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on other chemical reactions and especially on the luminescence of single pulses are necessary to elucidate the details of the chemical processes occurring during bubble nucleation and disappearance in ultrasonic fields. The present studies were limited to intensities below 3 W/cm2. Chemical effects at intensities of some IO W/cm2 have been observed by various authors, mainly by using transducers with a horn. It is not yet known quantitatively how the chemical yield responds to pulsing at these high intensities. Only two investigations at higher intensities have yet been carried out which showed that the time interval between pulses also
determines the efficiency of the pulses under these condition^.^^^' .4cknowledgment. The authors thank Dr. Eberhard Janata for construction of the electronic equipment. This work was supported by Deutsche Forschungsgemeinschaft and Fonds der Industrie. Registry KO. I-. 20461-54-5. ( 7 ) Henglein. A,, Gutitrrez. M.: Ulrich, R. In!. J. Radiat. Biol. 1988, 54,
123.
Square-Wave Voltammetry for ECE Mechanisms John J. O'Dea, K. Wikiel,+ and Janet Osteryoung* Department of Chemistry, State University of New York at Buffalo, Buffalo, New York 14214 (Received: January 31, 1989; In Final Form: December 6, 1989)
The theory of square-wave voltammetry is extended to include the ECE mechanism in which an intermediate chemical step occurs between sequential electrode reactions. Exemplary calculations show how the thermodynamic and kinetic parameters of this mechanism affect the shapes of square-wave voltammograms. The theory is used with a nonlinear least-squares technique to obtain first-order rate constants for the chemical step of the reduction of p-nitrosophenol. This procedure discriminates against several experimental artifacts and does not require the usual separate determination of a normalization current. The rate constants obtained for the intermediate chemical step are 0.46 s-l at 25 "C, 2.40 s-' at 50 "C, and 17.3 s-I at 80 "C (12% uncertainty at 95% confidence) in good agreement with previous studies.
The ECE mechanism, that is, electron transfer to generate an intermediate which subsequently reacts to produce a second electroactive species, is common in organic electrochemistry. Techniques for studying this mechanism, in particular by cyclic voltammetry, are well established. The modern pulse technique of square-wave voltammetry offers many advantages over cyclic voltammetry with respect to both the quality of the experimental signal and computation of theoretical curves. These advantages can be exploited readily by employing the COOL algorithm, which provides an unbiased way of comparing theory with experiment.' With the increased availability of automated instrumentation, the analytical and mechanistic capability of square-wave voltammetry continues to be extendede2J In this work we elaborate the theory of square-wave voltammetry to include ECE mechanisms and demonstrate its practical application for determination of the rate constant of the intermediate chemical step for the case of reduction of p-nitrosophenol. The general scheme of the ECE mechanism is A nle s B (1)
+
k
B-C C
+ n2e
(2) G!
D
(3)
in which the electron-transfer steps are characterized by potentials E," (eq 1) and EZ0 (eq 2) and we assume that the homogeneous chemical reaction (eq 2) is totally irreversible. In the present work we restrict ourselves to the case in which both electron-transfer steps are Nernstian. Then the homogeneous cross reaction, reaction 3 minus reaction 1 B + C s A + D is also at equilibrium at the electrode surface.
(4)
Permanent address: Institute of Precision Mechanics, Duchnicka 3,00967 Warsaw. Poland.
0022-3654/90/2094-3628$02.50/0
Of particular interest is the case in which E," 150 and the contribution of spherical diffusion to forward and reverse currents is less than 0.7%. A qualitative comparison of Figures 1 5 and 16 shows that the forward and reverse currents are less skewed at 100 Hz and 50 O C than at 10 Hz and 25 "C.Thus the product kr is smaller in the former case. From this one can estimate that the reaction must be somewhat less than 10 times faster at 50 OC than at 25
-0. 132
-0. 134
-0. 136
POTENTIAL VS. SCE IVOLTSI
Figure 17. Optimal values (+) and confidence regions for (k,E,,,) for p-nitrosophenol at 50 OC. f 2 (eqs 15, 20) = 0; confidence level = 95%.
OC. A detailed COOL analysis confirms that the rate is larger by about a factor of 5 at the higher temperature. Figure 16 also shows that the net peak current has moved about -25 mV from its position at 25 OC. This experimental artifact presents no special problem for the determination of the rate constant since the reversible half-wave potential is sought by COOL for each voltammogram. The solid lines in Figure 16 show the best fit obtained by COOL at 50 "C for the forward and reverse currents. The net current predicted by the fit is also compared with the observed net current. As is the case at 25 OC, the agreement between the experimental voltammogram and the theoretical model is excellent. Similar results were obtained also at 80 OC and 200 Hz. An Arrhenius plot of the rate constants in Table I suggests an activation energy of about 57 kJ/mol for the intermediate chemical step. The rather small rate constant places an upper limit on the frequency whereas the lower limit is constrained by spherical diffusion and perhaps convection. An example of the quality of the result at various frequencies is given in Figure 17. The computation of confidence regions as shown there has been described in detai1.l The main feature in this diagram is the dramatic increase in the confidence region in the k dimension a t higher frequencies. The values of log (kt,) from lowest to highest frequencies are -0.98, -1.28, -1.98, and -2.28, respectively. At the higher frequencies, the response is insensitive to the kinetic step so the value of k derived therefrom is less well-defined. The variation in reversible half-wave potential among replicate experiments is greater than that suggested by the confidence regions of Figure 17 because the temperature of the reference electrode is not constant. As described above, this uncertainty does not affect the derived values of k . We have not investigated the second-order homogeneous reaction mechanism for square-wave voltammetry. However, it is
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reasonable based on the similarity with the potentiostatic case to suggest that the square-wave response for the homogeneous and heterogeneous pathways would be indistinguishable for log (kt,) < -0.4.4 On the other hand, the charge-transfer step is Nernstian Z ]0.6. Combining these criteria together for log [ k l o ( t p / D ) ] l > with a reasonable value of D (9 X lod cm2/s), log ( k , o / k 1 / 2>) -1.7. This criterion should provide a general guide to the range of utility of this model for E l o
