Theory of Stationary Polarography

Single Scan and Cyclic Methods Applied to Reversible, Irreversible, and Kinetic Systems

RICHARD S. NICHOLSON and IRVING SHAIN Chemistry Department, University of Wisconsin, Madison, Wis.

The theory of stationary electrode tributed to the theory of stationary value of this approach. Sevcik (44) polarography for both single scan electrode polarography. The first were qualitatively discussed the method for and cyclic triangular wave experi- Randles (29) and Sevcik (44) who con- reversible reactions at a plane electrode ments has been extended to systems sidered the single scan method for a under steady state conditions—i.e., in which preceding, following, or reversible reaction taking place at a after many cycles when no further catalytic (cyclic) chemical reactions are plane electrode. The theory was changes in the concentration distribu- coupled with reversible or irreversible extended to totally irreversible charge tions take place in the solution from charge transfers. A numerical method transfer reactions by Delahay (4), and one cycle to the next. Later Matsuda was developed for solving the integral later, Matsuda and Ayabe (23) re- (22) presented the complete theory for equations obtained from the boundary derived the Randles-Sevcik reversible this multisweep cyclic triangular wave value problems, and extensive data theory, the Delahay irreversible theory, method, for a reversible reaction at a were calculated which permit con- and then extended the treatment to the plane electrode. The only other con- struction of stationary electrode polaro- intermediate quasi-re versible case. tributions to the theory of cyclic grams from theory. Correlations of Other workers also have considered both methods were those of Gokhshtein (13) kinetic and experimental parameters reversible (13, 22, 32, 35) and totally (reversible reactions), Koutecky (19) made it possible to develop diagnostic irreversible (12, 32) reactions taking (reversible and quasi-reversible re- criteria so that unknown systems can be place at plane . actions), and Weber (51) (catalytic characterized by studying the varia- In addition, the theory of the single reactions). A generalized function of tion of peak current, half-peak poten- scan method has been extended to time was included in each of these tial, or ratio of anodic to cathodic reversible reactions taking place at derivations, which thus could be ex- peak currents as a function of rate cylindrical electrodes (25) and at tended to cyclic triangular-wave voltam- of voltage scan. spherical electrodes (9, 31, 32, 35). metry. In the last two papers, however, Totally irreversible reactions taking the case actually considered was for a place at spherical electrodes (8, 32) also cyclic step-functional potential varia- ELECTRODE POLAROG- have been discussed. Further contribu- tion. STATIONARYRAPHY (6) ( with tions to the theory have included Because of the increased interest in linearly varying potential) has found systems in which the products of the stationary electrode polarography, it wide application in analysis and in the electrode reaction are deposited on an has become important to extend the investigation of mechanisms. inert electrode (2) ; the reverse reaction, theory to include additional kinetic For analysis, the method is more involving the dissolution of a deposited cases. Furthermore, many of the sensitive and faster than polarography film (26); and systems involving recent applications of cyclic triangular with the dropping electrode multi-electron consecutive reactions, wave voltammetry have involved only (37), and when used with stripping where the individual steps take place at the first few cycles, rather than the analysis, can be extended to trace different potentials (14, 15). steady state multisweep experiments. determinations (7, 17, 45). In studying Even in the cases involving reversible Therefore, a general approach was the mechanism of electrode reactions, reactions at plane electrodes, the sought, which could be applied to all the use of stationary electrodes with a theoretical treatment is relatively dif- these cases. In considering the cyclic potential scan makes it possible to ficult, ultimately requiring some sort mathematical approaches of other investigate the products of the electrode of numerical analysis. Because of this, authors, at least three have been used reaction and detect electroactive inter- the more complicated cases in which previously: applications of Laplace mediates (10, 11, 18). Furthermore, homogeneous chemical reactions are transform techniques, direct numerical the time scale for the method can coupled to the charge transfer reaction solution using finite difference tech- be varied over an extremely wide range, have received little attention. Saveant niques, and conversion of the boundary and both relatively slow and fairly and Vianello developed the theory for value problem to an integral equation. rapid reactions can be studied with a the catalytic mechanism (39), the The first approach is the most elegant, single technique. Various electrodes preceding chemical reaction (38, 41), but is applicable only to the simplest have been used in these studies, but the and also have discussed the case case of a reversible charge transfer most important applications have in- involving a very rapid reaction reaction (2, 19, 31, 35, 44), and also to volved the hanging mercury drop following the charge transfer (40). the catalytic reaction (51). Even in electrode [reviewed by Kemula (16) Reinmuth (32) briefly discussed the these cases, definite integrals arise which and Riha (36) ], and the dropping theory for a system in which a first can only be evaluated numerically. mercury electrode [reviewed by Vogel order chemical reaction follows the The second approach (8, 9, 25, 29) is (50)]. charge transfer. the least useful of the three, because Since the first application of the The mathematical complexity also has functional relations which may exist method by Matheson and Nichols (21), prevented extensive study of the cyclic between the experimental parameters numerous investigators have con- triangular wave methods, in spite of the are usually embodied in extensive

706 · ANALYTICAL CHEMISTRY numerical tabulations and are often where E¡ is the initial potential, v is the missed. Thus, the results may depend rate of potential scan, and X is the time on an extremely large number of at which the scan is reversed (Figure 1). variables. This is particularly so in the Equations 6a and 6b can be sub- more complicated cases involving stituted into Equation 5b to obtain the coupled chemical reactions, which may boundary condition in an abridged require the direct simultaneous solution form: of three partial differential equations = together with three initial and six Co/CR 6Sx(f) (7) boundary conditions. where 1. Wave form for tri- The third method possesses the Figure cyclic advantages of the first, and yet is more angular wave voltammetry = exp[(nF/RT)(Ei - E°)] (8) Several methods generally applicable. e~at for t can be used to convert the in of the extensive = boundary cases, spite previous Sx(i) - eat 2 for t value problem to an integral equation work. However, this makes it possible ^ (9) (33), and at least two methods of to discuss the numerical method pro- and solving the resulting Integral equations posed here in terms of the simplest have been used. The series solution possible case for clarity, and at the a = nFv/RT (10) proposed by Reinmuth (32) is very same time summarizes the widely If t is always less than X, then Equation but in cases scattered work in a form which straightforward, only previous 7 reduces to involving totally irreversible charge is most convenient for comparison of transfer does it provide a series which experimental results with theory. C0/CR = ee-‘ (11) is properly convergent over the entire which is the same boundary condition potential range. [This approach to I. REVERSIBLE CHARGE TRANSFER that has been used for obtaining series solutions is essentially previously Boundary Value Problem. For a theoretical studies of the scan the same as used by Smutek (47) for single reversible reduction of an oxidized method for a reversible transfer. irreversible polarographic waves. Series charge 0 to a reduced The direct use of the solutions of the same form can also be species species R, Laplace trans- form to solve this value obtained directly from the differential O ne^· R boundary + (I) is the form of equations, as was shown recently by problem precluded by a 7. the differential Buck (S)]. Reinmuth has outlined a taking place at plane electrode, the Equation However, can be converted into method for evaluating these series in boundary value problem for stationary equations integral electrode is the trans- regions where they aire divergent (33), polarography equations by taking Laplace form of 1 to for but attempts to use that approach in Equations 4, solving dC0 n d‘C0 the transform of the surface concentra- this laboratory (with a Bendix Model 0 (1) dt dr2 tions in terms of the transform of the G-15 digital computer) produced er- ratic results. surface fluxes, and then applying the dCR c)*Cr convolution theorem The methods most frequently used (2) (33): for the have solving integral equations 1 Cl f(r)dr = been numerical (4, 18, 13, 22, 23, 26, t = x Co(0,t) Co* 0, — ^0: and an of the * y/tD0 Jo y/t 39) adaptation approach Co = Co*·, Cr = Cr (~0) (3) suggested by Gokhshtein (IS) was used (12) was < z —> =0 : —> —> in this work. The method which ^ 0, C0 Co* ; Cr 0 developed is generally applicable to all (4) CR(0,t) = -J=, (13) f* J^t=- r of the cases mentioned above, all t > 0, x = 0: Vttdr Jo Vt additional first order kinetic cases of 'dCs\ where interest, and both single scan and cyclic (5a) dz triangular wave experiments. Except , /

- - m -D· (14> for reversible, irreversible, and catalytic Co/Cr = exp[(nF/RT)(E 5°)] the treatment is limited to (*/),.o‘iMFA reactions, (5b) plane electrodes because of the marked The boundary condition of Equation increase in complexity of the theory for where C0 and CR are the concentrations 7 now can be combined with Equations most of the kinetic cases if an attempt is of substances 0 and R, x is the distance 12 and 13, to eliminate the concentra- made to account for spherical diffusion from the electrode, t is the time, Co* tion terms and obtain a single integral rigorously. Conditions under which and Cr* are the bulk concentrations of equation, which has as its solution the derivations for plane electrodes can be substances 0 and R, D0 and Dr are flux of substance 0 at the electrode used for other geometries have been the diffusion coefficients, n is the surface: discussed by Berzins and Delahay (2). number of electrons, E is the potential 'l

f^dT _ Co* An empirical approach to making ap- of the electrode, E° is the formal VtD„ (15) proximate corrections for the spherical electrode potential, and R, T, and F Jofo y/t — r 1 + y6S\(t) contribution to the current will be have their usual The significance. where described elsewhere. The cases con- applicability of the Fick diffusion

sidered here all involve reductions for and the initial and = equations boundary 7 V D0/Dr (16) the first charge transfer step, but conditions has been discussed by Rein- extension to oxidations is obvious. muth (S3). Referring to Equation 10, it can be To present a logical, discussion, each For the case of stationary electrode noted that the term at is dimensionless of the kinetic cases was to the the in compared polarography, potential Equation = = — at nFvt/RT (nF/RT)(Ei E) corresponding reversible or irreversible 5b is a function of time, given by the reaction which would take place without relations (17) the kinetic it was is complication. Thus, — and to the 0 < í ^ X E = Ei vt (6a) proportional potential. necessar3r to include in this work a Since the ultimate goal is to calculate substantial discussior. of these two ^ t: E = Ei — 2v\ + vt (6b) current-potential curves rather than

VOL. 36, NO. 4, APRIL 1964 · 707 >n ( )

Table I. Current Functions V^xtof) for Reversible Charge Transfer (Case I) JoP y/n — (E — £1/2)71 (£ — £1/2)71 mv. Vtx(oí) 1f>(at) mv. VtX(o<) 1i>(at)

120 0.009 0.008 - 5 0.400 0.548 100 0.020 0.019 10 0.418 0.596 80 0.042 0.041 15 0.432 0.641 (29) 60 0.084 0.087 20 0.441 0.685 50 0.117 0.124 25 0.445 0.725 The integral on the right hand side of 45 0.138 0.146 -28.50 0.4463 0.7516 Equation 29 is a Riemann-Stieltj es 40 0.160 0.173 30 0.446 0.763 which can be its 35 0.185 0.208 35 0.443 0.796 integral, replaced by 30 0.211 0.236 40 0.438 0.826 corresponding finite sum (1). Eliminat- 25 0.240 0.273 50 0.421 0.875 ing the special points i = 0 and i = n 20 0.269 0.314 -60 0.399 0.912 from the summation, one obtains 15 0.298 0.357 80 0.353 0.957 10 0.328 0.403 100 0.312 0.980 n x($v)dv 5 0.355 0.451 120 0.280 0.991 = 2 x(l)V n + 0 0.380 0.499 150 0.245 0.997 Jof y/n — v 71—1 : To calculate the current - - y/n i [ ( + 1) x(i)] (30) (1) i = ¿(plane) + ¿(spherical correction). (2) = nFAVaDbCo* y/*x(at) + nFADoCo*(l/ „) ( ) (3) = 602 n3l2AVDov Co*[ Virx(o<) + 0.160( VDo/r<,Vnv)*(o<)], amperes. and substituting this result in Equation Units for (3) are: A, sq.cm.; Do, sq. cm./sec.; v, volt/sec.; Co*, moles/liter; r,, cm. 28

2 \/ x(l)Vn +

71—1 current-time it is useful to make furnish values of the current curves, ultimately - - + 1) all calculations with to at rather as a function of Vn [ ( x(t)] respect potential (Equations i = l than t. This can be a accomplished by 14, 19, 21): 1 change in variable (31) 1 + i = nFACo* y/irD0a x(at) (25) 70 ( ) = z/a (18) Equation 31 defines N algebraic equa- fit) = giat) (19) The values of x(at) are independent tions in the unknown function ( ), of the actual value of y6 selected, where each nth equation involves the pre- 15 becomes and Equation than — provided In y6 is larger perhaps 6, vious 1 unknowns. These equations Fat and a formal proof has been given by are then solved successively for the = Co* ·\/ 0 j(z)dz Reinmuth (35). This corresponds to the values of x(at)—i.e., ( ). When JO a — z 1 + V y/at yeS*\(at) usual experimental procedure of , the function »8$ ( ) is an initial anodic of — (20) selecting potential exp ( ), and when the point in the the foot of the wave, and in effect re- calculations is reached where > This equation can be made , integral — duces the number of variables involved the function is dimensionless (especially important if replaced by exp ( by one. In this way, single scan or cyclic numerical methods are used) by the 2 \). Numerical Solution. Although curves can be substitution current-potential cal- Equation 22 has been solved in culated easily, and the extension to several the numerical g(at) = Co* y/irDoa x(at) (21) ways, only multicycles is obvious. approaches appeared to be readily Analytical Solution. It is also and the final form of the integral applicable to the cyclic experiment. possible to obtain an analytical solu- equation is: The technique developed here involves tion to Equation 22. It is an Abel dividing the range of integration and the solution 1 integral equation (/t8), x(z)dz = = (22) from at 0 to at M into N equally can be written directly as y/ at — z 1 + y6Sa\(at) spaced subintervals by a change of variable, The solution to 22 x(at) ¿(0)_ + Equation provides 7r = Vat values of x{at) as a function of at, for a z (26) 'at 1 given value of y6. From Equations * (32) and the definition * — a = z 5b and 7, the values of at are related to -fJo V at _~(at) \Ja¡ the potential by = at/ (27) where L(at) represents the right hand E - E° — (RT/nF) In y + side of Equation 22. Performing the 5 is the of the subinterval In Here, length differentiation indicated (using Equa- (RT/nF) [In y6 + Sax(at) ] (23a) = and n is a serial number of ( M/N), tion 11 as the boundary condition) the or the subinterval. Thus, Equation 23 exact solution is becomes (E - Em)n = 1 n dv 1 x(at) + [In ye + In S„x(a<) ] (23b) ( ) (RT/F) · V at( 1 + 76) vr — Jo v 1 + 70Sjx(5n) where £1/2 is the polarographic half y/ dz (28) wave potential 4 where n varies from 0 to M in N inte- Em = E° + (RT/nF) In \/Dr/D0 The of (n (24) gral steps. point singularity (33) = v) in the kernel in Equation 28 Thus, values of x(at) can be regarded can be removed through an integration Equation 33 has been given previously as values of [( — Eu/)n], and will by parts to obtain by Matsuda and Ayabe (23), and by

708 · ANALYTICAL CHEMISTRY Gokhshtein (13). If it is assumed that 0—R mental condition, and will not be con- 70 is large, Equation 33 reduces to the sidered further.) Typical cyclic polaro- result obtained by Sevcik (44) and by grams are shown in Figure 2. By using Reinmuth (35), but such an assumption for the base line the cathodic curve is not required in this, case. Although which would have been obtained if the definite integral in Equation 33 there had been no change in direction of cannot be evaluated in closed form, potential scan, all of the anodic curves several numerical methods (such as the are the same, independent of switching Euler-Maclaurin summation formula or potential, and identical in height and Simpson’s rule) can be used, provided shape to the cathodic wave. Thus, the singularity at at = z is first removed when the anodic peak height is measured by a change of variable (23) or an to the extension of the cathodic curve, integration by parts (35). the ratio of anodic to cathodic peak Series Solution. If only the single currents is unity, independent of the scan method is considered, Equation switching potential. This behavior 22 can be solved in series form, and Figure 2. Cyclic stationary electrode can be used as an important diagnostic although the resulting series does not polarograms (Case I) criterion to demonstrate the absence all of the various for — (or unimportance) properly converge potentials, Switching to (Et/s E\)n of potentials correspond chemical reactions. Of those the form of the results obtained is very 64, 1 05, and 141 mv. for anodic scans coupled useful for comparison of the limiting considered in this paper, only the cases in the kinetic case can be obtained systems. 28.50/n millivolts cathodic of Em,— catalytic (which easily One possible approach is to expand distinguished from the reversible case the right hand side of Equation 22 as an by other behavior) gives a constant - In = exponential power series in at, as done (EP E°)n + (RT/F) y value of unity for the ratio of anodic to by Sevcik (44)j but this cannot be done — 28.50 ± 0.05 mv. (35) cathodic peak height on varying the for most cases involving coupled chemi- or switching potential. Experimentally, cal reactions. Reinmuth’s the cathodic base line can be obtained Thus, ap- ~ Ep = Em (1.109 ± 0.002) (RT/nF) proach (32, 33) is more general, and the by extending a single scan cathodic final result is (36) sweep beyond the selected switching Actually, the peak of a reversible potential, or if another reaction inter- stationary electrode polarogram is feres, by stopping the scan at some x(at) = -4= y/~j X fairly broad, extending over a range of convenient potential past the peak, and * V j = i several millivolts if values of x(at) recording the constant potential cur- curve cor- - rent-time exp[(-jnF/RT)(E Em)} (34) are determined to about 1%. Thus, (using appropriate it is sometimes convenient to use the rections for charging current). The half-peak potential as a reference point latter method of obtaining the base Single Scan Method. In every (24), although this has no direct line has been proposed for analytical case, the solution of Equation 22 thermodynamic significance. The half- purposes by Reinmuth (34). numerical evalua- the wave on ultimately requires peak potential precedes E1/2 by 28.0/ra The position of anodic tion which in the past has been carried mv., or the potential ax