THEORY AND APPLICATION OF CHRONOPOTENTIOMETRY FOR MEASURING HETEROGENEOUS ELECTRON TRANSFER KINETICS Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY» FLOYD HILBERT BEYERLEIN 1970 LIBRARY WEI“- . MichiganStato University ‘ This is to certify that the thesis entitled THEORY AND APPLICATION OF CHRONOPOTENTIOMETRY FOR MEASURING HETEROGENEOUS ELECTRON TRANSFER KINETICS presented by Floyd H. Beyerlein has been accepted towards fulfillment of the requirements for Ph.D. Chemistry degree in 7 i [’61, 71/5; ”(E—(:6 fligéi L1 - I Major professor I Date 0-169 I- ' * ““‘x ”We IINBING BY HUAB & sou3' mm meet me. unmnv muons i gyggplfllitillfllfll .fi— ‘1 *- 3" w...- v.- . 4—...» ABSTRACT THEORY AND APPLICATION OF CHRONOPOTENTIOMETRY FOR MEASURING HETEROGENEOUS ELECTRON TRANSFER KINETICS BY Floyd Hilbert Beyerlein A new method for measuring heterogeneous electron trans- fer kinetics is described and developed theoretically. The method is based on using chronopotentiometry with current reversal to observe directly the overpotential associated with a kinetically controlled redox reaction. A simple equa— tion is derived which relates observed overpotential to current density, bulk concentration of the depolarizer, and the standard rate constant, 58. It is estimated by calcula- tion that the method is useful for measuring rate constants in the range 4 x 10" < 33 < 2 x 10‘2 cm/sec. An experimental evaluation of the method for reduction of azobenzene is used to establish that the above Upper limit for £8 is correct, and that this limit is set by double-layer charging. These experiments also were used to demonstrate that in practice application of the new method is simple and straightforward. In an attempt to extend the range of the method to larger rate constants, the combined influence of double-layer Illll‘ll'lllllllllll. ll.TNIIlII-I Floyd Hilbert Beyerlein charging and electrode kinetics on chronOpotentiometry also was examined. In this case theory could only be obtained by numerical solution of nonlinear integral equations, and therefore results are exPressed as a family of working curves. The working curves relate observed overpotential to current density, concentration of depolarizer, and ks. Each working curve depends on the value of the double-layer capacitance, and therefore to identify the prOper working curve, double- layer capacitance must be evaluated independently. A pro- cedure for doing this is described in detail. Reduction of cadmium was used to evaluate use of the working curves, and also to compare calculated and eXperimental chronOpotentio- grams. The agreement between theory and eXperiment is excel- lent, and it is estimated that values of Es as large as 1.2 cm/sec can be determined with the working curves. To perform the eXperiments two different instruments were constructed from operational amplifiers. One of these instruments provides for automatic current reversal at a pre-selected Switching potential. The construction and opera- tion of this instrument is described in detail. THEORY AND APPLICATION OF CHRONOPOTENTIOMETRY FOR MEASURING HETEROGENEOUS ELECTRON TRANSFER KINETICS BY Floyd Hilbert Beyerlein A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1970 \ \ \\ I\P\usu Eoum mommmuocw mammaouuumam mswnsp mafia .mfimwaoupumao ustHso unnumsou madman snowman moonuomam onu Eoum Um>0§mu memEmm sowusaom Hmoflumnuommn How mamumoumHom Amy .H onsmfim a musmflm uncoo at which a faradaic reaction can proceed to consume the im- posed current. Hence, as soon as the current is imposed, the electrode potential will jump abruptly from Point A to Point B, somewhere near the half-wave potential for the depolarizer. As the electrolysis proceeds the concentration of depolarizer at the electrode surface is continually de- creased, so that at some later time the hypothetical polaro- gram will look like Curve 2. Here, the concentration of depolarizer is still large enough to sustain all of the con- stant current (3,3,, there is still 100% current efficiency with reSpect to depolarizer), and the only effect has been a small cathodic displacement of potential (from Point B to Point C). As the constant current electrolysis proceeds, however, eventually a condition is reached, represented by Curve 5, where the surface concentration of depolarizer is no longer great enough to maintain, by itself, all of the con- stant current. Thus, in Curve 5 the hypothetical limiting current for depolarizer is less than i and therefore -const.’ the electrode potential must shift to a point (Point D) where a new faradaic reaction (Egg), decomposition of the solvent or supporting electrolyte) can supply the additional faradaic reaction necessary to sustain the total constant current. Hence, at this particular time during constant current electrolysis there is an abrupt and rather large change of potential. It is important to note that since the new potential (Point D) corresponds to the limiting current region for reduction of depolarizer, it follows that the surface concentration of depolarizer has gone to essentially zero. If the potentials in Figure 1a (Points A, B, C and D) are now plotted gs, the times at which they were observed, the result is the chronOpotentiogram shown in Figure 1b. Hence, in general a chronopotentiogram consists of an ini— tially abrupt change of potential to a point near the polar- ographic half-wave potential, followed by a period of time where the potential is nearly constant. This period is then followed by another abrupt transition of potential, at which time the surface concentration of depolarizer also drops to zero. The time from the beginning of the eXperiment to the second sharp potential change is termed the transition time which is labeled T in Figure 1b. The first mathematical analysis of the chronOpotenti- ometry experiment was published by Sand in 1901 (54). .Sand derived an eXpression for the time dependence of surface con- centration of depolarizer based on a semi-infinite linear diffusion model. By defining T as the time at which surface concentration of depolarizer becomes zero, Sand obtained the following exPression for transition time (now known as the Sand equation): T é.= nFC*JnD /2i (1) O O 0 There g_is the number of electrons transferred in the elec- trode reaction, EDis the Faraday, 93 is the analytical . concentration of depolarizer, 20 is the diffusion coefficient for the depolarizer, and i0 is constant current density. An interesting feature of the Sand equation is that it is derived without assuming a model for the electron trans- fer reaction. Thus, the transition time is the same for both reversible and totally irreversible electrode reactions. The shape of the entire potential-time curve does, of course, depend on reversibility of the electrode reaction. For ex- ample, for reversible electron transfer, the chronOpotentio- gram is desdribed by substituting surface concentrations derived by Sand into the Nernst equation. In this case it is easily shown (16) that the potential at t = T/4 (the so—called quarter-wave potential) is identical with the classical polar— ographic half-wave potential. Although theoretically the transition time is a well— defined quantity, experimentally it is frequently found that chronopotentiograms exhibit distortions which make measurement of T ambiguous. As a result a number of empirical methods for measuring T, usually based on some graphical construction, have been developed over the years (48). One of these methods, which will be referred to frequently in the remainder of this thesis, is due to Berzins and Delahay (5), and is illustrated in Figure 2. Their method is derived from a similar construc— tion commonly used in the measurement of half-wave potentials on conventional polarograms. The method is applied in the following manner. Line AB is drawn tangent to the curve at .meu coHuflmcmuu mewsflauouop How mmnmama new mswuuom mo ponpoz .N ousmflm 10 N musmwm 11 Point A and CD is drawn tangent to the curve at Point D. Since the polarographic half-wave‘potential is at _t_ = T/4, Points E and F are located at one-fourth of the distance between A and C, and B and D respectively. A line joining Points E and F intersects the curve at the half—wave poten- tial, and the length of the line drawn through the point of intersection of Line EF with the curve, parallel to the time axis and bounded by Lines AB and CD, is taken as a measure of T. The instrumentation required for chronopotentiometry can be extremely simple, and is one of the virtues of the method. For example, many literature applications of chrono- potentiometry are based on the use of simply a battery and large resistor as a source of the constant polarizing current. Of course, more sophisticated circuitry can be used and is necessary in applications such as current reversal (gig; infra). Recording of potential-time curves is usually accomplished with a potentiometric recorder, or oscilloscope, depending on the time scale of the experiment. The applications of chronopotentiometry are extensive, ranging from purely analytical to measurement of kinetics and transport properties (14). The analytical applications in general are straightforward, and based on the fact that T is proportional to depolarizer concentration. Since T varies as the square of depolarizer concentration, the method in principle is more sensitive than techniques like polarography. 12 Analytical applications have been reviewed by Everett, Johns, and Reilley (24). The major applications of chronopotentiometry have been in the study of electrolysis mechanisms and the measurement of rates of chemical reactions coupled to the depolarizer or product of the electrode reaction. In these applications chronopotentiometry is especially useful diagnostically, since it in the absence of kinetic complications if is a constant independent of current density (cf. discussion of Sand equa— tion). For example, the presence of a chemical reaction preceding the electron transfer is easily detected by a decrease of 13% with increasing current density (28). In this case the test is eSpecially useful because it is un- affected by reversibility of the electron transfer (gig; supra). This is in sharp contrast with most other electro- chemical techniques where preceding chemical reactions and slow electron transfer behave similarly. Another important class of reactions where chrono- potentiometry has been used very successfully are following chemical reactions, i,g,, chemical reactions involving the initially formed product of the electron transfer. The ad- vantage of chronopotentiometry in this case derives from the fact that a stationary electrode is used, and therefore products of the electrolysis accumulate near the electrode surface and can be studied by re-oxidation (in the case of an initial reduction). The re-oxidation is accomplished by 15 abruptly reversing the direction of current flow. With this approach, known as current reversal chronOpotentiometry, chronopotentiometric waves are observed directly for both the reduction and oxidation processes. A typical example is shown in Figure 5 for reversible electron transfer, with cur- rent reversal at the first transition time. In this case the first transition, labeled T , corresponds to reduction of F depolarizer, and the second transition, T corresponds to R’ oxidation of the reduced form of depolarizer. When the re— duced form is chemically stable, it is easily shown (4) that the ratio of transition times, TR/TF, is one-third. Clearly, if the reduced form is chemically unstable, in general TR/TF will be less than one-third, since less of the reduced form will be available for oxidation. Moreover, the magnitude of the effect will be a function of current density, since, for example, if TR is small with reSpect to the life—time of the reduced form, then essentially all of the reduced form will still be oxidized. Thus, by observing the variation of TR/TF with current density the presence of this class of reactions is easily detected, and quantitatively correlated with the chemical lifetime of the reactant (22,56). Two other areas in which chronopotentiometry has been extensively employed are adsorption studies and measurement of diffusion coefficients. In the former area the technique enjoyed considerable popularity in the early part of the last decade, until the advent of chronocoulometry (2). .Since then 14 .Hmmum>ou ucounsu mo osvfisnowu may mcflumuumsaafi EcumOwucmuomosounu .m musmflm 15 m onsmflm 16 the use of chronopotentiometry for adsorption studies has been severely criticized (40), and presently has been essen- tially replaced by chronocoulometry. .Measurement of diffusion coefficients usually has been based on direct application of the Sand equation, under conditions where all parameters in Equation 1 are known except the diffusion coefficient. The meaning of diffusion coefficients measured in this manner recently was analyzed in detail by Laity and McIntyre (59), who were interested primarily in measuring tranSport pr0per- ties of fused salts. Finally, chronOpotentiometry has been used for measuring heterogeneous electron transfer rate constants, but these applications always involve data analysis and interpretation roughly as complicated (1) as those ascribed above to ac polarography. The approach to date is based on analysis of single cathodic (or anodic) polarization, so that the rate constant for the reverse reaction (anodic reaction in the case of cathodic polarization) must be estimated indirectly. It appeared, however, that by using current reversal, a more direct and conceptually simpler method could be devised, which apparently would satisfy the criteria stated at the beginning of this Introduction. .The essential ideas of this new method for measuring standard electron transfer rate con- stants (or exchange currents) are as follows. With current reversal chronOpotentiometry, the potentials at which oxidation and reduction occur are observed directly 17 (cf. Figure 5). For a reversible electrode process, by definition oxidation and reduction occur at the same (equilibrium) potential-fii.g., there is no so-called over- potential between the forward and reverse reactions. For example, the curve of Figure 5 corresponds to reversible electron transfer, since the forward and reverse quarter-wave potentials are identical and equal to the reversible half- wave potential. If'the reaction were not perfectly reversible the activation energy associated with the electron transfer would appear as a finite overpotential between the reduction and oxidation reactions. Now consider the following hypo- thetical experiment involving some simple redox system characterized by its standard rate constant, ks. For some given current density the rate of the electrochemical reaction will be large enough that even though electrode potential changes continually during the course of the chronOpotenti- ometric eXperiment, essentially equilibrium conditions will be maintained at the electrode surface. Under these condi- tions there will be no apparent overpotential between the forward and reverse chronopotentiograms. Now consider the effect on this same system of progressively increasing the current density. As current density is increased the transi- tion time decreases (cf. Equation 1), so that the rate at which the electrode potential changes increases. As this happens a point should be reached where the kinetics of the electron transfer are no longer rapid enough to maintain 18 electrochemical equilibrium at the electrode surface, and this condition should evidence itself as an overpotential between the forward and reverse chronOpotentiograms. Thus, as current density is increased one would eXpect the differ- ence between forward and reverse quarter—wave potentials to increase from zero. Hopefully, one should be able to corre- late this directly measured overpotential with the associated current density and the standard electron transfer rate con- stant for the redox system. This correlation was accomplished successfully, and a description of that theory constitutes the next major section of this thesis. As will be seen the theory proved to be as straight- forward as the conceptual basis of the method. Thus, it ap— peared that a truly simple method had been develOped which satisfied most of the criteria stated earlier. The sc0pe and practicability of the new approach were next evaluated ex- perimentally with some model chemical systems. In the course of this evaluation, the results of which are presented later in this thesis, it was concluded that the major limitation of the method was the interference of charging current, which set the upper limit of rate constants that could be measured. Charging current is a term used to indicate the fact that an electrode in an electrolyte solution behaves like a capa- citor, and therefore current (charging current) is required to change the electrode potential. Thus, in an electrolyte considerable structuring occurs at the interface of a charged 19 electrode (45). For example, if the electrode is negatively charged then the solution adjacent to the electrode consists predominately of cations, a structure referred to as the electrical double-layer. If the charge on the electrode is changed by an external source, then a restructuring of the double-layer occurs, and this movement of ions results in a net flow of current. At high current densities with chrono- potentiometry the relative importance of double-layer charging increases because the rate of change of electrode potential increases. Hence, since high current densities are required to measure large rate constants, it is apparent why the new method was limited by double-layer charging. Under conditions where double-layer charging is neglig- ible, chronopotentiometry theory is straightforward and results can be eXpressed in closed form, as mentioned above. Unfortunately, to include double-layer charging in the theo- retical model makes the mathematics virtually intractable. Thus, prior to 1968 all attempts to account for double-layer charging in chronOpotentiometry were empirical. In 1968 three groups independently published theoretical calcula- tions for chronopotentiometry based on a rigorous mathemati— cal model that included double-layer charging (21,50,55). These papers provided the first quantitative basis for evalu- ating the influence of double-layer charging on chrono- potentiograms. Unfortunately, none of these papers treated the case of kinetically controlled electron transfer, and 20 therefore none of the results was directly applicable to the case being considered here. Indeed, because of the mathe- matical complexity, it seemed unlikely that any theory could be developed to include double-layer charging and still satisfy the criteria of simplicity set forth above. On the other hand, Olmstead and Nicholson (50) showed that the method of Berzins and Delahay gives accurate values of the quarter- wave potential even in the presence of extensive double-layer charging. Thus, the possibility presented itself that using the method of Berzins and Delahay, or some modification there- of, the simple theory could be used directly, thereby con— siderably extending the range of applicability of the new method. Based on this possibility, and the fact that a more detailed analysis of the influence of double-layer charging should be useful pg£_§§J it was decided to attempt the appro- priate theoretical calculations. The last major section of this thesis presents this theory for a model which includes both kinetically controlled electron transfer and double- 1ayer charging. THEORY FOR THE CASE OF NEGLIGIBLE DOUBLE-LAYER CHARGING Once again, the objective is to develOp an expression for overpotential between the cathodic and anodic parts of a current reversal chronopotentiogram for a kinetically con- trolled redox reaction. Based on the conceptual idea of the prOposed method, it is anticipated that the overpotential will be proportional to current density, and inversely pro- portional to the rate of the redox reaction. The redox reaction can be symbolized as follows k 0 + ne —£~ R I A? where kf and Eb are the heterogeneous rate constants for electron transfer, and hence are functions of potential. The potential dependence of kf and Eb is given by the follow- ing well-known equations (58): k f ksexp[(-anF/RT)(E-EO)] (2) k ksepr(1-a)nF/RT)(E-EO)] (5) b There Es is the common value of 5f and Eb at the standard equilibrium potential (E?), and is directly proportional to the standard exchange current density (58). d is the 21 22 transfer coefficient and other terms have their usual mean- ing. The rate equation for Reaction I can be written in terms of the flux and surface concentration of O and R: FLUX = kaO(0,t) - kbCR(0,t) (4) where the first index on concentration represents distance from the electrode surface (zero in this case) and the second index represents time during electrolysis. The problem now is to substitute in this rate law ex- pressions for the temporal dependence of surface concentra- tions during the entire current reversal experiment, as well as expressions for potential dependence of kf and 5b, and from the result obtain a (hOpefully) simple expression for overpotential. Since the chronopotentiometry experiment is performed under conditions designed to make diffusion the only source of mass transport, the concentrations can be calculated by solving the apprOpriate Fick's law diffusion equations. These results already are available in the litera- ture (1), and could provide the starting point for the present treatment. However, to make the discussion more lucid and cohesive, the entire derivation starting with the Fick's law boundary value problem will be presented. Boundary7Value Problem To account for concentration polarization, linear diffu- sion is assumed to be the only source of mass transport. 25 The apprOpriate diffusion equations are 2 a C BCO_D'JO (5) a? ‘ 0‘5?! for the oxidized form of the couple, 0, and 2 SEE. = DR §_:§. (5) ct 8x for the reduced form, R. These partial differential equa- tions are to be solved for the concentration of 0 and R as a function of distance from the electrode surface, x, and time during the electrolysis, E, To obtain eXplicit solutions the following initial and boundary conditions will be assumed. Initially, the concentration of depolarizer at any point in solution is given by the bulk concentration value, which * will be represented as CO. It will be further assumed that substance R is generated ip_situ, and is therefore initially absent from the solution. Thus, stated mathematically, the initial conditions are t=0; x20 c =c - c =0 (7) For the first boundary condition it will be assumed that conditions of so-called semi-infinite diffusion prevail. In other words, it will be assumed that the thickness of the diffusion layer developed during electrolysis is much less than the dimensions of the entire solution. It is easily shown that this condition is satisfied whenever the walls of 24 the electrolysis cell are greater than a few millimeters from the electrode surface (52). Stated mathematically, this boundary condition is * tgo; x —+oo cO —>c0; cR—ao (8) The second boundary condition is a statement of mass balance for 0 and R at the electrode surface. The quantity of electroactive Species diffusing to the electrode can be expressed in terms of the surface flux, defined by the follow- ing eXpression 8C D g; xzo (9) FLUX Thus, the final boundary condition is simply aco = -0 5C3 (10) tZO; x==O DO 8x 8x Since it has been assumed that only substance 0 is initially present, the forward part of the chronopotentiogram consists of the reduction of 0 at a constant current density, iF' Arbitrarily, current reversal will be introduced when the forward transition time, T is reached. The current Fl density after reversal will be labeled iR' and the resulting anodic transition time will be designated TR. By convention ‘will be measured from the forward transition time, rather TR than from £_equal zero. 25 The flux is related to current density by Fick's first law, which for the present case takes the following form OgthF: x=0 aco DO W = IF/nF (11) Tth_TR; x=0 BCR DR —5§' = ' lR/nF (12) or written in terms of the flux of 0 TFSthR: x=0 'acR aco DR E - - DO $3: - RiIF/nF (‘13) where R1 = iR/iF (14) The above boundary value problem must now be solved for the surface concentration of 0 and R as a function of time. The only potential difficulties might come from the fact that the last boundary condition (Equations 11 and 12) is discon- tinuous, but as shown in the next section this problem is easily handled with the Laplace transform Operator. Solution of the Boundary Value Problem The discontinuous boundary condition is easily handled by first transforming Equations 5 and 6 into integral form, and then incorporating the boundary condition. Laplace transformation of Equations 5 and 6 leads directly to the following expressions (52): Co(t) = C0 - f (15) JNDO 0 Jt-x f (t)dx CR(t) = 1 ft 0 (16) 4'er 0 ~/t-x where the symbol f0(t) is used to represent the surface flux of substance 0. Prior to current reversal the expression for flux is given by Equation 11, and after current reversal by Equation 15. Thus, substituting Equations 11 and 15 in Equations 15 and 16, and performing the indicated integrations leads to the following expressions for surface concentration ogt (28) where pR.eXp[ag(E)] - 1 K(E) = 1 1 + eXp [g (E)] (29) it Equation 28 is of the form of a quadratic in the y , as can be seen by squaring both sides and tranSposing all terms to one side of the equality. The result is y[(1-Ri)2 — 1] - 2xuso “o.au u am .< m>uso .m.o u d can m.o u Q so£3 mamumowusouomoconno Hecauouomnu so .flm .mmauamcmp acouuso mo owumn mo uoommm .m ousmflm 54 m musmflm 0.6 oom cos 0 u m _ 3 7T( cos- m A com. 55 0$t2.5, the effect of a is given eXplicitly by Equations 55 and 56. For values of p between these limits, a affects both the symmetry of potential- time curves, and their position on the potential axis. These effects are illustrated in Figure 6 where theoretical chrono- potentiograms are plotted for three values of a. OverpotentialL7AE, as a Measure ofks. With current reversal chronopotentiometry the potentials at which oxida- tion and reduction occur are observed directly, and it is clear that the effect is the same as anticipated in the Introduction. Of course, to obtain a quantitative correlation between overpotential and the standard rate constant it is necessary to select a fixed reference time during the forward and reverse parts of the potential-time curves from which to calculate overpotential. An obvious choice for this reference point is the time at which the potential is equal to the reversible half-wave potential in the zero overpotential case--i,g,, the so-called quarter-wave potential defined earlier. To do this it is necessary to determine precisely the times relative to the respective transition times (T and TR) where the potential is equal to the half-wave F 57 a .o.dl n .m use 0.6 M Q conz .8 .ucoflowmmooo Hmmmsmuu mo uommmm mcw3onm mamumoflusouomosounu Hmowuouoone .m ousmwm 58 w musmwm a mi“ To - «.o o..o _ _ q _ _ _ 00m 03 . - o to L cos. m6 .L com- n6 u a l I con- Am ' (Ta-s) u 59 potential. Once this is known overpotential can be calculated as the difference of potential at these two times. Although the choice of reference is essentially arbitrary, this EQ- reference has the advantage that overpotential will be zero in the reversible case, so results will not be complicated by concentration overpotential. The times at which the potential is equal to the reversi- ble half-wave potential can be obtained from Equations 55 and 54 for the forward and reverse parts of the potential-time curve respectively. To evaluate these times it is necessary to recall that g(§) is zero at the half-wave potential (see Equation 26), which is equivalent to the argument of the logarithmic terms being unity. Thus, by setting the argument of the logarithm in Equation 55 equal to unity and solving for y, it is found that the half-wave potential occurs when y_= 0.25 (hence the name quarter-wave potential). The cor- responding time for the reverse part of the curve is obtained as follows. After setting the argument of the logarithm term in Equation 54 equal to unity and collecting like terms the following equation is obtained 1 + 2K(y-1)é - 2yé = 0 (57) where K = (1-Ri) (58) Equation 57 can be written in the form of a quadratic by transposing the (yr-1)é term to the right hand side of the 40 equality and squaring both sides of the eXpression to obtain 4(K2-1)y + 4y§ — (4K2 + 1) = o (59) After solving for y_by application of the quadratic equation, and retaining only the positive root, the following exPres- sion for the value of y_at which Eé occurs is obtained -i+{1+[(1-Ri)2-1][4(i-Ri)2+1]}§' (40) yg(E)=O - 2[(1—Ri)2-1] Thus, for example when 3i equals -1, from Equation 40 fig occurs at y = 1.0716. Since TR is to be measured from TF (where 2 equals 1) this corresponds to y_equal 0.0716 relative to TF. Or, since with 51 = -1, TF/TR is 5, the time on the reverse part of the potential-time curve corresponding to §§ is 0.215 TR. Interestingly, this time is not precisely a "quarterdwave" potential for the reverse part of the chrono- potentiogram. Potentials corresponding to these two times (0.25 and xg(E)=0) hereafter are referred to as EF and ER respectively. Ag will then be the difference between ER and e tan-s)- The effect of a on Ag is shown in Figure 7. Because for the mechanism being considered here a is typically about 0.5, and rarely outside the range 0.5-0.7, these data of Figure 7 show that for reasonable values of a, the parameter Ag tends to be independent of a, the dependence becoming less as p decreases. The explanation of this fact is that as a 41 .U .usofioammmoo Hommsmuu omumno so HQ mo mocmpcmmwn. .S wusmflm 42 m.o >.o h musmfim m.o m.o H.o O. HNQ >EON o.mua avu 45 varies both gF and gR shift in the same direction, and these shifts tend to cancel in terms of Ag. Nevertheless, for extreme values of a near 0 or 1, Ag is markedly dependent on a. The reason for this fact is that, for example, as a approaches 1, gF tends to be independent of a (see Equation 55), whereas gR tends to vary eXponentially with a (see Equation 56). The fact that for small values of p and reasonable values of a, Ag is independent of a is important for two reasons. First, when Ag is independent of a, Ag is determined uniquely by p, and therefore Ag is a simple measure of p, and hence 5% (see Equation 24). Second, for the special case of a equal 0.5, Equations 27 and 51 take the form of a quadratic equa- tion and both can be solved eXplicitly for g(§). This means that an equation can be derived for Ag_which always is valid when 0 equals 0.5, and which, depending on p, may be exact for any a between 0.5 and 0.7. Thus, by restricting the discussion of the case of a equal 0.5, it should be possible to derive an eXplicit eXpres- sion for overpotential. Introducing a of 0.5 into Equations 27 and 51 leads to the following equations yfi'x2 + p x + y%'- 1 = 0 (41) TthSTR tummy—1)" - y‘fixa - pRix + (1-Ri) we)“r - yin-.0 (42) 44 where x = exp[0.5 g(E)] (45) Equations 41 and 42 can be solved directly for g(§) by use of the quadratic equation. The positive roots are re- tained in both instances, since negative roots lead to un- defined values of 9(g). Thus, the pertinent solutions of Equations 41 and 42 are OgthF r .p +(92 + 4Y‘(1‘Y§)}§ 9(E) = 21n } (44) 2y ‘s TFSJZSTR 'PRi+[p2Ri-4 [yfi- (1-Ri) (y-1)é] [y'lr - 1 .. 9(E) = 21n (1‘Ri)(Y-1)é])é' 2 [yé - (1-Ri) (y-1)§] I (45; To calculate the overpotential, Ag, values of gF and gR are first determined by evaluating Equation 44 at y_= 0.25, and Equation 45 at xg(E)=0 given by Equation 40. The results when combined with the definition of g(§) (9;, Equation 26) are 0_(_t 03» How a cufl3 m4 m0 COMuMHnm> msq305m mo>usu mswxuoz .m musmwm 47 m onsmflm . a 9H NJ m.o «.o 0.0 _ _ a I _ _ _ 0 low L 13. l u m 48 m A l 18 m6- J looa OOH-" ll ONH \ I \J 48 The only assumption incorporated in the derivation of the equation for Ag is that a is 0.5. For Equation 48 to be generally useful, it is important to attempt to evaluate the errors that will result if Equation 48 is applied to a system where a is different from 0.5. By using the exact eXpressions it was found that for Ag of about 95/3 mV (p2:1.0). a value of p calculated on the basis of Equation 48 always is too small by about 10% if a is 0.5 or 0.7. For A§_about 50/g_mV (pG50.5) the error is reduced to about 6%. Because in terms of rate constants an error of 10% is not large, and often ‘within experimental error, it is concluded that Equation 48 can be applied whenever Ag is less than 95/g,mV. Thus, to apply Equation 48 eXperimentally, conditions (current density, etc.) are selected which give Ag less than 95/p.mV, and then from this experimental Ag, and Figure 8, p is determined. {The method due to Berzins and Delahay can be applied to the forward and reverse parts of the potential-time curve to O (59) 6t 0 ax2 2 6c 5 c R R at R ax2 The following initial and boundary conditions will be assumed in order to solve the above differential equations explicitly for concentration t = 0; x20 c0 = CO; CR = cR (61) * * t20, x—" 00 cO _"Co; cR —* CR (62) aco acR 1:207 X-O DO—g-X- — - DR—S; (63) These conditions are similar to those presented and discussed 55 previously (gg, the discussion of Equations 7, 8, and 10). The only difference between the two sets of conditions is that to preserve generality the analytical concentration of R, C;, is assumed to be finite in the present case (see Equations 61 and 62). As before, it will be assumed that the forward part of the chronopotentiogram consists of the reduction of 0 at a constant current density, ET , and that current reversal is F introduced when the forward transition time, is reached. TF, The current density after reversal will be labeled and it}; the resulting anodic transition time will be designated TR. Again, by convention, TR will be measured from the forward transition time. The major difference between the calculations presented here and those of the previous section is that previously it was assumed that the constant current was consumed entirely by the faradaic reaction, whereas in the present case the assumption will be made that the total current is partitioned between the faradaic and double-layer charging processes. Thus, rather than being a constant, the faradaic current dens- ity is actually some (unknown) function of time. The total current density, ;T(t), can be written as the algebraic sum of the faradaic, gf(t), and charging current, ;C(t). densities (Point 2 of model) iT(t) = if(t) + ic(t) ‘ (64) 56 where ;T(t) is written as time dependent to account for cur- rent reversal. The double-layer current in Equation 64 is given by dE(t) ic(t) = - CL dt (65) where El is the potential-independent differential double— layer capacitance referred to in Point 5. The negative sign is included so cathodic potential excursions correspond to positive current densities according to the usual sign con- vention. Thus, for the case of current reversal at the for- ward transition time, ;T(£) is given by the following expression 1T1? = 1f(t) + 1C(t) OithF iT(t) = (66) 1T = RilT = 1f(t) + 1c(t) TthETR R F where R = i /i (67) 1 TR TF The flux is related to current density by Fick's first law which for the present case takes the following form OgthF: x=0 aco DO-§;-= if(t)/nF = (iTF- iC(t))/nF (68) TFsfijfiR' X—O 8cR nR-§;-= if(t)/nF = (iTR- ic(t))/nF = (RiiTF- iC(t))/nF (69) 57 or written in terms of the flux of 0 - DO-§;'= (R111? - ic(t))/nF (70) The objective, now, is to solve this boundary value problem for the time dependence of surface concentrations of 0 and R, and then to combine these surface concentrations with the eXpressions for the potential dependence of 5f and Eb (Point 4 of the model), and the rate law eXpression (Equation 4). The results will define, within the framework of the model, the effects of double—layer charging on kinetically controlled electron transfer. Unfortunately, this boundary value problem cannot be solved analytically because of the nonlinearity associated with Equations 2, 5, 4, 68, and 69. The problem can, of course, be solved numerically. Numerical calculations are simplified by first transforming the problem to an integral equation, which can be solved without loss of generality by prOper dimensional analysis. These two tOpics are discussed next. Integral Equation Form of Boundary Value Problem An approach that has proved very successful in solving nonlinear boundary value problems is transformation to an integral equation (22). This approach has two major ad- vantages. First, the problem usually can be reduced to a 58 single equation involving a single unknown, rather than a set of partial differential equations and boundary conditions that must be solved simultaneously. And second, the numerical solution of integral equations is considerably more straight- forward and accurate than the numerical solution of partial differential equations. In the present case the problem can be transformed to two integral equations, one applicable prior to current re- versal and the other applicable after current reversal. Of course, these integral equations are necessarily nonlinear, and therefore they also cannot be solved analytically. The transformation to integral equations is readily accomplished with the Laplace transform Operator and is developed in Appendix A. The resulting equations are (Equations A15 and A16): Ogtng . “/2 1T (DR) i (t)(D )‘l/2 P c R * t -—- — = exp(-a,4D‘C /JD‘C )exp nFC;(DO)a/2ks nrcngW/ZkS O Q R R t . (anF/RTleO 1C(T)dT) * i (T)dT [1 — 2iT ny'nFCOflnDO + 1/nFCSJnDOft -2———--- exp(-nF/RTC1ft F o Jt - T O iC(T)dT) _( 2iTF¥EVbFCSflWDO)exp(JDBCS/VDEC;)exptnF/RTCIf: iC(T)dT) + (epryfiatS/JERC;)/npcanD5)exp(-nF/RTle: ic(T)dT)f: isiilgiil .t - T (71) 59 T ‘lr (1 - 121)) + exp(~/I_DBCO/ a» * * ‘ at it T @cR) (ZiTFJF/nrcod wo‘) - exp (JTS‘C /~/'5RCR)/nrco~/1rno‘ ff 0 0 ic(T)dT - , * * * t iC(T)dT ————t _ T exp(JBBcO/JB;cR)/nFcO~/1rno‘ fTF ———t _ T (72) Numerical Solution of Integral Equatggns To solve Equations 71 and 72 numerically without loss of generality it is essential to have the equations in dimension- less form so that results are described by the minimum number of independent parameters. Equations 71 and 72 can be made dimensionless with the following changes of variable * t = (nFCOJ—TTDO) 2y/4iT: (75) 60 ‘I r = (upchfifig)2x/4iT: <74) and the following definitions h(y) = ic(y)/iT(y) (75) exp(b) = WOO/CR (76) 41 C ‘Y = (RT/nF) T121 (77) (nE‘CO~/1r0‘0)2 _ . * a p — iTF/nrcov ks (78) . 2 4LT T Yf = F» F 2 (79) (nFcOJ—vrb'o) Equations 71 and 72 now become 0_<_t_<_TF y _ - Y - p(1 _ h(y)) _ e a(b 1/‘1’foh(x)dx)(1_e 1/‘l’fo h(x)dx) (80) e-a(b - 1/vgg h(x)dx)(1 + eb - 1/WJ: h(x)dx) =_J_y-+1/2fy 1M o J;‘:”;’ R. -aIb—l/‘ffzf h(x)dx- Y—lfiflflmdx) [1_e—1/nyf 0 TR Y i b-l/‘l’f-ofh(x)dx-= 9" Ripll-hiy) )ve Yf R1 Y e-a(b-1/on h(x)dx—~nyf h(x)dx)[1+e R1 Y h(X)dx - Tfyf h(x)dx ] = (y - yf)§I1 - Ri) - y§'+ fy h(x)dx ,y l y R- y , f E'f f h(x)dx + -£-f h(x)dx (81) 06—3 2 YfF—y-x - :E" U) #91 61 The only unknown in these equations is the function h(y). Once h(y) has been calculated it can be related to potential through the following eXpressions which are derived in Appendix B (Equations B5 and BB): OSISTF n[E(y) - 3%] = RT/F [b - 1/‘1’ [Y h(x)dx] (82) O TthgyR n[E(y) - 3%] = RT/F[b - 1/9 fyf h(x)dx - O R./Y fy h(x)dx] (85) i Yf Equations 80 and 81 were solved numerically by a step functional method (47),- To make a description of this method applicable to all of the integrals appearing in Equations 80 and 81, it is useful to employ the following general repre- sentation for these integrals Y f0 h(z)K(y-z)dz (84) where h(z) represents the unknown function, and K(y-g) is the kernel function, which is defined eXplicitly. To approximate this integral the range of integration from y = 0 to y = g is first divided into g_equally ppaced subintervals by the following change of variable n = y/é (85) 62 where 6 is the length of the subinterval (6 = g__), and g is a serial number of the subinterval. Thus, Integral 84 becomes [:6 h(z)K(n6 - z)dz (86) This integral can now be replaced by a finite sum of integrals which are integrated over each of the subintervals “ i6 2 i=1f(i-1)6 h(i)K(n6 - z)dz (87) At this point the unknown function can be approximated in several ways (57). The simplest of these is to assume that it is a constant over the i§h_subinterval. With this approxi- mation, Integral 87 becomes “ i6 2 h(i) f i=1 (i-1)6 K(n6 - z)dz (88) The notation can now be simplified by introducing the follow- ing change of variable w=né-z (89) which leads to n o n z h(i)f‘“’”1”x(w)dw= z h(i)A(n-i+1) (90) i=1 (n-i)6 i=1 Thus, to approximate the integrals of Equations 80 and 81 it is simply necessary to evaluate A§txP(-nhpin)) - ADEeXp(-Ehp(n)) + p+paE - p aEhI')(’n) + CEexp(-Eh (n))] (100) Bexp(aEhp(n)) p St SIR TF hp+1(n) = hp(n) - [V+GVexp(-RiEhp(n)) + g + GWexp(-RiEhp(n)) Rip — Riphp(n) Texp(aRiEhp(n)) + Ri(6)§hp(n) + GRi(6)§hp(n)exp(-R1Ehp(n)) - + 1-2exp(-Rishp(n))l/I-GVRiEexp(-Rishp(n))-RiEGwexp(-RiEhp(n)) 4- 4’ i- _ + Ri(6) - Rfsc(é) hp(n)exp(-RiEhp(n)) + GRi(6) exp( RiEhp(n)) an Rip + “R: pE - aRiEPhp(n) Texp(aRiEhp(n)) + RiEZexp(-R1Ehp(n))] (101) 66 where n-1 A = ((5)5 >3 h(i) [(n—i+1)é. - (n-i)§] (102) i=1 n-1 B = exp[ -0(b — 0/Y 3 h(i))l (105) i=1 n-1 C = exp(-5/Y Z h(i)) (104) i=1 n-1 D = exp(b - 0/Y E h(i)) (105) i=1 a = 6/9 (106) n1 n-1 G = exp(b - 0/9 2 h(i) - R.0/Y Z .h(i)) (107) i=1 1 i=n1+1 v = (n0 - n10)§(1 - ai) - (nai’ (108) n 1 w = (6)5 z h(i) [(n-i+1)’=" - (n-i)’<*] + aim)? i=1 n-1 §' §_ 2 h(i)[(n-i+1) -(n-i) I (109) i=n1+1 n1 n-1 T = exp[-a(b - 0/Y 2 h(i) - Rib/w z h(i))l (110) i=1 i=n +1 1 n1 n-1 z = 8Xp(-<5/‘i’ z h(i) - Rié/w z h(i)) (111) i=1 i=n1f1 To start the iterative procedure an arbitrary value of h(g) equal 0.0004 was always used as the initial guess from which h(y) was evaluated with y equal zero. Subsequent (\ n It ( l- E I HIM NM. v TN 9 VI. «A. 67 calculations were performed with the most recently calculated solution as an initial guess in the Newton-Raphson iteration. The iteration was continued until successive answers differed by a relative error less than 10'7. Calculations were per— formed on the Michigan State University Control Data 5600 digital computer, and the FORTRAN program is listed in Appendix C. Results of Calculations The numerical solution of Equations 80 and 81 provides values of h(y) (ratio of faradaic current to total current, Equation 75) as a function of y (dimensionless variable pro- portional to real time, see Equation 75). Alternatively, values of h(y) are directly related to potential (Equations 82 and 85) so that theoretical potential-time curves also result from the solution of Equations 80 and 81. Both h(y) and 2(g_- Efi) are tabulated by the computer program given in Appendix C. Theoretical Efx curves calculated in this way depend on several parameters, but most predominantly on W and p. The parameter Y is directly prOportional to the double-layer capacitance (see Equation 77) and therefore its magnitude determines the extent to which double-layer charging affects chronopotentiograms. The parameter p (Equation 78) has the same definition used previously (Equation 24), and therefore its magnitude determines the effect of electron transfer kinetics on chronOpotentiograms. Although these two parameters interact, their effect in general is readily 68 apparent from the theoretical chronopotentiograms of Figures 9 and 10. In Figure 9 curves were calculated for three values of Y and fixed p, whereas the curves of Figure 10 are for three values of p and fixed W. In particular the curves of Figure 9 illustrate the dramatic effect of in- creased double-layer charging on chronopotentiograms. When it is recalled that y_= 1 correSponds to the Sand equation transition time, it is apparent that transition time becomes ill-defined in the presence of appreciable charging current. Among other things, this effect obviously makes unambiguous definition of gF and gR impossible. Although the chronopotentiograms are determined largely by Y and p, they also depend to a lesser extent on the values of a, 0, 2, gi, and switching potential (potential where current reversal occurs). Thus, to interpret quantitatively the effect of Y and p, which is the primary goal, the effect of these other parameters also must be understood, and there- fore they are discussed individually in the following para- graphs. gggect of 0. It will be recalled that 0 is the width of the subintervals over which the unknown function is approxi- mated in the numerical integrations. »Since this is a fairly crude approximation, it is important to evaluate the effect of 0, and select a value which provides satisfactory accuracy. Obviously, as 0 approaches zero, the accuracy will improve, but at the same time the number of calculations involved will 69 .mo.o u a .o m>nao .m m>nzo “8.0 u a .a w>uso .m.on u .m u Q c033 .% ttnwumemumm Hmmeioansop cakonm mEmnmoHusmuomosonso pmumHsoamo .m onsmflm 70 0.0 ~§ oom .00fi OOdr OON) 00ml ( 3-3)u I Am 71 .m.m u Q mu 0>HDU “o.fi u Q .m 0>HDU “0.0 a Q .¢ 0>HDU .m.ou u .m cam .m.o u 8 .mo.o u a £033 .Q .Hmumfimumm vauoswx mo uowmwm.mna3onn mamumowusouomocouno pmumasoamo .oa enemas 72 o.¢ N.m oa 0.33m an «KN 0.0 fl _ , CON 00.“ J 87 l OONI cont A1n ' (ITS-3) u 75 increase. The Optimum value of 0 is one that provides accept- able accuracy without requiring inordinate amounts Of computer time. This value Of 0 was determined by varying 0 over the range 0.1 to 0.001. The value of 0 equal 0.01 was found to satisfy the above requirements, so that results are accurate to.i 0.5 mV, except for very small values of Y($0.005) and finite p. For Y§0.005, a value of 0 equal 0.001 was found to give results to the same degree of accuracy mentioned above. Effect of b. From its definition in Equation 76 parameter g_simply defines the initial equilibrium potential for the experiment. Normally, in a chronopotentiometric ex- periment only one oxidation state of the depolarizer is present, for example the oxidized form in the case Of an initial reduction. In this case the system is poised initial- ly at an equilibrium potential very anodic of gfi, It is Ob- served experimentally that under these conditions chrono- potentiograms are essentially independent Of this initial potential. This experimental fact was verified in the theo- retical calculations by Observing that gfy.curves are inde- pendent Of 2, except for very small shifts along the time axis, provided g.is greater than 6.5, (gi more than 165 mV positive of 2%). Thus, all calculations reported in this thesis were calculated for p,= 7.0, and results are independent ijg within the accuracy of the calculations. R“? “in _n is P: it, "v V. '11) (D l Th '1 (I) 74 Effect of Ri. Bi is the ratio of forward to reverse current densities, and its effect on gfiy curves in the present case is identical to that discussed earlier in this thesis (2;, discussion Of Figure 5). As pointed out in that discus- sion, the Optimum value from an experimental point Of view is about -0.5, and therefore, except where indicated, calculations reported here are for gi = -0.5. Effect Of a. The transfer coefficient, a, affects the symmetry of gfiy curves as illustrated by Figure 11, where theoretical chronopotentiograms are shown for three values of a at fixed Y and p. The effect is essentially the same as discussed earlier, and the limiting cases discussed there transpose directly to the present case. As shown by Figure 11 the overall influence Of a is fairly small, the effect de- creasing as p decreases. Since the curves become less de- pendent on a for smaller values of p, differences of potential (3,9,, Ag) on the forward and reverse parts of the gry_curve tend to be independent of a. This fact will be used below in develOping a relationship between Ag and p for the case Of finite Y. Effect of Switching Potential. The effect of current reversal potentials on Efx curves is illustrated in Figure 12, where theoretical curves are shown for two values of switch- ing potential at fixed Y and p. The overall effect Of cur- rent reversal potential is tO shift the anodic portion of the 75 . ohnou .U .a m>nso am.o u a .m mango “mac n 8 .4 w>uso. .m.ou n .m can .m.o u Q .mo.o u % c053 mEMHmOADCODOQOcOHSO OoumHDOHmo so .O.usmaowmmooo Hommcmnu omumno mo uoommm .es magmas 76 as magmas m.m «1N 0.N min min 0.0 ¢.0 0.0 m , __ese_q_.___0_,____ oom ,-)/, cos I. 00.7 Am ' (Ta-a) u 1 00m: oma J 00M) 77 nflfl amN >8 OHNI u Hawucouom mcwnouABm .m 0>Hso awn nmw >8 0am) u HmHucmuom mcwnouw3m .< m>uso .m.0u u .m use .m.0 n O .m.0 u Q .m0.0 w a £053 Hmflucmuom Hmmnm>mu ucmuuso mo poommm mCHBOnm mEmHmOwucmuomocousu OwumHSUHmO .3” 883m 78 ms magmas an 0.N A I. _ 4 0.0 00m L 00H L 03.. 1 00m) 00m”) Am ' (TS-3) U 79 chronopotentiogram on the time axis. It was found by calcu- lation that the shape of the reverse part of the curve be- comes independent Of current reversal potential, except for small differences near the anodic transition time, provided switching potential is greater than —210 mV yg, gé, Effect of Y and p. The effect of Y and p was considered briefly above, and will be analyzed now in greater detail. In particular, several limiting cases exist which can be evaluated and quantitatively defined. The case for p equal zero and finite Y already has been discussed in the litera— ture by several groups (21,50,55). Similarly, the case of zero Y and finite p was discussed in detail in the first part of this thesis. The calculations developed above permit conditions for which these limiting cases hold to be defined quantitatively. First, as Y and p both approach zero (small 91 and/Or * * AT , or large C , and large 58 and/or 1 , or large 90, 0 reZpectively) no kinetic or double-layechharging effects are eXpected. This condition corresponds to uncomplicated re- versible electron transfer, and was discussed earlier in connection with Equations 55 and 54. It was found from the theoretical gfy_curves that Equations 55 and 54 are obeyed within 2-5 mV whenever Y 5.0.0005 and p'S 0.01. A second limiting case arises when Y approaches zero, and p is finite. This situatiOn corresponds to the case where double-layer charging is negligible, and therefore is the case 80 discussed earlier in this thesis in connection with Equations 27 and 51. It was determined from gfy_curves that Equations 27 and 51 are obeyed within 2-5 mV whenever Y {_0.0005, regardless Of the value Of p. A third limiting case occurs when p approaches zero, and Y is finite. This situation corresponds to reversible elec- tron transfer where double-layer charging is prevalent, which is the case already discussed in the literature (21,50,55). Theoretical gfy curves were found to agree with these litera- ture data within 2-5 mV whenever pig 0.01, regardless Of the value Of Y. The final limiting case arises when p is sufficiently *- large (small 58 and/Or large gTF, or small go) that the pro- cesses of oxidation and reduction can be treated separately as the totally irreversible case (large overpotentials) (15). For zero Y this situation is described exactly by Equations 55 and 56. It was found that these equations are applicable in the present case within 2-5 mV whenever p.&_2.5 and Y S 0.0005. The case Of p 2 2.5 and Y 2 0.0005 has been dis- cussed in detail by Dracka (25), and Rodgers and Meites (55), and therefore was not investigated further. OverpotentiangAE, as a Measure Of ks, In the first part of this thesis a simple relationship between Ag_and 58 in the absence of double-layer charging was developed. It would be desirable to develOp a similar relationship for finite Y, but in view of the strong interaction discovered I' 5' :6.) Sets :QF‘L ivv ‘I'I' "I 4a . 1". Vln‘ \ 81 between Y and p this possibility is not very promising. Nevertheless, it is known that the Berzins-Delahay method accurately locates gé, and therefore the possibility of using this approach was investigated. TO do this a number of gfy.curves were calculated for many values Of Y and p, and values of gF and gR were determined by the method of Berzins and Delahay (to eliminate human prejudice the graphical con- structions were performed by the computer). From these values of gF and gR, Ag_was calculated and the results are summarized in Figure 15. The fact that a family Of curves is Obtained proves conclusively that the method of Berzins and Delahay does not satisfactorily compensate for double-layer charging when electron transfer is kinetically controlled. The reasons for this fact are apparent from the data of Figure 14. which illustrate the effect on Ag_of the strong interaction between Y and p. Figure 14 shows how the ratio if/iT (faradaic current efficiency) varies for three values of Y and a fixed value of p. Thus, as Y increases, the fraction of the total current which is consumed by the faradaic pro- cess decreases, as the charging current component increases (see Equation 64). Since overpotential is determined only by the faradaic current density, the value of Ag decreases as Y increases. Therefore, a new value of Ag results for each value Of Y, which in turn leads to a family of working curves 0 82 .m.0 n O can 0.0) Ham c0£3 as no mosam> Hmu0>mn How Q npfi3.u< mo sOwumHHm> mea3onm mo>uso mcaxuos .ma mucosa 85 “v...“ m...“ N?“ a...“ 0.H 0.0 0.0 ~30 0.0 0.0 To n.0 N.0I «.0 0. ma magmas Q q _ ___________L._s___504.__. L l. Lei l l nv0.0 l no.0 No.0 I l 11 l H0.0 000.0 [.1 It I 00.0 I m. 0 ONI 0d) 0d ON on Aux 'mu 0* om 00 0h 00 84 all .m.Ou u .m use .m.O u a .m.o u Q Hom.N nuaz_mucmaoammm ucmunsu mepmnmm mo soaumwnm> .dd ousmwm 85 I 2.0 l I I I I I I » , <2 «2 «a ‘2 .3 o. H <3 C) *0 C) 0 ((5)9 - I) (K)m,/(K)a, Figure 14 86 Although not as simple as hoped for, the working curves of Figure 15 do provide a basis for estimating 58. Thus, if Y can be determined independently, then the prOper working curve can be identified and measured values Of Ag correlated with p (and hence 58). Fortunately, Y can be determined eXperimentally, and therefore this approach is feasible. One method of estimating Y is as follows. The initial po- tential rise for a chronopotentiogram is almost entirely due to the charging current required to bring the working elec- trode to the potential at which discharge of the electro- active species occurs, as shown by the curves Of Figure 14. Thus, during this part of the eXperiment §c(t) is nearly equal to ET . Combining this fact with the definition of F ;C(t) (Equation 65) gives i = —c,s (112) where §_is ggfit)/§£, ;,g,, the lepe of the initial linear portion of the potential-time curve. Equation 112 combined with the definition for Y (Equation 77) and the Sand equa- tion yields the following expression 2 TF Y = -(RT/nF)(i /Srsi$ ) (113) 8 To use Equation 115 to estimate Y it is necessary to per- form two separate eXperiments on the same solution. One experiment is performed at a current density iT where no s 87 double-layer charging or kinetic effects are detectable, and the transition time corresponding to this current density is T8. The second experiment is performed at a current density iTF' where both double-layer charging and kinetic effects are significant, and g_is taken as the initial lepe of the resulting potential-time curve. Conditions Of Applicability of WOrking Curves. The work- ing curves of Figure 15 in general depend on the same para- meters as Efx curves. Thus, the working curves of Figure 15 were calculated for gi = -0.5, a = 0.5, and current reversal potential Of -240 mV yg, gfi, and they are rigorously applic- able only when these conditions are met experimentally. Additional calculations have shown, however, that for current reversal potentials more negative than -210 mV, the effect on Ag.is only of the order Of 2-5 mV. Thus, provided current reversal potential is more negative than ~210 mV, the curves of Figure 15 can be regarded as independent of current reversal potential. The working curves also depend on a, but for reasonable values of a and p, the dependence is not very great, as shown by the data Of Table I. These data also show that as p decreases the dependence of Afl,on 0 also decreases. Comparison Of values of p Obtained from Figure 15 with values of p calculated rigorously for a in the range 0.5 _<_ 0 _< 0.7, show that values Of 58 accurate within about 10% can be 88 Table I. Variation Of Ag,with Charge Transfer Coefficient for Several Values of Y and pa Y p nAE,mV a 0.005 0.00 -4.4 -4.4 -4.4 0.005 0.50 29.5 28.1 22.6 0.005 1.00 65.4 56.1 40.8 0.05 0.00 ~10.2 -10.2 -10.2 0.05 0.50 14.5 15.2 11.0 0.05 1.00 55.4 55.0 20.7 ‘JUICNQUICNNU‘ICNNUIOINUTCNNUTCN OOOOOOOOOOOOOOOOOO 89 Obtained from the working curves for any a in this range, provided Y 5 0.05 and p g 1.0. Limits Of Applicabigity O§_Working Curves of Figure 15. It is useful to determine how much the range of rate con- stants which can be measured can be extended by using the working curves of Figure 15. An estimate can be Obtained by combining Equations 77 and 78 and solving for 38 2 2 ._ * C1. k8 — n F WCODOY/4y RTpCl (114) Evaluating the above result at 250C for p_: 2 and y (= JD57DR ) = 1.0 gives k8 — 1.2 x 10 CODOY/Clp (115) By substituting the following typical values in Equation 115: = 1 x 10'5cm2/sec, and C = 5 x 10~s * = 1 x 10’8mole/cm3, D 1 530 0 farads/cma, the following eXpression is Obtained k8 = 5.9 Y/p (116) The largest value of 58 that can be measured is set by the restriction Y §_0.05 and p 2_0.1 to Observe kinetic effects. Equation 116 predicts that values of 55 $_1.2 cm/sec can be determined. This limit corresponds to extremely rapid elec- tron transfer, and thus a significant extension of the method has resulted, but at the cost of increased complexity of data analysis. OTHER POSSIBLE APPLICATIONS OF THE THEORETICAL DATA The main Objective of the theoretical calculations just presented was to determine the influence of double- 1ayer charging on correlations between overpotential (Ag) and electron transfer kinetics. There are, however, several other areas in which the calculations could be useful. For example, the results could be used to evaluate the several different literature methods for measuring transition times in a manner similar to that employed by Olmstead and Nicholson (50). Also, the results could be used to evalu- ate derivative chronOpotentiometry, to determine whether recent claims by Burden and Peters (10) for advantages of derivative chronopotentiometry are real. Finally, the re- sults could be used to define quantitatively the effects of linearization of the rate law. Of these possibilities the second was examined briefly (but will not be discussed here). and the third in some detail. Since results for this latter case are interesting, they will be discussed briefly. Because Of the functional dependence Of heterogeneous rate constants on potential, the rate law is nonlinear, which prevents analytical solutions for any boundary value 90 91 problems which use this rate law as a boundary condition. TO avoid these mathematical intractabilities it has been common practice in the electrochemical literature to employ a linearized form of the rate law (5,8,44,51). This approach usually permits closed form solutions to be Obtained, but necessarily limits the correSponding techniques to small departures from the equilibrium potential. Thus, it is usually estimated from the way in which the rate law is linearized, that results are applicable only if overpoten- tials are less than about 2-5 mV (5). The use Of such small overpotentials has obvious experimental disadvantages. Moreover, the way in which the limit of 2-5 mV is estimated does not take into account the fact that there may be compen- sating effects, and therefore the equations may actually be valid for larger overpotentials. This possibility has only been investigated rigorously in one case. Nicholson (46) found that for the coulostatic relaxation method, linearized equations were actually valid for overpotentials of about 25 mV, as compared to the 2-5 mV that had been previously claimed. This conclusion is very important, especially if it proves to be generally true for all electrochemical relaxation tech- niques. One of the popular relaxation techniques is called the galvanostatic method (5,6,7,9,54,55,56,57,45). With this method an electrode is initially at some equilibrium potential, usually near the half-wave potential (;,g,, both 0 and R are initially present at approximately equal concentrations). 92 The electrode is then perturbed by a constant current pulse, and overpotential is recorded as a function of time as the electrode potential is driven from its equilibrium value. Theory for this method was first derived by Berzins and Delahay (5), who used the linearized form of the rate law. Thus, according to these authors only the first 2-5 mV of overpotential on overpotential~time curves are interpretable in terms of their equations. On reflection it will become apparent to the reader that from an experimental point Of view the galvanostatic method is merely a special case of chronopotentiometry in which different initial potentials are used, and only the very first part of the chronOpotentiogram is recorded. When it is recalled that the theory Of the preceding section was derived * * 90 and QR. see parameter p, Equation 76) it is apparent that the computer for any arbitrary initial equilibrium potential (any program of Appendix C can be used to calculate theoretical galvanostatic curves, which do not encompass the assumption of a linearized rate law. Thus, it is possible to determine unambiguously the overpotential range for which the theory Of Berzins and Delahay is valid, and thereby determine if the conclusions Of Nicholson for the coulostatic method extend to the galvanostatic method. To make this comparison the final equation given by Berzins and Delahay [Equation 18 Of Reference (5)] could be used to calculate overpotential curves for the linearized 95 case. Unfortunately, this is difficult because for certain combinations of experimental variables, it requires evalua- tion of error functions which have imaginary arguments. Although it is possible to evaluate these functions, the computer calculations would be complex and therefore a simpler approach was sought. In this alternate approach the linear boundary value problem originally solved by Berzins and Delahay was trans- formed directly tO the following linear integral equation, which is equivalent to the lineariZed form of Equation 80: pexp(ab)[1-h(y)]= -[1+exp(b)]yé¥fill+exp(b)]L¥ ELELQE- m + 1/ny h(x)dx (117) o This integral equation was solved by the step—functional method, and results were in agreement with those calculated from the Berzins and Delahay equation. Galvanostatic curves were calculated from Equations 80 and 117 for identical values of Y, p, and p, and the results compared to determine the overpotential at which the rigorous nonlinearized theory begins to deviate significantly from the linearized theory. Typical results are shown in Figure 15, where the solid curve is Obtained from the linearized equa- tion, and the dashed curve from the nonlinearized equation. At least for the values Of Y, p and g_used in Figure 15, it is apparent that linearized theory is valid for consider- ably larger overpotentials than the 2-5 mV usually assumed. 94 .Omuflummcwacos .O>Hso Oonmma .pouwnmmcwa .0>HDU Oflaom .m.0 w O can .HO.0 n % .m.0 M Q cmn3 mw>uso Ufiumunosm>amm Uwumasoamo .md m.sm.t 95 we magmas a 00.0 °ba 3-3)u AuI '( 96 The range Of overpotential for which the two equations agree is, Of course, strongly dependent on the values of Y, p and p, This fact is illustrated by the data Of Table II, where the numerical quantities listed under various values of p are the overpotentials at which linearized and non- linearized results begin to differ by more than 1 mV. The data of Table II show that in general linearized theory can be used for overpotentials of about 50 mV, which is nearly an order of magnitude larger than assumed in the literature. This fact is significant in terms of the galvano- static method, because it means that it can be used over an experimentally more convenient range Of potentials, while still analyzing the results with the simple closed form theory of Berzins and Delahay. Moreover, together with the results of Nicholson for the coulostatic method, it is likely that linearized theories for all electrochemical techniques are valid for much larger overpotentials than previously assumed in the literature. 97 Table II. Variation of Overpotentiala with Y and p for b Several Values Of p for the Galvanostatic Method b Y 0.0 1.0 2.0 3.0 7.0 0.005 0.0 20.5a 11.9 10.3 12.4 >30.0 0.5 29.8 23.7 25.7 27.9 >30.0 1.0 >30.0C >30.0 >30.0 >30.0 >30.0 5.0 >30.0 >30.0 >30.0 >30.0 >30.0 0.01 0.0 20.8 12.2 11.3 14.0 >30.0 0.5 30.0 24.5 25.8 29.8 >30.0 1.0 >30.0 >30.0 >30.0 >30.0 >30.0 5.0 >30.0 >30.0 >30.0 >30.0 >30.0 0.03 0.0 21.8 13.9 15.6 17.8 >30.0 0.5 >30.0 25.5 26.1 29.5 >30.0 1.0 >30.0 >30.0 >30.0 >30.0 >30.0 5.0 >30.0 >30.0 >30.0 >30.0 >30.0 0.05 0.0 22.6 15.3 15.0 19.5 >30.0 0.5 >30.0 25.5 26.5 30.0 >30.0 1.0 >30.0 >30.0 >30.0 >30.0 >30.0 5.0 >30.0 >30.0 >30.0 >30.0 >30.0 0.05 0.0 22.6 15.3 15.0 19.5 >30.0 0.5 >30.0 25.5 26.3 30.0 >30.0 1.0 >30.0 >30.0 >30.0 >30.0 >30.0 5.0 >30.0 >30.0 >30.0 >30.0 >30.0 0.07 0.0 23.3 15.8 16.1 20.7 >30.0 0.5 >30.0 25.6 26.5 >30.0 >30.0 1.0 >30.0 >30.0 >30.0 >30.0 >30.0 5.0 >30.0 >30.0 >30.0 >30.0 >30.0 aNumerical values for overpotential arbitrarily defined as the potential at which linearized and nonlinearized results begin to differ by more than 1 mV. a = 0.5 and 6 = 0.001. CThe maximum departure from the equilibrium potential investi- gated was 50 mV, which explains why some values give agree- ment for greater than 50 mV. b EXPERIMENTAL The experimental work which constitutes the last major section Of this thesis was designed to evaluate the scope and limitations Of the theoretical calculations already described. To do this two different systems were studied for which the electron transfer rate constants had been measured by accepted methods. The first system studied was azobenzene in water-ethanol solvent. This system was selected because the apparent electron transfer rate is a function of pH, and therefore the measurable rate constant could easily be varied by changing pH. Thus, the new tech- nique could be evaluated for a fairly large range of rate constants without changing the depolarizer. The second sys- tem studied was the reduction of cadmium in aqueous solvent. This system was used primarily to evaluate the effect of double-layer charging, since the standard rate constant is sufficiently large to require experimental conditions for which double-layer charging is a significant factor. To perform the experiments a simple instrument was assembled from commercially available components. Although this equipment performed satisfactorily, it was found that with current reversal adjustment of the switching time by 98 99 trial and error was tedious, and also inaccurate at short times. Thus, a more SOphisticated instrument with automatic current reversal also was constructed. Since both of these instruments were used to collect data, and both have indi— vidual merits, a description of both instruments will be presented. instrumentation The first instrument was simply a constant current source which could be programmed to provide bipolar currents of independently variable magnitude and duration. A block dia- gram of the circuit is shown in Figure 16a. The control amplifier, CA, is a high gain differential amplifier (Wenking Potentiostat, Model 61RS, Brinkman Instru- ments) provided with negative feedback to maintain zero potential difference between the inverting (-) and noninvert- ing (+) inputs. Since in the circuit of Figure 16a the noninverting input is grounded, the inverting input is main- tained at virtual ground. Thus, if a voltage gate of ampli- tude g.is applied to the resistor, g, from the function generator (FG), a constant current given by g/g_flows from the output Of amplifier CA. Since the electrolysis cell is in series with the output of CA, the constant current g/g also passes between the counter electrode, gg, and the working electrode, fig, The chronOpotentiogram is Obtained by record- ing the potential of the reference electrode, ggypyg. ground 100 oumm ommuao> owpocm mo coflumusp mafia "Nu mumm mmmuao> pronumo mo soaumusn mafia "flu mpsuflamam Oumm wmmuao> pross “mm OODuHHmEm ovum ammuao> oaponumo “on Hmwucouom venouw “0mm .mcfinouASm ucouuso mom m nmsounu Omaammm Enomo>m3 pmaamumoum AQV noumumsmm mews ou mommaus mowammm 0cm mumm monuao> Ou ommuao>.mcammwn mmaammfi xon mocmumwmmu mpmooa ommuao> mean wanmumsncc Houmnmswm cofluossm oponuomam mcwxuoz OOOHDOOHO Houcsoo mpouuuoam mocmummmm usmcw msfluum>swcoz usmsw mswuum>sH umamwamam Honusoo 83388228“; .ucoEDHumCH usmunso ucmumsou mo Emnmmwv uwsouao Amv .md musmwm 101 we magmas mow>mo thAOuooom 102 (potential of fig) as a function of time on a suitable recording device. The voltage gate for programming the constant current generator was obtained from a commercial function generator (Exact Electronics, Inc., Model 255). This instrument is equipped with both a main function generator, which generates triangular, sine, and square waveforms, and a ramp generator. Timing for the two function generator sections is independ- ently adjustable, and a variety of different triggering modes can be selected from a single front-panel program switch. For the present application an external trigger was used to start the ramp generator, which in turn was programmed to trigger the main generator at the end of the ramp cycle. In this mode a time delayed square wave could be Obtained from the function generator (FG), the time delay being inversely prOportional to the ramp frequency. When switch S1 (Figure 16a) is closed a biasing voltage, gt, is applied to the cur- rent generator (causing a current gC[g_to flow through the cell), and simultaneously a trigger pulse is applied to the ramp generator. The frequency Of the ramp generator is ad- justed to give the time delay, £1, shown in Figure 16b. The time interval, 31, is adjusted by tFial and error to equal the forward transition time on the chronopotentiogram. After the interval, £1, the ramp generator internally triggers the main generator, which provides a square wave of amplitude gC + g3, and causes a current equal to ga/g_to flow through 105 the electrolysis cell. Since gC and ga are of Opposite polarity, current reversal occurs at this point, and elect trolysis proceeds at the constant current ga/g, By adjust- ing the amplitude of the main square wave, any value of gi (iR/i , see Equation 14) can be conveniently selected. The time interval, 32, is determined by the frequency of the main generator, and is simply adjusted to be large enough that the reverse transition time on the chronOpotentiogram is observed. Although this instrument functioned satisfactorily, adjustment Of by trial and error was very time consuming £1 and difficult at short times. Thus, a second instrument was designed to overcome these problems by automatic current reversal at the forward transition time. This was accom- plished with an electronic voltage comparator which provided automatic current switching when the potential of the work- ing electrode reached a preselected potential correSponding to the potential at the transition time. A description of this instrument is contained in the following paragraphs. The circuit for the instrument is shown in Figure 17. The amplifiers are solid state Operational amplifiers which are identified in the legend of Figure 17. Power for the Operational amplifiers was provided by a commercial power supply (Deltron Model 08 15-.5D, 1.15 V, i 500 ma). The circuit can be divided conceptually into three main parts. The current generating section involves amplifier CA, 104 moouuomam msflxuok OOOHuOOHO Houssoo Ononuuoao mucoummmm. xon oosmumanou mucosa umu080Husmuom sun» :09 mmmuao> mafia magnumsnpd mmmuao> consummeoo magnumsnpd mpoap oumum OHHOm mmOOaO HOCON seams Reannsanm .umamasmsm Honucoo 006m mOHms¢ .Hoamaamem umumoom apes mosses .nmsmasmsm 65mm Haaucmummmao. toms xoaunssam .umamasmsm umpmoom Dammm xownnawnm .HOHMAHQEM Houmummfioo Dcmmm xownnawnm .HOAMAHQEM Houum>sH seems xoaunsaam .umnmssmsm nmzoasom msmuao> “Hssgsssssssmsss V! D .msanouwam usmnuso owumeousm nus? ucoEOHumcfi ucouuso unnumsoo mo Emnmmwp pasoufio. .SH musmwm 105 as 655556 <0 OOH>OQ mcflpuoomm 106 and its operation is identical with the instrument described above--g,g,, a voltage g_applied through g causes a constant current g/R to flow through the electrolysis cell. Amplifier DA is the comparator, which compares the voltage of the work- ing electrode with a preselected transition time switching potential. Amplifiers I and DA provide signal conditioning described below for the comparator. The comparator is simply an Operational amplifier Operated in an open loop configura- tion. For example, whenever the noninverting input is at a potential more negative than the inverting input, the output of the amplifier is at its negative limit. Conversely, when the potential Of the inverting input becomes more negative than the noninverting input, the output of the amplifier swings to its positive limit. The zener diodes on the output of the voltage comparator clip the voltage swings Of the ampli- fier to provide symmetrical outputs. The booster (B1) is used to provide sufficient current to ensure prOper Operation of the zeners. Amplifier DG biases the output of the voltage comparator, and supplies the apprOpriate square waves to g. for generation of the cell current. Detailed Operation Of the circuit can be understood by considering the following example. Suppose that initially a reduction is to be performed requiring a cathodic current. The potential Of the working electrode will initially be at some equilibrium value, and then become progressively nega— tive as electrolysis proceeds, until it reaches some negative 107 potential, gc, corresponding to the forward transition time. When the electrode reaches this potential, current reversal should take place. Thus, the potential gC is selected and applied to the noninverting input of the voltage comparator. Since the potential of the reference electrode will become increasingly positive as electrolysis proceeds, this potential must be inverted before applying it to the other input of the voltage comparator. This inversion is accomplished with Amplifier I, which has a gain Of -1. Amplifier F is a voltage follower, which is used to provide impedance matching. Since initially gC is more negative than the inverting input of the voltage comparator (effectively the potential of the working electrode), the output of the voltage comparator is at its negative limit, and the circuit remains in this state as long as S1 is Open. When S1 is closed, amplifier DG adds the out- put Of the comparator to an adjustable bias voltage, g_ and B’ the sum, g, is applied through g generating the constant current g/g, As electrolysis continues the potential at the inverting input of the comparator becomes increasingly nega- tive until it passes the preselected transition time poten- tial, gc. When this happens, the output Of the comparator (amplifier DA) swings to its positive limit, and if gB is properly adjusted, the polarity of g changes causing current reversal. TO ensure that the circuit remains in this state until the reverse transition time is observed, when the out- put Of the comparator swings positive, diode D5 becomes 108 forward biased, applying a positive reference to the:com- parator through potentiometer P1. By simply adjusting gB. any ratio Of current densities, 31, can be Obtained. Thus, a current reversal chronopotentiogram is recorded by setting gC to the desired transition time potential, closing $1, and recording the potential at the output Of the follower (Amplifier F) yg, time. Cell and Electrodes. The electrolysis cell was a 200 ml Pyrex weighing bottle with a 60/12 standard taper joint. It was equipped with a tight-fitting Teflon lid, in which holes were provided for the various electrodes, nitrogen inlet, and a SCOOp used to transfer mercury drops. The working electrode was a hanging mercury drop, which was constructed according to the directions of Underkofler and Shain (59). Normally, two drOps of mercury from a DME capillary were collected and transferred to the working electrode. The counter electrode was a platinum wire in the form of a Spiral embedded in soft glass tubing, which was immersed directly in the solution under study. The reference electrode was a saturated calomel contained in a separate compartment, and connected to the cell through a double junction salt bridge ending in a Luggin capillary. The section Of the salt bridge adjacent to the SCE compart- ment contained 1 M sodium nitrate, while the Luggin capillary section was filled with the solution under study. 109 Chemicals. Zone refined azobenzene (Litton Chemicals, Inc.) was used without further treatment. Other chemicals were reagent grade with solids being dried at 1100C for several hours. RESULTS AND DISCUSSION OF EXPERIMENTS ggnetics of Reduction of Azobenzene The reduction of azobenzene in protic solvents has been studied extensively (11,12,25,26,50,51,52,55,42,49,60). The most detailed research is that of Lundquist (41), who investi- gated the reduction in both aqueous and nonaqueous solvents. In aqueous solvents he showed that the overall reaction in- volves two electrOns and two protons per molecule of azo- benzene. He found that the apparent reversibility Of the reduction is a function Of pH, with reversibility increasing as pH decreases. He used cyclic voltammetry to measure the apparent heterogeneous rate constant for electron transfer as a function of pH under conditions where the rate of the reaction is pseudo first order. Thus, the heterogeneous rate constants he Obtained are apparent 58 values with the pH dependence predicted by several possible mechanisms for the electrode reaction, he was able to arrive at a reasonable mechanism for reduction of azobenzene. He obtained further support for this mechanism from experiments in an aprotic solvent to which varying concentrations of acid were added. Azobenzene appeared to provide an ideal system for evaluating the simple overpotential equation for chronopo- tentiometry (Equation 48), since based on Lundquist's results 110 111 a range Of rate constants could be measured without changing the depolarizer (simply changing pH). Lundquist tabulated the rate constants as a function Of pH, and therefore direct comparisons could be made with his data. Hence, measurements were made on azobenzene for experimental conditions identical to those employed by Lundquist (see footnotes to Table III). The results of these measurements at several hydrogen ion concentrations are summarized in Table III, together with the values reported by Lundquist. A comparison of these two sets of data shows that at higher values Of pH, where 58 is small, agreement is good, but at lower pH the values of 58 determined by chronOpotentiometry are always larger than those obtained by Lundquist. It is interesting to note that these facts are consistent with the very approximate estimates made earlier for the upper limit Of 38 that could be determined with Equation 48. Thus, it was estimated that double-layer effects should prevent accurate measurement for 53 greater than about 0.02 cm/sec. This prediction is in good agreement with the data of Table III. Moreover, the direction Of the deviation for larger values of 38 is consistent with the effects predicted for double-layer charging. Thus, at low values of pH, where 58 is larger, it is necessary to employ higher current densities so that eXperimentally measurable values Of Ag are obtained. At these higher current densities charging current increases and the resulting current avail- able for the faradaic process decreases. Since the faradaic 112 .m4 m0 useEmuswmoE cw >6 0 H mo Hound ucmumsoo m mewEsmmm an OOCHEHOuOO mmcm m muasmmu mcwcwmamu on» How can >¢¢.0|uwm .Owom UwhOanoumm mchUm SQ Owumsnpd .mucoafluomxo H .ss4.ouu. .w< HOW poEsmmm nouns mswwmmu >8 0 H d .d 220 um 565.36 owcow samucwme Ou Omncm ODMHOHQUHOQ Esflnuwa ucmwOflmmsm nufi3 osmucmnoum a OIOH x 0.Hm m mo ommno>m on» ma woumwa .umwsvpssa SQ OmumHDQmu mosam>m mmm Ammv mo osam> noun. a .mosam> Owuumeoflucouomosounu Q U U Q Omlmm mm.Olmm.O mHmN ®.Nm mmlom O>.OIOd.O mHON m.0# dWINN Nm.OIH®.O meN m.mm H.© dNIMd mw.Olmm.O mHmH m.md 0m.0 m.mlfi.¢ m¢.dtmo.fi WHH¢ d.Hd ®.mlm.¢ Nd.Hlmh.O mHmm mm.m H.hlb.¢ mm.Otmm.O mflmm hm.b 0.6 m.mlm.n O>.OIO¢.O mHON om.m Od.o $.mtm.m mm.dION.d mflmd hm.h m.mlm.m mflodlmo.d mfififl mm.h m.MIm.N mO.HINh.O mHHm .Omim m.Nlm.N Om.OIO>.O meN mm.m N.H h.wl®.N mm.OI¢N.O mfldd mm.N no.0 dm.OImN.O #m.dlmm.d mflmd mam.o mm.OlmN.O ”$.Hlm0.d mfidfl omh.0 mm.OIom.O MH.dlmm.O mflmm mmm.0 0N.O dm.OI¢N.O O¢.HIOO.H mfimm mmm.0 H0.0 . mam ml . 68>... 469.661 .3 8.58 Nosxtmtmimfl .46 2.52 4.52.53} are mo COMuoadmm new ceaumuucmuc umumzuaoemnum profit: 00 coH cmmouvhm £uw3 as om cwmocmucmnoud may no tossesum> . HHH OHQMH. 115 current density determines overpotential, the magnitude of Ag should decrease as the faradaic current density decreases. Thus, if Ag decreases because of the influence of double- layer charging, the value of p obtained from Equation 48 will be smaller than if double-layer charging effects were negligible. Since p is inversely proportional to 58, the value of 38 calculated would be too large, which is precisely the trend observed in Table III. In summary, the experiments on azobenzene confirm the theory of the new method for measuring electron transfer rate constants. In addition, application of the technique proved to be as simple as anticipated, and therefore the method meets most Of the requirements set forth in the Introduction. The experimental results further show that double-layer charging sets the upper limit for measurement of 58, and that this upper limit is about 0.02 cm/sec. It‘was an attempt to extend this limit that prompted the calcula- tions for the influence of double-layer charging, and the eXperiments for this case, which are discussed next. Kinetics of Reduction of Cadmium As mentioned earlier reduction of cadmium was used to evaluate the effects of double-layer charging. These eXperi- ments were necessarily performed at higher current densities, and therefore the instrument with automatic current reversal was employed (see discussion of Figure 17). To ensure that 114 this instrument functioned prOperly, experiments initially were performed on reduction of cadmium under conditions devoid Of both double-layer and kinetic effects. These chronopotentiograms were diSplayed on an oscillosc0pe (Tektronix, Type 564), and photographed with a Polaroid camera attachment (Model C-12, and projected graticule, Model 100). The working electrode was a hanging mercury drop of radius 0.064 cm, and the current density was 9.99 x 10‘4 A/cmz. The'half-wave potential measured by the method Of Berzins and Delahay was -0.582 V yg, SCE. A diffusion co- efficient Of 2.8 x 10'5 cma/sec was calculated from the Sand equation, after T was determined by the procedure of Laity and McIntyre (59). These results compare with the values of -0.585 V yg, SCE and 6.5 x 10-6 cmz/sec obtained from the literature (27). Next, reduction Of cadmium was investigated under condi- tions where electron transfer is kinetically controlled, and where double-layer effects are necessarily prevalent. To evaluate the precision and sc0pe of the theory, experiments were performed at several different current densities, and the Observed values Of Ag recorded. In addition, values of the double-layer parameter were determined experimentally by the procedure described earlier (see discussion Of Equation 115). TheSe values Of Y were used to identify the prOper working curve in Figure 15, from which values of p correspond- ing to the Observed Ag_were Obtained. From these values Of p, 115 the correSponding values of 58 were calculated. Results of these experiments are summarized in Table IV, from which an average value of 38 equal 0.16.1 0.04 cm/sec is Obtained. The literature value Of 58 determined by ac polarography is 0.60 cm/sec, which is outside the estimated eXperimental error for the value reported here. There are several equally reasonable explanations for this discrepancy, and therefore no attempt will be made to justify the difference between the value reported above and the literature value. Because of the discrepancy with literature values for 58, it seemed appropriate to investigate the extent to which the theoretical model on which the calculations were based agreed with experiment for the entire chronOpotentiometric curve. To do this one of the experimental chronOpotentiograms used above in connection with Table IV was compared with a computer generated chronOpotentiogram. The eXperimental chronOpotentiogram was recorded with 51 = -0.5 and iTF= 2.6 x 10‘2 A/cme, and is represented by the points in Figure 18. For this curve pAg_equals 40 mV and Y = 0.015 (calcu- lated from Equation 115 for iT = 1.0 x 10_3 A/cma, T8 = s 0.85 sec, §.= -6.7 x 105 mV/sec). Using these data and the working curves of Figure 15, p equal 0.85 was Obtained, which corresponds to 58 equal 0.16 cm/sec. Next, the computer program of Appendix C was used to calculate a theoretical chronOpotentiogram for the same parameters as the experimental curve--i.e., Y = 0.015, p = 0.85, R1 = -0.5, a = 0.5, and 116 Table IV. 58 for Reduction Of Cadmiuma b c iTFx102,A/'cm2 nAE, mV p Y gsc' ,cm/sec 2.5 5115 0.57-0.76 0.014 0.16-0.21 2.6d 4215 0.78-1.00 0.015 0.15-0.17 2.6 4015 0.74-0.95 0.015 0.14-0.18 5.6 4815 1.01-1.50 0.019 0.14-0.18 a1.0 x 10‘3‘g_cadmium and 1.0 u_potassium nitrate. bA i 5 mV reading error assumed for gAg. C . . . Range determined by assuming a constant error of i 5 mV in measurement Of nAE. dNew solution eEach value of 58 listed is the average of 2 eXperiments. 117 g_= 2. This theoretical chronopotentiogram is represented by the solid line in Figure 18. Actually, to place the theoretical chronOpotentiogram on the potential axis of Figure 18, the theoretical curve was shifted to give the best fit between eXperiment and theory. This best fit corresponds to a half-wave potential of -0.590 V yg, SCE, which is in excellent agreement with the literature value for cadmium. The normalized transition time for the theoretical curve corresponds to a diffusion coefficient of 2.7 x 10'5 cm2/sec, which also agrees well with the value reported earlier. The excellent agreement between theory and experiment illustrated by Figure 18 demonstrates conclusively that the model assumed for the theoretical calculations is a reasonable one. Thus, it at least seems unlikely that the discrepancy between the 58 reported here and the literature value can be attributed to inadequacies of the theory. 118 .Hmucosflnmmxw .musflom .Hmoauouoonp .osfln .Esafivmo mo cofluosnou Mom EMHmOaucmuomosouno .me magmas 119 ms mnsmam 00m .moa x u 0.m ( m.# 0.6 m.m 0.0 m.m 0.N 0.6 0.H 0.0 0n0 __ ___4_44_____4.__mm.ou .0. O “.mfioOI o o o o o o _o o 1.6m.o- o o o o o o I o o o I56. 0 o I . lQNoOl O Jm®.0l LITERATURE CITED 4. 5. 6. 7. 8. 9. 10. 11. 12. 15. 14. 15. LITERATURE CITED Anderson, L. B., and Macero, D. J., Anal. Chem..§1, 522 (1965). Anson, F. C., and Barclay, D. J., Anal. Chem. 49, 1791 (1968). Berzins, T., and Delahay, P., J. Amer. Chem. Soc. 1§J 2486 (1955). gplg,, p. 4205. Berzins, T., and Delahay, P., 121g,, 11, 6448 (1955). Birke, R. L., and Roe, D. K., Anal. Chem. §1J 450 (1965). gpgg,, p. 455. Birke, R. L., and Roe, n. R., ipig,, gg, 1501 (1967). Blomgren, E., Inman, D., and Bockris, J. O'M., Rev. Sci. Instrum. gg, 11 (1961). Burden, S. L., and Peters, D. G., Anal. Chem. §§, 550 (1966). Castor, C. R., and Saylor, J. H., J. Amer. Chem. Soc. 1;, 1427 (1955). Chuang, L., Fried, I., and Elving, P. J., Anal. Chem. gl, 1528 (1965). Churchill, R. V., "Operational Mathematics," p. 55, McGraw-Hill Book CO., New York, 1958. Davis, D. G., in "Electroanalytical Chemistry," Bard, ed., Vol. I, Chap. 2, Marcel Dekker, Inc., New York, 1966. Delahay, P., "New Instrumental Methods in Electrochemistry," p. 58ff., Interscience, New York, 1954. 120 16. 17. 18. 19. 20. 21. 22. 25. 24. 25. 26. 27. 28. 29. 50. 51. 52. 55. 54. 35. 121 gpgg,, p. 182. ;Q;Q,, p. 182 and p. 196. gpig,, p. 186 and p. 197. gpig,, p. 207 ff. Delahay, P., "Double Layer and Electrode Kinetics," Interscience, New York, 1966. DeVries, W. T., J. Electroanal. Chem. 11, 51 (1968). Dracka, 0., Collect. Czech. Chem. Commun. gg, 558 (1960). Dracka, 0., ibid., g3, 2627 (1969). Everett, G. W., Johns, R. H., and Reilley, C. N., Anal. Chem. g1, 485 (1955). Florence, T. M., and Farrar, Y. J., Aust. J. Chem. 11J 1085 (1964). Foffani, A., and Fragiacomo, M., Ric. Sci. gg, 159 (1952). Frischmann, J., Ph. D. thesis, Michigan State University, East Lansing, Mich., 1966. Gierst, L., Juliard, A., J. Phys. Chem. §1, 701 (1955). Grahame, D. C., Chem. Rev. 2;, 441 (1947). Hillson, P. J., and Birnbaum, P. P., Trans. Faraday Soc. 2Q, 478 (1952). Holleck, L., and Holleck, G., Monatsh, Chem. gg, 990 (1964). Holleck, L., and'Holleck, G., Naturwissenschaften g;, 212, 455 (1964). Holleck, L., Shams-El-Din, A. M., Saleh, R.-M., and Holleck, G., Z. Naturforsch. 3g, 161 (1964). Inman, D., Bockris, J. O'M., and Blomgren, E., J. Elec- troanal. Chem. g, 506 (1961). Kooijman, D. J., and Sluyters, J. H., Electrochim. Acta 1;, 1147 (1966). 36. 57. 58. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. so. 51 O 52 O 55. 54. 122 Kooijman, D. J., and Sluyters, J. H., ibid., 1g, 1579 (1967). Kooijman, D. J., and Sluyters, J. H., J. Electroanal. Chem. gg, 152 (1967). Laitinen, H. A., "Chemical Analysis," p. 508, McGraw-Hill Book CO., New York, 1960. Laity, R. W., and McIntyre, J. D. E., J. Amer. Chem. Soc. fll. 5806 (1965). Lingane, P. J., Anal. Chem. g9, 541 (1967). Lundquist, Jr., J. T., Ph. D. thesis, Michigan State University, East Lansing, Mich., 1967. Markman, A. L., and Zinkova, E. V., J. Gen. Chem. U.S.S;R. 29. 5058 (1959) . Matsuda, H., Oka, S., and Delahay, P., J. Amer. Chem. Soc. §;, 5077 (1959). Mohilner, D. M., Hackerman, N., and Bard, A., Anal. Chem. Mohilner, D. M., in "Electroanalytical Chemistry," Bard, ed., Vol. I, Chap. 4, Marcel Dekker, Inc., New York, 1966. Nicholson, R. 8., Anal. Chem. él, 667 (1965). Nicholson, R. S., and Shain, 1., Anal. Chem. §§, 706 Noonan, D. C., Ph. D. thesis, Columbia University, New York, N. Y., 1967. Mygard, B., Ark. Kemi. 29, 165 (1965). Olmstead, M. L., and Nicholson, R. S., J. Phys. Chem. lg, 1650 (1968). Rangarajan, S. R., Anal. Chem. 49, 1582 (1968). Reinmuth, W. H., Anal. Chem. g2, 1446 (1962). Rodgers, R. S., and Meites, L., J. Electroanal. Chem. lg, 1 (1968). Sand, H. J. 5., Phil. Mag. _1_, 45 (1901). 55. 56. 57. 58. 59. 60. 125 Smith, D. E., in "ElectrOanalytical Chemistry," Bard, ed., Vol. I, Chap. 1, Marcel Dekker, Inc., New YOrk, 1966. Testa, A. C., and Reinmuth, W. H., Anal. Chem. gg, 1512 (1960) . Thomas, Jr., G. B., "Calculus and Analytic Geometry,” p. 585ff., Addison—Wesley, Reading, Mass., 1966. Ibid., p. 451. Underkofler, W. L., and Shain, 1., Anal. Chem. §§, 1966 (1961). Wawzonek, S., and Fredrickson, J. D., J. Amer. Chem. Soc. 11, 5985, 5988 (1955). APPENDICES APPENDIX A Reduction Of Boundary Valueggroblem to Integrainogg Equations 59 and 60 of the text can be integrated easily with the aid of the Laplace transformation, for which the following definition and notation will be adOpted Cb = fzglexp(-st)][CO(x,t)]dt (Co(x,t)) = COIx.s) (A1) Thus, the Laplace transformation Of Equation 59 with incorpor- ation of Equation 61 is E 2 =sC -c (A2) The general solution for Equation A2 is CO = Aexp(-x Js/DO') + Bexp(x'Js/DO‘) + Cg/s (A5) where A and B are integration constants. To satisfy Equa- tion 62, B must be zero, which reduces Equation A5 to Co = AexP(-x.Js/DO ) + Cg/s (A4) the value of A can be determined by evaluating Equation A4 at x = 0 124 125 - c /s (A5) Equations A4 and A5 can be combined to give 0 = (00 x=0 - CS/s)exp(-x.Js7DO ) + CS/s (A6) Since the remaining boundary conditions are in terms of flux it is necessary to derive a eXpression for the flux. This can be accomplished by differentiating Equation A6 with respect to g_and evaluating the resulting expression at g_= 0. The final result is 800 5;-' = (- JS/DO )(CO - 03/5) (A7) x=O x=0 Equation A7 can be rewritten in terms of concentration and flux of substance 0 as follows 50 F0 = cg/s - nofo(s)/~/‘s" (A8) where the function f0(§) is used to represent the surface flux of 0 (see Equation 9 Of the text) OCO(x,t) 0 dx fo(t) = D (A9) er The inversion of Equation A8 can be accomplished with tables of Laplace transform pairs and the convolution theorem (15) * t f (T)dT CO X=0 = C0 - 1/’JWDO f0 -:—:—: (A10) 126 At this point the final boundary condition for surface flux (Equation 68 Of the text) can be substituted in Equation A10, which gives * t if(T)dT. c _ = c - 1/nF ~1er I —~———-— (A11) 0 x-O 0 0 O t _ T Treatment Of the equation in CR(x,t) (Equation 60 Of the text) is identical to that for CO(x,t), and the final result is * t if(T)dT CR _ =c +1/nF Jm f —— (A12) X-0 R R o Jt—T? Equations A11 and A12 are general expressions valid for all times during electrolysis. In the case of current re- versal, however, the function if(t) is discontinuous (see Equations 68 and 69 Of the text). Therefore, in applying Equations A11 and A12 to times greater than TF it is con- venient to rewrite the integrals so that integration occurs over the intervals for which if(t) is defined explicitly. Thus, for TthfiflR Equations A11 and A12 can be written as * TF if(T)dT CO x=0 - C0 -1_/ nF’JTrDO f0 -—t—-—: - 1/nFV‘ITDO i (T)dT [t 4L— (A15) TF 4t - T and (.) T i T dT CR x=0 = c; + 1/nNTTDR f F -£-—— + 1/anIerR 0 Jt - T t if(T)dT F 'Jt - T' (A14) 127 At this point Equations A11 and A12, and A15 and A14, can be combined with Equation 4 of the text to give equations valid for‘all times during the experiment. The final results after simplification are ogtng a/z TFmR ) ic(t) (DREW2 ( */ *) - , T = exp -<1(JD C (JD C nFCO (Do ) H/2 nFC6(DO)a/Zk 0 0 R R S exp(anF/chlf: iC(T)dT) [1 - 2iTFJ'P/nrcgJfiS6‘+ 1/nrc34'1rTD—O‘ t i (T)dT t . , *- f0 55:5“:— - eXPI-nF/R'I‘C.’1 f0 1C(T)d1‘) - ( ZiTFfi/nrco JerO ) exP(J‘n‘ch/J5Rc;) eXp(-nF/RTC1f: iC(T)dT) + (exp(JFOCS/JDRCE)/ nFCgJ—N’Do)exp(-nF/RTC1f: iC(T)dT)f: ic(T)dT ] Jt - T (A15) TF_<_t_{_TR G/Z Ri i'I‘F‘mR) R 1C(t)(D) CV2 ,, * nrc0(no) k8 nFCO(DO)a k8 T exp[(anF/RTCl)(féFic(T)dT + 1/le: iC(T)dT)]{-1 + (2iT J?‘/ F F * . * é. . -)(- nrcodwoo‘ - ZITFm/DFCON/W'DO‘) (1 - R1) - ziTFJP/nrcodwno 128 4 I - i - + 1/nFC07/7IDOI.F 1c”)dT + 1/nFCd~/7TDo I: 1c(T)dT o J t -.T‘ EHJt - T‘ T - (exp[(-nF/RTC1)(I F iC(T)dT + 1/le: ia‘T’dT’JII 1 - o F - 21,1. JEF/nrcgd 77130))“; (exp (JFocg/J ch‘ 1:) (ZiTFJF/nrcgfirfic‘) r (1 - 31)) + exp(JDng/ngcg)(2iTéJEVhFC3JTDB) - exP(JDBCS/ T JDEC;)/nFCSJwD6 f F 1c(T)dT - eXp (JDBCS/VDEC;)/nFCSJTDb O Q?T:7? t i (T)dT J: 5: ,_ ] [A16] TF '(t _ T‘ APPENDIX B Relation of theygunction h(y) to Potential The relation of the function h(y) (see Equation 75 of the text) tO potential can be obtained from the following expression (Equation 65) iC(t) = -c1 Qfiéfl- (B1) This equation can be integrated (g(0)=gi) and combined with Equation 75 of the text to give E(t) = Ei - iT(t)/C1f: h(T)dT (32) The constant gi in Equation B2 can be related to the initial bulk concentrations Of 0 and R through the Nernst equation and the following definition of half-wave potential 5° = 2% + (RT/nF)ln ‘(D076R (B5) to give the following relationship valid prior to current reversal T 0_<_tg r n[E(t) - 1.3%] = RT/Flb - 1M]: h(x)dx] (B4) With the changes of variable (Equations 75 and 74 used in the 129 150 text, the final expression becomes ogthF n[E(y) - 8% ] = RT/Ffb - 1M]: h(x)dx] (B5) Similarly, using the approach discussed in Appendix A, the expression for times after current reversal is TthgTR Y n[E(y) - Efij = RT/F[b — 1/onf h(x)dx - Ri/Yfy h(x)dx] Yf (B6) APPENDIX C Computer_§rogram The numerical solution of Equations 80 and 81 of the text was performed on a Control Data 5600 digital computer with a program written in FORTRAN IV. »Since this language is compatible with most modern computers, the FORTRAN source program is listed below. The following data are read in: NRUN, which is the total number of sets Of Y, p, and a used; ERROR, which is the accuracy of the Newton-Raphson iteration; g, which is gi in the text; DELTA, which is 0 in the text; THETA, which is g_in the text; SWITCH, which is the potential at which the current is reversed; PSI, which is Y in the text; and ALPHA, which is a in the text. The output involves print- ing of the above data followed by the values Of N(E - E§)' y, H(y), IF/IT (the faradaic current efficiency), ITERATIONS (the number of Newton-Raphson iterations), QC/QF (ratio of the coulombs of electricity used by the charging process to that used in the faradaic process) and N X DERIVATIVE (derivative of the potential-time curve at each point on the curve). In addition, the program constructs tangents at the initial portion of the curve, and at a point just prior to current reversal, and it also constructs the Berzins and Delahay line 151 152 which intersects the curve at gF. 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