E , as a function of t is indeed a function of the mechanism concerned, although it does not allow one to gather information other than that obtained by application of the above-described criteria. This criterion was discarded, as variations of t resulted in rather small changes in E,; the criteria described above are much more sensitive. In Figure 4 is plotted the variation of (ItnW) as a function of (ItnaE)for all five schemes dealt with. From the figure it is apparent that for 2nAE values between 0 and 20 mV, %nW = 90 5 mV irrespective of the scheme
*
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ANALYTICAL CHEMISTRY, VOL. 60,NO. 20, OCTOBER 15, 1988
concerned. This allows one to estimate Gn to be roughly equal to 90/ W from a recording with a small enough M. This can be used to distinguish between EC and Ei schemes. Thus, a roughly whole Gn value would be indicative of an EC process, while a decimal number would be characteristic of an Ei process. In this case, &n can be calculated graphically with greater accuracy. This criterion cannot be used to distinguish between E, and CEC schemes because of the small separation between their corresponding curves in Figure 4. Finally, the potential criteria relying on the t dependence of W are less sensitive than those described above. From the reasoning above one might infer that E, and CEC are mathematically indistinguishable. A distinction is indeed possible by using criteria based on the t'dependence of the experimental parameters used. However, the identification of these schemes can be much easier if one takes into account that, in dc polarography, the constant diffusion current is defined as
-0.04
-0.06
ei
0.06
0.02
If D is assumed to be on the order of to 5 X lo4 cm2/s, then I will vary between 1.9 and 1.4 per electron exchanged in the process. This constant can also be defined in DPP as
For t = 0.5 and r = 0.1, F can be obtained graphically at a given GnAE value from Figure 1. For highly kinetic processes (CEC), IL
