852
Anal. Chem. 1981, 53, 852-856
Diagnosis of Reversible, Quasi-Reversible, and Irreversible Electrode Processes with Differential Pulse Polarography Ronald L. Blrke,* Myung-Hoon Kim, and Moses Strassfeld Department of Chemistry, City University of New York, The City College, New York, New York 10031
derived only qualitatively represent the behavior of the pulse current. The pulse peak current at constant concentration decreases as the standard heterogeneous rate constant is lowered for a quasi-reversible system and then becomes constant for an irreversible process (5); however, the numerical value of the current is overestimated by this treatment. For rate constants less than 10~6 cm s~\ the current-potential profile merely shifts toward higher overpotentials along the potential axis as the rate constant is lowered further. Similar conclusions were reached with digital simulation techniques (6, 7), and for these methods there is good numerical agreement between simulated and experimental results. The purpose of the present paper is to (a) investigate criteria for the diagnosis of a DPP wave for the various types of simple charge transfer and to (b) experimentally test the validity of the analytical solutions of the diffusion boundary problem. In order to adequately represent the case of a dropping mercury electrode, we make use of an expression which is derived for an expanding planar electrode using the treatment of Ferrier and Schroeder. This expression is slightly different from the one given by Ruzic (2) for the same case. Experimental results are fitted to the theoretical expressions with the simplex optimization curve fitting technique with the reversible half-wave potential, Ex/2r, the standard heterogeneous rate constant, ka, and the charge transfer coefficient, a, as variable parameters. Because the analytical theory of DPP has not received wide spread recognition, the method has not been used for studying the kinetics of electrode process except for the digital simulation approach (6, 7). Due to its great sensitivity the DPP method should have many applications especially for biological systems where only trace quantities of material are available. Furthermore, most laboratories using electroanalytical techniques have differential pulse instrumentation which up to now have been mainly used for chemical analysis. The desirability of using the method for kinetic studies prompted the present investigation which shows that the method can be used in a simple manner to obtain kinetic information about electrode processes.
Diagnostic criteria based on peak current and peak potential are discussed for the diagnosis of reversible, quasi-reversible, and irreversible charge transfer processes with the differential pulse polarographlc method. The diagnostic criteria are derived from theoretical expressions of the current-potential curve and plots of the diagnostic parameters vs. a kinetic parameter are given which can be used to determine heterogeneous rate constants. A more accurate procedure of fitting experimental data to the theoretical expressions with a simplex optimization technique Is also used. Several experimental systems are analyzed with the techniques. Comparison of the kinetic results with literature values obtained with other methods verifies the validity of the theoretical ex-
pressions.
The theory of differential pulse polarography, DPP, for simple charge transfer control coupled to diffusion has been given in the literature in several places and in various forms. The first analytical derivation for this case actually appeared in the theory for staircase voltammetry at a stationary planar electrode by Ferrier and Schroeder in 1973 (1). In this treatment the first step of the staircase represents the case of differential pulse polarography. An identical equation for DPP although given in a different form was later derived by Ruzic (2). The theory of Ferrier and Schroeder (1) for diffusion to a stationary planar electrode uses the linearized solution of the diffusion equation at the instant of the pulse application (t = r) as an initial condition for the solution of the boundary value problem during the pulse application, where t = r + and AE = The k¡s are the potential dependent forward heterogeneous rate constants and other parameters have their usual meaning (see ref 5). As pointed out by Ruzic (2) the influence of the electrolysis at (the dc component) on the pulse current is contained in the term F(t). For fast quasi-reversible charge transfer the factors exp(y2)erfcy can be approximated by the first term of an asymptotic series which gives and eq 1 becomes where k¡
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