momma Q mmvmsm A 0F. 6 :...... L i 1.6%}; QMEASUR i .5. mamas 31' V f; Er mam 551M fiN‘DiAPPLSCATa‘GNS r the .fiegrewf PM); 0 .5 S m.» ‘7‘“ mm M! THEORY , c-g-uo.~ ¢ ~.n» a... Lh!:¥q.;&.u)rnx {In .. {1/ a» 7.. "41% fro—I dfiflnm . . 7/}. link”. ryxm z, 5...; W; 134%.Ivar/a. . t .u 1...;..v.v.!.:b.v:v...hvol.x.:.(/.r1,—\L.t 1.:K damn. £4.11... , V... stuuuvhwf. fa This is to certify that the thesis entitled THEORY AND APPLICATIONS OF COULOSTATIC KINETIC AND ANALYTICAL MEASUREMENTS presented by JANET KUD IRKA has been accepted towards fulfillment of the requirements for _E\\. D;demem_flhm;§1rg @124; _ . . //1 V Major professor Date_Q Lei-1 519/. i 7/ U / 0-7639 : 'l '1' ‘ . 'vl ~ LIBRARX Michigan State University recur-1 : «magnum-W5 ABSTRACT THEORY AND APPLICATIONS OF COULOSTATIC KINETIC AND ANALYTICAL MEASUREMENTS By Janet Kudirka Coulostatic Data Analysis The errors involved in the various methods of analyzing the data of the current impulse and coulostatic techniques when the relaxations are neither charge transfer nor diffusion controlled are discussed as a function of the ratio of the charge transfer and diffusional time constants TC/Td and apparent rate constant k°a. The validity of the application of the simple charge transfer approximation is found to be dependent on the time at which experimental measurements are obtained as well as rC/rd. The accuracies of nomographic, complex plane, and curve fitting techniques of correcting the data for the influences of diffusion are discussed. The accuracy of the first is found to be dependent on an accurate knowledge of the capacitance and short time measurements. The second provides accurate results but requires a strong knowledge of mathematics to be useful. The third is found to be accurate providing a sufficient portion of the observed decay is partially charge transfer controlled. Graphs are presented which show the measurement or analysis requirements for experimental data and which can provide good error correction Janet Kudirka estimates in the mixed rate region. It is found that only a lower estimate of the exchange current can be obtained by even the most SOphisticated methods of analysis (322: curve fitting and graphical) if data are not obtained at times when the decay is partially charge transfer controlled. The linearized form of the overvoltage time relationship for coulostatic kinetic measurements, which in the past was known valid only for overpotentials of approximately 3/n mV, is shown to be applicable to overpotentials as high as 25/n mV. Experimental results with the use of high overpotentials on the hexacyanoferrate (III/II) couple are presented. Charge-Step Polarography Fundamentals of a new electroanalytical technique called charge-step polarography are presented. This method combines the advantages of the coulostatic analysis technique with the simplicity of operation and selectivity of dc polarography. The most favorable concentration range is 10"5 to 10-7 M, and the method appears applicable to any substance which is reduced or oxidized under polarographic conditions. A DME is held at a potential where there is essentially zero faradaic current. Late in the life of the drop, a small charge is quickly added to the electrode. The resulting potential-time curve is logged into a small digital computer which calculates the slope and intercept (E at t=0) for the first portion of the E X§.t1/2 curve. When the charge is sufficient to polarize the electrode at a potential approaching the polarographic El/Z for a substance in solution, the lepe will increase. A plot of the slope/intercept values for incrementally increasing charge steps applied to successive drops at a DME Janet Kudirka resembles a smoothed dc polarogram and allows both qualitative and quantitative analysis of mixtures of electroactive substances. Application of this method to zinc, cadmium, and nickel solutions shows that accurate results can be obtained for 6 M, but interference from the concentrations as low as 1 X 10- reduction of approximately 5 X 10-7 M residual oxygen in the background prevents obtaining accurate results at lower concentrations. Results of three variations of this method are also presented. In one case involving a mixture of substances, the electrode is polarized at a potential on the plateau region of the substance which is reduced first, and only the second wave is observed. Another case involves polarizing the electrode at a very cathodic potential so all substances in the solution are reduced and then pulsing anodically to observe their oxidation. A third variation involves incrementing the polarizing potential instead of the applied charge to obtain differential charge-step polarograms. THEORY AND APPLICATIONS OF COULOSTATIC KINETIC AND ANALYTICAL MEASUREMENTS By '\\F. ~\‘\r I Janet‘Kudirka A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1971 / '7 , , ’ r I " .‘r' l4 I ' “ V ; 1‘ I ACKNOWLEDGMENT The author wishes to express her appreciation to Professor Christie G. Enke for his guidance and encouragement throughout this study. Thanks are also given to Dr. Stan Crouch for acting as second reader and Dr. Bob Bleasdell for proofreading. She would also like to thank Dr. Joseph Cardarelli for the use of his computer interface and Dr. Roger Abel for his programming. Thanks are also given to Paul J. Kudirka, the author's husband, for his encouragement and understanding. ii TABLE OF CONTENTS I 0 INTRODUCTION 0 0 0 0 o o o o o o o o o o o o o o 0 II. THEORY FOR STUDY OF ELECTRODE KINETICS BY THE COULOSTATIC METHOD . . . . . . . . . . . . . III. A COMPARISON OF COULOSTATIC DATA ANALYSIS TECHNIQUES . A. Charge Transfer Approximation Errors in Capacitance by Charging Slope . Determination of Exchange Current. . Errors in Metal/Metal Ion System . . Errors in a . . . . . . . LIIJ-‘UJNH Nomographic Method of Coulostatic Analysis . . Complex Plane Analysis . . . Curve Fitting Method of Coulostatic Data Analysis . Graphical Method of Coulostatic Data Analysis NUOW IV. EXTENSION OF THE CURVE FITTING TECHNIQUE TO LONG TIMES AND HIGH OVERPOTENTIALS . . . . . . . . . . . . . A. Extension of Fit of Ferri-ferrocyanide Curves to Long Times- B. Use of High Overpotentials in the Coulostatic Method 0 . . . . . . . . . . . . . . . V. THEORY AND HISTORY OF ELECTROLYANALYSIS USING THE COULOSTATIC METHOD . . . . . . . . . . . . . . . . . . . VI. CHARGE-STEP POLAROGRAPHY . A. Introduction . . . B. Experimental - - - . . . . . . . . l. Instrumentation . . 2. Cell and Electrodes . 3. Reagents and Solutions . . . . . . . . . . . C. Results and Discussion LITERATURE CITED AND BIBLIOGRAPHY . APPENDICES . . . . . iii . Errors in Capacitance by Discharge Extrapolation . . 25 . 40 . 41 . 48 80 Page 12 13 18 21 31 34 39 53 53 . 57 . 63 71 71 74 74 81 81 99 .101 LIST OF TABLES Table Page I. Actual Bulk Concentration Compared to that Obtained by Curve Fit with Hg(I)/Hg System . . . . . . . . . . . . . 45 II. Capacitance and Exchange Current Determined from Curve Fit of Entire Decay Curve as Compared to those Obtained from Fit of Initial Portion of Decay Curve . . . . . . 56 III. Comparison of Linearized and Non-Linearized Coulostatic Decay Curves as n/nt=O . . . . . . . . . . . . . . . . . 59 IV. Variation of Slope in the Plateau Region for 1 X 10—5 M Cd++ with Changing Concentration of Supporting Electrolyte . . . . . . . . . . . . . . . . . . . . . . 89 -6 . E3/4-E1/4 for 5 X 10 M Concentration of Each of the Ions . . . . . . . . . . . . . . . . . . . . . . . . . . 90 iv Figure LIST OF FIGURES Page Electrical analog for a current impulse experiment. , , 11 Theoretical decay curves with CO = CR = 10-5m/cm3, D0 = D = 10-5cm2/sec, T = 300°K, n = l, and Cd = 2.0 X 15-5 F/cmz. Example 1 is for Tc/Td = 2.5 and shows how the measured exchange current and capacitance can vary with extrapolation from various time regions. Example 2 is for TC/Td = 25 and is essentially linear over the time range shown . . . . . . . . . . . l7 Ratio of discharge capacitance (determined by simple charge transfer approximation with extrapolation from various ratios of t/rd) to the actual capacitance as a function of TC/Td or C/kg. . . . . . . 20 Ratio of charge capacitance determined by obtaining the slope of the charging curge at various t/Id's as a function of TC/Td or C/ka. . . . . . . . . . . . . 24 Ratio of exchange current determined by the simple charge transfer approximation with extrapolation from various ratios of t/Td to the actual exchange current as a function of TC/Td or C/kg. . . . . . . . . 27 Exchange current determined by the simple charge transfer approximation and extrapolating between .25 and .45 usec as a function of apparent rate constant for CO = CR from 10'5 to 10'8 m/cm3 . . . . . . . . . . . . 30 Exchange current using the simple charge transfer approximation and extrapolating between .25 and .45 usec as a function of ap arent rate constant for the case CR = l and CO = 10’ to 10'8 m/cm3 . . . . . 33 Plot of log C0 3§_log I0 to determine a at various kg's for the case where CR = l and CO varies . . . . . 36 Exchange current using the simple charge transfer approximation and extrapolating between .25 and .45 usec as a function of apparent rate constant for the case where C = 10‘6 m/cm3 and CO = 10'5 to 10'8 m/cm3. . . I . . . . . . . . . . . . . . . . . . . 38 10. 11. 12. l3. 14. 15. l6. 17. 18. 19. 20. 21. 22. 23. Experimental and theoretical relaxation curves for the hexacyanoferrate (III)/(II) couple in 1 M KCl, C = 0.01 M, C = 0.01 M . . . . . . . . . . . 0 R Experimental and theoretical relaxation curves for the Hg(I)/Hg system in 1.0 M HC104, CO = 0.001 M . Ratio of exchange current determined by the simple charge transfer approximation with extrapolation from various ratios of t/Td to the actual exchange current as a function of (TC/T d)exp' Experimental and theoretical relaxation curves for the hexacyanoferrate (III)/(II) couple in 1 M KCl, C 1 x 10‘2 M, cR = 3 x 10‘3 M. . . . . . . . . . . .0 Experimental and theoretical relaxation curves for the hexacyanoferrate (III)/(II) couple in 1 M KCl for high overpotentials . . . . . . . . . . . Charge-step polarogram for a mixture of 5 X 10-6 M Cd++ and s x 10-6 M 2n++ in .01 M KCl. Circuit diagram for drop fall detector . Block diagram of circuit used in charge-step polarography . . . . . . . . . . Diagram of cell and reference electrode Charge—step polarograms for 5 X 10-.6 M Cd++, zd++, and Ni++ in .01 M KCl . . . . . . . . . . . Slope vs concentration for various metal ions Charge-step polarogram for 4-e1ectron reduction of oxygen 0 O O O O O O O O O O O O O I O O 0 O 0 O . Charge-step polarograms for oxidation of 5 X 10-6 M Cd++ and 5 x 10-6 M zu++ in 0.01 M KCl . . . . . Differential charge-step polarogram for 5 X 10“6 M Cd++ and 5 x 10-6 M 2n++ in 0.01 M KCl vi . 44 . 47 . 50 . 55 . 61 . 73 . 76 79 83 86 88 92 95 . 98 I. INTRODUCTION In 1962 Delahay (l) and Reinmuth (2) simultaneously introduced what they termed the "coulostatic impulse method" for studying electrode processes. (However, the method had been postulated by Barker (3) prior to this.) The principle of this method is as follows: The charge on the working electrode of an electrochemical cell is changed in a short time (a few tenths of a microsecond to a few milliseconds) with an instrument which allows flow of the charging current but prevents flow of the reverse current. Charging is achieved, for instance, by discharge of a capacitor, initially charged, at a known voltage, across the e1ectro~ chemical cell. The potential, E0 of the working electrode, before charging, is at its equilibrium value. The cell is at open circuit, for all practical purposes, after charging, and the faradaic current is entirely supplied by discharge of the dodble layer capacitance. The potential tends to return to its initial value before charging as the double layer capacitance is progressively discharged. Potential-time variations depend on the coulombic content of the charging current, the double layer capacitance, and the kinetics of the electrode process. Study of kinetic processes from overvoltage-time curves is therefore possible. This impulse technique offers several advantages over the step relaxation techniques commonly used to study rapid charge-transfer reactions. First,measurements are made at times during which no appreciable net current passes through the cell so that no corrections for ohmic potentials are needed in interpreting results. Second, there is no forcing function present during the kinetic measurement. The cell relaxes to its equilibrium value at its own rate unhindered by external factors so the same experimental conditions can be applied to relaxations of very different time constants. Third, the experimental requirements are relatively simple. Delahay (4) also proposed the use of this method to determine concentrations in the 10-5 to 10-7 mole/liter range. In this case the charge supplied to the working electrode is such that it brings the po- tential into a range in which the faradaic current quickly becomes diffusion controlled. Potential-time variations depend on the double layer capacitance and the magnitude of the faradaic current. Since the faradaic current generally depends on the concentrations of the substances involved in the electrode reaction, these concentrations can be determined, in principle, from the potential-time variations. In this method the difficulty resulting from the double layer, which prevents application of polarography to trace analysis, is entirely avoided. Instrumentation is much simpler than in some other electroanalytical methods for trace determination such as square— tmave polarography, pulse polarography, and faradaic rectification. The theory and general equations for application of the coulo— static method in each of these cases is presented below. II. THEORY FOR STUDY OF ELECTRODE KINETICS BY THE COULOSTATIC METHOD The rate of an electrode reaction represented by the symbolic equation, 0 + ne- + R, involving only a single rate-determining step is: if = nF{C0k;exp(-anFE/RT) - Cngexp[(l-o)nFE/RT]} (l) where the notations are as follows: if, the current density, which is positive for a cathodic reaction; n, the number of electrons in- volved in the electrode reaction; F, the faraday; R, the gas constant; T, the absolute temperature; CO and CR, the concentrations of substances 0 and R, respectively, at the electrode-solution interface; E, the electrode potential referred to the normal hydrogen electrode; a, the transfer coefficient; k; and kg, the formal rate constants at E = O for the forward and backward reactions, respectively. Equation (1) is written on the assumption that the number of electrons involved in the rate-determining step is equal to the number of electrons in the overall reaction. Furthermore, it is assumed that substance 0 is soluble in solution, and that the reduction product is soluble either in solution or in the electrode as in the decomposition of an amalgam- forming metal on a mercury electrode. The modification of Equation (1) éflld the subsequent treatment are trivial when R is an insoluble species Onetal). '4- r».. At the equilibrium potential Ee the current is equal to zero, and Equation (2) applies 0 o o o Cokf exp(-anFEe/RT) - CRkb exp[(l-a)nFEe/RT] (2) where C8 and c; are the concentrations at equilibrium. The quantities equated in (2) can also be written in the form 08(1-a) cgak; where k; is defined by k; = kgexp(-anFEo/RT) = kgexp(l—a)nFEo/RT (3) Eo being the standard potential for the system 0 + ne- = R. In view of Equations (2) and (3), the current-potential relationship (1) can now be written in a form containing only the rate constant k°, a = an2C0(l_a)Coa(CO/C8)exp[-anF(E-Ee)/RT] - (CR/C;)exp[(l-a)nF(E-Ee) 1 o R f /RT] (4) The product 0 o(l-a) oa _ o anaCO CR - I (5) is the exchange current density, and can be used to characterize the kinetics of electrode reactions. The main drawback in the use of 0 exchange current density results from the dependence of I on the equilibrium concentrations C3 and CE, i.e., on the potential. This is not the case for the rate constant k° which is therefore more character- a iSt1c of the kinetics of electron transfer. The total current density it is the sum of the faradaic component if corresponding to the electron transfer and ic, the current density used to charge, or discharge, the double layer capacitance. The faradaic component is given by Equation (4). The capacitor charging current density is 1C = —Cd d(E-Ee)/dt (6) where Cd is the differential capacity of the double layer. Since the sum of the faradaic and charging currents is constant 1 = i + i (7) The potential-time relationship is derived by solving Fick's equations for linear diffusion for the following initial and boundary conditions 0 _ o _ C0 = C0 and CR - CR for x :_O and t - 0 (8) C + Co and C + C0 for x + w and t > 0 (9) O O R R ‘— it = -Cd d(E-Ee)/dt + nFDO(8CO/BX)X=O (10) nFD (ac /ax) = I°{c /c°exp[-anF(E-E )/RT] - c /c°exp O O x=0 O 0 e R R [(l-a)nF(E-Ee)/RT]} (11) DO(3C0/8x)X=O = -DR(3CR/8x)X=O (12) where DO and DR are the diffusion coefficients for O and R, respectively, X, the distance from the electrode, and t, time. Condition (10), in Rfllich the second term on the right-hand side is the faradaic current density written in terms of the flux of substance 0 at the surface of the electrode, expresses the condition that the total current density is constant. Condition (11) expresses the equality of the flux of substance 0 at the electrode surface to the faradaic current density. Finally, condition (12) expresses the equality of the fluxes of substances 0 and R at the electrode surface. Relating the above expressions it is found that the general equa— tion describing the current-potential-time relations for a simple redox system, with both species soluble, with mass transport by semi- infinite linear diffusion, and with charge-transfer governed by the absolute rate theory expressions is: t — it + Cd(dE/Dt) = an;{c8 - [nF(flDO)l/2] 1 fo[it + Cd(dE/dt)](t-T) l/ZdT} ex [( -anF/RT)(E - E°)] - an°{c° + [nF(nD )1/21’1 It [i + c (dE/dt)] p a R R 0 t d (t - T)-l/2dr}exp[(1—a)nF/RT(E-EO)] (13) where T is a dummy variable of integration. In theory the equation for the exact form of the current pulse should be substituted into Equation (13) for exact solution. However, if times of observation are large compared with the impulse half— width (which is generally true experimentally), the current can be described in terms of the Dirac delta formalism,_ite., i = q6(6,t) where 6 = 0 for t k 8 and IOSdt I l (14) where 6 is the time of application of the impulse (for present pur- poses 6 = +0) and q is the charge passed during the impulse. The analytical solution of Equation (13), with the substitution of Equation (14), to yield an expression for the potential-time relation- ship in closed form is possible only when experimental conditions justify reduction of Equation (13) to the form of a linear integro- differential equation with constant coefficients. This is the case when the excursion of E during the experiment is sufficiently small that factors of the type exp(anFn/RT), (where n is the change of potential from its initial value, Ei’ i,§,, E = Ei + n) can be adequately represented by the first two terms of their Taylor expansions. In general, linearization is justifiable when nFq/CdRT<erfc(8:1/2) -BexP(Y2t)erfC(Ytl/2) <19) where B,Y = Ti/Z/ZTC i ("rd/4TC - l)l/2/T:/2 (20) where the plus sign in Equation (20) is associated with B and the minus sign with 7. Equation (19) is of relatively cumbersome form for direct applica- tion to experimental data, since 8 and y for many cases of experimental interest are complex quantities. Because of this, the two limiting extremes in the relaxation process (Efff’ Td >> TC or Td << Tc) are generally studied. In the case where Td >> TC Equation (19) reduces to: /2 exp(t/Td)erfc(t/Td)1 (21) n " nt=0 In this case the relaxation is entirely diffusion controlled, and no charge-transfer kinetic parameters can be determined so it is not of great interest. In the case where TC >> Td Equation (19) reduces to: o n - nt=0exp(-t/TC) - nt=0exp[(-RTCd/nFI )t] (22) Hence, in this case, the overpotential decays exponentially with time, and a plot of log (n) against t is linear. 10 can be computed from the slope of this plot, and the transfer coefficient, a, can be determined from the variations of I0 with the concentrations of O and/or R (Eqn. 5). The differential capacitance Cd’ which is needed in the computation of Io can be obtained from Equation (17) since the amount of charge injected is known and nt=0 can be obtained by extrapolation of the log (n) 3s t plot back to zero time. Thus, by assuming charge-transfer control, kinetic parameters can easily be determined from potential-time curves. Weir and Enke (5) modified the coulostatic impulse method to give a method termed the current impulse technique. The relaxation curves produced by this technique are of the same mathematical form as those produced by the coulostatic technique. However, the form of the perturbing impulse is a square wave, and this allows a separate measurement of the capacitance since the double layer charges linearly. From the slope of this charging curve and the magnitude of the applied current, the charge capacitance can be calculated. This method is shown in Figure 1. In both the coulostatic and current impulse techniques most systems will yield relaxation curves which are neither purely charge- transfer nor purely diffusion controlled. Thus Equation (19) should be used to analyze the data. Various methods exist for analyzing data of this type. These methods will be discussed and compared in this thesis. 10 Figure 1. Electrical analog for a current impulse experiment. 11 Current Pulse __1, _____.i 1' F_______. applied to Cd at RS W RAN-12m“) Rf Cd - Double layer capacitance Rf - Faradaic resistance RS - Solution resistance Zm(t) — Mass transport impedance Observed Response /(d7]/dt)= W00. 77 = 77t=oexp(-t/Rde) Figure l III. A COMPARISON OF COULOSTATIC DATA ANALYSIS TECHNIQUES In their original papers on the coulostatic method, Delahay and Reinmuth (1,2) proposed several conditions which must be met for the application of either the charge transfer (Eqn. 22) or the diffusion limiting (Eqn. 21) equations to experimental relaxation data. Unfor- tunately, some of the electrochemical reactions which have been studied with this technique produce relaxations which are neither purely charge transfer nor purely diffusion controlled, but a com— bination of the two. Weir and Enke (6) and Daum and Enke (7) have said that satisfactory estimates of the charge transfer parameters can be obtained from decays of this type by obtaining the slope of the log(n) vs. time curves at times sufficiently short to apply the simple charge transfer assumption. However, this assumption has not been subjected to a rigorous mathematical analysis, and, correspondingly, there has been some concern in the literature about the validity of some of the kinetic parameters which have been obtained by this method (8). There have been several attempts to correct relaxation data for mass transport phenomena (9,10), but all of these techniques require a significant amount of tedious calculation, since the arguments of the general function which includes mass transport become complex in the region of greatest experimental interest. The purpose of the following discussion is two-fold: 1) to clarify some of the ambiguities involved in obtaining kinetic data from this measurement technique by discussion of some of the determinate errors which are 12 13 present in the various methods of analyzing the relaxation data as a function of the ratio of the charge transfer and diffusional time constants; and 2) to suggest techniques which experimenters can use to minimize their errors and/or reduce the complexity of data analyses, or, alternatively, to determine the simplest analytical technique that can be applied to a given set of experimental data. This discussion of determinate errors is based on an analysis of the current impulse technique. A. Charge Transfer Approximation The simplest and most direct method of estimating the reaction rate from current impulse relaxation data is to assume the simple charge- transfer limiting equation. Application of this assumption to data which is simultaneously charge transfer and diffusion controlled obviously presents some difficulties, but these difficulties can be minimized by paying proper attention to the way in which the measure- ments are made. Tied inextricably to the accuracy of these kinetic measurements is the accuracy of the value of the double layer capacitance. There are a variety of methods available for the experimental determination of the capacitance under nonreactive conditions,_igg., when the electrode is ideally polarized. Many investigators have assumed that the capacitance for an electrode at a certain potential and in a specified supporting electrolyte should be the same regardless of whether the electroactive species is present or not. There is no a_priori reason to believe this is true in all cases. 14 For example, if specific adsorption of the electroactive species occurs, then the capacitance may differ widely from its value with no electroactive species. Or, if solutions of extremely high concen- trations of electroactive species are studied, the capacitance may change because of the addition of these ions to the double layer. Furthermore, in the case of solid electrodes, the value of the capacitance is dependent on small additions of oxide or other films to the electrode surface and is particularly sensitive to the adsorption of any foreign organic species in the solution. It is important, there— fore, to make the measurement of the double layer capacitance in the same solution and at the same time that the relaxation measurement is made. The value of the capacitance can be estimated from coulostatic data by extrapolation of the log(n) v§_ time curve to zero time and calculating from the relationship Cd = Aq/nt=0. The current impulse method provides an additional estimate from the slope of the charging curve, since Cd = it/(dn/dt), where it is the magnitude of the applied current. Both of these measurements can involve some very large determinate errors depending on the time at which the measurements are made, the definition of zero time, and the ratio Tc/T of the charge d transfer and diffusional time constants. For example, with CO = CR = 10-.5 m/cm3, D0 = DR = 10—5 cmz/sec, T = 300°K, and C = 2.0 x 10“5 farads/cmz, T is .1149 usec. If we d d assume an exchange current density of 1.8 A/cmz, corresponding to a rate constant of 1.86 cm/sec, and assume a = .5, then rC/rd = 2.5, 15 and, using the basic equation including both charge transfer and mass transport control, a relaxation curve can be calculated. The log n/nt=0 3§_ time plot for this relaxation curve is shown in Figure 2, example 1. Using the charge transfer approximation, lines are drawn to determine the capacitance from the intercept and the exchange current from the slope and the capacitance value. As is shown in Figure 2, the exchange current and capacitance will vary greatly depending on the time the line is drawn and extrapolated back to zero time. The line drawn between .056 and .1149 usec, or between .49 and 1.0 of dimensionless time t/Td, extrapolates back very close to what nt=0 should be, but data are not usually obtainable at these short times. If the extrapolation is from the section of the decay curve between .1149 and .2584 usec, or between 1.0 and 2.25 t/Td (which are about the shortest times at which data can presently be obtained), the result is still quite close to the actual value. If the extrapolation is between .2584 and .4594 usec, or between 2.25 and 4.0 t/rd, the error is considerably enhanced. Using the same example as above but with I° .18 A/cmz, correspond- ing to a rate constant of .186 cm/sec, and TC/Td = 25, the relaxation curve shown as example 2 in Figure 2 is obtained. For this example, even extrapolation of the line between 2.25 and 4.0 t/rd yields accurate values of slope and intercept, and the charge transfer assumption is valid. The determinate errors can also be determined in terms of the rate constant and concentration by relating the equation for TC/Td and the equation I0 = an2 Cé-acg, where I0 corresponds to exchange current 16 Figure 2. 10-5m/cm3, D O=DR= = 2.0 X 10--5 F/cmz. Example Theoretical decay curves with CO = CR 10-5cm /sec, T 300°K, n = 1, and Cd 1 is for TC/Td 2.5 and shows how the measured exchange current and capacitance can vary with extrapolation from various time regions. Example 2 is for TC/Td = 25 and is essentially linear over the time range shown. 17 com: 55:. Q m. w. m. o _ _ e _ m. BEESQ .I V. In. / I O. I 5. I'm. a x 1.0. a a. as m I O: 18 density in A/cmz. With CO = CR = C the resulting equation: rC/Td = (nzeD/4RTCd)(C/k:) shows the ratio TC/Td to be directly proportional to the ratio C/kZ. With D0 = DR = D = 10-.5 cmz/sec, a = .5, Cd = 20 MF and n = l, _ 5 o TC/Td - 4.6652 x 10 (C/ka). 1. Errors in Capacitance by Discharge Extrapolation First, the errors in determination of the capacitance using the extrapolation technique will be examined. Figure 3 shows the ratio of the measured discharge capacitance to the actual capacitance as a function of log (TC/rd) or log(C/k;). The ratios in this figure were calculated by first using a given value of capacitance and various values of exchange current to obtain various values of rc/r and the d basic equation involving both charge-transfer and diffusion control to calculate theoretical decay curves. Then, the slopes of the log(n) vs t curves from the theoretical decays were obtained between various times or ratios of t/rd and extrapolated to zero time to find the intercept. The value of the capacitance determined from this inter- cept was then compared to that used to calculate the theoretical decay curve. The errors in the capacitance become larger as the ratio (TC/Id) or C/ka decreases and, for given values of TC/Td or C/k;, the error is larger for extrapolation from larger t/rd. Thus, for the first example with Tc/Td = 2.5, the capacitance value is 1.025 to 1.2 times the actual value depending on the portion of the curve extrapolated. 19 Figure 3. Ratio of discharge capacitance (determined by simple charge transfer approximation with extrapolation from various ratios of t/Td) to the actual capacitance as a function of TC/Td or C/k;. 20 28.8.3 03 - ,. o chi—vquoS UhVVDhm: I3 new: mvuvnh I Gm thvvuvrtfi m I me I ow .2 _ m E: can own or... m 1.9.633 lenloe 3 /dxe 3 aouenoedeo 86.:erng 21 However, for the second example, with Tc/Td = 25, the capacitance is close to the actual value even on extrapolation from between 2.25 and 4.0 t/Td. 2. Errors in Capacitance by Charging Slope Determination of the double layer capacitance from the slope of the charging curve and the magnitude of the applied current is also subject to some rather large errors, the origin of which become obvious if a simple model of an electrochemical cell as shown in Figure l is considered. Phenomenologically, the presence of the fara- daic reaction in parallel with the double layer capacitance consumes some of the current which is being applied to the cell, and the slope of the charging curve becomes progressively less as time proceeds. These effects become more pronounced as the faradaic resistance becomes smaller. Thus, in order for a slope measurement to be valid, an insignificant amount of the charge must have been consumed by the fara- daic process up to the time of the slope measurement. The errors in the charge capacitance can be formulated in terms of the same variables as the errors of the discharge capacitance by invoking the methodology of the galvanostatic technique (10)- The basic equation derived by Delahay and Berzins to describe galvanostatic processes is: (2)+(2s(t/n>1/2 - 1)] / l2 n = it/cd(y-s>{y/eztexp(szc)erfck / + ...1, <26a> and for short times, 133: small values of k: n/ 1 - [(3 4.Y )2/sy1k2 + 4/3n1/2[(B + y)"/82y2]k3 + (26b) nt=0 = The right hand side of Equations (26) are then calculated for several values of (B-+ y)2/By = Td/Tc as a function of the parameter k = a2t = t/rd. These values are tabulated over a large range of (81+ Y)2/8Y or Ta/Tb and azt or t/Ia. To obtain a value of exchange current density 1/2 1/ R by using the table one must first determine (l/CODO 2) and + 1/CRD 40 Cd from other methods to obtain a value for a = Idl/Z. For a point of the rht curve the value of rVn can be calculated and compared with t=0 the value in the table. From the (8-+ y)2/8Y or Td/Tc value obtained, the exchange current density can be calculated from the relation: 1/2 R 1/2 0 >'1uao mHo:3 uHm o>uso mafia uuonm z .cowumuucmucou .w>uso known mo nowuuom HmHuH:H mo ufim aouw wmcwmuno mwonu ou woumafioo mm m>uso known muwucm mo ufim o>u=o aoum vmawahwuma ucmuuso owsmnoxm cam mocmufiomamo .HH magma 57 B. Use of High Overpotentials in the Coulostatic Method In deriving Equation (19) it was necessary to expand exponential terms in a Taylor series and retain only the first two terms. For this approximation to be valid at the 1% error level the argument of the exponentials must be less than 0.15. In terms of Equation (19) this corresponds to n values of less than 3/n mV from equilibrium, thus limiting the coulostatic method to small potential excursions. By solving the boundary value problem without introducting linearization, Nicholson (17) has found that for the case of pure diffusional relaxation the exponential terms are of minor importance in the final result so that larger overpotentials can be used without increasing the error level. The results agree with linearized theory for overpotentials as large as 25/n mV. To determine if the same is true of the case for mixed charge transfer and diffusional controlled relaxation, it was decided to solve the general equation for this case without introducing linearization and compare the results of this equation to those using Equation (19). By substituting Equations (10) and (11) into Equation (13) the general equation describing the potential-time relation for a simple redox system is obtained: 1/2 Tr1/2 / [fotdn/dt dT/Vt - T - nt=0ft06(6,t)dr//t - r - RT/nFexp[(nF/RT)n] - (28) Tc(dn/dt) - Tent=05(9.t) = exp[(-anF/RT)n]{RT/nF — To expl(nF/RT)anR1/2/n1/2[fotdqflt drl/E:? - nt=0fot6(e,t)dT/¢t-T} where Td1/2 = Tol/Z + TRl/Z and the other terms have been previously 58 defined. Now substituting ¢(t) = (nF/RT)n and wi = (nF/RT)nt=0 into Equation (28), replacing t with A and integrating from zero to t, a nonlinear integral equation is obtained: 2/twi - fotq;(}\)d)‘//t—Hf=nl/2cht expchMden/dxflx o TO1/2 + TR1/2 exp[W(A)] (29) _ magi“. expmowwam . “2 tieapmml — 11cm 0101/2 + TRl/Z exp[W(A)] o 101/2 + TRl/Zexp[¢(x)] By reducing this to dimensionless form and using the method of numerical solution (17), curves can be obtained which follow this equation. Typical results from computer solution of the nonlinear integral equation (Equation (29)) and Equation (19) are given in Table III for various ratios of Tc to Td. These examples cover the range from almost complete charge transfer control (TC/1d = 50) to almost complete diffusion control (Tc/Td = .05). For n = l and nt=0 = 25.6 mV the linear and nonlinear relaxation curves agree within 2%, thus, it appears that higher overpotentials can be used in the coulostatic method where either or both charge transfer and diffusional relaxation are occurring without introduction error. To test the use of the linearized equation for large overpotentials, the hexacyanoferrate (III)/(II) couple was studied. Experimental conditions were as given by Daum and Enke (6). Overpotentials from 4 to 25 meere used and capacitance and exchange current calculated using the computer curve fitting program for Equation (19). Figure 14 is an example of the N .19 .N «N ~33? 59 ooa. mom. me. mmm. mmq. mom. own. moo. oqw. mma. mH .ccm ummcfianco: mo. om p o moH. omH. me. mmm. mom. NNN. mNN. mmq. omN. mmm. mmm. “mm. omm. mom. mom. mac. mms. «Hm. MHm. Nsw. ame. Hoe. mom. «mm. mmm. NON. no“. smm. Noe. “mm. me. cam. mam. New. Hem. Nam. “mm. Nmm. Nmm. was. .cvm mH .ccm om .aum oH .cum mocaa pmmcwancoc umocwa ummcwalco: m. m Nmm. omq. mmm. woo. Hem. Now. 0mm. Cum. Nam. mam. o~ .ccm ummcwa mew. mww. on. mmm. owm. mwm. mmm. mmm. mom. mam. sH .aum ummswalcoc on New. www. mNm. nmm. 0mm. mmm. mmm. mmm. mmm. mom. om .cam ummcwa v oo.m m~.o oo.¢ mN.~ oo.H so. om. ca. «0. Ho. vp\u o p\ p nuc\s mm mo>uso known ofiumumoHsoo cmuaummcaguaoz vcm noNfiummcHA mo somfiummaoo .HHH maan 60 Figure 14. Experimental and theoretical relaxation curves for the hexacyanoferrate (III)/(II) couple in l M KCl for high overpotentials. _ -5 3 CO — CR - l X 10 m/cm Cd = 28.6 uF/cm2 1° = 0.235 A/cm2 Experimental calculated from Equation (19). 61 .ca musmfim Aw I ./ Glow 9mm mm 62 results which were obtained at higher overpotentials. Thus, overpotentials as high as 25/n mV can be used in the coulostatic method and accurate results obtained. V. THEORY AND HISTORY OF ELECTROANALYSIS USING THE COULOSTATIC METHOD The principle and theory of applying the coulostatic method to analysis of substances in the 10"5‘-10-7 M concentration range was deve10ped by Delahay (4,18). Equations for potential-time variations were derived for different types of electrodes and various electrolysis conditions. The basic example of a plane electrode on which a substance is reduced or oxidized with mass transfer controlled by semi-infinite linear diffusion will be presented here. The current-potential curve for any given time after the beginning of electrolysis of a substance generally exhibits a plateau at which the current density Id is generally prOportional to the bulk concentration of reducible or oxidizable substance. For diffusion control: Id = : nFC*(D/Trt)l/2 (30) where the + sign applies to a cathodic and the - to an anodic process, t is the time elapsed since the beginning of electrolysis, and the other terms have been previously defined. The potential of the electrode is initially set with a potentiometer at a value E1 on the foot of the I-E curve. A charge is supplied to the electrode in a short time such that the potential is brought to a value Ed on the plateau of the I-E curve. The cell is essentially at open circuit after charging, and consequently the faradaic current is only supplied by discharge of the double layer. The 63 64 charge on the electrode at potential E and time t is c(E)(E-Ez), where c(E) is the integral capacitance per unit area of the double layer at potential E, and E2 is the point of zero charge. If the time required for charging from E to Ed is negligible in comparison 1 with the values of t being considered, the variation of potential from Ed to a value E is c(E)(E - E2) - c(Ed)(Ed - E2) = foti(E,t)dt (31) If we consider the case where E is some value on the plateau of the I—E curve, i.e., the discharge of the double layer is occurring at constant current, we can substitute Equation (30) into Equation (31) and obtain: c(E)(E — Ez) - c(Ed)(Ed - E2) = i2nFC*D1/2tl/2/n1/2c (32) If c is independent of potential over the interval AB = E - Ed, Equation (32) reduces to: :AE = 2nFc*D1/2tl/2/n1/2c (33) 1/2 The potential varies linearly with t , and concentration C* can be determined from the slope of the AE vs tl/2 plot. The double layer capacitance can be computed from the shift of potential at t = 0 from E1 before charging to Ed just after charging. Thus Aq = C(Ed - E1) (34) where Aq is the known charge increment supplied to the electrode. 65 If a spherical electrode is used, the corresponding equation for diffusion current should be substituted in Equation (31). The E-t relation in this case is: 1/2 1/2/"1/2 :AE = 2nFD C*t c + nFDC*t/rc (35) where r is the radius of the sphere in cm. From Equations (34) and (35) it can be deduced that: AE __§222£s = 1 + 1,1/2131/2t1/2/2r plane (36) and hence experimental AE's for a spherical electrode can be reduced to the correSponding AE's for a planar electrode if D and r are known. The concentration can be obtained from the plot of the corrected shift of potential 3g t1,2 as for the planar electrode. This correction is only necessary for t exceeding a few tenths of a second with the usual drOp size of r N 0.05 cm. By analogy with the diffusion current constant of classical polarography, a decay constant, A, can be defined: x = 2nFDl/2/nl/2 (37) This decay constant will be characteristic of a substance for reduction or oxidation under given electrolysis conditions (supporting electrolyte, temperature, etc.). Equations (33) and (35) can then be written in a form which includes only quantities varying from one experiment to another. Thus, :_AE = A(c*/c)c1/2 (38) 66 — ‘ ““5— 41: m <39) This decay constant can be correlated to the polarographic diffusion current constant, K, which is 1/2 K = 607 nD (40) for the original form of the Ilkovic equation and for the average current during drOp life. Hence, A = 2FK/607nl/2 (41) :AE = l79KC*t1/2/C (42) :AE = 179 Kc*c1/2/c + 0.262 K2C*t/nrc (43) To generate a pulse, Delahay (4,19) used a simple device consisting of a small, high quality capacitor which was simply shunted across the cell through a system of relays after being charged with a battery of known voltage. Thus a charge of accurately known coulombic content, q=CV, where C is the capacitance of the capacitor, and V the voltage of the battery, was injected into the system. Delahay (19) first applied this technique to the reduction of iodate in alkaline solution (0.01 M NaOH) and to the reduction of zinc ion in 0.02 M KCl using a hanging mercury drOp electrode. Variations of E with time were obtained and the correSponding t1/2 plots determined. The distortion from sphericity of AB 22 the electrode was found to be negligible and the plots were essentially linear. After correction for the blank obtained with 67 supporting electrolyte alone, experimental AE's were compared with theoretical values. For both of these systems results were obtained that corresponded very well with theory until the concentration was decreased to approximately 5 X 10-7 M. The unsatisfactory agreement between theory and experiment at this concentration and below was attributed to the large blank correction. The blank was assumed to correspond to the four electron reduction of oxygen and a residual oxygen concentration of 5.3 X 10-.7 M was calculated from the blank curve, thus limiting determinations of lower concentrations. Results for mixtures of zinc and cadmium demonstrated the feasibility of analysis of mixtures by this method. Delahay postulated that the permissable ratio of concentrations should be similar to that in classical polarography - i.e., the concentration of the more easily reducible or oxidizable substance should not exceed approximately 10 times that of each of the other substances. The influence of cell resistance was also investigated. It was determined that cells of high resistance -_gzg., up to 0.3 megohm and probably higher resistance - can be utilized in this method. Because the presence of oxygen seems to limit the accuracy attainable by this method, Smith and Delahay (20) tried to devise relatively simple techniques for lowering the residual oxygen content to negligible proportions and then measure this concentration as accurately as possible using the coulostatic method. The method of applying the pulse in this case was the same as that used previously by Delahay, but a Kemula electrode was used. Both nitrogen and hydrogen gases were studied for their effective 68 saturation and oxygen removal. The results with these gases were approximately the same, $33,, - a residual oxygen concentration of about 5 X 10-7 M was meaSured. Light pre-electrolysis of the solutions was found to cause very little alteration of the measured values, but under some conditions the apparent oxygen concentration became much smaller and other substances made their appearance. These authors postulated the presence of a redox couple causing the phenomena, but were unable to remove it even after pre-electrolysis at -l.85 V vs. sce. Thus the presence of residual oxygen continues to be a problem in this method. To increase the sensitivity and overcome the oxygen removal problem, Delahay postulated the possibility of combining anodic stripping with a mercury electrode with the coulostatic method (21). This method termed "coulostatic anodic stripping" has the advantage that the decay of potential can be observed in a range of potentials in which oxygen is not reduced. No strenuous oxygen removal is therefore necessary. This method was applied for zinc determinations down to 5 X 10"8 M (22). A plating time of 200 seconds was used. The blank was found to be very low because the coulostatic pulse brought the potential during oxidation in the range (E = 0.05 V vs_ sce at t=0) in which oxygen and most other impurities are not reduced and few impurities are oxidized. Conditions are thus more favorable in general than is direct coulostatic analysis because of decrease in the blank correction. There is also an increase in sensitivity as in the ordinary stripping method. 69 In all of the above studies it was necessary to obtain a AE vs. t curve on an oscilloscope, photograph it, and plot the correSponding AE XE. tl/2 curve to obtain a concentration. For someone performing many analyses this could become very tedious. Thus, Delahay (22,23) developed a direct reading instrument for automatic coulostatic analysis. The circuit of this direct-reading instrument was composed of the coulostatic charging circuit which was the same as in previous works, and a circuit providing sampling, storing, and reading of the cell voltage. The cell voltage was stored in two high quality capacitors at two times, t1 and t2, during decay of potential. The sampling times were adjusted by means of a timing circuit. The difference of voltages across the two capacitors, which is equal to the variation of the cell voltage during the interval t2 - t1, was read with a vacuum tube electrometer. Readings were taken rapidly to minimize leakage in the capacitors. More than two time intervals could be selected by use of additional capacitors. This circuit was used for the analysis of zinc ion in 0.02 M KCl and very good results were obtained with concentrations as low as 5.8 x 10"7 M. Delahay (23) also proposed recording of potential-time curves with a pen-and-ink recorder by connecting a capacitor across the cell. A 3 uF capacitor was connected across the cell and potential- time curves recorded over a 10 second time interval for the zinc ion determination and excellent results were obtained. With the combination of the above two circuits in an instrument, Delahay believed coulostatic analysis could be used for routine 70 analysis, but this does not seem to be the case since there have been no publications using this method since 1965. The following discussion shows how a slight modification of this method and use of a small computer can make routine analysis much more probable. VI. CHARGE-STEP POLAROGRAPHY A. Introduction The technique of charge-step polarography combines the advantages of the coulostatic technique with the simplicity of operation and selectivity of dc polarography. A dropping mercury electrode (DME) is held at a potential where there is essentially zero faradaic current. Late in the life of the drOp, a small charge is quickly added to the electrode. The resulting potential-time curve is logged into a small digital computer which calculates the slope and intercept (E at t=0) for the first portion of the E !§_ tl/2 curve. The experiment is then repeated on the next drOp using a somewhat larger charge. When the charge is sufficient to polarize the electrode at a potential approaching the polarographic El/2 for a substance in solution, the slope will increase. Further increases in charge which bring the electrode potential to the polarographic plateau region of the substance should yield an increased, but potential-independent, slope value. In other words, a plot of the slope/intercept values for incrementally increasing charge steps applied to successive drops at a DME should resemble a smoothed dc polarogram and allow both qualitative and quantitative analysis of mixtures of electroactive substances. Such a curve for a mixture of 5 X 10-6 M Cd++ and 5 X 10-6 M Zn++ is shown in Figure 15, curve A. In this method the difficulty of double layer charging current which prevents the application of polarography 71 72 Figure 15. . -6 ‘++ Charge—step polarogram for a mixture of 5 X 10 M Cd and 5 x 10’6 M Zn++ ion in .01 M KCl. Curve A - initial polarizing potential of -.3 V.X§ sce. Curve B - initial polarizing potential of -.8 V !§_sce. 73 m.. > .mom 2 m- E m... 0.. m m. e. m. o p b . b n - . o I -. m 0330 .N.. < 9:30 rm. 1?. adols 74 to trace analysis is entirely avoided. Also measurements are made at times during which no appreciable net current passes through the cell so that no corrections for ohmic potentials are needed during the measurement or in interpreting the results. Thus, this method should have significant advantages over other polarographic techniques when using low concentrations of electro- active species in dilute solutions of supporting electrolyte. B. Experimental 1. Instrumentation The instrumentation used consisted of a Tektronix 535A oscillosc0pe with a type W plug in, an Intercontinental Instruments Model PG-33 pulse generator, and a PDP 8/I computer with an A/D converter and 8K of memory. The logic circuitry was designed using Heath EU 801 modules and logic cards. A Sargent Model XV polarograph was used in polarizing the electrode. Since a drOpping mercury electrode (DME) was used, a method of applying the charge at the same time in the life of each drop was needed. In order to obtain this the drOp fall detector shown in Figure 16 was used. The drop fall detector circuit was wired on a card for the Heath EU 801 module. The drop fall detector operates as follows: An oscillation is applied to the drop. The frequency and amplitude of this oscillation can be varied to obtain optimum results in solutions of various resistances and capacitances. The oscillation then passes through a tuned amplifier and a comparator. The great change in amplitude caused by the change in capacitance when the drOp falls causes a change in comparator output. This 75 Figure 16. Circuit diagram for drop fall detector. 76 QMEZQEG. QMZDF «o. x. 5. .o. m0h....~ + \.—k ..._I JEAI|<<<<¢ + 3.. <2: . .. 2 EN /—~ <92. h [Fax/>2... n"? 2 On #30 95 x. no A <02 :8 q 368 77 output is converted to a digital signal and the change in level which occurs with drop fall initiates a series of events which ultimately cause the charge to be applied at a chosen time in drop life. A block diagram of the components of the system and their interconnections is shown in Figure 17. All transitions except where noted take place on the falling edge of the logic signal. The sequence of events is initiated when drOp fall is detected by the drOp fall detector. F/F (l) is then set. The negative going transition of the 6 output of F/F (l) triggers MS (2) which initiates the timed delay by resetting and starting the time base and counter (4). MS (2) also sets F/F (3) which activates a relay disconnecting the drOp fall detector oscillator from the elctrode circuit. After the Prescribed delay, a pulse that is one second wide appears at the output of the counter. The leading edge of this pulse activates the relays, which remove the potentiostat from the circuit and set F/F (5), which provides a trigger signal t0 the sc0pe. After a short delay, the pulse appearing at the delayed trigger output of the oscilloscope is used to trigger the PUlse generator and, after suitable gating, provide a signal to the computer which initiates data acquisition by the PDP 8/1 computer. The rate at which data are acquired is determined by a programmable clock in the computer. Typically a data point was acChaired every millisecond for 50 milliseconds. It was necessary to measure the slope within about 2% of the drop life at the time of the pulse, to avoid distortion of the AB 33 t1/2 curve due to the increasing capacitance of the DME whose area was changing 78 Figure 17. Block diagram of circuit used in charge—step polarography. DROP FALL DETECTOR as c DIG our our 79 r4. m 4; PUL GENERATOR SE -o\ko—— .1 G __l fi‘fl TRIGG Q, TIME BASE AND (4') COUNTER ‘1 MS (8) SCOPE L DRIVER } COMPUTER 80 with time. The falling edge of the 1 second pulse from the counter deactivates the relays, which causes F/F (5) to be reset and the potentiostat to be reconnected to the circuit. This falling edge also triggers a 100 millisecond monostable, MS (8), which is used as follows: the Q output, which has an initial 1 to 0 transition, is used to reset F/F (3), which causes the drop fall detector oscillator to be reconnected to the electrode circuit. To insure that the relay which reconnects the drOp fall detector (ascillator has time to act before F/F (1) is reset, allowing the ssequence to start again, a delay of 100 milliseconds is introduced b)r using the Q output of MS (8). The 1 to 0 transition at this Orrtput occurs after the 100 millisecond delay and triggers MS (9), hfliich provides a pulse resetting F/F (1) and allowing the sequence tc> be reinitiated when the next drop fall is detected if desired. Provisions have been made in the computer program to allow ensemble averaging of the relaxation data acquired over a preselected liunfl>er of runs. After the experimental run or runs have been completed, the data are corrected for the injection time and a leaSt squares analysis is performed on the data to obtain an intercept and a slope proportional to the square root of time. 2. Cell and Electrodes A three electrode configuration was used in all experiments. 11153 ‘test electrode was a DME which consisted of a Sargent 2-5 second capillary and a glass and Teflon connection to a leveling bulb . A drop time of approximately 5 seconds was used. The Cloutlter electrode was a cylinder of platinum gauze of approximately E3 Q1112 area concentric to the DME. The reference electrode was a 81 saturated calomel electrode (sce) which was connected to the cell through a bridge of l M KCl, a glass frit, and then a solution of whatever substance was being studied (see Figure 18). The cell consisted of a weighing bottle with a standard taper rim. The lid consisted of a Teflon plate machined to fit the rim. Holes were machined to fit the DME, counter electrode, reference electrode, and a tube for deareating. Oxygen was removed from the cell by bubbling argon through for 20 minutes prior to each experiment. Argon flowed over the top of the solution during tfiie experiment. Before entering the cell, the argon was passed crver BTS deoxygenator from BASF at 200°C and through a wash bottle cxf distilled water. The temperature was 23 :.2° C. 3. Reagents and Solutions Analytical reagents were used without further purification. Purification by pre-electrolysis was not necessary since dilute Stqiporting electrolytes were used and blanks were not noticeably affected by variation of the supporting electrolyte concentration. Sciltitions were prepared with triply distilled water which had been PaSSed through a deionizer. Triply distilled mercury was used. C. Results and Discussion 1. Relation of Slope of AE vs, tl/2 Lines and Concentration To verify the proportionality between concentration and measured slope, a series of measurements were made using zinc, cadmium, and nickel ions at various concentrations between 1 X 10—6 M and 5 X 10-5 M in a -2 SupParting electrolyte concentration of l X 10 M KCl. The electrode was initially polarized at -.3 V _‘E sce. Typical charge-step polarograms 82 Figure 18. Diagram of cell and reference electrode. 84 for these ions are shown in Figure 19. The slopes of the various metal ions were obtained in potential regions where the reduction of the particular ion was diffusion controlled and are shown in Figure 20. Also shown in Figure 20 are the results from experiments run on solutions containing a mixture of zinc and cadmium ions in solution (see Figure 15 for typical curve). It would be expected that the results shown should lie on the same line since the diffusion coefficients of zinc, cadmium, and nickel are very similar (25) and the capacitance of the solutions should be similar. Background corrections were applied to these slopes before they were plotted, i.e., a charge-step polarogram for l X 10-2 M KCl alone was obtained and this curve subtracted from those obtained for the metal ion solutions. The background correction is necessary because of the presence of residual oxygen which at potentials cathodic of -.9 V.X§ sce is being reduced along with the metal ion. It was found here as in Delahay's work that the oxygen concentration amounts to approximately 5 X 10-7 M, making the background correction very appreciable at the lower concentrations. 2. Effect of Supporting Electrolyte Concentration Since one of the advantages expected to be achieved by using this technique was the possibility of using low concentrations of the supporting electrolyte, a series of experiments were performed to ascertain the effect of varying the concentration of supporting electrolyte at a constant concentration of electroactive species. For this series of measurements, a cadmium ion concentration of l X 10"5 M in KCl con- 1 centrations of 1 x 10' , 1 x 10’3, and 1 x 10-4 M were used. 85 Figure 19. Charge-step polarograms for 5 X 10-6 M Cd++, Zn++ and Ni++ in .01 M KCl. Electrode initially polarized at —.3 V X§_sce. Slope J" Slope 86 Cd” l I ' I .4 .6 .8 1.0 1.2 .. E vs SCH: V ‘2‘ Zn“ ' cl I I r I .s LO 1.2 L4 '-5 - E vs 565! V .2- Ni“ 0 a i 2 U) .l ‘ .‘ . ' ' 1.2 1.4 1.6 LO -E VS Sceov Slope‘gg 87 Figure 20. concentration for various metal ions. Cd++ alone Cd++ with Zn++ Zn++ alone Zn++ with Cd++ Ni'H' 88 m «0. x 0:00 N n b— .. 0.. 1 N._ r v.— w; 89 The results of these experiments are shown in Table IV. At these concentrations the values for the 810pes on the plateau region agreed to within 113% with each other and to the expected value of the slope calculated from Equation (33). Measurements veremade over a wide range of potentials to include the complete wave for cadmium. The slepe or sharpness of the wave was virtually independent of the supporting electrolyte concentration. It should be noted that the lowest c0ncentration of supporting electrolyte used, 1 X 10"4 M, does not represent the lower limit for this technique, but rather indicates the lowest limit that could be reached with the maximum voltage of the pulse generator used. Table IV. Variation of Slope in the Plateau Region for l X 10-5 M Cd++ with Changing Concentration of Supporting Electrolyte KCl concentration, M Slope at -.8 V vs_ sce 1 x 10-4 .300 1 x 10'3 .310 1 x 10’2 .320 1 x 10"1 .300 In all of the above experiments measurements were made at sufficient density and over a large enough potential range to give information about the shape of the wave obtained. Also in the measurements, only the first 30 to 50 mV of the decay curve that was obtained were used in the data analysis in order to improve 9O resolution. The waves for cadmium and zinc were similar in shape (Figures 15 and 19) while, as would be expected, nickel, which is more irreversible than zinc or cadmium, gave a more drawn out wave (Figure 19). The wave for the 4-electron reduction of oxygen (which is very irreversible) was also obtained. This is shown in Figure 21. The E3/4 - E1/4 values for these ions are shown in Table V. Table V. E3/4 -El/4 for 5 X 10-6 M Concentration of Each of the Ions. Species E3/4 - El/4’ mV 0dH 60 2nH 6O Ni'H' 90 02 180 Reversible Species 56/n in polarography 3. Charge—Step Polarography with Prepolarization The potential of —.3 V.X§ sce corresponds to a point on the diffusion Current plateau of the two electron reduction of oxygen. Thus, when a background charge-step polarogram is obtained only the wave for the four electron reduction is observed since at -.3 V the concentration gradient for the 2 electron reaction will be very steep. This elimination of a prior wave by polarizing at a potential on the plateau of this wave can also be applied to mixtures of substances. For example, if the solution of 5 X 10-6 M Cd++ and 5 X 10-6 M Zn++, the charge-step polarogram 91 Figure 21. Charge-step polarogram for the 4-electron reduction of oxygen. (0.01 M KCl with 8 minutes deareation) 92 w... > .mom m> m: m — _ a. adols 93 of which is shown in Figure 15, curve A, were initially polarized at —.8 V gs see, which corresponds to a potential on the diffusion current plateau of Cd++, only the wave for reducaion of Zn++ would be observed. This is shown in Figure 15, curve B. In this way an accurate value of the slope on the second wave can be obtained without having to extrapolate the plateau of the first wave to obtain a baseline as would be required in Curve A and could introduce error. This concept could be especially useful when there is a large difference in concentration of two substances in solution. The concept of prepolarizing to eliminate a wave could also be applied to eliminate the wave of the four electron reduction of oxygen from the background, but, unfortunately the plateau of this wave is at a potential so cathodic that anything in solution would be reduced along with the oxygen. To obtain data from Substances in solution, the electrode could be charged anodically to oxidize the substance from the electrode. Figure 22 is an example of the charge-step polarograms obtained when this method is applied to a solution of 5 X 10-6 M Cd++ and 5 X 10'6 M Zn++ in .01 M KCl. Although these results are qualitatively very good they have Inot been mathematically predicted. In this case Equation (33), which Egives the proportionality constant between concentration and slope, :is not applicable since the concentration of the substance in the cirop will depend on how long the electrode is polarized before it :18 pulsed. Also, diffusion within the mercury drop is not the 53«fine as diffusion in solution. The derivation of the mathematical relations for this case will put this case on a sounder quantitative fOuting. 94 Figure 22. . -6 ++ Charge—step polarograms for ox1dation of 5 X 10 M Cd and 5 x 10’6 M 2nH in .01 M KCl. A - prepolarization at -.8 V X§_sce. B - prepolarization at -l.2 V v§_sce. H1 95 >.mom s, m- ~01 96 4. Differential Charge—Step Polarography A slight variation of the charge-step polarographic method would be to set the charge pulse to give an overpotential of approximately 50 mV and increment the polarizing potential. As the polarizing potential approaches the half wave potential the slope should increase because of the occurrence of faradaic reaction. When the polarizing potential is increased cathodic of the half wave potential the slope should decrease because of depletion occurring at the electrode during polarization. An example of the curves which result from application 6 M Cd++ and 5 x 10'6 M ZnH in of this method on a solution of 5 X 10- 0.01 M KCl is shown in Figure 23. From the appearance of the curves this method could be termed "differential" charge-step polarography. Although the mathematical relationships for this method have not been derived, the slopes obtained at the peak of these curves are very close to those obtained in the plateau region of normal charge- step polarograms of the same solution. This method could prove very useful since in this case the background does not appear to affect the results. The results obtained by this technique and its variations demonstrate the feasibility of its use for rapid, automated analysis of low concentrations of electroactive species in fairly dilute concentrations of supporting electrolyte. Further work is under way to improve the instrumentation and more completely automate the experiment in order to obtain the entire slope X§_intercept curves automatically. With these further improvements, and derivation of the mathematical relationships for the anodic pulse and differential variations this technique is very promising for routine analysis. 97 Figure 23. . -6 ++ -6 Differential charge-step polarogram for 5 X 10 M Cd and 5 X 10 M 2n++ in 0.01 M KCl. ‘0 - Solution of Cd++ and 2n++ in KCl. X - KCl alone. 98 w._ .v._ N.— 0.. > .mom m> m .. m. em. v adols BIBL IOGRAPHY 10. ll. 12. 13. 14. 15. l6. 17. 18. 19. BIBLIOGRAPHY Delahay, P., J. Phys. Chem., 66, 2204 (1962). Reinmuth, W. H., Anal. Chem., 34, 1272 (1962). Barker, G. C., in "Transactions of the Symposium on Electrode Processes", Philadelphia, 1959, pp. 325-365. E. Yeager, ed., John Wiley & Sons, Inc., New York, 1961. Delahay, P., Anal. Chem., 34, 1267 (1962). Weir, W. D. and Enke, C. C., J. Phys. Chem., 21, 275 (1967). Daum, P. H., and Enke, C. C., Anal. Chem., 41, 653 (1969). Anson, F. C., in "Annual Review of Physical Chemistry", Vol. 19, H. Eyring, ed., Annual Review Inc., Palo Alto, Calif., 1968. Martin, R. R., Ph.D. Thesis, Louisiana State University, New Orleans, La., 1967. Kooijman, D. J., and Sluyters, J. H., Electrochim. Acta., 12, 1579 (1967). Berzins, T., and Delahay, P., J. Am. Chem. Soc., 17, 6448 (1955). Kooijman, D. J., and Sluyters, J. H., Electrochim. Acta, 12, 1579 (1967). Kooijman, D. J., J. Electroanal. Chem., 18, 81 (1968). Van Leeuwen, H. P., Kooijman, D. J., Sluyters-Rehbach, M. and Sluyters, J. H., J. Electroanal. Chem., 23, 475 (1969). Daum, P. H., Ph.D. Thesis, Michigan State University, East Lansing, Mich., 1969. Hartly, H. 0., Technometrics, 3, 269 (1961). Bleasdell, B. D., Ph.D. Thesis, Michigan State University, East Lansing, Mich., 1971. Nicholson, R. 8., Anal. Chem., 31, 667 (1965). Delahay, P., Anal. Chim. Acta, 31, 90 (1962). Delahay, P., and Ide, Y., Anal. Chem., 34, 1580 (1962). 99 20. 21. 22. 23. 24. 25. 100 Smith, F. R., and Delahay, P., J. Electroanal. Chem., 39, 435 (1965). Delahay, P., Anal. Chem., 34, 1662 (1962). Aramata, A., and Delahay, P., Anal. Chem., 33, 1117 (1963). Delahay, P., Anal. Chim. Acta, 31, 400 (1962). Delahay, P., and Ide, Y., Anal. Chem., 33, 1119 (1963). Kolthoff, I. M. and Lingane, "Polarography", Interscience, N. Y., 2nd edition, 1952, p. 52. APPENDICES Appendix A Calculation of Errors Resulting from Use of the Charge Transfer Approximation The procedure used to determine the error in exchange current and capacitance obtained when the simple charge transfer approximation was applied to curves resulting from simultaneous charge transfer and diffusion controlled relaxation was as follows: 1) A program was written to generate theoretical curves following Equation 19 in the text given values for capacitance and exchange current. This program is given at the end of this appendix. An example of the output is as follows: Time, usec I°, A/cm2 .ll486784E-08 .45012702E-Ol .10338106E-05 .45012702E-Ol T /T .OlOOOOOO d c 1/2 (t/td) n/nt=0 .10000000 .99990008 3.00000000 .91567516 101 102 This type output was obtained for a range of Td/TC values. 2) Then to determine the results that would be obtained from this curve using the charge transfer approximation the slope and intercept (and hence exchange current and capacitance) resulting from extrapolation /2 between various values of (t/Td)1 were calculated using a simple FOCAL program and the PDP—8/I computer. This program and an example of its output are given at the end of this appendix. 3) These values were then compared to the values used in calculating the curves. 4) Plots of log (Ioexp/Ioact) and log (Cap exp/Cap ) XE act log (TC/Id) were made. To obtain the plot of log (I°e /I°act).!§ log (TC/Td)exp new XP values of 1C and Id were calculated from the capacitance and exchange current determined from application of the charge transfer approximation. 103 .omsuiHH.u ooommmm.smmuHoH.u .cmHuioeu mmc-m¢m.mmu.aeu oc.¢mu.>.u ~mm-mcoHHNHceu oo.cuHm.u memommmmm.mniqeu o.mu.m.u ammosromm.HuHm.u oo.HuHH.u arbomaoooo.cumu mscmcsmooo.ouwu meHomchoc.on¢u mummcsmooc.ouro mchmmmmeo.oumu emsommmchc.anu .cchoum nsHm.mua .oomu» monmc.mno «abamnax Hxvhaomnham oce HZ<¢.NV\szhamnm .st.m.\an< Hazmhvbmownatm»um ozpatmzcnnrmh ooc Hose 0» O0 UIhzaatimmI<<.\o~uumox brimdlhxammnurhzaa NHom («\AUZZmQIoAVHHXQEQ Im\HUZZL1Io~.NH¥aU¢ mom Ham Op co HQ>¢¢m¢m~h._V\Hcm+.~vnuz:um HAAHAAHHHHHHHH.HNsoHM+ofilv¢N¢.om+.Hszsohmom .HIVstomN+.~v¢N¢.MN+.HI.¢N#.~N+.~.¢N¢.OH+.~I.smaofi~+ofiv¢Nscm~+o~I~ V¢N¢.m~+.~szt.-+oalvaN¢.O+.~.#Nso~+uHIV#N¢.m+.HV¢N¢.M+.~IV¢NVucm ANQ>¢.N.\.HHN mom Ham 0» cc a>uqmvima>voxmnuzaum i.efiivstHHAAHH©U¢a>+muvaa>+¢uv¢Q>+MUV¢Q>+NUVsa>+~uvsa>+efivn1>maw ¢om momoqomo¢om .o.mu&>.u~ com HQx¢m¢NN~.~.\Hc<+.HVuuzzuq HHHHHHHHHHHH.A.H3¢.HM+o~tvu3aoom+ofiv¢?s.nn+m .~0.¢3¢.mm+.~V¢3¢ommeo~I.¢3¢.—N+.~.03#.OH+oHOV¢3¢.~_+.~vs3n.mfi+.~I~ vo3som~+.~.a3a.-+.Huva3a.0+.Hv¢3¢.~+.~IV¢3¢.m+.Hv¢3soM+oHIV¢3VHC< Hmoxoem.\.~H3 mom com O» co meama.max.axmnuz:u< H.cHt.¢nHHHHHHCU¢QX+muvaax+¢U emx+muv¢QX+muvsax+duvsux+.fivuaxuam mom momomom.mom Aceriaxva mssa>nma> mssuxnmax Hawmmunx Haweqquux AXVFQCthQM rim 108 Ntaa>lmcta>umtsaxsoHm+masa>s¢¢aax¢.mMIarscssaxt.sump>wHa o¢¢Q>¢stosl¢¢¢a>sm¢¢axs.mm.m¢¢a>¢ma¢axvoHmlwsaaxumpxchm Abomzcuse¢v\ih>c~m#oMI."HOQIH> AFOmZOU#.¢.\Hhxo~m¢.mvuhmax msvh>omm+mtshxommuhcmzcu m¢¢1>+mass>amtsaxnooHua>vsasaxnemup>owm ¢¢sa>¢axsom+mssa>¢mstaxs.o~lmssaxuhxoam oxon\>onIno cmZOU\xo~mu4ax ms¢>me+m¢¢xOHmuom2CU Ntsnxsa>¢.clm¢ma>¢.mn>o~m masax¢.mtmata>aaxsocuxc_m Hmssa>+mssaxv\a>lu03hN Hmsva»+mtsaxv\axum2CN ~30¢ Oh Cc fiod Ch C0 Hex"! ¢cflesodomo~Hm0vauw wo~ewcHequHomtwo.filixuawim<.mm quX$A.m\x.¢HH+yv¢¢H~IV+Uuaxnuuax HXVU\©23wuxqu Arv0+623unozrm AHA filiemvsimlftmvV\H~IZVCvHN#¢QX\N*#Q>V¢AANIEsNVIXV¢HAVIxsvaXVIHHIVO xx.mnz He CC AHVGHOZDm afiixvmaux¢XtomnAfivw H+xnxy ~nx c.cuuu0x 3COM.Cgom.~Ho~HucmlHQx.Um<.u~ occr.rcom.-c~Hm.muia>.mm<+iax.wmVM$vm®qvuw mee~.moom.moemic.Huiqxiumqima OOOFoooomom~o~AOoMl ~como~comocccma~o ooom sea moH Nod ~¢ Ho“ HHoH mocm :oom moom Nocm 3% $1357: 33 siexemxecui mace 109 02w oc~ Oh 00 Hw.m~u.xr.m.m_m.xmowomdmvb+NO+Ho+C¢C3kNvOno smsmhh.~\Amwmax+mhgaxopqax+qax+wZONvum Hmcmzoua.cfiv\Hm»>wfiasomofilvuw042m> Hmcrzcu¢.bHvxnrhonde.moHenmesax neamw>cmm+m¢aMPx0~murcmzcu ossa>+hasu>¢mtauxseomla mssa>¢¢¢¢axnecm~+ci¢u>¢0#¢axs.dila>#x#¢ax#.OHMH>on msaa>aaxs.O+Csta>sm¢saxu.¢¢I~ sssa>¢masoxsoom~+N¢$Q>¢~¢¢Qxa.CMlmstaxurbxcHa ANCEIOU¢om.\Hmh>cha.m—inmo<7H> .mcrzcusom.\Amhxc~m#.m~lvuwhMax Nttmh>chr+wasmhxc_mumomzzu 110 FOCAL program for calculating capacitance and exchange current *3ol ASK ?0E TE? *302 SET L0=FLOG(OE)3SET LT=FLOG(IE)3$ET DE=LT'LO *303 SET 0T802584526435ET TT=o4S94713615ET DT=TT'0T *3-4 TYPE "SLOPE":DE/DT9! ’305 SET $=DE/DTISET Y=OT*SISET X=L0'Y}5ET I=FEKP(X) #306 TYPE "INTERCEPT":FEXP(X):! #307 TYPE "CAP RATIO":1¢0/lo! *308 SET A‘2506935ET 8:200 ‘3085 TYPE "EXCHANGE CURRENT"1A#(B/l)#b:! *309 SET IO=A#(B/I)*S #3095 ASK ?AI? *3.SM3'IYPE:"IO FUATICV'oAIJ'1039 #3097 TYPE "LOG IO RATIO":FLOG(10/Al)a! iK3098 QUIT tG 0E 8097900750 TE8¢JW63GSWM68 SLOPEg’ 0.0818 INTERCEPT= 0.9999 CAP RATIO= 1.0001 EXCHANGE CURRENT=- 42.0070 A1842.672043 IO RATIOz- 1.0158 L06 [0 RATIO=~ 0.0157 OE = overpotential at time OT TE overpotential at time TT AI actual exchange current Appendix B Program for Acquisition and Treatment of Data in Charge Step Polarography lll 7) 003003000 ~33 ’OO 6 17 C NOW TEST SR TO I2 'Jf'J‘ZTLRZflUIUZU’UIUIO 'q 'n -A ad .4- v- 'J? 112 THIS IS A COMRINATION FORTRAN SABR PROGRAM TO DO A CURRENT IMPULSE EXPERIMENT THERE ARE TWO SECTIONS THE FIRST THE SECOND IS THE ACTUAL REAL CELL EXPERIMENT ALL OPDEF OPDEF SKPDF SKPDF OPDEF nPan owns: SKPDF CPAGF MDATAC; IS FOR CALIRRATION 5/6/7l COMMON DA3ADA DIMENSION INPUT OF PARAMETERS FOR INPUT DA(2WW):ADA(PMW);ID(PWU) CALIHRAIION IS FLOATING POINT WRITF(I:IHM) READ(I:IMI) TI READ